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ASYMNIETRIC FLOWS OF NEWTONIAN AND NON - NEWTONIAN LIQUIDS
IN SYMNIETRIC PLANAR EXPANSIONS

By

Sanjay B. Mishra

A DISSERTATION
Submitted to
Michigan State University

in partial ﬁilﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY ,

Department of Chemical Engineering

1997

ABSTRACT

ASYMMETRIC FLOWS OF NEWTONIAN AND NON - NEWTONIAN LIQUIDS IN
SYMMETRIC PLANAR EXPANSIONS

By

Sanjay B. Mishra

The ﬂow bifurcation of Newtonian and inelastic, shear thinning liquids through an abrupt
planar expansion with an area expansion ratio of 16 has been investigated. Model
Newtonian and non - Newtonian liquids with power - law shear thinning indices spanning
0.5 to 1 were prepared and were carefully characterized. Streak photography and laser
Doppler velocimetry were used to explore the effect of ﬂuid rheology and inertia on the
ﬂow transitions. In particular, the detailed ﬂow patterns and the onset of asymmetric ﬂows
in the symmetric geometry were compared for the diﬂ‘erent ﬂuids. A continuation method
which used geometric perturbations was developed to analyze the ﬂow bifurcation for

different expansion ratios and ﬂuid rheology.

Experimental results with the 1:16 channel revealed that in Newtonian liquids, the onset of
asymmetric ﬂow occurred at a critical Reynolds number of 0.8, which is signiﬁcantly
lower than that reported for Newtonian ﬂow through planar channels with expansion ratio
of 2 and 3. Increasing pseudOplasticity delayed the onset of the ﬂow transitions but
resulted in a higher degree of asymmetry and a longer ﬂow length up to which asymmetry
persisted. An unexpected ﬂow transition involving the simultaneous presence of three

vortices close to the plane of expansion was observed with the shear thinning liquid.

Sanjay B. Mishra

Computations predicted the trends of delayed bifurcation with increased shear thinning.
Results of computations also showed that the onset of asymmetry occurred at much lower
Reynolds number with a high expansion ratio. The Newtonian vortex length prior to the
onset of asymmetry was related to the expansion ratio and the Reynolds number. The
present study will ﬁnd applications in many processes which involve ﬂow of shear thinning

ﬂuids through expansions such as the extrusion of food materials through dies.

ACKNOWLEDGMENTS

I am very thankful to my advisor, Dr. K. Jayaraman, for his steadfast support,
continuous guidance, patience and understanding, particularly during some of my rougher
times. I also thank my committee members - Drs. Charles Petty, John McGrath and
William Pratt - who devoted considerable time toward my dissertation.

Mike and Terry at the DER machine shop oﬁen went out of their way to help me
complete the seemingly endless task of perfecting the ﬂow conﬁguration that I used so
exhaustively in my experiments. The ever - smiling, never - reﬁrsing staﬂ‘ in the omce -
Julie, Candy and Faith really made administrative matters very simple. Thank you all!

I thank all my co - workers especially Ashim, Doug and Himanshu, who lent me
support whenever I needed it. Friends in my personal life - Venky, Shanti, Uma, Naresh
and many others helped me in numerous ways. It will be very difﬁcult to ﬁnd such a good
bunch of buddies.

My parents were a constant source of inspiration and motivation. They inspired me
to put my best efforts in everything I do in life. I have a lot to thank them for. Finally, this
work would never have been completed without the strong support and encouragement of
my wife Lata. She made life at home so relaxing that I was always fresh to tackle the

problems in grad school. Her contribution to my dissertation is really invaluable.

iv

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................ viii
LIST OF FIGURES ............................................................................................................. ix
CHAPTER 1
INTRODUCTION ............................................................................... : ............................... 1
1. Deﬁnitions ............................................................................................................ 1
2. Interaction of inertia, elasticity, extensional viscosity and pseudoplastic
behavior .......................................................................................................... 3
3. Review of previous work done .............................................................................. S
3.1 Contraction ﬂows .................................................................................... 6
3.2 Expansion ﬂows .................................................................................... 1 1
3.2.1 Experimental results ................................................................ 11
3.2.1.1 Flow transitions due to inertia in Newtonian
ﬂow through planar expansions ............................ 11
3.2.1.2 Non - Newtonian ﬂow through expansions ................ 16
3.2.2 Computational results in planar expansions .............................. 19
4. Objectives ........................................................................................................... 23
CHAPTER 2
FLOW VISUALIZATION OF THE ASYlvﬂWETRIC FLOW TRANSITIONS
IN AN ABRUPT PLANAR EXPANSION ........................................................................ 26
1. Introduction ........................................................................................................ 26
2. Backgron ........................................................................................................ 27
2.1 Contraction ﬂows vs Expansion ﬂows .................................................... 27
2.2 Eﬂ‘ect of geometry - Axisymmetric vs Planar .......................................... 29
2.3 Flow transitions due to inertia in Newtonian ﬂow through
planar expansions .............................................................................. 30
2.4 Experimental work on Non - Newtonian ﬂows through expansions ........ 32
2.5 Vortex grth prior to the onset of asymmetry ...................................... 34
3. Objectives ........................................................................................................... 34
4. Experimental ....................................................................................................... 36
4.1 Model ﬂow cell ...................................................................................... 36
4.2 Model ﬂuids .......................................................................................... 39
4.3 Flow visualization setup ......................................................................... 47

V

5. Results and Discussion ........................................................................................ 50

5.1 Flow patterns in the Newtonian liquid (Experimental) ............................ 50
5.2 Computations of the ﬂow biﬁrrcation in Newtonian liquids ..................... 55
5.3 Newtonian vortex growth prior to the onset of asymmetry ..................... 64
5.4 Flow patterns in the shear thinning liquids (Experimental) ...................... 72
5.5 Flow bifurcation in the non - Newtonian liquids ..................................... 77
5.6 Computed vortex growth in non - Newtonian liquids prior to the
onset of asymmetry ........................................................................... 80
6. Conclusions ........................................................................................................ 85
CHAPTER 3
LDV MEASUREMENTS OF NEWTONIAN AND NON- NEWTONIAN
LIQUIDS 1N ABRUPT PLANAR EXPANSIONS ............................................................. 88
1. Experimental ....................................................................................................... 88
1.1 Laser Doppler velocimetry ..................................................................... 88
1.2 Optical setup ......................................................................................... 90
2. Results and Discussion ........................................................................................ 93
2.1 Velocity proﬁles of the Newtonian liquids ............................................... 95
2.2 Velocity measurements of the shear thinning liquids (n = 0.7 and
n = 0.5) .......................................................................................... 105
2.3 Quantiﬁcation of the degree of asymmetry ........................................... 124
2.4 Biﬁrrcation curves ﬁ'om the experiments .............................................. 125
3. Computations of ﬂow biﬁrrcations ..................................................................... 130
3.1 Computed predictions ofthe decay ofﬂow asymmetry 133
3.2 Velocity proﬁles from the perturbation - continuation approach ........... 136
3.3 Comparison of experimental and computed bifurcation curves ............. 139
4. Conclusions ....................................................................................................... 142
CHAPTER 4
COMPUTATIONS OF SYMMETRY BREAKING FLOW BIFURCATIONS
IN ABRUPT PLANAR EXPANSIONS ........................................................................... 147
1. Introduction ...................................................................................................... 147
2. Background ...................................................................................................... 147
2.1 Newtonian ﬂow transitions in an abmpt planar expansion ..................... 147
2.1.1 Existing experimental results ................................................. 147
2.1.2 Prediction of Newtonian ﬂow biﬁircation using
computations .................................................................... 150
2.2 Non - Newtonian ﬂow through expansions ........................................... 154
3. Objectives ......................................................................................................... 156
4. Numerical scheme ............................................................................................. 157
4.1 Problem deﬁnition ................................................................................ 157
4.1.1 Scaled variables ..................................................................... 159
4.1.2. Boundary conditions ............................................................ 159
4.2 Perturbation - Continuation approach .................................................. 160

vi

4.3 ﬁnite elements grids ............................................................................ 164
5. Results and Discussion ...................................................................................... 165
5.1 Flow bifurcation in expansion ﬂows ..................................................... 165
5.1.1 Validation of the perturbation - continuation approach .......... 165
5.1.2 Use of the perturbation approach to study Newtonian
ﬂow biﬁrrcationsina 1:16 channel ................................... 171
5.1.3 Effect of non - Newtonian behavior on ﬂow
bifurcation in a 1:16 planar expansion .............................. 174
5.2 Flow patterns during transitions in planar expansions ........................... 178
5.2.1 Newtonian liquids .................................................................. 178
5.2.2 Shear thinning liquid .............................................................. 180
5.3 Velocity proﬁles in planar expansions ................................................... 183
5.3.1 Velocity proﬁles in the Newtonian liquid before and
after the onset of asymmetry ............................................ 183
5.3.2 Velocity proﬁles in shear thinning liquid P1 through
a 1:16 planar expansion ................................................... 188
5.4 Eﬂ‘ect of expansion ratio and rheology on the excess pressure
diﬁ‘erence during ﬂow through expansions ...................................... 188
5.5 Effect of expansion ratio and rheology on the ﬂow resistance ............... 198
6. Conclusions ....................................................................................................... 202
CHAPTER 5
CONCLUSIONS AND RECONﬂVIENDATIONS FOR'FUTURE STUDY ..................... 207
l. 1 Conclusions ......................................................................................... 207
1.2 Recommendations for ﬁxture work ....................................................... 213
APPENDIX A
COMPARISON OF ABRUPT EXPANSION FLOW WITH JEFFERY - HAMEL
FLOW .............................................................................................................................. 214
LIST OF REFERENCES ................................................................................................. 221

vii

LIST OF TABLES

Table 2.1 - Model ﬂuids used ............................................................................................... 40
Table 2.2-Parametersusedintheshearviscositymodels .................................................... 43
Table 2.3 -Fitting constantsusedinFigures 2.15, 2.16 and 2.17 .......................................... 72
Table 3.1 - Description of the components depicted in Figure 3.1 ........................................ 92
Table 4.1 Parameters used in the viscosity models .............................................................. 175

viii

LIST OF FIGURES

Figure 1.1 Schematic of ﬂow through expansions or contractions ..................................... 2
Figure 1.2a Vortices in contraction / expansion ﬂows ............................................................ 4
Figure 1.2b Asymmetrical ﬂow proﬁle in a planar expansion .................................................. 4
Figure 1.3 Vortex length in a 4:1 axisymrnetric contraction (Boger et al., 1986) ..................... 8
Figure 1.4 Flow patterns with water through a 1:3 planar expansion (Sobey, 1985).
(a)Re=55(b)Re=115 ..................................................................................................... 12
Figure 1.5 Vortex asymmetry in a Newtonian liquid through a 1:3 planar

expansion (Sobey, 1985) ...................................................................................................... 13
Figure 1.6 Effect of aspect ratio and expansion ratio on the ﬂow transition in

Newtonian ﬂow (Cherdron et al., 1978) ............................................................................... 15
Figure 1.7 Flow conﬁguration used by Townsend and Walters (1994) .................................. 18
Figure 2.1 Design of the model ﬂow cell .............................................................................. 37
Figure 2.2 Characteristic dimensions of the model ﬂow cell .................................................. 38
Figure 2.3 Eﬁ'ect of Methocel concentration on the viscosity of the polymer solutions ......... 42
Figure 2.4 Shear viscosity of the polymer solutions .............................................................. 44
Figure 2.5 Loss modulus and storage modulus of the polymer solutions ............................... 46
Figure 2.6 Flow visualization setup ..................................................................................... 48
Figure 2.7a Scanning mirror assembly for creating a laser sheet ............................................ 49
ﬁgure2.7bLasersheetcreationwithaglassrod ................................................................. 49

ix

Figure 2.8 Flow patterns of the Newtonian liquid. (a) Re = 0.001 (b) Re = 0 ........................ 52

Figure 2.8 Flow patterns of the Newtonian liquid (continued).

(c)Re=0.5 (d)Re=0.8 ..................................................................................................... 53
Figure 2.8 Flow patterns of the Newtonian liquid (continued).

(e)Re=1.5(t)Re=4 ..................................................................................................... 54
Figure 2.9 Experimental vortex growth in the Newtonian liquid ........................................... 56
Figure 2.10 Schematic of the ﬂow channel used in calculations ........................................... 57
Figure 2.11 Results from the perturbation - continuation approach for

Newtonian ﬂow through a 1:3 planar expansion ................................................................... 60
Figure 2.12 Effect of grid reﬁnement on biﬁrrcation curve. Newtonian ﬂow,

Expansion ratio = 3 ............................................................................................................. 62
Figure 2.13 Biﬁrrcation curve for the 1: 16 expansion ........................................................... 63
Figure 2.14 Positive halfof the bifurcation curve for Newtonian ﬂow .................................. 65
Figure 2.15 Experimental Newtonian vortex growth prior to the onset of

asymmetry. Solid line is from computation ........................................................................... 66
Figure 2.16 Newtonian vortex growth prior to the onset of asymmetry at high

Reynolds numbers. Lines are ﬁts using equation (2.9) .......................................................... 70
Figure 2.17 Newtonian ortex growth prior to the onset of asymmetry at

low Reynolds numbers. Lines are ﬁts using equation (2.9) ................................................... 71
Figure 2.18 Flow patterns of the polymer solution P1.

(a) Re" = 1.25 (b) Re“ = 1.9 ................................................................................................ 73
Figure 2.18 Flow patterns of the polymer solution P1 (continued).

(c) Re“ = 1.25 ((1) Re“ = 1.9 ............................................................................................... 74
Figure 2.19 Experimental vortex growth in the polymer solutions ........................................ 76
Figure 2.20 Predicted bifurcation curve for shear thinning liquid P1 ..................................... 79
Figure 2.21 Vortex growth in polymer solution P1. Solid line is ﬁ'om computation .............. 81
Figure 2.22 Vortex growth observed in P2. Solid line is from computation .......................... 82

Figure 2.23 Vortex growth before the onset of asymmetry, n = 0.7 ...................................... 83

Figure 2.24 Vortex growth before the onset of asymmetry, n = 0.5 ...................................... 84
Figure 2.25 Eﬁ'ect of shear thinning on the creeping ﬂow vortex length ................................ 86
Figure 3.1 Schematic of the LDV setup ............................................................................... 91
Figure 3.2 Nomenclature used to describe the LDV experiments .......................................... 94
Figure 3.3 Fully developed velocity proﬁle of the Newtonian liquid at X = 15.

Solid line is the plane Poiseuille ﬂow .................................................................................... 97
Figure 3.4 Symmetrical axial velocity proﬁles of the Newtonian liquid at Re = 0.6 ............. 98
Figure 3.5 Newtonian velocity proﬁles after the onset of asymmetry, Re = 1.5 ..................... 99
Figure 3.6 Newtonian velocity proﬁles aﬁer the onset of asymmetry, Re = 2. 5 ................... 101
Figure 3.7 Newtonian velocity proﬁles alter the onset of asymmetry, Re = 3 .5 .................. 102
Figure 3.8 Newtonian velocity proﬁles close to the expansion plane; X = 1 ........................ 103
Figure 3.9 Newtonian velocity proﬁles at X = 8 ............ ' .................................................... 1 04
Figure 3.10 Infared transverse ﬂow of the Newtonian liquid ............................................ 106
Figure 3.11 Fully developed velocity proﬁle of the 1% shear thinning liquid

at X = 45 ........................................................................................................................... 108
Figure 3.12 Velocity proﬁles of P1 at various axial locations before the

transition Re“ = l ............................................................................................................. 109
Figure 3.13 Velocity proﬁles ofPl aﬁerthe onset ofasymmetry. Re“ =2.0 ....................... 110
Figure 3.14 Velocity proﬁles ofPl aﬁerthe onset ofasymmetry. Re“ =2.5 ....................... 112
Figure 3.15 VelocityproﬁlesofPl aﬁertheonset ofasymmetry.Re* =3.0 ....................... 113

Figure 3.16 Presence of two vortices in the shear thinning liquid close to
Y=-l.Re*=2.5,X=1 ................................................................................................... 114

Figure 3.17 Effect of shear thinning on the velocity proﬁles at Reynolds
number = 2.5; X = l .......................................................................................................... 1 16

Figure 3.18 Effect of shear thinning on the velocity proﬁles at Reynolds

number = 2.5; X = 8 .......................................................................................................... 1 17
Figure 3.19 Velocity proﬁles ofshearthinningliquid P1 atX= 1 ...................................... 119
Figure 3.20 Velocity proﬁles of shear thinning liquid Pl at X = 8 ....................................... 120
Figure 3.21 Velocity proﬁles ofshear thinning liquid P1 atX= l6 ..................................... 121
Figure 3.22 Inferred transverse ﬂow in shear thinning liquid P1 ......................................... 122
Figure 3.23 Symmetric velocity proﬁles of P2 at Re“ = 1.5 ............................................... 123
Figure 3 .24 Growth of asymmetry with the Reynolds number in the

Newtonian liquid ............................................................................................................... 126
Figure 3.25 Growth of asymmetry with the Reynolds number in the

shear thinning liquid P1 ...................................................................................................... 127
Figure 3.26 Decay of asymmetry in the Newtonian liquid, with increasing

ﬂow length ....................................................................................................................... 128
Figure 3.27 Decay of asymmetry in the shear thinning liquid, with increasing

ﬂow length ........................................................................................................................ 129
Figure 3.28 Biﬁrrcation curve for the Newtonian liquid N .................................................. 131
Figure 3.29 Bifurcation curve for the shear thinning liquid P1 ............................................ 132
Figure 3.30 Computed decay of asymmetry in the Newtonian liquid ................................... 134
Figure 3.31 Computed decay ofasymmetryinthe shearthirming liquid .............................. 135
Figure 3.32 Computed velocity proﬁles of the Newtonian liquid at Re = 8 ......................... 137
Figure 3.33 Computed velocity proﬁles of the Newtonian liquid at Re = 25.3 ..................... 138
Figure 3.34 Computed velocity proﬁles of the shear thinning liquid P1 at Re“ = 20 ............ 140
Figure 3.35 Computed velocity proﬁles of the shear thinning liquid P1 at Re“ = 44.3 ......... 141
Figure 3.36 Comparison of the degree of asymmetry from LDV experiments

with that from the FEM computations for Newtonian ﬂow ................................................. 143
Figure 3.37 Degree of asymmetry for the polymer solution Pl ........................................... 144

xii

Figure 4.1 Schematic of the ﬂow channel used in calculations ............................................ 158

Figure 4.2a Schematic of the continuation - perturbation approach ..................................... 161
Figure 4.2b Steps involved in the continuation approach .................................................... 163

Figure 4.3 Eﬂ‘ect of grid reﬁnement on biﬁrrcation curve. Newtonian ﬂow,

Expansion ratio = 3 ............................................................................................................ 166
Figure 4.4 Demonstration of the results using the perturbation - continuation

approach for Newtonian ﬂow through a l :3 planar expansion ............................................ 167
Figure 4.5 Bifurcation curve for a Newtonian liquid in a 1:2 expansion .............................. 169
Figure 4.6 Bifurcation curve using center - line velocity as the degree

of asymmetry for a Newtonian liquid through a 1:2 expansion ............................................ 170
Figure 4.7 Bifurcation curve for Newtonian ﬂow in a 1:16 expansion ................................ 172
Figure 4.8 Effect of geometrical asymmetry on Newtonian ﬂow
biﬁrrcationinalzl6channel .............................................................................................. 173
Figure 4.9 Biﬁrrcation curve for n = 0.7, B = 16 ................................................................. 176
Figure 4.10 Bifurcation curve forn=0.5, B =16 ............................................................. 177
Figure 4.11 Flow transitions of a Newtonian liquid in a l :2 expansion ................................ 179
Figure 4.12 Flow transitions of the Newtonian liquid in a 1 :16 expansion ........................... 181
Figure4.13 Flowtransitionsofthe shearthinning liquid P1 ina 1:16 channel .................... 182
Figure 4.14 Velocity proﬁles prior to the onset ofasymmetry. B = 3, Re = 26 ................... 184
Figure4.15Velocityproﬁlesaﬁertheonsetofasymmetry. B=3,Re=34 ........................ 185
Figure 4.16 Computed velocity proﬁles with Newtonian liquid at Re = 8 ............................ 186
Figure 4.17 Computed velocity proﬁles with Newtonian liquid at Re = 25.3 ....................... 187

Figure 4.18 Computed velocity proﬁles of the shear thinning liquid P1
at Re" = 20 ........................................................................................................................ 189

xiii

Figure 4.19 Computed velocity proﬁles of the shear thinning liquid P1

at Re" = 44.3 ..................................................................................................................... 190
Figure 4.20 Eﬂ‘ect of expansion on excess pressure difference in Newtonian ﬂow ............... 192
Figure 4.21 Normalized excess pressure difference in Newtonian ﬂow ............................... 195
Figure 4.22 Eﬂ‘ect of pseudoplasticity on excess pressure difference, B = 16 ..................... 196
Figure 4.23 Excess pressure diﬂ‘erence in shear thinning liquid, n = 0.7 .............................. 197
Figure4.24lncreaseinﬂowresistanceatcritical Re. B=3, n=1 ..................................... 199
Figure 4.25 Increase in ﬂow resistance at critical Re. B = 16, n = 1 ................................... 200
Figure 4.26 Drop in ﬂow resistance due to expansion; Newtonian ﬂow .............................. 201
Figure 4.27 Eﬂ’ect of pseudoplasticity on ﬂow resistance. B = 16 ...................................... 203
Figure 4.28 Eﬂ‘ect of pseudoplasticity on ﬂow resistance in a 1:2 channel ........................... 204
Figure 4.29 Flow resistance in a 1:3 channel ...................................................................... 205
Figure 5.1 Three parameter plot summarizing the eﬂ‘ect of inertia, rheology

and expansion ratio on the occurrence of ﬂow transitions ................................................... 212
Figure A.1 Comparison ofvelocity proﬁlewithJHﬂow. X= 1,Re=1 .............................. 217
Figure A2 Comparison of channel velocity with JH ﬂow, Re = 1 ....................................... 218
FigureA3 Comparisonofvelocityproﬁlesatx= 1,Re=2 .............................................. 219
Figure A4 Comparison of channel velocity with 1H ﬂow, Re = 2 ..................................... 220

xiv

Chapter 1

INTRODUCTION

Flow of ﬂuids through converging and/or diverging geometries is very commonly
encountered in various engineering operations. Flow through oriﬁces, valves and pipe
joints involves ﬂow separations and local recirculations, which have an important impact
on the transfer of heat, mass and momentum. Separation ﬂows in expansions result in
advantages such as reduction in pressure losses and disadvantages like reduction in the
transfer rate. Other examples of diverging / converging ﬂows include the ﬂow of a resin
through a ﬁber bed, which is a complex network of contraction and expansion regions,
ﬂow of body ﬂuids through veins and arteries, etc. Very often the ﬂow is made even more
complicated by the ﬂuids shear thinning viscosity, visooelasticity or extensional viscosity.

Extrusion of shear thinning foods through dies of various shapes is one such operation.

1. Deﬁnitions

Flow through contractions or expansions is schematically shown in Figure 1.1. Flow
occurs around two corners - the salient (or 90°) comer and the non re—entrant (or lip or
270°) comer. Unless otherwise mentioned, the term “vortex" will be used to address the
salient corner vortex in this document. The vortex is measured in terms of the
reattachment length or vortex length (L,,). The ratio of the cross sectional areas of the two
sections (B) is termed as the expansion (or contraction) ratio. If the sections are circular,

I

I...

Salient corner

‘\

 

 

 

 

 

 

 

¢<
\
>

 

 

“I

 

Lip corner

 

 

\
7

 

Figure 1.1 Schematic of ﬂow through expansions or contractions

3

the geometry is referred to as axisymmetric whereas rectangular sections (channels) make

the geometry planar.

2. Interaction of inertia, elasticity, extensional viscosity and pseudoplastic behavior
In the study of ﬂow through contractions and expansions, the ﬂow features of interest
include (a) salient comer vortices (b) lip corner vortices and O symmetric to asymmetric
ﬂow transitions. Figure 1.2 illustrates these schematically. The salient corner vortices
have been the most commonly reported ﬂow feature in any study involving converging or
diverging ﬂows. It has now been conclusively proved, at least for Newtonian ﬂuids, that
these vortices do exist even at very low Reynolds numbers (Moﬁ‘att, 1964). Visualization
of these depend on the resolution of the experiment. Lip vortices are believed to arise in
viscoelastic solutions only, although the computations of Baloch et a]. (1995) predict their
presence in Newtonian liquids ﬂowing through a 1:80 planar expansion. In Newtonian
liquids ﬂowing through abrupt planar expansions, the transition ﬁ'om a symmetric to an
asymmetric ﬂow pattern has been shown to occur at high Reynolds numbers (Cherdron et
al., 1978, Sobey, 1985, Fearn et al., 1990 and Durst et al., 1993, ﬂowing through abrupt
planar expansions. The factors affecting these ﬂow features include inertia, elasticity,
extensional viscosity and pseudoplasticity. Results from experiments with Newtonian
liquids can isolate the contribution of inertia to these ﬂow features. Elasticity and
pseudoplasticity have similar effects on the grth of the salient corner vortices, both in
expansions and contractions. Hence, experiments with inelastic shear thinning polymer
solutions and elastic non-shear thinning polymer solutions are needed to clearly identify

the eﬂ'ects of pseudoplasticity and elasticity on the growth of the salient corner vortices.

Salient cornervortex

 

 

 

 

i

 

 

 

 

/

Lip corner

vortex

 

7

 

 

Figure 1.2a Vortices in contraction l expansion ﬂows

 

 

 

 

 

 

 

 

 

 

/\\\

 

 

Figure 1.2b Asymmetrical ﬂow proﬁle in a planar expansion

5

The importance of extensional viscosity on the occurrence of lip vortices has been shown
by Boger and co-workers (1986, 1994) in a 4:1 axisymmetric contraction, where a
solution with higher extensional viscosity was observed to produce lip vortices, a feature
not seen in the polymer solution with lower extensional viscosity but similar shear and
dynamic properties. Thus, solutions clearly delineating the eﬂ‘ects of inertia, elasticity,
pseudoplasticity and extensional viscosity need to be carefully prepared and characterized
to evaluate the ﬂow patterns through contractions or expansions. The present work
attempts to study the effect of inertia and pseudoplasticity on the ﬂow patterns

through planar expansions with high expansion ratios.

3. Review of previous work done

During the last two decades, considerable attention has been focused on the ﬂow of
polymer solutions and melts through geometries with abrupt changes in cross section, with
more emphasis on contraction ﬂows. A variety of interesting ﬂow characteristics have
been observed in converging ﬂows (Nguyen and Boger, 1978, Walters and Rawlinson,
1982, Boger et a1, 1986, Lawson et al., 1986, Evans and Walters, 1986, McKinley et al.,
1991, Chiba et al., 1992,1995, Quinzani et al., 1995) leading researchers to devote more
attention to contraction ﬂows than to expansion ﬂows. It should be pointed out that only
in the case of creeping ﬂow of Newtonian liquids will the ﬂow features in an expansion be
a mirror-image of that in a contraction ﬂow. This is so because ﬂow of polymers is
governed by non-linear equations with the nonlinearities appearing due to presence of

inertial terms, a shear rate dependent viscosity or polymeric contributions to the stress

6

tensor. Hence, no direct inference for polymers ﬂowing through expansions can be

made based on corresponding contraction ﬂow results.

3.1 Contraction ﬂows
Although the focus of the present work is on planar expansion ﬂows, a description of

contraction ﬂows has been included here for completeness.

Axisymmetric contractions have so far been the favored conﬁguration to be studied, both
in experiments and computations. Experiments with Newtonian liquids through such
geometries have indicated that inertia decreases the salient comer vortex size (i.e. reduces
L,,) irrespective of the contraction ratio (Boger et al., 1986, Abdul-Karem et al., 1993).
When the vortex length is normalized by the upstream section diameter, the value obtained
for creeping ﬂow (Reynolds number < 0.1) is 0.18. This has been veriﬁed computationally
by Christiansen et a1. (1972) and Vrentas and Duda (1973). The effect of elasticity on the
ﬂow in axisymmetric contractions has been studied extensively by Boger and co-workers
and Armstrong and co-workers. Boger et a1. (1986) have concluded that characterization
of the polymer solutions based on the steady and the dynamic shear properties is not
sufﬁcient to explain the differences in ﬂow behaviors of elastic solutions through
axisymmetric contractions. They have reported that differences in the extensional viscosity
of the polymer solutions may lead to completely diﬂ‘erent ﬂow features. To demonstrate
this, they chose two polymer solutions (now called Boger ﬂuids) - a 0.4% solution of
polyacrylarnide (PAA) in water and corn syrup and a 0.1% polyisobutylene (PIB) solution

in kerosene and polybutene. These ﬂuids are constant, high-viscosity ﬂuids with a high

7
degree of elasticity. The eﬂ’ect of elasticity in absence of inertia and shear thinning in any
type of ﬂow may be evaluated using the Boger ﬂuids. Although both these solutions have
similar steady and dynamic shear properties, the PIB solution has a higher extensional
viscosity than the PAA solution. While ﬂowing through a 4:1 axisymmetric contraction,
the PAA solution showed a vortex enhancement beyond a Deborah number (De) of 1,
where Deborah number is deﬁned as the ratio of a characteristic ﬂuid time to a

characteristic ﬂow time.

De=ly

Here A is the relaxation time of the ﬂuid and y is the wall shear rate in the downstream
section. De is a measure of the elasticity of the solution and is increased by increasing the
ﬂow rate in experiments. For an increase in De ﬁ'om 1 to 5, the normalized vortex length
with the PAA solution increased from 0.2 to 0.65. The PIB solution through the very
same geometry, showed a vortex of length 0.2 (the Newtonian, creeping ﬂow value) until
a De of about 1.6 and then reduced to almost zero at higher De. At around De of 2,
vortices were observed on the lip corners. These grew in size as the ﬂow rate was
increased. For De between 2 and 2.2 both the vortices coexisted. This eﬂ‘ect is
summarized in Figure 1.3. As the contraction ratio was increased from 4 to 16, it was
observed that both the ﬂuids behaved in the same fashion i.e. like the PAA solution in a
4:1 contraction. These observations proved that the contraction ratio was also an
important parameter in evaluating the ﬂow patterns. More importantly, this work also
brought out (probably for the ﬁrst time) the importance of extensional properties of the

ﬂuids in entry ﬂows. The solution with the higher extensional viscosity (PIB) showed a

 

0 PAA/Corners» (n - 0'96)
08 . 4 P18! Poiybutens (n I 099)

‘ I I I I

- Lip vortex growth.

or .

O
05 - «l
o-si- J!
0.4..

i
0'
0'3 - /

62- /°/

\ l

 

 

 

Figure 1.3 Vortex length in a 4:1 axisymmetric contraction (Boger et al., 1986).

9

reduction in the salient corner vortex length and an enhancement in the lip corner vortices
with elasticity. The PAA solution exhibited an enhancement of salient comer vortices with
no lip vortices at all. Boger and Binnington (1994) have mentioned that a "true"
comparison of elastic eﬁ‘ects needs the Deborah numbers to be evaluated using a shear
rate dependent relaxation time and not a constant, zero-shear relaxation time as used by
Boger et a1. (1986). Thus, to study elastic effects regardless of the geometry used, both
shear and extensional properties of the ﬂuids are essential. Further, a priori assumption
about the effect of the area ratio on the vortices based on results obtained using diﬂ‘erent
ﬂuids might lead to incorrect conclusions. Similar results with the PIB Boger ﬂuids have
been presented by Lawson et al. (1986). McKinley et a1. (1991) have reproduced the lip
vortices in the PIB Boger solution through a 4:1 contraction. In addition, they have shown
that as De is increased beyond 2, the lip vortices expand outward toward the 90° comer,
until ﬂow instability leads to periodic oscillations of the large vortices. This time-
dependent, 3-dimensional ﬂow near the contraction lip is observed only between
contraction ratios of 2 to 5 and ﬂow remains steady for higher contraction ratios. At very

high De (~ 15), ﬂow becomes aperiodic.

A shear thinning, elastic liquid (1.5% PAA in aqueous corn syrup solution) has been
observed to enhance vortex growth (Boger et al., 1986) in a 4:1 contraction. The range of
De beyond which ﬂow becomes unstable is also increased. Shear thinning, thus, seems to

have a stabilizing effect on the ﬂow patterns produced in an axisymmetric contraction.

10

The ﬂuid rheology and contraction ratio are not the only important parameters in
determining the ﬂow behavior in complex geometries. The mode of contraction, i.e. planar
or axisymmetric assumes signiﬁcance too. Walters and Rawlinson (1982) have shown that
the vortex size decreases with increasing inertia in a 13.3:1 planar contraction (from a
square channel to a slit) for a Newtonian liquid. Boger ﬂuids (aqueous PAA solutions)
which show vortex enhancement in axisymmetric contraction, show very small or no
vortices even at very high ﬂow rates. A 80:1 planar contraction was needed by Evans and
Walters (1986) to observe lip vortices. With 1-2% aqueous PAA solutions (shear thinning,
elastic), a considerable increase in the vortex size was observed with increase in the ﬂow
rate. These solutions exhibited lip vortices in a 16:1 planar contraction. Recently Chiba et
a1. (1992,1995) have reported ﬂow instabilities in a 0.1% PAA solution through a 5:1
planar contraction. At high ﬂow rates, the salient corner vortices were observed to grow
and decay alternately, giving an impression of a moving bundle along the width of the ﬂow

cell. Effect of shear thinning on such ﬂow instabilities has not been reported.

Based on all the existing data on contraction ﬂows, White et al. (1987) in their review on
entry ﬂows mention that vortex growth in general, is more readily observed in
axisymmetric contractions than in planar contractions. A possible explanation of this may
be obtained from the work by James and Saringer (1980, 1982) wherein they measured the
pressure drop for ﬂow of polyethylene oxide solution through axisymmetric and planar
oriﬁces. Since ﬂow through an oriﬁce is essentially shear-ﬂee, the pressure dr0p gives a
relative measure of the extensional stress in the contraction region. The pressure drop in

the axisymmetric case was several times higher than in the planar case leading to the

11
conclusion that large extensional stresses are developed in axisymmetric ﬂow as compared
to planar ﬂow. Since vortices are a means for the ﬂuid to relieve stress, it would be
expected that larger vortices would be formed in axisymmetric conﬁgurations at

essentially similar ﬂow conditions.

3.2 Expansion ﬂows

3.2.1 Experimental results

3.2.1.1 Flow transitions due to inertia in Newtonian ﬂow through planar expansions
Sobey (1985) has studied the ﬂow patterns developed in a Newtonian liquid (water)
ﬂowing steadily through a 1:3 two-dimensional planar expansion. He observed that at low
ﬂow rates (or low Reynolds number, Re, deﬁned using the smaller channel half height and
the average velocity in the upstream channel), the corner vortices were symmetric and that
they increased in size as Re was increased. Beyond a certain critical Re (around 25 for an
abruptly expanding channel), ﬂow became asymmetric and one vortex grew more rapidly
than the other. This is shown in Figures 1.4 and 1.5. As Re was raised ﬁrrther, a third
recirculation region far downstream of the expansion plane and on the wall with the
smaller corner vortex was observed too. The critical Reynolds. number for the onset of the
ﬂow transitions was observed to be higher for channels with gradual expansions. By using
long rectangular channels of large aspect ratios (5 and above) Sobey and others have
ensured that the ﬂow far away from the expansion plane is one dimensional. Experiments
showing similar ﬂow transitions in Newtonian ﬂuids through abrupt expansions in 1:2 and
1:3 channels have been reported by Durst et al. (1974) and Cherdron et al. (1978) and

conﬁrmed later by Feam et al. (1990) and Durst et al. (1993). The phenomenon wherein

12

 

Figure 1.4 Flow patterns with water through a 1:3 planar expansion (Sobey, 1985).
(a) Re = 55 (b) Re = 115.

13

5.6

LJD

 

0.8

 

 

 

Figure 1.5 Vortex asymmetry in a Newtonian liquid through a 1:3 planar expansion
(Sobey, 1985).

l4
ﬂow transitions lead to diﬂ‘erences in the ﬂow patterns beyond a certain critical Reynolds
number is commonly referred to as “ﬂow bifurcation” and the critical Reynolds number is

oﬁen called the bifurcation point.

Cherdron et al. (1978) have investigated the eﬂ‘ect of the expansion ratio and the aspect
ratio (ratio of the ‘width of the channel to its height) on the ﬂow patterns developed in a
Newtonian liquid ﬂowing through a planar expansion. They have reported that the critical
Re at which ﬂow transitions occur goes down with an increase in the expansion ratio as
well as an increase in the aspect ratio (see Figure 1.6). For example, with a channel of
aspect ratio 8, the critical Re goes down from about 150 to 30 in going from an area
expansion ratio of 1 :2 to 1 :3. The mechanism of initiation of such ﬂow transitions, even
for Newtonian liquids, is not well understood. However, the Coanda effect does seem to
explain how it is sustained. Flow asymmetry is a result of the momentum transfer in the
direction normal to ﬂow between the shear layers. As a result, the boundary layer adjacent
to one wall is richer in momentum and consequently at a lower pressure as compared to
the other. A transverse pressure gradient in the channel is created and as more liquid
enters, ﬂow of the liquid from the wall at higher pressure to that at the lower pressure
takes place. This increases the momentum at the already richer wall even more, resulting
in a more pronounced effect of the ﬂow asymmetry. Interestingly, no ﬂow asymmetry has
been reported in axisymmetric expansions. In such conﬁgurations, ﬂow induces concentric
shear layers, as opposed to the stacked shear layers in planar ﬂow. This provides a
continuity all along the plane of expansion in the former, preventing any occurrence of

asymmetrical ﬂow patterns.

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33. a... .3 eta.— ..eiiaﬁ can eta.— 325. he tuba w; 8:»...—

 

 

 

 

 

 

 

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16

Although trends between the ﬂow transitions occurring in expansion channels are
available now, all the existing results are limited to low expansion ratios of 2 and 3.
Experiments with large expansion ratios will help to answer questions like what
happens to the ﬂow patterns at much higher expansion ratios and what is the rate of
fall of the critical ﬂow rate (or an equivalent critical Reynolds number) when

extreme expansion ratios are involved?

3.2.1.2 Non - Newtonian ﬂow through expansions

Abdul-Karem et al., (1993) have investigated ﬂow behavior of 0.05% polyacrylarnide
solution in glucose syrup (Boger ﬂuid) through a 1:6.8 axisymmetric expansion and a
6.8:] axisymmetric contraction. Although the vortex size observed in expansions was
lower than that in contractions, the trend of vortex enhancement with increasing ﬂow rate
was observed in the expansion ﬂow too. No lip vortices or instabilities were observed in
either conﬁguration. Halmos et al. (1975) have conducted experiments using aqueous
carboxymethyl cellulose solution (inelastic, shear thinning power-law index of 0.73) in a
1:2 abrupt axisymmetric expansion. They have measured the radial and axial velocity
proﬁles downstream of the expansion plane using ﬂash photography. Their experimental
results on velocity proﬁles and reattachment lengths agree well with their calculations for
Reynolds numbers between 11 to 125. The authors do not observe any ﬂow asymmetries,

oscillations or lip vortices.

 

17
Townsend and Walters (1994) have studied the ﬂow of a 0.15% aqueous solution of
polyacrylamide (slightly shear thinning, strongly elastic) through an expansion from a
square section to a rectangular slit (area expansion of 13.3) and ﬁ'om the rectangular slit
to another square section (area expansion of 13.3). A schematic of the geometry used by
them is shown in Figure 1.7. With a Newtonian liquid, signiﬁcant vortex activity was
observed due to ﬂuid inertia. The vortex size increased with the ﬂow rate. The polymer
solution, on the other hand, produced small comer vortices, the size of which decreased
with the ﬂow rate. Fluid elasticity thus, was observed to inhibit vortex growth. It should
be pointed out that their study with Newtonian liquids has not shown any evidence of ﬂow
asymmetry, in contrast to those reported by Durst et al. (1974), Cherdron et al. (1978),
Sobey (1985) and Feam et al. (1990), despite a wide range of Reynolds numbers used.
The square channels used by Townsend and Walters (1994) approximate axisymmetric
ﬂow which has never been known to show the ﬂow asymmetry mentioned above. In
addition, based on the ﬁndings of Cherdron et al. (1978) that the critical Reynolds number
for the onset of ﬂow asymmetry increases with decreasing channel aspect ratio, one might
conclude that at an aspect ratio of 1, ﬂow asymmetry might be observed, if at all, only at
very high Reynolds numbers. One notable contribution of the Townsend and Walters work
was to clearly bring out the effect of ﬂuid elasticity on the vortex growth in planar
expansions. The observed trends in the effects of ﬂuid inertia and elasticity on secondary
ﬂows was in contrast to what had been observed in various contraction geometries, where
ﬂuid inertia reduced the vortex size while elasticity led to vortex enhancement. This
reversal of roles may be the reason for lack of further investigation of viscoelastic eﬂ‘ects

in expansion ﬂows.

18

 

Flow

 

 

 

A\
Ky
\/

 

 

 

 

Figure 1.7 Flow conﬁguration used by Townsend and Walters (1994).

l9

Conspicuous by its absence from literature is any indication whether non -
Newtonian ﬂows will show ﬂow bifurcation at high Reynolds numbers. If they
indeed do so, the effect of ﬂuid rheology on the bifurcation point and on the ﬂow

patterns created needs to be studied.

3.2.2. Computational results in planar expansions

Various numerical procedures have been used to predict the ﬂow biﬁircation in a planar
sudden expansion. Computations involving ﬁnite elements have always needed some kind
of perturbation - either to the geometry or to the boundary conditions - to yield a
biﬁrrcated ﬂow structure. Durst et al. (1993) have made transient calculations to predict
the onset of ﬂow transitions for a Newtonian liquid. They followed a perturbation
approach which involved the following steps:

At time t = 0, a fully developed velocity proﬁle was imposed at the inlet of the smaller
channel. After n time iterations and during m time iterations, the inlet velocity conditions
were modiﬁed such that the axial velocity component (U) was a ﬁrnction of time and the
normal velocity component (V) was a ﬁrnction of both time and space. Subsequently, the
original symmetric ﬁrlly developed velocity proﬁle was restored as the inlet boundary
condition. The solution was allowed to achieve steady state. Using this approach a critical

Reynolds number of 42 was predicted for ﬂow through a 1:2 expansion.

20

F earn et al. (1990) used numerical biﬁrrcation techniques applied to a ﬁnite element
discretization of the two - dimensional Navier - Stokes equations and predicted ﬂow
bifurcation in a 1:3 planar expansion at Re,l of 27. Their work sheds more light on the
sensitivity of the bifurcation point to experimental uncertainties. They suggest that the
critical transition point is structurally unstable. The slightest imperfection in the
experiment may lead to a discontinuity in the bifurcation curve, with a shift in the original
biﬁrrcation point. They showed that a vertical shift in the grid of the larger channel by
0.42% with respect to the axis of symmetry provided much closer match with the
experimental data. Their work seems to explain the possible reason for the scatter
observed in the experimental results on ﬂow biﬁrrcations through planar expansions.
Indeed , experimental imperfections could have many sources such as geometrical
asymmetry, upstream ﬂow asymmetry, momentary mechanical shock, temperature

variation, etc.

Teschauer (1994) employed an asymmetric upstream channel to compute the bifurcated
branch of the solution for various expansion ratios between 1.5 to 3. A commercial ﬁnite
element code was used. She predicted critical Reynolds numbers of 72 and 27 for

expansion ratios of 2 and 3 respectively.

In a diﬂ‘erent ﬂow conﬁguration involving Newtonian ﬂow into a rectangular channel
through two symmetric openings, Goodwin and Schowalter (1996) have shown the
possibility of the simultaneous existence of numerous solutions beyond a critical Reynolds

number. To access the asymmetric solutions, they have followed a continuation approach

21

coupled with a perturbation in the ﬂow rates in the two channels. Using a ﬁnite element
discretization, solutions with the asymmetrical ﬂow rates were computed for increasing
Reynolds numbers. Results at the lower Reynolds numbers were used as initial iterates for
computations at higher Reynolds numbers. At signiﬁcantly high Reynolds numbers, the
ﬂow was rebalanced and the same continuation approach was used for successively
decreasing Reynolds numbers. Such an approach detected the various asymmetric
branches of the solutions. A similar approach using continuation and a perturbation in the
geometry to compute the ﬂow bifurcation in planar expansions with a single inlet has been

devised in the present work. This will be discussed later in Chapter 4.

Very recently numerical results using a ﬁnite volume approach for ﬂow through planar
expansions have been presented. A striking feature of such an approach is that ﬂow
biﬁrrcation has been computed without providing any kind of perturbation to the ﬂow
conditions. Drikakis (1997) has presented biﬁrrcation curves from several schemes for
expansion ratios up to 6. Various numerical schemes based on an overall ﬁnite volume
approach have been tested. His results for the onset of asymmetry with expansion ratios of
2 and 3 match the corresponding results of Teschauer (1994) and the results ﬁ'om the
linear stability analysis of Shapira et al. (1990). Foumeny et al. (1996) have shown that a
commercial code, FLUENT, can be effectively utilized to study the ﬂow bifurcations. This
code employs a ﬁnite volume discretization procedure, in which linear approximation
ﬁrnctions are used to interpolate values of the ﬂow variables throughout the solution

domain. Their focus is on obtaining results ~on channels with more than one inlet opening.

22

A characteristic observation ﬁom the computational results is that ﬂow biﬁrrcation
calculations almost invariably predict a higher critical Reynolds number than the
corresponding value obtained in experiments. This is consistently observed in the work

described above on channels with low expansion ratios of 2 and 3.

Sobey (1985) and Sobey and Drazin (1986) have attempted to relate the experimentally
observed biﬁrrcation to that in Jeffery - Hamel (JH) ﬂows. In the latter, radial ﬂow occurs
between two inclined plates intersecting at an angle 2a. Flow is generated by a source
located on the line of intersection. JH ﬂows have been shown to bifurcate subcritically,
leading to no stable ﬂow above the critical Reynolds number. Planar expansion ﬂows, on
the other hand, display a supercritical biﬁircation which is conﬁrmed by several reports of
stable asymmetric patterns in planar expansions. Sobey, therefore, concluded that JH

ﬂows are not of any signiﬁcance in explaining expansion ﬂow after biﬁrrcation.

Shapira et al. (1990) have performed a linear stability analysis of ﬂow in symmetric
expansions. They have considered a gradual expansion with a constant slope of the
channel walls. Although their predictions of Recl = 27.5 for a 1:3 expansion are in
agreement with the computational results of F earn et al. (1990), whose computations
suggest the onset of asymmetry at Re of 27, they over predict Recl of 72 for a 1:2
expansion. More importantly their computations show that the symmetric branch of the
solution obtained above the critical Reynolds number is unstable and therefore unlikely to

be ever witnessed in experiments.

 

23
Alleborn et al. (1997) have performed a linear stability analysis on ﬂows through abrupt

expansions. Finite diﬂ'erence approximations of the equations of motion in the stream
ﬁrnction - vorticity form have been used. The biﬁrrcation picture is extended by computing
the subsequent biﬁrrcation point (Red) and solution branches. Solutions for inﬁnite
expansion ratio have been compared with Jeﬂ'ery - Hamel ﬂow. They conclude that away

ﬁom the expansion plane ﬂow proﬁles do approach the proﬁles of JH ﬂow.

Computational results on ﬂow bifurcation of Newtonian liquids through planar
expansions are limited to low expansion ratios. Further, no results demonstrating
the effect of non - Newtonian ﬂow behavior on the ﬂow bifurcations, even for low

expansion ratios, are available in literature.

4. Objectives
Attempts have been made to ﬁll some of the missing links within the literature for ﬂow

through planar expansions. The objectives of the present study are:

e To carry out experiments to study the ﬂow transition of Newtonian liquids ﬂowing

through planar channels of large expansion ratios.

.To compare the ﬂow bifurcation results with those available for low expansion ratios.

24

.To use experimental techniques to explore the eﬂ'ect of non - Newtonian viscosity on the

ﬂow bifurcation and on the ﬂow patterns produced by the ﬂow of shear thinning, inelastic

liquids through planar expansions with large expansions ratios.

.To devise a protocol to compute the ﬂow biﬁrrcation for Newtonian and non -

Newtonian liquids, ﬂowing through planar expansions.

.To evaluate the eﬁ‘ect of ﬂuid rheology and expansion ratio on the ﬂow bifurcation of

liquids through planar expansions.

Model channels have been designed with large aspect and expansion ratios. Newtonian
model ﬂuids and inelastic, non-Newtonian model ﬂuids have been prepared and the eﬂ‘ects
of ﬂuid rheology and injection rate on the ﬂow patterns have been experimentally
investigated using streak photography and laser Doppler velocirnetry. A novel approach to
compute the ﬂow bifurcations for Newtonian and non - Newtonian liquids has been
devised and then tested for accuracy. Flow bifurcation results using this approach have
been presented for Newtonian and non - Newtonian ﬂuids ﬂowing through planar

expansions.

Following is the organization of the various results in this document:
(1) Chapter 2: Critical Reynolds numbers for the onset of ﬂow asymmetry have been

identiﬁed for the model ﬂuids using streak photography. The detailed ﬂow patterns and

25

ﬂow transitions for the Newtonian and non - Newtonian liquids have been presented.
Relationships between the Newtonian vortex length prior to the onset of ﬂow transitions,
Reynolds number-and the expansion ratio have been proposed and tested. A brief
description of the computations performed to determine the bifurcation curves for
Newtonian and non - Newtonian ﬂow through 1:2, 1:3 and 1:16 planar expansions has
been included. Flow prior to the onset of asymmetry has been studied in detail for channels
with expansion ratios 2, 3 and 16 with Newtonian and inelastic shear - thinning liquids.
Experimental and computational results on the trends of vortex growth for non-Newtonian

liquids have been compared.

(2) Chapter 3: Laser Doppler velocity measurements with both Newtonian and non -
Newtonian liquids have been presented. Details of the vortex grth and the decay of
asymmetry with axial position have been reported. Flow biﬁrrcation curves using the

center line shear rate as the measure of asymmetry have been generated for the liquids.

(4) Chapter 4: The approach developed to compute the ﬂow bifurcation for Newtonian
and non - Newtonian liquids has been described in detail. Using this approach, biﬁrrcation
curves, streamlines, velocity proﬁles, excess pressure difference and ﬂow resistances have
been evaluated for ﬂuid shear thinning indices of 0.5, 0.7 and 1 and channel expansion

ratios of 2, 3 and 16.

(5) Chapter 5: Conclusions from the present study have been listed and recommendations

for future work have been made.

Chapter 2
FLOW VISUALIZATION OF THE ASYMMETRIC FLOW TRANSITIONS

IN AN ABRUPT PLANAR EXPANSION

1. Introduction

Flow of ﬂuids through diverging geometries is very commonly encountered in various
engineering operations. Flow through oriﬁces, valves and pipe joints involves ﬂow
separations and local recirculations, which have an important impact on the transfer of
heat, mass and momentum. Separation ﬂows in expansions result in advantages such as
reduction in pressure losses and disadvantages like reduction in the transfer rate. Another
example of diverging ﬂows includes the ﬂow of a resin through a ﬁber bed, which is a
complicated network of contraction and expansion regions. Very often ﬂow is made even
more complicated by the shear thinning viscosity, viscoelasticity or extensional viscosity.

Extrusion of shear thinning foods through complex shaped dies is one such operation.

Another class of ﬂow through expansions is the unsteady state ﬁlling operation observed
in injection molding and the exit ﬂow of a polymer extrudate from an extruder die. In
injection molding, ﬂow of polymer melts into the mold cavity oﬁen results in ﬂow
transitions above a certain critical injection rate. In extrusion, ﬂow of the extnrdate ﬁom
the die to the ambient results in many instabilities (like wavy, bamboo, shark skin, ripple,
severe distortion, etc.) only at higher injection rates and with smaller die sizes (Baird and

26

27

Collias, 1995). A clear understanding of such ﬂow phenomena is very complicated due to
three - dimensional ﬂows, non - isothermal melt temperatures, complex ﬂow geometries
and non - Newtonian properties of the melt. Continuous, steady ﬂow of Newtonian ﬂuids
through abrupt planar expansions has also revealed ﬂow transitions above a critical ﬂow
rate of the ﬂuid (Cherdron et al. 1978, Sobey, 1985, Feam et al., 1990 and Durst et al.,
1993). This critical ﬂow rate is known to vary with the area expansion ratio ([5) and the
aspect ratio (ratio of the width to the height) of the channel. Attention has been restricted
to obtain and analyze results for the ﬂow of a Newtonian liquid (water) through abruptly
expanding channels with area expansion ratios of 2 and 3. Although trends between the
ﬂow transitions occurring in these channels are available now, certain questions still need
to be resolved. For example, what happens to the ﬂow patterns at much higher expansion
ratios? What is the rate of fall of the critical ﬂow rate when extreme expansion ratios are
involved? As far as this author is aware, there has been no attempt - experimental or
computational - to gather evidence of such ﬂow transitions with non - Newtonian ﬂuids
and to study the effect of ﬂuid rheology on the occurrence of ﬂow transitions, assuming
that non - Newtonian liquids also behave similarly. The present study has been devoted to
answering these queries related to the steady ﬂow of Newtonian and non - Newtonian

liquids through planar expansions.

2. Background
2.1 Contraction vs Expansion ﬂows
To understand the ﬂow transitions of Newtonian liquids through planar expansions, a

systematic, fundamental study is needed to understand more clearly the effects of ﬂuid

28
nature, ﬂow geometry and inertia on the ﬂow characteristics. Unfortunately, there has
been relatively less focus on expansion ﬂows compared to the considerable attention that
has been devoted to the ﬂow of polymer solutions and melts through converging
geometries. One possible reason for this could be the occurrence of various interesting
ﬂow characteristics in steady converging ﬂows reported by Nguyen and Boger, 1979,
Walters and Rawlinson, 1982, Boger et a1, 1986, Lawson et al., 1986, Evans and Walters,
1986, McKinley et al., 1991, Chiba et al., 1992,1995, Quinzani et al., 1995, etc. These
features include the presence of lip vortices, enhanced comer vortex growth due to ﬂuid
viscoelasticity, oscillating ﬂow and Goertler vortices. White et al. (1987) and Boger
(1987, 1997) have presented a detailed review of such entry ﬂows, while a few relevant
ones have been discussed in Chapter 1. It is now accepted that in contractions, inertia
decreases the size of the salient corner vortex (Boger et al., 1986, Abdul - Karem et al.,
1993) whereas viscoelasticity enhances the vortex growth (Boger et al., 1986, Lawson et

al., 1986 and McKinley et al., 1991).

Fluid viscoelasticity and inertia produce the opposite effect in expansions (Abdul - Karem
et al., 1993). Shear thinning seems to have a stabilizing effect on the ﬂow patterns
produced in an axisymmetric contraction by delaying the onset of ﬂow instabilities (Boger
et al., 1986). The effect of shear thinning on expansion ﬂows has not been reported
yet. The ﬂow transitions of Newtonian liquids through planar expansions reported by
Cherdron et al., 1978 and others have not been observed in contraction ﬂows. It should be
pointed out that only in the case of creeping ﬂow of Newtonian liquids will the ﬂow

features in an expansion be a mirror-image of that in a contraction ﬂow because only in

29
this case is the problem governed by linear equations. Inertial eﬂ‘ects with any liquid would
lead to differences in ﬂow patterns. With non - Newtonian liquids, additional nonlinearities
ﬁom the liquid properties such as shear thinning viscosity, viscoelasticity, extensional
viscosity and yield stress would make the ﬂow patterns in expansions diﬁ‘erent from that in
contractions. Hence, no direct inference for polymers ﬂowing through expansions can be

made based on corresponding contraction ﬂow results.

2.2 Effect of geometry - Axisymmetric vs Planar

Boger ﬂuids (viscoelastic ﬂuids with a constant viscosity) which show vortex enhancement
in axisymmetric contractions, show very small or no vortices in planar contractions, even
at high ﬂow rates (Walters and Rawlinson, 1982). Lip vortices observed commonly in
viscoelastic liquids ﬂowing through axisymmetric contractions, are seen in planar
contractions only at very high contraction ratios (Evans and Walters, 1986). Halmos et al.
(1975) have shown that increasing shear thinning leads to increasing size of the vortices in
a 1:2 axisymmetric expansion. Corresponding results for planar expansions are non -
existent. White et al. (1987) in their review on all the existing data on contraction ﬂows
mention that vortex growth in general, is more readily observed in axisymmetric
contractions than in planar. Differences in the ﬂow features of Boger ﬂuids between
axisymmetric and planar contractions have been explained by White et al. on the basis of
the work by James and Saringer (1980, 1982). In their work, James and Saringer have
examined planar and axisymmetric sink ﬂow of polyethylene oxide solutions. Sink ﬂow
may be approximated essentially as a shear-free ﬂow and therefore the pressure drop

through the oriﬁce is a relative measure of the extensional stress in the contraction region.

30

It was observed that the pressure drop in the axisymmetric case was several times higher
than in the planar case. This behavior was numerically predicted by Chakraborty and
Metzrrer (1986). A similar analogy could be derived for expansion ﬂows too, to explain

diﬂ‘erences between axisymmetric and planar expansion ﬂows.

2.3 Flow transitions due to inertia in Newtonian ﬂow through planar expansions
Sobey (1985) has. studied the ﬂow patterns developed in a Newtonian liquid (water)
ﬂowing steadily through a 1:3 two-dimensional planar expansion. He observed that at low
ﬂow rates (or low Reynolds number, Re, deﬁned using the smaller channel half height and
the average velocity in the upstream channel), the corner vortices were symmetric and that
they increased in size as Re was increased. Beyond a certain critical Re (around 25 for an
abruptly expanding channel), ﬂow became asymmetric and one vortex grew more rapidly
than the other. As Re was raised further, a third vortex was observed downstream of the
expansion plane on the wall with the smaller corner vortex. The critical Reynolds number
was observed to be higher when channel with gradual expansions were used. By using
long rectangular channels of large aspect ratios (5 and above) Sobey and others have
ensured that the ﬂow far away ﬁ'om the expansion plane is one dimensional. Experiments
showing similar ﬂow transitions in Newtonian ﬂuids through abrupt expansions in 1 :2 and
1:3 charmels have been reported by Durst et al. (1974) and by Cherdron et al. (1978) and
conﬁrmed later by Fearn et al. (1990) and by Durst et al. (1993). The phenomenon
wherein ﬂow transitions lead to diﬂ‘erences in the ﬂow patterns beyond a certain critical
Reynolds number is commonly referred to as “ﬂow bifurcation” and the critical Reynolds

number is called the biﬁrrcation point.

31

Cherdron et al. (1978) have investigated the eﬁ‘ect of the expansion ratio and the channel
aspect ratio (ratio of the width of the channel to its height) on the ﬂow patterns developed
in a Newtonian liquid ﬂowing through a planar expansion. They have reported that the
critical Re at which ﬂow transitions occur goes down with an increase in the expansion
ratio as well as an increase in the aspect ratio. For example, with a channel of aspect ratio
8, the critical Re goes down from about 150 to 30 in going from an area expansion ratio
of 1:2 to 1:3. The mechanism of initiation of such ﬂow transitions, even for Newtonian
liquids, is not well understood. However, the Coanda eﬁ‘ect does seem to explain how it is
sustained. Flow asymmetry is a result of the momentum transfer in the direction normal to
ﬂow between the shear layers. As a result, the boundary layer adjacent to one wall is
richer in momentum and consequently at a lower pressure as compared to the other. A
transverse pressure gradient in the channel is created and as more liquid enters, ﬂow of the
liquid from the wall at higher pressure to that at the lower pressure takes place. This
increases the momentum at the already richer wall even more, resulting in a more
pronounced effect of the ﬂow asymmetry. Interestingly, no ﬂow asymmetry has been
reported in axisymmetric expansions, where the ﬂow induces concentric shear layers, as
opposed to the stacked shear layers in planar ﬂow. A continuity all along the plane of

expansion in the former, prevents any occurrence of asymmetrical ﬂow patterns.

Although results on the ﬂow transitions of Newtonian ﬂows through expansion
channels are available now, all the existing results are limited to low expansion

ratios of 2 and 3. Experiments with large expansion ratios will help to answer

32

questions like what happens to the ﬂow patterns at much higher expansion ratios
and what is the rate of fall of the critical ﬂow rate (or an equivalent critical

Reynolds number) when extreme expansion ratios are involved?

2.4 Experimental work on Non - Newtonian ﬂows through expansions

Abdul-Karem et al., (1993) have investigated ﬂow behavior of 0.05% polyacrylanride
solution in glucose syrup (Boger ﬂuid) through a 1:6.8 axisymmetric expansion and a
6.8:1 axisymmetric contraction. Although the vortex size observed in expansions was
lower than that in contractions, the trend of vortex enhancement with increasing ﬂow rate
was observed in the expansion ﬂow too. No lip vortices or instabilities were observed in
either conﬁguration. Halmos et al. (1975) have conducted experiments using aqueous
carboxyrnethyl cellulose solution (inelastic, shear thinning power-law index of 0.73) in a
1:2 abrupt axisymmetric expansion. They have measured the radial and axial velocity
proﬁles downstream of the expansion plane using ﬂash photography and have compared
those with their calculations for Reynolds numbers between 11 to 125. A good agreement
was observed between their experimental and computational results. The authors do not

observe any ﬂow asymmetries, oscillations or lip vortices.

Townsend and Walters (1994) have studied the ﬂow of a 0.15% aqueous solution of
polyacrylarnide (slightly shear thinning, strongly elastic) through an expansion ﬁ'om a
square section to a rectangular slit (area expansion of 13.3) and ﬁ'om the rectangular slit
to another square section (area expansion of 13.3). With a Newtonian liquid, signiﬁcant

vortex activity was observed due to ﬂuid inertia. The vortex size increased with the ﬂow

33

rate. The polymer solution, on the other hand, produced small comer vortices, the size of
which decreased with the ﬂow rate. Fluid elasticity thus, was observed to inhibit vortex
growth. It should be pointed out that their study with Newtonian liquids has not shown
any evidence of ﬂow asymmetry, in contrast to those reported by Durst et al. (1974),
Cherdron et al. (1978), Sobey (1985) and Fearn et al. (1990), despite a wide range of
Reynolds numbers used. The square channels used by Townsend and Walters (1994)
approximate axisymmetric ﬂow which has never been known to show the ﬂow asymmetry
mentioned above. In addition, based on the ﬁndings of Cherdron et al. (197 8) that the
critical Reynolds number for the onset of ﬂow asymmetry increases with decreasing
channel aspect ratio, one might conclude that at an aspect ratio of 1, ﬂow asymmetry
might be observed, if at all, only at very high Reynolds numbers. One notable contribution
of the Townsend and Walters work was to clearly bring out the eﬂ‘ect of ﬂuid elasticity on
the vortex growth in planar expansions. The observed trends in the effects of ﬂuid inertia
and elasticity on secondary ﬂows was in contrast to what had been observed in various
contraction geometries, where ﬂuid inertia reduced the vortex size while elasticity led to
vortex enhancement. This reversal of roles may be the reason for lack of further

investigation of viscoelastic effects in expansion ﬂows.

Conspicuous by its absence from literature is any indication whether non -
Newtonian ﬂows will show ﬂow bifurcation at high Reynolds numbers. Ifthey
indeed do so, the effect of ﬂuid rheology on the bifurcation point and on the ﬂow

patterns created needs to be studied.

 

34
2.5 Vortex growth prior to the onset of asymmetry
Acrivos and Schrader (1982) have shown that prior to the onset of asymmetry, a linear
relation exists between the reattachment length, the expansion ratio and the Reynolds
number. They have derived an analytical expression that successﬁrlly predicts the
symmetric vortex length at high Reynolds numbers for Newtonian ﬂow in 1:2 and 1:3
planar expansions. Their expression, however, is valid only at high Reynolds numbers and
does not account for the Moﬂ‘att eddies (Moﬂ‘att, 1964), which exist even under creeping
ﬂow conditions. Baloch et al. (1995) have numerically computed the reattachment length
and have reported vortex growth curves for a wide range of Re for 1:4, 1:133 and 1:80
planar expansions. They indicate that the vortex growth is not linearly related to Reynolds
number, particularly for channels with higher expansion ratios. As far as this author is
aware, no study has addressed this issue of trying to relate the vortex length and the
expansion ratio at both low and high Reynolds numbers in planar channels, before

the onset of asymmetry.

3. Objectives
The present work attempts to ﬁll some of the missing links within the literature for ﬂow

through planar expansions. The objectives of the present chapter are:
c To study the ﬂow transition of Newtonian liquids ﬂowing through planar channels of
large expansion ratios.

.To compare the ﬂow bifurcation results with those available for low expansion ratios.

35

e To relate the Newtonian vortex grth before the onset of the ﬂow transitions to

expansion ratio and the Reynolds number.

.To explore the effect of non - Newtonian viscosity on the ﬂow bifurcation and on the

ﬂow patterns produced by the ﬂow of shear thinning, inelastic liquids through planar

expansions with large expansions ratios.

Model channels have been designed with large aspect and expansion ratios. Newtonian
model ﬂuids and inelastic, non-Newtonian model ﬂuids have been chosen and the effects
of ﬂuid rheology and injection rate on the ﬂow patterns have been experimentally
investigated using streak photography. Critical Reynolds numbers for the onset of ﬂow
asymmetry have been identiﬁed for the model ﬂuids. Brief descriptions of the
computations performed on 1:2, 1:3 and 1:16 planar expansions have been included. The
details are reported in Chapter 4. Flow prior to the onset of asymmetry has been studied in
detail for channels with expansion ratios 2, 3 and 16 with Newtonian and inelastic shear-
thinning liquids. Relationships between the Newtonian vortex length and Re for the whole
range of Reynolds number have been proposed. Experimental and computational results
on the trends of vortex growth for non-Newtonian liquids have also been presented. Laser
Doppler velocity (LDV) measurements have been presented in Chapter 3. Detailed

computations of the bifurcation curves have been performed and are reported in Chapter

4.

36
4. Experimental
4.1 Model ﬂow cell
A model ﬂow cell. with an abrupt area expansion of 16 has been constructed by using
inserts to join two rectangular channels of the same width. An area expansion ratio of 16
is much higher than the ratios of 2 and 3, results for which are available in literature. A
schematic of this transparent ﬂow cell is sketched in Figure 2.1. Figure 2.2 shows the
characteristic dimensions involved. An aspect ratio of 10 or more in either channel
ensures that the upstream and downstream ﬂows far from the expansion plane are one
dimensional. Flow at the entry of the smaller channel is spread uniformly in the width of
the channel by using a distributor plate. The inner inserts may be easily modiﬁed to
incorporate a gradual expansion between the two channels. The height of the larger
channel may be varied to change the area expansion ratio. Thus, the present "model
conﬁguration" permits ﬂow observations on both abruptly and gradually expanding
channels for more than one expansion ratio. Since the ﬂow cell is constructed ﬁ'om
plexiglass, optical studies may be performed all along the length. During the assembly of
the cell, care was taken to monitor the deﬂection of the plates while tightening the screws.
This was achieved by using a Mag Base Indicator (NDI - MX50), which could detect
deﬂections as small as 0.001 ". The thickness of each of the plates was measured by a
screw gauge at least 12 locations. The maximum variation that was inherent to these

plexiglass plates was about 7-8%.

During ﬁlling of the cell, the vents were kept open to facilitate the ﬂow of entrapped air.

The cell was positioned such that the ﬂow direction was aligned vertically against gravity.

37

     
  

 

 

Outer Plate
(15” x 19” x 1”)

Window
(15” x 1.6” x 0.5” )

A

 

 

 

 

 

 

 

 

 

 

 

/ \\\\\\\V\\\\\\\\ \ j
A
’(6135 36” x 0.75”)
I< 19,, >1

Figure 2.1 Design of the model ﬂow cell.

38

 

 

 

l6” ’
F >1 i”: 1...

Left Side View Front View

Figure 2.2 Characteristic dimensions of the model ﬂow cell.

39

After the removal of the trapped air, the vents were shut oﬁ‘ and a valve connecting the
central circular exit on the end-plate to a ﬂow meter was turned on and steady state was
allowed to be established. It must be pointed out that the weight of the ﬂow cell (due to an

exceptionally large size) and its contents did hinder easy movement of the assembly.

4.2. Model ﬂuids
Any model ﬂuid used in the present study needs to be optically clear, non-toxic and non-

degradable due to shear. Table 2.1 lists the model ﬂuids and some of the relevant

properties.

Aqueous solutions of glycerol were used as Newtonian ﬂuids. They were suitably diluted
to yield solutions of viscosities between 12 to 1350 mPa-s. Viscosity measurements were
performed on a Brookﬁeld Viscometer (Model LVF-DVIII). A spindle with a diameter of
1.2 cm (Model SC-31) was used in the tests. The temperature of the sample was

controlled by a heater in which the cup containing the sample was placed.

Different ﬂuid viscosities were needed to span a wide range of Reynolds number, which

was deﬁned as

h U
Re = A3 (2.1)

Here h is the separation between the walls of the upstream channel, p and p are the ﬂuid

density and viscosity respectively and U”. is the average velocity in the upstream channel.

40

The latter was evaluated ﬁ'om the ﬂow rate of the liquid collected at the exit of the
downstream end. The Reynolds number of ﬂow may be varied from 0.001 to 6, covering

regimes of creeping ﬂow as well as low Re ﬂows with non-negligible inertia.

Table 2.1: Model ﬂuids used

 

 

 

 

Solution Designation Viscosity (mPa-s) n Relaxation
(zero shear) time (s)
Aqueous glycerol N 12 - 1350 1 -
1 wt% aqueous
hydroxypropyl cellulose P1 350 0.7 5.4 x 10"
ether
2 wt% aqueous
hydroxypropyl cellulose P2 63 00 0.5 7.8 x 10'3
ether

 

 

 

 

 

 

 

Inelastic polymer solutions have been prepared and used in the study. These shear thinning
solutions used are aqueous solutions of hydroxypropyl cellulose ether (Methocel K4M,
Dow). The shear thinning index (or pseudoplasticity) of the solution may be varied from
0.5 to 1 depending on the concentration of Methocel. The shear viscosity of the solutions

was measured at 25 °C on a Fluids Rheometer (Rheometrics, RFS 8400). These steady

41

tests were performed at shear rates ranging between 0.1 to 100 s-l. A 0.02 rad cone and a
5 cm diameter plate were used for this purpose. A water-bath was used to maintain
isothermal conditions in the sample holder. Figure 2.3 shows the eﬂ‘ect of concentration
on the shear viscosity of the solution. As the amount of the polymer was increased ﬁom
0.5 wt% to 2 wt%, the zero shear viscosity increased by 2 orders of magnitude. The shear
thinning index went down ﬁ'om 0.86 to 0.5. To prepare the solutions, about a quarter of
the total water required was heated to 90° C. Methocel was then added to it with
continuous stirring. When the temperature of this paste reached 40° C, the remainder of
the water (in the form of water at room temperature and ice) was added and the mixture
was stirred for about 30 minutes. The solution was allowed to stand for at least 24 hours
with intermittent stirring. For lower stirring times, it was observed that the consistency of
the prepared solution was very inhomogeneous and its appearance was cloudy and misty.
This rendered the solution useless. The present study uses 1% (wt/wt) and 2% (wt/wt)
Methocel solutions yielding shear thinning indices of 0.7 and 0.5 respectively. No

signiﬁcant diﬂ‘erences in properties were observed due to repeated shearing of the ﬂuids.

The shear viscosity of the polymer solutions was ﬁtted to a Carreau - Yasuda model of the

form (2.2)

111 = [1+(bY)"]("'”/° (2,2)

0

and to the generalized Oswald-dc Waele power-law model (2.3).

Shear viscosity, mPa-s

42

 

 

 

 

 

104 ' ' "r'rl v ﬁ vvwwvv' 1 . 1,,fVT;
AAAA AAA 3
AAA A 1
AAA w
A
0000 000 4
000 00 00A A
103: 0
t 0
(boooooooooocpc,0000
L O
101: M0% 1
i 01.5% :
013% nunuunnnnnuu‘
no.5% T
101 a .r.r . . ....r.r . . ...,..7
lo-1 10° 101 101
Shearrate,s'l

Figure 2.3 Effect of Methocel concentration on the viscosity

of the polymer solutions.

43

n = k (r)‘""’ (2.3)

Here n, is the viscosity at zero shear rate and a, b and k, n are ﬁtting constants. The
constant n is called the degree of shear thinning or the pseudoplastic index. Figure 2.4
compares the measured shear viscosity and the values predicted by using equations (2.2)
and (2.3). Table 2.2 lists the ﬁtting constants used. Equation (2.2) provides a better ﬁt of

the viscosity since it incorporates the Newtonian viscosity behavior at low shear rates.

Table 2.2: Parameters used in the shear viscosity models

 

 

 

 

 

Model Solution 1],, (Pa-s) a b (s) n k (Pa-s“)

Carreau 1% 0.35 2 0.05 0.7 -

Carreau 2% 6.4 2 0.24 0.5 -
Power-law 1% - - - 0.7 0.751
Power-law 2% - - - 0.5 12.784

 

 

 

 

 

 

 

 

 

Since the viscosity of the polymer solution is dependent on the shear rate, equation (2.4)

was used to deﬁne the Reynolds number of the shear thinning solutions. This took into

Shear viscosity, mPa-s

104

103 ,

 

— Carreau model

  

O 2% solution
0 l % solution

°°°°° Power - law model

 

A A I I ‘4 A J L A A I

  

 

 

100 101

Shear rate, 5'1

Figure 2.4 Shear viscosity of the polymer solutions.

45

account the power-law eﬂ‘ect of shear thinning.

n 2-n
h Um p

Re. - 2(1l2n)],,.l (2.4)

2H

 

The wall shear rate in the upstream channel was used as the characteristic shear rate for
the computation of the viscosity. The ranges of Re“ achievable for P1 and P2 were 0.1 to
4 and 0.1 to 1.6 respectively. The upper bound was related to the pressure that the

apparatus could withstand.

The polymer solutions were characterized by using a Fluids Spectrometer (Rheometrics
RFS 8400) for both steady and dynamic shear properties. Dynamic measurements were
made at ﬁequencies between 0.1 to 100 rad/s by using a 0.02 radian cone and a 5 cm
diameter plate. The sample temperature was maintained at 25 °C by placing it in a
thermally controlled water-bath. Figure 2.5 shows the storage modulus (G’) and the loss
modulus (G”) curves for both the polymer solutions. For the 1% solution, the modulus
could be reproducibly detected only at frequencies above 6 rad/s. The relaxation time of
the polymer solutions was calculated ﬁ'om the zero-ﬁequency limit of the G’ curve as

GI

71009

A = (lim mm)

 

(2.5)

Relaxation times for the 1% and 2% solutions were 0.54 and 7.8 milliseconds respectively.
The ﬂuid elasticity was evaluated from the Deborah number. This product of the

relaxation time and the wall shear rate at the highest ﬂow rate indicated that the 1%

G'(Pa), G" (Pa)

 

 

 

 

101. - m . "r": 'r'"! ' ' '
I A;
l A
r 06' P1 ago"
’ A‘e ‘
p .G' P2 . q
p AAA . 4
AG”P1 ‘ .
100- AG”P2 A“ ' (i
D ‘ . 6 1
: ‘A .. 88 f
. ‘A . AA .
A A O
I . A 1
O
A . AA 1
O A O
'1. q
10 E e 0 3
I ' 0 I
s . o
_ O
O O
10.2 A . .r...r . r x .mrIrr A r . .L...
10:1 10° 101 102

Frequency, rad/s

Figure 2.5 Loss modulus and storage modulus of the polymer solutions.

47

solution was inelastic (Dem < 10"). The 2% solution, however, had a Dem of ~ 0.3 and

was believed to exhibit elasticity under the conditions of the present study.

4.3 Flow visualization setup

In order to carry out ﬂow visualization experiments, the model ﬂuids were seeded with
0.03 gm of neutrally buoyant Silicon Carbide particles (45 - 70 microns, Aldrich
Chemicals). Using a sheet of light, a plane encompassing the expansion plane was
illuminated. Particles passing through this sheet reﬂected light which was collected by a
CCD camera (Nikon, F3). Using long exposure times (4 seconds to a few minutes) the
path of these particles was traced. The size and the shape of the corner vortices and the
ﬂow patterns in the rest of the cavity could be picked up by this technique. The setup used
for the ﬂow visualization experiments is portrayed in Figure 2.6. Compressed nitrogen
was used to force the model ﬂuid ﬁom the reservoir to the ﬂow cell and then back to the
reservoir. A regulator was used to maintain constant ﬂow rate conditions in the ﬂow cell.
By increasing the pressure in the reservoir, ﬂow rates up to 400 ml/s were achievable in
the ﬂow cell. Two techniques were used to create a laser sheet ﬁom a laser beam. Figure
2.7 schematically demonstrates these methods. In the ﬁrst technique, a laser sheet was
created by passing a laser beam ﬁ'om a 20 mW He-Ne laser source (Newport ULM )
through a cylindrical glass rod. By altering the position and to some extent the size of the
rod, it was possible to get sheets of various widths and intensities. Although this
procedure was simpler to set up and to manipulate than the other one, it resulted in
signiﬁcant intensity losses. Also, the sheet was non-uniform in intensity, with the center

being much brighter than the periphery. Another indigenously developed technique that

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horses“ 3 3.933—

 

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£338“

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mac‘— couch: Z

.8555

 

 

 

 

 

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3.52

49

Rotating shaft

    

pseudo-sheet

Figure 2.7a Scanning mirror assembly for creating a laser sheet.

     
   

Refracted
laser sheet

Cylindrical glass rod

Figure 2.7b Laser sheet creation with a glass rod.

50

was used involved reﬂecting the laser beam oﬂ‘ a well-polished mirror rotating at around
2000 rpm such that the reﬂected beams scanned through a cross section of the ﬂow cell.
The latter approach created a sheet of higher intensity since it minimized loss of power.
The former technique was the choice at lower ﬂow rates and the latter yielded better
results at higher ﬂow rates. Flow visualization studies were then carried out using the 35
mm CCD camera mounted on a tripod close to the illuminated plane. Photographs of the
seed particles were taken with exposure times varying between 4 seconds to a few minutes
and apertures between f/ll to f/2.8. Use was made of a timer that allowed up to 16

minutes of exposure time.

5. Results and Discussion

5.1 Flow patterns in the Newtonian liquid (Experimental)

Flow patterns at low Reynolds numbers of 0.01, 0.1 and 0.5 are shown in Figure 2.8a, b
and c. Streak lines symmetric about the channel center-line are observed, with the
reattachment lengths on both walls being equal. The angle of entry of the liquid is very
close to 90°. This is expected because of the low Reynolds number and a large degree of
expansion. With increasing inertia, the vortices were observed to grow in size. The ﬂow
proﬁles were always symmetric for Re between 0.001 and 0.6. Reproducibility was
conﬁrmed by repeating the experiment twice. To obtain the reattachment length, the
negatives of the photographs were projected onto a screen and measurements were made
using a meter ruler. These were then normalized with the step height. This non-

dimensional reattachment length will be referred to as L,.

51
At Re = 0.8 (Figure 2.8d), a ﬂow transition is observed. Close to the expansion plane,
the streak lines in the center of the larger channel droop toward one wall and then ﬁrrther
downstream shoot toward the other wall. This leads to an asymmetry in the ﬂow patterns
with a large vortex reattaching itself to one wall and a small one to the other. Sobey
(1985) observed a ﬂow transition with water through a 1:3 planar expansion at a Reynolds
number of 25. Experiments performed by Durst et al. (1974) on a 1:2 planar expansion
showed that the ﬂow turned asymmetric at a Reynolds number of around 42. Thus, the
interesting feature of the ﬂow through a channel with expansion ratio 16 is the occurrence
of ﬂow transition at Reynolds numbers 2 orders of magnitude lower than those observed
with the smaller expansion ratios of 2 and 3. This extends the ﬁndings of Cherdron et al.
(197 8) who reported that increasing expansion ratios from 2 to 3 resulted in lowering the

critical Reynolds number at which the ﬂow asymmetry started.

As Re is increased ﬁrrther, the larger vortex grows quite rapidly. The smaller vortex length
decays slightly or remains constant. Figure 2.8e and f show the ﬂow patterns at Re of 1.5
and 4 respectively. Some of the photographs showed the crossing of some streaklines
close to the walls. These were initially assumed to be due to unsteadiness in the ﬂow but
later conﬁrmed to be a result of slight movement of the camera timer. Laser Doppler
velocirnetry measurements in the larger channel conﬁrmed no ﬂow unsteadiness at these
Reynolds numbers. The LDV results have been reported in Chapter 3. The presence of a
large vortex in Figure 2.8e and f results in a very narrow jet of the ﬂuid in the core
between the vortices. The swiveling motion of this jet is very similar to the jetting

phenomenon observed in injection molding at high injection rates and low gate land

52

 

(b)

Figure 2.8 Flow patterns of the Newtonian liquid. (a) Re = 0.001 (b) Re = 0.

53

v
i

t‘mﬁﬂ,‘ »...~T . ' 5-

v {Mu-"Q! '

 

((1)

Figure 2.8 Flow patterns of the Newtonian liquid (continued). (c) Re = 0.5
(d) Re = 0.8.

54

 

Figure 2.8 Flow patterns of the Newtonian liquid (continued). (e) Re = 1.5 (f) Re = 4.

55
lengths (White and Dec, 1974 and Oda et al., 1976). In the experiments conducted by

Sobey (1985), Durst et al. (1978, 1993) and Feam et al. (1990) for expansion ratios of 2
and 3, a second ﬂ0w transition showing the presence of a third vortex has been reported at
Reynolds numbers higher than the critical Reynolds number for the onset of the ﬁrst ﬂow
transition. Unfortunately, due to a much larger channel size being used in the present
experiments, practical limitations prevented the achieving of ﬂow rates corresponding to
higher Reynolds numbers. The second transition for the Newtonian liquid was therefore
not observed. Figure 2.9 shows the vortex growth as a ﬁrnction of the Reynolds number.
At Re of 0.8 and above, the salient vortices are diﬂ‘erent in size. The diﬁ‘erence grows as
the Reynolds number is increased. The larger corner vortex grows more rapidly than the

smaller corner vortex.

5.2 Computations of the ﬂow bifurcation in Newtonian liquids

Steady, two.dimensional ﬁnite element calculations were performed to predict the critical
Reynolds number for the onset of asymmetry. A commercial code FIDAP (Fiuent, Inc.)
was employed for this purpose. Only a brief description of the technique used and the
numerical grids is included in this section. The details follow in Chapter 4. Figure 2.10
shows a schematic of the ﬂow channel used. No slip velocity on the walls and a ﬁrlly
developed velocity proﬁle at x = -l, were imposed. l1 and I2 chosen were 5 and 65 step
heights respectively. Reducing their values by 50% changed the solution only by 0.01%.
The upstream channel height and the half height of the downstream channel were used to

non - dimensionalize the axial and normal distance respectively.

Reattachment length

56

 

 

 

O
_ O
O
O
O
. o o 0
0
no 0 °° °
0 1 2 3 4
Reynolds number

Figure 2.9 Experimental vortex growth in the Newtonian liquid.

 

57

 

Figure 2.10 Schematic of the ﬂow channel used in calculations.

58

The ﬁrst goal was to validate the computational ﬁndings on Newtonian ﬂow in a 1:3
channel by comparing them with the experimental and computational results of Feam et al.
(1990) for a similar ﬂow conﬁguration. Their experiments and calculations have shown the
onset of asymmetry at Reynolds numbers of around 23 and 27 respectively. Steady state
computations on a symmetric grid failed to show any ﬂow asymmetry ﬁ'om Re = 0 to 40.
The calculation protocol was then modiﬁed by incorporating a perturbation to the grid. An
asymmetry was introduced in the geometry of the conﬁguration by shortening one of the
steps by about 10%. Stokes solution was then computed for this asymmetric channel.
Using the Stokes solution as the initial iterate for the next run, the equations of motion
were solved for a higher Reynolds number. For every subsequent Re, the result of the
previous computation was used as the initial guess. This procedure was continued until Re
of 40. At this point, the computational domain was made symmetric again and
computations were now performed on this symmetric grid. For each subsequent run, Re
was lowered gradually and the results of the previous runs were used as the initial iterate.
This procedure was followed until a Reynolds number much lower than the biﬁrrcation
point was reached. This approach will be called a perturbation - continuation approach.

The step size of Re was anywhere ﬁ'om 0.1 to 1.

To evaluate the reattachment length, the sign of the axial velocity was monitored at the
nodal layer closest to the walls and two successive points with negative and positive u-
velocity were identiﬁed. A simple linear interpolation of the form of equation (2.6) then
yielded the reattachment length. This was then normalized with the step height and was

referred to as L,

59

(Jr+1(xrrl - xi)
x = x , - 2.6
f I (1",! _ (It ( )

 

Figure 2.1 1 demonstrates the three branches of the bifurcation curve which result ﬁom
the perturbation -‘ continuation approach for a channel with expansion ratio 3. The
diﬂ'erence in the reattachment lengths on the two walls normalized by the step height is
used as the index of the degree of asymmetry in Figure 2.11 (and in subsequent
bifurcation curves). Up to Re = 27, no asymmetry exists (i.e. the difference in the
reattachment length is zero). This stable branch (1) is the one that is picked up
experimentally. At Re greater than 27, the vortex lengths become unequal. The degree of
asymmetry increases with increasing Reynolds number. Either of the branches 2 or 3 may
be followed at Re > 27. At a critical Reynolds number of 27, the unique solution (1) is
said to have biﬁrrcated into multiple solutions (2 and 3). The branch of the solution
representing symmetry beyond the critical Re (branch 4) is the one that will be picked up
by computations without using the perturbation - continuation approach. Durst et al.
(1997) have recently performed a stability analysis and have shown that the branches 1, 2
and 3 are indeed stable solutions and branch 4 is the unstable solution and therefore would
not be observed experimentally. The predicted onset Reynolds number of 27 for
Newtonian ﬂow through a 1:3 planar expansion using the perturbation - continuation
approach is in very close agreement with the computations of Feam et al. (1990), which

also predict the onset of ﬂow asymmetry at the same Reynolds number.

Difference in reattachment length

 

8 U ' l I '
x Asymmetric grid 3
4 ' OSymmetric grid 3 -. '

 

 

 

 

 

0 ..... 4
4 .
-8 . r r r r r r
0 10 20 30

Reynolds number

Figure 2.11 Results from the perturbation - continuation approach for

Newtonian ﬂow through a 1:3 planar expansion.

 

61

To investigate the eﬂ‘ect of grid resolution on the outcome of our calculations, three grids
- Gl, GZ and G3 were designed with 2351, 5871 and 11431 nodes respectively. The mesh
was graded ﬁnely in all the corners and was less dense away from the comers. Figure 2.10
schematically shows the variation of the grid density in the channel. In the axial direction
(i.e., along the x axis), the mesh density was the maximum close to the x = 0 plane. In the
transverse direction, the mesh was more dense near both the 90° corners (y= t 1) and the
270° corners (y = :h 1/ B) and was less dense in between. The bifurcation point for a
channel with an area expansion ratio of 3 was used to compare the performance of the
three grids. Figure 2.12 illustrates the result. GI and G2 produced results to within 0.1%.
The performance of G2 and G3 was nearly identical. Further results have been computed

on the grid G3.

The computations for a Newtonian liquid ﬂowing through a 1:16 planar channel revealed
that the critical Reynolds number was around Re = 6.3. The corresponding biﬁrrcation
curve is shown in Figure 2.13. The computations predicted a higher critical Reynolds
number than that observed in the experiments. Such an over prediction has also been
observed by other workers with 1 :2 and 1:3 channels. The lowering in the experimental
critical Reynolds number for the onset of ﬂow biﬁrrcations could be brought about by
uncertainties in the experimental conﬁgurations. Computations for a 1:16 channel are also
signiﬁcantly diﬂ‘erent from the corresponding 1:2 or 1:3 channel in the far severe corner

singularity and the required more dense mesh grading close to the corners in the former.

Difference in reattachment length

62

 

 

AG]
0G2
X G3

 

 

1"
,1 .
7 ' .
.‘\
\C . d
-i
1 4 I -
30 35
Reynolds number

Figure 2.12 Effect of grid refinement on bifurcation curve.

Newtonian ﬂow, Expansion ratio = 3.

Difference in reattachment length

ﬂ

s
y-s

63

 

 

 

b

 

 

an

6 8

Reynolds number

Figure 2.13 Bifurcation curve for the 1:16 expansion.

10

64

Since the critical Reynolds numbers obtained ﬁ'om experiments and computations are
diﬂ'erent, a modiﬁed Reynolds number is deﬁned to facilitate a comparison between the

experiments and computations.

Rem, = Re - Rem,

The experimental critical Reynolds number is used to evaluate the experimental modiﬁed
Reynolds number and the computational critical Reynolds number is used to evaluate the
computational modiﬁed Reynolds number. Figure 2.14 compares the biﬁrrcation curves
using the diﬂ‘erence in the reattachment lengths on the two walls as the index of ﬂow
asymmetry. Only the positive half of the biﬁrrcation curve has been illustrated for purposes
of clarity. At Re close to the critical Re, computations predict a lower degree of
asymmetry than the experiments. This trend is reversed at Re much higher than the critical

Reynolds number.

5.3 Newtonian vortex growth prior to the onset of asymmetry
Figure 2.15 shows the trend of vortex grth with inertia, up to Re of 0.6. The data
points used are ﬁom the ﬂow visualization experiments. The solid line is from the

computations described earlier. Equation (2.7) provided a good ﬁt between LV and Re.

L» = aRe + b (27)

Values of 0. 18 and 0.425 were used for a and b respectively. Baloch et al. (1995) have
computed the Newtonian creeping ﬂow reattachment length (i.e., b in equation 2.7) for a

1:4 planar expansion to be 0.385, which is close to our value for the 1:16 expansion. They

Difference in reattachment length

65

 

3 r U V I V ﬁ' 1

0 Experiment

V

—- Computation O

 

 

Remod = Re - Rec,“

Figure 2.14 Positive half of the bifurcation curvefor Newtonian ﬂow.

 

Reattachment length

 

0.6 u u u

e
in

 

 

 

 

0.40.0 0.2 0.4 0.6 0.8
Reynolds number

Figure 2.15 Experimental Newtonian vortex growth prior to the onset

of asymmetry. Solid line is from computation.

67

have also presented the reattachment lengths normalized with the downstream channel
height for 1:133 and 1:80 planar expansions. The asymptotic reattachment length in the
creeping ﬂow limit for the 1213.3 channel is slightly higher than that for the 1:4 expansion.
The vortex growth curves reveal two distinct regions, thus rendering a simple relation like
equation (2.7) unsuitable to model the reattachment length for the entire range of
Reynolds number used in their study (0 to 50). Acrivos and Schrader (1982) have
proposed a linear relation between the vortex length and the Reynolds number based on
the momentum ﬂux. This leads to the following proportionality relating the reattachment

length, expansion ratio ([3) and the Reynolds number.

Re

LV (1
(B -1)"2

 

(2.3)

This expression is valid for large Reynolds numbers and predicts a zero vortex length for
creeping ﬂows. The presence of a ﬁnite corner vortex (Moffatt eddy) even under creeping
ﬂow conditions has been conclusively demonstrated in Moﬂ'att’s analysis (1964). To
account for this anomaly and also to model the vortex length at low Reynolds numbers,

the following is proposed:

At low Re (the deﬁnition of low will be left ambiguous for the present), the growth of the
normalized vortex length from creeping ﬂow is proportional to the average inlet velocity

of the channel, i.e.,

Lv-LoaRe

68

or

L, = mRe + L0 (2.9)

L, is the creeping ﬂow reattachment length and m is the vortex growth coeﬁcient. At

higher inertia levels, ﬂow behaves like a “jet” entering a stagnant ﬂuid ﬁom a small oriﬁce.
This would require the reattachment length to depend on the ﬂux of the momentum of this
jet. Acrivos and Schrader (1982) have used this line of logic to come up with the model in

equation (2.8) which is now modiﬁed to

kRe kRe
L, - L, = (0-1)!” = (l) (2.10)

 

 

At high Reynolds numbers the assumption that L, << I.v leads to the equation (2.11).

L = kRe = kRe
v (ii-1)” (l) (2.11)

 

 

Acrivos & Schrader have showed that their model is very effective with k = 0.137 (L, = 0
in their model) predicting the Newtonian reattachment lengths in a 1:2 and 1:3 planar
expansion respectively. At the high Reynolds numbers at which this model has been tested,
L, is indeed << L, For the data obtained at low Reynolds number in a 1:16 channel, such
a relationship fails, as is obvious ﬁ'om Figure 2.15. Clearly L, - L, is not negligible here
and our earlier equation (2.9) is a better relationship to be used at low Reynolds numbers.
To test this hypothesis, the reattachment lengths were computed for 1:2, 1:3 and 1:16

expansions for a wide range of Reynolds numbers, below the critical Reynolds numbers.

69

For B = 2, 3 and 16, the computations were carried out between Re of 0 - 65, 0 - 25 and 0
- 5.5 respectively. The upper bound is slightly lower than the critical Reynolds number for
the onset of asymmetry for the channels. To compare the computed results with those of
Acrivos and Schrader, L, was plotted as a function of Re / d), for B = 2 & 3. Only the
reattachment lengths at high Reynolds numbers were considered. Figure 2.16 predicts
slopes of 0. 135 and 0.137 respectively which are in very close agreement with 0.137, the
prediction of Acrivos and Schrader (1982). The y-intercepts of 0. 14 and 0.2 for B = 2 and
3 respectively support the hypothesis that at high Reynolds numbers, L, << L, and that it
may be neglected to yield equation 2.11. Such a relation, however, fails to predict the
reattachment lengths accurately at lower Re, at which equation (2.9) is a more accurate
representation of I.,,. Figure 2.17 shows the corresponding ﬁts for the reattachment
lengths in 1:2 and 1:3 channels at low Reynolds numbers. Table 2.3 reveals the ﬁtting
constants used in Figures 2.15, 2.16 and 2.17. Interestingly, the creeping ﬂow
reattachment length predicted for all the channels lies between a narrow range of 0.425 -

0.465.

The vortex growth in the 1 :16 channel was already illustrated in Figure 2.15 for Re

between 0 to 0.6 . L, predicted is 0.43 and is close to the values obtained for the 1:2 and
1:3 channels, thereby conﬁrming the general applicability of equation (2.9) at low inertia
levels. Excellent agreement between the computational results and the experimental data

collected through streak photography is also evident in Figure 2.15.

70

 

 

 

 

12 ' I ' l ' I ' I ' U ' I
OB=2
XB=3 ..
too.
a. s- .. -
r: .e'
.2 . ‘
Fl , .0" ,
g ,0"
g c a
3 4. ' '
I
00 10 20 30 40 50 60 70
Rel(B-1)m

Figure 2.16 Newtonian vortex growth prior to the onset of asymmetry

at high Reynolds numbers. Lines represent'ﬁts using equation (2.11).

Reattachment length

71

 

 

 

 

4 ' I ' I j w I
OB = 2
. ,s>"' .
x B = 3 "0"
55'?
0‘0
2 - 0o .
00
O
O
O
10 15 20 25

 

Reynolds number

Figure 2.17 Newtonian vortex growth prior to the onset of asymmetry

at low Reynolds numbers. Lines are fits using equation (2.9).

72

Table 2.3: Fitting constants used in Figures 2.15, 2.16 & 2.17

 

 

 

 

 

 

B Re range k m L,
2 0 - 22 - 0.12 0.45
2 22 - 65 0.135 - -

3 0 - 7 - 0.15 0.465
3 7 - 25 0.137 - -
16 O - 0.6 - 0.18 0.425

 

 

 

 

 

 

 

5.4 Flow patterns in the shear thinning liquids (Experimental)

Figures 2.18a - d show the effect of increasing inertia on the ﬂow patterns developed in
the 1% shear thinning ﬂuid (n = 0.7). Symmetric streak lines are observed at Reynolds
numbers of 1.25. The ﬂow patterns at lower Reynolds numbers were qualitatively similar
to that shown in Figure 2.18a. The corner vortices increased in size with increasing
Reynolds numbers. At Re“ = 1.9, a ﬂow transition leading to asymmetrical ﬂow patterns is
observed. Thus, the critical Reynolds number for the onset of ﬂow transition for the 1%
shear thinning liquid is between 1.25 and 1.9 and is higher than the corresponding value
for the Newtonian case (between 0.5 and 0.8). Shear thinning, therefore, seems to delay
the ﬂow transition. An interesting feature observed in Figures 2.18b through 2.18d is the
presence of a bunch of closely-spaced streak lines, starting from the center of the

expansion plane and curving toward one of the walls. These bent streak lines resemble the

73

 

(b)

Figure 2.18 Flow patterns of the polymer solution P1. (a) Re* = 1.25 (b) Re“ = 1.9.

74

 

Figure 2.18 Flow patterns of the polymer solution P1 (continued). (c) Re' = 2.4 (d)
Re" = 3.1.

75
asymmetric ﬁlling patterns observed in injection molding at a high ﬁlling rate (Re* =
0.02, Dee and White, 1974), wherein the melt instead of radially ﬁlling the mold near the
gate tends to shoot toward the opposite wall and then piles upon itself. This phenomenon
of jetting was not observed by Dee and White at a low injection rate (Re‘ ~ 104 ). It
would seem that both of these “diﬂ‘erent” ﬂow conditions i.e., the unsteady state mold
ﬁlling during injection molding and the steady state channel ﬂow in our present
experiments show similar ﬂow transitions beyond a critical injection rate. Another
interesting observation above the critical Reynolds number is the simultaneous presence of
three vortices. The largest vortex attaches itself to the transverse wall, close to the lip
corner and it grows the fastest with increasing inertia. The ﬁrst ﬂow transition from the
symmetric two-vortices state to the asymmetric two-vortices condition is not captured in
the present experiments. Pseudoplasticity hastens the second transition which was
observed by Durst et al. (1993) and Fearn et al. (1990) at much higher Reynolds numbers
in their Newtonian experiments (214 and 85 for 1:2 and 1:3 expansions respectively). The
simultaneous presence of three vortices has been conﬁrmed by laser Doppler velocimetry
measurements (results are reported in Chapter 3). Figure 2.19 shows the vortex growth
with increasing Re“ for the 1% solution. Only the two 90° corner vortices have been
presented in the ﬁgure because it was very difﬁcult to get all of the three vortices on to
one single photograph. While the smallest vortex length remains essentially constant, the

larger vortex is seen to grow much faster at Re‘ > 1.9.

Streak photography experiments carried out with the more shear thinning liquid (2%

Methocel, n = 0.5) did not reveal any ﬂow asymmetry until the highest achievable Re“ of

Reattachment length

76

 

 

 

 

1.0 ' j r 1 '
0 P1
0.8 " 0 p2
0.6 " o
O
0.4 r
o o o 0
O O O
0.2 '-
[ 0 O O O
0'0 o 1 2
Reynolds number

Figure 2.19 Experimental vortex growth in the polymer solutions.

77

1.6. The only conclusion that may be made is that the critical Reynolds number for this
liquid is above 1.6. This is consistent with the trend of increasing critical Reynolds number
(from 0.8 to 1.9) with increasing shear thinning (for n = l to n = 0.7) observed in our
earlier experiments. In addition, characterization of the 2% polymer solution did reveal
that it possessed a ﬁnite elasticity at the higher ﬂow rates. Studies performed on elastic
polymer solutions by Townsend et al. (1995) have shown that elasticity decreases the
vortex activity in a 1:133 planar expansion. The eﬂ‘ects of higher shear thinning and
higher elasticity would thus translate to an even higher critical Re“ for the 2% Methocel

solution. The reattachment lengths for the 2% solution have been included in Figure 2.19.

5.5 Flow bifurcation in the non - Newtonian liquids

A power - law model was used to describe the viscosity of the polymer solutions. The
same perturbation - continuation approach described earlier for the Newtonian liquids was
used and ﬂow bifurcation curves were obtained for the shear thinning liquids. A critical
Reynolds number of about 18 was predicted for the shear thinning liquid with n = 0.7. As
with the Newtonian liquid, computations for the shear thinning liquid also over - predicted
the critical Reynolds number. The non-idealities that are inherent to the experiments are
probably responsible for an earlier transition. The computations did reveal both the ﬂow
transitions for the pseudoplastic liquid. Interestingly, the ratio of the experimental critical
Reynolds number for n = 1 and n = 0.7 and the computational critical Reynolds number

are very close in magnitudes.

78

= 0.39

 

[Re‘ lap!

C

and

[——‘—],0 = 0.35
Re a c "'P
For a shear thinning index of 0.5, a higher critical Reynolds number of about 27 was
obtained. This conﬁrmed that shear thinning delays the onset of ﬂow transitions in a 1:16
channel. The bifurcation curves and other associated details for the shear thinning liquid

have been presented in Chapter 4.

To facilitate a comparison between the experimental and the computed biﬁrrcation curves

for the shear thinning liquid with n = 0.7, a modiﬁed Reynolds number was deﬁned as

Rum = Re* — Re-rcm

For the experiments and the computations, the respective critical Reynolds numbers were
used to evaluate the modiﬁed Reynolds numbers. Figure 2.20 compares the positive half
of the bifurcation curves. The trends observed are essentially similar to those described
earlier for the Newtonian liquid. Computations seem to deviate much more ﬁ'om the

experiments as the Reynolds number is increased.

Difference in reattachment length

79

 

1.5 u

0 Experiment
— Computation

1.0

0.5

 

0.00.0 0.5

Figure 2.20 Predicted bifurcation curve for shear thinning liquid Pl.

 
  
   

1.0 1.5

Re*m,d = Re“ - R6,,“

 

2.0

80
5.6 Computed vortex growth in non-Newtonian liquids, prior to the onset of
asymmetry
Figure 2.21 compares the experimental and the calculated vortex lengths for P1 before the
onset of asymmetry. The solid line represents a linear ﬁt of the computed vortex lengths.
A good agreement between the calculated and the experimental values is observed. A
zero-Re vortex length of 0.23 is predicted, which is almost halfof that for the Newtonian
liquid. A similar comparison for the 2% solution is done in Figure 2.22. Here the
computational predictions are within 5.5% of the experimental L, data. The predicted
vortex length under creeping ﬂow conditions is around 0.136, which is lower than that for
both the other liquids. Thus shear thinning consistently suppresses the growth of the

corner vortices.

The eﬁ‘ect of expansion ratio on the vortex lengths before the onset of asymmetry for n =
0.7 is brought out in Figure 2.23. Here the results for B = 3 have been compared with
those for B = 16. At low Reynolds numbers, L, for B = 16 is lower than that for B = 3. In
both the channels, L, rises monotonously with Re', with the grth being higher in the
larger expansion channel. At Re' of around 7, a cross-over occurs. This cross - over is
merely an artifact of the higher scaling term (step height) used for L, at B = 16 as
compared to that at B = 3. The step height in the former is 7.5 times that in the latter. The
dimensional vortex lengths will, therefore, be much larger for the higher expansion ratio.
For the more shear thinning liquid with n = 0. 5, no cross-over is observed for the entire
range of Reynolds numbers studied (Figure 2.24). The non - dimensional L, for the

channel with B = 16 is always less than that for the one with B = 3. The vortex lengths for

Reattachment length

81

 

 

 

 

0.4 . . . .
os - .
0.2 i ..
0.1 - .
Moo T 0:5 A 1:0

Reynolds number

Figure 2.21 Vortex growth in polymer solution P1. Solid line is from

computation.

1.5

Reattachment length

82

 

 

 

 

 

0016 ' I ' r ' U V
, 0 d
0.15
0.14
(moo A 0.5 A 1.0 ‘ 15 A 2.0
Reynolds number

Figure 2.22 Vortex growth observed in P2. Solid line ,is from computation.

Reattachment length

 

3 O

0B

' "[3:16 0

 

 

 

o 5 A 10 15

Reynolds number

Figure 2.23 Vortex growth before the onset of asymmetry, n = 0.7.

 

Reattachment length

 

 

 

 

006 ' f j I ' I I
OB=3
0 "13:16
0.4 ~ ° -
O
O
x x
0.2 " x x x x x -
x x x X x x x
0'00 2 4 6 8
Reynoldsnumber

Figure 2.24 Vortex growth before the onset of asymmetry, n = 0.5.

10

85

n = 0.5 are shorter than those obtained for n = 0.7. Figure 2.25 reveals the eﬂ‘ect of shear
thinning and expansion ratio on the creeping ﬂow vortex length, L,. With an increase in
the expansion ratio from 3 to 16, the normalized vortex length goes down at any shear
thinning index. Attempts made to model the reattachment lengths for the shear thinning
liquids at high Reynolds numbers using equation (2.11) met with little success. The values
of k for the best linear ﬁt ranged from 0.01 to 0.08 for the 1:3 and 1:16 channels. There is
deﬁnitely a need to address this issue further and to propose new relationships between L,

and Re“ for the shear thinning liquids.

6. Conclusions

Flow visualization was used to study the onset of asymmetry in a 1:16 planar channel. One
Newtonian and two shear thinning solutions were employed as the test ﬂuids. Flow
transition in the Newtonian liquid occurred at Red of 0.8, which is signiﬁcantly lower than
the critical Reynolds numbers reported previously for 1:2 and 1:3 planar channels. This
extends the ﬁndings of Cherdron et al. (1978) who reported that increasing expansion
ratios from 2 to 3 resulted in the lowering of the critical Reynolds number at which ﬂow
transitions were witnessed. With the shear thinning liquid P1 (n = 0.7), ﬂow transition was
observed between 1.25 - 1.9. Shear thinning, therefore, delayed the onset of asymmetry.
No ﬂow transition was observed with the more shear thinning liquid P2 (n = 0.5), for the
entire range of Reynolds numbers studied. Higher shear thinning and/or higher elasticity
could be responsible for the absence of the expected ﬂow transitions. With P1, the ﬂow
transition occurred from a state of two symmetrical vortices to a state of three

asymmetrical vortices. The intermediate step of two asymmetrical vortices was not

Creeping flow reattachment length

86

 

 

 

 

0.5 r ' j
0
O x
0.4 r '
0.3 ' '
O
x
0.2 - "
X 0 B = 3
0.1 r x B = 16 ‘
0.00.4 0.6 0.8 1.0 1.2
Shear thinning index

Figure 2.25 Effect of shear thinning on the creeping ﬂow vortex length.

87

observed. Others have reported a similar second ﬂow transition ﬁ'om two asymmetrical
vortices to three asymmetrical vortices with Newtonian liquids, at much higher Reynolds
numbers. Therefore, it may be said that pseudoplasticity hastens the second transition with
a 1:16 planar expansion. It may be summarized that a combination of high enough
expansion ratio, a fairly non-shear thinning liquid and high injection rates will lead to early

ﬂow transitions in planar expansions.

A perturbation - continuation approach was developed to compute the bifurcation curves
for the liquids. The predictions of the critical Reynolds numbers were higher than the
experimentally obtained values. This could be due to the uncertainties inherent to the
experimental conﬁguration. The computations of the vortex lengths prior to the onset of
asymmetry showed excellent agreement with the experimental data. Relations between the
Newtonian reattachment lengths, Reynolds number and the expansion ratio were proposed
for the entire range of Re prior to the onset of asymmetry. These were conﬁrmed by our
computations with 1:2, 1:3 and 1:16 planar channels. A Newtonian non-dimensional
vortex length of about 0.44 step heights was predicted for creeping ﬂow, for all the three
channels investigated. The creeping ﬂow reattachment length increased with B and with n.
More work needs to be done to obtain a modiﬁed relationship that successﬁrlly predicts

the non-Newtonian reattachment lengths.

Chapter 3
LDV MEASUREMENTS OF NEWTONIAN AND NON - NEWTONIAN LIQUIDS

IN ABRUPT PLANAR EXPANSIONS

This chapter discusses the laser Doppler velocimetry experiments performed to acquire
detailed velocity proﬁles in Newtonian and non - Newtonian solutions, particularly aﬁer
the onset of ﬂow asymmetry, in a 1:16 planar channel. The occurrence of ﬂow transitions
in the Newtonian and the non - Newtonian model ﬂuids ﬂowing through a 1:16 planar
expansion were revealed in Chapter 2. The objectives of this chapter are to make detailed
velocity measurements with particular emphasis on the velocity proﬁles at Reynolds
numbers much higher than the critical Reynolds number for the onset of ﬂow transitions.
These measurements will conﬁrm some of the ﬁndings reported in the earlier chapter.
They will then be used to determine the rate of decay of the ﬂow asymmetry with axial
location and also to evaluate the ﬂow bifurcation curves using the center line shear rate as
a measure of asymmetry. These trends will be compared with the predictions developed in

the present study.

1. Experimental
1.1 Laser Doppler velocimetry
A single component laser Doppler velocimetry (LDV) system was used to measure the

velocity along the direction of bulk ﬂow. A detailed description of the principles of LDV is

88

89

given in texts by Drain (1980), Durst et al. (1976), Watrasiewics and Rudd (1976), etc.
LDV (or LDA, laser Doppler Anemometry) is a non-obtrusive technique which measures
the velocity of non-buoyant particles suspended in a ﬂow. Ifthese particles are small, they
behave like ﬂow tracers and their velocity is equal to the velocity of the medium. When
the suspended particles are illuminated with a laser beam, the particles scatter light at a
particular ﬁequency, diﬂ‘erent ﬁ'om that of the incident beam. This difference in the
ﬁ'equencies, called the Doppler shift, is linearly proportional to the particle velocity. In the
commonly employed ﬁinge mode, LDV is implemented by splitting a laser beam and
making the resultant beams intersect at the focal point of a transmitting lens, thus forming
interference ﬁinges. As the suspended particles cross these ﬁinges, they scatter light which
is intensity modulated. The scattered light is collected and focused onto a photodetector,
whose output is a Doppler signal with frequency independent of the collection angle. Ifthe
transmitting lens itself is used to collect the scattered light, the scattering is called
backward mode. The scattering is forward mode if a separate lens is used to focus the
scattered light onto the receiving optics. The Doppler frequency is estimated using a signal

processor and the velocity is obtained from the following relation:

A

U=u -f)d =(f-f)
D S f D S 2Sin(%) (31)

 

where U is the component of velocity in the plane of the two laser beams and
perpendicular to their bisector, fD and f9, are the Doppler and the shiﬁ frequencies

respectively, df is the ﬁinge spacing, A is the wavelength of the laser light and 6 is the

beam intersection angle.

90

When the ﬁinges are stationary in the measurement volume, particles moving in opposite
directions result in identical photodetector signals. Diﬂ‘erentiation of the direction of
motion then becomes impossible. This problem is commonly encountered in the
measurement of reversal of ﬂows, which form part of the present study. Diﬂ'erentiation is
accomplished by introducing a ﬁequency shiﬁ to one of the laser beams using an acousto-
optic modulator called Bragg cell. This results in a bias motion of the ﬁinges, which makes
the resulting signals direction sensitive. Another distinct advantage of using frequency
shifting is its ability to measure the small magnitudes of the velocities, often encountered

close to the walls.

1.2 Optical setup

Figure 3.1 shows a schematic of the setup used in the present study. Table 3.1 details the
various components. A 35 mW He-Ne laser emitting light at a wavelength of 632.8 nm
was used to make measurements of the axial velocities in the forward scatter mode. The
unusually long width of the ﬂow cell and the extremely small magnitudes of the velocity
component normal to the bulk ﬂow, prevented their acquisition. The major optical
components consist of a beam splitter, which divides the incident beam in two equal
intensity beams separated by 50 mm, and transmitting and receiving lenses, each having a
focal length of 250 mm. This lens system produces a beam halfangle of 5. 17° and an
ellipsoidal measuring volume ofO. 18 x 0.18 x 1.9 mm’. The ﬁinge spacing is 3.18 pm,
resulting in 64 ﬁinges across the measuring volume. The velocities measured in this study
typically are in the range of 0.005 to 20 cm s", which correspond to Doppler frequencies

01“ approximately 0.16 to 640 kHz. To measure the ﬂow reversal and the slower velocities

91

.953 >A: a... ._e BEES—um ﬁn 95!..—

 

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

U
U

 

On.

 

 

 

 

 

 

 

 

E

 

 

2

 

 

 

 

 

 

 

 

 

 

 

 

92

Table 3.1 Description of the components depicted in Figure 3.1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

No Model Description
' Number
1 9126-105A 35 mW He-Ne laser, Wavelength = 632.8 nm
2 9126—255 Laser power supply
3 TSI 9115-2 Beam Splitter, 50 mm separation
4 TSI 9182 Bragg Cell
5 TSI 9186 Frequency Shifter Electronics
6 TSI 9118 Transmitting lens, focal length = 25 cm
7 TSI 9118 Receiving lens, focal length = 25 cm
8 TSI 9126, Optical rails
9121
9 TSI 9160A Photomultiplier
10 TS] 9165 Photomultiplier power supply
1 l Compudyne Computer
486 DX2
12 DANTEC N Signal Processor
Burst
Spectnrm
Analyzer
13 465 M Oscilloscope

 

 

 

 

93
near the wall, frequency shifting was employed by using a Bragg cell. The power supply to
the Bragg cell contains circuitry for downmixing the detector signal, so that the effective '
shift ﬁ'equency may be adjusted to yield the optimum Doppler signal ﬁequencies. After the
signal ﬁ'om the photodiode detector is downmixed by the frequency shift electronics, it
passes to a signal processor, which obtains essentially real time results for the Doppler
ﬁequency by using a dedicated FFT processor and a four-bit quantization of the real and
imaginary parts of the complex Doppler signal. The DANTEC software BURST was used
to perform statistical analysis on the measurements. The standard deviations were always
less than 0.5% of the mean values in the measurements reported here. The transmitting
and the receiving optics were mounted on mechanical traverses which had a range of 10
cm in all the three dimensions and a resolution of 0.025 mm. Care was taken to ensure that

no mechanical shocks were imparted to the setup during data acquisition.

2. Results and Discussion

LDV experiments were performed in the larger channel of the ﬂow cell. Figure 3.2
deﬁnes the nomenclature to be followed in the next few sections. The central plane of the
ﬂow channel along the z - axis (the axis normal to the x-y plane) was chosen to be
investigated. This ensured that end-effects were avoided. The axial velocity component,
U, was measured at various x - locations, traversing the laser beams along the y - direction
from one wall to the other. The upstream channel height, h, and the downstream channel
half - height, I-I/2 were used to normalize x and y respectively. The velocity was
normalized by Um, the maximum inlet velocity in the upstream channel. This was

estimated from the measured ﬂow rate and by assuming a fully developed velocity proﬁle

94

 

 

 

 

 

 

 

I I

I I

I I Y=I

r A I

I I

I I

I I

H

I I

I I

I I

I I

I I

I I

I V r Y=_1
X=0 I I

r r

X=l X=5

Figure 3.2 Nomenclature used to describe the LDV experiments.

95

at the inlet of the smaller channel. In the present setup, measurement of the normal
velocity component (V) along y = 0 could not be accomplished due to two reasons - the
unusually long length of the channel in the direction normal to the x-y plane and the very
small magnitudes of V which resulted from the low Reynolds numbers investigated
(maximum Reynolds number was 3.5). The presence of thick plexiglass walls and
extremely small magnitudes of U close to the wall resulted in zero - velocities about 2 mm
from either wall. These measurements have not been considered in the present work.

Reproducibility was conﬁrmed by repeating each experiment twice.

2.1 Velocity profiles of the Newtonian liquids

Velocity was measured at X =1, 8 and 15 at various points between Y = -1 to +1, and at
Reynolds numbers of 0.6, 1.5, 2.5 and 3.5 for the Newtonian liquid N. Recall that the
streak photography experiments reported in Chapter 2, revealed that the critical Reynolds
number for the onset of ﬂow asymmetry was between 0. 5 - 0.8. The Reynolds number
referred to here is deﬁned in Chapter 2 and is reproduced below in equation (3 .2).

_ ”Um?
211

Re

 

(3.2)

Here h is the upstream channel height, U,,,ll is the average velocity of the ﬂuid in the

upstream channel and p and u are the ﬂuid density and viscosity respectively.

96

To validate the LDV measurements, velocity proﬁles were measured far downstream of
the plane of expansion and compared with the plane Poiseuille ﬂow. Figure 3.3 shows the
comparison of the velocity proﬁles at X = 15 for various Reynolds numbers. The solid line
represents the known analytical solution. By X = 15, ﬂow is fully developed for all the
Reynolds numbers investigated. The maximum deviation is about 3%. Figure 3.4
illustrates the axial velocity proﬁles at Re = 0.6. The solid lines were obtained from
computations, which will be described later. Symmetry in the ﬂow is obvious at all the
three locations. At X = 1, the expansion eﬁ‘ect is clearly seen when the incoming
momentum ﬁ'om the upstream channel results in a peak velocity of only about 40% of
Um. At X =1, negative velocities are observed between Y = i 0.8 to i 1. The point at
which the negative velocity crosses over to positive values, indicates the center of the
recirculating vortex. This ﬂow reversal region spans more than 20% of the channel. As the
monitoring plane is moved downstream, the peak velocity is reduced drastically from
about 0.09 at X = 5 to the fully developed value of 0.0625 at X = 15. No negative
velocities were detected at X = 5 and 15. The overall ﬂow proﬁle ﬂattens out as the axial
distance is increased. The ﬂuid excluding the portion in the recirculation zones will be

called the core. This core continuously expands from X = 1 to X = 15.

Considerable diﬁ‘erenoes in the velocity proﬁles are observed (Figure 3.5) when the
Reynolds number is increased to 1.5, i.e., above the onset of asymmetry. At X = 1, a peak
velocity of about 0.43 occurs at Y = 0.2, which is clearly far away ﬁ'om the center line Y =
0. The asymmetry is also seen in the recirculation zones, with one being almost thrice the

other. Signiﬁcant motion in the corner vortices is detected, with the peak negative velocity

Axial Velocity

 

 

 

 

0.0a - . - . . . . .
0.06 - , ; ‘ - ..
0.04 - 3 Z .
0 cm = 0.6
i 0 Re = 15 ' ‘
0.02 - : ' ‘Re = 25 ; r
a Re = 35
0 s r s r L I a A
.1 -0.6 .0.2 0 2 0 6 r
Y

Figure 3.3 Fully developed velocity profile of the Newtonian liquid at X = 15.

Solid line is the plane Poiseuille ﬂow.

Axial Velocity

98

 

 
  
  
  
 

0.5 . r . I n r
Ax=l
0.4 "
OX=5
I'-|X=15

0.3 r
Solid line - Calculations

 

 

 

 

 

 

0.2

0.1

0

'0'1-1 A 0 o .02 03 0.6

Figure 3.4 Symmetrical axial velocity profiles of the

Newtonian liquid at Re = 0.6..

Axial Velocity

 

 

 

 

 

0.5 ' U V 1 U T F j
A Ax=l
0.4 " '
OX=8
0.3 _ I3X=15 ‘
A
0.2 " a
O O A
01 l- O A o A d
n n D D II a
an a D a a O a a I g 8
0.0-qoooo<>°AA 8!.
A A
A A A
A A A
'0°1-l.0 -0.6 -0.2 0.2 0.6 1.0

Figure 3.5 Newtonian velocity profiles after the onset of asymmetry, Re = 1.5.

100

being about -0.07. At X = 8, the smaller vortex has disappeared but the eﬂ'ect of the larger
one can still be seen. The ﬂow becomes symmetric again, with a ﬁllly developed proﬁle
observed at X = 15. With an increase in the Reynolds number to 2.5 and then to 3.5,
similar trends are observed. These are shown in Figures 3.6 and 3.7. The peak velocities
at X = 1 and X = 8 increase only slightly with increasing Reynolds numbers. For the range
of Reynolds numbers used in the present study, the second ﬂow transition that has been
reported in Newtonian ﬂows through 1:2 and 1:3 planar channels (by Cherdron et al.,
1978, Sobey, 1985, Fearn et al., 1990 and Durst et al., 1993) was not observed in the 1:16

channel. Higher Reynolds numbers would be required to witness such a transition.

Figure 3.8 compares the velocity proﬁles at X = 1. At Re = 0.6, a symmetrical proﬁle
with a peak at Y = 0 and with weak recirculation zones is observed. A drastic change in
the proﬁle occurs at Re = 1.5. Velocity now peaks at Y = 0.2. As Re is increased to 3.5,
the peak shifts slightly toward Y = l and the motion in the vortices also increases. Some
experiments revealed the larger vortex clinging to Y = 1 during a run and then showed the
same vortex attaching itself to Y = -1, when the experiment was repeated. The vortices
did not seem to have preference to attach to any particular wall. In our presentation,
however, we have always retained the smaller vortex at Y = 1, as a matter of convention.
At X = 8 (Figure 3.9), the smaller vortex has disappeared at all the Reynolds numbers.
The proﬁle is still asymmetric at Re = 1.5 and 3.5 but has started approaching the
symmetric proﬁle. This may be concluded by the observation that the peaking of the

velocity now occurs closer to the center line than at X = 1.

Axial Velocity

101

 

 

 

 

 

0.5 v u w w I u v
04 A AX=1
' ox=s
0.3 _ A DX=15
0.2 ’ "
A
X o
0.1 a a I: n o a
a u u a u
an O a O O 2 B a
0.0rllnooooO A A3...
A A A A:
A
A 9 A
.0.1 s 4 l l a l A
-l.0 -0.6 -0.2 0.2 0.6 1.0
Y

Figure 3.6 Newtonian velocity profiles after the onset of asymmetry, Re = 2.5.

102

 

 

 

 

 

0.5 I I I II
A
AX = l
0.4 ' "
A OX = 8
03 . a x = 15 .
5’
.3.
9 .. A J
z 0.2 A
O
3 o
0.1 ' o O o A d
a D D u D u D a
an a u u o 8 B B 3&3
A A A A A M
.0 1 LA A l l a l A
. -l.0 -0.6 -0.2 0.2 0.6

1.0

Figure 3.7 Newtonian velocity profiles after the onset of asymmetry, Re = 3.5.

Axial Velocity

103

 

 

 

 

 

0.5 + u u r ﬂ 1
D
°Re=0.6 O
0.4 ' ‘
0Re=l.5 a
DR =3.5
0.3 - e «
o
0.2 - o D -
o
0.1 L a -
o 0
o o 9
. o o .
0.0 W°°°°°003 099883
B 9 0 ° 0 n a I!"
.01 P a P l 4 a l A
° -l.0 -0.6 -0.2 0.2 0.6

1.0

Figure 3.8 Newtonian velocity profiles close to the expansion plane; X = 1.

Axial Velocity

104

 

 

 

 

 

0018 I V I I V I
°Re=0.6 g
0.14 - ORe=1.5 "
B
URB=3.5 q
5 a J
0.10 - o
o o
o o '3 ‘
o o o
p 0 cl
0.06 o o o
o o 8 ‘
. o 0 o
0.02 o a 0 0 § 1
9 Ia.
* to 0 ° 9 8 1
a U u
.o.oz_w .0 6 -o.2 0.2 o 6

Figure 3.9 Newtonian velocity profiles at X = 8.

1.0

105

The area under the ﬂow curve from our experimental data was evaluated using Simpson’s
rule. A parameter A using the diﬁ‘erence between this area and the measured ﬂow rate per
unit width in the z - direction is deﬁned in equation (3.3). This parameter may be

considered as a measure of the inferred transverse ﬂow.

(%)actual " (‘9wa

0%,»ch

A

 

x 100 (3.3)

Here (Q/W),cum is the measured ﬂow rate per unit width in the z - direction, (Q/W),_Dv is
the area under the curves shown in Figures 3.4 through 3.7. The inferred transverse
ﬂows at various X - locations and for all the Reynolds numbers are shown in Figure 3.10.
These range from 0.2% to 9%. Signiﬁcantly high values of A close to the expansion plane
indicate that the transverse ﬂow is maximum close to X = O and that it reduces with

increasing axial distance.

2.2 Velocity measurements of the shear thinning liquids (n = 0.7 and n = 0.5)

LDV measurements were performed with the shear thinning solution, Pl at four locations
- X = l, 8, l6 and 45, and at Reynolds numbers of 1, 2, 2.5 and 3. Since the viscosity of
the polymer solution is dependent on the shear rate, equation (3.4) was used to deﬁne the
Reynolds number (denoted by Re“) of the shear thinning solutions. This took into account
the power-law effect of shear thinning. The half height of the upstream channel, the
upstream average velocity and the wall shear rate in the upstream channel were taken as

the characteristic parameters to deﬁne the Reynolds number.

106

 

10 - u - . - . . . .
' o °Re=0.6
8 ~ ‘
ORe=1.5
’ ARe=2.5
6 . ﬂRe=3.5
U

A,%
6

 

 

 

4..
O
D
A
2' c
O
A
0 - n J a 1 a 4 4
0 4 8 12 16

Figure 3.10 Inferred transverse flow of the Newtonian liquid.

107

n 2-n
h Um,g p

2"[2(1;2n)],,.1 (3-4)

Re" =

Flow visualization experiments (Chapter 2) had indicated that the ﬂow transition from
symmetric to asymmetric ﬂow occurred at Re“ between 1.25 and 1.9. With the shear
thinning liquid the development of the ﬂow proﬁles was observed to be much slower than
in the Newtonian liquid. At the higher Reynolds numbers, ﬂow became fully developed at
X = 45, as Opposed to X = 15 for the Newtonian liquid. Figure 3.11 shows the velocity
proﬁles at X = 45. The solid line is the analytical solution for ﬂow between parallel plates
of a power - law liquid. An excellent agreement indicates the reliability of the
measurements and the fact that the ﬂow at X = 45 is not aﬁ‘ected by the abrupt changes in
the geometry at X = 0. The velocity proﬁles at Re“ = 1 (before the onset of asymmetry)
are shown in Figure 3.12. The solid lines are ﬁom computations which are described
later. Velocity at all X - locations is maximum at Y = 0. At X = 1, the peak velocity is
about 0.375, which is less than the peak of the Newtonian liquid, even at Re = 0.6. At X =

8, the velocity proﬁle has approached the fully developed ﬂow observed at X = 45.

The velocity proﬁle above the onset of asymmetry is illustrated in Figure 3.13 at Re" = 2.
At X = 1, strong vortex activity is observed in both the corners. The peak velocity in the
recirculation zone is about -0.04. A narrow core indicative of the presence of large
recirculation eddies is clearly seen. Diﬂ‘erences in the sizes of the recirculation zones are

not very obvious. The peak velocity appears to be only slightly off - centered, a behavior

Axial Velocity

108

 

0008 ' I ' 1 ‘ I V U

    
    
    
 

0.06 " ‘
°Re* = 1.0

0.04 " t
0 Re“ = 2.0
ARe“ = 2.5 i

0.02 _ D Re"I = 3.0 ..

 
 
 

Solid line - FEM computation

   

 

     

 

 
    

I l A l

4.0 A -0.6 -o.2 ‘ 0.2 0.6 A 1.0

 

Figure 3.11 Fully developed velocity profile of the 1 % shear thinning liquid

at X = 45.

Axial Velocity

109

 

0.5 V I ' T U ‘7 I
A X = l
0.4 ' '
O X = 8
ﬂ =
03 _ x 45
Solid line - FEM computation
0.2 " .

 

 

 

 

 

Figure 3.12 Velocity profiles of P1 at various axial locations before

the transition. Re“ = 1.

1.0

Axial Velocity

110

 

 

 

 

 

05F I I . I I I I ﬂ
°X=l
04' OX=8 ‘
AX=16
0.3 ’ ‘
0
0.2 b o O '
o o A
A
0.1 r 9 "
AA 0
A A 80° ‘ 9
A A A 0 ° 0 6 A
0.0 " 0 O o A "
I: : : r c . a
0.1.1.0 -0.6 -0.2 0.2 0.6 1.0

Figure 3.13 Velocity profiles of P1 after the onset of asymmetry.

Re“ = 2.0.

111

in contrast to what was observed with the Newtonian liquid. At X = 8, the maximum
velocity occurs signiﬁcantly away from the center line at Y = 0.3. The core also expands
considerably, demonstrating the decay of the corner vortices at X = 8. Portions of the
recirculation zones are still visible. At X = 16, the peak has moved slightly toward the
center and the proﬁle has started acquiring symmetry. The velocity proﬁle is ﬁrlly
symmetric at X = 45. Figures 3.14 and 3.15 show essentially the same behavior at Re“ =
2. 5 and 3, with the peak velocities increasing marginally with an increase in the Reynolds

number.

In Chapter 2, the presence of three simultaneous vortices was reported in the shear
thinning liquid, at Re“ above the critical Reynolds number. The intermediate stage wherein
ﬂow transition occurs from a state of two symmetric vortices to a state of two asymmetric
vortices in the Newtonian liquids was not detected in the ﬂow visualization experiments
with the shear thinning liquid. The present LDV measurements also do not show a ﬂow
transition to a state of two asymmetrical vortices. Instead the existence of three vortices is
conﬁrmed from Figure 3.16, which shows the presence of two comer vortices in the
portion of the channel close to Y = -l, at Re" = 2.5. The dimensional velocities
corresponding to those shown in Figure 3.16 are in the range of 0.2 to -7 mm s ", which
are well above the minimum velocities that can be measured reliably by the laser Doppler
equipments used in the present study. Very close to the walls, faint signals with positive
velocities were detected. The velocity changes sign and becomes negative at Y = -O.8 and
then turns positive again at around Y = -0.25. The occurrence of three cross - overs at X

= 1 clearly demonstrates the simultaneous existence of three vortices in the larger channel.

Axial Velocity

112

 

 

 

 

 

0.5 V I I 7 I w I '
0X=1 l
04' OX=8 ‘
O
AX=16 °
03 b o ..
0.2 - 0 ° .
o O
0 6 A0
0.1 r o ‘
A
A A
A A A A A 306 o 2
0.0 ’ 68 9 o O 0 9° 0 6 AM .
'0'1-1.o -0.6 -o.2 0.2 o 6 1.0

Figure 3.14 Velocity profiles of P1 after the onset of asymmetry.

Re“ = 2.5.

Axial Velocity

113

 

 

 

 

 

0.5 U ' ﬁ U f j 7
°X=1
0.4 ' 0X=8 O ‘
AX=16 o
0.3 P c:
00 O
0.2 ’ O .
A
0 A0
A 6 J
0.1 ' A 0
A6
A A A A A go A 2
9 o o 8’
g o o
.o'l-L0 -0.6 -0.2 0.2 0.6 1.0

Figure 3.15 Velocity profiles of P1 after the onset of asymmetry.

Re“ = 3.0.

Axial Velocity

114

 

 

 

 

 

0.01 r - . s
o 0
our - ° «
o ‘
.0.03 - ° -
o I
9
'°'°§o.9 .0 7 .0.5 .03
v

Figure 3.16 Presence of two vortices in the shear thinning liquid

close to Y = -1. Re" = 2.5, X = l.

-0.1

115

Such a phenomenon has been reported as a second ﬂow transition in Newtonian liquids at
Reynolds numbers much higher than the critical Re for the ﬁrst ﬂow transition (Durst et
al., 1993 and Feam et al., 1990). No other results are known to be reported for any kind
of ﬂow transition in inelastic, shear thinning liquids. The presence of three simultaneous
vortices close to the expansion plane results in a thin core of the order of the upstream
channel height. As a result, a quick look at the velocity proﬁle close to the expansion

plane (X = 1 in Figure 3.13) creates an impression of essentially symmetric ﬂow.

Figures 3.17 and 3.18 bring out the contrasting behavior between the Newtonian and the
non-Newtonian liquid, at Re = Re“ = 2.5. At X = 1 (Figure 3.17), the recirculation zone
close to Y = l is much larger in P1, indicating the vortex is more stretched in the y -
direction. A wider core is observed in the Newtonian liquid. This is a manifestation of the
three vortices in the shear thinning liquid taking up most of the channel height, close to the
expansion plane, as is evident ﬁ'om our ﬂow visualization results in Chapter 2 (Figures
2.18 b, c and d). The additional vortex attaches itself to the smaller corner vortex, thereby
resulting in a core which is almost symmetrically distributed about Y = 0. Further
downstream at X = 8 (Figure 3.18), the largest vortex shrinks in the y - direction as does
the vortex close to Y = 1. This results in a wider core in P1. The off - centered peaking is
now more obvious because the recirculation zone close to Y = 1 has almost died out but

the large central vortex (closer to Y = -l) is still present, resulting in signiﬁcant

asymmetry.

Axial Velocity

0.5

0.4

0.3

0.2

0.1

0.0

-0.1

116

 

 

ﬂ

0 Non - Newtonian

O Newtonian l o

 

 

 

Figure 3.17 Effect of shear thinning on the velocity profiles

at Reynolds number = 2.5; X =. 1.

1.0

Axial Velocity

117

 

 

 

 

 

 

0.3 . . . . . . .
ONewtonian
ONon-Newtonian
0.2 - ' ' -
e
o
e 0 '
0.1 - ° ° ..
o O
o
e. 0
a 0
e °azo
0.0 - ‘9 3 a 8 8 0 r
'v
'o'l-m ‘ -0.6 ‘ -o.2 0.2 0.6 1.0
Y

Figure 3.18 Effect of shear thinning on the velocity proﬁles

at Reynolds number = 2.5; X = 8.

118

Figures 3.19 - 3.21 compare the velocity proﬁles of the shear thinning liquid at X = l, 8
and 16 respectively. The evidence of three simultaneous vortices is seen at Re“ = 2, 2.5
and 3 in Figure 3.19. Based on the oﬁ‘ - centered peaking, ﬂow appears to be quite
symmetric at X = 1. At X = 8 (Figure 3.20) though, peaking at signiﬁcantly large oﬁ‘ -
centered locations is obvious. The smallest vortex has disappeared and the vortex along Y
= 1 has shrunk in the y - direction. Strangely, the peak appears to be more oﬁ‘ centered at
Re" = 2 than at Re“ = 3. This would indicate that the eﬁ‘ect of inertia on the largest vertex
is weaker than that on the vortex along Y = 1. At X = 16, all the vortices have
disappeared. Flow is still asymmetrical about Y = 0. As shown earlier (Figure 3.11),

symmetrical ﬁrlly developed ﬂow was conﬁrmed at X = 45.

The inferred transverse ﬂow was calculated for the shear thinning liquid using equation
3.3. The results are shown in Figure 3.22. A ranges from 0.5% to 7%. Contrary to what
was observed with the Newtonian liquid, A is larger at X = 8 and X = 16 than at X = 1.
This could be a manifestation of the three vortices close to the expansion plane hampering

flow in the normal direction at X = 1.

Results from the streak photography experiments reported in Chapter 2, demonstrated the
absence of ﬂow asymmetry in the more shear thinning liquid P2, even up to the highest
achievable Reynolds number of 1.6. This was conﬁrmed from our LDV measurements

too. The results are shown in Figure 3.23. As mentioned in Chapter 2, the solution P2 is

Axial Velocity

119

 

 

 

 

 

005 U ﬁ' I I I
°Re*=l.0
0.4 ' n -
ORe*=2.0
U I"= .
03 Re 30 I: oo .
O u
0.2 - a o ‘
0° 0
a
0.1 " 9 9° '
08 08
0.0’ 0.800330 .880099.
a ° 0 u ° ° ° 8§
a u D I: u
.0°l-1.0 -0.6 -0.2 0.2 0.6
Y

Figure 3.19 Velocity profiles of shear thinning liquid P1 at X = 1.

1.0

Axial Velocity

120

 

 

 

 

 

 

0.25 ' I 7 U U ' I V
D
°Re*=l.0 °
F o J
ORe*=2.0 a
O
IllRe"'=3.(l
0.15 - a 0 -
D
O
. o .
3
0 0
o o oo o °°o
0.05 - o o 0 o 3 -
o o 0 83° 0 o
°¢ O o O O ‘
Q0 0 a u o ‘
a u n a
“3
'0'05-1.o A .045 -o.2 0.2 0.6 A

Figure 3.20 Velocity profiles of shear thinning liquid P1 at X = 8.

1.0

Axial Velocity

121

 

 

 

 

 

 

0.20 ' 1 V I I f I 1
°Re* = 2.0
A
0 Re" = 2.5 O
0.15 - o .
ARe“ = 3.0 a A
o
o o
A
0.10 - 0 o -
° A
t o .
6 o
t A .
0 §
. § § 9 Q .
Q l
0.00 J? . . . . . . 2g—
-1.0 -0.6 ~0.2 0.2 0.6 1.0
Y

Figure 3.21 Velocity profiles of shear thinning liquid P1 at X = 16.

A,%

122

 

 

 

 

» o ORe“ = 1.0 <
_ one = 2.0 .
° ARe“ = 25
D * - 1
ll Re - 3.0 l
. a o .
0 A
b o 1
’0 ° ‘
.n a a .
A
o 10 20 so 40
x

Figure 3.22 Inferred transverse flow in shear thinning liquid P1.

50

Axial Velocity

123

 

0.6

U

0.5

0.4 '

0.3

 

  
  
  
 

 

 

AX=1 .
OX=8
DX=45
Solid line - Calculations .

 

 

 

0.2 0.6 A 1.0

Figure 3.23 Symmetric velocity proﬁles of P2 at Re“ = 1.5.

124

viscoelastic at high Reynolds numbers. Townsend and Walters (1995) have shown that
viscoelasticity dampens the vortex growth in planar expansions. Also based on our
experiments, shear thinning too delays the ﬂow transitions. The combined eﬁ‘ect of ﬂuid
viscoelasticity and higher pseudoplasticity could explain why ﬂow transitions were not

observed in P2.

2.3 Quantiﬁcation of the degree of asymmetry

The normal velocity component at Y = O has been used as an index of the degree of
asymmetry by Feam et al. (1990) in their experiments with Newtonian ﬂow through a 1:3
planar expansion. Durst et al. (1993) have used the diﬂ‘erence in the reattachment lengths
along the two walls as a suitable measure of asymmetry. Another parameter that may be
used is the diﬁ‘erence in the wall shear stresses. As described earlier, practical limitations
prevented the measurement of the center line V component of velocity in the present
work. The diﬁ‘erence in reattachment lengths requires LDV measurements at very close x -
positions to locate the exact reattachment point. The current work uses the center line
shear rate as the index to quantify the degree of asymmetry. Non - zero center - line shear
rates are indicative of ﬂow asymmetry. Figure 3.24 plots the center - line shear rates for
varying Reynolds numbers, for the Newtonian liquid at X = l, 8 and 15. At X = 15 the
symmetric ﬂow is obvious from the zero degree of asymmetry observed at all the
Reynolds numbers. At Re = 0.6, the center - line shear rate is zero at all axial locations.
For Re 2 1.5, a ﬁnite asymmetry exists at both the locations X = 1 and X = 8, and it
increases with increasing Re. As the ﬂuid moves downstream, the asymmetry reduces.

Since the vortices have no preference to attach to any particular wall, there is an equal

125

likelihood of the shear rate being negative. So a mirror - image of the positive shear rates
has been used to sketch the whole curve shown in Figure 3.24. A similar plot for the
shear thinning liquid at X = 8 and X = 16 is shown in Figure 3.25. The shear rates at X =
1 for P1 have not (been reported because the inﬂection point of the velocity proﬁles at X =
1 is very close to Y = O. This has the potential of introducing signiﬁcant errors. Although
the trends observed are similar to those described above for the Newtonian liquid, the

magnitudes of the asymmetry in P1 is always higher than that in the liquid N.

Figures 3.26 and 3.27 reveal the axial variation of the center - line shear rate for N and
P1 respectively. At any x - location, the asymmetry increases with increasing Reynolds
number. At Re 2 1.5, Figure 3.26 shows that as the Newtonian liquid ﬂows ﬁ'om X = 1 to
X = 8, the center - line shear rate falls down by an order of magnitude before the ﬂow
ﬁnally becomes symmetric at X = 15. With the shear thinning liquid, the ﬂow is still
asymmetric at X = 16 and zero asymmetry is observed only at X = 45. This clearly

indicates that the ﬂow length over which asymmetry persists is also higher.

2.4 Bifurcation curves from the experiments
Equation (3.5) shows a modiﬁed parameter called the degree of asymmetry, D,” which is

deﬁned using the center - line shear rate and the axial location.

D =—| -X” (3.5)

U is the non - dimensional axial velocity and p is a ﬁtting constant selected such that the

Center - line shear rate

126

 

 

 

 

 

1.0 T 1 V r v I v
' 0N, X = 1
.. _ o d
0.6 ON, X — 8 o
> AN, X = 15 o
0.2 r .
o O °
-——0 r r t——
O o o
-0.2 - '-
0
0
-0.6 " o "
‘1'0 o A 2 3 4
Reynolds number

Figure 3.24 Growth of asymmetry with the Reynolds number
in the Newtonian liquid.

Center - line shear rate

127

 

 

 

 

 

 

1.0 . . - . s .
ORX=8 ’
e
0.6 . 0 P, X = 16
. 0
AP, x = 45 '
O
0.2 b
a a—+——a
-o.2 -
O
O
O
-0.6 ~ '
o
’ e
'1'" o A 1 2 3
Reynolds number

Figure 3.25 Growth of asymmetry with the Reynolds number

in the shear thinning liquid P1.

Center - line shear rate

0.8

0.6

0.4

0.2

0.0

128

 

 

 

 

' oRe=0.6

. ORe= 1.5
A ARe=2.5

. URe=3.5

P O

0"" J 5 W in ‘ *5

X

Figure 3.26 Decay of asymmetry in the Newtonian liquid,

with increasing flow length.

Center - line shear rate

1.0

0.8

0.6

0.4

0.2

0.0 ‘

129

 

 

 

Figure 3.27 Decay of asymmetry in the shear thinning liquid,

with increasing flow length.

 

n ORe“ = 1.0
0Re* = 2.0
P A 4
ARe" = 2.5
' a Re“ = 3.0 i
0 1
A
O
——e—._l—.e.¢ 1 e 1 l -.__
0 10 20 30 40 50

130

degree of asymmetry at any Reynolds number is the same at both axial positions.
Bifurcation curves relating the degree of asymmetry, D,” and the Reynolds number are
shown in Figures 3.28 and 3.29 for the Newtonian liquid and the shear thinning liquid
respectively. p = 0.9 and 0.6 have been used for N and P1 respectively. Interestingly, p
lies very close to the shear thinning index of the liquids used (i.e., 1 and 0.7 respectively).
For the Newtonian liquid, ﬂow occurs along branch 1 until Re = 0.6. Flow bifurcates
between Reynolds numbers of 0.6 and 1.5 and either of the paths shown by branches 2 and
3 could be followed at higher Re. Branch 4, which is an extension of the symmetric state,
is not observed in the experiments. Shapira et al. (1990) and Allebom et al. (1997) have
performed a linear stability analysis of Newtonian ﬂows through expansions and have
conclusively demonstrated that the branch 4 is indeed unstable and will not be realized in
experiments. The polymer solution also exhibits a qualitatively similar bifurcation curve.
The bifurcation point is located between Re“ = 1 and 2. At comparable Reynolds
numbers, the shear thinning solution exhibits a higher degree of asymmetry than the

Newtonian liquid.

3. Computations of ﬂow bifurcation

Flow bifurcation curves were computed by using a perturbation - continuation approach.
Only a brief description of the method has been included here. Refer Chapter 4 for a
description of the technique followed, the details of the solution method and the grid
resolution. Finite element calculations were performed on an asymmetric domain for
Reynolds numbers spanning O to Re,, where Re, is a Reynolds number greater than the

critical Reynolds number. A continuation method was used wherein the results at lower

Degree of asymmetry

131

 

 

 

 

 

1.0 - u . I u
9N, X = 1
ON, X = 8
o 2
“-5 ’ AN x = 15
’ c
1
0.0 _t 1 1 JA——
0
-0.5 " o
3
.1.0 o 1 2 3
Reynolds number

Figure 3.28 Bifurcation curve for the Newtonian liquid N.

Degree of asymmetry

132

 

 

 

 

 

3 ' I 1 j—'—
OP, x = s ’ 2
2 OP, X = 16 0
AP, X = 45
l
0 t h—i—i
1
-l
O
.2 3
0
. 1 . n 4 .
3 0 l 2 3
Reynolds number

Figure 3.29 Bifurcation curve for the shear thinning liquid P1.

 

133

Reynolds numbers were used as initial iterates for computing the solution at higher Re. At
Re = Re,, the domain was made symmetric and continuation solutions were computed on
the symmetric grid by successively reducing Reynolds numbers. The results from this
technique on Newtonian ﬂow through channels with expansion ratios of 2 and 3 showed
excellent agreement with the computational results available for such channels.
Computations were then performed for planar channels with expansion ratio of 16. A
power - law model was used to describe the viscosity of the shear thinning liquid. The
computed axial velocity proﬁles before the onset of asymmetry for the Newtonian and non
- Newtonian liquids are compared with the experimental data in Figures 3.4, 3.12 and
3.23. Excellent agreement validates the approach used in the computations. Critical
Reynolds numbers of 6 and 18.5 for the onset of ﬂow transitions were obtained for n = l
and n = 0.7 respectively. These are higher than the corresponding values observed

experimentally.

3.1 Computed predictions of the decay of ﬂow asymmetry

Figure 3.30 shows the variation of the center - line shear rate with increasing axial
positions, at locations X = 1 and 8 for the Newtonian liquid. The asymmetry decays
downstream of the expansion plane for both the Reynolds numbers considered. At the
same position, higher Reynolds number results in a higher asymmetry. These trends are
consistent with the ﬁndings from the LDV experiments reported in Figure 3.26. A similar
decay was computed for the shear thinning liquid with n = 0.7. The results at X = 8 and at
X = 16 are shown in Figure 3.31, for Re“ = 26, 31 and 48. The computed critical

Reynolds number for the ﬁrst and second ﬂow transitions were 18.5 and 42 respectively.

Center line shear rate

134

 

 

 

 

’ ORe = 8.3
ARe = 13.3
.. A 4
o
b- d
‘ A
O
0 2 4 6 8
X

Figure 3.30 Computed decay of asymmetry in the Newtonian liquid.

10

Center line shear rate

135

 

 

ORe"=26
ARe*=3l
uRe*=48

 

Figure 3.31 Computed decay of asymmetry in the shear thinning liquid.

 

136

For all the three Reynolds numbers reported in Figure 3.31, the ﬂow asymmetry reduces
as X increases. At any location, higher Re“ results in a higher ﬂow asymmetry. As
opposed to the Newtonian ﬂow where ﬂow asymmetry had subsided by X = 15, ﬂow of
the shear thinning. liquid is considerably asymmetric even at X = 16, conﬁrming the earlier
experimental observation that shear thinning results in a slower decay of ﬂow asymmetry

and consequently a longer flow length up to which asymmetry in the ﬂow persists.

3.2 Velocity proﬁles from the perturbation - continuation approach

Figure 3.32 shows the computed velocity proﬁles at various axial positions for the
Newtonian liquid at Re = 8. At X = 5, the two recirculation regions are clearly seen. The
velocity peaks slightly away from the center line. At X = 20 the smaller vortex is no longer
present and the velocity peaks much closer to Y = 1 than at X = 5. Flow is ﬁrlly developed
at X = 75. The velocity proﬁles shown in Figure 3.32 is qualitatively similar to the ones
obtained experimentally (Figures 3.5 through 3.7). The velocity proﬁles at Re = 25 are
depicted in Figure 3.33. At this Reynolds number, the second ﬂow transition has already
occurred and three vortices are present. At X = 5, the two recirculation regions
representing the two comer vortices are seen. The smaller corner vortex is no longer
present at X = 30 and at X = 60 the third vortex is inferred on the wall on which the

smaller comer vortex had attached itself close to the expansion plane.

Figure 3.34 illustrates the computed velocity proﬁles of the shear thinning liquid at Re“ =

20. A ﬂow transition ﬁ'om a state of two symmetrical vortices to two asymmetrical

Axial velocity

137

 

0.5

e
in

0.1 *

 

DX=5
OX=ﬂl
D
AX=75 a
D
a G
u
D

 

 

 

 

1.0

Figure 3.32 Computed velocity profiles of the Newtonian liquid at Re = 8.

Axial velocity

138

 

0.7 . a . . . . .
ax=5 #1
OX=30 D
05' AX=60 D ‘
D

 

 

 

 

 

 

”41.0 ‘ -0.6 A -o.2 0.2 ‘ 0.6 1.0

Figure 3.33 Computed velocity profiles of the Newtonian liquid at Re = 25.3.

139

vortices is observed at this Reynolds number. This ﬂow transition was not detected in the
corresponding LDV experiments. The ﬂow transition is obvious with the two asymmetric
corner vortices coexisting at X = 5. By X = 15 and X = 20, only the larger corner vortex
exists. The computed velocity proﬁles at Re“ = 44 corresponding to the second ﬂow
transition in the shear thinning liquid are shown in Figure 3.35. At X = 5, two
recirculation regions are observed. Only the larger comer vortex exists at X = 30 and at X
= 45 the third vortex is conﬁrmed on the wall on which the smaller corner vortex had
attached itself close to the expansion plane. Streamlines from the computations always
showed the third vortex to attach itself far downstream of the expansion plane. The LDV
measurements (and the ﬂow visualization), however, revealed that the third vortex was

located on the walls at X = O.

3.3 Comparison of experimental and computed bifurcation curves

The degree of asymmetry was calculated at Re = 8.33 and 13.33 for the Newtonian liquid
and at Re" = 26 and 31 for the shear thinning liquid with n = 0.7. The same values of p as
that used in the experiments were employed to determine the degree of asymmetry from
the computations. To account for the differences in the experimental and the predicted

critical Reynolds numbers, a modiﬁed Reynolds number was deﬁned as -

Remod=Re-Re - Re‘mod=Re‘-Re‘. (3.6)

Cf" ’ Cf"

The experimental critical Reynolds number was used to evaluate the experimental
modiﬁed Reynolds number and the computational modiﬁed Reynolds number was

calculated from the computational critical Reynolds number. Figure 3.36 illustrates the

Axial velocity

140

 

 

 

 

 

 

0.5 - 1 . u u
- D

a X — 5 r0

O X = 15 I:

AX = 20 D
0.3 L a

D
b D

0.1 "
.0J-l.0 -0.6 -0.2 0.2

Figure 3.34 Computed velocity profiles of the shear thinning liquid Pl

at Re“ = 20.

 

Axial velocity

141

 

 

 

 

 

 

0.8 u
D X = 5
0.6 " OX = 30 ..
AX = 45
0.4 ' "
0.2 " ‘
AAA ‘ ‘
0.0
an O a a an .
0° 0
'02-].0 -0.6 '02 0.2 0.6

Y

1.0

Figure 3.35 Computed velocity proﬁles of the shear thinning liquid Pl

at Re“ = 44.3.

142

modiﬁed biﬁircation curve for the Newtonian liquid. Only the positive half is shown for
purposes of clarity. Increasing the Reynolds number increases the degree of asymmetry for
both the computations and the experiments. D. evaluated from the experiments seem to
level oﬂ‘ in comparison with that obtained from the computations. In the shear thinning

liquid, however, a continuity is apparent between the experimental and the computed

degrees of asymmetry (Figure 3.37).

4. Conclusions

Laser Doppler velocimetry was used to obtain detailed information on the velocity proﬁles
of Newtonian and non - Newtonian liquids flowing through a 1:16 planar channel.
Reynolds numbers between 0.6 to 3 .5 and 1 to 3 were spanned for the Newtonian liquid
and the shear thinning polymer solution with a power - law index of 0.7 respectively. The
velocity proﬁles at various axial locations were used to make detailed observations on the
dynamics of vortex growth at Reynolds numbers above the onset of ﬂow transitions. The
center - line shear rate was used as a measure of the degree of asymmetry and plots were

drawn to show the ﬂow bifurcation in these liquids above the critical Reynolds number.

The Newtonian liquid experienced flow transition between Re = 0.6 and 1.5. Results from
the ﬂow visualization experiments had indicated that at Re = 0.8, ﬂow transition had set in
in the Newtonian liquid. At Re 2 1.5, ﬂow was asymmetric close to the expansion plane

(X = l) and also at X = 8. All asymmetries decayed by X = 15. The second ﬂow transition

to a state of three vortices was not observed for the range of Reynolds numbers studied.

143

7 v u . u w u

 

 

 

 

0 Experiment
6 - A -
A Computation
ca 5 I- u
E“
a 4- -
8
.5 3 b d
A
i
so
a 2 P .
l ' ..
0 ° 0
a A l A l A l
0 0 2 4 6 8

Remod = Re - Rec,“

Figure 3.36 Comparison of the degree of asymmetry from LDV experiments

with that from the FEM computations for Newtonian ﬂow.

Degree of asymmetry, D,ls

144

 

 

 

 

A
_ OExperiment .
A Computation
b A d
D O C
0

ho a
a l l
0 5 10

Remod = Re - Rec,“

Figure 3.37 Degree of asymmetry for the polymer solution P1.

15

145

At Re"I = 1, the shear thinning liquid showed symmetric velocity proﬁles at all the axial
locations. It also showed a flow transition to the simultaneous presence of three vortices
at Re“ = 2 and above. The ﬂow transition to the state of two asymmetrical vortices
occupying the comers of the larger channel was not detected by the experiments with the
pseudoplastic liquid. Shear thinning, therefore, seems to hasten the onset of the second
ﬂow transition experienced by the Newtonian liquids at very high Reynolds numbers.
Asymmetry in the ﬂow in terms of the center line shear rate was observed to decrease
downstream of the plane of expansion for both liquids. However, the degree of asymmetry
in the shear thinning liquid was greater than that in the Newtonian liquid and it also
persisted for a longer ﬂow length. The shear thinning liquid with a power - law index of
0.5 (liquid P2) revealed symmetric proﬁles up to the maximum achievable Reynolds
number of 1.5. Fluid viscoelasticity at high Reynolds numbers and a higher

pseudoplasticity could be responsible for the delayed flow transitions.

Flow biﬁircation curves were computed using a perturbation - continuous approach for
liquids with power - law shear thinning indices of l and 0.7. The predicted bifurcation
points were higher than those obtained from experiments. However, the computed trends
of higher bifurcation points and of a greater degree of asymmetry with increasing shear
thinning were the same as that observed experimentally. Also, computations predicted the
trends of decaying ﬂow asymmetry with increasing axial positions. The computed velocity
proﬁles demonstrated both the ﬂow transitions for the Newtonian and shear thinning

liquids.

146
The eﬂ‘ects of shear thinning may be summarized as:

(1)to delay the onset of the ﬁrst ﬂow transition,
(2) to hasten the onset of the second ﬂow transition,
(3) to raise the degree of asymmetry at comparable Reynolds numbers and

(4) to cause the asymmetry to persist for a longer ﬂow length.

Chapter 4
COMPUTATIONS OF SYMMETRY BREAKING FLOW BIFURCATION

IN ABRUPT PLANAR EXPANSIONS

1. Introduction

Flow of ﬂuids through geometries with abrupt changes in cross section is commonly
encountered in various engineering operations. Such ﬂows are ﬁrrther complicated by
inertia or by ﬂuid nonlinearities such as pseudoplasticity, stretch thickening, yield stress or
viscoelasticity. While the literature is replete with the description of converging ﬂows (see
reviews by White et al., 1987 and Boger, 1987 and 1997), the corresponding diverging
ﬂows have not evoked comparable attention. Only in the case of creeping ﬂow of
Newtonian liquids will the ﬂow features in an expansion be a mirror-image of that in a
contraction ﬂow. Due to the nonlinearities in other cases, no direct inference for polymers

ﬂowing through expansions can be made based on corresponding contraction ﬂow results.

2. Background

2.1. Newtonian ﬂow transitions in an abrupt planar expansion

2.1.1. Existing experimental results

Sobey (1985) has studied the ﬂow patterns developed in a Newtonian liquid (water)
ﬂowing steadily through a two - dimensional 1:3 planar expansion. He observed that at

147

148
low ﬂow rates (or low Reynolds number, Re, deﬁned using the smaller channel half height
and the average velocity in the upstream channel), the corner vortices were symmetric and
that they increased in size as Re was increased. The ﬁrst ﬂow transition was observed
beyond a certain critical Reynolds number, Rem (around 25 for an abruptly expanding
channel), when ﬂow became asymmetric and one vortex grew more rapidly than the other.
In mathematical terms, the ﬂow is said to have biﬁrrcated at Recl into two possible stable
solutions (both exhibiting asymmetric ﬂow), with the pre-biﬁrrcated symmetric solution
now becoming unstable. As Re was raised further, a second ﬂow transition took place at a
critical Reynolds number, Red, leading to three vortices, two along one wall and one
along the other. The steady ﬂow became periodic at higher Re before ﬁnally turning
turbulent. The critical Reynolds numbers were observed to be higher for channels with
gradual expansions. By using long rectangular channels of large aspect ratios, i.e., the
ratio of the width of the channel to its height, Sobey ensured that the ﬂow far away from
the expansion plane was one dimensional. Experiments showing similar ﬂow transitions in
Newtonian ﬂuids through abrupt expansions have been reported by Durst et al. (1974) and

Cherdron et al. (1978) and conﬁrmed later by Feam et al. (1990) and Durst et al. (1993).

Cherdron et al. (1978) have investigated the effect of the expansion ratio (B) and the
aspect ratio on the ﬂow patterns developed in a Newtonian liquid ﬂowing through a planar
expansion. They have reported that the critical Re at which the ﬁrst ﬂow transition occurs
goes down with an increase in the expansion ratio as well as an increase in the aspect
ratio. For example, with a channel of aspect ratio 8, the critical Re goes down from about

150 to 30 in going ﬁom an area expansion ratio of 1 :2 to 1:3. The mechanism of the

149

initiation of such ﬂow transitions, even for Newtonian liquids, is not well understood.
However, the Coanda eﬂ‘ect does seem to explain how it is sustained. Flow asymmetry is a
result of the momentum transfer between the shear layers in the direction normal to ﬂow.
As a result, the boundary layer adjacent to one wall is richer in momentum and
consequently at a lower pressure as compared to the other. A transverse pressure gradient
in the channel is created and as more liquid enters, ﬂow of the liquid from the wall at
higher pressure to that at a lower pressure takes place. This increases the momentum at
the richer wall even more resulting in a more pronounced eﬂ‘ect of the ﬂow asymmetry.
No ﬂow asymmetry has been reported in axisymmetric expansions, where the ﬂow induces

continuous concentric shear layers, as opposed to the stacked shear layers in planar ﬂow.

Some scatter in the critical Reynolds numbers reported is evident in the literature. The

Reynolds number used in the present work is deﬁned in equation (4.1).

=hUmp
2n

Re

(4.1)

Here, h is the upstream channel height, U" is the average velocity in the upstream channel
and p and u are the ﬂuid density and viscosity respectively. Cherdron et al. (1978)
reported that Recl was around 62 for ﬂow in a channel with a 1:2 expansion ratio and with
an aspect ratio of 8 or more in the larger channel. Mth a similar channel, Durst et al.
(1993) observed ﬂow asymmetries at Re = 42. In a channel with an area expansion ratio of
3, Durst et al. (1974), Sobey (1985) and Feam et al. (1990) reported that Newtonian ﬂow

became asymmetric at Re of 38, 25 and 22 respectively. In our experiments (reported in

150

Chapters 2 and 3) with a much higher expansion ratio of 16, asymmetric ﬂow sets in at

Reynolds number of 0.8.

2.1.2. Prediction of Newtonian ﬂow bifurcation using computations

Various numerical procedures have been used to predict the ﬂow bifurcation in a planar
sudden expansion. Computations involving ﬁnite elements have always needed some kind
of perturbation - either to the geometry or to the boundary conditions - to yield a
bifurcated ﬂow structure. Durst et al. (1993) have made transient calculations to predict
the onset of ﬂow transitions for a Newtonian liquid. They followed a perturbation
approach which involved the following steps:

At time t = O, a ﬁrlly developed velocity proﬁle was imposed at the inlet of the smaller
channel. After n time iterations and during m time iterations, the inlet velocity conditions
were modiﬁed such that the axial velocity component (U) was a function of time and the
normal velocity component (V) was a function of both time and space. Subsequently, the
original symmetric fully developed velocity proﬁle was restored as the inlet boundary
condition. The solution was allowed to achieve steady state. Using this approach a critical

Reynolds number of 42 was predicted for ﬂow through a 1:2 expansion.

F earn et al. (1990) used numerical biﬁrrcation techniques applied to a ﬁnite element
discretization of the two - dimensional Navier - Stokes equations and predicted ﬂow
bifurcation in a 1:3 planar expansion at Re,1 of 27. Their work sheds more light on the
sensitivity of the bifurcation point to experimental uncertainties. They suggest that the

critical transition point is structurally unstable. The slightest imperfection in the

15 1
experiment may lead to a discontinuity in the bifurcation curve, with a shift in the original
bifurcation point. They showed that a vertical shift in the grid of the larger channel by
0.42% with respect to the axis of symmetry provided much closer match with the
experimental data. Their work seems to explain the possible reason for the scatter
observed in the experimental results on ﬂow biﬁircations through planar expansions.
Indeed , experimental imperfections could have many sources such as geometrical

asymmetry, upstream ﬂow asymmetry, momentary mechanical shock, temperature

variation, etc.

Teschauer (1994) employed an asymmetric upstream channel to compute the bifurcated
branch of the solution for various expansion ratios between 1.5 to 3. A commercial ﬁnite
element code was used. She predicted critical Reynolds numbers of 72 and 27 for

expansion ratios of 2 and 3 respectively.

In a different ﬂow conﬁguration involving Newtonian ﬂow into a rectangular channel
through two symmetric openings, Goodwin and Schowalter (1996) have shown the
possibility of the simultaneous existence of numerous solutions beyond a critical Reynolds
number. To access the asymmetric solutions, they have followed a continuation approach
coupled with a perturbation in the ﬂow rates in the two channels. Using a ﬁnite element
discretization, solutions with the asymmetrical ﬂow rates were computed for increasing
Reynolds numbers. Results at the lower Reynolds numbers were used as initial iterates for
computations at higher Reynolds numbers. At signiﬁcantly high Reynolds numbers, the

ﬂow was rebalanced and the same continuation approach was used for successively

152

decreasing Reynolds numbers. Such an approach detected the various asymmetric
branches of the solutions. A similar approach using continuation and a perturbation in the
geometry to compute the ﬂow bifurcation in planar expansions with a single inlet has been

devised in the present work. This will be discussed later.

Very recently numerical results using a ﬁnite volume approach for ﬂow through planar
expansions have been presented. A striking feature of such an approach is that ﬂow
biﬁrrcation has been computed without providing any kind of perturbation to the ﬂow
conditions. Drikakis (1997) has presented biﬁrrcation curves ﬁ'om several schemes for
expansion ratios up to 6. Various numerical schemes based on an overall ﬁnite volume
approach have been tested. His results for the onset of asymmetry with expansion ratios of
2 and 3 match the corresponding results of Teschauer (1994) and the results from the
linear stability analysis of Shapira et al. (1990). Foumeny et al. (1996) have shown that a
commercial code, FLUENT, can be effectively utilized to study the ﬂow bifurcations. This
code employs a ﬁnite volume discretization procedure, in which linear approximation
ﬁinctions are used to interpolate values of the ﬂow variables throughout the solution

domain. Their focus is on obtaining results on channels with more than one inlet opening.

A characteristic observation from the computational results is that ﬂow biﬁrrcation
calculations almost invariably predict a higher critical Reynolds number than the
corresponding value obtained in experiments. This is consistently observed in the work
described above on channels with low expansion ratios of 2 and 3 and also for a channel

with a more severe expansion ratio of 16, the computational results for which are reported

153
in the present work. The corresponding experimental ﬁndings for the 1:16 channel have

been presented earlier in Chapters 2 and 3.

Sobey (1985) and Sobey and Drazin (1986) have attempted to relate the experimentally
observed bifurcation to that in Jeffery - Hamel (II-I) ﬂows. In the latter, radial ﬂow occurs
between two inclined plates intersecting at an angle 2a. Flow is generated by a source
located on the line of intersection. JH ﬂows have been shown to biﬁircate subcritically,
leading to no stable ﬂow above the critical Reynolds number. Planar expansion ﬂows, on
the other hand, display a supercritical bifurcation which is conﬁrmed by several reports of
stable asymmetric patterns in planar expansions. Sobey, therefore, concluded that 1H

ﬂows are not of any signiﬁcance in explaining expansion ﬂow after biﬁrrcation.

Shapira et al. (1990) have performed a linear stability analysis of ﬂow in symmetric
expansions. They have considered a gradual expansion with a constant slope of the
channel walls. Although their predictions of Recl = 27.5 for a 1:3 expansion are in
agreement with the computational results of Feam et al. (1990), whose computations
suggest the onset of asymmetry at Re of 27, they over predict Rec, of 72 for a 1:2
expansion. More importantly their computations show that the symmetric branch of the
solution obtained above the critical Reynolds number is unstable and therefore unlikely to

be ever witnessed in experiments.

Allebom et al. (1997) have performed a linear stability analysis on ﬂows through abrupt

expansions. Finite difference approximations of the equations of motion in the stream

154

function - vorticity form have been used. The biﬁrrcation picture is extended by computing
the subsequent bifurcation point (Re,,_) and solution branches. Solutions for inﬁnite
expansion ratio have been compared with Jeﬂ‘ery - Hamel ﬂow. They conclude that away

ﬁom the expansion plane ﬂow proﬁles do approach the proﬁles of JH ﬂow.

2.2. Non - Newtonian ﬂow through expansions

Still much work remains to separate the contributions of inertia, shear thinning and stretch
thickening viscosities, elasticity, expansion ratio and the mode of expansion (axisymmetric
vs. planar) on the ﬂow patterns in expansions. Abdul-Karem et al., (1993) have
investigated ﬂow behavior of 0.05% polyacrylamide solution in glucose syrup (Boger
ﬂuid) through a 116.8 axisymmetric expansion and a 6.8:] axisymmetric contraction.
Although the vortex size observed in expansions was lower than that in contractions, the
trend of vortex enhancement with increasing ﬂow rate was observed in the expansion ﬂow
too. No lip vortices or instabilities were observed in either conﬁguration. Halmos et al.
(1975) have conducted experiments using aqueous carboxymethyl cellulose solution
(inelastic, shear thinning power-law index of 0.73) in a 1:2 abrupt axisymmetric expansion.
They have measured the radial and axial velocity proﬁles downstream of the expansion
plane using ﬂash photography. Their experimental results on velocity proﬁles and
reattachment lengths agree well with their calculations for Reynolds numbers between 11

to 125. The authors do not observe any ﬂow asymmetries, oscillations or lip vortices.

Townsend and Walters (1994) have studied the ﬂow of a 0.15% aqueous solution of -

polyacrylamide (slightly shear thinning, strongly elastic) through an expansion from a

155

square section to a rectangular slit (area expansion of 13 .3) and from the rectangular slit
to another square section (area expansion of 13 .3). With a Newtonian liquid, signiﬁcant
vortex activity was observed due to ﬂuid inertia. The vortex size increased with the ﬂow
rate. The polymer solution, on the other hand, produced small corner vortices, the size of
which decreased with the ﬂow rate. Fluid elasticity thus, was observed to inhibit vortex
growth. It should be pointed out that their study with Newtonian liquids has not shown
any evidence of ﬂow asymmetry, in contrast to those reported by Durst et al. (1974),
Cherdron et al. (1978), Sobey (1985) and Feam et al. (1990), despite a wide range of
Reynolds numbers used. The square channels used by Townsend and Walters (1994)
approximate axisymmetric ﬂow which has never been known to show the ﬂow asymmetry
mentioned above. In addition, based on the ﬁndings of Cherdron et al. (1978) that the
critical Reynolds number for the onset of ﬂow asymmetry increases with decreasing
channel aspect ratio, one might conclude that at an aspect ratio of 1, ﬂow asymmetry
might be observed, if at all, only at very high Reynolds numbers. One notable contribution
of the Townsend and Walters work was to clearly bring out the effect of ﬂuid elasticity on
the vortex growth in planar expansions. The observed trends in the eﬂ‘ects of ﬂuid inertia
and elasticity on secondary ﬂows was in contrast to what had been observed in various
contraction geometries, where ﬂuid inertia reduced the vortex size while elasticity led to
vortex enhancement. This reversal of roles may be the reason for lack of further

investigation of viscoelastic effects in expansion ﬂows.

No results demonstrating the effect of pseudoplastic flow behavior on the bifurcation

phenomenon, even for lower expansion ratios are available in literature.

156
3. Objectives
Experimental evidence of Newtonian ﬂow transition at expansion ratios higher than 3 is
non - existent. Also conspicuously absent is any indication whether non - Newtonian ﬂows
will show ﬂow biﬁrrcation at high Reynolds numbers. The present study seeks to extend
the ﬁndings of Cherdron et al. (1978) by experimentally locating the ﬂow biﬁrrcation in
channels of expansion ratio 1:16. Our detailed observations on the ﬂow visualization and
laser Doppler velocimetry experiments carried out with Newtonian and non - Newtonian
liquids have already been reported in Chapters 2 and 3 respectively. In the present

Chapter, the corresponding computational results have been discussed.

The unique feature of this work is the determination of the effect of non - Newtonian
viscosity on the occurrence of ﬂow bifurcation in planar channels with a high expansion
ratio of 16. A power - law model has been chosen to represent the shear thinning viscosity
of the ﬂuids. The parameters have been based on the properties of the polymer solutions
used in our earlier experiments. Two shear thinning indices of 0.5 and 0.7 have been
investigated. A commercial ﬁnite element code, FIDAP (Fluent, Inc), has been employed
for solving the equations of motion. A perturbation - continuation approach has been
developed to study the ﬂow biﬁrrcation in channels with expansion ratios of 2, 3 and 16.
Results showing the effect of pseudoplasticity and expansion ratio on the excess pressure
drop and on the ﬂow resistance have been presented. A comparison of ﬂow through a

1:16 expansion with that of Jeffery - Hamel ﬂow has been included in Appendix A.

157

4. Numerical scheme
Steady, twoodimensional ﬁnite element calculations were performed to predict the critical

Reynolds number for the onset of asymmetry. A commercial code FIDAP (Fluent, Inc.)

was employed for this purpose.

4.1 Problem definition
The steady, two - dimensional, isothermal ﬂow of an incompressible ﬂuid through an

abrupt expansion (Figure 4.1) is described by

 

 

 

 

(BUx 6U), O 42
+ _ =
6x 8y ( ' )
(U 6U): U aux) [61:10: 61”] 3p 4 3
+ = + _ - _ +
p ‘ ax ’ 6y 8x 8y 5x pg, ( ' )
6U 6U 61: at a
U—1 + U y = xx + W - P + 4.4
.1. = k1" (4.5)

where x and y are the distances along and normal to the direction of bulk ﬂow
respectively, U,I and Uy are the axial and normal components of velocity, rii are the

components of the stress tensor 1:, y is the shear rate tensor, p is the pressure, p is the

158

X

 

Figure 4.1 Schematic of the ﬂow channel used in calculations.

159

density and g is the gravitational constant. k and n are the consistency index and the shear
thinning index used to deﬁne the pseudoplastic viscosity of the non - Newtonian ﬂuid.

Expansion ratios of 2, 3 and 16 and shear thinning indices of 0.5, 0.7 and 1 were

 

 

investigated.
4.1.1 Scaled variables
U U
U: ”,1/= y ,X=£;y=.l_
Um Um h _B_}1
2

Um is the maximum velocity at the inlet of the upstream channel and h is the height of the
upstream channel. [3 is the expansion ratio. The domain of the problem is now -11 < X < 12
and -1 < Y < 1. l1 and 12 chosen were 5 and 65 step heights (i.e. (B - 1)/B) respectively.

Reducing their values by 50% changed the solution only by 0.01%.

4.1.2 Boundary conditions

U(O,:h%sYsil) = O

U(X,:L~%) = V(X,:t-[1;) = U(X,i1) = V(X,:+:l) = 0

1+—

U(-11,Y) = 1 - Y "; V(-I,,Y) = o

160

4.2 Perturbation - Continuation approach
Our steady state computations for Newtonian ﬂow through a 1:3 planar expansion using a
symmetric grid failed to show any signiﬁcant ﬂow asymmetry ﬁ'om Re = 0 to 40. Recall
that the critical Reynolds number for this case is around 27. The calculation protocol was
then modiﬁed by incorporating a grid perturbation method. Figure 4.2a illustrates
schematically the procedure followed. An asymmetry was introduced in the geometry of
the conﬁguration by shortening one of the steps from .111 to IH. This resulted in IH being
about 10% shorter than the step ON. The resulting asymmetrical domain A is shown in
Figure 4.2a. The Stokes solution was then computed for this asymmetric channel. Using
the Stokes solution as the initial iterate for the next run, the equations of motion were
solved for a higher Reynolds number. For every subsequent Re, the result of the previous
computation was used as the initial guess. This procedure was continued until point b,
where the Reynolds number was much above the expected Red. The resulting path is
shown in dashed lines in Figure 4.2a. At point b, the computational grid was made
symmetric again (Domain S). The solution was then computed for the symmetric domain
using the results at point b on Domain A as the initial iterate. Subsequently the
continuation method was repeated on the symmetrical grid with the Reynolds number now
being reduced gradually ﬁom point c to e. A point d, called the bifurcation point (or the
critical Reynolds number) was crossed over, below which the ﬂow was essentially
symmetric. We call this whole approach a perturbation - continuation approach. The step
size of Re was anywhere from 0.25 to 2. The solid lines (edc and edc’) reveal the
bifurcation curve obtained by following the continuation - perturbation approach. At low

Reynolds numbers, the degree of asymmetry is zero. Above the critical Reynolds number,

Degree of Asymmetry

161

 

 

 

J K
I II_II_II—II—II-II.L
G
F o
N M

Domain A: Asymmetric (FGHILMNOF)
Domain S: Symmetric (FGHJKMNOF)

A

- .. _ .. Asymmetric grid

Symmetric grid b

 

 

 

 

Reynolds number (Re)

Figure 4.2a Schematic of the continuation - perturbation approach.

162

the degree of asymmetry starts increasing with Reynolds number. The curve edc’ would be
obtained if the perturbation in the channel wall were made such that JH would be longer
than ON. The ﬁnal solution was not dependent on the amount of perturbation provided to
the grid. Checks made with 2%, 10% and 20% asymmetries resulted in identical
bifurcation curves. Figure 4.2b shows the various steps involved within the continuation
approach. A continuation approach carried out at Reynolds numbers above points c or c’
on a symmetric grid yielded the second ﬂow transition reported by Feam et al. (1990) and
Durst et al. (1993), etc. Interestingly, our ﬂow calculations revealed that the branch of the
solution representing this second ﬂow transition which shows the simultaneous presence
of three asymmetric vortices could be reached by a steady state, two dimensional
computation on a symmetric grid alone, without using the perturbation approach and by

starting from Stokes solution on a symmetric grid.

The degree of asymmetry may be represented by several indices. These include the
difference in the reattachment lengths on the two walls, the difference in the wall shear
stresses, the center line normal velocity component or the shear rate in the center of the
channel. The difference in the reattachment lengths at the two walls normalized by the step

height was used as the measure of asymmetry in the present work.

To evaluate the reattachment length, the sign of the axial velocity was monitored at the
nodal layer closest to the walls and two successive points with negative and positive u-
velocity were identiﬁed. A simple linear interpolation of the form of equation (4.6) then

yielded the reattachment length. This was then normalized with the step height and was

163

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Post . Compression of
Processing key files
FORTRAN
Program
UNIX Sh ll

 

 

 

 

 

Figure 4.2b Steps involved in the continuation approach.

164
referred to as L,

U.+ x+ -x,.
x =x'.+1 " ’2; i 1 U ) (4.6)

1+1 i

 

4.3 Finite elements grids

Nine node isoparametric quadrilateral elements were used in the computations.
Biquadratic and discontinuous linear interpolating functions were chosen for the velocity
and pressure representation. A mixed discretization for the pressure was used. The
Newton-Raphson method was employed to obtain the nonlinear steady solution.
Successive substitution was also tried and it yielded the same results. However, the rate of
convergence for the latter method was slower due to its asymptotically linear convergence
rate (as opposed to the quadratic rate of Newton—Raphson method). The solution was
assumed to have converged when the maximum difference in the velocity between two
iterations was less than 10“ times the velocity in the earlier iteration. The convergence

criterion for the residual was also set to 10" .

To investigate the effect of grid resolution on the outcome of our calculations, three grids
- Gl, GZ and G3 were designed with 2351, 5871 and 11431 nodes respectively. The mesh
was graded ﬁnely in all the comers and was less dense away from the comers. Figure 4.1
schematically shows the variation of the grid density in the channel. In the axial direction
(i.e., along the x axis), the mesh density was the maximum close to the x = 0 plane. In the
transverse direction, the mesh was more dense near both the 90° corners (y= i 1) and the

270° corners (y = :1: 1/13) and was less dense in between. The bifurcation point for a

165

channel with an area expansion ratio of 3 was used to compare the performance of the
three grids. Figure 4.3 illustrates the result. 61 and 62 produced results to within 0.1%.
The performance of 62 and G3 was nearly identical. Future results have been computed

on the grid G3.

5. Results and Discussion

5.1 Flow bifurcation in expansion ﬂows

5.1.1 Validation of the perturbation - continuation approach

To test the accuracy of the continuous - perturbation approach, our ﬁndings were
compared with the existing computational results of Newtonian ﬂow through 1:2 and 1:3
planar expansions. Figure 4.4 demonstrates the three branches of the biﬁrrcation curve
which result ﬁ'om the perturbation - continuation approach for a channel with expansion
ratio 3. The difference in the reattachment lengths on the two walls normalized by the step
height is used as the index of the degree of asymmetry in Figure 4.4 (and in subsequent
biﬁrrcation curves). Up to Re = 27, no asymmetry exists (i.e. the diﬂerence in the
reattachment length is zero). This stable branch (1) is the one that is picked up
experimentally. At Re greater than 27, the vortex lengths become unequal. The degree of
asymmetry increases with increasing Reynolds number. Either of the branches 2 or 3 may
be followed at Re > 27. At a critical Reynolds number of 27, the unique solution (1) is
said to have bifurcated into multiple solutions (2 and 3). The branch of the solution
representing symmetry beyond the critical Re (branch 4) is the one that will be picked up
by computations without using the perturbation - continuation approach. Durst et al.

(1997) have recently performed a stability analysis and have shown that the branches 1, 2

Difference in reattachment length

166

 

 

Acr
0G2
x G3
I, ‘.
r .
i .
' K
s
‘c

 

 

30
Reynolds number

Figure 4.3 Effect of grid refinement on bifurcation curve.

Newtonian flow, Expansion ratio = 3.

35

Difference in reattachment length

167 .

 

8 ' U T I I r
x Asymmetric grid
4 ’ O Symmetric grid '

 

 

 

 

 

 

4
0 .....
~4- -
'8 A J a l a l 4 7
0 10 20 30 40

Reynolds number

Figure 4.4 Demonstration of the results using the perturbation - continuation
approach for Newtonian flow through a 1:3 planar expansion.

168

and 3 are indeed stable solutions and branch 4 is the unstable solution and therefore would
not be observed experimentally. The predicted onset Reynolds number of 27 for
Newtonian ﬂow through a 1:3 planar expansion using the perturbation - continuation
approach is in very close agreement with the computations of Feam et al. (1990), which

also predict the onset of ﬂow asymmetry at the same Reynolds number.

The corresponding computations for an expansion ratio (B) of 2 showed ﬂow bifurcation
to occur at 72. This bifurcation point is in close agreement with the computational results
of Shapira et al. (1990) and Drikakis (1997) but is higher than the experimental ﬁndings of
Cherdron et al. (1978) and those of Durst et al. (1993). Figure 4.5 shows the biﬁrrcation
curve obtained in this work. The diﬂ‘erence in the reattachment lengths of the two vortices
has been used as the degree of asymmetry. Another commonly used measure of the degree
of asymmetry is the normal velocity at the center line (Y = O). The bifurcation curve using
the V - component is illustrated in Figure 4.6. Non - zero V at X = -O.5 indicates that the
ﬂuid anticipates the enlargement even before reaching the expansion plane. For the
purposes of clarity, only half of the bifurcation curves are shown at various axial locations.
The degree of asymmetry increases ﬁ'om X = O to X = 1.25 and then falls down further
downstream. Between Re of 70 to 85, symmetry was regained at X = 15 when the ﬂow

became ﬁrlly developed.

Difference in reattachment length

169

 

 

 

 

10 u u u I
6 .
2 ..
.2 - c ........
-6 -
'1070 72 7l4 7.6 71
Reynolds number

Figure 4.5 Bifurcation curve for a Newtonian liquid in a 1:2 expansion.

Normal velocity at center - line Y = 0

170

 

 

 

 

 

 

 

0.06 . r . u . r T r
x =
b 1.25
2.5
0.04 "
i
5
0.02 -
0
’ -o.s
0“0070 74 A 78 A 82 86
Reynolds number

Figure 4.6 Bifurcation curve using center - line velocity as the degree of

asymmetry for a Newtonian liquid through a 1:2 expansion.

17 1
5.1.2 Use of the perturbation approach to study Newtonian flow bifurcations in a
1:16 channel
The computations for a Newtonian liquid ﬂowing through a 1:16 planar channel, similar to
the one used in our optical experiments (reported in Chapters 2 and 3), revealed that the
critical Reynolds number was around Re = 6. The corresponding biﬁrrcation curve is
shown in Figure 4.7. The computations predicted a higher critical Reynolds number than
that observed in our experiments. As mentioned earlier, such an over prediction has also
been observed by other workers with 1 :2 and 1:3 channels. The lowering in the
experimental critical Reynolds number for the onset of ﬂow biﬁrrcations could be brought
about by uncertainties in the experimental conﬁgurations. Computations for a 1:16 channel
are also signiﬁcantly different from the corresponding 1:3 channel in the far severe comer

singularity and the required more dense mesh grading close to the corners in the former.

The effect of geometrical asymmetry on the lowering of the critical Reynolds number in a
1 :16 channel is shown in Figure 4.8. Only half of the biﬁrrcation curve has been included
for purposes of clarity. In this case, one of the steps forming the expansion has been”
shortened and continuation calculations have been made on such an asymmetric domain.

Grid asymmetry is deﬁned as

Grid asymmetry = 100 %
‘2"

where H is the height of the larger channel and AH is the reduction in the step height.

The results indicate a ﬁnite degree of asymmetry even at low Reynolds numbers. A grid

Difference in reattachment length

172

 

 

 

 

 

3 . . .
1.
.1-
'32 :1 a i

Reynolds number

Figure 4.7 Bifurcation curve for Newtonian ﬂow in a 1:16 expansion.

Difference in reattachment length

173

 

 

 

18 . .
J
03.1%
15 ' 01.25% '
' 110.6% ‘
12 - 0% «
9 - .
6 n- -r
l
3 P q
D a u 4
o M3

 

 

 

Reynolds number

Figure 4.8 Effect of geometrical asymmetry on Newtonian ﬂow

bifurcation in a 1:16 channel.

10

174

asymmetry of 3% results in shifting the critical Reynolds number to values lower than 2,

an almost 70% reduction from the value of 6.3 calculated with zero grid asymmetry.

5.1.3 Effect of non - Newtonian behavior on flow bifurcation in a 1:16 planar
expansion

As far as this author is aware, no results - experimental or computational - indicating the
effect of shear thinning on the ﬂow bifurcation in planar channels have ever been reported
previously. The present computational work brings out the eﬂ‘ect of non - Newtonian
behavior on the critical Reynolds number for the onset of ﬂow asymmetries in planar
channels. The corresponding experimental ﬁndings have been discussed in Chapters 2 and

3.

A generalized Oswald-dc Wale power - law viscosity model ( equation (4.7)) was used to
describe the viscosity of the shear thinning liquid. The parameters used are shown in

Table 4.1.

n = k (t)‘"‘ " (4.7)

Here n is the shear viscosity, 7 is the shear rate and k and n are rheological constants

called the consistency index and the shear thinning index respectively.

175

Table 4.1: Parameters used in the viscosity model

 

Designation n k (Pa-s')
P1 0.7 7.82
P2 0.5 77.46

 

 

 

 

 

 

 

Since the viscosity of the polymer solution is dependent on the shear rate, equation (4.8)
was used to deﬁne the Reynolds number (denoted by Re") of the shear thinning solutions.
This took into account the power-law effect of shear thinning. The halfheight of the
upstream channel, the upstream average velocity and the wall shear rate in the upstream
channel were taken as the characteristic parameters to deﬁne the Reynolds number.

n 2-n
h Um,g p

Re. z 2(1+2n)]n-1 (4‘8)

 

2H

The computations to obtain the ﬂow bifurcation curve for n = 0.7 in a 1:16 channel
yielded about 18.5 as the critical Re’. Shear thinning, therefore, delayed the onset of
asymmetry in a 1:16 planar expansion. The bifurcation curve for n = 0.7 is shown in
Figure 4.9. Until Re" of 18.5, ﬂow is symmetric and a zero degree of asymmetry is
observed. At higher Reynolds number, ﬂow becomes asymmetric and the degree of
asymmetry rises to around 3 for almost a 100% increase in Reynolds number above the

critical value. With increasing shear thinning (n = 0.5), biﬁrrcation was observed at an

Difference in reattachment length

176

 

 

 

 

 

5 - . . . . . a
3.
1.
.1-
.3-
'510 15 ‘ 20 25 A 30 35

Reynolds number

Figure 4.9 Bifurcation curve for n '= 0.7, B = 16.

Difference in reattachment length

177

 

 

 

 

45

b c : : : : : : : : : : : 1 o; t
15 20 25 30 35 40
Reynolds number

Figure 4.10 Bifurcation curve for n = 0.5, B = 16.

178

even higher Reynolds number of around 27.5. Figure 4.10 demonstrates the
corresponding bifurcation curve. Thus shear thinning clearly results in delaying the onset

of the ﬂow asymmetry in abrupt planar expansions.

5.2 Flow patterns during transitions in planar expansions

5.2.1 Newtonian liquids

The various ﬂow transitions in a Newtonian liquid ﬂowing through a channel with B = 2
are shown in Figures 4.11a through 4.11d. Figure 4.11:: represents the symmetric state
at Reynolds number of 38, which is less than Re,l (which for this case is 72). Figure 4.11b
reveals the ﬂow patterns at Re = 77. A distinct asymmetry is observed in terms of the
diﬁ‘erence in the reattachment lengths. This asymmetry increases at higher Reynolds
numbers (seen in Figure 4.1 1c) before the second ﬂow transition occurs (at around Re =
200) leading to an additional vortex on the wall with the smaller vortex (Figure 4.11d).
Computations at higher Reynolds numbers were very expensive, with convergence
requiring anywhere between 8 - 12 hours on a SPARC Ultra workstation. The nature of
the second ﬂow transition reported for Newtonian ﬂows through 1:2 and 1:3 channels at
Rec2 is different from that of the ﬁrst one in at least one aspect. Our ﬂow calculations
revealed that the branch of the solution representing the simultaneous existence of three
asymmetric vortices (the streamlines for which have been shown in Figure 4.11d) could
be reached by a steady state, two - dimensional computation on a symmetric grid alone,
without using the perturbation - continuation approach. Such a ﬂow transition in

experiments would therefore not be dependent on the uncertainties introduced by the

 

 

 

 

 

 

 

 

=38

(a) Re

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b) Re = 77

 

 

 

 

 

 

 

 

 

 

 

 

 

= 150

(c) Re

 

 

 

 

 

 

 

 

 

 

 

 

 

(d) Re = 250

Figure 4.11 Flow transitions of a Newtonian liquid in a 1:2 expansion.

180

geometrical conﬁguration or other aspects of the experimental procedure. The streamlines
showing the eﬂ‘ect of increasing Reynold numbers on the ﬂow patterns in a more severe
expansion of 1:16 are shown in Figure 4.12a through 4.12d. The more severe expansion
ratio results in essentially similar trends in the transition. The corner vortices, however, ﬁll
up most of the transverse length of the downstream channel close to the expansion plane,
resulting in a thinner core between the recirculation zones. In the 1:16 channel, the second
ﬂow transition resulting in three vortices was detected at around Re = 13. Reynolds
numbers above 26, revealed the formation of more than three vortices in the 1:16 channel.

No experimental evidence of such ﬂow patterns has been reported.

5.2.2 Shear thinning liquid

Figure 4.13a through d depict the ﬂow patterns in a shear thinning solution (P1) through
the 1:16 channel. At Re“ = 18, the two comer vortices are of equal lengths. Figure 4.13b
shows that at Re” = 20, the reattachment lengths are unequal and the difference in the
reattachment lengths increases at higher Reynolds numbers (seen in Figure 4.13c). At
high Reynolds numbers, a thin jet of liquid in the core is seen to move toward one wall and
then toward the other. At around Re“ = 44, the second flow transition was detected,
wherein three vortices coexist in the channel. Our flow visualization and laser Doppler
velocimetry experiments were not able to pick up the ﬁrst ﬂow transition. Instead the
shear thinning liquid P1 went from a state of two symmetric vortices to a state of three
asymmetric vortices directly. Computations, however, show the ﬂow transitions in the

shear thinning liquid to be similar to that demonstrated by the Newtonian liquid.

181

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) Re=6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(d) Re = 24

Figure 4.12 Flow transitions of the Newtonian liquid in a 1:16 expansion.

182

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) Re“ = 18

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b) Re* = 20

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(d) Re" = 44.

Figure 4.13 Flow transitions of the shear thinning liquid P1 in a 1:16 channel.

1 83
5.3 Velocity proﬁles in planar expansions
5.3.1 Velocity proﬁles in the Newtonian liquid before and after the onset of
asymmetry
Figures 4.14 shows the velocity proﬁles at various axial locations in a 1:3 planar
expansion before the onset of asymmetry. At Re = 26 (critical Reynolds number for this
case is 27), the proﬁles at X = l, 5.2 and 9.3 are all observed to be symmetric about the
center line Y = 0. Closest to the expansion plane, negative velocities clearly indicating the
presence of the recirculation regions are seen between Y = :L- 0.6 to :t 1. At X = 5.2, the
smaller vortex is no longer present and negative velocities corresponding to the larger
vortex only are observed. At X = 9.3, ﬂow has approached the ﬁilly developed proﬁle (U
= 0.33) expected far downstream of the expansion plane. The velocity proﬁles at Re = 34
(higher than the critical Re“) are shown in Figure 4.15. The asymmetry in the ﬂow is
obvious at X = 1 and X = 5.2. At the former location, the velocity peaks slightly oﬂ‘ the Y
= 0 axis and shifts toward the wall with the smaller vortex. At X = 5.2, the shift toward Y
= 1 is stronger. The ﬂow is still not fully developed at X = 9.3, indicating that higher
Reynolds numbers result in a longer ﬂow length over which asymmetry persists. However,
the proﬁle has acquired considerable symmetry by then. Flow proﬁles in channels with
expansion ratios of 2 and 16 were qualitatively similar. Figure 4.16 demonstrates the
computed velocity proﬁle after the onset of asymmetry for a Newtonian liquid in a 1:16
expansion at Re = 8. The corresponding streamlines are shown in Figure 4.12b. At X = 5,
asymmetry in terms of off-center peaking and difference in the sizes of the recirculation
zones are seen. By X = 20, the smaller vortex ceases to exist and the velocity proﬁle is

markedly asymmetric. Fully developed ﬂow is observed at X = 75. The velocity proﬁles

Axial velocity, U

184

 

1.0

 

 

 

Figure 4.14 Velocity proﬁles prior to the onset of asymmetry.

B=3,Re=26.

 

1.0

Axial velocity

185

 

 

 

 

 

 

1.0 I w u u u

p — x = l 1
0.8 - - X = 5 -

°°°° X = 20 .

0.6 ' .
0.4 "
0.2 -

.
0.0
'0.2-1.5 -1.0 -0.5 0.0 0.5 1.0

Y

Figure 4.15 Velocity proﬁles after the onset of asymmetry.

B=3,Re=34.

1.5

Axial velocity

186

 

 

 

 

 

 

0.5 1 I ' I V I r I '
DX=5
OX=ﬂl
D
AX=75 a
D D
05 - g a .
II
D
D D
D
a 00000
O
0.1 - ° ° ° -
a...“ "I ”use... a
'o'l-m -0.6 .02 02 0.6 1.0

Figure 4.16 Computed velocity proﬁles with Newtonian liquid at Re = 8.

Axial velocity

187

 

0.7 ' I ' I V I v I v
UX=5 &
ox=3o °
0'5 ' AX=60 1:1 ‘
D

 

 

 

 

 

 

'0'311.o A -0.6 A -o.2 0.2 ‘ 0.6 I 1.0

Figure 4.17 Computed velocity profiles with Newtonian liquid at Re = 25.3.

188
corresponding to the streamlines shown in Figure 4.12d are shown in Figure 4.17. At X
= 5, two recirculations regions corresponding to the two comer vortices are seen. At X =
30, only the larger vortex exists along one wall and at X = 60 the vortex on the other wall

is detected.

5.3.2 Velocity proﬁles in shear thinning liquid P1 through a 1:16 planar expansion
Figure 4.18 shows the velocity proﬁles at Re“ = 20. The corresponding streamlines are
shown in Figure 4.13b. At X = 5, both the comer vortices are obvious. The velocity
peaks at around Y = 0.05. Further downstream at X = 15 and 20, the velocity peaks closer
to the wall with the shorter vortex. The smaller vortex has disappeared at these locations.
Figure 4.19 illustrates the velocity proﬁles corresponding to the streamlines shown in
Figure 4.13c. The trends followed are very similar to that described above for the

Newtonian ﬂow at Re = 25.3.

In Chapter 2, the effect of expansion ratio and inertia on the reattachment lengths has been
careﬁrlly laid out. Expressions have been proposed to relate the three in the low and high

inertia regime for Newtonian ﬂow. These will not be discussed here.

5.4 Effect of expansion ratio and rheology on the excess pressure difference during
flow through expansions

To determine the effects of vortex enhancement, expansion ratio and inertia, the excess
pressure difference was calculated for the entire ﬂow channel. This is deﬁned in equation

(4.9).

Axial velocity

189

 

 

 

 

 

 

0.5 ' I ' I r U ' I
_ D
uX—S r: a
» ox=15 a n r

AX=20 a D

05 - a a ‘
D

0.1 -
'0'1-1.o

 

Figure 4.18 Computed velocity proﬁles of the shear thinning liquid P1

at Re“ = 20.

Axial velocity

189

 

 

 

 

 

 

0.5 ‘ I ‘ s r I v 3 v
_ D
aX—S u a
> OX=15 a u ‘
AX=20 g D
0.3 - u a '
D
0.1 '
.0'1-1.0

 

Figure 4.18 Computed velocity proﬁles of the shear thinning liquid P1

at Re“ = 20.

Axial velocity

190

 

 

 

 

 

1.0

0.8 . . . . .
DX=5
0.6 - ox=3o .
AX=45 °
U
0.4 - -
as a
0.2 - '3 °°°°° -
.5 f
AAA D
A
0'0 manual: 3333:“
00000
”24.0 41.6 ‘ 4; 2 0:2 0:6

Figure 4.19 Computed velocity proﬁles of the shear thinning liquid P1

at Re" = 44.3.

191

Ape = (“Amnesia ' (’AP)fd,u ‘ (-AP)fd,d (4-9)

GAP)“ = Pi - Po, Pi is the computed wall pressure at the inlet of the upstream channel
and P, is the computed wall pressure at the outlet of the downstream channel, (~AP),¢, is
the pressure drop in the upstream channel assuming fully developed ﬂow conditions and (-
AP)“ is the corresponding ﬁrlly developed pressure drop in the downstream channel. As
per our deﬁnition of the pressure diﬂ‘erences, all of the three terms in round braces on the
right hand side of equation (4.9) are positive. For ﬁrlly developed ﬂow through a channel
with B = 1 (plane Poiseuille ﬂow), the excess pressure diﬂ‘erence should theoretically be
zero. For expansion ratios greater than one, the behavior of (AP), is shown for Newtonian
liquids in Figure 4.20. Here the viscous scaling has been used for the pressure using Um
and the step height as the characteristic velocity and length respectively. With an increase
in the Reynolds number, the excess pressure diﬂ‘erence decreases. As the expansion ratio
is raised ﬁom 2 to 16, the excess pressure difference also rises. For B = 16, a distinct
ﬂattening is observed above the critical Reynolds number. A similar leveling oﬁ‘ at higher

Reynolds numbers is expected for the channels with lower expansion ratios.

To understand the effect of B and inertia on the development of the shown pressure
diﬂ‘erence, a careful breakdown of the components of (AP), reveals that the excess
pressure difference consists of two parts - the ﬁrst is the pressure diﬁ‘erence downstream
of the expansion plane between x = 0 to x = 1,, where l, is the portion of the larger channel
that experiences non-fully developed ﬂow due to the presence of the vortices. This part is

positive at low Reynolds numbers but becomes negative at high Re. The second part

192

 

 

 

 

10 ' I ' I v y w ' 1
D :2
0 B "
éA AB=3
8 P a“ 16 ‘
O =
§ .10 - 0AA 5 .
é a
b ’ ”MA
2 u A
:r -20 - .
i: » .
8 .30 - ,
S
Er.) U 4
‘40 ’ 4
“a:
n
'500 20 4o 60 so 100
Reynoldsnumber

Figure 4.20 Effect of expansion on excess pressure difference in Newtonian flow.

193

which is always negative is the fully developed pressure diﬁ‘erence corresponding to length
1,. The magnitude of this part is proportional to 1,. It has been shown earlier in Chapter 2
that the vortex length is proportional to the Reynolds number. Therefore, it may be
assumed that 1, follows a similar trend. For any ﬁxed expansion ratio, low Reynolds
numbers result in a small 1, which in turn translates into a small negative term. As inertia
increases, 1, increases and the magnitude of the second part increases. Simultaneously the
pressure rise across the expansion falls down resulting in reducing the ﬁrst part. The

combined effect then is the reduction in (AP), with increasing Reynolds numbers.

At the same Reynolds number, an increase in the expansion ratio increases (AP),
consistently. Since it has been already demonstrated that the reattachment length scales
with the step height and the step height is the same in all our computations, it follows that
the magnitude of the second part of the excess pressure diﬂ‘erence decreases with
increasing expansion ratio. Based on a macroscopic momentum balance across an
expansion (Bird et al., 1984), the magnitude of the pressure rise component also reduces
with increasing expansion ratio. Since this component is also negative at high Reynolds
numbers, the effect of increasing the expansion ratio is to raise the excess pressure

difference.

To determine the effect of shear thinning on the excess pressure difference a suitable
parameter was used to normalize AP, This shown in equation (4.10). This positive

normalization quantity represents the fully developed pressure drop between parallel

plates.

194

_ = _2_k{ 2(1+2n) ~
( AP)...” H [ "H, 0...,1 (4.10)

U" is the average velocity of the ﬂuid in the downstream channel and H and l are the
height and length of the downstream channel. k and n are the rheological parameters of the
ﬂuid and have been deﬁned earlier. The excess pressure diﬁ‘erence divided by (AP)mull will
be called the normalized excess pressure diﬁ‘erence and will be represented by (AP),,.
Figure 4.21 shows the effect of varying expansion ratio on the normalized pressure
difference for a Newtonian liquid. The crossing over of the curves is due to the reduction

in the normalization parameter with increasing expansion ratios.

The eﬁ‘ect of varying shear thinning on the excess pressure diﬁ‘erence is shown in Figure
4.22, for a 1:16 channel. It was observed that the reattachment length in the shear thinning
liquid changed very slowly at low Reynolds numbers. This shows up as a plateau in
Figure 4.22 at low Re. At high Reynolds numbers the normalized excess pressure
difference reduces with decreasing pseudoplasticity. In this sense, increasing the shear
thinning is similar to reducing the expansion ratio. The rate of fall of (AP),u is much higher
for a less shear thinning liquid. Figure 4.23 compares the normalized excess pressure
difference for all the three expansion ratios for n = 0.7. Data for expansion ratios of 2 and
3 are limited to small Reynolds numbers before the onset of ﬂow biﬁrrcation. Trends
observed are similar to those seen in the Newtonian liquid. The magnitudes of the pressure

differences are considerably lower than those observed for Newtonian ﬂow.

Normalized Excess Pressure Difference

195

 

 

 

 

2.5 ‘ I ' I f T ' I
P
D B = 2
1.5 .9 a
A B = 3
.0 o p = 16
o
0.5 - s
o
o
. A" ..
-0.5 a A AI: 0
a o “A n d
6 a
-1.5 - 0 .
0 Ch;
0 u u .
'2'5 o 20 4o 60 80
Reynolds number

Figure 4.21 Normalized excess pressure difference in Newtonian flow.

Normalized Excess Pressure Difference

196

 

 

 

 

2.5 w s . s 1 .
,o
on=1
15 - ° An=0.7
_ o
0.5 " “AA
0 A
AA
0 W
.05 - Mu
\ AAAAAAAAAA
o
%
-1.5 ' 0
o
o
'250 10 20 30
Reynolds number

Figure 4.22 Effect of pseudoplasticity on excess pressure difference, B = 16.

Normalized Excess Pressure Difference

un

 

 

 

L0 . . u . u -
03:2
0.6 ° ‘5 = 3
O B = 16
c2-
». o,
02 a
- . A 0000
g o
0000
ﬁﬁ’ 0000
O O O O o
.1.0 o 10 20 30
Reynolds number
Figure 4.23 Excess pressure difference in shear thinning liquid, 11 = 0.7.

 

198
5.5 Effect of expansion ratio and rheology on the flow resistance
To evaluate the ﬂow resistance in a planar abrupt expansion, a parameter called fr was
deﬁned. This non-dimensional parameter utilizes the pressure drop downstream and
upstream of the expansion plane and it may be used as a modiﬁed overall pressure drop in

the channel.

(—’;—(-AP)). +( ’; (-AP».

_ 2R8 Uavg avg (411)
f' - [ ]
p 11 + 12

 

The subscript d and u refer to the channels downstream and upstream of the plane of
expansion respectively, h refers to the channel height, U", is the average velocity in and -
AP is the pressure drop. For plane Poiseuille ﬂow, fr would be 12. The Newtonian ﬂow
resistance for B = 3 and 16 are shown in Figures 4.24 and 4.25 respectively. Increasing
inertia reduces the ﬂow resistance. Interestingly, both the curves show a discontinuity at
Re close to the critical Reynolds number. Just above the biﬁircation point, there is a
sudden increase in the ﬂow resistance. At higher Reynolds numbers, fr reduces again.
Figure 4.26 compares the Newtonian flow resistances for the three diﬁ‘erent expansion
ratios. As expected, the ﬂow resistance increases as the expansion ratio is decreased. At
very low Reynolds numbers, the ﬂow resistance approaches the plane Poiseuille ﬂow
resistance. In diverging channels, the expansion effect results in a negative pressure drop
for part of the channel and a positive drop in the ﬁrlly developed region. This results in a
net pressure drop which is lower than that in plane Poiseuille ﬂow, thereby yielding a
lower ﬂow resistance. Increasing inertia results in a longer separation length and

consequently even more lowering of the net pressure drop.

Flow resistance

12

199

 

 

 

D
O O
O
O
' o
O

.. O
O
00 o

O
0 10 20 30
Reynolds number

Figure 4.24 Increase in ﬂow resistance at critical Re. B = 3, n = 1.

 

Flow resistance

12

 

 

 

 

o o
O
O
O
I- o d
O
0
00000
°o
O
O
O
0 3 6 9 12
Reynolds number

Figure 4.25 Increase in ﬂow resistance at critical Re. B = 16, n = 1.

15

Flow resistance

201

 

 

 

 

 

1
....... B = 2

--_- 5:3 +

—— B= 16 .

0 - .

-4 l' cl

.3 1' .
'12 o 20 4o 60 so 100

Reynolds number

Figure 4.26 Drop in ﬂow resistance due to expansion; Newtonian flow.

202

Figure 4.27 shows the ﬂow resistance in a 1:16 channel for the two Newtonian and the

shear thinning liquid. fr for ﬁrlly developed ﬂow with a power-law viscosity is given by

Ur)” = 40 +2")

 

(4.12)

For n = 0.7 (ﬁ'),, is about 13.7, which is higher than that for n = 1. At low Reynolds
numbers, ﬂow resistance approaches the value corresponding to fully developed ﬂow, thus
indicating negligible expansion effect. The ﬂow resistance increases with the lowering of
the shear thinning index. The discontinuity observed close to the biﬁrrcation point in the
Newtonian liquids is also seen in the pseudoplastic liquid. Figures 4.28 and 4.29 compare
the ﬂow resistances in a l :2 and 1 :3 planar expansion respectively. Similar trends as in the

1:16 channel are observed even for the lower expansion ratio channels.

6. Conclusions

A perturbation - continuation approach was developed and used to compute the
bifurcation curves for Newtonian and non - Newtonian liquids ﬂowing through abrupt
planar expansions. Detailed ﬂow patterns depicting the various ﬂow transitions and
velocity proﬁles prior to and after the onset of ﬂow asymmetry were presented for the
Newtonian and non - Newtonian liquids. With Newtonian ﬂuids, the onset of asymmetry
occurred at a critical Reynolds number of 72, 27 and 6 for a 1:2, 1:3 and 1:16 planar
expansion respectively. Calculations for the lower expansion ratios showed an excellent
agreement with the available results in the literature. Excess pressure drop was calculated

for all three expansion ratios. For any particular channel, increasing inertia reduced the

203

 

15 7 ' I ' I ' I '
LAAAA on= 1

 

 

 

A An=OJ
A
0000 m
10 ' o A
g o AAAAAAA
AA
.9 0 AAA A
f ' 0
i a».
E %
5 ’ o
O
' o
0 0 10 20 30
Reynolds number

Figure 4.27 Effect of pseudoplasticity on flow resistance. B = 16.

Flow resistance

 

 

 

 

15 ' I V I I I
A
00 A On=l
10- O ‘
An=0ﬂ
O
5- ° .
O
0' a
O
.5 - d
% o
O
40' ‘
'15 o 20 40 so so 100
Rumdﬂnmmhn

Figure 4.28 Effect of pseudoplasticity on flow resistance in a 1:2 channel.

Flow resistance

205

 

 

 

 

’ A
D An=0J
o o A
O
10 - o
O A
i O
O
5 r-
000 o
O
0 0 10 20 30
Reynolds number

Figure 4.29 Flow resistance in a 1:3 channel.

206

excess pressure drop. At a ﬁxed Reynolds number, the excess pressure drop increased
with increasing expansion ratio. Flow resistance in a channel was deﬁned. Higher
expansion ratios resulted in lowering the ﬂow resistance. At the biﬁrrcation point, the ﬂow
resistance shot up and then gradually reduced at higher Reynolds numbers. This

discontinuity was observed for all the expansion ratios investigated.

A power - law model was used to describe the shear thinning behavior of a ﬂuid ﬂowing
through a 1:16 planar expansion. Two shear thinning indices - 0.7 and 0.5 - were studied.
Biﬁrrcation curves showed the onset of asymmetry at Re“ of 18.5 and 27.5 respectively.
Shear thinning, therefore, delays the onset of asymmetry in ﬂow through planar abnrpt
expansions. Pseudoplasticity resulted in a higher normalized excess pressure difference at
high Reynolds numbers, for all three expansion ratios. Flow resistance curves for non -
Newtonian liquids also showed discontinuities at the critical Reynolds number. Increased

pseudoplasticity led to a higher resistance in ﬂow.

Chapter 5

CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE STUDY

1.1 Conclusions
Symmetry breaking ﬂow transitions were investigated in Newtonian and non - Newtonian
liquids ﬂowing through abrupt planar expansions. The eﬂ‘ects of ﬂuid rheology, expansion

ratio and inertia on the ﬂow transitions in planar expansions was studied in detail.

A model conﬁguration with an aspect ratio of 10 in the downstream channel and an area
expansion ratio of 16 was designed. Newtonian model ﬂuids and inelastic shear thinning
model ﬂuids were prepared. The present study was divided in three parts. In the ﬁrst part
streak photography was used to visualize the ﬂow patterns at Reynolds numbers spanning
0.001 to 4, 0.1 to 3.5 and 0.1 to 1.6 for the model ﬂuids with power - law shear thinning
indices of 1, 0.7 and 0.5 respectively. Appropriate Reynolds numbers (Re‘) were deﬁned
for the non - Newtonian liquids to incorporate the shear rate dependent viscosity.
Bifurcation curves using the differences in reattachment lengths on the two walls as an
index of the degree of asymmetry were presented. The second part employed laser
Doppler velocimetry to measure the axial velocities of all the model ﬂuids at various
locations in the downstream channel. The center - line shear rate was used as a measure of

the degree of asymmetry and the ﬂow bifurcation in these liquids above the critical

207

208

Reynolds numberwas illustrated. The rate of decay of asymmetry with increasing axial
positions was evaluated. In the third part of the study, an innovative protocol involving
grid perturbation and continuation was devised to compute the ﬂow biﬁrrcation curves for
Newtonian and non - Newtonian liquids through channels with expansions ratios 2, 3 and
16. Velocity proﬁles, ﬂow patterns, critical Reynolds numbers for the onset of ﬂow
transitions, excess pressure diﬁ‘erences and ﬂow resistances were evaluated and the eﬂ'ects
of ﬂuid rheology and expansion ratio on these were determined. Relations between the

Newtonian reattachment length, expansion ratio and Reynolds number were proposed.

Flow transition in the Newtonian liquid (liquid N) occurred at a critical Reynolds number
(Reﬁ) of 0.8, which is signiﬁcantly lower than the critical Reynolds numbers reported
previously for 1:2 and 1:3 planar channels. This extends the ﬁndings of Cherdron et al.
(1978) who reported that increasing expansion ratios ﬁ'om 2 to 3 resulted in the lowering
of the critical Reynolds number at which ﬂow transitions were witnessed. With the shear
thinning liquid Pl (n = 0.7), ﬂow transition was observed between 1.25 - 1.9. Shear
thinning, therefore, delayed the onset of asymmetry. No ﬂow transition was observed with
the more shear thinning liquid P2 (n = 0.5), for the entire range of Reynolds numbers
studied. Higher shear thinning and/or higher elasticity could be responsible for the absence
of the expected ﬂow transitions in P2. With P1, the ﬂow transition occurred ﬁ'om a state
of two symmetrical vortices to a state of three asymmetrical vortices. The intermediate
step of two asymmetrical vortices was not observed. Others have reported a similar
second ﬂow transition from two asymmetrical vortices to three asymmetrical vortices with

Newtonian liquids, at much higher Reynolds numbers. Therefore, it may be said that

209

pseudoplasticity hastens the second transition in a 1:16 planar expansion.

For the Newtonian liquid, LDV was used to measure the axial velocities at normalized
axial positions ofX = 1, 8 and 15 at Reynolds numbers (Re) of0.6, 1.5, 2.5 and 3.5.
Corresponding measurements for the shear thinning liquid were made at X = 1, 8, 16 and
45 and at Re“ = 1, 2, 2.5 and 3. Detailed observations on the dynamics of vortex growth
at Reynolds numbers above the onset of ﬂow transitions were presented. At Re = 0.6, the
Newtonian liquid demonstrated symmetric velocity proﬁles at all the axial locations. At
Re 2 1.5, ﬂow was asymmetric close to the expansion plane (X = l) and also at X = 8. All
asymmetries decayed by X = 15. The second ﬂow transition to a state of three vortices
was not observed for the range of Reynolds numbers studied. At Re“ = 1, the shear
thinning liquid showed symmetric velocity proﬁles at all the axial locations. A ﬂow
transition to the simultaneous presence of three vortices was observed at Re“ = 2 and
above. The intermediate ﬂow transition resulting in two asymmetrical corner vortices was
not seen in the experiments with the pseudoplastic liquid. Shear thinning, therefore, seems
to hasten the onset of the second ﬂow transition experienced by the Newtonian liquids at
very high Reynolds numbers. The shear thinning liquid with a power - law index of 0.5
(liquid P2) revealed symmetric proﬁles at the maximum achievable Reynolds number of
1.5. Fluid viscoelasticity at high Reynolds numbers and a high degree of pseudoplasticity
could be responsible for the delayed ﬂow transitions. Asymmetry in the ﬂow in terms of
the center line shear rate decreased downstream of the plane of expansion in the liquids N
and P 1. However, the degree of asymmetry in the shear thinning liquid was greater than

that in the Newtonian liquid and it also persisted for a longer ﬂow length.

210
Computations performed with a grid perturbation - continuation approach for Newtonian
ﬂuids, predicted the onset of asymmetry at Re,,-, = 72, 27 and 6 for a 1:2, 1:3 and 1:16
planar expansion respectively. The calculations for the lower expansion ratios of 2 and 3,
showed an excellent agreement with the available computational results in the literature.
Excess pressure drop was calculated for all three expansion ratios. For any particular
channel, increasing inertia reduced the excess pressure drop. At any Reynolds number, the
excess pressure drop increased with increasing expansion ratio. The ﬂow resistance for a
diverging channel was deﬁned using the friction factor for ﬂow between parallel plates.
Higher expansion ratios resulted in lowering the ﬂow resistance. At the biﬁrrcation point,
the ﬂow resistance shot up and then gradually reduced at higher Reynolds numbers. This

discontinuity was observed in all the expansion ratios investigated.

A power - law model was used to describe the shear thinning viscosity of the
pseudoplastic ﬂuids. Two shear thinning indices - 0.7 and 0.5 - were considered. The
perturbation - continuation approach was then used to compute the ﬂow biﬁrrcation of the
shear thinning liquids. The onset of asymmetry was identiﬁed at Re" = 18.5 and 27.5 for
shear thinning indices of 0.7 and 0.5 respectively. This was consistent with the
experimental observation that shear thinning delayed the onset of asymmetry in ﬂow
through a 1:16 abrupt planar expansion. Pseudoplasticity resulted in a higher normalized
excess pressure difference at high Reynolds numbers, for all three expansion ratios. Flow
resistance curves for non - Newtonian liquids also showed discontinuities at the critical

Reynolds number. Increased pseudoplasticity led to a higher resistance in ﬂow.

211

The computations revealed both the ﬂow transitions for the Newtonian and the shear
thinning liquids. As already mentioned, the ﬁrst ﬂow transition ﬁ'om a state of two
symmetrical corner vortices to the state of two asymmetrical comer vortices was not
observed in the experiments with the shear thinning liquids. Although the predicted
bifurcation points were higher than the corresponding experimental observations, trends of
increasing degree of asymmetry with increasing Reynolds numbers and of decaying
asymmetry downstream of the expansion plane were conﬁrmed by the computations. The
computed Newtonian and non - Newtonian vortex reattachment length prior to the onset
of ﬂow asymmetry showed excellent agreement with the experimentally obtained data on a
1:16 channel. Using the results from computations performed on channels with expansion
ratios of 2, 3 and 16, relations between the Newtonian reattachment length, Reynolds
number and the expansion ratio were proposed for the entire range of Re prior to the
onset of asymmetry. A Newtonian vortex length of about 0.44 step height was predicted
for creeping ﬂow, for all the three expansion ratios investigated. The creeping ﬂow
reattachment length decreased with increasing pseudoplasticity. More work needs to be
done to obtain a modiﬁed relationship that successfully predicts the non-Newtonian

reattachment lengths.

Figure 5.1 shows a three parameter plot which summarizes the effect of ﬂuid rheology,
expansion ratio and inertia on the occurrence of ﬂow transitions from symmetric to
asymmetric ﬂow patterns. An approximate division between the conditions resulting in
ﬂow asymmetry and those retaining symmetry is shown by a dotted line for the Newtonian

liquid. For the sake of completeness, the experhental ﬁndings of others has also been

Critical Reynolds number

212

 

 

 

75 . n l 1 ﬂ
0
‘ D n = 0.7
1 An=05
60 X‘ G - 0 n = l
l
l
45 l
x I
“ ASYMMETRY
30 l A
G \
x \ \
x \
\ \ \ El
15 ‘ \
SYMMETRY \ \
\ \ e
1 A r . 1 l
0 10 15
Expansion Ratio, B

 

Figure 5.1 Three parameter plot summarizing the effect of inertia,
rheology and expansion ratio on the occurrence of flow
transitions. Filled - Experiments, Unﬁlled - Computations,
X - Others' work.

213

included. A combination of high enough expansion ratio, a fairly non-shear thinning

viscosity and high injection rate will lead to early ﬂow transitions in planar expansions.

1.2 Recommendations for future work
(1) Experiments and/or computations to determine the bifurcation points for shear
thinning liquids through channels with low expansion ratios of say 2 and 3 are needed to

complete the ﬁndings reported in Figure 5.1.

(2) Appropriate relations for the vortex reattachment lengths, analogous to those
developed in the present work for Newtonian ﬂows, still need to be developed for shear

thinning liquids.

(3) Borgas and Pedley (1990) have used a boundary - layer analysis to predict ﬂow
bifurcations in geometries with very gradual expansions. Such an approach is semi -
analytical and if adapted for abrupt expansions could increase our understanding of

biﬁrrcations in diverging ﬂows.

(4) Studies on injection molding (Dee and White, 1974, Serrano et al., 1995, etc.) have
shown that during the ﬁlling of cavities by polymer melts, ﬂow transitions occur above a
critical injection rate, which depends on the polymer rheology and the geometry of the
cavity and gates. The present study which deals with steady ﬂows through expansions has
also revealed the occurrence of ﬂow transitions above a critical ﬂow rate. A possible

connection between these two different types of ﬂows needs to be explored.

APPENDIX A

Appendix A

COMPARISON OF ABRUPT EXPANSION FLOW

WITH JEFFERY - HAMEL FLOW

Sobey (1985) and Sobey and Drazin (1986) have shown the biﬁrrcation in Jeﬂ‘ery - Hamel
(II-I) ﬂow to be subcritical and that in the planar expansion ﬂow to be supercritical. They,
therefore, conclude that Jeﬁ‘ery - Hamel ﬂow may not be used to explain the biﬁrrcation in
abrupt expansion ﬂows. Very recently Allebom et a1. (1997) have performed a linear
stability analysis on abruptly diverging ﬂows and have compared the ﬂow in an inﬁnitely
diverging channel with JH ﬂow. In this section, the comparison of JH ﬂow with the ﬂow
in a 1:16 abrupt planar channel has been presented. The objective is to explore any

similarity in the two ﬂows at low Reynolds numbers prior to the onset of ﬂow transitions

in abrupt channels.

JH ﬂow may be described as the radial ﬂow between two plates, with an angle of 2a in
between them. Using cylindrical coordinates with the origin being at the intersection of the

plates, the ﬂow may be described as -

.U = (U,, 0. 0); X = (7.3.2)

214

215

The equations of continuity and motion yield a fourth order ordinary diﬁ‘erential equation

shown in equation (A. 1).

f’”’ + 4f” + 6Refff” = 0 (Al)

Here the non - dimensionalization parameters used to normalize the locations and
velocities are the same as in Chapter 4. f is the stream ﬁrnction and is related to the

velocity by equation (A 2).

U, = L33)- (A2)

The boundary conditions are:
l
feta) = 4:3; f’(=*=a) = 0 (A3)

a = ‘lt/Z represents the case of abrupt expansion. The boundary conditions on f imply that
the normalized net volume ﬂux is equal to 2/3. The ode system (equation A] and A3)
was solved by using an IMSL routine BVPFD, where the basic discretization is the
trapezoidal rule and the resulting nonlinear algebraic system is. solved by Newton’s
method. A tolerance (the ratio of the error to the maximum f) of 10“ was used. The
corresponding velocities were translated to planar coordinates using the following

relations:

(1,412» = mow—jigs); Y =Xtan<6> (A.4)

216
In Figure A.1, the velocities in the 1:16 expansion are compared with the JH ﬂow at X =
1 at Re = 1. At locations close to the expansion plane, r approaches very small values and
blows up the velocity. Since the upstream channel is of a ﬁnite height, the 1H ﬂow closer
to the expansion plane behaves as if expanding into a channel with expansion ratio less
than 16. This results in higher velocities than in the abrupt expansion. As X is increased
(Figure A.2), the two proﬁles show more agreement (X = 10) and ﬁnally the 1H ﬂow
resembles ﬂow in a channel with a higher expansion ratio and consequently under -
predicts the velocity (X = 15). Further downstream, the ﬂow in the channel will not
change much because fully developed proﬁle will be achieved. However, the JH ﬂow will
continue to approach lower magnitudes with increasing X and therefore the agreement
with the channel velocity will go on deteriorating. The disparity close to the expansion
plane will be reduced if a much higher expansion ratio is used to build the abrupt channel.
Higher Reynolds numbers, result in a higher maximum velocity in the JH ﬂow (similar to
that in the channel). This is evident from Figures A.3 and A.4, where the velocities are
plotted at Re = 2. A better agreement is observed at X = 5 and 10 at Re = 2 than at Re =
1. As mentioned above, Sobey and Drazin (1986) have already shown conclusively that
the bifurcation structure of JH ﬂow is diﬁ‘erent ﬁ'om that in abrupt planar ﬂow. For a
above 73°, they have shown that JH ﬂow is unstable. For an abrupt expansion (for which a
= 90°), therefore, the critical Reynolds number of 0 is predicted for JH ﬂow. As the
expansion ratio of an abruptly expanding channel increases, the critical Reynolds number
for the onset of biﬁrrcation goes down. It may be hypothesized that at inﬁnite ratios, Re,

approaches zero and ﬂow will approximate IH ﬂow.

Urn

217

 

0.6 ﬁ 1 . i . v
— J H flow

- - - Channel flow

 

 

 

 

0'0-10 A .5 o 5 ' 10

Figure A.1 Comparison of velocity proﬁle with J H flow. X = 1, Re = 1.

U111

218

 

   

' V ' V ' 1

—JH,X=5 AChanneLX=5
’ "-‘JH,X=10 OChannel,X=10
----- in,x=ls xChanne1,X=15

 

 

 

Figure A.2 Comparison of channel velocity with JH ﬂow, Re = 1.

 

Urn

219

 

 

 

 

 

 

0e6 ' I V A v I
> — JH flow
- - ' Channel ﬁow
0.4 ' ‘
0.2 " .
0'0-10 .5 o 5 10
Y

Figure A.3 Comparison of velocity proﬁles at X = 1, Re = 2.

220

 

' r I ' I

—JH,X=5 AChannel,X=5
. "’JH’X=10 ' 0Channel,X=10
°°°° JH,X=15 , . XChannel,X=lS

 

 

 

 

Figure A.4 Comparison of channel velocity with JH ﬁow, Re = 2.

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