AZIMUTHAL MODE INTERACTION IN
COUNTER-ROTATING COUElTE FLOW
A Dissertation
Presented to
Tile Faculty of the Department
of Mechanical Engineering
University of Houston
In Partial
Fulfillment
of the Requirements for the Degree of
Doctor of Philosophy
by
Catalina Elizabeth Stern
December 1988

---

/'
\\6 \\
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AZIMUTHAL/MODE INTERACTION IN
COUNTER-ROTATING COUETIE FLOW
Catalina Elizabeth Stern
Approved:
Chairman of the Committee
F. Hussain, Distinguished Professor
Mechanical Engineering
: '
Committee Members:
M. Golubititsky, P;;ofessor
Mathematics
S.J.
IS, Associate Professor
Mechanical Engineering
Kunin,~'-r
lA
-=..."C----
Mechanical Engineering
R.W. Metcalfe, Professor
Mechanical Engineering
Charles Dalton, Associate Dean
L. C. Witte, Professor and Chairman
Cullen College of Engineering
Mechanical Engineering
ii

---

·.-'
•
To all the Sterns
and
S.K.B.
iii

---

ACKNOWLEDGEMENTS
I would like to express my gratitude to my advisor Or. F. Hussain for his
guidance and support during all the years I was a graduate student at the
University of Houston.
I would like to thank Mr. A. Leitko for the design of the facility; without his
help this research would not have been possible.
Mr. G. Hoffman built the
facility, and Mr. F. Robinson, Mr. J. Koncaba and Mr. J. Baklik repaired it
innumerable times and machined most of the cylinders. The alignment was
made by Mr. X. Li. I am most thankful for their excellent disposition, good
craftmanship and patience. Also, I would like to thank Or. H. Swinney -and his
collaborators at the Institute of Nonlinear Dynamics at Austin for their
recommendation on the computer-control of the cylinders.
I am specially grateful to Dr. S. Kleis whose help and encouragement
throughout my research is immeasurable; and whose example as a teacher
and researcher has been an inspiration.
I would like to thank Dr. P. Chossat for a most stimulating and enjoyable
collaboration; and Dr. M. Golubitsky for his help and direction in the numerical
part of this thesis, and for allowing me to use his program to solve the
amplitude equations.
The interaction with Mr. G. Broze has contributed enormously to the
success of this research. I want to thank all my laboratory colleagues for their
collaboration, support and friendship; I am very proud of having been part of
ATL.
Many friends have helped to put the final version of this thesis together.
I would like to thank mostly Dr. L. Sanchez, but also Mr. J. Jenkinson, Mr. S.
Jain, Ms. C. Williams, Ms. A. Bascaran and Mr. L. Guazzone for their help.
iv

---

AZIMUTHAL MODE INTERACTION IN
COUNTER-ROTATING COUETIE FLOW
Abstract of a Dissertation
Presented to
The Faculty of the Department
,..",,- f!'
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of Mechanical Engineering
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,
~'{\\\\
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<'";',Ie",
In
Partial
Fulfillment
\\.,'.,
of the Requirements for the Degree of
Doctor of Philosophy
by
Catalina Elizabeth Stern
December 1988
v

---

Abstract
The stability of counter-rotating Taylor-Couette flow has been
investigated using flow visualization and laser-Doppler velocimetry.
Two
critical points (in parameter space) have been studied: CP1 due to the
interaction between azimuthal wavenumbers m=1 and m=2; and CP2 due to
m=2 and m=3. These findings are compared with the predictions of Chossat,
Demay & looss (1987, hereafter cited as COl).
In the neighborhood of CP1, only Couette flow and periodic flows
were found to be stable. Quantitative disagreements with COl include: i) the
region of stability for m=2 spirals (in parameter space) is smaller than
predicted; and ii) experiments show hysteresis in the transition from Couette
flow to spirals.
Near CP2, in addition to two transitions predicted by COl, a transition
to a chaotic regime was observed.
The flows found chaotic here have
broadband spectra, low dimension ($6) and a positive Lyapunov exponent.
Experiments also show a nonperiodic amplitude modulation of the velocity
signal almost everywhere around CP2 and a first transition to a doubly-
periodic flow.
Calculations using the COl model show that it captures much of the
behavior observed experimentally.
Also discussed are new phenomena discovered in this study: Taylor
vortices that travel axially, intermittent behavior between two (sometimes
three) flows and a flow that appears to be the "superposed ribbons"
described by COl. In addition, the exact critical parameters for the two-cell to
four-cell primary flow exchange process were measured.
vi

---

CONTENTS
page
Acknowledgements
iii
Abstract
vi
Table of Contents
vii
List of Figures
ix
List of Tables
xiv
Nomenclature
xv
Glossary
xvi
CHAPTER 1: Introduction and Objective
1
1.1
Introduction
1
1.2 Backg rou nd
2
1.3 Motivation
5
1.4 Objective
12
1.5 Organization of Dissertation
12
CHAPTER 2: Apparatus and Experimental Procedures
14
2.1
Apparatus
14
2.2
Experimental Techniques and Procedures
15
CHAPTER 3: Results
18
3.1
CP1: Interaction between m = 1 and m = 2
18
3.2 CP2: Interaction between m = 2 and m = 3
25
3.3 Topological Characteristics
38
3.4 Summary
50
CHAPTER 4: Summary and Conclusions
52
4.1
Summary
52
4.2 Conclusions
53
4.3
Recommendations for Further Investigation
56
vii

---

APPENDIX A: COl Model, Numerical Calculation andComparison
with Experiments
60
A.1
COl Model
60
A.2 Numerical Calculations
70
A.3 Comparison with Experiments
111
APPENDIX B: Other Significant Results in this Research
113
B.1
Primary Flow Exchange Process
11 3
B.2 Low [' in Counter-Rotating Flows
125
B.3 Intermittent Behavior
127
B.4 Higher-Order Transitions
127
APPENDIX C: Suggestions for Future Investigation
133
C.1 Suggestions Based on Experimental or Numerical Observations
133
C.2 Other Suggestions
135
APPENDIX 0: Diagnostics in Nonlinear Dynamics
137
0.1 Summary of Concepts
137
0.2 Experimental Implementation
143
viii

---

FIGURES
Capti 0 n
pag e
1.1
Stability diagram showing the first transition in TCF
6
1.2
CP1: a) Stability diagram, (J.!,v) space; b) Bifurcation diagram
8
1.3
CP2: a) Stability diagram, (J.!,v) space; b) Bifurcation diagram
10
3.1
Photographs of spirals: a) m=1, J.!=0.04, v=0.45 ; b) m=2,
J.!=0.04, v=-0.45
19
3.2
. Experimental stability diagram for CP1 ; M flow in Fig. 3.3a; and
N flow in Fig. 3.3b
20
3.3
Power spectra of flows close to CP1 : a) m=1, J.!=0.04, v=0.45,
Q s 1=0.302, point M in Fig. 3.2; b) m=2, J.!=0.04, v=-0.45,
Q s2=0.603, point N in Fig. 3.2
22
3.4
Flow visualization cannot resolve flows near CP2
27
3.5
Experimental stability diagram for CP2: a) 0.0 $ L $ 0.05;
b) 0.05 $ L $0.10
29
3.6
a) Periodic flow, J.!=0.045, v=-0.040, Q s2=0.931 ;
b) Doubly-periodic flow, J.!=0.08, v=O.O, Q s1=0.780, Q s2=0.923;
c) Doubly-periodic flow with broadening, L $ 0.05
30
3.7
Evolution of the spectrum with time: a) from t=O to t=256 s,
Q s 2=0.943; b) from t=768 to t=1 024 s, Q s2=0.961
32
3.8
Broadband spectra: a) J.!=0.05, v=0.033; b) J.!=0.04, v=0.045;
c) J.!=0.02, v=0.0562
34
3.9
Slow nonperiodic amplitude variation of an m=2 flow; J.!=O.O,
v=0.08, Q s1=0.756: a) Time series; b) Power spectrum
35
3.10
Photographs of flows far enough from CP2 to be resolved
visually: a) m=3, J.!=0.217, v=0.156; b) m=2 and 3, J.!=0.217,
v=0.103
37
ix

---

3.11
a) Correlation dimension for the function (cos t + cos 2 "2t);
b) Correlation dimension for the function (cos t + cos 0.809524t)
(32,768 points were used in all three cases)
40
3.12
Periodic flow: a) Correlation dimension 0=1.2;
b) Nearest-neighbor dimension 0=1.3
43
3.13
Two-frequency flow: a) Correlation dimension, no scaling;
b) Nearest-neighbor dimension 0=2.8
45
3.14
Beginning of spectral broadening, dimension of flow in Fig. 5.6a:
a) Correlation dimension 0=1.9; b) Nearest-neighbor dimension
0=3.2
.46
3.15
Dimension of flow in Fig. 5.6b: a) Correlation dimension, no
scaling; b) Nearest-neighbor dimension, no saturation
47
3.16
Dimension of flow in Fig. 3.8c: a) Correlation dimension, no
scaling; b) Nearest-neighbor dimension 0=3.9
49
3.17
Correlation dimension of the flow of Fig. 3.7, 0=1.8
51
A.1
Calculation of Ric from the neutral stability CUNes
65
A.2
a) Numerical stability diagram around CP2; b) Expanded view
72
A.3
Phase portraits, 11.=0.0275, v=-0.029, trajectories in all four
subspaces go to a fixed point: a) X1; b) X2; c) X3; d) X4
73
AA
Phase portraits, 11.=0.028, v=-0.0286: a) X1 trajectories go to a
fixed point; b) )(4 trajectories go to a limit cycle with frequency
Q s2=0.907
74
A.5
Periodic flow, 11.=0.028, v=-0.028: a) Time series;
b) Power spectrum, Qs2=0.907
75
A.6
Phase portraits, 11.=0.031, v=-0.0253: a) X1 , limit cycle with
frequency 0.753; b) X2 fixed point; c) X3, fixed point; d) X4,
limit cycle with frequency Qs2=0.907
76
x

---

A.7
Doubly-periodic flow, Jl=0.031 O. v=-0.0253: a) Time series;
b) Power spectrum, Q s2=0.907, Q s1=0.753
77
A.8
Phase portraits, trajectories in all subspaces are limit cycles:
a) X1, Q s1=0.675; b) X2, Q s1=0.753; c) X3, Q s2=0.907;
d) X4, Q s2=0.907
78
A.9
Doubly-periodic flow, Jl=0.03266, v=-0.02309: a) Time series;
b) Power spectrum, Q s2=0.907, Q s1=0.753
80
A.10
Phase portraits, Jl=0.03267, v=-0.02308, all subspaces have
limit cycles: a) X1, Q s1=0.753; b) X2, Q s1 =0.753; c) X3,
Q s2=0.907; d) X4, Q s2=0.907
.81
A.11
Phase portraits, Jl=0.03268, v=-0.02307, all trajectories are
two-torii: a) X1, Q s1=0.754, Q s3=3.938; b) X2; c); X3; d) X4,
Q s2=0.908, Q s4=4.095
82
A.12
Four-frequency flow, Jl=0.03268, v=-0.02307: a) Time series;
b) Power spectrum, Q s3=3.938. Q s4=4.095
83
A.13
Phase portraits, 1-1=0.0327, v=-0.02304, trajectories in all
subspaces go to two-torii: a) X1; b) X3
84
A.14
Phase portraits, Jl=0.0328, v=-0.0230: a) X1 ; b) X3
85
A.15
Power spectra: a) 1-1=0.04327, v=-0.02304; b) Jl=0.04328,
v=-0.02300
86
A.16
Phase portraits, Jl=0.03285, v=-0.02282. more than four
frequencies: a) X1; b) X3
88
A.17
Phase portraits, Jl=0.04329, v=-0.02280, several modulations in
each subspace: a) X1 ; b) X3
89
A. 18 Phase portraits, Jl=0.3295, v=-0.02268, quasi-periodic regime
with several frequencies: a) X1; b) X3
90
A. 19 Spectra of quasi-periodic flows: a) Jl=0.03285, v=-0.02282;
xi

---

b) 1..l=0.0329, v=-0.02280; c) 1..l=0.03295, v=-0.02268
91
A.20 Phase portraits, 1..l=0.0330, v=-0.0226,quasi-periodic behavior
with several frequencies: a) X1; b) X3
92
A.21
Quasi-periodic behavior, 1..l=0.0330 v=-0.0226: a) Time series;
b) Power spectrum
93
A.22
Phase portrait, 1..l=0.0331, v=-0.02246, nonperiodic behavior:
a) X1; b) X3
94
A.23
Nonperiodic behavior, 1..l=0.0331, v=-0.02246: a) Time series;
b) Power spectrum
95
A.24
Spectrum of each subspace is broad: a) X1; b) X3, X2
and X4 are like X1 and X3 respectively
97
, A.25
Phase portraits, 1..l=0.0332, v=-0.02231, quasi-periodic behavior
in all subspaces: a) X1; b) X3
98
A.26
Phase portraits, 1..l=0.0333, v=-0.02216, quasi-periodic behavior
with several frequencies: a) X1 ; b) X3
99
A.27
Phase portraits, 1..l=0.04335 v=-0.0219, quasi-periodic behavior:
a) X1; b) X3
100
A.28
Quasi-periodic behavior: a) 1..l=0.0332, v=-0.02231 ;
b) 1..l=0.0333, v=-0.02216; c) 1..l=0.0335, v=-0.0219
101
A.29
Phase portraits, 1..l=0.04661 v=-0.0450, nonperiodic behavior,
L=0.08: a) X1; b) X3
102
A.30
Broadband spectra for L=0.08; 1..l=0.0661, v=-0.0450:
a) X1 +X3; b) X3 only
103
A.31
Phase portraits, 1..l=0.01634, v=-0.01152, nonperiodic, L=0.02:
a) X1; b) X3
105
A.32
Periodic flow: a) Correlation dimension;
b) Nearest-neighbor dimension
107
xii

---

A.33
Doubly-periodic flow: a) Correlation dimension;
b) Nearest-neighbor dimension
108
A.34
Quasi-periodic regime with several frequencies: a) Correlation
dimension; b) nearest-neighbor dimension
109
A.35
Nonperiodic regime with broad spectrum: a) Correlation
dimension; b) Nearest-neighbor dimension
110
B.1
a) Pitchfork bifurcation; (b)-(e) Unfolding of a pitchfork bifurcation;
cases (b)-(d) correspond to different values of r
114
B.2
Cusp in Rj vs. r due to the unfolding
116
B.3
Velocity measurements showing hysteresis in the transition
CF~TVF for r =3.7
122
B.4
Photographs of the hysteresis: a) Ri=115 (up); b) Ri=125 (up);
c) Ri=125 (up); d) Rj=125.5 (up); e) Ri=124.5; f) Ri=123 (down);
g) Ri=122 (down)
123
8.5
Intermittent behavior: a) Time series alternates between two
different states; b) Time series alternates among three different
states
127
B.6
Photograph of rhombuses
129
B.7
Region of stability of rhombuses
131
D.1
Capacity of a Cantor set
139
D.2
The correlation dimension is given by the slope of
IOg2 C vs IOg2 £
141
D.3
The nearest-neighbor dimension is given by the inverse of the
slope of log2 <dk> vs IOg2 k; n is the order of the
nearest-neighbor
146
xiii

---

TABLES
1.1
Parameters at the multicritical point
7
1.2 Theoretical slopes for the stability diagram of CP1
9
1.3 Theoretical slopes for the stability diagram of CP2
11
3.1
Slopes in the experimental stability diagram about CP1
21
3.2 Slopes in the experimental stability diagrams about CP2
33
A.1
Slopes of lines in the numerical stability diagram about CP2
104
xiv

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NOMENCLATURE
a
inner radius
b
outer radius
Qj
speed of inner cylinder
ne
speed of outer cylinder
h
height of the lluid column
d = b-a
gap between cylinders
r= hid
aspect ratio
11= alb
radius ratio
u
kinematic viscosity
Rj =d a Qi/U
inner cylinder Reynolds number
Ra = db Oo/u
outer cylinder Reynolds number
W = ne/Qj
angular velocity ratio
a
axial wave number
m
azimuthal wavenumber
m1' m2
interacting azimuthal wavenumbers
Jl =(103/3) (1 IRjc - 1/Rj)
coordinates in parameter space·
v = W - Wc
L = [Jl2 + v2J112
radius in parameter space (Jl,v)
Q s1
frequency of spirals with m1
Q s2
frequency of spirals with m2
o
dimension of the attractor
CF
Couette flow
TVF
Taylor vortex flow
S1
spirals with m1
S2
spi rals with m2
SI
interpenetrating spirals
QP3
quasi-periodic flow (three frequencies)
LDV
laser Doppler velocimetry
• subscript c refers to critical values.
xv

---

GLOSSARY
state space: space of the variables required to describe the system. A point
in state space represents the state of the system at a given instant in
time.
dynamical system: vector field that determines the evolution of points in the
state space; e.g., a differential equation defines a dynamical system.
trajectories: curves that describe the time-evolution of a system in state
space.
phase portrait: set of all trajectories in state space.
control parameter: parameters external to the system that can produce
changes in the system; e.g., Ri, R J.l or v.
OI
bifurcation diagram: set of points (X,A) that satisfy the equation g(X,A) = 0,
where x is the state variable and A the control parameter.
bifurcation: abrupt changes at certain values of the parameters in the
qualitative structure of solutions.
Each point at which such changes
occur is called a bifurcation point.
neutral stability curve: curve that describes the relationship between
parameters when Re(cr) = 0, where cr is the eigenvalue of the linearized
equation (for example, Ri as a function of the wavenumber a).
critical point: the point in parameter space at which a solution first loses
stability as a control parameter is varied quasi-statically.
It is the
minimum of the neutral stability curve.
critical parameter: value of the parameter at the critical point (for example
Ric is the value of Ri at the critical point.
xvi

---

mode:
eigenvector corresponding to a single eigenvalue of the Iinearized
equation.
Mode is used also to denote a flow with a specific
wavenumber.
critical mode: a mode whose eigenvalues cr lie on the imaginary axis (Re(cr)
= 0). There are two types of critical modes: steady-state (Im(cr) = 0) and
Hopf ( Im(cr)"# 0).
multicritical point: critical point at which a solution loses stability to two or
more modes simultaneously.
mode interaction: non linear interaction between two or more modes near a
multicritical point.
xvii

---

CHAPTER I
INTRODUCTION AND OBJECTIVE
1.1
Introduction
One of the most difficult problems yet to be solved is transition of a
laminar flow to a turbulent state. Experimental and mathematical techniques
developed recently in the field of nonlinear dynamics may lead to a better
understanding of this complicated phenomenon. Taylor-Couette flow (TCF) is
the perfect ground for testing and applying these techniques. For example,
experiments have shown that when the outer cylinder is stationary, a chaotic
flow develops at a high Reynolds number.
Investigations of the counter-
rotating case have revealed a rich variety of flows. Pre-chaotic, periodic and
quasi-periodic flows in time have even been observed close to the boundary
of the region in the parameter space where Couette flow is stable (Coles
1965; Snyder 1969; Andereck, Liu & Swinney 1986; Langford et al. 1988). In
parallel, new mathematical techniques in nonlinear stability have proven to
be successful not only in describing much of the experimental observations
but also in predicting new phenomena.
In the present investigation, experiments were conducted to explore the
dynamics of flows near two critical points at which the Couette flow (CF)
becomes unstable simultaneously with respect to two nonaxisymmetric
azimuthal modes.
Nonlinear interaction between the modes leads to more
complicated behavior than would be expected when each mode is
considered individually.
Chossat, Demay & looss (1987), referred to
hereafter as COl, studied this problem theoretically. (The interaction between
an axisymmetric and nonaxisymmetric azimuthal modes was studied by
Golubitsky & Stewart (1986) and Golubitsky & Langford (1987).)
1

---

1.2
Background
An enormous amount of experimental and theoretical work has been
done on TCF. The reader interested in tile details of this work is referred to the
only two reviews available: Di Prima & Swinney (1981) and Drazin & Reid
(1982).
In this dissertation only a brief review relevant to the present
investigation will be presented.
The physical system being studied is the flow of a Newtonian fluid
between two concentric cylinders which can rotate independently. Cylindrical
coordinates (r,S,z) are the most appropriate to describe the flow. The .velocity
is denoted by V = ( vr, vs, vz)' the inner radius by a, the outer radius by b, and
their corresponding speeds of rotation by Qj and Qo.
The nondimensional
a
o
parameters of the problem are: the radius ratio 11= b ; the speed ratio 00= Q ;
Qj
the inner Reynolds number Ri = \\f(d a Qi,U) ; the outer Reynolds number, Ra =
d b Q 0 ; and the aspect ratio r _~. The gap width is given by d = b-a; U is the
u
kinematic viscosity and h is the height of the fluid column. The equations of
motion are the Navier-Stokes equations.
For a system with infinitely long cylinders (r--7 00), the Navier-Stokes
equations admit an exact laminar solution, representing purely azimuthal flow.
This solution, called "Couette How", is of the form
11
1
VO = (O,vO(r), 0),
rE [ - , - ] ,
1-11 1-11
B
where VO = Ar + -r '
(1.1 )
2

---

00 - 11 2
a2 1 - (t)
with A =
and B - -
- - -
1 - 112
- d 2 1 - 11 2
All quantities have been nondimensionalized with d and Qj -1 as the length
and time scales.
The instability of flows between two cylinders due to the effects of
rotation, i.e., centrifugal force, was first studied theoretically by Lord Rayleigh
in 1916 (see Drazin & Reid 1982). He considered axisymmetric disturbances
in an inviscid fluid with angular velocity Q(r) and derived what is now known
as Rayleigh's criterion for stability. This criterion states that a necessary and
sufficient condition for stability to axisymmetric disturbances in a rotating flow
is that the square of the circulation does not decrease anywhere in the field of
the flow.
The stability of the viscous now was first studied experimentally for a
stationary inner cylinder by Couette in 1890, and for a stationary outer cylinder
by Mallock in 1896.
However, it was Taylor's (1923) experiments and
theoretical analysis that resulted in a better understanding of the stability of the
flow. The deviation between his experimental and theoretical results was less
than 5%. In Tay/or's apparatus, the two cylinders rotated independently. He
showed that the motion that appeared at the onset of instability was
axisymmetric (as was assumed by Rayleigh), but that the Rayleigh criterion did
not apply when the cylinders rotated in opposite directions. Although Taylor's
theoretical analysis considered only axisymmetric disturbances, he observed
three-dimensional disturbances in his experiments.
Much has been learned about TCF in the last five decades. When the
outer cylinder is stationary (00=0), the following sequence of transitions occurs
as the control parameter Ri is increased quasi-statically from rest. The first
3

---

transition is to an axisymmetric flow known as Taylor vortex flow (TVF), in
which toroidal vortices (called cells) with alternate circulations are stacked on
top of each other. TVF loses stability to a nonaxisymmetric disturbance, and
each cell acquires a periodic pattern called wavy vortex flow (WVF). When
WVF becomes unstable, a modulation of the wavy pattern occurs, creating a
doubly-periodic flow called modulated wavy vortex now (MWVF).
Finally,
small-scale motions start to appear in each cell.
Eventually the small scale
motions fill up the cells and the flow becomes turbulent. It has been shown by
Brandstater & Swinney (1987) that, at the onset of the small scale behavior the
system is chaotic, and the flow dynamics can be described by a strange
attractor
(see
Appendix
D).
This
sequence
(for
w =0) ,
TVF~WVF~MWVF~chaos, is a typical Ruelle-Takens-Newhouse scenario
for transition to chaos. The series of transitions sequence is the same for all
values of 11 (provided that r is sufficiently high, say r>20), but the values of Ri
at which transitions occur are strongly dependent on 11.
When both cylinders rotate, two control parameters which characterize
the rotations of the cylinders are required. The first parameter has traditionally
been Ri; the second can be either 00 or Ro. The direction of Ri is taken as
positive.
For the co-rotation case (00 > 0), the first transition is always to TVF.
Higher-order transitions depend on 11 and on Ro (or, equivalently, w).
For the counter-rotating case (00 < 0), the first transition is to Tay/or
vortex flow if 0 ~ w ~ -0.8 (this value changes slightly with 11). For more
negative values of 0>, Couette flow becomes unstable with respect to a
nonaxisymmetric disturbance which constitutes both a rotating wave in e and
a travelling wave in z. This results in a helical pattern called spiral vortex
4

---

flow.
The pitch of the spiral indicates the azimuthal wavenumber m of the
flow.
1.3
Motivation
Figure 1.1 shows a typical stability diagram in (Ri, Ra) space. Only the
first transition is shown because higher-order transitions depend strongly on
11. Notice that the azimuthal wavenumber m increases as Ro becomes more
negative.
There are points in the parameter space where Couette flow loses
stability to two azimuthal wavenumbers m1 and m2 simultaneously: These
points are called multicritical points.
Nonlinear interaction between m and
1
m2 can lead to interesting dynamics.
In Figure 1.1 the multicritical point
labelled CPO corresponds to the interaction of m =0 and m
1
2=1. CP1 and CP2
correspond to the interactions of m1 =1 and m2=2, and m1=2 and m2=3,
respectively.
One of the goals of this work was to describe experimentally the stability
of flows near CP1 and CP2 and to compare the observations with theoretical
predictions.
The corresponding theory, derived by COl, is reviewed in
Appendix A. Only the theoretical predictions for the radius ratio 11=0.75 will be
summarized here.
To study the stability of flows in the neighborhood of the critical points,
circular paths around CP1 and CP2 are followed in (Il,v) space, where
1
1
103
I l = ( - - - } -
and
Ric
Ri
3
V=ro-~.
(1.2)
Ric and roc are the values of Ri and ro at the multicritical points.
5

---

Spirals
m=3
TVF
CPO
Couette Flow
Figure 1.1 Stability diagram showing the first transition in TCF
6

---

Note that (ll,v)=(O,O) at Ri=Ric and 0) =O)c
1.3.1
CP1: Interaction between m1 =1 and m2=2.
The values of the critical parameters and the frequencies of the flows at
the multicritical points predicted by COl are given in Table 1.1 below.
All
quantities are nondimensionalized with d and Qr1 as the length and time
scales; a is the most unstable axial wavenulTlber, Q s1 and Q s2 are the
frequencies of Hows with m1 and m2 respectively.
11
0)
Ri
a
m2
Q s1
Qs2
CP1
0.75
-0.701
135.7
3.87
2
0.36747
0.65259
CP2
0.75
-1.1558 200.1
4.33
2
3
0.74832
0.90483
Table 1.1 Parameters at the Multicritical Point.
The stability of flows around CP1 is shown in Figure 1.2a in (11, v)
coordinates.
The straight lines are tangent (at CP1) to the boundaries
between two different stable flows. The slopes of these lines were calculated
by COl and are given in Table 1.2. S1' S2 and CF denote, respectively, the
regions where spirals with m=1, spirals with m=2 and Couette flow are stable.
The frequencies of rotation of the spirals are Q s1 and Q s2 respectively. The
region of stability for each type of spirals changes depending on the direction
in which the two parameters are varied. If the parameters are varied following
a circular path in the clockwise direction, the sequence of transitions is as
follows: Starting on line A, CF is stable. (Upper case bold letters will be used
to designate predicted lines in the stability diagram.)
On line B there is a
transition to S1, then a transition to S2 on C, and back to CF on D. If the same
path is followed in the counter-clockwise direction, CF becomes unstable to S2
7

---

0.15 r
\\
v
S1
S1 or S2
0.1
m=1
m=1 or 2
S1 or S2
0.05
B....
m=1 or 2
S2
m:=2
of--- - A-· -- -- -
CF
CF
C'
C
co
-0.05
8
o
A
A
(b)
-0.1
CF
-0.15 I
I
I
J
I
I
I
-U.15
-0.1
-0.05
0
0.05
0.1
0.15
Il-
Figure 1.2 CP1 : a) Stability diagram, (Il,v) space; b) Bifurcation diagram
I

---

on 0, there is a transition to 8 1 on C', and finally the flow goes back to CF on
B. The hysteresis in the transitions 8 1 ~ 8 2 actually corresponds to a "jump"
from one branch of solutions to another, as can be seen in the bifurcation
diagram shown in Figure 1.2b. Note the locations of C and C'.
line
B
C
C'
o
slope
-0.3819
-0.3348
-2.0330
-0.5550
Table 1.2 Theoretical slopes for the stability diagram of CP1
1.3.2
CP2: Interaction between m1=2 and m2=3.
The
predicted values of the critical parameters and frequencies of
rotation of the flows at CP2 are given in Table 1.1.
The stability diagram
predicted theoretically is shown in Figure 1.3a; only the transitions to -stable
flows are shown. In the neighborhood of CP2, 8 1 refers to spirals with m1=2
and 8 2 to spirals with m2=3. Following a circular path in the counter-clockwise
direction, with CP2 as the center, the transition sequence is as follows: CF is
stable between A and B; Couette flow becomes unstable to 82 on B; and on C
there is transition to interpenetrating spirals 81 12 (Interpenetrating spirals are
doubly-periodic flows that bifurcate from spirals and can be represented as a
superposition of two spirals, each with different wavenumber, and travelling in
the same or opposite directions). In this particular case, the interpenetrating
spirals have wavenumbers 2 and 3 and the two spirals travel in opposite axial
directions.
Table 1.3 gives the predicted values of the slopes indicating
transitions to stable solutions in the stability diagram.
9

---

0.1.
I
v
..... ......
Sl
'"
"-
m=2
,/
; '
not accessible 10
"
/
0.05
the model
/
",,
A
,
\\
I
\\
,
/
,
\\
,
I
,
\\
I
\\
\\
o
I
I
I
CF
I
.......
CP2~
L
o
A
B
(b)
~~I"
-0.05
B 'C.
CF
o
0.1
(a)
11
-"
I..,.: ,I
Figure 1.3 CP2: a) Stability diagram, (Il,v) space; b) Bifurcation diagram
(dotted line represents unstable solution)

---

line
A
B
c
slope
-0.5239
-1.0136
-0.8322
-0.6301
Table 1.3 Theoretical slopes for the stability diagram of CP2
At some line 0 between C and I, one can expect a transition to a triply-
periodic 1'Iow QP3 with a frequency given by
(1.3)
The exact location of this transition could not be calculated; furthermore, it is
not known whether or not this solution is stable.
After this transition, the
analysis in COl does not proceed any further.
For the sake of simplicity, in the remainder of this dissertation, the
distinction between m1 and m2 will be indicated by the numerical value of m.
Figure 1.3b shows the corresponding bifurcation diagram. Only one of the
unstable solutions is shown as a dotted line in this diagram to illustrate that
an 8 1 solution (with m = 2) is not stable near this critical point.
As can be seen (Figures 1.2b and 1.3b), the bifurcation diagrams in the
neighborhood of these two critical points are quite different.
In the
neighborhood of CP1, no quasi-periodic flow (two or more frequencies) is
expected to occur, while in that of CP2, a regime of interpenetrating spirals
should be observed.
In the second case, a higher-order analysis would be
required to determine the stability of the three-frequency flows and the theory
fails to predict any stable flow in a large sector around the critical point.
1 1

---

1.4
Objective
The purpose of this investigation is to study experimentally the stability
of flows in the neighborhood of CP1 and CP2.
Specifically, the objectives
are: a) to provide the experimental stability diagrams in the neighborhoods of
CP1
and CP2; b) to compare these diagrams with those predicted
theoretically; and c) to characterize the various flow regimes in terms of their
power spectra, dimension and largest Lyapunov exponent.
To achieve these goals, two facilities with the appropriate experimental
parameters for comparison with the theoretical predictions have been
designed.
The values of these parameters are: a radius ratio 11=0.75; an
aspect ratio r that would allow 20 axial wavelengths (calculated from the most
unstable wavenumber predicted theoretically); and Reynolds numbers in the
range of [0,1000] to ensure that a wide range of (Il,v) around the critical points
could be attained.
To provide a spatial description of the flow, Kalliroscope was chosen for
flow visualization.
The calculations of dimension and largest Lyapunov
exponent require a continuous time series, thus LDV (laser-Doppler
velocimetry) with a tracker was chosen as the most convenient technique.
1 .5
Organization of Dissertation
This dissertation is organized as follows: Chapter 2 will describe the
apparatus and experimental procedures; Chapter 3 will present the results;
and Chapter 4 will summarize the conclusions and recommendations.
Appendix A will present a review of the theory derived in COl and numerical
calculations that were completed during the course of this investigation to
complement the experimental results.
Appendix B will describe new
experimental observations in regions of parameter space different from those
12

---

of Chapter 3. Appendix C will list more specific recommendations for future
investigation, and Appendix D is a review of concepts and tec~Jniques in
nonlinear dynamics.
13

---

CHAPTER 2
APPARATUS AND EXPERIMENTAL PROCEDURES
The stability diagrams of the multicritical points have been studied
experimentally using LDV and flow visualization.
A brief description of the
flow facility will be given in this chapter followed by the relevant details of the
equipment, instumentation and experimental procedures.
2.1
Apparatus
Two versatile and precise facilities were built in which Taylor-Couette
flow could be studied over wide ranges of values of the various parameters.
The outer cylinders were made of precision pyrex tubing; the inner cylinders
were machined from bronze and then coated with black oxide.
The inner
cylinders can be easily removed; thus, the radius ratio allowed by the fourteen
possible combinations ranges from 0.50 to 0.95. (The gap between the two
cylinders ranges from 0.250-2.540 cm in the smaller facility and from 0.381-
3.810 cm in the larger one.) Nylon or Plexiglas rings can be attached to either
cylinder at a chosen height to fix the aspect ratio. The maximum allowable
height h is =76 cm.
The two cylinders in each facility can rotate independently; their motion
is controlled by microstepping motors (25,000 steps/rev), which utilize
microprocessor-based, programmable pulse generators.
The accuracy is
0.001 rev/sec in the speed of rotation and 0.01 rev/sec2 in acceleration. The
maximum motor velocity is 20 rev/sec and the maximum acceleration 99
rev/sec2. The pulse generators can be computer-controlled or used manually.
The use of this type of control was recommended by Dr. Swinney and
14

---

collaborators from the Center of Nonlinear Dynamics of the University of Texas
at Austin. The author is very grateful for this recommendation.
The radius ratio for the mode interaction experiments was 11=0.75. The
radii were 5.715 ± 0.003 cm and 3.810 ± 0.003 cm for the inner cylinders and
for 7.620 ± 0.004 cm and 5.080 ± 0.004 cm for the corresponding outer
cylinders. The height of the fluid column h was adjusted to allow twenty
wavelengths.
The wavelength was calculated from the theoretical
wavenumber (see Table 1.1). The values of h corresponded to r=32.5 for
CP1 and r =29 for CP2. In these experiments, end nylon rings were attached
to the inner cylinder.
2.2
Experimental Techniques and Procedures
Two different kinds of techniques were used to study the different flow
regimes: flow visualization and laser-Doppler velocimetry (LDV).
The flow
was visualized using mixtures of either glycerine, water and Kalliroscope or
silicon oil and pearl maid. Kalliroscope is an aqueous solution of a synthetic
polymeric substance. The particles in the solution are platelets that align with
the shear, causing flow patterns to appear in various shades of gray.
Pearlmaid is an oil-soluble powder (obtained from mother-of-pearl) having
properties very similar to those of Kalliroscope. Glycerine and water seeded
with silicon carbide particles or silicon oil seeded with titanium dioxide were
used for LDV; particles were approximately 5~m in size. The viscosity was
measured using a Cannon-Fenske viscometer, which has a stated accuracy of
0.355.
The motion of the cylinders was computer controlled to change the
Reynolds numbers quasi-statically following prescribed circular trajectories
around the critical point in (~,v) space. The flows were allowed to settle for
15

---

h2
times of the order of
These times ranged between half an hour to four
6u'
hours.
The source of largest error in these experiments was the uncertainty in
the measurement of viscosity. This gave a maximum overall error of about
0.5% in ~ and v, with some variation depending on the distance from the
critical point.
2.2.1
Flow Visualization
Flow visualization was performed in both facilities; thus, the results
have been checked in independent observations and facilities. In the larger
facility, the glass cylinder was surrounded by a square pyrex box fixed in the
laboratory frame. The gap between the outer cylinder and the box was filled
with water. In general, this has proven convenient for photographing the flows
and for visualization of their cross sections.
In addition to the real-time observations, a video camera was used to
record the flows. Still photographs of the flows with a scale attached to the
pyrex box were used to determine the cell size.
The accuracy of these
measurements was ::3.5%.
Flow visualization was used primarily to study
CP1 (see Figure 1.1); its limitations in studying CP2 will be apparent in
Chapter 3.
2.2.2
Laser-Doppler Velocimetry
CP2 was studied mostly by velocity measurements in the larger facility.
When silicon oil was used as the working fluid, it was also used to fill the gap
between the outer cylinder and the box. The index of refraction was 1.47 for
16

---

pyrex glass, 1.43 for silicon oil, and varied between 1.32 and 1.38 for mixtures
of water and glycerine.
The LDV used is a modular Dantec system, operated in backscattering
dual mode. It consists of a 15 mW He-Ne laser (632.8 nm), a frequency shifter,
a spatial filter, an 80mm focusing lens and a photomultiplier. Depending on
the index of refraction of the working fluid, the length of the scattering volume
varied between 0.25 and 0.53 mm; the diameter was 0.059 mm; and the
number of fringes varied between 45 and 30.
The backscattering mode
proved to be the most convenient to avoid spurious reflections from the outer
cylinder. For the few experiments done with the outer cylinder stationary (see
Appendix B), detection at ninety degrees gave the highest quality signal.
A Dantec tracker (model
55N20)
was used to process the
photomultiplier signal. The signal was then fed either to a computer through a
12-bit NO converter or to an Ono-Sokki spectrum analyzer (model CF-290).
Spectra were calculated using ILS library routines.
The techniques to extract nonlinear dynamical measures of the signals
are discussed in detail in Appendix D.
17

---

CHAPTER 3
RESULTS
The stability of flows in the neighborhoods of CP1 and CP2 has been
established experimentally by following circular paths around each
multicritical point in (~,v) space and describing the behavior of the flow at each
point in the circular paths. The observations around CP1 will be described in
section 3.1; those around CP2 in section 3.2.
Diagnostics from non linear
dynamics have been used to characterize the different flows; these results will
be presented in section 3.3.
3.1
CP1: Interaction between m=1 and m=2
The motion of the two cylinders was controlled with a computer to follow
quasi-statically circular paths at chosen radii from CP1 in (~,v) space. Each
path was followed both in the clockwise and counter-clockwise directions.
The experiments were performed mostly using flow visualization. Figures 3.1 a
and 3.1 b show photographs of 8, and 8 2. (In this section, 8, refers to spirals
with m=1 and 8 2 to spirals with m=2). It was quite simple to determine visually
the azimuthal wavenumber (given by the pitch of the spiral). After the stability
regions were established, LDV measurements were used to obtain the
velocity spectra of some chosen flows and determine the frequencies Qs' and
Q s2 of the spirals. To complete the analysis, still photographs were used to
measure the axial wavelength and thus, determine the corresponding
wavenumber a of the flow.
Circular paths around CP1 were followed for radii L= 0.015, 0.030,
0.045, 0.060, 0.075, 0.09, 0.100 and 0.120 (L2=~2+v2). The stability diagram is
shown in Figure 3.2. (Lower case bold letters will be used to designate
18

---

(a)
(b)
Figure 3.1 Photographs of spirals: a) m=1, p=O.040, v=-O.045; b) m=2,
~L=O.040, v=O.045
19

---

0.15
TVF
v
+
+
+
0.1
~ + +
+
51
+
)/
m=1
0.05
a - - -
o
I\\)
o
c'
-0.05
------
~ S1 orS2
~
=1 or 2
_-:---_,rl .....~';.';"_._.
10
'(
.,-....
.{
. l':
'" 1.V~'
'd
4".3
.. '"
l/)fO'-,n \\
J>.
0;~\\,P
S2
-0.1
::»
n
. :l
::::..
1>-
m=2
'~
\\\\.f'
rn
y
-
CF
./
~;.o
r.;:;
. (ii
't,
r $
~
,,",
l
",..~'''I'.
,~. ',.p'
.........
f
.. ' ' "
~~~;...:~:. ~...:+'
-0. ~~. 15
-0.1
-0.05
o
0.05
0.15
j.l
Figure 3.2 Experimental stability diagram for CP1; M flow in Fig. 3.3a~ N flow in
Fig. 3.3b ("+" represent locations where TVF was found to be stable)

---

experimental lines in the stability diagrams.) Following the clockwise direction,
the transition CF ~ S1 occurs on line b. S1~ S2 occurs on c and S2 ~ CF on
d. In the counter-clockwise direction, the transition CF ~ S2 occurs on d', a
different location than the S2 ~ CF transition. The S2 ~ S1 transition occurs
on c', and on b' the flow goes back to CF. The S1 ~ CF transition occurs at a
different location that the CF ~ S1 transition. As it was mentioned in Chapter
1, the transition boundaries are not straight lines. In the theoretical predictions
of Figure 1.3, the lines are tangent to the transition boundaries at the
multicritical point.
In the experiments the boundaries are "wiggly" and the
lines in Figure 3.2 indicate the approximate location of such boundaries. With
these considerations Table 3.1 below can be compared with Table 1.2. Note
that even though the sequence of transitions is as predicted, the locations at
which transitions occur are very different.
line
b
c
d
b'
c'
d'
slope
-0.25
-0.51
-1.91
-0.19
-0.32
-0.82
Table 3.1. Slopes in the experimental stability diagram about CP1.
Typical spectra of flows close to CP1 are shown in Figures 3.3a and
3.3b.
These two flows are identified in Figure 3.2 as points M and N
respectively. Note that for the m=1 flow (point M) the frequency of the spiral is
Q s 1=0.302, and for the m=2 flow it is Q s2=0.603. These values are close to
0.367 and 0.653 in Table 1.1. The frequencies of the spirals change slightly
as the values of (Il,v ) are varied around CP1.
21

---

53.33
ns1
(a)
40.00
2Qs1
26.66
3 Q s1
dB
13.33
4Qs1
0.0
-13.33
0j
10
53.33
°S2
(b)
40.00
26.66
2Q
dB
s2
13.33
0.0
-13.33
1
~
10
Figure 3.3 Power spectra of flows close to CP1:
a) m=1, 1.1=0.04, v=-0.45,Qs1=0.302, point M in Fig. 3.2;
b) m=2, 1.1=0.04, v=-0.45, Qs2=0.603,point N in Fig. 3.2
22

---

Hysteresis
It was explained in Chapter 1 (see the bifurcation diagram in Figure
1.2b) that there is an apparent hysteresis effect due to the fact that the "jump"
between branches occurs at different locations when a circular path is
followed in opposite directions.
Although this "jump" was detected
experimentally, it occurs in a smaller region of parameter space than predicted
by COl, as can be seen by comparing the distances between lines c and c' in
Figure 3.2 with the distance between lines C and C' in Figure 1.2a. In the
experiments, the "jump" is observed as a sudden change in the inclination of
the spirals all along the fluid column. For this case, spirals of different m do
not coexist.
In the experiments the transition from CF to S1 or S2 does not occur at
the same location as the transition from S1 or S2 to CF. This hysteresis was
not predicted by the theory.
Note that in the theoretical stability diagram
(Figu re 1.2a), the transition CF H
S1 occurs on line B, while in the
experimental diagram CF --7 S 1 occurs on b, but S1 --7 CF occurs on b'.
Similarly, there is only one line 0 in Figure 1.2a, but two lines, d and d', in
Figure 3.2.
It is possible that the transition from S1 or S2 to CF is detected at a
different value of (Il,v) than the transition from CF to S1 or S2 because the
settling time ts is not sufficient for the flow to attain steady-state. After each
change of parameters the system is allowed to settle for a time ts =~ ; ts is
4 u
the diffusion time for the semi-height of the liquid column.
It is important to mention that experimentally the transition CF to either
S1 or S2 does not occur suddenly at given values of the parameters, but
gradually as (Il,v) are varied around CP1. The spirals start to form close to the
23

---

ends; there are values of (~,v) for which the spirals do not fill the entire fluid
column even after long periods of time. As (~,v) values are varied, the spirals
propagate towards the center (in z) of the fluid column until they eventually fill
up the entire column. The determination of the precise values of the critical
parameters is thus, quite subjective. It is customary to define the experimental
critical parameters as the values of the parameters at which the entire fluid
column is filled with spirals. However, different methods of detection can lead
to different critical parameters: LDV can detect the time dependence at values
of (~,v) different than those values at which the eye can detect spiral motion.
(The same kind of phenomenon, gradual changes and hysteresis, occurs in
the transition from CF to TVF when 0)=0 for short r, see Appendix S.)
In the ideal case of infinite cylinders, the transition from CF to either 81
or 82 is a Hopf bifurcation.
To date, there is no explanation as to why a
gradual transition instead of a bifurcation is observed experimentally.
Critical wavenumber
Measurements from still photographs show that when the cylinders are
brought from rest to (~,v)=(O,O), the experimental axial wavenumber (3.90 ±
0.13) is very close to the predicted most unstable value 3.86; see Table 1.1. It
was observed that not all cells in the fluid column have the same size, even
after allowing the flow to settle for long periods of time (=3 ts). Cells are
usually larger at the ends than at the center; sometimes the cells at one or
both ends look like TVF (flat boundaries and no inclination) having a much
smaller wavelength than the corresponding spirals. Four or five cells near the
semi-height (in z) of the facility were used to determine the average cell size.
It is reasonable to compare the cell size predicted theoretically only with the
cells at the center of the facility where the ends have less effect.
24

---

The internal structure of the spirals is not known. Cross-sections of the
flow are difficult to see because the instability occurs very close to the inner
cylinder. The higher the pitch of the spiral, the more difficult it is to observe.
New flow visualization experiments (for example, illuminating from inside)
might give more insight into their spatial structure and contribute to the
understanding of their hydrodynamical origin.
Boundary of the model
Taylor Vortex Flow (TVF) was observed at a radius of 0.12 from CP1 in
(Il,v) coordinates.
The "+" symbols in Figure 3.2 represent TVF.. These
observations are not unexpected.
They indicate the proximity of the critical
point (Iabeled CPO in Chapter I) where interaction between m=O (TVF) and
m=1 (S1) occurs.
Summary
The qualitative behavior in the neighborhood of CP1 is very similar to
the theoretical predictions by COL However, the experiments show a smaller
region of stability for S2 and hysteresis in the first transition.
3.2
CP2: Interaction between m=2 and m=3
The same procedure described in Section 3.1 was followed for CP2.
The critical point was not found where it was predicted by COl. The location of
CP2 in (Rj,Ro) space had to be determined experimentally.
The following
procedure using flow visualization was tried first. From previous experiments
it was known that at Ro=-375 when Ri is increased quasi-statically from rest,
the first transition is to an m=3 flow. Thus, at this Ro, the value of Rj at which
the first transition occurred was determined. The system was then stopped and
25

---

the same procedure was followed, each time fixing a less and less negative
value of Ro and increasing Ri quasi-statically to the first transition. It was
hoped that, in this way, the point at which the first transition is no longer to an
m=3 flow could be determined.
Location of CP2
Surprisingly, flow visualization was not sensitive enough to resolve the
spatial structure of the flow. Some structure could be detected close to the
ends but the center of the facility appeared featureless for a wide range of
parameters (see Figure 3.4).
LDV measurements in this same range of
parameters revealed a variety of behaviors.
Thus, CP2 could not be
determined visually, and it was decided to repeat the procedure following the
same variations of the parameters using LDV to determine the flow behavior.
A velocity signal obtained by placing the measuring volume of the
velocimeter at the center (in z) of the facility was fed to a spectrum analyzer.
The type of flow was determined from the spectrum. Table 1.1 shows that at
the critical point an m=2 flow has a frequency of 0.748 and an m=3 flow a
frequency of 0.905. Spirals near the critical point should have frequencies
close to these values.
Thus, a peak in the spectrum close to 0.9 was
considered an m=3 flow, one close to 0.75 as an m=2 flow.
At Ro=-250, as Ri was increased from rest, a first transition to an m=3
flow occurred. However, after some time, a second peak corresponding to an
m =2 flow also appeared.
The second peak was unstable, appearing and
disappearing with time, probably due to mode competition between a periodic
flow (with m=3) and a doubly-periodic flow (with m=2 and 3). This point was
chosen as CP2. The critical inner Reynolds number Ric which corresponds to
this point is the same as predicted, Le., 200.1. However, Wc was found to be
26

---

.....'..:,~:;;~.~' :., .
. .,:;#i;::~
.. 1-;';...
•
"-i¥
.::'!~::.;
~ - -,~
Figure 3.4 Flow visualization cannot resolve flows near CP2
27

---

-1.312 (Ra = -350) instead of -1.1559 (Ra = -308.4), a value about 12% higher
than expected. Note that consistent with the prediction by COl, no periodic
flow with m=2 was observed.
The stability regions were established by following circular paths in the
parameter space and determining the velocity spectrum at each point with the
spectrum analyzer and with the IL8 routines. It should be pointed out that the
characteristics of the flows have been determined by using the frequency
content of the LDV signal and not by visual observation; thus, their spatial
pattern is not known.
This is important because, even though frequent
reference is made here to azimuthal wavenumbers, they were not measured.
The relationship between the measured frequency and the corresponding
azimuthal wavenumber was made using Table 1.1.
In this region of parameter space, the instability occurs in a small region
very close to the inner cylinder. When the flow is illuminated from outside,the
light is lost due to reflections on the glass walls, and absorption and scattering
by the Kalliroscope.
New kinds of flow visualization experiments should be
designed to solve this problem. Perhaps illumination or injection of dye from
the inner cylinder will help to identify the spatial structure.
L~0.05
Figure 3.5a shows the results obtained at distances L ~ 0.05 from CP2
in (~,v) space; Figure 3.5b shows those obtained for 0.05 < L ~ 0.10. Note that
lines band c are the same in both diagrams. CF is stable between a and b.
Following a circular path in the counter-clockwise direction, CF undergoes
transition to a periodic flow on b with a frequency close to 0.9.
This was
presumed to be an 82 flow. Figure 3.6a shows an m=3 flow with frequency
0.931. On c in Figure 3.5a, the flow goes to a doubly-periodic flow whose
28

---

O~D51
0.1
'\\
" \\
~1
J
spectra
SI12
I
V
m=2
e
broadband
a,~~
m=2,3
0.025 /-
''-¥
I
0.05
"""
"
/
,,
I,
, I
, ,
OI-
-
I
"
' -
I
I
01-
,,-:.- - _..
CP2':: - - _
I\\)
I
\\
...........
,. ~--- I
I
\\
" ...
CD
-{I.025 t-
"'-.......
.-./
0,
I l l - -
I
-0.05
!
CF
CF
.......
I
I
I
I
!
I
-{I·~t05
-0.025
0
0.025
0.05
~·Jo.1
-0.05
0
0.05
0.1
(a)
11
(b)
11
Figure 3.5 Experimental stability diagram for CP2: a) 0.0 :; L:; 0.05;
b) 0.05:; L:; 0.10

---

60
dE)
30
o
O.I
60 r---~--.----r----.----.-------"''----'-------,
(b)
c1B
30
o·
10
I
Figure 3.6 a) Periodic flow, 1-l=0.045, v=-0.040, 0 52=0.931;
b) Doubly-periodic flow,I-l=0.08, v=O.O, 0 5 1=0.780, 0 5 2=0.923;
c) Doubly-periodic flow with broadening, L ~ 0.05
30

---

frequencies correspond to SI 12 with m=2 and m=3. Figure 3.6b shows the
spectrum of a doubly-periodic flow with frequencies 0.780 and 0.923.
For
circular paths with small radius (L::; 0.05), this doubly-periodic flow persists all
around the circular path until it goes back to CF on line a (see Figure 3.5a).
The only change along the circular path is that, as Figure 3.6c shows, the
background level of noise increases in the first quadrant of the stability
diagram. This increase in the background level of noise is not understood.
0.05 < L ::; 0.10
For larger radii (0.05 ::; L ::; 0.10) near (below) line d in Figure 3.5b,
sometimes a third frequency appears in the spectrum; its value is very close to
the difference (Qs2-Qs1). When Q s3 is calculated from equation 1.3, its value is
very close to (Q s2-Q S1) and cannot be distinguished from this difference in
these experiments. It is not known if this is a triply-periodic flow or not.
As the circular path is continued in Figure 3.5b, Q s1 and Q s2 get closer
to each other and the flow is not stationary. The value of the frequency Q s2
shifts with time. When flows are not stationary, instead of obtaining a long-
time average spectrum, it is convenient to study the changes in the spectrum
with time. To do so, the acquired time series is divided into smaller intervals
and the spectrum of each interval is calculated. Figures 3.7a and 3.7b show
such an analysis for a time series 15365 sec long (12288 points).
The
spectrum in Figure 3.7a was obtained from the first 256 sec (2048 points).
Note that Q s1 =0.784 and Q s2=0.943. Figure 3.7b shows the spectrum from
768 to 1024 sec. Q s1 remains the same, but Q s2 is now 0.961. The spectrum
calculated from the last 256 sec (not shown) has the same frequencies as
Figure 3.7a.
31

---

60
(a)
dB
30
o
-30
10 .
4S
(b)
30
dB
lS
o
-1 S
-30 L..ILJILL.L..I'---'-LULIIJ.&.
n·
10
I
Figure 3.7 Evolution of the spectrum with time: a) from t=O to t=256 sec, Us2=0.943;
b) from t=768 to t=1 024 sec, o.s2=0.961
32

---

In the region of parameter space where this "shifting" of the Qs2 peak
occurs, a broadband component at a very low frequency (=0.005 Hz) appears
in the spectrum. The spectrum is calculated after subtraction of the mean so
that the dc component cannot be the origin of the peak. The source of the
slow nonperiodic amplitude variation of the velocity signal is unknown. The
signal seems to "ride" on a slow frequency and also have a slow amplitude
modulation.
Between lines d and e in Figure 3.5b, the velocity spectrum of the flows
has broadband peaks centered at Qs1 and 2Qs1. Sometimes there are other
broadband peaks centered at higher harmonics, and the level- of the
background noise increases (Figures 3.8a, b and c). On line e (Figure 3.5b),
the peaks become narrow and the flow appears to be periodic with m =2.
However, the slow nonperiodic amplitude variation does not disappear
(Figures 3.9a and 3.9b) until the path reaches line a (Figure 3.5b), where the
flow goes back to CF. It is important to note that the velocity signals close to
CP1 did not show this slow nonperiodic amplitude variation.
The slopes of the lines in the stability diagram are shown in Table 3.2.
As for CP1, the lines indicate approximately the transition boundaries. The
only lines that can be compared with the theory are band c because a, d and
e occur in regions of (J.!.,v) space inaccessible to the theory. The locations of
the transitions to S2 and S12 are very different from the predicted values.
line
a
b
C
d
e
slope
-1.25
-0.82
-0.29
0.07
9
Table 3.2. Slopes in the experimental stability diagrams about CP2.
33

---

60 . . . . . . - - - - - - - - - - - - - - - - - - - - - 1
dB
(a)
30
Q.
10
I
60
dB
(b)
30
o
Q.
10
I
Figure 3.8 Broadband spectra: a) J..l=O.05, v=O.033; b) J..l=O.04, v=O.045;
c) J..l=O.02, v=O.0562
34

---

6~----r--........---~----,---..,....--.,....-----,--:-:--,
(a)
4
2
amplitude
o
-2
-6 L.._----L-_--l~ _
___J__
_ _ _ 1_ _J _ __
_ _ L_ _....L..__
___l
o
1/0.
230
I
60
(b)
dB
30
o
-30
Figure 3.9 Slow nonperiodic amplitude variation of an m=2 flow, J.l.=O.O, v=O.08.
0s1=O.756: a) Time series; b) Power spectrum
35

---

Clockwise direction
When circular paths are followed in the clockwise direction, the same
sequence of flows in the opposite order is observed.
There is a region
between v=O.O and 0.05 in Figure 3.5a (Ra = -330 and -350) where the first
transition is to a doubly-periodic flow. At distances larger than 0.05 from CP2
in (I1,V) space a first transition from Couette flow to m=2 spirals has been found
(Ri - 196, co - -1.263 and Ra - -330). Recall that the theory predicts that spirals
with m=2 are not stable close to CP2. Therefore, the region where m=2 spirals
have been observed experimentally is out of the domain of validity of the
theory. Spirals with m=2 appear closer to the CP1 when clockwise paths in
(I1,V) space than when counter-clockwise paths are followed. The fact that a
region where a doubly-periodic flow exists as a first transition implies that
another multicritical point must exist, between CP1 and CP2, which has a
quasi-periodic flow on one side and a periodic flow with m=2 on the other.
This multicritical point is not mentioned in the theory.
It would be interesting to study the variation of the location of the critical
points with aspect ratio to show whether end effects can be the source of the
discrepancy between the predicted and the experimental critical parameters.
At the same time, care must be taken because large scale modulations due to
the simultaneous excitation of several axial modes may occur at high aspect
ratio (a continuum of wavenumbers is excited for infinite cylinders). This effect,
which may result in complicated "slow dynamics" (Iooss, Coullet & Oemay
1986), was not taken into account by COl or in this work.
Far away from the critical point, spirals with m=3 (11=0.217 and v=0.158)
and interpenetrating spirals with m=2 and 3 (11=0.217 and v=0.128) can be
visualized. Figure 3.10 shows photographs of such flows.
36

---

(a)
(b)
Figure3.10 Photographs of flows far enough from CP2 to be resolved
visually: a) m=3, 1l-=0.217, v=0.156; b) m=2 and 3,
1l-=0.217, v=0.103
37

---

Summary
Some of the qualitative behavior described by COl has been confirmed
by the experiments: the transitions to spirals (with m=3) and interpenetrating
spirals (with m=2 and m=3), and the absence of spirals with m=2 very close to
CP2. The locations of CP2 and of the transitions occur at values different than
predicted.
It was also found that there is a region in which doubly-periodic
flows are not stationary. In addition, a first transition to a doubly-periodic flow
and a complicated behavior identified by broadband spectra have been found
experimentally in the region of (Il,v) space inaccessible to the theory.
This complicated behavior was analyzed in more detail using some
diagnostics from nonlinear dynamics and will be described in the next section.
3.3
Topological Characteristics
The latest developments in nonlinear dynamics have influenced not
only the theoretical approach to Taylor-Couette flow, but also the experimental
analysis. Traditional methods of diagnostic, such as the power spectrum, are
excellent to determine whether a flow is periodic or quasi-periodic. However,
the occurrence of broadband spectra could indicate either deterministic chaos
or random behavior.
It is now common to study experimentally the asymptotic behavior of
dissipative systems by reconstruction of the phase portrait from the
measurement of one variable.
Then, different new diagnostics like the
measurement of the dimension 8 and largest Lyapunov exponent are used to
characterize the attractor.
All these concepts are defined and their
applications to experimental data are described in Appendix D.
From a nonlinear dynamics approach, the broadening of the spectrum
very close to the critical point was a surprise. Thus, it was decided to study
38

---

this region of parameter space in more detail in order to investigate chaotic
behavior.
3.3.1
Testing of the Diagnostics
The dimensionality of the flows was determined using both a
Grassberger & Procaccia algorithm for the correlation dimension and a Badii &
Politi nearest-neighbor method (following Kostelich & Swinney 1987).
The
largest Lyapunov exponent was calculated following Wolf et al. (1985). (Only
the largest exponent was calculated. Hereafter, the term Lyapunov exponent
will refer to the largest.) The decay of the velocity power spectrum ·at high
frequencies, as suggested by Sigeti and Horsthemke (1987) has been
considered also in the analysis. For a detailed description of these methods,
see Appendix D. All methods were tested first on well-defined quasi-periodic
time series and in one well known experimental situation.
The correlation dimension in Figure 3.11 a was calculated from the time
series A=(cos ~t + cos 2...)2 ~t). The dimension in Figure 3.11 b from B=(cos ~t
+ cos 0.809524 ~t). (The latter function was chosen from a typical doubly-
periodic flow in the experimental data.) Note that in Figure 3.11 a, there is a
scaling region and the calculated dimension is 2. In Figure 3.11 b, there is no
scaling. The algorithm fails to find a scaling region, even in the absence of
noise, when the two frequencies are very close to each other. The correlation
dimension was used in this investigation, in spite of the negative result
described above, because when there is scaling, additional information about
the attractor can be obtained (see Appendix D).
The Lyapunov exponents
calculated for both time series is :=0.
39

---

8
6
dim
4
2
0
4.5
5.0
5.5
6.0
6.5
7.0
8.5
log k
3.0
2.5
(b)
2.0
dim
1.5
1.0
0.5
0.0
4
5
6
7
8
9
10
11
12
13
logk
Figure 3.11 a) Correlation dimension for the function (cos t + cos 2 "J2t);
b) Correlation dimension for the function (cos t + cos 0.809524t)
(32,768 points were used in all three cases)
40

---

WVF
The first experimental test was done using a velocity time series from a
wavy vortex flow. The correlation dimension in this simple set-up was 1.08.
As the flow situation becomes more complicated (the outer cylinder rotating
and the unstable region closer to the inner cylinder), the noise level in the
signal increases and the value of the calculated dimension is farther from the
integer values expected for periodic and doubly-periodic flows.
In such
situations, the value of the dimension should be considered only as an
indicator of the high- or low-dimensionality of the flow, and not as a measure
of its fractal nature (see Appendix D).
In most experimental signals, a non-
integer value of the dimension is measured as a consequence of instrument
noise; this does not necessarily imply that the flow is chaotic.
To calculate the dimension, the attractor is reconstructed from the
measured time series.
The choice of the embedding dimension for the
reconstruction and the number of points in the data file are important factors in
the dimension calculation.
Besides noise, embedding dimension, length of
the time series and evolution time, prior knowledge of the scaling regions is
required for the calculation of the Lyapunov exponent (see Appendix D).
When IJsing numerical data or when the systems have only two or three
dimensions, this information is easily obtained from the correlation dimension
and/or the phase portraits. However, for higher dimensions and/or very noisy
time series, phase portraits obtained from the reconstruction are "messy", and
the correlation dimension does not always show scaling.
It has been
observed that in those cases when scaling occurs in the correlation
calculation, the largest scaling region of the attractor is always between 0.75
and 1.5 of the standard deviation of the time series. This criterion was used in
the calculation of the Lyapunov exponent when no other was available.
No
41

---

equivalent criterion has been found for the smallest scale. Its choice remains
relatively arbitrary, and several values usually need to be tested.
Small
variations in each of these input parameters can give relatively large changes
in the value of the calculated exponent. The exact value should be regarded
with skepticism.
However, when the exponent is ::0.1 or higher, it may be
considered as an indication that neighboring trajectories indeed diverge
exponentially in phase space.
From these tests it was concluded that to characterize a flow both
topological characteristics (dimension and largest Lyapunov exponent) and
the spectrum (width of the peaks and decay at high frequencies) should be
taken into account. No single characteristic is sufficient to fully describe the
flow.
3.3.2
CP1
Only one flow close to CP1 was analyzed.
Brandstater & Swinney
suggest that the number of points required for the calculation of dimension is
of the order of (110.03)0. For Ibis periodic flow a data file with 12,288 points
was used. Figure 3.12a shows the correlation dimension for the m=2 'flow in
Figure 3.3b, and Figure 3.12b shows the nearest-neighbor dimension.
As
expected, this one frequency flow has a dimension very close (but not equal)
to 1 and a Lyapunov exponent ::0.
3.3.3
CP2
Similar results to those described above should be expected for
periodic flows near CP2. Thus, they were not analyzed.
42

---

6
(a)
,
:::I
4
dim
3
2
1
0
3
5
6
7
8
9
10
11
12
logk
8r---------------------~
7
(b)
B
5
dim
2
30
70
110
150
190
230
270
310
order
Figure 3.12 Periodic flow: a) Correlation dimension 0=1.2;
b) Nearest-neighbor dimension &=1.3
43

---

Quasi-periodic -flows
Figure 3.13 shows the dimension calculation for a doubly-periodic flow
whose time series has 32,768 points. Note that even though the correlation
dimension does not show scaling, the trend of the slopes seems to indicate a
dimension slightly under 2.
The nearest-neighbor dimension saturates
between 2 and 3. The latter is probably a better indicator of the true dynamics
because there is always a slow nonperiodic amplitude variation of the flow
that should affect the calculation.
Spectrally broadband flows
Figure 3.14 shows the dimension calculation for the flow in Figure 3.8a.
The dimension calculated with the correlation algorithm is close to 2; the one
using the nearest-neighbor method is approximately 3.
The maximum and
minimum scales necessary for the calculation of the largest Lyapunov
exponent were chosen from Figure 3.14a; the exponent was found to be =0.2.
Note that even though the peaks are broad, there is no increase in the
background level of noise.
Figure 3.15 shows a flow for which both techniques fail to calculate the
dimension. This is not surprising because the dimension of this flow is larger
than 4.
A data file with 106 points would be necessary for an attractor of
dimension 4.
Thus, even the largest experimental data files with 131,072
points are not sufficient to determine such a high dimension.
The largest
embedding dimension used in the calculation was 10 for the correlation
dimension and 8 for the nearest-neighbor dimension, not large enough
according to Taken's embedding theorem.
For this particular flow, the
spectrum at high frequencies shows an exponential decay as suggested by
Sigeti & Horsthemke (see Appendix D). Because the choice of scales for the
44

---

g
(a)
7
6
5
dim
4
3
2
1
0
3
4
5
7
8
9
10
11
12
log k
e . - - - - - - - - - - - - - - - - - - - - - - - - - - ,
(b)
7
6
5
dim
4
3
2
OL-_----l_ _---L._ _-..L_ _-l-_ _....l....-_ _....L.-_ _.l....-_ _
30
70
110
150
190
230
270
310
order
Figure 3.13 Two-frequency flow: a) Correlation dimension, no scaling;
b) Nearest-neighbor dimension 0=2.8
45

---

6
(a)
5
...
dim
3
2
1
0 L----1...--.L--..l..------:-7--....I.- - - :---::O::--""---:1~1-~12
3
5
8
9
1
logk
(b)
1
6
5
dim
4
s
- - - - -
2
30
70
1 10
ISO
1 Q D
230
270
300
order
Figure 3.14 Beginning of spectral broadening, dimension of flow in Fig. 5.6a:
a) Correlation dimension &=1.9; b) Nearest-neighbor dimension 0=3.2
46

---

, ?
(a)
6
5
4
dim.
:3
2
1
0
3
4
5
6
7
8
9
10
11
12
log k
8
7
(b)
~
6 ~
-
5 ~- -
- --
dim.
4 ~
3~
2 ~
1~
0
30
70
110
150
190
230
270
310
order
Figure 3.15 Dimension of flow in Fig. 5.Gb: a) Correlation dimension, no scaling;
b) Nearest-neighbor dimension, no saturation
47

---

largest Lyapunov exponent calculation could not be obtained either from the
correlation dimension or from the phase portraits, the criterion described in
Section 3.3.1 was used.
The smallest scale was chosen arbitrarily, but
several values were tested. The calculated Lyapunov exponent was between
0.5 and 0.9.
The time series was acquired using a 12 bit A/D converter.
If the
Lyapunov exponent is 0.9 bits/sec, this means that all the information is lost
after approximately 13 seconds. The main frequency of this flow is £2s 1=1.1
Hz; it will take =14 cycles to lose all the initial information.
Figure 3.16 shows a flow with a lower dimension. Both techniques give
a value of approximately 4. The number of points in this data file is 131,072;
the number required for a four dimensional attractor is -106. The embedding
dimension used in the correlation dimension calculation was 10; in the
nearest-neighbor dimension calculation, 8 was used.
From the Takens'
theorem the embedding dimension should be approximately 9.
[Taken's
requirement of the embedding dimension for the reconstruction gives a
theoretical upper-limit guarantee.
Scaling has often been observed at
embedding dimensions smaller than 28+1]. The largest Lyapunov exponent
of this flow is about 0.4.
Three different types of broad band spectra have been analyzed. In
two cases (Figure 3.8a and Figure 3.8c) a low dimension and a positive
Lyapunov exponent have been calculated from the corresponding time series.
Nothing can be said about the decay of the spectrum at high frequencies
because it is below instrument noise. Nevertheless it can be concluded that
the flows are chaotic. In the third case (Figure 3.8b), the dimension could not
be calculated. However, the Lyapunov exponent is positive and the decay of
48

---

(a)
dim
5
6
7
8
9
10
11
logk
fl
(b)
7
6
5
dim
4
3
:2
30
70
110
150
190
230
270
310
order
Figure 3.16 Dimension of flow in Fig. 3.8e: a) Correlation dimension, no scaling;
b) NeareSl-neighbor dimension &=3.9
49

---

the power spectrum at high frequencies is exponential. These observations
indicate that the flow is probably chaotic of dimension higher than 4.
Spirals with m=2
The power spectrum of the m=2 flows found at larger radii from CP2
has sharp peaks at Q51 and 2Q51, and a slightly broadband peak very close to
zero. This last peak comes from the nonperiodic amplitude variation of the
signal which, very loosely speaking, could be understood as a chaotic
modulation of the flow (see Figures 3.9b and 3.17). The dimension is ::1.8, the
Lyapunovexponent ::0.
3.4
Summary
The stability of flows in the neighborhood of two multicritical points has
been
established
experimentally.
Qualitative
agreeme nt between
experiments and COl has been observed in the vicinity of CP1.
However,
spirals with m=2 have been found to be stable in a smaller region of
parameter space than predicted and an unexpected hysteresis in the primary
transition was observed.
In the neighborhood of CP2, in addition to the first two transitions
predicted by the theory, a region of chaotic behavior has been observed.
Flows in this region have a broadband spectrum, are low dimensional and
have a positive Lyapunov exponent.
Also, experiments have been able to
describe the stability of flows in regions of parameter space inaccessible to the
theory.
The nonperiodic amplitude modulation that is present everywhere
around CP2 has not be explained.
50

---

8
7
5
5
dim
4
scaling
3
region
2
1
0 5
6
7
8
9
10
11
12
log k
Figure 3.17 Correlation dimension of the flow of Fig. 3.7,0=1.8
51

---

CHAPTER 4
SUMMARY AND
CONCLUSIONS
4.1
Summary
The stability of flows in counter-rotating Taylor-Couette flow has been
studied experimentally using laser-Doppler velocimetry and flow visualization.
Studies were focused on the neighborhood (in parameter space) of two
multicritical points, CP1 and CP2, at which Couette flow loses stability to two
nonaxisymmetric modes simultaneously.
Findings were compared with the
theoretical predictions by COl.
Two versatile and precise facilities were designed in which various
regions of parameter space can be studied.
The mode interaction
experiments were carried out using a radius ratio 11=0.75. The height of the
fluid column was chosen to allow twenty axial wavelengths (calculated from
the predicted most unstable wavenumber). This resulted in r=32.5 for CP1
and r=29 for CP2.
In addition to the spectral analysis of the velocity signals, the calculation
of dimension and largest Lyapunov exponent of the different flows was used to
investigate chaotic behavior.
To complement the experimental studies, the predictions by CDI were
extended by solving numerically the reduced amplitude equations that
describe the stability of the flows in the neighborhood of the multicritical points.
These results are presented in Appendix A.
During the course of this investigation, various regions of parameter
space other than those in which mode interaction occurs have been studied;
the results are presented in Appendix B.
52

---

4.2
Conclusions
4.2.1
CP1
Experimental observations near CP1 are in close qualitative agreement
with the theoretical predictions by COl. However, there are some quantitative
disagreements: i) the hysteresis in the transition between the two spiral flows
due to the "jumps" between two different branches of solutions was detected in
a much smaller region of parameter space than predicted; and ii) hysteresis
was observed in the primary transition from Couette How to spirals.
The quantitative discrepancies in the primary transitions might be
explained by various factors: (i) a subjective element due to the gradual
changes in the transitions from Couette flow to spirals; (ii) the possible effect of
variations in the axial wavenumber along the fluid column on the value of the
critical parameters at transition; and (iii) the possibility that the transition does
not correspond to a Hopf bifurcation in the case of finite cylinders. End effects
are probably the root source of (i) and (ii) also.
The good agreement between the experimental and predicted values of
the most unstable wavenumber show that the choice of a periodic (height is
chosen as a multiple of the number of cells) but finite boundary condition in
the theory is reasonable.
The m=1, m=2 interaction seems to determine flows up to a radius of
0.12 from CP1 in (Jl,v) space. At this distance, TVF (not a spiral flow) is stable.
4.2.2
CP2
The location of CP2 in the parameter space was established
experimentally at a value of Wc approximately 12% higher than predicted. The
value of Ric was as predicted.
Flows were determined by their frequency
composition which was then related to the azimuthal wavenumber using Table
53

---

1.1.
That is, the true spatial structure of the now is not known and the
azimuthal wavenumbers were deduced rather than observed. The fact that
flow visualization could not resolve the spatial structure of the flows near CP2
is a surprising result for the experimentalist used to working in TCF; flows have
always been visualized. This result is probably due to the loss of light intensity
by reflection and scattering before the instability is illuminated.
It was confirmed that no spirals with m=2 are stable very close to CP2
[0::;L::;0.05 in (Il,v) space]. Instead, a first transition to a doubly-periodic flow
with m=2 and m=3 was detected. Spirals with m=2 are stable at distances
larger than 0.05 from CP2. This observation implies the existence of -another
multicritical point between CP1 and CP2 at which interpenetrating spirals (with
m=2 and m=3) and spirals (with m=2) interact.
To analyze and characterize the flows, the dimension, largest Lyapunov
exponent and frequency power spectrum were calculated.
It was confirmed
that the Lyapunov exponent and the dimension calculations are extremely
sensitive to input parameters which cannot always be chosen in a perfectly
objective manner. In addition, it was found that the correlation dimension has
difficulty identifying two torii whose two frequencies are very close to each
other, as is the case in the study of CP2. The calculation of dimension was
used only as an indicator of the high- or low-dimensionality of the flow, not as
a measure of its fractal nature. A positive Lyapunov exponent (=0.1 bits/sec or
larger) was used as an indicator of exponential divergence of two nearby
trajectories in state space. No single diagnostics; neither spectrum, dimension
nor largest Lyapunov exponent was found to be sufficient, in the analysis of
experimental data, to characterize fully the flow.
In addition to the first two transitions expected from COl, a region of
chaos was found experimentally. Some of the flows with broadband spectrum
54

---

have low dimension (=4) and a positive Lyapunov exponent.
However,
nothing can be said about the decay of their power spectrum at high
frequencies because it is "hidden" by instrument noise. In other broadband
flows, no dimension could be measured but their Lyapunov exponent is
positive and the power spectrum has an exponential decay at high
frequencies. All these flows were considered chaotic in this investigation.
Two different transition sequences were observed as circular paths
were followed around CP2.
For 0 ~ L ~ 0.05, after the quasi-periodic
transition, the two main frequencies could always be identified even in the
region of parameter space where the background level of noise increased.
For 0.05 ~ L ~ 0.10, the transition after quasi-periodicity is to flows whose
broadband peaks are centered at .0. 5 1.
The latter series of transitions
CF~S2~SI12~chaos seems to be a Ruelle-Takens-Newhouse sequence to
chaos.
In the experiments, a nonperiodic amplitude variation of the time series
was found to be present in most of the domain around CP2. Also, small shifts
in the frequency value of .0.52 in the doubly-periodic flows have been detected.
Neither one of these phenomena is understood.
To complement the experimental data, the COl equations that describe
the stability of flows close to the multicritical point have been solved
numerically. (The results and discussions are presented in Appendix A.) The
main conclusion from these calculations is that the amplitude equations seem
to describe adequately the quasi-periodic transition to chaos that was
observed experimentally.
55

---

4.2.3
General Conclusions
In this work the validity of the analytical solutions found by COl has
been confirmed. The limitations of the theory and the disagreements between
theoretical predictions and experiment have also surfaced.
In addition, the
experiments have given a comprehensive characterization of the dynamics
around the critical points, describing behavior that cannot be explained by the
theoretical model. It has been shown that the application of techniques from
nonlinear dynamics to experimental data can provide useful qualitative
information about the stability and asymptotic behavior of flows.
4.3
Recommendations for Further Investigation
From the results and conclusions of these investigation the following
extensions are suggested:
1) Better flow visualization techniques that would allow cross-sectional views
of the flows should be designed. In counter-rotating flows, the motion of the
fluid particle in the unstable region is difficult to see when flows are illuminated
from the outside because the instabilities occur very close to the inner
cylinder.
Illumination and/or dye injection from the inner cylinder should be
considered.
2) The
velocity field
should be mapped using
phase-locked
LDV
measurements in those cases for which flow visualization fails to resolve the
spatial structure of the flow.
3) The effect of r in the location of the multicritical points and in the
nonperiodic amplitude variation of the flows should be investigated.
This
study will reveal whether end effects are the main source of discrepancy
between theoretical predictions and experiments.
56

---

4) Higher embedding dimensions (=13) should be used in the dimension
calculations of the flows with broadband spectrum. Also, at least one test with
a time series with 106 points should be acquired and analyzed to con'firm the
dimensions found in this investigation.
57

---

BIBLIOGRAPHY
Andereck, C.D., Liu, S.S., Swinney, H.L.; J. Fluid Mech. 164(1986) p.155.
Benjamin, T.B.;Proc. R. Soc. London; A359(1978)p.1 Parts I and 11.
Brandstater, A. and Swinney H. L.,Phys. Rev. A35(1987)
Broze, J.G.; personal communication; 1988.
Chossat, P.; C. R. Ac. Sc. Paris 302(1986) p.624.
Chossat, P. and looss G.; Japan J. Appl. Math 2(1985) p.37.
Chossat, P., Demay V. and looss G.; Arch. Rat. Mech. Anal. 99(1987)p.213.
Cliffe, K.A.; report T.P.1060; Theoretical Physics Division, UKAEA Harwell
Laboratory; Oxford; U.K. ; 1988.
Coles, D.; J. Fluid Mech. 21(1965) p. 385.
Demay, V. and looss, G.; J. Mec. Theor. Appliq.; No. special (1984) p.183.
Di Prima, R.C. and Grannick R.C.; IUTAM Symp. on Instabllity of Continuous
Media (ed. H. Leipholz); Springer; 1971.
Di Prima, R.C. and Swinney, H.L.; Instabilities and Transitions in Flow
Between Concentric Rotating Cylinders; Hydrodynamic Instabilities and
the Transition to Turbulence (ed. H.L. Swinney and J.P. Gollub);
Springer-Verlag; 1981.
Drazin, P.G. and Reid, W.H.; Hydrodynamic Stability; Cambridge University
Press; 1982.
Golubitsky, M.; personal communication; 1988.
Golubitsky, M., Stewart, I. and Schaeffer, D.G.; Singularities and Groups in
Bifurcation Theory; volll, Springer-Verlag, N.V.; 1988.
Golubitsky, M. and Stewart, J.; SIAM J. Math. Anal. 17(1986) p.249.
Golubitsky, M. and Langford, W.; Pre-print UH/MD-25; Department of
Mathematics, University of Houston, 1988.
58

---

Grassberger, P. and Proccaccia, I.; Phys. Rev. Letters 50(1983) p.346.
Guckenheimer, J. and Holmes P.; Nonlinear Oscillations, Dynamical Systems
and Bifurcations of Vector Fields; Springer-Verlag, N.Y.; 1983.
Hirsch, M.W. and Sma.le, S.; Differential Equations, Dynamical Systems and
Linear Algebra; Academic Press; 1974.
looss, G.;J. Fluid Mech. 173(1986) p.273.
looss, G., Coullet, P. and Demay, Y.; Preprint No.89; Universite de Nice
(1986).
Kostelich, E. and Swinney H. L., Chaos and Related Nonlinear Phenomena
(ed. I. Proccaccia and M. Shapiro, Plenum, N.Y., 1987).
Langford, W., Tagg, R., Kostelich E.J., Swinney, H.L. and Golubitsky, M.; Phys.
Fluids 31 (1988) p. 776.
Lorenz, E.N.; J. of the Atmospheric Sciences 20(1963)p. 130.
Mullin, T.; J. Fluid. Mech. 121 (1982) p.207.
Mullin, T., Pfister G. and Lorenzen A; Phys. Fluids 25(1982) p.1134.
Schaeffer, D.G.; Proc. Camb. Phi!. Soc. 87(1980) p.307.
Sigeti, D. and Horsthemke, W.; Phys. Rev. A35(1988) p. 2276.
Snyder, H.A; J. Fluid Mech. 35(1969a) p.337.
Snyder, H.A; J. Fluid Mech. 35(1969b) p.273.
Streett, C.L. and Hussaini, M.Y.; AIAA 87-1444; 1987.
Takens, F.; Dynamical
Systems
and
Turbulence, Lecture Notes in
Mathematics; vol. 898 (ed. D.A Rand & L.S. Young, D Springer, NY,
1981 )
Taylor, G.I.; Phil. Trans. R. Soc. London 223A(1923) p.289.
Wolf, A, Swift, J.B., Swinney, H.L. and Vastano, J.; Physica 160(1985) p.285.
59

---

APPENDIX A
COl MODEL, NUMERICAL CALCULATION AND COMPARISON
WITH
EXPERIMENTS
A.1
COl Model
The stability of flows near multicritical points at which Couette flow loses
stability to two nonaxisymmetric wavenumbers simultaneously was studied
theoretically by COl. First, using the techniques of bifurcation with symmetry,
they obtained a set of reduced amplitude equations that describe the stability
of the flows in the neighborhood of these points; then they deduced many of
the possible solutions. Finally, they applied the theory to the particular case of
11=0.75 for which they studied the stability of CP1 and CP2. They calculated
the coefficients of the equations for each multicritical point and were able to
predict locally the stability and bifurcation diagrams. Their results led to the
theoretical predictions described in Chapter 1. However, they never solved
explicitly the amplitude equations.
In Section A.1, the theoretical approach of COl is summarized.
In
Section A.2 the results from the numerical solutions of the amplitude
equations near CP2 are presented.
In Section A.3 numerical and
experimental results are compared.
A.1.1
Equations of Motion
The mathematical study of multicritical points starts with the usual linear
analysis to determine the value of the control parameter at which the basic
flow loses stability.
Next, it applies a combination of techniques from
nonlinear dynamics and group theory. In COl, the analysis is focused on the
interaction between two nonaxisymmetric wavenumbers m1 and m2.
60

---

Consider a three-dimensional velocity perturbation .u. = (ur, Ue, uz) to the
basic flow Vc (eq. 1.1) with pressure perturbation q.
The perturbation
equations are then
yr·.\\J.=o,
(A.1 )
where all quantities have been nondimensionalized with d and Qj as length
and time scales.
11
1
The boundary conditions are u = ° at r = -
and r = - .
1-11
1-11
In an attempt to make a more realistic approximation of the finite length
of the system, periodic boundary conditions in the z direction were introduced,
with the height h of the fluid column fixed a priori. This limits the possible axial
wavenumbers a to a discrete set, and the wavenumber that is most unstable
for two different azimuthal modes simultaneously can be determined.
A.1.2
Symmetries
When periodic boundary conditions are imposed, a remarkable feature
of Couette flow is its symmetries.
It is invariant to translations along z, to
reflections with respect to the plane z=o (at mid-height) and to rotations about
the z axis. These properties are represented by the operators t s , Sand Rtl>
respectively. The action of the operators on the velocity field is given by
61

---

['ts!lJ (r, e, z) = U (r, e, Z + s)
[8ill (r, e, z) = ( Ur (r, e, -z), Us (r, e, -z), -Uz (r, e, -z) )
(A.2)
[R<jl ill (r, e, z) = U (r, e+ <1>, z)
with an analogous action on the pressure.
The equations for the perturbations inherit these symmetries; thus, eq.
(A.1) commute with 'ts, 8 and R<jl.
The transformations that leave a circle invariant are rotations by an
angle <1> and reflections along any line that goes through the center.
These
transformations form a group called 0(2).
When only the rotations are
considered, the transformations form a subgroup of 0(2) called 80(2).
Because of the assumption of periodic boundary conditions in the axial
direction, the action of 'ts and 8 together is isomorphic to 0(2). The action of
R<jl is isomorphic to 80(2). Then it can be said that the total symmetry group of
the problem is isomorphic to 0(2) x 80(2).
A.1.3
Conditions for mode interaction
The linear analysis starts by assuming that the perturbations are of the
form
.lJ. = y' (r)
exp [ i (az + me) + 01 ] ,
q = q' (r) exp [ i (az + me) + 01 ] ,
(A.3)
where m is an integer and where, to preserve the periodicity in h, 0: must have
the form
62

---

21tno
a=-h- ,
with no integer.
(AA)
If Lo is an operator representing the linear part of the equations, that is,
the Jacobian, and the perturbations can be expressed in terms of normal
modes as in (2.3), then the eigenvalue problem is of the form
Lo.l.!' = (j .I.!'.
(A.5)
Due to the rel'lectional symmetry, it must also be true that
Lo (S!,!') = (j (S!,!') .
(A.6)
The eigenvalues (j are thus double. This is typical of systems with symmetry
and is one of the main characteristics of the TCF problem.
A summary of the results by CDI follows. For given" and (0, the neutral
stability, Re (j =0, has to be determined. This curve is given by
(A.7)
The critical Reynolds number Ric is obtained from the minimum of this curve. It
is well known (see Di Prima & Grannick 1971; Demay & looss 1985), that for (0
sufficiently negative, m is different from zero at Ric, and that the corresponding
eigenvalue (jc is purely imaginary (Hopf bifurcation).
Because of the
symmetry, the eigenvalue is double.
In the particular case of mode interaction, it is required that Ric
correspond simultaneously to two different azimuthal wavenumbers m1 and
63

---

m2 but to the same axial wavenumber u. Because of the assumption of fixed h
and periodic boundary conditions, eq. (AA) is required.
Ric is then obtained from
(A.8)
Figure A.1 illustrates the procedure. Note that the condition (A.8) can
be required because u can have only discrete values.
If the minima of two
neutral stability curves (for different m) do not coincide, they can still
correspond to the same <Xc if they are within a distance 2rc/h from each other.
The multicritical point where m1 and m2 occur at the same time is given
by Ric and Wc- The value of u at this point is U c. To each m corresponds a
purely imaginary double eigenvalue.
In the numerical calculation by COl, m2=m1+1. The theory requires only
m2 >m1 with m1, m2 relatively prime.
(For a description of the numerical
calculations see Demay & looss 1987.)
A.1.4
Amplitude equations
So far, at Rj =Ric and W=Wc • there are by hypothesis two pairs of purely
imaginary eigenvalues 0': ±i1C1 and ±i1C2 (corresponding to m1 and m2
respectively). The associated eigenvectors for +i1C1 are
-'1-= y'(r) exp [ i (m1 e + uz)]
k =[Sy'(r)] exp [ i (m1 e- az) ] ;
for +i1C2' they are
(A.9)
~ = w'(r) exp [ i (m2 e+ az) ]
-'1-= [Sw'(r)] exp [ i (m2 e- uz) ] .
64

---

o
U c 1 U c2
U
IUc2 - Ue21 < 2rc/h
Figure A.1 Calculation of Ric from the neutral stability curves
65

---

The associated eigenvectors for -iK1 and -iK2 are the corresponding
complex conjugates. The eigenvectors corresponding to the eigenvalues of
the Jacobian are called modes. In a problem like this, several modes become
unstable simultaneously; hence the name mode interaction.
To simplify
notation, a change of variables is convenient
and
v =(l) -~.
(A.10)
Thus, at the critical point (ll-,v)=(O,O), La has four double- purely
imaginary eigenvalues, with all other eigenvalues having negative real part.
Close to (0,0), the total nonlinear problem with the appropriate boundary
conditions can be considered in an eight-dimensional center manifold.
A center manifold (CM) is an invariant manifold tangent to the center
eigenspace. [The center eigenspace is defined by the eigenvalues with zero
real part of the Jacobian. The manifolds tangent to the stable (Re (j < 0) and
unstable (Re (j > 0) eigenspaces are called stable and unstable manifolds
respectively.]
The behavior transverse to the center manifold is relatively
simple because it is determined by the exponentially contracting or expanding
flows of the stable and unstable manifolds.
The center manifold reduction
isolates the complicated behavior in a center manifold, thus reducing the
dimensionality of the problem but maintaining the dynamic characteristics of
the interesting behavior.
It can be shown that a locally attracting center manifold can be found for
the present problem. The solution to the nonlinear problem is
66

---

!J.= x + y
(A.11 )
with
X = X1'1+ X2~ + X3~ + Nt'1 + X1* '1*+ X2* ~* + X3* ~* + Nt* ~*,
(A.12)
and
=L ~p yq <I>pqR X1 r1 X1*s1 X2r2 X2"s2 X3 r3 X3*s3 X4r4 X4*s4
(A.13)
pqR
where ",," denotes complex conjugate, R=(r1, s1. r2. s2, r3, s3, r4, s4) and
<I>pqo=O, <I>oOR=O if I R 1=1.
As t---700, the dynamics of the original problem close to (0,0) is
asymptotically identical to that of the system,
dX
dt =F (Il,Y, X).
(A.14)
This last equation has been obtained by projection of the velocity field
on the center manifold.
The Taylor expansion of (A.13) can be obtained
explicitly close to (0,0) by identifying the terms IlP yq X1 r1 X1 *s1 X2r2 X2*s2 X3 r3
X3"S3 X4r4 X4"s4. For a description of this method, see Demay & looss (1985).
The symmetry properties are transmitted to eq. (A.14). Their action on
the eigenvectors is as follows
R<jl'tS'1 = exp [i (m1 <l> + as)] '1
R<jl'ts S2 = exp [ i ( m1 <l> - as )] S2
R<jl 'ts ~ = exp [ i ( m2 <l> + as )] '1
R<jl 'ts ~ = exp [ i ( m2 <l> - as )] S4
(A.15)
S'1 =~
S~=~
67

---

Equation (A.14) can be written in the form
j = 1,2,3,4.
(A.16)
The same symmetries that complicated the problem initially by forcing
eigenvalues to be double can be used to simplify the equations.
(For a
thorough understanding of these techniques, the reader is referred to
Golubitsky, Stewart & Schaeffer 1988.) One of the methods used to simplify
the analysis is called fixed-point subspace analysis.
In this technique, the
symmetry of the solutions is determined in advance, and a subspace of the
kernel of the Jacobian is associated with each symmetry. Then, the precise
structure of F restricted to those subspaces can be obtained.
With another
method called invariant theory, the Taylor expansion of the {fj} can be
calculated from eq. (A.15). It can be shown that the expansions are generated
by a finite number of terms of degree at most 2(m1+m2)+1 in x1, X1*, X2, X2*, x3,
X3*, x4, X4*.
These generators have been calculated by Chossat (1986).
Finally, eq. (A.16) can be written in the form
(A.17)
with a1 = a2, a3 = ~, and the qj given by
and
With rj2 = IXd2, the Pi are given by
P1( Jl,V, r1 2, r22,r32, r42) =iK1 + U1 Jl + P1 V + b1 r1 2 + C1 r22 + d1 r32 + e1 r42 + '" ,
68

---

P2( ~,V, r,2, r22,r32, r42) = P,( ~,v, r22, r,2,r42. r22),
P3( ~,v, r,2, r22,r32, r42) =iK2 + a3~ + P3V + b3 r,2 + C3 r22 + d3 r32 + e3 r42 + ... ,
P4( ~,V, r,2, r22,r32, r42) = P3( ~,V, r22, r,2,r42, r22).
These amplitude equations describe the stability of the solutions near
(0,0) and are one of the most important results of COl.
A.1.5
Conclusions
Using the fixed-point subspace analysis described above, many
solutions to the problem were obtained by COl.
Four types of -primary
branches of solutions were found: "spirals" and "ribbons", each with two
different azimuthal wavenumbers. Both spirals and ribbons are periodic flows.
Spirals are waves that rotate azimuthally and travel axially, like a lead screw.
The azimuthal wavenumber m represents the "pitch" of the spiral. Ribbons are
rotating waves that preserve the "flip" symmetry; this implies that the
boundaries between cells has to be flat (Chossat & looss 1985). In this flow, m
represents the number of undulations around the circumference of the
cylinder.
Doubly-periodic secondary branches bifurcate from spirals and
ribbons in the form of interpenetrating spirals and superposed ribbons
respectively. Interpenetrating spirals can be of two types, with the two spirals
travelling in the same or in opposite directions. Superposed ribbons are also
of two types; one preserves the "flip" symmetry, the other does not. Triply-
periodic flows can be expected to bifurcate from the secondary branches.
The stability of all these types of solutions depends on the values of the
coefficients in the restricted equations. These coefficients were calculated by
COl for the two critical points CP1 and CP2 of Figure 1.1, for a radius ratio
T] =0. 75.
(The coefficients were determined numerically using the method
69

---

described in Demay & looss 1985.) Their results led to the theoretical
predictions described in section 1.3.
A.2
Numerical Calculations
COl derived the amplitude equations that describe the stability of tlows
around the critical points CP1 and CP2, calculated numerically the coefficients
of the Taylor expansion and determined the stability of the flows close to CP1
and CP2 using fixed-point subspace analysis.
However, they never solved
the amplitude equations. The equations are valid in a larger domain than the
one that can be attained using fixed-point subspaceanalysis.
To complement our experimental study of the stability of flows around
CP2, eq. (A.17) was integrated using a fourth order Runge-Kutta method and
double precision.
The method was first used by Golubitsky (1988); his
program was used in these calculations.
The author is grateful to Dr.
Golubitsky for his help and direction in this numerical work, and for allowing
her to use his program.
The initial conditions were chosen arbitrarily; the
same were used for all the calculations. The step size was chosen to obtain
70 points per orbit.
(This is known to be an appropriate choice for the
calculation of dimension from the time series.) The numerical integration uses
the coefficients calculated by COl.
The four complex quantities Xj=Vj+ iWj (j =1,2,3,4) in eq. (A.17) are
obtained through this integration.
Each Xj describes the behavior in a
subspace of the center manifold. The four subspaces corresponding to the
four Xj will be called Xj hereafter. The total velocity of the flow is given by a
linear combination of the (Xj, xt).
From eq. (A.12) it can be seen that the
imaginary parts cancel so that the four real parts Vj should be sufficient to
describe the temporal behavior of the flow.
70

---

A.2.1
Numerical Results
As in the experiments, circular paths were followed around CP2 in
parameter space. The objective of this study is to compare the sequence of
transitions and the characteristics of the flows obtained numerically with the
experimental results of Chapter 3.
The first series of results were calculated following a circular path of
radius 0.04. The stability diagram is given in Figure A.2. (For clarity only the
fourth quadrant is shown in Figure A.2a, an expanded section is shown in
figure A.2b.) Before 8, the solution is CF (the perturbation .l.! in eq. (A. 1) goes
to zero) and the trajectories in all four subspaces decay to fixed points. Figure
A.3 shows the phase portraits in each one of the four subspaces for
Jl=0.02750 and v=-0.02905. (The phase portrait is Vj vs. Wj.) On line 8 (see
Figure A.2) the fixed point in subspace X4 becomes unstable, and the solution
settles into a limit cycle. The trajectories in the three other subspaces go to
fixed points. Figure AA shows the phase portraits in X1 and X4, Figure A.5a
the time series of V4 and A.5b the corresponding spectrum. The frequency of
these solutions is n s2=0.907. From eqs. (A.9) and (A.12) it can be deduced
that these solutions correspond to spirals with m2. On C, the fixed point in X1
becomes unstable and the solution settles into a limit cycle with a frequency
ns1=0.753. Solutions in X2 and X3 still decay to fixed points; see Figure A.6.
The total solution, obtained from the superposition of v1, v2, V3, and V4, has two
frequencies (Figures A.7a and A.7b).
This solution corresponds to
interpenetrating spirals with m1 and m2 .
As the circular path is continued in parameter space, subspaces X2 and
X3 become unstable on line D and the phase portraits show also limit cycles
(Figure A.B). The trajectories in X2 have the same frequency as the
71

---

O~
I
-0.03 r
"
'< '<
'< '"
'<
I
v
V
-0.025
I
I
I
c
I
1-
_
-0.05
-0.04
.......
I\\)
8
-0.075
I
I
I
I
), I
-0. 1
-0.oa.03
0
0.025
0.05
- - -
-
0.04
J.1 0.05
J.1
(b)
(a)
Figure A.2 a) Numerical stability diagram around CP2; b) Expanded view
I

---

W1 10
W260Q
/
X2
80
I
/
X1
/
,~
!lOO
7ot--
,
50\\-
/
; "
/
~-
!lOl-
l
/
"
,
-«11-
/
3llO
./
/~
.-
JOI-
,
zoo
2D
/
/
10
100
0
-10
I
I
!
I
J
I
1
!
I
0
40
60
,fl(I
100
120
140
160
180
ZOO
Vo
20
40
60
80
100
L2lI
140
(a)
V1
(d)
V2
(",)
:r'
X3
w4!101
X4
""'.J
W3
/
1
0
JOG
-5D
2ID
-100
2DO
-1!10 ••
1!l1l
-lOO' .
100
-i!lO
!IO
-300 ••
0
-s~
-ss
0
50
100
1!l1l
2.00
l5U
0
50
''1bO
l!1O
2.00
eo
(c)
V3
(b)
V4
Figure A.3 Phase portraits, 11=0.0275, v=-0.029, trajectories in all four
subspaces go to a fixed point: a) X1 ; b) X2; c) X3; d) X4

---

~I
w1
X1 I
~
w4
!
I
100
~Or
;
I I
50
150 I-
I I
0
100 f-
r i - 5 0
I
J
50 [
I
I
-100
~
~
/
- 1 5 0 '
o
(
~~~~~)
/
'____.
-200
~'"
/
~.........
-50 I-
" " -
~
.
-250 -
-.....,-
-100 I
I
I
I
I
I
I
:
I
I
I
-300
=-60
-40
-20
0
20
40
60
80
100
120
140
-·150
v1
Figure AA Phase portraits, 1.1=0.028, v=-0,0286: a} X1 trajectories go to a fixed
point; b) X4 trajectories go to a limit cycle with frequenc .os2=0.907

---

150
(a)
100 l-
SO I-
.Amplitude
0 ,-
-50
-100
-1 SO
0
10
. 20
1/0·I
(b)
dB
)
10
Figure A.S Periodic flow, !.1=0.028, v=-0.028: a) Time series;
b) Power spectrum, 0s2=0.907
75

---

1IIlI
aoo
w1
X1
w2 k
X2
100
250--
!Ill
.-
ZOO
0
150
,
/'
-lIll
-100 f--
"-
. /
I
~'-l
.c9.!I!l!I!5
fixed
point
~
,
-1!iO L
1
r
I
1
I
v"
-1!10
-100
~.~70
~
0
50
100
150
200
20
~
60
110
(a)
v1
(b)
v2
0>
w~(
600
W4
-....J
Xli
~
«lO
I
~.
:~
j
zoo
J
0
1001"-
/
I
/
-200
50 f-
fIxed
...-
~
point
~.
-.«lO
or ~I-~! I I I I
-lIll.
° 20 40 ao 110 100 120 1<40 160 I
180
~
-..00
-aoo
0
200
(c)
V3
(d)
V4
Figure A.6 Phase portraits, 1.1=0.031, v=-0.0253: a) X1, limit cycle with
frequency 0.753; b) X2 fixed point; c) X3, fixed point;
d) X4, limit cycle with frequency 0s2=0.907

---

(a)
Amplitude
o
10
20
10
1/ rl .I
Figure A.7 Doubly-periodic.flow, Jl=O.031 0, v=-O.0253: a) Time series;
b) Power spectrum,Qs2=O.907,Qs1=O.753
77

---

W~l
300
(.A'"g,. '- ~
....->
W;ol'~
X1 ,
S
'.
X2,
\\
2llO
"',.
300
100
t l\\
i!5lI ,-
~ ~
I
2llO
-tOO
"
150 .-
-iOO
100 ,-
-300
!lO~ ~;
-
~-
....
\\ ..._~1-
0
C 3>
~4lO -DJ -200 -lOO 0
100
2llO
300
a
!lOO
-!IO'40'~
0
~ 40 &ll 1IO 100 l2ll 140 180
(a)
v1
(b)
V2
J!IID
500 - - - - - - - - - - - - - - -
W3
X3
W4
f
X4
'.J
•
'\\
400
( X ) :
\\
:
~
,J
100
100
0
S
-~
o
-200
1 )
-300
:
:1
,,/~/"
..... - --
[
- " . -
~ -15lI -100 -!Ill 0
!lO
100
l.!lO
2llO
i!SD
~ -.400 -~ -200 ..100 0 1110 2llO 100 •
(~V3
(d)
V4
Figure A.a Phase portraits, trajectories in all subspaces are limit cycles:
a) X1, (251=0.675; b) X2, (251=0.753; c) X3, (252=0.907;
d) X4, (252=0.907

---

trajectories in X1 but different amplitudes. Solutions in X3 and X4 also have
equal time dependence but different amplitudes.
Even though there was a
transition and now solutions in all four subspaces are periodic, no new
frequencies are detected (see the time series and the spectrum in Figure A.9).
This behavior, in which the frequency content in X2 is the same as in X1, and
in X3 the same as in X4, prevails in all the domain where the amplitude
equations are valid, and is a surprising result. It shows that the knowledge of
the time dependence in X1 and X4 is sufficient to describe the temporal
behavior of the flow.
The solutions take longer and longer to settle in a limit cycle as the
angle around CP2 is varied in the counter-clockwise direction (see Figure
A.10).
On E, solutions in all four subspace settles into two-torii. The second
frequency in each torus is different and roughly five times the value of the first
(see Figure A.11). The corresponding spectrum is shown in Figure A.12; all
the peaks in this Figure can be explained in terms of four frequencies and their
linear combinations (Qs1=0.754, Qs2=0.908, Qs3=3.938 and Qs4=4.095). This
is another surprising result because a triply-periodic solution was expected
instead of one with four frequencies and it is not clear to which solution this
would correspond to. The frequency values of Q s3 and Q s4 decrease as the
parameters are varied in the counterclockwise direction.
This can be
observed from the phase portraits in Figures A.13 and A.14, and their
corresponding spectra in Figures A.15a and A.15b. Only a few points were
used to trace the phase portraits in order to show clearly the "wrapping"
frequency in each torus. If many more points are used, the two-torii would be
covered. The four frequency regime remains stable between lines E
79

---

-
(a)
I-
n
l-
Ii
f
1\\
.Amplitude
11
~
o
10
20
30
1I [l.I
150
120
(b)
90
dB
60
30
0
-30
10
[l.I
Figure A.9 Doubly periodic flow, 1.1=0.03266, v=-0.02309: a) Time series;
b) Power spectrum, Qs2=0.907, Qs1 =0.753
80

---

w![
~Th_.
X1 I
.coo
w:~
X2 ,
/
:[rT
" I
/
250
Of-l'..n
<
200r
;/
:t
150
l~t
.p
-300
/ '
1IO-
~
:r
"""-/
I
I
I
I
,,50
I
c.
or
I
.~
!
I
I
I
=.coo
-
-300
-2110
-100
0
100
i!OO
300
!iOO
-!il!..4Q
-20
0
2Q
.co
IiO
III
100
120
1-010
1110
(a)
v1
(b)
v2
wl/JO
......
'\\3
!lOO I
X4
.- ----
CXl
300
«10
2!10
300
i!OO
)
200
1!IlI
I
100
100
../
o~ f
1IO
-100
0
.-iOO ••
....
-300
-UIO
-at
. C <
--
-iSI
\\
~
~
~ ~ -300 -200 -100 0
100
200
300
(c)
v3
(d)
v4
Figure A.1 0 Phase portraits, 11=0.03267, v=-0.02308, all subspaces have
limit cycles: a) X1, 051 =0.753; b) X2, 0s1=0.753; c) X3, Os2=0.907;
d) X4, 0s2=0.907

---

-£Sx_'\\
I
..coo
w:l
X1
w~~
X2
r · ,~../-
I
DI
2!0
:t
/
1CC1
/
.'
. 4 ~ I
501-
, ./
. /
- - t ' ~
-iIlIO
I
I
i
;
!
I
I
I
or
-'5lI,-iO
.'-lOO
··lIDO
-200
-100
° 100 2llO DI 400 SIlO
5~
-ill
0
2D
.w 60- Il)
100
120
1.(0
160
(a)
v1
(b)
v2
!!ID
w3
wr,
)3
~
.........
X4
]DO
.\\00
co
N
2SI
300 ,-
200
200 ,.
!!Ill
~
100
0
50
-100
°
--i'110
-5l
-300
-100
~
-!!Ill
-!lOO
~
~L...-..L-
_
-t!IO
-100
-!Ill
0
100
l!lO
o -.«lO ··300 -200 -lOO
0
100
200
3OO~- 500
(c)
v3
(d)
V4
Figure A.11 Phase portraits, 11=0.03268, v=-0.02307, all trajectories are
two-torii: a) X1, 051 =0.754,°53=3.938; b) X2; c); X3;
d) X4, ° 52=0.908,°54=4.095

---

(al
I-
.Amplitud e
o
10
20
3-0
1/0·I
Figure A.12 Four-frequency flow, Jl=0.03268, v=-0.02307: a) Time series;
b) Power spectrum, 053=3.938, 054=4.095
83

---

w1
r
X1 I
500
w'00 I
X3
400
_~
3
L
(
~ \\
~
300
"
300
200 I
(~
1/
I
200
iOO
Or
~
~ y
I
100
co
-iOO
~
I
/
I
\\
--->c
I
0
-200
-300 I
-
--A "V "j--J
I
-100
-400
-509500
Figure A.13 Phase portraits, 1l=O.0327, v=-O.02304, trajectories in all
subspaces go to two-torii: a) X1; b) X3

---

500
400
w1
X1
I
X3
W3
400
300
300
200
~
~
200
10: ~ ;
-100
~ 1000
a>
(J1
-200
~
-100
-300
-200
-400
-50.9500
-300
500
-250
-200 -150 -100
-50
0
50
100
150
200
250
v3
(b)
Figure A. 14 Phase portraits, ~=O.0328. v=-O.0230: a) X1 ; b) X3

---

Figure A.15 Power spectra: a) 11=0.04327, v=-0.02304;
b) 11=0.04328. v=-Q.02300
86

---

(~=0.03268, v=-0.02307 and Ri=204.10) and F at ~=0.03285, v=-0.02282 and
Ri=204.13.)
Starting on line F, a series of transitions to quasi-periodic regimes with
several frequencies begins. Figure A.16 shows the first signs of modulation in
X3. The modulation increases in Figures A.17 and A.18. The presence of
several discrete frequencies is evident from the increased number of sharp
peaks in the spectra of Figures A.19a, A.19b and A.19c.
Figures A.20 and
A.21 show another quasi-periodic flow. Note that from the time series alone it
would be impossible to determine the regularity of the flow, but this regularity
is apparent in the phase portraits and in the spectrum.
All these new
frequencies appear in a range of parameter space that corresponds to Ri
between 204.13 and 204.16. (It would be very difficult to set up an experiment
with the resolution required to detect any of these quasi-periodic flows with
four or more frequencies.)
So far, the flows have undergone transitions from CF--7Sr-7S112--7neW
doubly-periodic--7four-frequency flow --7 several-frequency flow.
Note in
Figure A.2 that the transitions get closer to each other as a circular path is
followed around CP2.
On G (~=0.0331 and v=-0.02246), the behavior becomes irregular, as
shown in Figure A.22.
The "irregularity" arises from the inability to predict
when the trajectory will follow a large or a small loop in the phase portrait.
Figure A.23a shows the corresponding time series and A.23b the spectrum.
From the time series alone, it would be impossible to differentiate between a
quasi-periodic regime with several frequencies (Figure A.21 a) and a
nonperiodic one (Figure A.23a).
The spectrum shows an increase in the
background level of noise but the two main frequencies remain clearly
87

---

400 I
X3
w3
I
300
200
200 I
100
0
0
en
en
-200
-100
-400
-200
-60P600
-309300
600
-200
-100
0
100
200
300
v1
(a)
(b)
V2
Figure A,16 Phase portraits, 11=0.03285, v=-0.02282, more than four
frequencies: a) X1; b) X3

---

4 0 0 r -
X1
W3
I
X3
300 [I
200
200
100
o
o
CD
<.D
-100
-200 ~-
-200
-400
-300
L
I
!
!
~O\\OO
-400
-200
0
200
400
600
-409400
··300
-200
-100
0
100
200
300
400
v1
(b)
v3
Figure A.17 Phase portraits. 11=0.04329. v=-0.02280, several modulations in
each subspace: a) X1; b) X3

---

BOO I
X1 I
800
w3
I
X3
w1
6001-
,......",.
A
I
600 ~,I
400 I-
~ \\"'l V '\\\\ /rVf --'
I
400 L
II
200
200
0
0
(0
-200
-200
0
-400
G
I -400
-600
-600
---l.
I
I
I
I
-809800
-80~800
0
200
400
600
800
-600
-400
-200
0
200
400
600
800
(a)
v1
(b)
V3
Figure A. 18 Phase portraits, 1.1.=0.3295, v=-0.02268, quasi-periodic regime
with several frequencies: a) X1; b) X3

---

150
120
(c)
dB
90
60
30
IJ
o
,
~
~
v
I
l~
-30
"
\\t
lr
n·
10
I
Figure A. 19 Spectra of quasi-periodic flows: a) J.1=O.03285, v=-O.02282;
b) J.1=O.0329, v=-O.02280; c) J.1=O.03295, v=-O.02268
91

---

800
800
X1
I
X3
w1
w3
600
600
400
400
L
200
200
I
Of-
~
~
I
-
to
-200 I
I'\\)
////I/ffA
II
-20: r
~ ~
I
-400
-400
-600 .
-600
-809800
-aoQaoo
600
--600
-400
-200
0
200
400
600
800
(a)
v~
(b)
v3
Figure A. 20 Phase portraits, 1l=0.0330, v=-0.0226,quasi-periodic behavior with
several frequencies: a) X1; b) X3

---

(a)
Amplitude
o
10
20
. 30
11 O.I
150
(b)
120 t-
90
dB
60
30
~
o -
V
\\...
~
J
'-J
'--
hJ U
I
-30
~
1
o.
10
I
Figure A.21 Quasi-periodic behavior. Jl=O.0330 v=-O.0226: a) Time series;
b) Power spectrum
93

---

BOO
800
w1
X1
w3
I
X3
600
600
400
400
200
200
0
0
CD
-200
-200
.f:>..
-400
-l~~
-400
-600
-600
/
-809800
·BO~800
BOO
'-600
-400
-200
0
200
400
600
BOO
.. -
(a)
Figure A.22 Phase portrait, 11==0.0331, Y==-0.02246, nonperiodic behavior:
a} X1; b) X3

---

.Amplitude
o
10
20
30
11 (1.I
Figure A.23 Nonperiodic behavior, Jl=O.0331, v=-O.02246: a) Time series;
b) Power spectnJm
95

---

identifiable. The spectrum of solutions in each subspace separately is also
broadband (see Figure A.24).
On H (Figure A.2), 11=0.0332 and v=-0.02231, the solutions become
regular again as the phase portraits in Figures A.25 to A.27 and their
corresponding spectra in Figure A.28 indicate.
Finally on I, the equation can no longer describe the behavior of the
flow and the solutions blow up.
To proceed any further in the theoretical
analysis, higher order terms in the Taylor expansion in eq. (A.17) would be
required.
So far it has been observed that as a circular path is followed in the
parameter space at a distance L=0.04 from CP2, there is a quasi-periodic
sequence of transitions. Then there are flows with broadband spectra. After
the broadening, as the circular path is continued, the flow reorganizes
following an inverse quasi-periodic sequence until line I. From line Ion, the
model can no longer describe the behavior of the flow.
When a circular path is followed at larger radii (L=0.08), the sequence
of transitions is the same as for 0.04, except at the point of spectral broadening
(on G). Figure A.29 is an example of the nonperiodic behavior observed for
11=0.0661 and v=-0.0450, and Figure A.30a shows the corresponding
spectrum. Note that, besides the increase in the background level of noise,
the broadband peaks are centered at ns1. (At L=0.04 two frequencies could
be identified always.)
The spectrum of each subspace separately is also
broad (see Figure A.30b).
After the broadening, the process is the same as for radius 0.04: the
level of noise starts to decrease and the spectrum consists of several sharp
peaks. The number of peaks continues to decrease until I, where the equation
is no longer valid and the solutions blow up.
96

---

Figure A.24 Spectrum of each subspace is also broad: a) X1; b) X3, X2
and X4 are like X1 and X3, respectively
97

---

W1 800
600
X1
w3
I
y;;;/\\/,/'//:~
X3
,4',. 'J A/., . .'I,.{
_
~
600
400
400
200
200
0
-200
CD
co
-400
-600 •..
I
I
I
-809800
J
I
I
I
I
-609
-600
-~OO
-200
0
200
400
600
BOO
600
-400
-200
O.
200
400
600
(a)
v1
(b)
V3
Figure A.25 Phase portraits, 11=0.0332, v=-0.02231, quasi-periodic behavior
in all subspaces: a) X1; b) X3

---

6 0 0 r - - - -
600 I
,
X3
w1
w3
(~,\\,"l"t'l , "
(-4.
....\\ ~rl$\\
400.f-
..
x1 I
',,\\
',,- ,\\.\\ '
\\
... ' I
400
,
., .. ,'
\\
\\,. ".,~\\".;\\<~\\\\\\. // /J1··
, " " " : ,'.',
/ ..,
..c"
/
200 l-
200
'. '""
"
'.
,11'. II //'/ '/"
'
/..../--:,'
/ - - - , , 'J'"
/ - .,
."
/ " /
,.......
.
'" .",
, , / --:::-
,/'
.//
..
.----- _."
_..
. / , / . - /
O~
o
--:::~--~a': o~
I
//"'/'-
... - : : . - - - - / /
r
,,' ~
/. .
'. '-
(D
/
(D
;"·'7~/.\\' .,~,'
-200 .....
-200
S-··_~-~· .~ '.
/',-/-1'/'/,'
\\':' \\"'-
"
Lj/;r'/.'> '>/;\\\\\\\\"\\\\~<\\ .
Lj-"" /
,'\\
,,'- ',\\
-400 f-
I
/ ' "
\\
' \\ \\ '. \\ : . ' . ,
\\-....;.
)
-400
~-../
L - -
'\\',
\\ ,
\\ .........
-
,
\\~\\\\..\\,.~
I
I
.~
I
I
-609
I
800
-600
-400
-200
0
200
AOO
600
-60~00
600
(a)
v1
v3
Figure A,26 Phase portraits, ~=O.0333, v=-O.02216, quasi-periodic behavior
wnh several frequencies: a) X1; b) X3

---

~i
_
•
600 i
i
W1
w3
<100
400
200
200
o
o >-
-200
.......
o
-200
o
-400
-400
-600
-BOllaoo
-600
-400
-200
0
200
400
6~0
-60~600
-400
-200
o
200
400
600
v1
v3
(b)
Figure A.27 Phase portraits. 1.1=0.04335 v=-0.0219. quasi-periodic behavior:
a) X1; b) X3

---

1SO r - - - - - - - - - - - - - - - - - - - - - - - - ,
(a)
120
90
dB
60 f-
n.I
1SO r - - - - - - - - - - - - - - - - - - - - - - - .
(b)
120
90
dB
60
30
o
-30 L...----l'---
--=::::::::::::::::.....---.l
10
n.I
150 r - - - - - - - - - - - - - - - - - - - - - - - .
120
(c)
90
dB
60
30
o
-30 '--_.1....-
~::..._---...:~~
n.
10
I
Figure A.28 Quasi-periodic behavior: a} ~=O.0332, v=-O.02231;
b} ~=O.0333, v=-O.02216; c} ~=O.0335, v=-O.0219
101

---

1000
- - - - - - - - - .
1000
w,
X1
w3
I
t"'1 ....
X3
800
BOO
600
600
400
400
200
200
0
0'-
.....
-200
-200
.....-
0
I'\\)
I
-400
I
-400
I
-soo
-600
-800
-800
-100~0
I
I
I
!
I
!
I -100q
-200
0
200
400
600
BOO
1000
-...000 -BOO --600 -400 -200
0
200
400
600
BOO
1000
" .
-- -
v3
Figure A.29 Phase portraits, Il=O.04661v=-O.0450, nonperiodic behavior,
L=O.08: a) X1; b) X3

---

Flgure A.30 Broad spectra for L==0.08; 11==0.0661 t v==-0.0450: a) X1 +X3;
b) X30nly
103

---

The slopes of the stability diagram for all the numerical calculations are
given in Table A.1 below.
The values of the slopes of lines B, C and I
coincide with those in the theoretical predictions, Table 1.3; the other values
were not calculated by COl.
line
B
C
o
E
F
G
H
slope
-1.0136 -0.8322
-0.7070 -0.7060
-0.694
-0.6786 -0.6720 -0.6301
Table A.1
Slopes of lines in the numerical stability diagram about CP2.
For completeness, some calculations were done at a radius of 0.02,
with results very similar to those obtained at 0.04.
Figure A.31 shows
nonperiodic behavior observed at L=0.02.
Notice that no triply periodic flow was found to be stable anywhere in
the region surrounding CP2.
There are two interesting results.
The first is on line 0 where
trajectories in the four subspaces settle into a limit cycle but only two
frequencies are different. The results are unexpected and it is not understood
if it corresponds to a phase shift from the interpenetrating spirals, a bifurcation
from them or a "jump" to another branch of solutions.
The second is the
unexpected transition to a four frequency flow. It is not clear yet to what kind of
flow this solution corresponds to.
Before these numerical results can be compared with the experiments,
the dimension and Lyapunov exponent of the solutions will be calculated.
104

---

500
~Oi
1
1
I
W3
W1
400
G -
300
300
200
200
100
100
o
o
......
o
CJl
-100
-200
-200
-300
-300
-400
-40P400
-300
-2Q';)' -100 =0
lll0
200
300
400
500
-5°PsOO
-400 -300
-200 -100
0
100
200
300
400
500
(a)
v1
(b)
v3
Figure A.31 Phase portraits, 11=0.01634, v=-0.01152, nonperiodic, L=0.02:
a) X1; b) X3

---

A.2.2
Topological Characteristics of the Solutions
The dimension and Lyapunov exponents of the numerical solutions
were calculated using the same techniques as used for the experimental data.
Figures A.32a and A.32b show the correlation and nearest neighbor
dimensions for the periodic solution.
Figures A.33a and A.33b Sl10W the
corresponding results from the doubly-periodic flow; the Lyapunov exponents
are =0. Notice that no scaling is observed in the correlation dimension of the
doubly periodic flow. As described earlier, this is often the case when the two
frequencies are very close to each other. The frequency around the attractor
is almost the same as the "wrapping frequency", and the technique does not
seem capable of identifying a two-torus. For the calculations of the Lyapunov
exponents, the maximum and minimum scales were chosen from the phase
portraits.
The dimension increases from two to about four, and then appears to
saturate at around 0=5 in the quasi-periodic regimes (Figures A.34a and
A.34b) and up to 0=6 in the chaotic regimes (Figure A.35). This means that in
the quasi-periodic regimes, most of the sharp peaks correspond to linear
combinations of five or six frequencies at the most.
As was the case with
experimental data, none of the numerical files has enough points to measure
dimensions of four or more. The value of the embedding dimensions was also
limited by the software. It is not surprising then that, at the highest dimensions,
no saturation was observed in the nearest neighbor calculations and no
scaling region was found in the correlation dimension.
The Lyapunov
exponent corresponding to the quasi-periodic regime in Figure A.32 is -0.05
(almost zero), while that of the nonperiodic flow in Figure A.35 is 0.42. Even
though this should have been expected theoretically, it is a positive result that
106

---

3.5
(a)
3.0
2.5
2.0
dim
1.5
1.0
0.5
0.0
-16
-14
-12
-10
-8
-6
-4
-2
log k
8
7
(b)
6
5
dim
4
3
2
o L------l.0-----l7-0--11l....0--1...L- --1---.l.9-0--23.L..O--2..J...70-~3:-:00
3
5 0
order
Figure A.32 Periodic flow: a) Correlation dimension;
b) Nearest-neighbor dimension
107

---

3.5
(a)
3.0
2.5
2.0
dim
1.S
1.0
0.5
0.0
2
4
6
8
10
12
14
log k
8
7
(b)
6
5
dim 4
3
2
o '-__-'--__L...-._ _L...-._ _...L.-__-'--__.J.
. . J - _ - - - - - - I
30
70
110
150
190
230
270
300
order
Figure A.33 Doubly-periodic flow: a) Correlation dimension;
b) Nearest-neighbor dimension
108

---

7
(a)
6
5
dim
4
3
2
0
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
log k
8
(b)
7
6
5
dim
4
~
3
2
O~------''-_----L_ _---..I..._ _~_ _--L-_ _. . l -_ _..L...-_----.J
30
70
110
150
190
230
270
300
order
Figure A.34 Quasi-periodic regime with several frequencies:
a) Correlation dimension; b) nearest-neighbordimension
109

---

7 r - - ; ; - - - - - - - - - - - - - - - - - - - - - - .
(a)
6
5
dim
4
3
2
OL..-.J._ _
__
....L...-_ _......L...-_ _----L.._ _--L--=~
l____.._ _
-7
-6
-5
-4
-3
-2
-1
log k
8
7
-------
(b)
6
5
dim
4
3
2
30
70
110
150
190
230
270
300
order
Figure A.35 Nonperiodic regime with broad spectrum:
a) Correlation dimension; b) Nearest-neighbor dimension
110

---

the Lyapunov exponent calculation can differentiate between a quasi-periodic
regime with several 'frequencies and a nonperiodic one.
A.3
Comparison with Experiments
Much of the behavior observed experimentally is captured in the
numerical solutions.
In both, the numerics and the experiments, there is a
transition sequence from CF~S2~SI12 expected from the predictions by COl.
In the numerics, after the previous sequence, there is a series of quasi-
periodic transitions that are not detected in the experiments. It is not surprising
because they occur in a range of Ri between 204.10 and 204.16.
The
experiments described in Chapter 3 did not have the resolution required to
detect different flows in such a small range of the parameters.
Numerics and experiments show the same kind of chaotic behavior.
The broadening of the spectrum of flows very close to CP2 (0 ~ L ~ 0.05) is
such that the two main frequencies can always be identified (compare Figure
3.6c with A.23b).
In flows farther away from CP2 (0.05 ~ L :5; 0.10) the
broadband peaks are centered on Q s1 (compare Figures 3.8b and A.30a).
The peaks from the numerical calculations are broader than those from the
experiments. The region in parameter space where broadband spectra were
detected experimentally is much larger than the one in the numerics.
The dimension measured in the numerical quasi-periodic regimes with
four or more frequencies is approximately 4; then at the broadening, the
dimension increases to approximately 6. In the experiments, flows with
broadband peak but no increase in the background level of noise have
dimension of approximately 4. The dimension of the flows with exponential
decay at high frequencies in the power spectrum could not be calculated but
was definitely larger than four.
111

---

It is not surprising that after the broadening, the flows become regular
again. In fact, if the equations were valid everywhere, the solutions would be
expected to go back to Couette flow as a circular path is completed around
C1::>2. The reorganization in the experiments and the numerics does not follow
the same sequence. In the numerics, it is an inverse quasi-periodic sequence
with fewer and fewer frequencies at each step. In the experiments it is not so.
It cal'! be concluded that the amplitude equations do not describe adequately
the behavior of the flow after the broadening.
It would be worthwhile in the future to analyze the transition on line D
and determine whether it is due to a phase shift, a bifurcation from the
interpenetrating spirals or to another branch of solutions. Also, the unstable
solutions should be calculated to determine whether the triply-periodic flow
exists or not. It might be convenient to solve the equations with a more robust
method and make sure that none of the broadening is due to round-off error.
The fact that numerics and experiments yield a very similar transition to
chaos is an important conclusion.
It indicates that the reduced amplitude
equations might be an adequate model to describe chaos in real flows. It also
suggests that the chaos observed experimentally is a consequence of the
mode interaction.
112

---

APPENDIX B
OTHER SIGNIFICANT RESULTS IN THIS RESEARCH
Studies of TCF in various regions of parameter space other than those
described in Chapter 3 have led to either discoveries of new phenomena or to
a better understanding of the flow behavior.
A summary of these studies
follows.
B.1
Primary Flow Exchange
B.1.1
Background
The primary flow exchange between a two-cell and a four-cell TVF as
the heig ht of the fluid column is varied was first observed by Benjamin (1978)
when studying the transition from CF to TVF in a system in which the outer
cylinder was stationary and the aspect ratio r was varied between 3.5 and 4.
He suggested that the effect of the finite length of the fluid column on the
morphogenesis and stability of the flow could be understood as a perturbation
of the ideal infinite cylinder model.
The perturbation gives rise to an
"imperfect" pitchfork bifurcation with the following characteristics:
i) There is a smooth transition between the basic CF and the primary
TVF as the Reynolds number Ri is increased quasi-statically from rest.; that is,
there is no bifurcation (see Figure B.1 b).
ii) The original pitchfork bifurcation diagram (Figure B.1 a) is decoupled;
that is, one of the branches is disconnected from the basic flow. The branch
which remains attached to the basic CF is called the primary flow.
The
disconnected branch corresponds to a secondary flow that can only be
accessed by a sudden change in Ri (see Figure B.1 b).
113

---

L
(a)
Ncells
r
basic
...~,,'
.. -..........--
p<--
......
,
----~,
p,,"- N±2cells
secondary
(c)
(b)
secondary
~ Ncells
p
p{
~----
~--------
N±2 cells
primary
(d)
(e)
Figure B.1 a) pitchforK bifurcation; (b)-( e) Unfolding of a pitchforK bifurcation;
cases (b)-(e) correspond to different values of r
114

---

iii) If, for a given value of r. an N-cell flow can be obtained as a primary
flow, then a TVF with N+2 or N-2 cells (depending on r) exists as a secondary
flow. As r is increased (or decreased), the N+2 or (N-2) cell flow becomes the
primary flow and the N-cell TVF becomes the secondary flow (see Figure
B.1 e). This process is known as the primary flow exchange.
iv) In a range of r where the exchange between the primary and
secondary flows takes place, the transition as Ri is varied shows hysteresis.
Hysteresis occurs when the bifurcation diagram has an "S" shape; it can be
shown that the part which bends to the left is unstable (see Figures B.1 c and
B.1d).
The aspect ratio r is thus the unfolding parameter of the bifurcation.
The word "unfolding" comes from the fact that in the three-dimensional space
with Ri and r included, the bifurcation diagram is indeed folded. As a result of
the fold, the projection in r -Ri plane has a cusp (see Figure B.2). Fold and
cusp are both indicators of hysteresis. The lines AB and DE show all the points
P from Figures A.1 a and A.1 d where the secondary flow becomes unstable as
Ri is decreased quasi-statically, and the flow settles into a primary state. For
the particular experiment of Benjamin, the line AB shows where the secondary
two-cell flow jumps to a primary four-cell flow. The line DE shows where the
secondary four-cell flow jumps to a primary two-cell flow. The shaded region
B-C-D is where hysteresis is observed.
In the range r1 $ r $ r2, as Ri is
increased quasi-staticaily from rest, the flow settles into a "degenerate" four-
cell flow.
The two middle cells in the degenerate four-cell flow are much
smaller than the end cells.
If the process is continued, the flow settles
smoothly into a two-cell flow along CD. If Ri is now decreased quasi-statically,
the 'flow will go back to the four cell flow at BC, thus, the hysteresis.
I
115

---

r
A
2-cell
r2 - - -
degenerate
four-cell
flow
c
4-cell
E
o
Figure B.2 Cusp in Ri vs. r due to unfolding
116

---

Figure B.2 was confirmed experimentally by Benjamin (1978) for 3.5 $ r
$
4.
Much work has been done on finite-length cylinders since then.
Schaeffer (1980) used a mathematical model to show that the behavior
predicted by Benjamin was indeed justified. Schaeffer's model applies to the
exchange between 2N and 2(N+ 1) cell flows for N > 2.
(Notice that
Benjamin's experiment in particular was not included in the model!) Mullin
(1982) confirmed experimentally the hysteretic behavior of the exchange
process for the changeover between four and six, six and eight, eight and ten,
and ten and twelve cellls. The main difference between his results and those
of Benjamin is that the cusp in all these other cases is directed up in~tead of
down.
Mullin, Pfister & Lorenzen (1982) have verified experimentally the
hysteresis in the four-cell to six-cell exchange using LDV measurements.
8.1.2
Controversy
Cliffe (1988) studied the problem numerically using a finite element
method to solve the axisymmetric Navier-Stokes equations with appropriate
boundary conditions. He found that in the two-to-four exchange, outside of the
hysteresis region, the Reynolds numbers at which transition occurs are about
5-10% higher than in Benjamin's experiment.
He also found hysteresis for
such a small range of r that it could not be observed experimentally. For the
changeover of flows with more (four or more) cells, his results compare
favorably with Mullin's (1982).
Numerical simulations of the two-to-four cell exchange process by
Streett and Hussaini (1987, referenced as SH) using spectral collocation
methods showed not only quantitative but also qualitative disagreements with
Benjamin's results.
They found that outside the hysteresis region as the
Reynolds number is decreased, the secondary flow changed to the primary
117

---

flow at Ri values about 5-10% higher than those measured by Benjamin. This
is similar to Clitte's predictions. However, their controversial result is that they
found no hysteresis in the region between r1 and r2. As Ri is increased from
rest, a four-cell now is formed.
As Ri is increased further, the flow starts to
change until it becomes a two-cell flow. As Ri is decreased from this two-cell
flow, changes occur and the flow settles into a four-cell flow again.
Even
though the changes occur over a finite range of Ri
,no hysteresis was
observed. The rate of change of Ri in these numerical calculations was 0.5/ts
h2
or slower (ts= -).
4u
B .1.3
New Experiments
The main goal of these experiements is to determine wwhether or not
there is hysteresis in the changeover process (described above) between a
two-cell and a four-ceIl1'l0w. The parameters were the same as those used by
Benjamin: 11=0.615 and 3.5 ~ r < 4. The working fluid was silicon oil with a
viscosity of 50.6 cs; pearl maid was used for visualization.
For this particular
investigation, Rj=Qj a2/u ; ffi=O. The two parameters of the process are Ri and r.
The speed and acceleration of the inner cylinder were computer controlled
using a micro-stepping motor.
The Reynolds number was changed by
increments ~Ri=0.1 at a rate less or equal to 0.5/ts ; this procedure was
considered quasi-static.
However, a time of about 3ts was allowed to pass
after each change in Ri to make sure that the flow had achieved steady-state.
The height of the fluid column was determined with a traverse and a laser
beam focused on the edge of the rings which bounded the fluid above and
below. The error due to the width of the laser beam is about 0.2mm (less than
118

---

0.2% of the height of the fluid column). As in Benjamin's experiments, the end
rings were attached to the outer cylinder and thus remained fixed in space.
The goal of the experiment was not to reproduce Figure B.2, but to
describe the behavior of the transitions at chosen values of r. These values
were chosen such that the observations could be compared to Benjamin's,
Cliffe's and SH's results. This limited number of experiments was enough to
show that the transitions occur at higher Reynolds numbers than those
observed by Benjamin, but that there is indeed an hysteresis region.
A
summary of the results follows.
Outside the hysteresis region
i) r < r1
Experiments were performed at r=3.5.
(From Benjamin, r1=3.63)
When Ri was increased slowly from rest, the flow settled smoothly into a two-
cell flow that remained stable up to very high values of Ri (even to the right of
DE). However, if the speed of the cylinder was suddenly increased from rest
to values of Ri' much higher than on DE, a four-cell flow could be obtained.
Once this flow was obtained, as Ri was decreased quasi-statically, the four-cell
flow "jumped" to a two-cell flow at Ri=173.2. This number is extremely close to
SH's simulation where the "jump" occurs for Ri between 172.8 and 173.6. The
value from Cliffe's calculation is Ri=175, and from Benjamin's figure Ri=165.
Clearly, this experiment does not support Benjamin.
ii) r > r2
Experiments were performed at r=3.8.
(From Benjamin, r2=3.72)
When Ri was increased quasi-statically from rest, a four-cell flow was smoothly
established. The "jump" from the secondary two-cell 'flow to the primary four-
119

---

cell flow as Ri was decreased occured at R;=131.7.
In SH's simulation, the
"jump" occured for Ri between 131.2 and 131.9. From Cliffe's figure, Ri=134,
and from Benjamin's Ri=122.
This result does not support Benjamin's.
Hysteresis region
Two values of r were chosen to study this region: 3.65 and 3.70. Both
can be quantitatively compared with SH.
A degenerate four-cell flow was
formed at Ri=115 by increasing Ri quasi-statiically. In this four-cell flow, the
two middle cells were much smaller than the end cells. This was referr~d to as
a degenerate four-cell How by Benjamin.
i) r=3.65
At Rj=120 the two middle cells got even smaller, but it was clearly a
four- cell flow. At Rj=123, after a few minutes, the flow changed to two-cell flow.
The time is important here because even though the Reynolds number was
changed slowly enough to assume a quasi-static process, the flow took an
additional time of about 3ts to reach steady state. For any Reynolds number
higher than this, the flow had two cells.
When the Reynolds number was decreased quasi-statically, some
changes started to occur at Rj=122.5. A band appeared between the two cells,
looking almost like a three-cell flow, but the motion inside could not be
distinguished by flow visualization.
A clear four-cell flow was not observed
until Rj=121. Summarizing, when the Reynolds number is increased, a clear
"jump" from four cells to two cells could be observed but when the Reynolds
number was decreased, the changes occured in a much greater range in
Reynolds number and no jump could be detected visually.
In Benjamin's
120

---

experiments, the hysteresis occurred for Ri between 113 and 115. In SH's
calculation, the transition takes place smoothly with no hysteresis for values of
Ri between 120.5 and 124 at this value of r.
Cliffe's results indicate no
smooth transition at this value of r.
ii) r==3.7
The transition process was similar to the one described above, but
occurred at different Reynolds numbers. The flow changes from four to two
cells at Rj=125.5. As before, it took a few minutes for the now to achieve
steady-state even though the Reynolds number was changed at a rate of
0.5/ts. When Ri is decreased, a band between the two cells was formed at
Rj=125, but it was not until Rj=122.5 that a clear four-cell flow was observed.
The axial component of the velocity was measured with LOA and
processed by an Intelligent Flow Analyzer from TSI, which was able to detect
very small velocities.
The sudden increase in the axial velocity when the
Reynolds number was increased and the smooth process when the Reynolds
number was decreased were confirmed by the measurements (see Figure.
8.3). The photographs in Figure BA show this same process. In Benjamin's
experiments, the hysteresis occured between 117 and 120.
In SH, the
transitions occur with no hysteresis for values of Ri between 122 and 126.
Experiments show that hysteresis does exist in a range of Ri 5% higher
than the previous measurements by Benjamin.
8.1.4
Conclusions
The qualitative description of Benjamin of the 2-4 cell exhange
is
correct.
However, his values for Ric are not. SH's results outside the
hysteresis region are the same as those determined experimentally; however,
they were unable to detect hysteresis. Cliffe's results are slightly higher.and
121

---

2,.....------------------:~-__,
vel
1
down
o
-1 -+-"T""'"""'l....--T"""'T"~"T""'"""'l~"""T'" ........T__.___,.......,.....,........._.~......,.....,.......__..~__1
110
115
120
125
130
Ri
Figure B.3 Velocity measurements showing hysteresis in the transition
CF~TVF for 1=3.7
122

---

i\\)
(a)
(b)
(c)
(d)
(e)
w
iii:;;
(j)
(i)
(h)
(g)
(f)
Figure B.4 Photographs of the hysteresis: a) Ri=115 (up); b) Ri=125 (up);
c) Ri=124 (up); d) Ri=125 (up); e) Ri=125.5 (up); f) Ri=12.4.5 (down);
g) Ri=124 (down); h) Ri=123.5 (down); i) Ri=123 (down);
j) Ri==122 (down)

---

he did not find hysteresis at the values of r studied in this investigation. It was
found experimentally that settling time is -3ts .
8.2
Low r in counter-rotating flows.
Experiments were conducted for r=4.0, 11=0.5 and the end rings were
attached to the outer cylinder. The outer cylinder was set to a fixed rotation
rate. Then, when a steady Couette flow was established, the rotation of the
inner cylinder was increased quasi-statically from rest.
Two interesting
observations using flow visualization were made.
8.2.1
Ro=-100
TVF occurs at Rj=150 with the direction of circulation such that at the
boundary between the end rings and the cells the now goes towards the inner
cylinder. This is the direction of rotation that would be expected for 00=0, and is
referred to as "normal". The primary flow has six cells. At this same aspect
ratio, the primary flow for (j)=0 has only four cells. (This is not very surprising
because, for counter-rotating flows, the domain where Rayleigh's criterion is
valid is smaller than the gap between the two cylinders, and the Taylor cells
are expected to scale on this new length.)
At Rj=220, the flow becomes
unstable and a new pair of cells appears close to the center (in z) of the
facility. The cells at the ends are "pushed" towards the end rings; they become
smaller and smaller until they disappear. The final result is a steady six-cell
flow with the circulation in each cell opposite to the initial one.
There is
hysteresis in this change of circulation; when the value of Ri is decreased
quasi-statically, the change in direction of rotation occurs at Rj=180.
The change in the direction of circulation in the flow can probably be
explained in terms of the velocity profile near the end rings. What is surprising
124

---

is that this effect has not been observed for higher aspect ratios. To date it
seems that, at higher aspect ratios, the flow becomes unstable to non-
axisymmetric disturbances (WVF) before the change in the direction of rotation
occurs.
However, careful experiments have not yet been performed to
document the phenomenon for different values of Ra , Ri and r. The change in
the direction of rotation is definitely a finite-length effect that cannot be
addressed with the assumption of infinite cylinders or periodic boundary
conditions.
8.2.2
Ro =-200
TVF starts to appear Ri=200. However, new pairs of Taylor cells are
constantly being formed near (but not at) z=O by small jets ejecting from the
inner cylinder. As these new pairs grow, the cells close to the ends become
smaller and disappear.
The final flow consists of Taylor vortices which
propagate very slowly in the axial direction, from the center towards the ends,
giving the impression of a flow pattern that alternates between six, eight and
ten cells. Hysteresis has been observed in this transition, and the passage of
cells seems to be periodic in some regions of (Ri, Ra) space.
However,
accurate measurements have not yet been performed.
These time-dependent Taylor vortices have not previously been
reported in the literature, and should be documented in detail. For given r, the
number of cells in the primary flow increases as Ra becomes more negative. It
would be interesting to analyze whether ATV is a result of axial mode
interaction between two steady states with different axial wavenumbers.
125

---

8.3
Intermittent 8ehavior
Time series that alternate between two (sometimes three) different
kinds of behavior (see Figure 8.5) have been observed in the region of
parameter space Ri between 230 and 250 and Ra between 370 and 400. The
"alternating" behavior is not transient; that is, it repeats itself for long periods of
time (hours). The phenomenon has not been documented in detail and has
not been visualized.
This kind of behavior might indicate homoclinic or
heteroclinic curves in state space. The subject is interesting and should be
pursued.
8.4
Higher-Order Transitions
In Chapter 1 it was mentioned that the first transition is always to either
TVF or spirals for all values of ll. but that the values of the critical parameters is
strongly dependent on ll.
Higher-order transitions can result in completely
different flow as II is varied. In this section, a summary of observations on two
such now regimes is presented, and their study is recommended for future
research.
8.4.1
Spiral turbulence
For ll=0.883 and Ra fixed at some value ~ -500, spiral turbulence (SpT)
is found as a fourth transition as Ri is increased quasi-statically from rest. SpT
consists of alternating patches of laminar and turbulent flow spiralling along
the fluid column like the bands of a "barber pole". Tl1is flow is a small gap
phenomenon, it does not occur for ll=0.75 or 0.50.
SpT is particularly
interesting to fluid dynamicists because it has some characteristics also
present in transitional pipe flow.
SpT is stationary in a rotating reference
frame. This indicates that, as a fraction of the turbulent flow laminarizes,
126

---

(a)
~Jt_ .......:"...,-~-"""'--t----t-, .-+----t"""t"--
.'
Figure 8.5 Intermittent behavior: a) Time series alternates between two
different states; b) Time series alternates among three different states
127

---

another fraction of the laminar flow becomes turbulent, creating interfaces in
which transition from laminar to turbulent behavior occurs and others in which
transition from turbulent to laminar behavior occurs.
Similar behavior has
been observed in the evolution of equilibrium puffs in pipes.
However, SpT
should be easier to study, as it remains "captive" between the two cylinders.
Preliminary studies show that the velocity field is very complicated. The
spiral turbulence is superposed on interpenetrating spirals.
In the regions of
space where the flow is turbulent, it occupies the whole gap between the two
cylinders.
The laminar part consists of a purely azimuthal flow close to the
outer cylinder and interpenetrating spirals close to the inner cylinder. The time
series of the velocity measured at a point shows periodically alternating
turbulent and laminar behavior superposed on .a periodic sig nal.
The
frequency of rotation of SpT scales either on Qo or on (Qi+Qo) /2 depending on
(Rj,Ro)·
The study of this flow might lead to the understanding of the dynamics
of entrainment and relaminarization.
8.4.2
Rhombuses
At T1=0.75, the sequence of transitions differs from that at 0.883. For-
650 ~ Ro ~ 360 , there is a transition to a wavy flow with the following
characteristics (see Figure 8.6): i) the boundaries between cells are Hat; ii) the
number of undulations in each cell varies depending on the values of Ri and
Ro; iii) "valleys" on one cell are aligned with "crests" in the neighboring cell; (iv)
the flow preserves the "flip" symmetry; and v) there is a very slow modulation
of the amplitude of these rotating waves such that valleys become crests and
crests become valleys (the reflectional symmetry is maintained in spite of the
128

---

Figure 8.6 Photograph of rhombuses
129

---

modulation). For higher values of Ri, there is a shift in the rotating waves, all
crests align along the fluid column, and the reflectional symmetry is broken.
Even though the region of stability for this flow has been documented
for r=40 (see Figure B.7), its temporal behavior has not been studied in detail.
This flow might be related to the superposed ribbons described by COl.
Note
that this flow recovers the "flip" symmetry that was lost when Couette flow
became unstable to spirals. This symmetry is lost again in the next transition.
At this radius ratio, the transition to turbulence does not occur by bursts
as it does for Tl=0.883.
Instead, small-scale fluctuations begin to appear
everywhere in the flow at once and as Ri is increased they eventually. "fill-up"
the entire fluid column.
130

---

i
.400
350
•······
300
250
200
:D
......
rhombuses
-'-
w
150
-'-
100
50
o
-900
-800
-700
-600
-500
-400
-300
-200
o
Aa
Figure B.7 Region of stability of rhombuses

---

APPENDIX C
SUGGESTIONS FOR FUTURE INVESTIGATION
This appendix summarizes all the recommendations for future
investigations that have been suggested in nlis work based on the
experimental and numerical observations described in Chapter 3 and in
Appendices A and B.
In addition, other directions of investigation are
suggested based either on the existing literature or on the experience
acquired while studying TCF.
C.1
Suggestions
Based
on
Experimental
or
Numerical
Observations
C.1.1
Mode Interaction
The spatial structure of most flows that occur when the cylinders are
counter-rotating is difficult to visualize because the instability occurs very close
to the inner cylinder. When the flow is illuminated from outside of the facility,
much of the light is lost on reflections by the walls and scattering by the fluid.
New methods of flow visualization should be designed.
For example,
illumination or injection of dye from the inner cylinder might be useful.
It would be interesting to know the spatial structure of the spirals at the
ends and to understand how the spatial superposition of the interpenetrating
spirals occurs.
Also, some information about the spatial structure of flows
around CP2 would be useful.
To complete the experimental study on mode interaction, the effect of r
on the critical parameters, on the hysteresis in the transition from Couette flow
to spirals and in the nonperiodic modulation of the flow should be
132

---

documented.
High values of r might result in slow periodic modulations
which, if detected, should also be documented.
The position of the end rings (whether attached to the inner or the outer
cylinder) might also have an effect on the exact values of the critical
parameters. This effect should be checked.
C.1.2
Numerical Calculations
In the numerics the transition to a state where the solutions in all four
subspaces is periodic but only two frequencies are detected is particularly
interesting. It should be determined whether this transition corresponds to a
phase shift bifurcation from the interpenetrating spirals or to a "jump" to a
different branch of solutions.
This analysis can be done by studying the
behavior of the solution in a neighborhood of the point (in parameter space)
where this transition occurs.
After this transition is understood, the next
transition to a four-frequency solution will be easier to understand.
C.1.3
Axially Travelling Taylor Vortices (ATV)
The region of stability of the ATV in (Rj,Ro) space, its time dependence
and possible hysteresis as the parameters are changed in opposite directions
should be documented. The neighboring flows (in parameter space) should
also be described.
This flow is axisymmetric and occurs at low Reynolds
numbers; it might be an interesting problem for the numerical analyst.
C.1.4
Spiral Turbulence
The study of spiral turbulence as a coherent structure and the
understanding of relaminarization and entrainment are fascinating subjects.
More information about SpT than the one already available can be obtained
133

---

only by mapping the velocity field with phase-locked LDA measurements and
then studying the regions of laminarization and entrainment in detail.
Even
though the experiments are complicated, they are still less difficult to do than
in any open flow.
C.1.5
Rhombuses
The spatial structure of this flow is known and can be easily visualized.
However, its time dependence needs to be determined.
Only then, the
relationship between this flow and the theoretical studies can be determined.
C.2
Other Suggestions
C.2.1
End Effects
Most disagreements between theory and experiments have been
blamed on end effects. However, they are usually not documented.
Most instabilities are triggered at the end cells.
More insight into the
effect of r on transition and on pattern selection in higher order transitions will
lead to better models and a better understanding of TCF.
C.2.2
Routes to Chaos
Besides the Ruelle-Takens-Newhouse sequence to chaos, period
doubling and intermittency have been observed in TCF. This shows that TCF
is again the appropriate ground to link ideas from nonlinear dynamics and
fluid dynamics.
C.2.3
Effects of 11
For counter-rotating flows, the sequence of transitions is completely
different for different values of 11.
Also, the transition to turbulence shows
134

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different characteristics for different 11. For 11=0.883 and Ra < -SOD, the
transition is similar to that of flow in pipes: First, random bursts appear; then,
alternate patches of larninar and turbulent fluid form a periodic pattern. This
flow becomes completely turbulent as Ri is increased. For 11=0.75 and Ra <
500, the transition occurs gradually.
First, small scales start to appear all
along the fluid column; then, as Ri is increased. the small scales fill the entire
flow.
More insight as to why this difference in the transitions depends on 11
might lead to the understanding of some aspects of transition in open flows.
C.2.4
Pattern Formation in Co-Rotating Flows
Higher order transitions in co-rotating flows usually lead to random
spatial patterns and might be the appropriate ground for studies in spatial
chaos.
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APPENDIX D
DIAGNOSTICS FROM NONLlNEAR DYNAMICS
This appendix will provide a summary of the different concepts and
tools that have been used to analyze experimental data. It is not intended to
provide rigorous definitions.
For a more complete treatment, the reader is
referred to Hirsch and Smale (1974), Guckenheimer and Holmes (1983).
D. 1
Summary of Concepts
It is often convenient to represent the behavior of a physical system in a
multi-dimensional state space (space of the variables) in which each point
represents the entire state of the system at a given instant of time. The "rule"
that determines the behavior of the system, say a differential equation, defines
a dynamical system. The solution curves or trajectories describe the history of
the system in state space. The set of all solutions is called the phase portrait
of the dynamical system.
Stable solutions in dissipative systems tend (as t~oo) to limit sets called
attractors.
To study the behavior of the trajectories in the attractors is
equivalent to study the asymptotic behavior of the system.
The simplest kind of attractor is a fixed point, which represents a steady-
state. For example, Couette flow is represented by a fixed point in state space.
The attractor that represents a periodic solution is a limit cycle. A spiral, which
is a singly-periodic solution (described in Chapter 3) is described by a limit
cycle.
A two-torus represer)ts a doubly-periodic flow; a three-torus a triply-
periodic flow, and so on for all quasi-periodic behavior. (Quasi-periodic refers
to two or more incommensurate frequencies.) The dimension of the attractor
increases with the number of independent frequencies.
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There is a particular kind of attractor, first described by Lorenz (1963), in
which the behavior is nonperiodic.
(A behavior is nonperiodic if it is neither
steady, periodic, nor quasi-periiodic.) Such attractors are called "strange" or
"chaotic".
Their main characteristic is a sensitive dependence on initial
conditions. This means that the distance between two nearby trajectories, no
matter how close at some initial time, increases exponentially with time in at
least one direction.
To keep the trajectories bounded in the attractor, there
must be exponential contraction in other directions.
These expansions and
contractions in several directions, result in the very complicated internal
structure of the chaotic attractors.
The exponential rate at which two trajectories either diverge or
converge in the attractor is measured by the Lyapunov exponents. A useful
way to define these exponents is the following, taken from Wolf et al., (1985):
Consider a continuous dynamical system in an n-dimensional state space and
an infinitesimal n-sphere of initial conditions. The n-sphere will become an n-
ellipsoid as it evolves in time. The i-th one-dimensional Lyapunov exponent
is then defined in terms of the length of the corresponding ellipsoidal principal
axis Pi(t) by
1
..e.J!.L
Ai=limt~ooT log 2 Pi(O)'
(D.1 )
There is an exponent for each direction of motion in state space. The
set of all Lyapunov exponents forms the Lyapunov spectrum, in which the
exponents are usually ordered from largest to smallest.
Any continuous time-dependent dynamical system without a fixed point
has at least one Lyapunov exponent equal to zero (in the direction of the flow).
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Any dissipative system will have at least one negative exponent. This means
that a chaotic attractor can exist only in a state space of dimension at least 3.
The signs of the exponents give a qualitative description of the flow
behavior. Their magnitudes quantify the loss or gain of information during the
process. Exponents are usually measured in units of bits/sec.
The complicated internal structure of strange attractors can be
measured by another property, dimension.
The Euclidean concept of
dimension refers basically to the number of coordinates (variables) required to
model the system, and is always an integer. The irregular winding and folding
in strange attractors requires a more general definition.
Many different dimensions have been defined in the literature in the last
few years.
They can be considered equivalent for the purpose of this
investigation. The simplest definition was given by Kolmogorov and is called
the capacity.
Consider a subset of a p-dimensional Euclidean space.
Suppose N(£) is the number of p-dimensional cubes of side £ needed to cover
the given set. Then the capacity de is given by
log N(£)
de = Iim
(0.2)
E~oo
1
log (-)
£
1
For a point, N(£) - 1 and de = 0; for a line, N(£) - - and de = 1; for an
£
1
area, N(£) - -
and de = 2. In these examples, the capacity measures the
£2
usual Euclidean dimension.
However, for more general sets, it can give a
non-integer value.
Consider a Cantor set obtained by removing the middle
third at each step (see Figure 0.1). If £ - ~n' then N = 2n and eq. (0.2) yields
138

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~~
o 0
"
~:o0
N
~
CX)
11
11
8
11
11
z
i
z
z
z
w
.... I~
....I'M
.... 1%
.... I~
E
11
11
11
11
W
W
W
W
...-... "
w
..........
I
w
I
z
CJ)
CJ)
0
(j)
I
0
(fJ
.....
8
0
i
cC1l
w
0
E
C1l
I
I
'0
.~
I
u
C1l
CL
C1l
0
c:i
<lJ
.....
::J
Ol
iL
I
I
I
I
I
I
139

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log 2
de = log 3 - 0.63 .
(D.3)
The capacity provides a lower bound for the number of variables
required to model the dynamics. It also provides information about the specific
location of points in the attractor within an accuracy of E.
Even though the capacity definition is easier to understand, other
definitions provide simpler or less time-consuming methods for the calculation
of the dimension. For example, the correlation dimension can be defined as
follows (Grassberger & Procaccia 1983 and Kostelich & Swinney' 198?).
Given N points in an attractor, the correlation integral is given by
C (N, E) = ~2 X { number of pairs of points which are less than E apart}. (D.4)
It can be shown that under certain conditions, C(N ,E} - E Ii for large Nand
small E, where 0 is the correlation dimension. The slope of log2 C vs IOg2E
gives the dimension. The range of values of E for which this slope does not
change is called the scaling region, and the value of the slope in the scaling
region is the dimension of the attractor (see Figure D.2).
Still another approach to dimension has been examined by Badii &
Politi (1985) and implemented by Kostelich (1988). Consider N points in the
attractor, and let x be a reference point. For k < N, let { Yi } be a set of k distinct
points (different from x) chosen at random from the original set of N points. Let
(D.5)
140

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scaling
region
Figure D.2 The correlation dimension is given by the slope of 1092 C vs 1092 E
141

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The calculation is repeated for all other reference points. Several sets of k are
chosen, each time with a larger number of points. Badii & Politi suggest that
the average value of dk over all reference points is given by
< dk > = k -1/0 ; 8 the nearest-neighbor dimension.
(0.6)
In both calculations of dimension, the power law is a limit as the number
of data points becomes infinite.
0.2
Experimental Implementation
0.2.1
Phase Portraits
Probably the most important result for the nonlinear analysis of
experimental data comes from Taken's embedding theorem (1981).
This
result states that a multi-dimensional phase portrait can be reconstructed from
the measurement of a single variable as follows. For almost every observable
v(t) and time delay T, the n-dimensional phase portrait constructed from the
vectors {v(t), v(t k + T), ...... , v(t k + (n-1)T)} (k=1, ...... , 00) will have the same
(topological) properties as one constructed from the measurement of the N
independent variables provided n ~ 2N+1.
The reconstruction is an
embedding of the original state space.
In principle, the theorem is valid for any T. However, it has been shown
that there are better choices of T that allow "angles of view" of the
reconstructed object that are easier to work with.
Once the state space is reconstructed, the largest Lyapunov exponent
and the dimension can be measured using the following techniques.
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D.2.2
Lyapunov Exponent
The program used is the one described in Wolf et al., (1985) for fixed
evolution.
The procedure is as follows:
First, the phase portrait is
reconstructed from the experimental time series.
A point on the attractor is
then given by v(t) = {v(t), v(t k + T),
, v(t k + (n-1 )T)}. The closest point (in the
Euclidean sense) to the initial v(to) is located. The distance between these two
points is called L(to). At a later time t1. the length would have evolved to L'(t1).
The length is followed for a short period of time to make sure that the correct
scales in the attractor are being examined. (If the evolution time is too long,
then L' might actually get smaller if the two trajectories pass through a folding
region.) A new point V(t1) is then chosen such that both the new distance L(t1)
and the angular separation between L(t1) and L'(t1) are small. The procedure
is repeated until the trajectory has traversed the entire data 1'IIe. At that point,
the largest Lyapunov exponent is calculated by
1
~
L'(t k)
A,1 =-tM---t-o LJ log 2 L'(t k-1) ,
(D.7)
where M is the total number of replacement steps. In this program the time
step is kept constant.
To use this program, several input parameters must be specified: length
of the data file, the time T for the reconstruction, sampling frequency,
embedding dimension, largest and smallest scales in the attractor and the
evolution time. The authors suggest a replacement step size of about one to
one-and-a-half orbits; this was taken into account in the calculations in
Chapter 3.
In many cases, the largest and smallest scales can be obtained
directly from the calculation of correlation dimension or from the phase
143

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portraits.
However, when the phase portraits are complicated or the
correlation dimension provides no scaling, there is no criterion available.
0.2.3
Dimension
Two of the dimensions defined above have been calculated: the
correlation dimension and the nearest-neighbor dimension.
The calculation of the correlation dimension is long, but straightforward.
To reduce the computation time (Brandstater & Swinney 1987), the correlation
integral was calculated by forming pairs with one thousand reference points
(randomly, but uniformly distributed in the attractor) and all the points in the
attractor. The method was used following Broze (1988). A fit to the correlation
integral is performed using a cubic spline. The slope of the correlation integral
vs. distance' in state space was calculated analytically from the spline
coefficients.
The range of values of E for which the slope does not vary is
called the scaling region. If the value of this constant slope does not increase
with increasing embedding dimension, it is said to saturate.
The saturated
slope is the correlation dimension. The number of points in the data file, the
sampling rate and the embedding dimension were chosen taking into account
therecomendations of Brandstater & Swinney whenever possible.
This method has the disadvantages of being long and unable to
calculate properly dimensions higher than three. Furthermore, as explained in
Chapter 3, it fails to provide scaling for doubly-periodic nows when the two
frequencies are very close to each other. In those cases in which it gives good
results, the correlation dimension has the enormous advantage of providing
the scaling information required for the calculation of the Lyapunov exponents.
The calculation of the nearest-neighbor dimension was done using
Kostelich's program.
Because of noise in experimental data, the distances
144

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between points and their nearest neighbors might not reflect the fractal
structure of the attractor. A power law can be obtained if only neighbors that
are sufficiently far away are considered. The distance dk is then measured
from the reference point to the n-tl1 nearest neighbor, where n is called the
order.
In these calculations, n can be chosen between 10 and 300; the
maximum embedding dimension that can be used is eight.
The only other
parameter that must be known is the time delay T used in the reconstruction of
the phase portrait. Usually, a power law between < dk > and k is obtained for
large n and large k. When the power law between < dk > and k is the same for
increasing order n, the slope is said to saturate. Figure 0.3 shows log2 < dk >
vs log2 k. When a power law exists in this plot, the slope of these lines gives
the inverse of the nearest-neighbor dimension, as can be seen from eq. (0.6).
The dimension is often plotted as a function of n. When the dimension does
not change with increasing n, it is said to saturate. This method of calculation
of dimension is very convenient because it does not require long computation
times and works for dimensions higher than three.
However, it has some
inconveniences. A small error in the calculation of the slope of log2 < dk > vs.
log2 k results in a large error in the calculation of dimension o. Besides, the
algorithm does not provide the scale information required for the calculation of
the largest Lyapunov exponent.
D.2.4
Spectra
Most of the information obtained from power spectra is well known and
will not be discussed.
Recently, however, Sigeti & Horsthemke (1988) have
shown that the spectral power of random processes decays as a power law at
high frequencies.
They suggest that the power spectrum of a deterministic
process described by a Coo function, should decay faster than a power law.
145

---

-1
1092 <dk>
-1. 5
-2
-2.5
-3
n=300
-3.5
.......
~
(j)
-4
-4.5
-5
-5.5
n=10
-6 9
10.5
11
11.5
12
12.5
13
13.5'
14
1092 k
Figure 0.3 The nearest-neighbor dimensi0r:' is given by the inverse of the slope
of log2 <dk> vs 1092 k; n is the order of the nearest-neighbor

---

Thus, the decay should be at least exponential.
This means that the high-
frequency spectrum of a deterministic process in semi-log plots should show
linear behavior. This idea has been considered in Chapter 3, in the analysis
of the broadband spectra.
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