Studies of vortex breakdown and its stability in a confined cylindrical container 5

CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES
*
Part of this work has also appeared in Phys. Fluids. 18, 2006.
68

CHAPTER 5
*


COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES


5.1 Introduction
The competition organized by double Hopf bifurcations between an axisymmetric
mode (m
1
= 0) and a non-axisymmetric mode (m
2
 0) has been studied numerically by
Marques et al. (2002) and Lopez and Marques (2003). In both the m
1
 m
2
 0 and m
1

= 0, m
2
 0 the double Hopf bifurcations, the competition has been between modes that
reside in orthogonal subspaces and the quasi-periodic mixed mode (which may be
stable or unstable) evolved in the product space.

In this chapter, the mode competition between two axisymmetric limit cycles in the
neighborhood of a double Hopf point was investigated experimentally and
numerically, so that the mode competition takes place wholly in the axisymmetric
subspace. This work is motivated partly by the linear stability analysis of Gelgat et al.
(1996, 2001),which showed the existence of an axisymmetric double Hopf bifurcation,
and the purpose of this experiment is to see if the dynamics associated with this double
Hopf bifurcation can be captured in laboratory conditions. A close examination of the
linear stability results of Gelfaget et al (2001) shows that in the range of the aspect
ratio between about 1.6 to 2.8, where the steady basic state loses stability to
axisymmetric time-periodic flow, there are two distinct Hopf bifurcations leading to
axisymmetric states with different frequencies. The crossover point between these two
Hopf bifurcations (the double Hopf bifurcation point) is at   1.72 and Re  2665.
CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

69
However, the linear stability analysis itself says nothing about the nonlinear behavior
of the flow following the bifurcation. In particular, it says nothing about the
competition between the two limit cycles. In order to say anything in this regard, one
must do some nonlinear analysis, as done by Guckenheimer and Holmes (1983),
Marsden and McCracken (1976) to perform a center manifold reduction and a normal
form analysis in the neighborhood of the double Hopf bifurcation. Doing such an
analysis gives a multitude of possible nonlinear scenarios describing all possible
competition dynamics. In order to pin-point which scenario corresponds to a particular
flow problem requires detailed spatio-temporal information about the bifurcating limit
cycles, which may be obtained from quantitative experimental measurements or fully
nonlinear computations. Thus, in this study, a combined experimental and numerical
study was performed. The experimental measurements provide, for the first time, the
laboratory evidence of the existence of an axisymmetric double Hopf bifurcation,
involving the competition between two stable coexisting axisymmetric limit cycles
with periods (non-dimensionalized by the rotation rate of the endwall) of

approximately 31 and 22. The dynamics is also captured in our nonlinear
computations, which clearly identify the double Hopf bifurcation as “type I simple”
with the characteristic signatures that the two Hopf bifurcations are supercritical and
that there is a wedge-shaped region in [, Re] parameter space where both limit cycles
are stable, delimited by Neimark-Sacker bifurcation curves.

5.2 Experimental Method
The experiments presented here were carried out in the test rig described in
Chapter 2, and only the essential features are presented here. The working fluid was a
CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

70
mixture of glycerin and water (roughly 76% glycerin by weight) with kinematic
viscosity  = 0.404 ± 0.002 cm
2
/s at a room temperature of 23.0°C. The aspect ratio 
tested was between 1.67 and 1.81, and Reynolds number between 2600 and 2800.
To capture the oscillatory behavior of the flow, flow visualization and hot-film
measurements were used. Note that all flow visualization photos were inverted for ease
of comparison with numerical results.
5.3 Numerical Method
The numerical simulation results presented here were provided by Lopez J.M.
using the spectral code described in Lopez et al. (2002) and previously used to explore
the nonlinear dynamics of confined vortex breakdown flows (Marques and Lopez
2001, Marques et al. 2002, Lopez 2006). The solutions presented here have 48
Legendre modes in the radial and axial directions, and up to N = 16 (resolving up to
azimuthal wavenumber m = 16; these were used to test the stability of the
axisymmetric solutions to three-dimensional perturbations); the time-step used is t = 5
× 10
3

, which is much smaller than needed for stability of the code in the parameter
regime investigated.

5.4 Results and Discussions
5.4.1 Basic state
Before we present the results on mode competition between two axisymmetric
limit cycles in the neighborhood of a double Hopf point, it is useful to briefly review
some aspects of the issues regarding the basic steady state and its bifurcation to
unsteady state.
CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

71
For the range of aspect ratio of interest ( from 1.64 to 1.80 and 2600  Re 
2800), the basic steady state from which the limit cycles bifurcate does not have
recirculation zones on the axis. There is however a very pronounced radial deflection
of the flow from the axis which may still be considered a vortex breakdown region.
Figure 5.1 shows contours of the streamfunction ψ and the three components of
velocity for the basic state at Re = 2600,  = 1.75. For comparison, the contours of Re
= 1850 are also attached.

Fig. 5.1 Contours of , u, v, and w for the axisymmetric steady-state solution at  =
1.75 and (a) Re = 1850, and (b) Re = 2600. There are 20 positive and 20 negative
contours quadratically spaced, i.e. contour levels are [min|max] (i/20)
2
with i = 1 20,
and ψ
∈[-0.0078, 0.000045], u∈[-0.16, 0.16], v∈[0, 1], and w∈[-0.16, 0.16]. The solid
(broken) contours are positive (negative). The left boundary is the axis and the bottom
is the rotating endwall.

CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

72
The Re = 1850 case has a very distinct recirculation zone (vortex breakdown
bubble) on the axis near the top stationary endwall, together with the corresponding
reversed axial flow (w component of velocity). The Re = 2600 case does not have such
a recirculation zone, but it does have a comparable radial deflection of the flow in the
same region. The Re = 1850,  = 1.75 case has a notorious history stemming from the
experimental results of Spohn et al. (1998) in which they reported that the flow at this
point in parameter space was not axisymmetric. Linear stability analysis of Gelfgat et
al. (2001) however, showed theoretically that the steady axisymmetric state is stable to
general three-dimensional unsteady perturbations, and numerous direct numerical
simulations using the three-dimensional unsteady Navier-Stokes equations with small
random perturbations in the initial conditions also show evolution to the steady
axisymmetric basic state. Subsequently, Sotiropoulos et al. (2002) have conducted
experiments at this point in parameter space and have shown that the degree to which
the observed flow is three-dimensional can be reduced by reducing the level of
imperfection in the apparatus. They obtained flow visualization results very similar to
those Spohn et al. (1998) by tilting the stationary endwall by approximately 0.4º from
horizontal (see their figure 3). Thompson and Hourigan (2003) also investigated the
same flow numerically and were also able to produce streak-lines in very close
agreement with the dye visualization results of Spohn et al. (1998) by numerically
imposing a small misalignment of the rotating endwall (misalignment angle of about
0.1º). They also re-did the linear stability analysis of the basic state using a different
numerical method to that of Gelfgat et al. (2001) and reached the same conclusion: the
axisymmetric state is stable to all three-dimensional perturbation at  = 1.75 for Re <
2800 and unsteady axisymmetric flow appears at about Re = 2650. It would thus seem
CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

73

that the three-dimensional nature of the flow visualization at low Re = 1850 is not
intrinsic to the flow but is due to extrinsic imperfections.
That the flow visualizations at Re = 1850 are extremely sensitive to small
imperfections is not surprising. The basic state has stagnation points on the axis; in the
language of dynamical systems these hyperbolic fixed points are structurally unstable
(Guckenheimer and Holmes 1983), and as pointed out by Holmes (1984) “…certain
degenerate invariant manifolds of homoclinic or heteroclinic orbits connecting the
stagnation points can be expected to break up under arbitrary small perturbations.” The
question, of course, is whether these arbitrarily small imperfections have any effect on
the dynamics of the flow (rather than on the kinematics, such as the dye steaks).
(a) (b)


Fig. 5.2 Flow visualization at Re = 1853,  = 1.75, using (a) fluorescent dye
illuminated by a laser sheet and (b) food dye with ambient lighting.
CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

74
Great effort was made in the present experiments to reduce the level of
imperfections in the apparatus, but of course, no apparatus is perfect. Figure 5.2 shows
flow visualizations of the breakdown region at  = 1.75, Re = 1853, using both
florescent dye illuminated by a laser sheet and food dye with ambient lighting. The
level of imperfection is apparently lower than in the experiments of Spohn et al. (1998)
and somewhere in between the “perfect” and the 0.4º tilted cases of Sotiropoulos et al.
(2002), although the causes of the imperfections in the various apparatus are probably
different. Note that the pictures in Fig. 5.2 have been reversed from the original blue
dye with white background in order to improve their contrast. The dye sheet seen in
the figure is steady, it does not precess, and there is a clear m = 1 azimuthal
wavenumber associated with it. That being said, it must be remembered that although
the dye port is located at the center of the stationary endwall, there is no certainty that

the dye will emerge axisymmetrically about the axis r = 0, and so even in a perfectly
axisymmetric flow, a dye sheet that is released an arbitrarily small distance off-axis
will have a non-axisymmetric appearance with a predominately m = 1 azimuthal
wavenumber (see Lopez and Perry1992).
As Re is increased from 1850, the level of non-axisymmetry observed in the dye
visualizations is gradually reduced. This is consistent with the fact that the basic state
does not have stagnation points on the axis at higher Re (see Fig. 5.1 for Re = 2600),
and hence the flow does not have the structural instability associated with the
stagnation points commented on by Holmes (1984). Figure 5.3 shows dye
visualizations of steady states for  = 1.75 with Re varying from Re = 2020 to Re =
2605, showing how the recirculation bubble disappears with increasing Re, and how
dye released very close to the axis develops into a spiral kink in the vortex breakdown
CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

75
region when the recirculation bubble no longer exists (see the discussion in Hourigan
et al. (1995), regarding spiral dye streaks in axisymmetric vortex flows).

Re = 2020 2212 2306 2423 2452 2508 2605


Fig. 5.3 Dye visualization of steady states at  = 1.75 and Re as indicated.



On increasing Re to 2688, the flow becomes time-periodic and the spiral dye
filament undergoes an oscillatory excursion in the axial direction. The basic steady
state (SS) bifurcates to axisymmetric limit cycle solutions LC1 and LC2 with different
frequencies, and details will be presented in the following sections.

5.4.2 Hopf bifurcations of the basic state
Over the parameter range reported here (
∈[1.64, 1.80] and Re  2800), the
numerical solutions show that the basic steady state (SS) bifurcates to axisymmetric
limit cycle solutions LC1 and LC2 with different frequencies, and that these are
nonlinearly stable to three-dimensional perturbations. For several cases throughout the
parameter regime in question, the three-dimensional governing equations resolving up
to the m = 16 azimuthal wave number have been solved, using the axisymmetric
solution together with small random perturbations in all m  0 modes as initial
conditions. In all cases, the m  0 components of the flow decay toward machine zero.
CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

76
At larger Reynolds number (Re > 3000), some m  0 modes do grow and non-
axisymmetric solutions become stable (Lopez 2006).
To characterize the axisymmetric time-periodic states LC1 and LC2, we use the
oscillations in their kinetic energy as a global measure of their amplitudes, denoted as
E
0
, and the period in E
0
(t), where

rdrdzuE
z
z
r
r
∫∫
Γ=

=
=
=
=
0
1
0
2
00
2
1
(5.1)
u
0
is the 0-th Fourier mode of the velocity field. Figure 5.4 shows computed time-
series of E
0
for LC1 (solid curve) and LC2 (dashed curve), both at Re = 2750 and  =
1.72.



Fig. 5.4 Computed time-series of E
0
for LC1 (solid curve) and LC2 (dashed curve),
both at Re = 2750 and  = 1.72.



Note that near the onset, the spatial characteristics of LC1 and LC2 are very similar

to those of the steady state, SS, from which they bifurcate (see Fig. 5.1 for SS at Re =
2600,  = 1.75). For both limit cycles, the oscillations consist of pulsations in the
vortex breakdown region. Figures 5.5 and 5.6 show contours of the axial component of
velocity w for LC1 and LC2, respectively, both at Re = 2700 and  = 1.72. A
CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

77
significant feature in these sequences is that for a short time interval during the period
of each limit cycle, there exists a small region of reversed flow on the axis precisely in
the vortex breakdown region. Fluid particles originating near the center of the top
stationary disk (where dye is released in the experiment) form a kinked spiral in the
steady flow and in these time-periodic flows the kink oscillates up and down. Note that
if the Hopf bifurcation were to break axisymmetry, then the kink would process in the
azimuthal direction without change of form.

Fig. 5.5 Contours of w for the axisymmetric time-periodic state LC1 at Re = 2700,  =
1.72 at six phases over one oscillation period (T  31.89); there are 20 positive and 20
negative contours quadratically spaced, i.e. contour levels are ± 0.15(i/20)
2
with i =
120. The solid (broken) contours are positive (negative). The left boundary is the
axis and the bottom is the rotating endwall.





Fig. 5.6 Contours of w for the axisymmetric time-periodic state LC2 at Re = 2700,  =
1.72 at six phases over one oscillation period (T  22.01); there are 20 positive and 20
negative contours quadratically spaced, i.e. contour levels are ± 0.15(i/20)

2
with i =
120. The solid (broken) contours are positive (negative). The left boundary is the
axis and the bottom is the rotating endwall.

CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

78
For experiments, Figure 5.7 shows a dye visualization sequence over approximately
one period. The first image in the sequence is very similar to the last. The temporal
variation of the cross correlation coefficient Cr over several periods is shown in Fig.
5.8, which also includes Cr(t) for the steady states at Re = 2395 and Re = 2660. It is
clear that for Re between 2660 and 2688, the steady state has lost stability and a time-
periodic state results. The power spectrum corresponding to Cr(t) for Re = 2688 is also
included in the figure, showing a fundamental frequency of about 0.42 Hz plus a
number of harmonics. This frequency corresponds to a non-dimensional period T  33,
indicating the flow state being the limit cycle LC1. Figure 5.9 presents another typical
time sequences of the dye filament at  = 1.78 and Re = 2704. In this case, the flow
pattern also exhibited up-down oscillation, but this time with the frequency of 0.635
Hz (or non-dimensional period of about 23.2, which is the limit cycle LC2) as shown
in Fig. 5.10. These results are in good agreement with numerical predictions.

t = 0 s 0.333 s 0.667 s 1.000 s 1.333 s 1.667 s 2.000 s 2.333 s


Fig. 5.7 Dye sequence of LC1 at  = 1.75 and Re = 2688, at times as indicated in
seconds (time for the first frame is arbitrarily set to zero).

CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

79





Fig. 5.8 (a) Time series of the cross-correlation coefficient Cr of dye sequences for
steady state at Re = 2395 (dash line) and Re = 2660 (dot-dash line), and for LC1 at Re
= 2688 all at  = 1.75 and (b) the power spectrum for Re = 2688.






t = 0 s 0.25 s 0.50 s 0.75 s 1.00 s 1.25 s 1.50 s 1.625s


Fig. 5.9 Dye sequence of LC2 at  = 1.78 and Re = 2704, at times as indicated in
seconds (time for the first frame is arbitrarily set to zero).

CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

80

.

Fig. 5.10 (a) Time series of the cross-correlation coefficient Cr of dye sequences for
LC2 at  = 1.78 Re = 2704 and (b) the power spectrum of (a).


Though flow visualization can capture the oscillating frequency of the flow as
shown above, hot-film was chosen in the following study as it has an added advantage,
particularly in the vicinity of Hopf and Neimark-Sacker bifurcations, where the flow
exhibits long transients during which dye diffusion limits visualization-based
techniques.

To further verify the symmetry of the flow, we present in Fig. 5.11 time-series from
the two hot-films placed 180º apart on the stationary endplate. Part (a) of the figure is
the output for an LC1 state at Re = 2760 and  = 1.704, and part (b) is for an LC2 state
at Re = 2750 and  = 1.780 (each state is asymptotically stable, and reached from
different conditions). Notice that in both cases, the hot-film outputs are synchronized
(peaks match in time), providing further experimental evidence of the axisymmetric
nature of the limit cycles LC1 and LC2.


CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

81


(a)
0102030
-0.2
-0.1
0.0
0.1

0.2
t
Hot-Film Output


(b)
0102030
-0.2
-0.1
0.0
0.1
0.2
t
Hot-Film Output

Fig. 5.11 Hot-film data (time series over 1 minute) from the two hot films placed 180º
apart on the stationary endplate. Showing (a) an LC1 state at Re = 2760 and  = 1.704,
and (b) an LC2 state at Re = 2750 and  = 1.780, (each state is asymptotically stable).


Now we examine the Hopf bifurcation lines determined by numerical calculations.
Figure 5.12 shows how the amplitudes of oscillation (measured by E
0
) of LC1 and
LC2 vary with Re for various values , indicating the onset of oscillatory flow is via
supercritical Hopf bifurcations. The solid (open) symbols in the figure correspond to
the primary (secondary) limit cycle to bifurcate from the steady state as Re is increased
CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

82

for fixed ; for  < 1.72, LC1 is the first to bifurcate and for  > 1.72, LC2 bifurcates
first.


Fig. 5.12 Variation of E
0
with Re for (a) LC1 and (b) LC2, at various values of  as
indicated. The solid curves with filled symbol indicate that for the corresponding
values of , the limit cycle solution results from a primary supercritical Hopf
bifurcation; for the dotted curves with open symbols, the limit cycle solution at the
corresponding  bifurcates at a second Hopf bifurcation from the basic state, and
becomes stable at a Neimark-Sacker bifurcation at a higher Re. All the symbols (both
open and filled) correspond to stable limit cycles.



It is virtually impossible to distinguish between LC1 and LC2 based on a snapshot
of the flows. The primary characteristic distinguishing them is their period of
oscillation. Figure 5.13 shows how the periods of LC1 and LC2 vary with Re for
various . As is typical for a Hopf bifurcation, the period does not vary much with the
parameter near the bifurcation. For both limit cycles, the period is virtually
independent of Re and has only slight variation with . Figure 5.14 shows how the
period and corresponding frequency, averaged over Re, <T> and <> = 2/<T>, vary
with . The frequencies <> agree very well with the Hopf frequencies reported by
Gelfgat et al. (2001) from their linear stability analysis.

CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

83

Fig. 5.13 Variation of the period T with Re for the periodic states (a) LC1 and (b) LC2,
at various values of  as indicated. The solid curves with filled symbol indicate that
for the corresponding values of , the limit cycle solution results from a primary
supercritical Hopf bifurcation; for the dotted curves with open symbols, the limit cycle
solution at the corresponding  bifurcates at a second Hopf bifurcation from the basic
state, and becomes stable at a Neimark-Sacker bifurcation at a higher Re. All the
symbols (both open and filled) correspond to stable limit cycles.






Fig. 5.14 Variation with  of (a) the periods and (b) the frequencies of LC1 and LC2,
averaged over Re.


The overall dynamic behavior in a neighborhood of the double Hopf bifurcation
point is shown in Fig. 5.15. The parameter space is divided into six regions, delimited
by bifurcation curves. The number of solutions and their stability is different in each
CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

84
region. The curves of H1 and H2 are numerically determined Hopf bifurcation curves,
where upon crossing these curves, LC1 and LC2 bifurcate from the steady basic state,
respectively; where NS1 and NS2 are Neimark-Sacker curves, where upon crossing
these LC1 and LC2 respectively change their stability, but continue to exist either side
of the curves. The double Hopf point at 

dH
 1.723 and Re
dH
 2661 is where these
four curves meet. For the two Hopf bifurcation curves, only H1 is for   
dH
and H2
is for   
dH
. These were determined numerically by extrapolating in Re the
amplitude of the oscillations of the limit cycles LC1 and LC2 to zero at various fixed
values of  as shown in Fig. 5.12. For the other halves of the two Hopf curves, the
respective limit cycles become unstable before their amplitudes vanish, and so good
extrapolations were not obtained. However, the solution of (, Re) at which they
become unstable gives good estimates of the Neimark-Sacker curves NS1 and NS2. In
the coexistence region between NS1 and NS2, there is also a quasi-periodic mixed
mode, but in this problem it is unstable and so we only observe it indirectly in the early
transients of the flow evolutions. Above the Hopf curves, all initial conditions
eventually evolve to either LC1 or LC2. This is a characteristic signature of a “type I
simple” double Hopf bifurcation (using the nomenclature of Kuznetsov 1998) case.
CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

85




Fig. 5.15 State diagram in (, Re) space; × axisymmetric limit cycles LC1 with period
of T
1

 31; •, axisymmetric limit cycles LC2 with period of T
2
 22. The curves H
1
and
H
2
are supercritical Hopf bifurcation curves at which LC1 and LC2 bifurcate from the
steady basic state SS. The curves NS
1
and NS
2
are Neimark-Sacker bifurcation curves
at which LC1 and LC2 lose stability and a quasiperiodic mixed mode QP is spawned.
In the wedge-shaped region between the curves NS
1
and NS
2
, four states co-exist: LC1
and LC2, which are both stable, and SS and QP which are both unstable.


5.4.3 Detailed experimental results
Before presenting the experimental results in detail, it is important to reiterate that
the hot-films used in the present study were not calibrated. Nevertheless, the range of
Re (and therefore, velocity) covered in the present study is quite small (less than 5%).
Within this small range of velocity variation, the temporal variation in the hot-film
signal is approximately proportional to the temporal variation in velocity.
Re
CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

86

To experimentally establish the characteristics of the double Hopf bifurcation and
the associated Neimark-Sacker bifurcations, the same strategy as in the numerical
work of Marques et al. (2002) was adopted in this study, i.e., once a desired state was
obtained at a particular point in (, Re) parameter space, a quasi-static parameter
sweep was carried out either by fixing  and varying Re or vice versa. Fixing  and
varying Re allow us to estimate the critical Re for the Hopf bifurcations at particular
values of , and by fixing Re and varying , we estimate Neimark-Sacker bifurcation
curves. Following this approach, the results obtained from the hot-film measurements
are summarized in a state diagram Fig. 5.16.

Although the figure may appear confusing, we find that this is still the best way of
presenting the data (after trying several other methods) as it gives an overall picture of
the states and their stability boundaries in the neigborhood of double-Hopf point as
predicted by numerics as shown in Fig. 5.15. There are three distinct states that are
observed: the steady axisymmetric basic state SS which is stable below both Hopf
curves, H1 and H2; the two axisymmetric period states, LC1 (stable above Hopf curve
H1 and to the left of Neimark-Sacker curve NS1) and LC2 (stable above Hopf curve
H2 and to the right of Neimark-Sacker curve NS2), so that in between NS1 and NS2
both LC1 and LC2 are stable, between H1 and NS2 both LC1 and LC2 exist but only
LC1 is stable, and between H2 and NS1 both LC1 and LC2 exist but only LC2 is
stable. How these various bifurcation curves were determined experimentally is
described in the following subsections.


CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

87

Fig. 5.16 Experimentally determined state diagram. The square symbols correspond to
stable limit cycle solutions LC1 (filled square) and LC2 (open squares) obtained from
sweeps in Re at fixed  and the triangle symbols correspond to stable LC1 (filled
triangles) and LC2 (open triangles) obtained from sweeps in  at fixed Re. The filled
stars are LC1 states that evolved from an LC2 initial condition on crossing the
Neimark-Sacker curve NS2, and the open stars are LC2 states that evolved from an
LC1 initial condition on crossing the Neimark-Sacker curve NS1. The experimentally
determined Hopf curves, H1 and H2, and Neimark-Sacker curves, NS1 and NS2, are
shown as dashed curves. The cross symbols correspond to Hopf bifurcation points.

5.4.3.1 Fixed Λ, variable Re
The procedure adopted for this part of the experiment is as follows: For a given ,
the rotation speed of the disk was started impulsively from rest to a speed
corresponding to a predetermined Reynolds number. Upon reaching the preset
condition, the flow was allowed to stabilize over a period of time, usually more than
two viscous times (a viscous time unit corresponds to R
2
/, which is about 250 s,

CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

88
depending on the temperature and glycerin/water ratio). Once the hot-film signal
remained unchanged for more than one minute, the flow was presumed to have
reached an asymptotic state, and the hot-film signal was then sampled by a PC for
subsequent analysis. Next, a parameter sweep was made along constant  by reducing
Re (the rotation speed of the disk) very slowly, but stopped at each predetermined Re

to obtain the required hot-film measurements in a similar manner as discussed above,
until the vicinity of the Hopf bifurcation was reached. Figure 5.17 shows the temporal
evolution to an LC1 state when the rotating disk was impulsively starting from rest at
 = 1.704. The left column of the figure is the time series of hot-film output, and the
right column is the corresponding amplitude of FFT results.
0204060
-0.4
-0.2
0.0
0.2
0.4
t (s)
Hot-film output
0 0.31 0.46 1 2
0.00
0.04
0.08
f
Amplitude

0 204060
-0.4
-0.2
0.0
0.2
0.4
t (s)
Hot-film output
0 0.31 0.46 1 2
0.00

0.04
0.08
f
Amplitude

0 204060
-0.4
-0.2
0.0
0.2
0.4
t (s)
Hot-film output
00.31 1 2
0.00
0.04
0.08
f
Amplitude


Fig. 5.17 Hot-film data (time series over 1 minute and corresponding the amplitude of
FFT results) taken (a) 4 minutes, (b) 7 minutes, and (c) 16 minutes after start-up from
rest with Re = 2760 and  = 1.704, showing evolution to an LC1 state.

(a)

(b)






(c)
CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

89
It can be seen from the figure that at early stages of the flow development (4
minutes  0.86 viscous time units), the hot-film signal is quasi-periodic with two
dominant frequencies 0.31 Hz and 0.46 Hz, corresponding to the non-dimensional
periods T
1
of  31.9 (LC1) and T
2
of  21.5 (LC2), respectively (time has been scaled
by 1/). With increasing time, the LC1 mode becomes progressively more dominant,
until about 16 minutes after start-up (3.44 viscous time units) when LC1 prevailed. At
another nearby point in parameter space,  = 1.76 and Re = 2806 (see Fig. 5.18), the
initial evolution again has a quasi-periodic character typical of the competition
between the two limit cycles, but in this case LC2 eventually wins.


0 204060

-0.4
-0.2
0.0
0.2
0.4
t (s)
Hot-film output

0 0.29 0.42 1 2
0.00
0.04
0.08
f
Amplitude


0 204060
-0.4
-0.2
0.0
0.2
0.4
t (s)
Hot-film output

0 0.29 0.42 1 2
0.00
0.04
0.08
f

Amplitude


0 204060
-0.4
-0.2
0.0
0.2
0.4
t (s)
Hot-film output

00.42 1 2
0.00
0.04
0.08
f
Amplitude

Fig. 5.18 Hot-film data (time series over 1 minute and corresponding the amplitude of
FFT results) taken (a) 9 minutes, (b) 12 minutes, and (c) 18 minutes after start-up from
rest with Re = 2806 and  = 1.76, showing evolution to an LC2 state.
(a)

(b)







(c)
CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

90
5.4.3.2 Coexistence of the two limit cycles LC1 and LC2
In this part of the investigation,  was kept constant and measurements were taken
at different Re. The results obtained in this manner are represented by open and filled
squares in Fig. 5.16. For  >1.739 it is found that LC2 can be obtained by impulsively
starting the rotating disk from rest, and likewise for  < 1.728 LC1 can be obtained in
the same manner. Depending on the initial starting condition, it is found that both LC1
and LC2 can be obtained at the same  and Re. Figures 5.19 and 5.20 show this very
clearly for  = 1.733, where the first figure shows the evolution to LC2 following an
impulsive start of the bottom rotating disk from rest to a predetermined speed
(corresponding to Re = 2750), and the second figures shows how LC1 was generated,
also from rest, but with Re increasing gradually to Re = 2750 at the rate Re/t  50/s.
This is just one of many experimental observations we have made of the coexistence of
the two limit cycles LC1 and LC2 at the same point in parameter space (, Re); which
one dominates depends on the initial starting condition. The coexistence region is in
between the two Neimark-Sacker curves NS1 and NS2 (see Fig. 5.16).
0204060
-0.4

-0.2
0.0
0.2
0.4
t (s)
Hot-film output
0 0.30 0.43 1 2
0.00
0.04
0.08
f
Amplitude

0204060
-0.4
-0.2
0.0
0.2
0.4
t (s)
Hot-film output
0 0.30 0.43 1 2
0.00
0.04
0.08
f
Amplitude

(a)







(b)

CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

91


0204060
-0.4
-0.2
0.0
0.2
0.4
t (s)
Hot-film output
00.43 1 2
0.00
0.04
0.08

f
Amplitude

Fig. 5.19 Hot-film data (time series over 1 minute and corresponding the amplitude of
FFT results) showing the evolution to an LC2 state at  =1.733 taken (a) 17 minutes,
(b) 23 minutes, and (c) 32 minutes after start-up impulsively from rest to Re = 2750.




0 204060
-0.4
-0.2
0.0
0.2
0.4
t (s)
Hot-film output
0 0.30 0.43 1 2
0.00
0.04
f
Amplitude

0 204060
-0.4
-0.2
0.0
0.2
0.4

t (s)
Hot-film output
00.30 1 2
0.00
0.04
0.08
0.12
f
Amplitude

0 204060
-0.4
-0.2
0.0
0.2
0.4
t (s)
Hot-film output
00.30 1 2
0.00
0.04
0.08
0.12
f
Amplitude

Fig. 5.20 Hot-film data (time series over 1 minute and corresponding the amplitude of
FFT results) showing the evolution to an LC1 state at  =1.733 taken (a) 10 minutes,
(b) 20 minutes, and (c) 25 minutes after starting gradually from rest to Re = 2750 at a
rate of Re/t  50/s.

(a)








(b)








(c)
(c)
CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

92
5.4.3.3 Determination of critical Reynolds numbers for the Hopf bifurcations
With  fixed, reducing Re toward the Hopf bifurcation point led to a reduction in
the amplitude of the hot-film signal (or velocity variation) as the flow approached the
steady state. This behavior can be clearly seen in Fig. 5.21 for an LC2 mode at  =
1.769. The critical Reynolds number is then determined by extrapolating to zero the
peak to peak amplitude of the hot-film. Figure 5.22 shows such an extrapolation from

the data in Fig. 5.21, giving an estimated Re
c
 2647 for LC2 at  = 1.769. Figures
5.23 and 5.24 present a similar results for LC1 with  = 1.728. Using this procedure,
the critical Reynolds numbers for other aspect ratios were also obtained to produce the
Hopf bifurcation curves H1 for the onset of LC1 and H2 for the onset of LC2; their
loci in (, Re) parameter space is presented in Fig. 5.25 as filled and hollow circles,
respectively. The figure also shows the computed H1 and H2 curves. The experimental
measurements of the critical Re for the Hopf bifurcations at the various  are within
2% of the computed values.

0246810
-0.3
-0.2
-0.1
0.0
0.1
0.2
0
.
3
Re = 2804 Re = 2776 Re = 2746 Re = 2724
Re = 2698 Re = 2660 Re = 2650
t
Hot-film output


Fig. 5.21 Hot-film outputs over 10 seconds, taken once flow transients had died down,
of the LC2 state at  = 1.769 for various Re as indicated.