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

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 stabil

CHAPTER 5*

COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES

5.1 Introduction

The competition organized by double Hopf bifurcations between an axisymmetric

mode (m1 = 0) and a non-axisymmetric mode (m2 ≠ 0) has been studied numerically by

Marques et al (2002) and Lopez and Marques (2003) In both the m1 ≠ m2 ≠ 0 and m1

= 0, m2 ≠ 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 Λ ≈ 1.72 and Re ≈ 2665

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 [Λ, 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

mixture of glycerin and water (roughly 76% glycerin by weight) with kinematic

viscosity ν = 0.404 ± 0.002 cm2/s at a room temperature of 23.0°C The aspect ratio Λ

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

For the range of aspect ratio of interest (Λ 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, Λ = 1.75 For comparison, the contours of Re

= 1850 are also attached

Fig 5.1 Contours of ψ, u, v, and w for the axisymmetric steady-state solution at Λ =

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

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, Λ = 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 Λ = 1.75 for Re <

2800 and unsteady axisymmetric flow appears at about Re = 2650 It would thus seem

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, Λ = 1.75, using (a) fluorescent dye

illuminated by a laser sheet and (b) food dye with ambient lighting

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 Λ = 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 Λ = 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

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 Λ = 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 (Λ∈[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

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

ΔE0, and the period in E0(t), where

E z u rdrdz

z

r r

∫ ∫==Γ

=

=

=0

1 0

2 0 0

2

1

(5.1)

u0 is the 0-th Fourier mode of the velocity field Figure 5.4 shows computed

time-series of E0 for LC1 (solid curve) and LC2 (dashed curve), both at Re = 2750 and Λ =

1.72

Fig 5.4 Computed time-series of E0 for LC1 (solid curve) and LC2 (dashed curve),

both at Re = 2750 and Λ = 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, Λ = 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 Λ = 1.72 A

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, Λ =

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, Λ =

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

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 Λ = 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 Λ = 1.75 and Re = 2688, at times as indicated in

seconds (time for the first frame is arbitrarily set to zero)

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 Λ = 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 Λ = 1.78 and Re = 2704, at times as indicated in

seconds (time for the first frame is arbitrarily set to zero)

Fig 5.10 (a) Time series of the cross-correlation coefficient Cr of dye sequences for

LC2 at Λ = 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 Λ = 1.704, and part (b) is for an LC2 state

at Re = 2750 and Λ = 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

(a)

-0.2 -0.1 0.0 0.1 0.2

t

(b)

-0.2 -0.1 0.0 0.1 0.2

t

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 Λ = 1.704,

and (b) an LC2 state at Re = 2750 and Λ = 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 ΔE0) of LC1 and

LC2 vary with Re for various values Λ, indicating the onset of oscillatory flow is via

supercritical Hopf bifurcations The solid (open) symbols in the figure correspond to

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

first

Fig 5.12 Variation of ΔE0 with Re for (a) LC1 and (b) LC2, at various values of Λ as

indicated The solid curves with filled symbol indicate that for the corresponding

values of Λ, 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 Λ 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 Λ 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 Λ Figure 5.14 shows how the

period and corresponding frequency, averaged over Re, <T> and <ω> = 2π/<T>, vary

with Λ The frequencies <ω> agree very well with the Hopf frequencies reported by

Gelfgat et al (2001) from their linear stability analysis

Fig 5.13 Variation of the period T with Re for the periodic states (a) LC1 and (b) LC2,

at various values of Λ as indicated The solid curves with filled symbol indicate that

for the corresponding values of Λ, 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 Λ 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 Λ 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

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 ΛdH ≈ 1.723 and RedH ≈ 2661 is where these

four curves meet For the two Hopf bifurcation curves, only H1 is for Λ ≤ ΛdH and H2

is for Λ ≥ ΛdH 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 Λ 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 (Λ, 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

Λ

Fig 5.15 State diagram in (Λ, Re) space; × axisymmetric limit cycles LC1 with period

of T1 ≈ 31; •, axisymmetric limit cycles LC2 with period of T2 ≈ 22 The curves H1 and

H2 are supercritical Hopf bifurcation curves at which LC1 and LC2 bifurcate from the

steady basic state SS The curves NS1 and NS2 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 NS1 and NS2, 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

Re