Studies of vortex breakdown and its stability in a confined cylindrical container 1

CHAPTER 3 NUMERICAL SIMULATION METHOD 333.2 GOVERNING EQUATIONS AND BOUNDARY CONDITIONS 34 CHAPTER 4 SPIRAL VORTEX BREAKDOWN STRUCTURE AT HIGH ASPECT RATIO CONTAINER 49 4.3.1 Generatio

STUDIES OF VORTEX BREAKDOWN AND ITS STABILITY IN A CONFINED CYLINDRICAL

CONTAINER

CUI YONGDONG

NATIONAL UNIVERSITY OF SINGAPORE

2008

STUDIES OF VORTEX BREAKDOWN AND ITS STABILITY IN A CONFINED CYLINDRICAL

CONTAINER

CUI YONGDONG

(B Eng., M Eng.)

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF MECHANICAL ENGINEERING

2008

ACKNOWLEDGEMENTS

First and foremost, I would like to express my sincere gratitude to my adviser, Professor T T Lim for his constant advice, encouragement, and guidance that have contributed much toward the formation and completion of this thesis

I wish to thank Professor J M Lopez, Department of Mathematics and Statics of the Arizona State University, for his invaluable suggestions for this study and allowing me

to use his numerical results in this thesis

I am deeply indebted to A/P S T Thoroddsen of Department of Mechanical Engineering for his help in the analysis of image cross-correlation and Dr Lua Kim Boon for his assistance in the Labview programming

I am also grateful to the Technical Staffs of the Fluid Mechanics Laboratory for their valuable technical assistance and for setting up the experimental apparatus

Deep thanks go to every member of my family and my many friends for their encouragements and their confidence in me

Last but not least, I would like to express my appreciation to Temasek Laboratories of the National University of Singapore for supporting me to do my Ph.D at the Department of Mechanical Engineering

TABLE OF CONTENTS

Pages ACKNOWLEDGEMENTS i

1.2.2 Flow structures and dynamic behaviors in unsteady flow regime 10

1.2.3 Mode competition in unsteady flow regime 15

1.2.4 Vortex breakdown control and flow under modulation 16

CHAPTER 3 NUMERICAL SIMULATION METHOD 33

3.2 GOVERNING EQUATIONS AND BOUNDARY CONDITIONS 34

CHAPTER 4 SPIRAL VORTEX BREAKDOWN STRUCTURE AT

HIGH ASPECT RATIO CONTAINER

49

4.3.1 Generation of an S-shape vortex structure and a spiral-type vortex

5.4.3.1 Fixed Λ, variable Re

5.4.3.2 Coexistence of the two limit cycles LC1 and LC2

5.4.3.3 Determination of critical Reynolds numbers for the Hopf

bifurcations 5.4.3.4 Oscillation periods of LC1 and LC2

5.4.3.5 Fixed Re, variable Λ

8790929496

CHAPTER 6 QUENCHING OF UNSTEADY VORTEX BREAKDOWN

OSCILLATIONS VIA HARMONIC MODULATION

105

6.4.2 Harmonic forcing of LCN: Temporal characteristics 1126.4.3 Harmonic forcing of LCN: Spatial characteristics 124

CHAPTER 7 HARMONICALLY FORCING ON A STEADY

ENCLOSED SWIRLING FLOW

SUMMARY

A combined experimental and numerical study on vortex breakdown structure and its dynamic behavior in a confined cylindrical container driven by one rotating endwall has been performed The experiments included flow visualization and hot-film measurements, and the numerical simulation included the solution by solving axisymmetric and the three-dimensional Navier-Stokes equations The thesis offers

detail investigations on the following specific issues: spiral vortex breakdown

structure, mode competition between two axisymmetric limit cycles, vortex breakdown oscillation control and effects of modulation of the rotating endwall on the flow state

The first issue to be addressed is whether an S-shape vortex structure and a type vortex breakdown can be produced under laboratory condition as predicted by numerical simulations Our experiments with flow visualization confirm the existence

spiral-of an S-shape vortex structure and a spiral-type vortex breakdown for the aspect ratio

of H/R as low as 3.65 The results also further show that a bubble-type vortex breakdown in a low aspect ratio container is extremely robust and introducing flow asymmetry merely distorts the bubble geometry without it transforming into an S-shape vortex structure or a spiral-type vortex breakdown

The second issue to be addressed is the mode competition between two axisymmetric limit cycles in the neighborhood of a double Hopf point, with the mode competition taking place wholly in the axisymmetric subspace A combined experimental and numerical study was performed Hot-film measurements provide, for the first time, experimental 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 arealso 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

Another motivation of this study is to explore vortex breakdown oscillations’ control through predetermined harmonic modulation and to investigate the effects of modulation on the flow state As far as we are aware, this study has not been attempted before The experimental and numerical results show that the low-amplitude modulations can either enhance the oscillations of the vortex breakdown bubble (for low frequencies) or quench them (for high frequencies) Enhancing the oscillations can

be beneficial in some applications where mixing is desired, such as micro-bioreactors

or swirl combustion chambers Suppressing the oscillations can be a potential means in other applications where unsteady vortex breakdown is prevalent, such as the tail buffeting problem

Overall, the objectives of this study have been fulfilled, and the present study has made some valuable contributions to our understanding of the flow physics in the confined cylindrical container with one rotating endwall Each topic holds a tremendous challenge to the author as it has not been attempted before, and much meticulous attention has been paid to capture the flow behavior, particularly in the experiment It is the laboratory observations that make the complicated dynamic behavior able to be understood more tangibly

NOMENCLATURE

A relative amplitude of modulation

CCD charged-coupled device

CTA constant temperature anemometry

DNS direct numerical simulation

LDA laser Doppler anemometry

H height of the flow domain (height of the stationary cylinder)

MRW modulated rotating wave

PIV particle image velocimetry

R radius of the flow domain (inner radius of the stationary cylinder)

Re Reynolds number = ΩR2/ν

SO(2) A system has SO(2) symmetry if it is invariant under rotation about

its rotation axis

T Period of the oscillating flow

t time scaled by 1/Ω

t* dimensional time, seconds (s)

u velocity in radial direction

v velocity in azimuthal direction

w velocity in azimuthal direction

r radial direction in cylindrical coordinate

z vertical direction in cylindrical coordinate

Greek Symbols

Λ aspect ratio = H/R

θ azimuthal direction in cylindrical coordinate

μ viscosity of the working fluid

ν kinematic viscosity of the working fluid

ρ density of the working fluid

τ non-dimensional period of the oscillating flow = ΩT

Ω angular frequency of the rotating endwall

Ωf angular modulation frequency of the rotating endwall

ωf non-dimensional modulation frequency = Ωf /Ω

Fig 1.2 Laser cross-section of vortex breakdown structures at Λ = 2.5

for different Reynolds numbers Flow images were captured using florescent dye

3

Fig 1.3 Vortex breakdown structures at Re = 1900 for different aspect

ratios Flow images were captured using food dye

4

Fig 1.4 Stability boundaries for single, double and triple breakdowns,

and boundary between oscillatory and steady flow from Escudier (1984)

7

Fig 2.1 Schematic drawing of the overall experimental setup 23

Fig 2.2 Schematic drawing of the confined cylinder setup 24

Fig 2.3 A photograph of function generator with a modified knob

Fig 2.4 Typical dye sequence of flow structures at Γ = 1.75 and Re =

2688, at times as indicated in seconds (time for the first frame is arbitrarily set to zero)

28

Fig 2.5 (a) Time series of the cross-correlation coefficient Cr of dye

sequences for Re = 2688 and Λ = 1.75, with also steady state at

Re = 2395 (dash line) and Re = 2660 (dot-dash line) (b) Correspondingpower spectrum for Re = 2688

29

Fig 2.6 Schematic of electronic circuit for constant temperature

anemometer (CTA)

31

Fig 3.1 Flow configuration in a confined cylinder with one rotating

Fig 3.2 Time history of kinematic energy Ek with various grid densities

for Λ = 2.5, Re = 2494

43

Fig 3.3 Contours of ψ, Γ and η for the axisymmetric steady-state

solution at H/R = 2.5 and Reynolds number as indicated; there are 20 positive and negative contour levels determined by c-level (i) = [min/max]x(i/20)3 respectively

45

Fig 3.4 Time history of kinematic energy Ek at Λ = 2.5, Re = 2765,

showing the time-periodic flow state The filled squares and the alphabets correspond to the images in Fig 3.5

46

Fig 3.5 Instantaneous streamline contours of ψ, for the axisymmetric

time-periodical solution at Λ = 2.5, Re = 2765; there are 20 positive and negative contours determined by c-level (i) = [min/max] x (i/20)3, with ψ ∈[-0.007, 0.0002]

48

Fig 4.1 Schematic drawing of the first set of apparatus 52

Fig 4.2 Schematic drawing of the second set of apparatus 52

Fig 4.3 Results of Escudier (1984) showing the initiation and evolution

of a bubble-type vortex breakdown with increasing Re for H/R

= 3.5 HI denotes a “helical instability” which is a manifestation

of an offset dye injection as highlighted by Hourigan et al (1995)

53

Fig 4.4 Generation and evolution of vortex structures with increasing

Re for Λ = 3.5 obtained in the present study HI denotes

“helical instability” of dye filament Note that the results of Escudier appear larger because the radial distances are uniformly stretched by about 8% due to the refraction at the various interfaces The vertical distances of separation between the two bubbles in both Figs 4.3 and 4.4 matched each other

54

Fig 4.5 Generation and evolution of vortex structure with increasing Re

for Λ = 4.0 Note the formation of an S-shaped structure in (c) and a spiral-type breakdown in (f)

56

Fig 4.6 Generation and evolution of vortex structure with increasing Re

for Λ = 3.75 Note the formation of a helical instability in (a) and (b), S-shaped structure in (e), and a spiral-type breakdown

in (f)

56

Fig 4.7 Generation and evolution of vortex structure with increasing Re

for Λ = 3.65 Note the increasing size of the bubble-type vortex breakdown with the Reynolds number before it disintegrated into an S-shaped structure

58

Fig 4.8 Evolution of vortex breakdown with increasing aspect ratio Λ

for a fixed Re = 3149

60

Fig 4.9 Close up view of the evolution of downstream vortex

breakdown structure with increasing aspect ratio Λ for a fixed

Re = 3149

60

Fig 4.10 Vortex breakdown generated in the presence of an eccentric

rotating cone with Λ = 2.5 and Re = 2500 (a) ε/R = 0.05 (b) ε/R = 0.10 Photographs depicted in (i) are the negatives of the vortex structures obtained using food dye and those in (ii) are the corresponding laser cross sections using fluorescent dye The rotating endwall is located at the top of each photograph, and notice how the dye filament is displayed from the axis of symmetry of the container in the proximity of the cone

63

Fig 4.11 Vortex structures obtained using different eccentricity of the

cone on the base plate with Λ = 2.5 and Re = 2000 Sequence (a)-(d) show the effect of increasing eccentricity These pictures are the negatives of the original pictures captured using food dye released from the apex of the cone The rotating endwall is located at the top of each picture, and the cone is at the stationary bottom wall

65

Fig 4.12 Vortex structures obtained using identical flow conditions as in

Figs 4.11(b) and 4.11(d) Here, some background dye has been introduced in the flow domain prior to the experiment These pictures clearly show the presence of a bubble-type vortex breakdown Note how the dye filament in each picture spirals around the peripheral of the vortex breakdown

65

Fig 4.13 Laser cross sections of the vortex structures generated using

identical flow conditions as in Fig 4.11, Λ = 2.5 and Re = 2000 The pictures were captured using laser induced fluorescent dye illuminated with a thin laser sheet The distortion of the vortex breakdown with increasing eccentricity can be seen in (c) and (d)

66

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 is the rotating endwall

71

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

73

Fig 5.3 Dye visualization of steady states at Λ = 1.75 and Re as

indicated

75

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

(dashed curve), both at Re = 2750 and Λ = 1.72 76 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

77

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

77

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)

78

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

79

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)

79

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)

80

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)

81

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 symbols 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

82

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 symbols 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 Γ 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

83

Fig 5.14 Variation with Λ of (a) the periods and (b) the frequencies of

LC1 and LC2, averaged over Re

83

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

85

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 Λ and the triangle symbols correspond to stable LC1 (filled triangles) and LC2 (open triangles) obtained from sweeps in Λ

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

87

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

Λ = 1.704, showing evolution to an LC1 state

88

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

Λ = 1.76, showing evolution to an LC2 state

89

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 Λ =1.733 taken (a) 17 minutes, (b) 23 minutes, and (c)

32 minutes after start-up impulsively from rest to Re = 2750

91

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 Λ =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

91

Fig 5.21 Hot-film outputs over 10 seconds, taken once flow transients

had died down, of the LC2 state at Λ = 1.769 for various Re as indicated

92

Fig 5.22 Variation with Re of the peak-to-peak amplitudes of the time

series shown in Fig 5.21

93

Fig 5.23 Hot-film outputs over 10 seconds, taken once flow transients

had died down, of the LC1 state at Λ = 1.728 for various Re as indicated

93

Fig 5.24 Variation with Re of the peak-to-peak amplitudes of the time

series shown in Fig 5.23

94

Fig 5.25 Bifurcation curves The curves H1, H2, NS1, and NS2 are the

numerically determined Hopf and Neimark-Sacker bifurcation curves The filled and hollow circles are experimental estimates

of the Hopf bifurcations H1 and H2, determined by fixing Λ, measuring the amplitude of the oscillation at various Res and extrapolating in Re to zero amplitude The symbols + are experimentally observed LC1 states that evolved from an LC2 initial condition on crossing the Neimark-Sacker curve NS2 as

Λ was quasi-statically reduced, and the symbols × are experimentally observed LC2 states that evolved from an LC1 initial condition on crossing the Neimark-Sacker curve NS1 as

Λ was quasi-statically increased

94

Fig 5.26 Variation with Re of the oscillation period T (scaled by1/Ω) of

(a) LC1 and (b) LC2, for Λ as indicated

96

Fig 5.27 Variation with Λ of the non-dimensional period averaged over

Re, <T> for LC1 and LC2; the open symbols are experimentally measured, the filled symbols are numerically computed, and the lines are best fits to the computed data

96

Fig 5.28 Hot-film data (time series over 1 minute and corresponding the

amplitude of the FFT results) taken after transients have died down (more than two viscous time units), showing an LC1 state

at Re = 2750 and (a) Λ = 1.693 and (b) Λ = 1.780

98

Fig 5.29 Hot-film data (time series over 1 minute and corresponding the

amplitude of the FFT results) taken after transients have died down (more than two viscous time units), showing an LC1 state

at Re = 2700 and (a) Λ = 1.716 and (b) Λ = 1.757

99

Fig 5.30 Hot-film data (time series over 1 minute and corresponding the

amplitude of the FFT results) taken at times as indicated, starting with an LC1 state at Re = 2700 and Λ = 1.757 and increased to Λ = 1.780

100

Fig 5.31 Hot-film data (time series over 1 minute and corresponding the

amplitude of the FFT results) taken after transients have died down (more than two viscous time units), showing an LC2 state

at Re = 2700 and (a) Λ = 1.739, (b) Λ = 1.728 and (c) Λ = 1.716

101

Fig 5.32 Hot-film data (time series over 1 minute and corresponding the

amplitude of the FFT results) taken at times as indicated, starting with an LC2 state at Re = 2700 and Λ = 1.716 and decreased to Λ = 1.704

101

Fig 6.1 (a) Time series of hot-film output at Λ = 2.5 and Re = 2800,

and (b) variation with Re of the peak-to-peak amplitude of the hot-film output, both for the natural (unmodulated) limit cycle state LCN

110

Fig 6.2 Dye flow visualization of the central core region of LCN at Λ =

2.5 and Re = 2800 at various times; the period is about 36.2 (the time for the first frame has been arbitrarily set to zero)

113

Fig 6.3 Hot-film output time series and corresponding power spectral

density for Λ = 2.5, Re = 2800 with forcing frequency ωf = 0.1 and forcing amplitude A as indicated In (b) and (d) the hot-film outputs from both channels are plotted

114

Fig 6.4 Phase portraits (with Wa and Ww as the horizontal and vertical

axes,respectively) of the numerical solutions at Re = 2800, Λ = 2.5, ωf = 0.10 (ωf/ω0 ≈ 0.576) and A as indicated

115

Fig 6.5 Critical forcing amplitude, Ac, versus the forcing frequency ωf,

and versus ωf /ω0, at Re =2800 and Λ = 2.5; (b) is an enlargement of (a) highlighting some of the resonance horns The small solid symbols are the numerically determined loci of Neimark–Sacker bifurcations (the curve joining these symbols

is only to guide the eye), and the open diamonds are the corresponding experimental estimates Below the Neimark–

Sacker curve the QP state is observed, above it LCF is observed

In the regions enclosed by the dotted curves and open circles (there are three, near ωf /ω0 ≈1/3, 4/3, and 2/1) the flow is locked to a limit cycle with frequency 0.5ωf , and the star symbols are experimentally determined edges of the period-doubled region near ωf /ω0 =1.33

117

Fig 6.6 Phase portraits (with Wa and Ww as the horizontal and vertical

axes, respectively) for Re = 2800, Λ = 2.5, A = 0.02 and ωf /ω0

as indicated

118

Fig 6.7 Power spectral density of hot-film output for Λ = 2.5, Re =

2800 with forcing amplitude A = 0.08 and forcing frequency ωf

as indicated

119

Fig 6.8 Enlargement of Figure 6.5 near the 2:1 resonance horn There

are three bifurcation curves separating regions where the locked

LCL, the forced LCF, and the quasi-periodic state QP are found The solid curves with filled circles are the Neimark–Sacker bifurcation curves separating QP and LCF, the dashed curve with filled triangles is the period-doubling bifurcation curve separating LCF and LCL, and the solid curves with filled squares are saddle-node-on-invariance circle (SNIC) bifurcation curves on which the QP state synchronizes to the LCL state The other symbols are loci of experimentally observed QP (open circles), LCL (filled diamonds) and LCF (open squares) The two dotted curves at ωf /ω0 = 1.96 and 2.0 are one-parameter paths along which the variation with A in the power at ωn and

ωf are shown in Fig 6.11

121

Fig 6.9 (a) Phase portraits in the neighborhood of the 2:1 resonance for

QP at ωf /ω0 ≈ 0.1965 and A = 0.005 just outside the resonance horn and for LCL at ωf /ω0 = 2.0 and A = 0.005 inside the resonance horn; and (b) are the corresponding Poincáre sections

122

Fig 6.10 Phase portraits in the neighborhood of the 2:1 resonance at ωf

/ω0 = 2.0 showing a reverse period-doubling bifurcation of limit cycles as A is increased

124

Fig 6.11 Variation of the experimentally measured power (normalized

by the power of LCN) with A in the neighborhood of the 2:1 resonance horn: the open symbols correspond to the power at the natural frequency ω0 and the filled symbols correspond to the power at the forcing frequency ωf; the circles correspond to

LCL inside the horn at ωf /ω0 ≈ 2 and the triangles correspond to

QP just outside the horn at ωf /ω0 ≈ 1.96

124

Fig 6.12 Phase portraits (with Wa and Ww as the horizontal and vertical

axes, respectively) at Re = 2800, Λ = 2.5, ωf = 0.5 (ωf /ω0 ≈ 2.88) and A as indicated The dashed circle in the four panels is

LCN, included for reference

126

Fig 6.13 Dye flow visualization of the central core region of a forced

state at Λ = 2.5, Re = 2800, and A = 0.04 at roughly equispaced times over one forcing period for (a) ωf = 0.1 Tf = 2π/ωf = 62.84 (b) ωf = 0.2 Tf = 2π/ωf = 31.42 (the time for the first frame has been arbitrarily set to zero)

127

Fig 6.14 Power spectral density of hot-film output for Λ = 2.5, Re =

2800 with forcing frequency ωf = 0.2 and forcing amplitude A

as indicated

128

Fig 6.15 Dye flow visualization of the central core region of a forced

state at Λ = 2.5, Re = 2800, ωf = 0.5 and A = 0.04 at various times; the forcing period Tf = 2π/ωf = 12.57 (the time for the first frame has been arbitrarily set to zero)

129

Fig 6.16 Power spectral density of hot-film output for Λ = 2.5, Re =

2800 with forcing frequency ωf = 0.5 and forcing amplitude A

as indicated

130

Fig 6.17 Fluorescent dye illuminated with a laser sheet through a

meridional plane of LCF at Re = 2800, Λ = 2.5, A = 0.04, and

ωf = 0.5 at various times over about two forcing periods Note the spatial variation with time of the dark region in the bottom left corner

130

Fig 6.18 Computed streamlines of LCF over one forcing period 2π/ωf

(time increases from left to right) at Re = 2800, Λ = 2.5, A = 0.04, for increasing values of ωf: (a) ωf = 0.1 (ωf / ω0 ≈ 0.576), (b) ωf = 0.2 (ωf / ω0 ≈ 1.15), (c) ωf = 0.3 (ωf / ω0 ≈ 1.73), (d) ωf

= 0.4 (ωf / ω0 ≈ 2.31), (e) ωf = 0.5 (ωf / ω0 ≈ 2.88), (f ) ωf = 0.6 (ωf / ω0 ≈ 3.46)

132

Fig 6.19 Computed azimuthal vorticity contours LCF over one forcing

period 2π/ωf at Re = 2800, H/R = 2.5, A = 0.04, for increasing values of ωf: (a) ωf = 0.1 (ωf / ω0 ≈ 0.576), (b) ωf = 0.2 (ωf / ω0

≈ 1.15), (c) ωf = 0.3 (ωf / ω0 ≈ 1.73), (d) ωf = 0.4 (ωf / ω0 ≈ 2.31), (e) ωf = 0.5 (ωf / ω0 ≈ 2.88), (f ) ωf = 0.6 (ωf / ω0 ≈ 3.46)

133

Fig 6.20 Streamlines (left two figures) and contours of the azimuthal

vorticity (right two figures) for LCF at Re = 2800, Λ = 2.5, A = 0.04, and ωf = 0.5 (showing a snapshot in time) and for the (unstable) basic state which was computed in Lopez et al (2001) at Re = 2800, Λ = 2.5

135

Fig 7.1 The steady axisymmetric basic state at Re = 2600 and Λ = 2.5

(a) flow visualization using food dye (only the axial region is shown), (b) computed streamlines ψ, and (c) computed azimuthal component of vorticity η There are 10 positive (red) and negative (blue) contours quadradically spaced, i.e contour levels are [min/max] x (i/10)2, and ψ ∈[-0.00702, 8.305 x 10-5],

η ∈[−4.12, 21.68]

138

Fig 7.2 Power spectral density from time-series of hot-film output for

flows with Λ = 2.5, Re = 2600, ωf = 0.2 and forcing amplitudes

A as indicated

139

Fig 7.3 Power spectral density from time series of hot-film output for

flows with Λ = 2.5, Re = 2600, A = 0.02 and forcing frequency

Fig 7.5 Peak-to-peak amplitudes of hot-film output with varying

forcing frequency ωf at Λ = 2.5, Re = 2600, and A = 0.01 and 0.02 The three dotted vertical lines indicate the Hopf frequencies of the three most dangerous modes of the basic state at Re = 2600, as determined by Lopez et al 2001, where

ωH1 = 0.1692, ωH2 = 0.1135 and ωH3 = 0.2181

142

Fig 7.6 Computed variation with ωf of the peak-to-peak amplitude of

the kinetic energy relative to the kinetic energy of the basic state, ΔE/E0, and scaled by ωf0.5, of the synchronous state for A

= 0.01, Λ = 2.5 and various Re as indicated The three dotted vertical lines indicate the Hopf frequencies of the three most dangerous modes of the basic state at Re = 2600, as determined

by Lopez et al 2001, where ωH1 = 0.1692, ωH2 = 0.1135 and

ωH3 = 0.2181

144

Fig 7.7 Time sequences of contours of ψ at Re = 2600, Λ = 2.5, A =

0.01 and ωf as indicated; there are 16 positive (red) and negative (blue) contours quadradically spaced, i.e contour levels are [min/max]x (i/16)2, and ψ ∈ [-0.00757, 0.0002972]

145

Fig 7.8 Snap-shots of the azimuthal vorticity modulation, η(t)-η0

(where η0 is the steady η for A = 0), at various ωf as indicated, all at Re = 2600, Λ = 2.5, A = 0.01 and at the same phase in the forced modulation There are 15 positive (blue) and 15 negative (red) contour levels with η∈[−0.2, 0.2]; some clipping particularly for the ωf = 0.171 case is clearly evident

146

Fig 7.9 Snap-shots of the azimuthal vorticity modulation, η(t)-η0

(where η0 is the steady η for A = 0) for (top row) the natural limit cycle at Re = 2800 and Λ = 2.5, and (bottom row) the synchronous state at Re = 2600, Λ = 2.5, A = 0.01 and ωf = 0.171 There are 15 positive (blue) and 15 negative (red) contour levels with η ∈[−0.2, 0.2]

147

Fig 7.10 Snap-shots of the azimuthal vorticity modulation, η(t)-η0

(where η0 is the steady η for A = 0), at various ωf as indicated, all at Re = 800, Λ = 2.5, A = 0.01 and at the same phase in the forced modulation There are 10 positive (red) and 10 negative (blue) contour levels with η ∈ [−0.01, 0.01]

148