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

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 my Ph.D at the Department of Mechanical Engineering. i TABLE OF CONTENTS Pages ACKNOWLEDGEMENTS i TABLE OF CONTENTS ii SUMMARY v NOMENCLATURE vii LIST OF FIGURES ix LIST OF TABLES xx CHAPTER INTRODUCTION 1.1 MOTIVATION 1.2 LITERATURE REVIEW 1.2.1 Steady flow structures 1.2.2 Flow structures and dynamic behaviors in unsteady flow regime 1.2.3 Mode competition in unsteady flow regime 1.2.4 Vortex breakdown control and flow under modulation 6 10 15 16 1.3 OBJECTIVES AND APPROACH 19 1.4 ORGANIZATION OF THESIS 20 CHAPTER DESCRIPTION OF EXPERIMENT 22 2.1 EXPERIMENTAL SETUP 22 2.2 FLOW VISUALIZATION 27 2.3 CROSS-CORRLATION OF IMAGES 27 2.4 HOT-FILM MEASUREMENT 30 ii CHAPTER NUMERICAL SIMULATION METHOD 33 3.1 INTRODUCTION 33 3.2 GOVERNING EQUATIONS AND BOUNDARY CONDITIONS 34 3.3 METHOD OF SOLUTION 37 3.4 METHOD VERIFICATION 42 CHAPTER SPIRAL VORTEX BREAKDOWN STRUCTURE AT HIGH ASPECT RATIO CONTAINER 49 4.1 INTRODUCTION 49 4.2 EXPERIMENTAL SETUP AND PROCEDURE 50 4.3 RESULTS AND DISCUSSIONS 4.3.1 Generation of an S-shape vortex structure and a spiral-type vortex breakdown 4.3.2 Can spiral-type vortex breakdown be produced in a low aspect ratio container? 52 52 CONCLUDING REMARKS 66 4.4 61 CHAPTER COMPETITION OF AXISYMMETRIC TIMEPERIODIC MODES 68 5.1 INTRODUCTION 68 5.2 EXPERIMENTAL METHOD 69 5.3 NUMERICAL METHOD 70 5.4 RESULTS AND DISCUSSION 5.4.1 Basic state 5.4.2 Hopf bifurcations of the basic state 5.4.3 Detailed experimental results 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 70 70 75 85 87 90 92 5.5 CONCLUDING REMARKS 94 96 103 iii CHAPTER QUENCHING OF UNSTEADY VORTEX BREAKDOWN OSCILLATIONS VIA HARMONIC MODULATION 105 6.1 INTRODUCTION 106 6.2 NUMERICAL METHOD 107 6.3 EXPERIMENTAL METHOD 109 6.4 RESULTS AND DISCUSSIONS 6.4.1 The nature limit cycle LCN 6.4.2 Harmonic forcing of LCN: Temporal characteristics 6.4.3 Harmonic forcing of LCN: Spatial characteristics 109 109 112 124 6.5 CONCLUDING REMARKS 134 CHAPTER HARMONICALLY FORCING ON A STEADY ENCLOSED SWIRLING FLOW 136 7.1 INTRODUCTION 136 7.2 EXPERIMENTAL METHOD 137 7.3 NUMERICAL METHOD 137 7.4 RESULTS AND DISCUSSIONS 137 7.5 CONCLUDING REMARKS 149 CHAPTER CONCLUSIONS AND RECOMMENDATIONS 152 8.1 CONCLUSIONS 152 8.2 RECONMMENDATIONS 157 PUBLICATIONS 159 ACHIEVEMENT 161 REFERENCES 162 iv 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 spiraltype vortex breakdown can be produced under laboratory condition as predicted by numerical simulations. Our experiments with flow visualization confirm the existence 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 Sshape 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 v 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 are 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. 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. vi NOMENCLATURE A relative amplitude of modulation CCD charged-coupled device CTA constant temperature anemometry DNS direct numerical simulation LC limit cycle LDA laser Doppler anemometry ff frequency of modulation 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 RW rotating wave SO(2) A system has SO(2) symmetry if it is invariant under rotation about its rotation axis T Period of the oscillating flow Tf Period of modulation 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 vii r radial direction in cylindrical coordinate z vertical direction in cylindrical coordinate Greek Symbols Γ angular momentum = vr Λ aspect ratio = H/R ψ Streamfunction 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 viii CHAPTER INTRODUCTION 1.2 Literature Review In this section, a literature survey on the studies of the flow in a confined cylinder with one rotating endplate will be presented. Attention is focused on vortex breakdown structure, dynamical behavior, vortex breakdown control and modulation of the rotating endplate. 1.2.1 Steady flow structures Vogel (1968, 1975) was the first to identify regions of recirculation bubble with a stagnation point on the center axis (commonly referred to as vortex breakdown) under certain combinations of Λ and Re. Subsequent LDA (laser Doppler anemometry) measurements by Ronnenberg (1977) confirmed the existence of a steady vortex breakdown structure. A more comprehensive study was conducted by Escudier (1984) who used a flow visualization technique to map out a “stability” diagram for the occurrence of one, two or three bubbles type vortex breakdown for the aspect ratio of up to 3.5 (see Fig. 1.4). He also found an upper limit of the Reynolds number, above which the flow is oscillatory, before becoming turbulence at even higher Reynolds number. He postulated that the vortex breakdown is inherently axisymmetric and any departures from axisymmetry are the result of instabilities not directly associated with the breakdown process. CHAPTER INTRODUCTION Fig. 1.4 Stability boundaries for single, double and triple breakdowns, and boundary between oscillatory and steady flow from Escudier (1984) [with permission from Springer]. In parallel with the experimental studies, numerical investigations have also been conducted. For example, the computational studies based on the solution of the steady and unsteady axisymmetric Navier-Stokes equations were carried out by Lugt and Huassling (1982), Lugt and Abbound (1987), Neitzel (1988), Lopez (1990), Brown and Lopez (1990), Tsitverblit (1993), Sørensen and Christensen (1995), and Gelfgat et al. (1996). Lopez (1990) depicted precisely the inner structure of vortex breakdown using axisymmetric simulations, and based on these results Brown and Lopez (1990) further examined the underlying physics of vortex breakdown. They highlighted the important features of divergence of core flow, and correspondingly the ‘wavy motion’ phenomenon, which suggests that the appearance of the recirculation bubble is associated with it. They postulated that under some conditions, the tilting and CHAPTER INTRODUCTION stretching of vorticity would lead to the production of an azimuthal component of vorticity which induces a reversed flow on the axis, i.e. the vortex breakdown of the central vortex. On the other hand, Gelfgat et al. (1996), who conducted a numerical study and linear stability analysis on the flow characteristics up to the aspect ratio Λ = 3.5, concluded that vortex breakdown in such a confined container is a continuous evolution of the stationary meridional flow, and not a result of instability. They explained that the change of signs of the azimuthal vorticity is a necessary, but not sufficient condition for vortex breakdown, and the concave shape of the streamsurfaces of a confined swirling flow may be considered as an additional characteristic of vortex breakdown. They argued that the swirling flow, which is created by the centrifugal and the Coriolis forces, may lead to the appearance of recirculation bubbles when the local maximum of the centrifugal force is sufficiently large. Though the steady flow structures can be well determined by both the experiment and the numerics, there are still some arguments about the fundamental aspects of the flow structure, i.e. if the basic state (steady state) is axisymmetric and how it breaks the symmetry. The results of Lopez (1990) showed that the streamlines inside the breakdown bubbles are closed for steady flow. Lopez and Perry (1992) further analyzed the unsteady flow using non-linear dynamical system theory combined with axisymmetric numerical simulations. They indicated that the observation of streaks downstream of the vortex breakdown was due to the oscillation of the flow, rather than the true flow structures. Hourigan et al. (1995) studied the streaklines just before the formation of the vortex breakdown bubble, and argued that the emergence of spiral streaklines was due to an artifact of the flow visualization (small offsets of the dye injection from the central axis) and not the occurrence of three-dimensional flow. On the contrary, Spohn, et al. (1998) using the flow visualization technique (i.e. CHAPTER INTRODUCTION electrolytic precipitation technique) showed that the stationary vortex breakdown bubbles in the container flow are open and asymmetric at their downstream end. They attributed the asymmetry of the flow structure to the existence of asymmetric flow separations on the container wall, which suggests that the asymmetries of the breakdown bubbles are not artifacts of the visualization technique as has been argued by Hourigan et al (1995) and Stevens et al (1999), but rather very real flow features (see Sotiropoulos and Ventikos 2001). Unlike those who have conducted numerical simulations with axisymmetric assumption, Sotiropoulos and Ventikos (1998, 2001) presented their results by solving directly the unsteady, three-dimensional Navier-Stokes equations and analyzing the Lagrangian characteristics of the flow fields. They stated that the asymmetric features of the stationary vortex breakdown bubbles along the axis are related to asymmetries that originate inside the stationary-cover Ekman layer and the sidewall Stewartson boundary layer, and that in the unsteady regime the spiral separation lines are due to the emergence of counter-rotating pairs of spiral vortices related to the centrifugal instability of the boundary layer on the container wall. They argued that the apparent discrepancy between the solutions of the axisymmetric Navier-Stokes equations and laboratory experiments may be due to the fact that axisymmetric assumption simulates the flow in an ideal environment (perfect condition), and that imperfections are unavoidable in laboratory experiments. Nevertheless, they also indicated that the threedimensionality is excited by the distorted structure of the curvilinear mesh in their numerical simulations. On the contrary, recently Blackburn and Lopez (2000, 2002) and Marques and Lopez (2001), with their three-dimensional direct numerical simulation (DNS) results, believe the symmetry breaking is due to instability of the swirling jet produced by the turning of the Ekman layer on the stationary wall. CHAPTER INTRODUCTION 1.2.2 Flow structures and dynamic behaviors in unsteady flow regime Though the controversy remains over the questions of the symmetry breaking of the flow and of vortex breakdown, more attentions have recently turned to the transition from steady to unsteady, from axisymmetry to three-dimensional aspects of the flow structures for Reynolds numbers beyond some critical values at certain aspect ratios. Lopez and Perry (1992a and 1992b) reported two distinct modes of oscillation for Λ = 2.5 by solving axisymmetric Navier-Stokes equations. The mode with Re < 3500 had pulsation of the vortex breakdown bubbles by periodically coalescing and separating with period τ1 = ΩT1 ≈ 36, corresponding to the branch that bifurcates from the basic state at Re ≈ 2700. The other mode with Re > 3500 consisted of wave traveling on the central axis from the stationary endwall to the rotating endwall with the period τ3 ≈ 28. Westergaard et al. (1993) studied the oscillating states for Λ = 2.0 and Re ranging from 3000 to 8000 using particle image velocimetry (PIV) and hot-film measurements. Their PIV measurements showed that the period of the oscillating was about 23.6 for Re = 3000, 22.6 for Re = 5000, and were further confirmed by the hot-film measurements but without detailed descriptions. However, they noticed that hot-film measurements exhibited a complicated time behavior with a periodic or quasi-periodic behavior between Re = 2500 and 6500. Later, Tsitverblit and Kit (1998) used linear stability analysis to further demonstrate that the axisymmetric oscillations in a cylindrical container are not the result of a local instability of the central vortex core. Instead, the onset of unsteady 10 CHAPTER INTRODUCTION flow is due to the occurrence of traveling Taylor-Gőrtler vortices along the container wall. In related work with numerical and experimental studies on the vortex breakdown generated by a rotating cone, Pereira and Sousa (1999) noted that when the Reynolds number exceeded a critical value, the oscillatory regimes are caused by a supercritical Hopf bifurcation. Their numerical simulations also revealed that at lower Reynolds number (Re = 2700) for Λ = 3, the breakdown still exhibits axisymmetric geometry, however, when the Reynolds number reaches 3100, the re-circulating bubbles are seen merging and separating during the oscillation process with a fully non-symmetric behaviour. Based on their numerical calculations, they attributed this behavior to a shift from a bubble to a spiral breakdown. Stevens et al. (1999) further demonstrated through both axisymmetric numerical and experimental studies (flow visualization) the existence of multiple oscillatory states in the flow for Λ = 2.5 over a range of Re far from the onset of the oscillation; two of the states were found to be periodic and the third one was quasi-periodic. For 2700 < Re < 3400, the state had a period of τ1 ≈ 36, and when 3400 < Re < 3500, the period switched to τ3 ≈ 28, which existed beyond Re ≈ 5000. On the other hand, the quasi-periodic state had a primary period of τ2 ≈ 57 over 3200 < Re < 3700, with lowfrequency modulation of two orders of magnitude smaller. In another approach, the multiplicity of time-dependent states was analyzed by Lopez et al. (2001) with the linear stability analysis of the basic state, including the subsequent bifurcations of the basic state, to axisymmetric perturbations. They predicted the first and third periodic branches which were observed in experiments by Stevens et al. (1999). This multiplicity of time-dependent state behavior was further confirmed by three-dimensional direct numerical simulation (DNS) results of 11 CHAPTER INTRODUCTION Blackburn and Lopez (2000). Later, Blackburn and Lopez (2002) presented detailed analysis with direct numerical simulation (DNS) results for Λ = 2.5 and with Reynolds number up to 4300. They revealed that the first axisymmetric and periodic branch bifurcated at Re ≈ 2707 with the period of τ1 ≈ 36, and remained stable to threedimensional perturbations up to Re ≈ 3500. Another branch had a characteristic period of τ3 ≈ 28.5, which was stable to axisymmetric perturbations and unstable to azimuthal perturbations with either m = or m = wavenumbers, leading to modulated rotating wave (MRW) states. Each of these two co-existing MRW states had two frequencies related to them; one being the frequency of the underlying axisymmetric oscillation, which was close to τ3, and the other being the precession frequency of the azimuthal wave structure. The third branch had the characteristics of a very-low-frequency modulation, which was stable to three-dimensional perturbations over the middle range of Re for which it existed (3250 < Re < 3683). And, lost stability to MRWs with m = azimuthal modes via supercritical Hopf-type bifurcations from the axisymmetric quasiperiodic state. Using linear stability analysis with three-dimensional perturbations to the basic state, Gelfgat et al. (2001) showed that the transition from a steady to an axisymmetric oscillatory state is a result of a supercritical Hopf bifurcation for aspect ratio 1.63 < Λ < 2.76. This confirmed previous results in the related range of aspect ratio. For < Λ < 1.63, a rotating wave (RW) state with azimuthal wavenumber m = was bifurcated through a symmetry-breaking Hopf bifurcation. This has been subsequently confirmed by DNS results of Marques et al. (2002). For Λ > 2.76, a rotating wave state with azimuthal wavenumber m = or was bifurcated through a symmetry-breaking Hopf bifurcation. This has also been confirmed by DNS results of Marques and Lopez (2001). 12 CHAPTER INTRODUCTION However, Gelfgat et al. (1996) and Tsitverblit and Kit (1998) both noted a discrepancy between their predicted critical Reynolds number and that obtained experimentally by Escudier (1984) for the aspect ratio Λ ≥ 3. Gelfgat et al. (1996) postulated that, for a large aspect ratio (Λ > 3), the discrepancy was due to threedimensional effects, and could not be explained by a subcritical axisymmetric bifurcation. Tsitverblit and Kit (1998) also believed that “the precession of the lower breakdown structure” as noted by Escudier (1984) signifies the onset of threedimensionality, however they did not analyze the flow stability with respect to threedimensional disturbances. Marques and Lopez (2001) further analyzed this discrepancy using fully nonlinear three-dimensional Navier-Stokes computations, and attributed the disagreement to the different views of the same flow that takes place in different spatial domains. They pointed out that in experiments, the main focus is often on the axis region, and therefore wave number = mode is not noticed, only the processing m = mode is observed at Re = 3,000. They also indicated that the state m = was an instability via a supercritical Naimark-Sacker bifurcation of the m = RW state, and not an instability of the steady axisymmetric basic state. This is further strengthened by the linear stability analysis of Gelfgat et al. (2001) which shows that for Λ > 3, the basic state is not unstable to mode m = for Re < 4000. With a free surface set-up, Hirsa et al. (2002) also demonstrated experimentally that for Λ = the instability arises from a supercritical symmetry breaking Hopf bifurcation to a rotating wave with azimuthal wave number m = 4, which is due to the instability of the shear layer. More recently, Lopez (2006) investigated numerically the complicated spatiotemporal transition behavior at the double Hopf bifurcation point for Λ = 1.72 with variation of Re up to 7000, where there are two axisymmetric time-periodic modes 13 CHAPTER INTRODUCTION (referred to as limit cycles, LC1 and LC2). The results revealed that LC1 lost stability via a symmetry-breaking Neimark-Saker bifurcation at Re ≈ 5500, and LC2 lost stability via saddle-node bifurcation and jumped to another branch LC3 at Re ≈ 4400. This LC3 was unstable to a saddle-node of limit cycles bifurcation at Re ≈ 4265 and jumped back to LC2, and experienced a period-doubling bifurcation to LC3PD at Re ≈ 5840. This LC3PD became unstable to a symmetry-breaking Hopf bifurcation at Re ≈ 6050. Apart from these axisymmetric limit cycle states, Lopez also detected a nonsymmetric state - a rotating wave RW2 with azimuthal wavenumber m = 2, being stable for Re ≥ 3425. This branch of RW2 lost stability at higher Re to modulated rotating waves. The experiments of Sørensen et al. (2006) investigated in detail the transition scenario from steady and axisymmetric flow to unsteady and three-dimensional flow in the range of ≤ Λ ≤ 3.5. Combining the high spatial resolution of PIV with the temporal accuracy of LDA, they captured different modes of the dominant azimuthal waves and the route of transition from steady and axisymmetric flow to unsteady and three-dimensional flow. Their results show that experimentally determined neutral, critical Reynolds numbers and associated frequencies of the perturbed velocity field, and azimuthal periodicities, patterns are in good agreement with numerical stability analysis of Gelfget et al (2001). They also found the existence of a stable triplet of helical vortices that may explain the steady behavior of the k = mode in the range of 3.3 ≤ Λ ≤ 3.5. While unlike usual investigations conducted at lower aspect ratio, Serre and Bontoux (2001, 2002) carried out numerical studies of vortex breakdown structure for high aspect ratio with Λ = 4.0, where the axial confinement is believed to be reduced and the flow approaches a more unstable state. Their numerical solutions based on 14 CHAPTER INTRODUCTION three-dimensional N-S equations, showed the emergence of a periodic regime at Re = 3500 through the axisymmetric Hopf bifurcation. They also found that for Re > 3500, the flows were characterized by spiral arms evolving in helical structures in the central region of the flow, followed by a rotating wave of azimuthal wavelength m = through a secondary Hopf bifurcation. When Re = 4500, the downstream vortex breakdown (closer to the rotating endwall) evolved into an “S-shape vortex structure”, and is accompanied by a precessing motion about the axis, before a spiral-type vortex breakdown is generated on the upstream side (i.e. closer to the stationary endwall) at a higher Reynolds number. The term “S-shape vortex structure” was used by Serre and Bontoux (2002) and referred to a wavy vortex filament that resembles an “S” and which gives an indication of “the first step of the vortex breakdown shift from the bubble to the spiral type”. 1.2.3 Mode competition in unsteady flow regime Competition between distinct modes in rotating flows has been and continues to be of both fundamental and practical interest. With rotating flows, the basic state is often axisymmetric, and for the most part, the competition has been studied between modes which break the symmetry (via Hopf bifurcations) to different azimuthal wave numbers, m1 ≠ m2 ≠ 0, where mi are the azimuthal wavenumbers of the competing modes (e.g., Iooss and Adelmeyer 1998; Marques, et al. 2002; Lewis and Nagata 2003; Lopez and Marques 2004). Mode competitions between multiple coexisting states have a long history in, for example, Taylor-Couette flow (Coles 1965; Golubitsky and Stewart 1986; Chossat and Iooss 1994) and in differentially rotated systems (Hide and Titman 1967; Früh and Read 1999). More recently, Marques et al. (2002) investigated the mode interactions at the double Hopf bifurcation points (Λ = 1.583, Re = 2627) by 15 CHAPTER INTRODUCTION using a three-dimensional Navier-Stokes solver. They identified that an axisymmetric limit cycle (m = 0) and a rotating wave bifurcate simultaneously at the double Hopf point, where the two-tori states are also spawned there. They indicated that the two bifurcated stable solutions, the m = limit cycle and the m = rotating wave, lost stability by colliding with the two-torus at their respective Naimark-Sacker curves. They attributed the collision of the jet, which is formed when the Ekman layer on the rotating endwall is turned by the stationary sidewall, with the top stationary endwall to the instability of the rotating wave (m = 2); while for the axisymmetric limit cycle state, the instability is due to how the jet turns at the top and collides with itself near the axis. 1.2.4 Vortex breakdown control and flow under modulation Having reviewed the studies on the flow structures and their dynamic behavior, we now switch to the phenomenon of vortex breakdown control. Interest in controlling vortex breakdown has received significant attention for swept-wing aircraft at high angles of attack, where the lift is primarily generated by the vortices shed from the delta wings or leading-edge extensions. A characteristic of these vortices is that above a critical angle of attack, they suffer vortex breakdown, and the associated unsteadiness induces unsteady buffet loads on the aircraft’s vertical stabilizers, leading to premature fatigue failure. There have been numerous studies on how to control and/or modify the breakdown of these vortices in order to alleviate the tail buffeting (see Mitchell and Délery 2001, for a comprehensive review), but for the most part the strategies have focused on either shifting the location of the vortex breakdown or on inhibiting its occurrence. And yet, it seems that the problem is not that the vortex has broken down per se, but rather that the temporal characteristics of the unsteady vortex 16 CHAPTER INTRODUCTION breakdown excite the tail buffeting. There have been very few investigations of control strategies on vortex breakdown where the goal has been to affect its temporal characteristics. Unsteady blowing on model aircraft has been explored, but such problems have so many variables due to the complex geometry that more generic fundamental studies are called for. The recent experimental investigations into the effects of periodic axial pulsing in a more idealized flow (Khalil et al. 2006) have also focused on shifting the location of the breakdown. Following the experimental study of Escudier (1984), the vortex breakdown of the swirling flow in the axis region of an enclosed cylinder driven by the rotation of an endwall has been a very popular setting for fundamental investigations of vortex breakdown. The confined geometry leads to a well-posed problem where steady axisymmetric vortex breakdown recirculation zones are readily realized in laboratory experiments and simulated numerically (e.g. Lopez 1990). More recently, this flow geometry has been used to investigate a number of strategies for the control of the steady axisymmetric vortex breakdown. Herrada and Shtern (2003) numerically investigated the thermal suppression of steady axisymmetric vortex breakdown by means of an imposed axial temperature gradient which induces via centrifugal convection a large scale meridional circulation opposing (or enhancing, depending on the sign of the temperature gradient) the recirculation on the axis. Earlier, using the Boussinesq approximation, Lugt and Abboud (1987) also showed that an imposed axial temperature gradient could suppress steady axisymmetric vortex breakdown. Husain, Shtern and Hussain (2003) experimentally studied the control of steady axisymmetric vortex breakdown by the co- or counter-rotation of a slender rod placed along the axis. Mununga et al. (2004) showed experimentally that a small differentially rotating disk embedded into the stationary endwall could be used to affect a similar 17 CHAPTER INTRODUCTION control of the steady axisymmetric vortex breakdown. All of these open-loop control studies were restricted (by design) to parameter regimes where in the absence of the control strategies, the flow was steady and axisymmetric. Gallaire et al (2004) conducted a closed-loop control study of vortex breakdown in an idealized pipe flow with the objective being to stabilize the steady axisymmetric columnar vortex, i.e. to suppress the onset of vortex breakdown. On the other hand, there has been much interest in the swirling flow in an enclosed cylinder driven by the rotation of an endwall for applications where a high degree of mixing is desired, such as in micro-bioreactors, albeit at a low level of shear stress (see Yu et al. 2005a, 2005b, 2007; Dusting et al. 2006; Thouas et al. 2007). The interest stems from the very good mixing properties when the flow is operated above the threshold for self-sustained oscillations as these provide chaotic mixing (Lopez and Perry 1992). The concern is, of course, that the chaotic mixing is only present when the Reynolds number is above a critical level for the Hopf bifurcation, and so one would like to achieve comparable oscillations at lower Reynolds numbers, thereby subjecting the biological material to lower damaging stress levels. While the studies above are constrained to constant rotation speed, the modulated rotation problem needs to be addressed since the imposed modulation of the rotating speed can either destabilize an otherwise stable state or stabilize an otherwise unstable state by variation of the amplitude or frequency of the modulation, as reflected in the earlier study of the stability of time-periodic flows for general case by Davis (1976), Donnelly (1990), and the modulated Taylor-Couette problem by Barenghi (1995) to name just a few. 18 CHAPTER INTRODUCTION 1.3 Objectives and Approach From the above literature review, it can be seen that the flow behavior in the confined cylindrical container with one rotating end has been investigated extensively by experimental and numerical or theoretical methods. Although many advances have made, there are still interesting and challenging issues that need to be addressed and to be further explored. The objectives of this study are thus as follows: To study whether the spiral vortex breakdown structure exists under laboratory condition: The numerical study by Serre and Bontoux (2002) at H/R = 4.0 showed that under some conditions, an S-shape vortex structure followed by a spiral-type vortex breakdown could be produced. This finding is most interesting since a spiral-type vortex breakdown in an enclosed cylindrical container has not been produced previously in experiments. To study mode competition between two axisymmetric limit cycles in the neighborhood of a double Hopf point: For the setup used in this study, the system possesses SO(2) symmetry, being invariant to rotation about the axis. It may be expected that there exists the points where the two types of Hopf coincide and interact as is reflected in the numerical study by Marques et al. (2002). Their results indicated the coexistence and competition of the mode with m = limit cycle and the mode with m = rotating wave. A close examination of the linear stability analysis of Gelfgat et al. (1996, 2001) shows that there are two distinct Hopf bifurcations leading to the axisymmetric state with different frequencies at approximately Λ = 1.72. As far as we are aware, no numerical or experimental verification of this finding has been conducted. 19 CHAPTER INTRODUCTION To study vortex breakdown oscillation control via modulation: The dynamics of the system can be manipulated by variation of the amplitude or frequency of the modulation as studied in other apparatus. Thus, vortex breakdown oscillations at supercritical conditions may be suppressed, or the steady vortex breakdown at subcritical conditions can be promoted via modulation. This area has not been investigated before in the literature. To investigate the above mentioned topics, the approach adopted is combining experimental and numerical study to better understand the temporal and spatial flow behavior. 1.4 Organization of Thesis The layout of this thesis is as follows: Chapter briefly introduces the motivation of this study, and presents a detailed literature survey. Based on this literature review, the objectives and approach for this study are formed. Chapter describes the experimental set-up, instrumentation, visualization and measurement techniques employed in the experimental study except for Chapter 4, where a different test-rig was used and the corresponding experimental details are presented in that chapter. Chapter presents a brief description of the numerical scheme for solving the axisymmetric Navier-Stokes and continuity equations in streamfunction / vorticity / circulation forms using a predictor-corrector finite difference method. The quality of the numerical simulation code has also been verified in this chapter by comparison 20 CHAPTER INTRODUCTION with numerical simulation results obtained by independent calculations in other studies and with experimental results. Chapter reports the results of the experimental investigation on the existence of the S-shape vortex structure and a spiral-type vortex breakdown under laboratory conditions. Chapter discusses the experimental and numerical results obtained on the mode competition between two axisymmetric limit cycles in the neighborhood of a double Hopf point (Λ ≈ 1.72 and Re ≈ 2665). Chapter presents the study on the control of unsteady vortex breakdown by a forced harmonic modulation of the rotating endwall (sinusoidal modulation) at a supercritical condition Λ = 2.5, Re = 2800. In Chapter 7, the effects of modulation on the steady flow condition are presented. Finally, in Chapter 8, the main achievements and conclusions arising from the present study are highlighted. This is followed by some recommendations and possible directions for further research on this unique setup. 21 [...]... 2700 and (a) Λ = 1. 739, (b) Λ = 1. 728 and (c) Λ = 1. 716 10 1 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 10 1 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. .. and analyzing the Lagrangian characteristics of the flow fields They stated that the asymmetric features of the stationary vortex breakdown bubbles along the axis are related to asymmetries that originate inside the stationary-cover Ekman layer and the sidewall Stewartson boundary layer, and that in the unsteady regime the spiral separation lines are due to the emergence of counter-rotating pairs of. .. container is a continuous evolution of the stationary meridional flow, and not a result of instability They explained that the change of signs of the azimuthal vorticity is a necessary, but not sufficient condition for vortex breakdown, and the concave shape of the streamsurfaces of a confined swirling flow may be considered as an additional characteristic of vortex breakdown They argued that the swirling flow,... 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 12 1 Fig 6.9 (a) Phase portraits in the neighborhood of the 2 :1 resonance for QP at ωf /ω0 ≈ 0 .19 65 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 12 2 xvi... 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... Herrada and Shtern (2003), Husain, et al (2003), and Mununga et al (2004), to name just a few Although significant advances in understanding the flow behavior in this configuration have been made over the years, a full knowledge of the flow field is still lacking This motivated us to employ this particular configuration - a confined cylindrical container with one rotating endplate, to study vortex breakdown. .. 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 10 0 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... 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 (19 84) 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 (19 95) 53 Fig 4.4 Generation... in different spatial domains They pointed out that in experiments, the main focus is often on the axis region, and therefore wave number = 4 mode is not noticed, only the processing m = 1 mode is observed at Re = 3,000 They also indicated that the state m = 1 was an instability via a supercritical Naimark-Sacker bifurcation of the m = 4 RW state, and not an instability of the steady axisymmetric basic... stretching of vorticity would lead to the production of an azimuthal component of vorticity which induces a reversed flow on the axis, i.e the vortex breakdown of the central vortex On the other hand, Gelfgat et al (19 96), who conducted a numerical study and linear stability analysis on the flow characteristics up to the aspect ratio Λ = 3.5, concluded that vortex breakdown in such a confined container . STUDIES OF VORTEX BREAKDOWN AND ITS STABILITY IN A CONFINED CYLINDRICAL CONTAINER CUI YONGDONG NATIONAL UNIVERSITY OF SINGAPORE 2008 STUDIES OF VORTEX BREAKDOWN. 1 minute and corresponding the amplitude of the FFT results) taken at times as indicated, starting with an LC2 state at Re = 2700 and  = 1. 716 and decreased to  = 1. 704. 10 1 Fig. 6 .1. 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). 81 xiii Fig. 5 .12