Flow-Mediated Interaction Between a Vibrating and an Elastically-Mounted Cylinder Zhonglu Lin Department of Engineering University of Cambridge This dissertation is submitted for the degree of Doctor of Philosophy Jesus College September 2019 I would like to dedicate this thesis to my loving parents. Declaration I hereby declare that except where specific reference is made to the work of others, the contents of this dissertation are original and have not been submitted in whole or in part for consideration for any other degree or qualification in this, or any other university. This dissertation is my own work and contains nothing which is the outcome of work done in collaboration with others, except as specified in the text and Acknowledgements. This dissertation contains fewer than 65,000 words including appendices, bibliography, footnotes, tables and equations and has fewer than 150 figures. This thesis includes the content already published in 3 journal papers and 3 conference papers, where the I am the first author and the major contributor. The included publications are listed below: Journal papers Lin, Z., Liang, D. & Zhao, M. Effects of Reynolds Number on Flow Mediated Interaction Between Two Cylinders. J. Eng. Mech. - ASCE (2019). (Accepted) doi:10.17863/CAM.38404 Lin, Z., Liang, D. & Zhao, M. Flow-mediated interaction between a vibrating cylinder and an elastically-mounted cylinder. Ocean Eng. (2018). doi:10.1016/j.oceaneng.2018.04.019 Lin, Z., Liang, D. & Zhao, M. Effects of Damping on Flow-Mediated Interaction Between Two Cylinders. J. Fluids Eng. - ASME (2018). doi:10.1115/1.4039712 Conference proceedings Lin, Z., Liang. D. & Zhao, M. Numerical simulation of fluid coupling between two elastic support cylinders. In Proceedings of the 29th China National Hydrodynamics Symposium (Volume 1) (2018). vi Lin, Z., Liang, D. & Zhao, M. Interaction Between Two Vibrating Cylinders Immersed in Fluid. in The 27th International Ocean and Polar Engineering Conference (International Society of Offshore and Polar Engineers, 2017). Lin, Z., Liang, D. & Zhao, M. Numerical Study of the Interaction between Two Immersed Cylinders. in The 12th International Conference on Hydrodynamics (2016). Zhonglu Lin September 2019 Acknowledgements I would like to thank my supervisor, Dr. Dongfang Liang, for his academic support on my research. My PhD study could not be successful without his support. I would like to express my sincere gratitude to Prof. Ming Zhao from Western Sydney University for his support over my works. I would like to thank my advisor, Dr. Fehmi Cirak, for his critical review over my PhD works. I would like to thank all my friends who supported me during this once seemingly unend- ing journey of PhD study. I cannot go this far without your help. The study was primarily funded by Fuzhou Nuocheng Construction Project Manage- ment Ltd. This work has been performed using resources provided by the Cambridge Tier-2 system operated by the University of Cambridge Research Computing Service (http:// www.hpc.cam.ac.uk) funded by Engineering and Physical Sciences Research Council (EP- SRC) Tier-2 capital grant (EP/P020259/1). The computing at Cirrus is funded by EPSRC Tier-2 Open Access Call. This study used the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk). This study used the Cambridge Service for Data Driven Discovery (https://www.hpc.cam.ac.uk) hosted by the Research Computing Services at the University of Cambridge. I am very grateful for the technical support from the team at the University of Cambridge’s Research Computing Services. Thanks are due to my beloved parents. I thank them dearly for their support and encour- agement for my pursuit of PhD. Abstract This study investigates the interaction between two cylinders of an identical diameter im- mersed in still fluid: the active cylinder is subject to forced vibration, while the adjacent passive cylinder is elastically-mounted with a damper and has only one-degree-of-freedom along the centreline of the two cylinders. The hydrodynamic interaction is simulated with an extensively-validated 2D Navier-Stokes solver that is based on the finite element method and the Arbitrary Lagrangian-Eulerian method. In total, 23,400 simulation cases are conducted for a range of combinations of parameters. The active cylinder’s oscillation frequency f1/ fn ranges from 0.05 to 3.2; the amplitude of the active cylinder A1/D varies from 0.025 to 1.432; the mass ratio of the passive cylinder m⇤ takes the value of 1.5, 1.7, 2.0, 2.2 or 2.5; the structural damping factor of the passive cylinder ranging from 0 to 1.4; the Reynolds number Rem varies from 10 to 315; the gap ratio G/D ranges from 0.2 to 3. This parametric space is chosen to reflect the values usually seen in engineering applications. With considered ranges of parameters, the solid-solid contact does not occur. The flows corresponding to regimes A, A*, C, E, F and G as classified by Tatsuno and Bearman (1990) are investigated. In all these regimes, harmonic frequency components in the response of the passive cylinder are found to persist in all regimes, causing the major and minor resonance. Even though the flow instability in regimes E, F, G causes the significant irregular frequency components to appear in low frequency. In the periodic regimes, the vibration centre of the passive cylinder can be attracted or repelled away from the active cylinder by varying the Reynolds number. The phase difference between the active cylinder and the passive cylinder’s fundamental frequency component experiences a 180 shift with the increase of the active cylinder’s oscillation frequency. All 6 non-dimensional parameters influence the the behaviour of the passive cylinder in different ways and will be discussed in detail in this thesis. Flow patterns in the current two-cylinder case are at large similar to the single-cylinder case, although at regime C, the flow pattern is fundamentally different due to a pulse beating passive cylinder. Overall the existence of the passive cylinder adds irregularity to the flow. Table of contents List of figures xiii List of tables xxvii Nomenclature xxix 1 Introduction 1 1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Single Cylinder Oscillating in Still Fluid . . . . . . . . . . . . . . 3 1.1.2 A Oscillating Cylinder Actuating a Free Cylinder in Still Fluid . . . 5 1.2 Aims and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Methodology 11 2.1 Problem Setup and Dimensional Analysis . . . . . . . . . . . . . . . . . . 11 2.2 Computational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.2 Streamline Upwind Petrov-Galerkin FEM . . . . . . . . . . . . . . 15 2.2.3 Arbitrary Lagrangian-Eulerian Method . . . . . . . . . . . . . . . 18 2.3 Mesh Independence Study . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.5 Data Processing Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 Effects of Parameters Associated With the Active Cylinder in Periodic Regimes 31 3.1 Resonance of Passive Cylinder . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Phase Lag Shift from Anti-phase to In-phase . . . . . . . . . . . . . . . . . 36 3.3 Effects on Flow Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 xii Table of contents 4 Effects of Parameters AssociatedWith the Passive Cylinder in Periodic Regimes 45 4.1 Structural Damping Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.1.1 Approximating the Passive Cylinder’s Motion to be Harmonic in the Context of Varied Structural Damping . . . . . . . . . . . . . . . . 46 4.1.2 Effects of Structural Damping on the Vibration of the Passive Cylinder 48 4.1.3 Force Acting on the Passive Cylinder . . . . . . . . . . . . . . . . 52 4.1.4 Effect of Structural Damping on Flow Field . . . . . . . . . . . . . 55 4.2 Mass Ratio of the Passive Cylinder . . . . . . . . . . . . . . . . . . . . . . 59 4.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5 Effects of Intermediate Fluid and Gap in Periodic Regimes 65 5.1 Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.1.1 Effects of Reynolds Number on the Passive Cylinder’s Vibration Centre Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.1.2 Effects of Reynolds Number on Passive Cylinder’s Vibration Amplitude 70 5.2 Flow Fields Around the Two Cylinders . . . . . . . . . . . . . . . . . . . . 72 5.3 Gap Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6 Effects of Active Cylinder’s Amplitude and Frequency in Less Regular Regimes 93 6.1 Overview of KC Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.2 Regime C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.3 Regime E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.4 Regime F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.5 Regime G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 7 Conclusions 127 7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.2 Parametric Study in Periodic Regimes A and A* . . . . . . . . . . . . . . . 128 7.3 Interactions in Less Regular Regimes C, E, F and G . . . . . . . . . . . . . 128 7.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 References 131 Appendix A Data Processing Scripts 137 A.1 Generation of Time History and Amplitude Spectra . . . . . . . . . . . . . 137 A.2 Mesh Generation by Gmsh . . . . . . . . . . . . . . . . . . . . . . . . . . 142 List of figures 1.1 An offshore oil drilling platform with multiple cylindrical structures (Credit: Chesroc Nigeria Limited) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 (a) Classical Huygens’ pendulum clocks setup (Huygens, 1660) (b) Idealised model by Peña Ramirez et al. (2014). . . . . . . . . . . . . . . . . . . . . 2 1.3 Flow regimes classified by Tatsuno and Bearman (1990). In this thesis, the values of KC and b lies well within regimes A and A⇤, and the flow characteristics in all the simulations do coincide with the regimes A and A⇤: two symmetric vortices shed per half cycle. . . . . . . . . . . . . . . . . . 4 1.4 Simulated flow patterns represented by streaklines in different flow regimes for oscillatory flow past a single cylinder, conducted under the same condition as the experiments carried out by Tatsuno and Bearman (1990). It was simulated by Zhao and Cheng (2014) using the code of current study. (a) Regime A* atKC= 3.14 and b = 52.8 (b) Regime A atKC= 11 and b = 7.4 (c) Regime D at KC = 6.28, and b = 18 (upper) and b = 22.1 (lower) (d) Regime E at KC = 6.28 and b = 25.6 in the 72nd period (left) and 200th period (right). (e) Regime F at KC = 8.16,b = 27 (upper), KC = 12.6,b = 17.8 (lower). (a-e) The oscillation direction is horizontal in all the sub-figures. 6 1.5 (a) The setup sketch of Case B (Gazzola et al., 2012). The active cylinder is forced to oscillated harmonically along the centreline of two cylinders, whereas the passive cylinder is free. This is different from the present study, where the passive cylinder is attached to a spring. (b) Threshold Reynolds number illustrated as a function of d/Dm and Ds/Dm (Gazzola et al., 2012). 7 xiv List of figures 1.6 In inviscid fluid, time history of the passive cylinder’s normalised displace- ment from its initial position Dx/Dm at initial phase of the active cylinder vibration at (a) 0 and (b) 180, as demonstrated by Nair and Kanso (2007). (c)In viscous fluid, time history of slave’s normalised displacement from its initial position Dx/Dm at initial phase of the active cylinder vibration at 0 (solid line) and 180 (dashed line), as illustrated by Gazzola et al. (2012). The initial condition of the active cylinder can affect the drifting direction of the passive cylinder in inviscid fluid, but not in viscous fluid. . . . . . . . . 8 2.1 A sketch of interaction between the two cylinders: While the active cylinder undergoes harmonic forced vibration along the centreline of the two identical rigid cylinders, the passive cylinder is elastically mounted with a damper, vibrating in response to the active cylinder with 1DOF along the centreline as well. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Computational meshes for interaction between two cylinders withG/D= 0.2 and Dr  1.38⇥103 and Nc 152. . . . . . . . . . . . . . . . . . . . . . 20 2.3 Effect of Dr on the responding amplitude A2 with Nc = 116 and Effect of Nc on the responding amplitude A2 with Dr = 1.25⇥ 103, for the case G/D= 0.2,A1/D= 0.1, f1/ fn = 0.825,m⇤ = 2.5. . . . . . . . . . . . . . . 21 2.4 Comparison of the passive cylinder’s displacement time histories from different meshes for G/D = 0.9,A1/D = 0.477,m⇤ = 1.5,z = 0 and (a) Rem = 10, f1/ fn = 3.2, where the passive cylinder drifts towards the near side (b) Rem = 150, f1/ fn = 3.2, where the passive cylinder drifts towards the far side (c) Rem = 150, f1/ fn = 0.725, where the vibration amplitude is large due to the resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5 Computational meshes for interaction between two cylinders withG/D= 0.9 with Dr  1.30⇥103 and Nc 152. . . . . . . . . . . . . . . . . . . . . 24 2.6 Vorticity contours of the mesh independence case atKC= 9,G/D= 2.5,A1/D= 1.432, f1/ fn = 2.8,m⇤ = 2,z = 0,Rem = 315. (a-d) Coarse mesh (e-h) Nor- mal mesh (i-l) Dense mesh. . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.7 Comparison of fluid velocity distribution between the numerical simulation results and the experimental data by Dütsch et al. (1998). The horizontal & vertical velocity components (u & v) of the simulation results (continuous lines) and measured data (discrete symbols) are compared along four hori- zontal lines (i.e. y/D= 0.6,0,0.6 and 1.2) at three different phases (i.e. f = 180,210 and 330). . . . . . . . . . . . . . . . . . . . . . . . . . . 27 List of figures xv 2.8 Computational meshes for validation case of a single cylinder vibrating in an otherwise still fluid with Nc = 134 and Dr = 1.83⇥103. . . . . . . . . . 28 2.9 (a) Threshold Reynolds numbers as function of the normalised centre-to- centre distance d/D, where m⇤ = 1, A1/D= 1, and the active and the passive cylinders have identical diameters. The present simulation output agree well with the results from Gazzola et al. (2012). (b) Passive cylinder’s non- dimensional displacement Y2/D versus the time normalized by the active cylinder vibration frequency t f1, for a series of Reynolds numbers (from top to bottom curves Rem = 100,60,50,40,30,27,25,20,19,18,15,10), with A1/D= 1,d/D= 3.5. Here, the threshold Reynolds number is 20. . . . . . 29 3.1 Convergence time history for a case in periodic regime A/A⇤. After a few initial steps, the vibration amplitude of the passive cylinder converges. . . . 32 3.2 Variation in A2/A1 with f1/ fn at G/D= 0.4,A1/D= 0.0250.1,m⇤ = 1.5, and (a) undamped z = 0 (b)damped z = 0.2. A2 and A1 are linearly correlated unless when the resonance occurs or the damping ratio z >= 0.2. . . . . . 33 3.3 Amplitude spectra showing the response of the passive cylinder with various f1/ fn at G/D = 0.4, A1/D = 0.075, m⇤ = 1.5. (a) FFT spectra at f1 = fw/3⇡ 0.25 (b) FFT spectra at f1 = fw/2⇡ 0.375 (c) FFT spectra at f1 = fw ⇡ 0.75 (d) An overview of FFT spectra. It can be seen that, at f1 = fw/2, there are two distinct frequencies while at f1 = fw there is only one. The red dashed thin line tracks the dominant frequencies. The red dashed thick squares highlight the resonating frequency component. . . . . . . . . . . . 34 3.4 Amplitude spectra showing (a) responding displacement of the passive cylin- der and (b) force coefficient upon the passive cylinder at 0.95  f1/ fn  1.1, G/D = 0.4, A1/D = 0.075, m⇤ = 1.5, z = 0, Rem = 100. The domi- nant frequencies for displacement and force are different. . . . . . . . . . . 35 3.5 Variation of the oscillation phase difference Df21 between the passive and the active cylinder (for the fundamental frequency components) with f1/ fn at A1/D = 0.025 0.100,m⇤ = 1.5,G/D = 0.4,Rem = 100 and (a) z = 0 (b) z = 0.2. The increase of the active cylinder oscillation frequency causes the phase difference to shift from anti-phase to in-phase at the immersed natural frequency. Df21 decreases with A1/D. At z = 0, A1/D affects more the phase difference at low frequencies compared with that at z = 0.2. . . 36 xvi List of figures 3.6 Velocity vectors on the left of the passive cylinder at G/D = 0.4,A1/D = 0.075,m⇤ = 1.5,z = 0,Rem = 100, and f1 = 45 with (a) f1/ fn = 0.74, Df21 = 90.7 and (b) f1/ fn = 1.00, Df21 = 0.58. The vortex is always attached to the surface of the cylinders on their two sides, but it is dissipated before they can detach from the surface. Df21 is found to have significant influence on the flow field. The velocity vector is drawn on every grid point and the vector scale factors are (a) 0.06 grid units/magnitude and (b) 0.6 grid units/magnitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.7 Velocity vectors (u⇤i = ui/ fnD) at f1/ fn = 1.00, f1 = 0. A pair of small vortices show up in the wake of the passive cylinder when the passive cylinder reaches its highest or lowest position. These tiny vortices only exist for a short period of time in each cycle. The velocity vector is drawn on every grid point. The vector scale factor is 1.6 grid units/magnitude. . . . . . . . . . . 38 3.8 Velocity vectors at f1/ fn = 0.74 and f1 = 135 with G/D = 0.4,A1/D = 0.075,m⇤ = 1.5,z = 0,Rem = 100. At exactly 90 < f1 < 180, the zero- velocity point travels from the bottom of the passive cylinder (at f1 = 90) to the top of the active cylinder (at f1 = 180). The velocity vector is drawn on every grid point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.9 Non-dimensional pressure p⇤ contours at f1 = (a & e & i) 0, (b & f & j) 90, (c & g & k) 180, (d & h & l) 270. (a-d) f1/ fn = 0.55, Df21= 168.71; (e-h) f1/ fn = 0.74, Df21 = 90.673; (i-l) f1/ fn = 1.0, Df21 = 0.582 withG/D= 0.4,A1/D= 0.075,m⇤ = 1.5,z = 0,Rem = 100. The motion of the passive cylinder is found to have great influence on the distribution of pressure. Streamlines are plotted to show the status of the fluid. . . . . . . . . . . . . 39 4.1 (a) Variation in magnification factor of the passive cylinder A2/(F0/k) with f1/ fn, and (b) variation in force-displacement phase difference for the passive cylinder (for frequency components with f/ fn = f1/ fn) with f1/ fn atG/D= 0.2, A1/D= 0.1, m⇤= 2.0, and z = 01.4. The red dashed line is the locus of maxima by assuming harmonic force input. . . . . . . . . . . . . . . . . 46 4.2 (a) Variation of A2/A1 and (b) variation of phase difference between the two cylinders Df21 with f1/ fn (for the frequency components with f/ fn = f1/ fn) with f1/ fn at G/D = 0.2 ,A1/D = 0.1, m⇤ = 2.0, and z = 0 1.4. A2/A1 is negatively correlated with z , particularly within the regime of resonance. The phase difference curve converges at Df21 ⇡ 80, f1/ fn ⇡ 0.775. The increase in damping slows down the 180 phase shift of the passive cylinder as f1/ fn reaches its immersed natural frequency. . . . . . . . . . . . . . . . 49 List of figures xvii 4.3 Amplitude spectra showing the responding displacement of the passive cylinder with various f1/ fn at G/D = 0.2, A1/D = 0.1, and m⇤ = 2.0. (a) An overview of the FFT spectra (b) the FFT spectra at f1 = fw/2⇡ 0.375 . The red dashed thin line tracks the dominant frequencies. It can be seen that, at f1 = fw/2, the high frequency component is always smaller than the fundamental component, whereas for the undamped cases, a high-frequency component becomes larger than the fundamental frequency, thus causing a minor peak of resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.4 (a) Zoom-in at peaks for variation of A2/A1 at z = 0.2, and (b) zoom- in at peaks for variation of A2/A1 at z = 0.4 for G/D = 0.4, A1/D = 0.025 0.1, m⇤ = 1.5. A threshold damping ratio of the passive cylin- der is discovered, beyond which its peak amplification factor decreases with the active cylinder vibration amplitude, and under which the peak increases with the amplitude. In this scenario, the threshold lies in z = 0.20.4. . . . 52 4.5 (a–d) Non-dimensional pressure contours p⇤= p/r f 2nD2 atG/D= 0.4,A1/D= 0.075,m⇤ = 1.5, f1/ fn = 0.74,z = 0,Rem = 100, and Df21 = 90.673. The dashed lines indicate negative values of non-dimensional pressure. The colour map shows corresponding colour for each value of non-dimensional pressure. The velocity vectors are plotted every 20 points. The vector scale factor is 0.3 grid units/magnitude. (e) The y-direction force coefficient upon the passive cylinder and its shear and pressure components were examined. Pressure rather than viscosity is the main contributor to the force acting on the passive cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.6 Non-dimensional pressure contours p⇤ = p/r f 2nD2 at G/D= 0.2; A1/D= 0.1; m⇤ = 1.5; f1/ fn = 0.78; and f1 = 0 (a, e, and i), 90 (b, f, and j), 180 (c, g, and k) and 270 (d, h, and l). (a–d) z = 0, Df21 = 46.6; (e–h) z = 0.1, Df21 = 50.0928; (i–l) z = 0.8, Df21 = 68.309. The cylinder on the top is the passive passive cylinder. The cylinder on the bottom is the active active cylinder. The dashed lines indicate negative values of non-dimensional pressure. The color map shows corresponding color for each value of non- dimensional pressure. With the increase in damping, the main driver of the passive cylinder changes from the pressure fluctuation at the far side of the passive cylinder to that within the gap. The black sticks and the dots indicate the displacement of the cylinder in one cycle. . . . . . . . . . . . . . . . . 56 xviii List of figures 4.7 Velocity vectors on the left of the passive cylinder at G/D= 0.2, A1/D= 0.1, m⇤= 1.5, and f1/ fn = 0.75, with (a) z = 0, Df21 = 108.4; (b) z = 0.1, Df21 = 95.62; (c) z = 0.8, Df21 = 68.309 and (1) f1 = 90; (2) f1 = 180. The velocity vector is drawn on every grid point and the vector scale factors are 0.15 grid units/magnitude. The lifespan and the strength of vortices both decrease with damping. The black sticks and the dots indicate the displacement of the cylinder in one cycle. . . . . . . . . . . . . . . . . . . 57 4.8 Velocity vectors in the gap at G/D = 0.2, A1/D = 0.1, m⇤ = 1.5, and f1/ fn = 0.75, with (a) z = 0, Df21 = 108.4; (b) z = 0.1, Df21 = 95.62; and (1) f1 = 158; (2) f1 = 189. The velocity vector is drawn on every grid point and the vector scale factors are 0.011 grid units/magnitude. The zero-velocity point (see dashed circle in b1) travels from the bottom of the passive cylinder to the top of the active cylinder. For the damped case with z = 0.1, the stagnant flow point appears earlier than the undamped case and has a longer lifespan. For both the damped and undamped cases, the stagnant flow point disappears at f1 = 189. The black sticks and the dots indicate the displacement of the cylinder in one cycle. . . . . . . . . . . . . . . . . 58 4.9 Variation of A2/A1 with f1/ fn and m⇤ at G/D = 0.2,A1/D = 0.1,Rem = 100,z = 0. m⇤ affects the resonance frequency. The red dashed rectangle highlights the convergence point. . . . . . . . . . . . . . . . . . . . . . . . 60 4.10 Variation of the oscillation phase difference between the two cylinders with f1/ fn at A1/D= 0.1,m⇤= 1.52.5,G/D= 0.2,Rem = 100,z = 0 the phase change occurs. G/D does not affect much the phase difference . . . . . . . 61 4.11 (a) Variation in A2/A1 with f1/ fn, and (b) variation in displacement phase difference Df21 between the passive cylinder and the active cylinder (for the frequency components with f/ fn = f1/ fn) with f1/ fn atG/D= 0.2,A1/D= 0.1,m⇤ = 1.5 2.5,Rem = 100 and z = 0 0.2. The color and line type denote z , whereas the marker type denotes m⇤. With a non-zero damping, the peak A2/A1 decreases with m⇤. The curve of Df21 disperses at resonance, which follows the same pattern as the undamped cases. . . . . . . . . . . . 62 5.1 Convergence time history for a case with Re= 140. After a few initial steps, the vibration amplitude and the centre drift of the passive cylinder converges to constant values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 List of figures xix 5.2 (a) Variation in vibration centre drift DY 2 with f1/ fn at G/D= 0.9,A1/D= 0.477,m⇤ = 1.5,z = 0 and Rem = 10150. (b) Zoom-in at the secondary and the primary resonance regime (c) Further zoom-in at the primary reso- nance regime (d) Variation in vibration centre drift DY 2 with Rem at G/D= 0.9,A1/D= 0.477,m⇤ = 1.5,z = 0 and f1/ fn = 13.2. . . . . . . . . . . 67 5.3 (a) Variation in DY 2/A1 with f1/ fn and (b) Zoom-in for its resonance regime at G/D= 0.9,A1/D= 0.477,m⇤ = 1.5,Rem = 10,70 and z = 00.2. The marker type denotes z , whereas the line type denotes Rem. . . . . . . . . . 68 5.4 (a) Variation in DY 2/A1 with f1/ fn and (b) Zoom-in for its resonance regime atG/D= 0.9,A1/D= 0.477,z = 0.2,Rem = 10,70 andm⇤= 1.52.5. The marker type denotes m⇤, whereas the line type denotes Rem. . . . . . . . . . 69 5.5 Amplitude spectra showing the responding displacement of the passive cylinder with f1/ fn = 0.053.2 atG/D= 0.9,A1/D= 0.477,m⇤= 1.5,z = 0 with (a) Rem = 10 (b) Rem = 30 (c) Rem = 50 (d) Rem = 110. The dashed thin line tracks the dominant frequencies. . . . . . . . . . . . . . . . . . . 71 5.6 (a) Variation of passive cylinder’s amplification factor A2/A1 with active cylinder’s oscillation frequency f1/ fn at G/D = 0.9,A1/D = 0.477,m⇤ = 2.0,Rem = 10 110 and z = 0 0.2. (b) Zoom-in at resonance regimes. The marker type denotes damping factor z , whereas the line type denotes the Reynolds number Rem. . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.7 (a) Variation of amplification factor A2/A1 with active cylinder’s oscillation frequency f1/ fn (b) Zoom-in at primary and secondary resonance regimes and (c) Variation of phase difference between the two cylinders Df21 with f1/ fn (for the frequency components with f/ fn = f1/ fn) with f1/ fn at G/D = 0.9 ,A1/D = 0.477, m⇤ = 1.5,z = 0,Rem = 10 150, i.e. KC = 3,b = 3.350. A2/A1 is positively correlated with Rem, particularly within the regime of resonance, and the resonance frequency increases with Rem. The resonance amplitude at Rem = 150 is as large as 1.6 times of that at Rem = 10. The phase difference is shifted towards the positive side with the increase of Rem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.8 (a) Variation of passive cylinder’s amplification factor A2/A1 with active cylinder’s oscillation frequency f1/ fn at G/D = 0.9,A1/D = 0.477,z = 0,m⇤ = 1.5 2.5,Rem = 10 110 and (b) Zoom-in at resonance regimes. The marker type denotes mass ratio m⇤, whereas the line type denotes the Reynolds number Rem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 xx List of figures 5.9 Contours of pressure coefficient Cp and velocity vectors (1) in the gap at f1 = 180 and (2) in the far side of the active cylinder at f1 = 0, given G/D= 0.9,A1/D= 0.477,m⇤= 1.5, f1/ fn= 2.8, with (a) Rem= 10, Df21= 30.6; (b) Rem= 50, Df21=16.8; (c) Rem= 110, Df21=12.22. The velocity vector is drawn on every two grid points and the vector scale factors are 0.1 grid units/magnitude. . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.10 Pressure coefficient contours and velocity vectors in the gap at G/D = 0.9,A1/D= 0.477,m⇤= 1.5,z = 0, and f1= 180 with (a) Rem= 10, f1/ fn= 0.65,Df21 = 27.05, (b) Rem = 50, f1/ fn = 0.7,Df21 = 64.66, (c) Rem = 110, f1/ fn = 0.72,Df21 = 71.42, The velocity vector is drawn on every two grid points and the vector scale factors are 0.1 grid units/magnitude. . . . . 76 5.11 Pressure coefficient contours and streamlines atG/D= 0.9,A1/D= 0.477,m⇤= 1.5,z = 0, and (1) Rem = 10, f1/ fn = 2.8,Df21 = 30.64, (2) Rem = 50, f1/ fn= 2.8,Df21=16.77, (3) Rem= 110, f1/ fn= 2.8,Df21=12.22, (a) f1 = 0 (b) f1 = 90 (c) f1 = 180 (d) f1 = 270. The numbers [1-4] identify the case with different Re, whereas the alphabets [a-d] indicate the instances in one cycle of the active cylinder oscillation. The dots on the sticks indicate the position of the cylinders in one cycle of motion. . . . . . 78 5.12 Evolution of non-dimensional vorticity contours at G/D = 0.9,A1/D = 0.477,m⇤ = 1.5,z = 0, f1/ fn = 3.2, and (1) Rem = 10, (2) Rem = 150, (a) f1 = 0 (b) f1 = 90 (c) f1 = 180 (d) f1 = 270. The numbers [1-2] identify the case with different Re, whereas the alphabets [a-d] indicate the instances in one cycle of the active cylinder oscillation. . . . . . . . . . . . . . . . . 79 5.13 Streamlines and non-dimensional vorticity contours at G/D= 0.9,A1/D= 0.477,m⇤ = 1.5,z = 0, f1/ fn = 3.2,f1 = 180, and (a) Rem = 10, (b) Rem = 150. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.14 Time histories of hydraulic force acting on the passive cylinder at primary or secondary resonance with G/D = 0.9,A1/D = 0.477,m⇤ = 2, and (a) z = 0, Rem = 10, f1/ fn = 0.7 (b) z = 0, Rem = 50, f1/ fn = 0.745 (c) z = 0, Rem = 110, f1/ fn = 0.765 (d) z = 0.2, Rem = 10, f1/ fn = 0.73 (e) z = 0.2, Rem = 50, f1/ fn = 0.775 (f) z = 0.2, Rem = 110, f1/ fn = 0.795 (g) z = 0, Rem = 10, f1/ fn = 0.39 (h) z = 0, Rem = 50, f1/ fn = 0.39 (i) z = 0, Rem = 110, f1/ fn = 0.39 (j) z = 0.2, Rem = 10, f1/ fn = 0.39 (k) z = 0.2, Rem = 50, f1/ fn = 0.39 (l) z = 0.2, Rem = 110, f1/ fn = 0.39. . . 82 List of figures xxi 5.15 Pressure coefficient contours and velocity vectors at G/D = 0.9,A1/D = 0.477,m⇤ = 2,z = 0 and (1) Rem = 10, f1/ fn = 0.7, (2) Rem = 50, f1/ fn = 0.745, (3) Rem = 110, f1/ fn = 0.765, (a) f1 = 0 (b) f1 = 90 (c) f1 = 180 (d) f1 = 270. Primary resonance occurs at around f1/ fn = 0.75 and the damping factor is zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.16 Pressure coefficient contours and velocity vectors at G/D = 0.9,A1/D = 0.477,m⇤= 2,z = 0.2 and (1) Rem= 10, f1/ fn= 0.73, (2) Rem= 50, f1/ fn= 0.775, (3) Rem = 110, f1/ fn = 0.795, (a) f1 = 0 (b) f1 = 90 (c) f1 = 180 (d) f1 = 270. Primary resonance occurs at around f1/ fn = 0.75 and the damping factor is relatively high at z = 0.2. . . . . . . . . . . . . . . . . . 84 5.17 Pressure coefficient contours and velocity vectors at G/D = 0.9,A1/D = 0.477,m⇤ = 2,z = 0, f1/ fn = 0.39 and (1) Rem = 10, (2) Rem = 50, (3) Rem = 110, (a) f1 = 0 (b) f1 = 90 (c) f1 = 180 (d) f1 = 270. Secondary resonance occurs at f1/ fn = 0.39 and the damping factor is zero. . . . . . . 85 5.18 Pressure coefficient contours and velocity vectors at G/D = 0.9,A1/D = 0.477,m⇤ = 2,z = 0.2, f1/ fn = 0.39 and (1) Rem = 10, (2) Rem = 50, (3) Rem = 110, (a) f1 = 0 (b) f1 = 90 (c) f1 = 180 (d) f1 = 270. Secondary resonance occurs at f1/ fn = 0.39 and the damping factor is relatively high at z = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.19 Variation of A2/A1 with f1/ fn and G/D at m⇤ = 2.5,A1/D = 0.05,Rem = 100,z = 0. A2/D decreases with G/D. . . . . . . . . . . . . . . . . . . . 88 5.20 Variation of the oscillation phase difference between the two cylinders with f1/ fn at A1/D= 0.05,m⇤ = 2.5,G/D= 0.21.0,Rem = 100,z = 0. G/D does not affect much the phase difference. . . . . . . . . . . . . . . . . . . 88 5.21 (a) Variation of A2/A1 with f1/ fn, and (b) variation of displacement phase difference Df21 between the passive cylinder and the active cylinder (for frequency components with f/ fn = f1/ fn) with f1/ fn at G/D = 0.3 0.9,A1/D = 0.05,m⇤ = 2.5, and z = 0.2. A2/A1 decreases with G/D, but the initial gap distance has little effect on Df21. This pattern is exactly the same as the undamped cases. . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.1 Typical time histories of passive cylinder’s vibration for G/D= 2.5,m⇤ = 2,z = 0,b = 35, f1/ fn = 2.8, (a) KC = 1,Rem = 35,A1/D = 0.159, (b) KC = 4,Rem = 140,A1/D= 0.637, (c) KC = 5,Rem = 175,A1/D= 0.796, (d) KC = 6,Rem = 210,A1/D = 0.955, (e) KC = 7,Rem = 245,A1/D = 1.114, (f) KC = 8,Rem = 280,A1/D= 1.273 . . . . . . . . . . . . . . . . 94 xxii List of figures 6.2 Amplitude spectra for the displacement histories of the passive cylinder at G/D= 2.5,m⇤ = 2,z = 0,b = 35, (a) KC = 3 in regime A, (b) KC = 4 in regime A, (c) KC = 5 in regime C, (d) KC = 6 in regime E, (e) KC = 7 in regime F, (f) KC = 8 in regime F . . . . . . . . . . . . . . . . . . . . . . . 96 6.3 At structural damping z = 0, (a) Variation of amplification factor A2/A1 for only the harmonic components with f1/ fn (b) Zoom-in for resonance (c) Variation of oscillation phase difference between the two cylinders Df21 with f1/ fn (for the frequency components with f/ fn = f1/ fn) with f1/ fn at G/D= 3.0,A1/D= 0.1591.114,m⇤ = 2.0,z = 0,Rem = 35245 and KC= 17,b = 35. The amplitude of the harmonic components is extracted by filter out the low-frequency components by FFT. The data processing method is discussed in Section 2.5. . . . . . . . . . . . . . . . . . . . . . 97 6.4 At structural damping z = 0.02, (a)Overview and (b)Zoom-in for varia- tion of amplification factor A2/A1 with f1/ fn and KC. (c) Variation of oscillation phase difference between two cylinders Df21 (for the harmonic component with f/ fn = f1/ fn) with f1/ fn and KC at G/D = 3.0,A1/D = 0.1591.114,m⇤ = 2.0,Rem = 35245 and KC = 17,b = 35. . . . . . 98 6.5 Displacement time history of the passive cylinder in regime C with KC = 5,b = 35,G/D= 2.5,A1/D= 0.796,m⇤= 2,z = 0,Rem= 175 with (a) f1/ fn= 0.05, (b) f1/ fn = 0.35, (c) f1/ fn = 0.77, (d) f1/ fn = 0.82, (e) f1/ fn = 1.0, (f) f1/ fn = 1.2, (g) f1/ fn = 1.4, (h) f1/ fn = 1.8, (i) f1/ fn = 2.0, (j) f1/ fn = 2.4, (k) f1/ fn = 2.8, (l) f1/ fn = 3.2. KC = 5 is a critical value that above which the irregularity of the displacement time history is significantly in- creased. With the increase of frequency ratio, the vibration of the passive cylinder gradually changes from a repetitive pattern to pulsed beating. . . . 100 6.6 Amplitude spectra of the passive cylinder’s displacement In regimeC with KC = 5,b = 35,G/D= 2.5,A1/D= 0.796,m⇤ = 2,z = 0,Rem = 175 with (a) f1/ fn= 0.05, (b) f1/ fn= 0.35, (c) f1/ fn= 0.77, (d) f1/ fn= 0.82, (e) f1/ fn= 1.0, (f) f1/ fn= 1.2, (g) f1/ fn= 1.4, (h) f1/ fn= 1.8, (i) f1/ fn= 2.0, (j) f1/ fn= 2.4, (k) f1/ fn = 2.8, (l) f1/ fn = 3.2. . . . . . . . . . . . . . . . . . . . . . 101 6.7 Vorticity contours In regimeC with (a-d) f1/ fn = 0.05, (e-h) f1/ fn = 0.77, and (i-l) f1/ fn = 2.8 with (a & e & i) f = 0 (b & f & j) f = 90 (c & g & k) f = 180 (d & h & l) f = 270 at KC = 5,b = 35,G/D = 2.5,A1/D = 0.796,m⇤ = 2,z = 0,Rem = 175. . . . . . . . . . . . . . . . . . . . . . . . 102 List of figures xxiii 6.8 Non-dimensional Vorticity Contours in Regime C with f1/ fn = 0.05 and KC = 5,b = 35,G/D= 2.5,A1/D= 0.796,m⇤ = 2,z = 0,Rem = 175, cor- responding to instants (a)A (b)B (c)C (d)D in Fig. 6.9. The shape of vorticity contours switch the direction of skewing for approximately every 10 periods. 104 6.9 (a)Displacement time history and (b) amplitude spectra of the passive cylin- der in RegimeC with f1/ fn = 0.05 and KC= 5,b = 35,G/D= 2.5,A1/D= 0.796,m⇤ = 2,z = 0,Rem = 175. The marked locations in (a) corresponds to the instants ad in Fig. 6.8. . . . . . . . . . . . . . . . . . . . . . . . . 105 6.10 Displacement time history of the passive cylinder in Regime C with f1/ fn = 3.2 and KC = 5,b = 35,G/D = 2.5,A1/D = 0.796,m⇤ = 2,z = 0,Rem = 175. The marked locations AF corresponds to the Figs. 6.11a-6.11f. . . . 106 6.11 Vorticity Contours for a typical cycle of beating in Regime C, corresponding to instants (a)A (b)B (c)C (d)D (e)E (f)F in Fig. 6.10 with f1/ fn = 3.2 and KC = 5,b = 35,G/D= 2.5,A1/D= 0.796,m⇤ = 2,z = 0,Rem = 175. The passive cylinder is pushed further away when the streaming direction is pointing at it. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.12 Displacement time history of the passive cylinder in Regime E with KC = 6,b = 35,G/D= 2.5,A1/D= 0.955,m⇤= 2,z = 0,Rem= 210 with (a) f1/ fn= 0.05, (b) f1/ fn = 0.35, (c) f1/ fn = 0.77, (d) f1/ fn = 0.82, (e) f1/ fn = 1.0, (f) f1/ fn = 1.2, (g) f1/ fn = 1.4, (h) f1/ fn = 1.8, (i) f1/ fn = 2.0, (j) f1/ fn = 2.4, (k) f1/ fn = 2.8, (l) f1/ fn = 3.2. At KC = 6, the time histories become more irregular compared with KC = 5, especially at high frequency f1/ fn. 109 6.13 Amplitude spectra of the passive cylinder’s displacement in Regime E with KC = 6,b = 35,G/D= 2.5,A1/D= 0.955,m⇤ = 2,z = 0,Rem = 210 with (a) f1/ fn = 0.05, (b) f1/ fn = 0.35, (c) f1/ fn = 0.77, (d) f1/ fn = 0.82, (e) f1/ fn = 1.0, (f) f1/ fn = 1.2, (g) f1/ fn = 1.4, (h) f1/ fn = 1.8, (i) f1/ fn = 2.0, (j) f1/ fn = 2.4, (k) f1/ fn = 2.8, (l) f1/ fn = 3.2. . . . . . . . . . . . . . 110 6.14 Vorticity contours in Regime E with f1/ fn = 0.05 at KC= 6,b = 35,G/D= 2.5,A1/D = 0.955,m⇤ = 2,z = 0,Rem = 210. The flow pattern changes between one pattern and its mirror-image intermittently. With the increase of the f1/ fn, the interaction remains steady for less cycles of the active cylinder oscillation and the passive cylinder’s vibration becomes more irregular as well.111 xxiv List of figures 6.15 Vorticity contours in Regime E with (a-d) f1/ fn = 0.05, (e-h) f1/ fn = 0.77, and (i-l) f1/ fn = 3.2 at KC = 6,b = 35,G/D = 2.5,A1/D = 0.955,m⇤ = 2,z = 0,Rem = 210. Sub-figures in the same column are at the similar state of vortex shedding process. With the increase in f1/ fn, the vortex shedding at the temporary steady state becomes more curved, and the streaming more circular. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.16 Typical time histories of passive cylinder vibration for G/D = 2.5,m⇤ = 2,z = 0,b = 35 (a)KC = 5 with z = 0 and z = 0.02, (b)KC = 6 with z = 0 and z = 0.02. In (a), we see that the time histories with z = 0 and z = 0.02 almost exactly overlap each other despite the difference of damping factor, whereas, in (b), we see that the minor difference of initial condition, i.e. damping factor, leads to dramatic deviation of time history after t fn = 18, which is an indication of chaos. . . . . . . . . . . . . . . . . . . . . . . . . 114 6.17 Displacement time history of the passive cylinder in Regime F with KC = 8,b = 35,G/D= 2.5,A1/D= 1.273,m⇤= 2,z = 0,Rem= 280 with (a) f1/ fn= 0.05, (b) f1/ fn = 0.35, (c) f1/ fn = 0.77, (d) f1/ fn = 0.82, (e) f1/ fn = 1.0, (f) f1/ fn = 1.2, (g) f1/ fn = 1.4, (h) f1/ fn = 1.8, (i) f1/ fn = 2.0, (j) f1/ fn = 2.4, (k) f1/ fn = 2.8, (l) f1/ fn = 3.2. At KC = 8, irregularity is apparent even at very low frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.18 Amplitude spectra of the passive cylinder’s displacement in Regime F with KC = 8,b = 35,G/D= 2.5,A1/D= 1.273,m⇤ = 2,z = 0,Rem = 280 with (a) f1/ fn = 0.05, (b) f1/ fn = 0.35, (c) f1/ fn = 0.77, (d) f1/ fn = 0.82, (e) f1/ fn = 1.0, (f) f1/ fn = 1.2, (g) f1/ fn = 1.4, (h) f1/ fn = 1.8, (i) f1/ fn = 2.0, (j) f1/ fn = 2.4, (k) f1/ fn = 2.8, (l) f1/ fn = 3.2. At KC = 8, irregularity is apparent even at very low frequency. . . . . . . . . . . . . . . . . . . . . 117 6.19 Vorticity contours for a typical regime F flow with only one forced oscillating cylinder at Re= 150, KC = 12 and b = 12.5. (Zhao and Cheng, 2014). The flow oscillates on the horizontal direction. . . . . . . . . . . . . . . . . . . 118 6.20 Vorticity contours for vortex shedding patterns in Regime F with f1/ fn = 0.05,KC = 8,b = 35,G/D= 2.5,A1/D= 1.273,m⇤ = 2,z = 0,Rem = 280 (a-d) Pattern F1, one vortex pair shed from both sides convected upward and another pair shed from the left side convected downward (e-h) Pattern F2, one pair + one negative vortex shed upward and one pair shed downwards from the left side (i-l) Pattern F3, one pair shed upwards from the left side, and one pair + one negative vortex shed downward. The mirror-images of these patterns can also occur. . . . . . . . . . . . . . . . . . . . . . . . . . 119 List of figures xxv 6.21 Displacement time history of the passive cylinder in Regime G with KC = 9,b = 35,G/D= 2.5,A1/D= 1.432,m⇤= 2,z = 0,Rem= 315 with (a) f1/ fn= 0.05, (b) f1/ fn = 0.35, (c) f1/ fn = 0.77, (d) f1/ fn = 0.82, (e) f1/ fn = 1.0, (f) f1/ fn = 1.2, (g) f1/ fn = 1.4, (h) f1/ fn = 1.6, (i) f1/ fn = 1.8, (j) f1/ fn = 2.0, (k) f1/ fn = 2.4, (l) f1/ fn = 2.8. . . . . . . . . . . . . . . . . . . . . . 122 6.22 Amplitude spectra of the passive cylinder’s displacement in Regime G with KC = 9,b = 35,G/D= 2.5,A1/D= 1.432,m⇤ = 2,z = 0,Rem = 315 with (a) f1/ fn = 0.05, (b) f1/ fn = 0.35, (c) f1/ fn = 0.77, (d) f1/ fn = 0.82, (e) f1/ fn = 1.0, (f) f1/ fn = 1.2, (g) f1/ fn = 1.4, (h) f1/ fn = 1.6, (i) f1/ fn = 1.8, (j) f1/ fn = 2.0, (k) f1/ fn = 2.4, (l) f1/ fn = 2.8. The dominant frequency of the passive cylinder’s vibration becomes equivalent to its immersed natural frequency at f1/ fn 1.0. . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.23 Non-dimensional vorticity contours for regime GwithKC= 9,b = 35,G/D= 2.5,A1/D= 1.432,m⇤ = 2,z = 0,Rem = 315, f1/ fn = 2.0. . . . . . . . . . 124 6.24 Passive cylinder’s vibration time history in Regime G with KC = 9,b = 35,G/D= 2.5,A1/D= 1.432,m⇤ = 2,z = 0,Rem = 315, f1/ fn = 2.0. The instants A and P corresponds to Figs. 6.25a and 6.25p, respectively. . . . . . 124 6.25 Non-dimensional vorticity contours in Regime GwithKC= 9,b = 35,G/D= 2.5,A1/D = 1.432,m⇤ = 2,z = 0,Rem = 315, f1/ fn = 2.0. (a-h) Multiple negative vortices shed from the active cylinder merges around the passive cylinder and pushes it away. (i-p) The vortex pairs shed from the active cylinder goes around the passive cylinder without causing its significant displacement. Instants A and P in Fig. 6.25 corresponds to Figs. 6.25a and 6.25p, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 List of tables 2.1 Non-dimensional groups and the range of value in this thesis . . . . . . . . 13 2.2 Comparison of the passive cylinder’s displacement with different meshes for KC = 3,G/D= 0.9,A1/D= 0.477, f1/ fn = 3.2,m⇤ = 1.5,z = 0,Rem = 10. 22 2.3 Comparison of the passive cylinder’s displacement with different meshes for KC = 9,G/D= 2.5,A1/D= 1.432, f1/ fn = 2.8,m⇤ = 2,z = 0,Rem = 315. 22 Nomenclature Non-dimensional Groups z = c 2 p mk = c 4p fnm , damping ratio of the passive cylinder A1/D amplitude ratio of the active cylinder f1/ fn frequency ratio of the active cylinder G/D initial clear gap distance ratio m⇤ = mc/mdisp, mass ratio of the passive cylinder Rem =UmD/n , Reynolds number regarding the maximum oscillation velocity Symbols b = Rem/KC, Stokes number Df21 = f2f1, the vibration phase of the passive cylinder minus that of the active cylinder Dr⇤ the minimum non-dimensional mesh size in the radial direction of the cylinder Dt⇤ non-dimensional computational time step n fluid kinematic viscosity w arbitrary velocity of a moving reference frame in arbitrary Lagrangian-Eulerian description f vibration phase for the single cylinder oscillation f1 phase of the vibrating active cylinder f2 phase of the vibrating passive cylinder z damping ratio of the passive cylinder mass-spring system A amplitude of a displacement frequency component obtained from fast Fourier trans- form A2 = (YmaxYmin)/2, passive cylinder vibration amplitude during the steady state xxx Nomenclature CA added mass coefficient, the added mass divided by the displaced fluid mass by the cylinder CY2A amplitude of a force coefficient frequency component obtained from fast Fourier transform CY2 the force coefficient on the passive cylinder along the y-axis D diameter for both the active and the passive cylinders d initial centre to centre distance between the two cylinders f frequency of a frequency component obtained from fast Fourier transform f1 frequency of the active cylinder oscillation f2 frequency of the passive cylinder vibration during the steady state fn = (1/2p)⇤ p k/mc, structural (vacuum) natural frequency for the elastically mounted passive cylinder fw = fn ⇤ p m⇤/(m⇤+CA), the immersed natural frequency of the passive cylinder in water fpeak the resonance frequency obtained directly from the numerical simulations FY2 the force acting on the passive cylinder along the y-axis G initial clear gap distance between the two cylinders k stiffness of the passive cylinder mass-spring system KC =UmT/D, Keulegan-Carpenter number mc mass of the passive cylinder mdisp = rpD2/4 mass of the fluid displaced by the cylinder Nc the number of computational elements along the circumference of a cylinder p pressure p⇤ = p/r f 2nD2, normalised pressure T the period of forced vibration t time t⇤ = fnt, normalised time u horizontal fluid velocity component u⇤j,mesh the non-dimensional velocity of the mesh movement ui fluid velocity component in the xi direction Nomenclature xxxi u⇤i = ui/ fnD, normalised velocity Um = 2pA1 f1, maximum velocity of the active cylinder relative to the fluid v vertical fluid velocity component xi Cartesian coordinates with x1 = x and x2 = y x⇤i = xi/D, normalised Cartesian coordinates Y1 the displacement of the active cylinder Y2 the displacement of the passive cylinder Ymax,min the maximum and the minimum of the passive cylinder displacement Y2 in the last 50 periods of steady oscillations Ymax maximum of the Y2 values in the last 50 periods of steady oscillations Ymin minimum of the Y2 values in the last 50 periods of steady oscillations Acronyms ALE Arbitrary Lagrangian-Eulerian method CFD Computational Fluid Dynamics DEM Discrete Element Method DNS Direct Numerical Simulation DOF Degree Of Freedom FEM Finite Element Method FVM Finite Volume Method HPC High-Performance Computing PDE Partial Differential Equations VIV Vortex-Induced Vibration Chapter 1 Introduction This chapter presents the background of this study by discussing the most relevant research. The aims and objectives of this work are stated. An outline of the entire thesis is provided at the end of this chapter. 1.1 Literature Review Flow-mediated interaction is a phenomenon concerning objects moving in a fluid which interact with the nearby objects via the perturbed fluid. The ensuing movement of the neighbouring objects is more complicated compared with traditional scenarios with objects under a steady or a periodic flow. The significance of researching flow-mediated interaction relies on its wide occurrence in both natural and artificial situations, for example, sperm-egg interaction (Riffell and Zimmer, 2007), fish schooling (Liao, 2003; Gazzola et al., 2011, 2016), dolphin drafting (Weihs, 2004), swimming of micro-organisms (Gyrya et al., 2010; Ishikawa et al., 2006; Koch and Subramanian, 2011), formation of solid particle clusters (Voth et al., 2002), and dispersion of particle clouds (Metzger et al., 2007), interference between immersed risers (Bampalas and Graham, 2008), seismic pile water interaction (Pang et al., 2015). Extensive and elaborate studies have been carried out on this topic, for the purpose of comprehending the mechanism of the interactions. The revelations from these studies can thus be exploited in engineering applications, e.g. vortex-induced vibration energy harvesting (Bernitsas et al., 2008), self-propulsion device design (Van Rees et al., 2015), and the cyber octopus being developed by US Department of Defense Office of Naval Research. Figures 1.1 and 1.2 show some examples of phenomena and applications that is relevant to the present study. This thesis focuses on the flow-mediated interaction between two rigid cylinders im- mersed in a quiescent incompressible fluid. One cylinder, the active cylinder, carries out 2 Introduction Fig. 1.1 An offshore oil drilling platform with multiple cylindrical structures (Credit: Chesroc Nigeria Limited) Fig. 1.2 (a) Classical Huygens’ pendulum clocks setup (Huygens, 1660) (b) Idealised model by Peña Ramirez et al. (2014). 1.1 Literature Review 3 sinusoidal forced vibration to perturb the surrounding fluid, and the generated periodic flow thus interacts with the other cylinder, the passive cylinder, which is elastically-mounted with a damper. Despite the basic geometry, this problem involves a rich spectrum of physics and carries practical implications in many fields of research and engineering. Due to the necessity to check the flow-mediated interactions between multiple cylinders, various studies have already been conducted over the past decades to analyse the fluid- structure interactions involving solely one periodically oscillating cylinder. 1.1.1 Single Cylinder Oscillating in Still Fluid The scenario of a cylinder oscillating in a fluid is physically equivalent to a cylinder immersed in an oscillatory flow, which can be determined by two dimensionless parameters, i.e. the Keulegan-Carpenter number and Reynolds number. The Keulegan-Carpenter number can be defined as KC =UmT/D (Keulegan and Carpenter, 1958), where Um is the amplitude of the oscillatory flow velocity, T the period of oscillatory flow, and D the diameter of the cylinder. Given a sinusoidal flow we have KC= 2pA1/D. The Reynolds number is defined as Rem =UmD/n , where n is the fluid kinematic viscosity. The Stokes number, b , is dependent on the Reynolds number and the Keulegan-Carpenter number and is defined as b = Rem/KC. In engineering applications, the prediction of fluid-induced drag forces for a circular cylinder in oscillating or wave flow is predicted by the Morrison Equation (Morison et al., 1950), which represent the in-line force per unit length as F = 1 2 rDCDU |U |+ 14prD 2CmU˙ (1.1) where r is the fluid density, D is the body diameter, U is the fluid velocity, Cm = 1+Ca is the inertia coefficient, Ca the added mass coefficient, and CD is the drag coefficient. Cm and CD are found to depend on both KC (Keulegan and Carpenter, 1958) and b = Re/KC (Sarpkaya, 1977). Williamson (1985) further studied the effects of vortex motions on the forces upon a single cylinder in the range of 0< KC < 35 with b fixed at 730. Williamson (1985) reported reasonable symmetry at KC < 4. Several regimes were classified according to the vortex patterns and KC. Specifically, at 0< KC < 7 a pair of small attached vortices are observed, which is similar to the flow pattern generated by the active cylinder in the present study. Based on this pattern, five flow regimes were defined according to KC. While Williamson (1985) carried out experiments at a fixed b , Tatsuno and Bearman (1990) further explored in the range of KC < 15 and b < 160. They classified eight flow regimes based on flow visualization, which were referred to as regimes A⇤, A, B, C, D, E, F, G. The flows in regimes A and A⇤ are entirely two-dimensional (2D). The flow regimes discussed 4 Introduction Fig. 1.3 Flow regimes classified by Tatsuno and Bearman (1990). In this thesis, the values of KC and b lies well within regimes A and A⇤, and the flow characteristics in all the simulations do coincide with the regimes A and A⇤: two symmetric vortices shed per half cycle. in this thesis are within regimes A, A⇤, C, E, F , G. Based on the useful flow regime map from Tatsuno and Bearman (1990), Elston et al. (2004, 2006) further examined the symmetry breaking phenomenon, and identified the parameters related to the breakage of different types of symmetries for this case, including the onset of both 2D and 3D instability. Many researchers investigated the flow transition from 2D to 3D as well (Honji, 2006; Hall, 1984; Sarpkaya, 1986, 2002, 2005, 2006). The present study is within regimes A and A⇤, and for all the involved cases, the flow field is symmetric. Dütsch et al. (1998) carried out laser Doppler anemometry measurements of a laminar flow generated by the harmonic oscillation of a circular cylinder in otherwise still water. By comparing with these experimental results, an in-house 2D Navier-Stokes model is validated in Lin et al. (2017), which is the basic tool for the present study. In addition to the experimental efforts, with the continuous enhancement of computational power, fluid mechanics problems are increasingly studied in a numerical manner. The problem of oscillatory flow past a circular cylinder is not an exception. Many researchers have used 2D simulations to predict the flow field. Justesen (1991) conducted numerical simulation on this problem for b = 196,483,1035 and 0 < KC < 26. By solving 2D NS equations, they achieved good agreement with the experimental data. Lin et al. (1996) utilised a 2D discrete vortex method to carry out numerical simulation at b = 76 and KC < 30. They successfully reproduced all the major vortex-shedding regimes found in previous experiments. Iliadis and Anagnostopoulos (2002) focused on an aperiodic flow case of oscillatory flow 1.1 Literature Review 5 past a circular cylinder at Rem = 200 and KC = 20 using 2D finite element method. They observed intermittent changes between different vortex shedding modes. Also, Zhao and Cheng (2014) numerically studied the oscillatory flow past two circular cylinders. They found that, in the side-by-side arrangement with small gap ratios, the vortex shedding from the gap of the two cylinders dominates, resulting in the unique gap vortex shedding (GVS) regime, which cannot be found for a single cylinder case. In the tandem arrangement with a very small gap between the two cylinders, the flow regimes are similar to that of a single cylinder. A strong interaction between the vortex shedding flows from the two cylinders makes the flow notably irregular at large KC values in both side-by-side and tandem arrangements. Zhao and Cheng (2014) used the same code used in this study to simulate the single-cylinder case and achieved very good agreement with the experimental results by Tatsuno and Bearman (1990), as seen in Fig. 1.4. 1.1.2 A Oscillating Cylinder Actuating a Free Cylinder in Still Fluid In contrast to the abundant research on cylinders vibrating in a still fluid, the research related to the flow-mediated interaction between multiple immersed cylinders is relatively scarce. Lamb (1932) studied the interaction between two spheres immersed in inviscid fluid. One sphere is forced to oscillate along the centre-line, whereas the other nearby sphere of neutral buoyancy responds freely to the disturbed fluid. By theoretical analysis, Lamb (1932) stated that the free sphere is "on the whole" attracted towards the forcedly oscillating sphere due to the imbalanced pressure force. By both analytical and numerical methods, Nair and Kanso (2007) studied an identical configuration in greater detail, but for the case of two circular cylinders rather than two spheres. One cylinder is started impulsively and is forced to oscillate along the centre-line between two cylinders, whereas the second responds freely without any constraints. By contrast, in the present study, the second (passive) cylinder is constrained by the spring and the damper. Nair and Kanso (2007) discovered that the free cylinder can be either repelled away or attracted towards the forcedly oscillating cylinder, depending on the initial velocity direction of the oscillating cylinder. They further suggested that this should also be the case for the sphere scenario analysed by Lamb (1932), who only captured the attraction due to a mathematical mistreatment. The flow-mediated interaction between two cylinders was further investigated by Gazzola et al. (2012), as seen in Fig. 1.5a, but the fluid was taken to be viscous rather than inviscid. They found a threshold Reynolds number, beyond which the passive cylinder is repelled by the active cylinder, and under which the passive is attracted to the active cylinder. An increase in Reynolds number, i.e. , a decrease in viscosity, slows down the dissipation of the secondary flow, which favours the repulsion of the passive cylinder by the active cylinder. 6 Introduction Fig. 1.4 Simulated flow patterns represented by streaklines in different flow regimes for oscillatory flow past a single cylinder, conducted under the same condition as the experiments carried out by Tatsuno and Bearman (1990). It was simulated by Zhao and Cheng (2014) using the code of current study. (a) Regime A* at KC = 3.14 and b = 52.8 (b) Regime A at KC = 11 and b = 7.4 (c) Regime D at KC = 6.28, and b = 18 (upper) and b = 22.1 (lower) (d) Regime E at KC = 6.28 and b = 25.6 in the 72nd period (left) and 200th period (right). (e) Regime F at KC = 8.16,b = 27 (upper), KC = 12.6,b = 17.8 (lower). (a-e) The oscillation direction is horizontal in all the sub-figures. 1.1 Literature Review 7 (a) (b) Fig. 1.5 (a) The setup sketch of Case B (Gazzola et al., 2012). The active cylinder is forced to oscillated harmonically along the centreline of two cylinders, whereas the passive cylinder is free. This is different from the present study, where the passive cylinder is attached to a spring. (b) Threshold Reynolds number illustrated as a function of d/Dm and Ds/Dm (Gazzola et al., 2012). When the repelling effect prevails, a secondary flow structure is discovered between the two cylinders. They further discovered that the threshold Reynolds number is indifferent to the initial phase of the active cylinder, i.e. , the direction of the initial velocity of the active cylinder. This conclusion is different from that drawn in Nair and Kanso (2007) concerning the inviscid flow, where the initial phase of the movement governs the repulsion or attraction. Fig. 1.6 compares the effect of initial phase between inviscid and viscous fluid. With the increase in the initial gap, the threshold Reynolds number decreases exponentially, whereas it is less sensitive to the size difference between the two cylinders, as seen in Fig. 1.5b. Based on these observations, Gazzola et al. (2012) concluded that the flow features have a greater influence than the inertia of the passive cylinder. As a result, in the present study, the diameters of the passive cylinder and the active cylinders are identical. They also found that, given a very small vibration amplitude of the active cylinder, the level of repulsion or attraction is significantly reduced. Therefore, in the present study, the active cylinder oscillates at an amplitude of 0.477D, i.e. KC = 3, allowing convenient observation and study of the repulsion and attraction. In summary, a large amount of research has been carried out on oscillatory flow past a single cylinder, but there is only limited research on the flow-mediated interaction between immersed objects despite the rich physics involved and the potential engineering applications. 8 Introduction (c) Fig. 1.6 In inviscid fluid, time history of the passive cylinder’s normalised displacement from its initial position Dx/Dm at initial phase of the active cylinder vibration at (a) 0 and (b) 180, as demonstrated by Nair and Kanso (2007). (c)In viscous fluid, time history of slave’s normalised displacement from its initial position Dx/Dm at initial phase of the active cylinder vibration at 0 (solid line) and 180 (dashed line), as illustrated by Gazzola et al. (2012). The initial condition of the active cylinder can affect the drifting direction of the passive cylinder in inviscid fluid, but not in viscous fluid. 1.2 Aims and Objectives 9 1.2 Aims and Objectives The interaction between two vibrating cylinders immersed in a fluid has never been carefully studied in the past, despite many relevant natural phenomena and engineering applications. The aim of this work is to understand the mechanism of the flow-mediated interaction among multiple vibrating cylinders. To investigate this mechanism, this thesis focuses on a representative case with two cylin- ders immersed in otherwise static fluid. An active cylinder is forced to oscillate harmonically, whereas a nearby passive cylinder is elastically-mounted with a damper and responds to the actuated flowing fluid. Both cylinders vibrate only along the centreline connecting the two cylinders. The achievements of this study are listed here: • Review the existing literature • Validate the numerical method • Non-dimensionalise the representative case • Design the parametric space to be explored • Run extensive amount of simulation cases on high performance computing facilities • Process huge amount of data by scripting, and analyse the processed data • Summarise the discoveries and contribute to the physical understanding 1.3 Outline of Thesis Chapter 2 describes computational methods for the in-house finite element method (FEM) code applied in this study. The dimensional analysis and governing equations are presented followed by the mesh independence study and validation of the numerical model. Chapters 3 to 5 present the parametric studies for the non-dimensional groups associated with the active cylinder, the passive cylinder, and the intermediate fluid, respectively. Chapter 3 focuses on the effects of the parameters relevant to the active cylinder, i.e. the frequency and amplitude of its forced oscillation. Chapter 4 investigates the influence parameters related to the passive cylinder, i.e. its damping ratio and mass ratio. Chapter 5 studies the roles played by the parameters related to the intermediate fluid, i.e. the Reynolds number and the gap distance between the two cylinders. 10 Introduction In Chapters 3 to 5, the representative cases selected from 23,400 simulations are demon- strated and interesting phenomena are discussed, including the primary and secondary resonance, drifting of the passive cylinder, the shift of phase difference between the two cylinders and the characteristic flow field around the cylinders. Chapter 2 Methodology This chapter discusses the methodology of this thesis. Section 2.1 demonstrates the problem setup for the interested physical case of the multi-cylinder interaction in fluid and also the corresponding dimensional analysis. Simulation of this physical problem was carried out using an in-house FORTRAN code. The two-dimensional (2D) Navier-Stokes equations are directly solved to simulate the flow-mediated interaction, as seen in Section 2.2.1. The simulation code applies the streamlined upwind Petrov-Galerkin finite element method (FEM), as described in Section 2.2.2, with arbitrary Lagrangian-Eulerian (ALE) method, as demonstrated in Section 2.2.3. The mesh independence study and the validation were conducted in Section 2.3 and Section 2.4, respectively, to check the validity of the model. 2.1 Problem Setup and Dimensional Analysis In this study, two identical rigid cylinders are immersed in otherwise still fluid, as seen in Fig. 2.1. At time zero, the active cylinder impulsively starts and vibrates harmonically disturbing the surrounding fluid, whereas the passive cylinder vibrates correspondingly with 1 degree of freedom (1DOF) along the y-axis in response to the imbalanced hydrodynamic force due to the actuation of the active cylinder. In all the studied cases, the passive cylinder and the active cylinder are always separated by the fluid, with no occurrence of solid-solid contact. Here, dimensional analysis is carried out to simplify the targeted problem by reducing variables involved, as seen in Figure 2.1. Buckingham theorem is applied to find out the dimensionless groups influencing the simulation results. The detailed procedure is as follows: For this case, 9 quantities and 3 dimensions are involved: 12 Methodology Fig. 2.1 A sketch of interaction between the two cylinders: While the active cylinder undergoes harmonic forced vibration along the centreline of the two identical rigid cylinders, the passive cylinder is elastically mounted with a damper, vibrating in response to the active cylinder with 1DOF along the centreline as well. 2.1 Problem Setup and Dimensional Analysis 13 26664 D fn m A1 f1 G r n c 0 0 1 0 0 0 1 0 1 1 0 0 1 0 1 2 2 0 0 1 0 0 1 0 0 1 1 37775 ML T where D is the diameter of the cylinder; fn = p k/m/2p is structural natural frequency of the passive cylinder; m is mass of the passive cylinder; k is the stiffness of the spring mounted to the passive cylinder; A1 is the oscillation amplitude of the active cylinder; f1 is oscillation frequency of the active cylinder; G is initial gap distance between the two cylinder; r is density of fluid; n is kinematic viscosity of fluid; c is viscous structural damping coefficient of the passive cylinder;M, L and T represent dimensions of mass, length and time, respectively. According to Buckingham theorem, these quantities can be reduced to 93= 6 indepen- dent dimensionless groups. If D, fn, and m are selected as repeating variables, the 6 groups can be written as follows: p1 = A1 D , p2 = f1 fn , p3 = G D , p4 = r mD2 = 4 pm⇤ , p6 = c 4p fnm = z , p5 = n fnD2 = 2p(A1/D)( f1/ fn) Rem . In other words, any case can be determined by 6 independent dimensionless groups: A1/D, f1/ fn, G/D, m⇤, Rem and z as seen in Table 2.1: Table 2.1 Non-dimensional groups and the range of value in this thesis Frequency ratio of active cylinder f1/ fn 0.053.2 Amplitude ratio of active cylinder A1/D 0.0250.477 Damping ratio of passive cylinder z c4p fnm 01.4 Mass ratio of passive cylinder m⇤ mcrD2p/4 1.52.5 Reynolds number regardingUm Rem 2pA1 f1D n 10315 Gap ratio G/D 0.21.0 Simulations are conducted for a range of combinations in both regular and irregular regimes. As seen in Table 2.1, the active cylinder’s frequency f1/ fn ranges from 0.05 to 14 Methodology 3.2; the amplitude of the active cylinder A1/D varies from 0.025 to 1.432; the damping ratio ranges from 0 to 1.4; the mass ratio m⇤ takes the value of 1.5, 1.7, 2.0, 2.2 or 2.5; the Reynolds number Rem varies from 10 to 315; the gap ratio G/D ranges from 0.2 to 2.5. The corresponding Keulegan-Carpenter number and the Stokes number of the active cylinder can be calculated as KC = 2pA1/D= 0.1579 and b = Rem/KC = 3.3110, respectively. In total, 21224 combinations of periodic regime are examined. The tested range of the active cylinder’s oscillation frequency allows clear demonstration of all the resonating frequencies of the passive cylinder’s response, which will be discussed later in Chapter 3. The range of Reynolds number Rem involves both the repelling and attraction regimes for the passive cylinder, which will be discussed later in Chapter 5. The range of the active cylinder’s amplitude and the Reynolds number together constrain the involved flow regimes to A⇤,A,C,E,F,G as defined by Tatsuno and Bearman (1990). The mass ratios are typical for engineering structures immersed in water, e.g. the mass ratio of concrete in water is about 2.5. 2.2 Computational Method This section describes the governing equations of the current numerical model. The streamline upwind Petrov-Galerkin finite element method was implemented to discretise the governing equations. Arbitrary Lagrangian-Eulerian method was used to handle the moving boundary. 2.2.1 Governing Equations The two-dimensional Navier-Stokes equations are solved by the Streamline Upwind Petrov- Galerkin finite-element method (Brooks and Hughes, 1982). The moving cylinder boundaries are handled through the use of an arbitrary Lagrangian-Eulerian (ALE) scheme, as seen in Section 2.2.3. In the governing equations, the length, time, velocity and pressure are non-dimensional quantities according to: x⇤i = xi D , t⇤ = t fn, u⇤i = ui fnD , p⇤ = p r f 2nD2 (2.1) where x⇤1 = x/D and x⇤2 = y/D are the Cartesian coordinates, as shown in Fig. 2.1. t is time, fn = p k/m/2p is the structural natural frequency in vacuum determined by the passive cylinder’s mass and spring stiffness as explained before, ui is the fluid velocity component in the xi direction, and p is the pressure. The variables with stars represent the non-dimensional quantities. The non-dimensional incompressible two-dimensional Navier-Stokes equations in the ALE framework are 2.2 Computational Method 15 ∂u⇤i ∂x⇤i = 0, (2.2) ∂u⇤i ∂ t⇤ +(u⇤j u⇤j,mesh) ∂u⇤j ∂x⇤j + ∂ p⇤ ∂x⇤i = Um Rem fnD ∂ 2u⇤i ∂x⇤j∂x⇤j (2.3) where u⇤j,mesh is the velocity of the mesh movement. The motion equations for the active and the passive cylinders are Y1 = A1 sin(2p f1t) (2.4) ∂ 2Y2 ∂ t2 +4p fnz ∂Y2 ∂ t +4p2 f 2nY2 = 2 p U2m m⇤D CY2 (2.5) where Y1 and Y2 are the y direction displacements of the active and passive cylinders, respec- tively, z stands for the damping factor,CY2 = FY2/(0.5rDU2m) is the force coefficient for the passive cylinder, FY2 is the y direction force on the passive cylinder. These equations can also be written in a non-dimensional form: Y ⇤2 = Y2/D (2.6) Y1 D = A1 D sin(2p f1 fn t⇤) (2.7) ∂ 2Y ⇤2 ∂ t⇤2 +4pz ∂Y ⇤2 ∂ t⇤ +4p2Y ⇤2 = 8p m⇤ ( A1 D )2( f1 fn )2CY2 (2.8) 2.2.2 Streamline Upwind Petrov-Galerkin FEM Finite element method (FEM) is often exploited to solve the problems where the governing equations are known, but analytical solutions are difficult or impossible to achieve. The popularity and fame of FEM can be attributed to its various advantages. First and foremost, FEM can accurately represent very complex geometries and can include dissimilar material properties, which enables FEM to deal with a wide range of engineering problems, for example, solid mechanics, fluid dynamics, heat transfer and electrostatic problems. FEM can also handle complex restraints and complex loadings, for example, nodal loading, element loading and time or frequency dependent loading (Weck et al., 2016). FEM discretises the governing equations into small elements and uses a weighted residual formulation to obtain a system of matrix equations. An approximate solution of the original problem can thus be achieved by FEM. 16 Methodology Bubnov-Galerkin method is the most common weighted residual formulation. The Galerkin method is capable of achieving good results while applying it to most structures or heat conduction problems, since it can minimise the difference between the finite element solution and the exact solution (Brooks and Hughes, 1982). When it comes to fluid flow simulations, Bubnov-Galerkin method was initially applied as well, but sometimes, significant unphysical oscillation of the flow velocity were discovered. This problem can only be fixed by excessive refinement of the computational mesh, which is not desirable in most cases. This disadvantage of the Bubnov-Galerkin method demands a new formulation without such unphysical oscillations at any level of mesh refinement. The upwind difference method can produce simulation results without the unphysical oscillations, but has the disadvantaged of only first degree of accuracy. To solve this problem, Brooks and Hughes (1982) developed a new Navier-Stokes solution algorithm with streamline upwind Petrov-Galerkin formulation, which possesses both the robust qualities of an upwind method and the accuracy of Petrov- Galerkin method. The principal idea of this formulation is to add a streamline upwind perturbation to the standard Bubnov-Galerkin weighting functions, acting only in the flow direction. This formulation is employed in the present simulation code (Zhao et al., 2007, 2009; Zhao, 2013). In the present study, a Galerkin finite element method is used to discretise the governing equations in space. The conventional Galerkin discretisation equals to the central-difference approximation in finite difference method. Since central-difference discretisation has negative artificial diffusion, numerical instabilities often take place in central-difference solutions of the convective-diffusive equations. In the present study, by introducing an artificial diffusion term to the momentum equations, a streamline upwind scheme is applied. In the momentum equations, the added diffusion term can be written as (Brooks and Hughes, 1982): k˜ ∂ ∂xk (u0ju0k ∂ui ∂x j ), (2.9) where u0i = ui/kuk, kuk is the magnitude of the velocity defined as kuk2 =Â i uiui, k˜ is defined as (Brooks and Hughes, 1982; Jester and Kallinderis, 2003): k˜ = x˜kuekhe/2, (2.10) x˜ = 8<:Reh/3, Reh  3,1, Reh > 3, (2.11) 2.2 Computational Method 17 where kuek is the magnitude of the velocity at the element centroid, he is the element’s average edge length, Reh(= kuekhe/n) is the cell Reynolds number. For the time integration of the momentum equation Eq. (2.3), a fractional step formulation is applied (Meting et al., 1997; Meneghini et al., 2001). First, by neglecting the pressure gradient terms, an intermediate velocity is computed. The pressure field is then obtained by solving the pressure Poisson equation. The final velocity is achieved by including the pressure effect. In the intermediate velocity computation, the streamline upwind scheme is applied. During a time increment from t = nDt to (n+1)Dt, with Dt being the computational time step, the algorithm of the time advancement scheme for the equations is as follows: 1. Solving the momentum equations without pressure terms, the intermediate velocity u˜n+1i is computed at time t = (n+1)Dt. The diffusion terms are considered implicitly whereas the convection terms are considered explicitly from the previous time step. The parameter k˜ is evaluated at the time level nDt. The equation for computing u˜n+1i is: u˜n+1i Dt ∂ ∂x j [ 1 Re ( ∂ u˜n+1i ∂x j + ∂ u˜n+1j ∂xi )]Dtk˜ ∂ ∂xk (unju n k ∂ u˜n+1i ∂x j ) = uni Dtunj ∂uni ∂x j , (2.12) 2. In general, the intermediate velocity field does not satify the continuity equation. At the time t = (n+1)Dt, the pressure is computed to enforce the continuity in the final velocity field. The pressure is computed by solving the following pressure Poisson equation, ∂ 2pn+1 ∂x j∂x j = 1 Dt ∂ u˜n+1j ∂x j , (2.13) 3. The final velocity is obtained by including the pressure terms un+1i = u˜ n+1 i Dt ∂ pn+1 ∂xi . (2.14) The computational domain W is divided into linear finite elements. In an element, the velocity components, pressure and the turbulence quantities can be approximated as: ui ⇡ NU i, p⇡ NP, (2.15) where N is the shape function vector,Ui and P are the nodal values of ui and p, respectively. By selecting the shape function as the weighting function and applying a weighted residual 18 Methodology formulation to Eqs. (2.12–14), the following set of matrix equations is obtained: [M+Dt(Dn+Dnu)]U˜ n+1 i =MU n i +DtCnUni , (2.16) SPn+1 = 1 Dt AiU˜ n+1 i , (2.17) MUn+1i =MU˜ n+1 i DtAiPn+1, (2.18) where M = Z W NTNdW, Dn = Z W k˜nuni u n j ∂NT ∂xi ∂N ∂x j dW, Ai = Z W NT ∂N ∂xi dW, S = Z W ∂NT ∂xi ∂N ∂xi dW Dnu = Z W n ∂N T ∂x j ∂N ∂x j dW, Cn = Z W unjNT ∂N ∂x j dW In summary, the computational procedure is: 1. Calculate the intermediate velocity by Eq. (2.16); 2. Solve the pressure Eq. (2.17) 3. Obtain the final velocity by Eq. (2.18) 4. Return to step 1 and repeat the procedure till the flow is fully developed. 2.2.3 Arbitrary Lagrangian-Eulerian Method The current numerical model uses the Arbitrary Lagrangian-Eulerian (ALE) method to deal with the moving boundaries due to the fluid-structure interaction. ALE method was originally developed for inviscid flows in finite difference formats by Noh (1964), Trulio (1966) and Hirt et al. (1974). Hughes et al. (1981) later presents the theoretical framework of ALE for incompressible viscous flows. Donea et al. (1982) proposed the ALE method in FEM format for the fluid-structure interaction problems. The Lagrangian and the Eulerian description are the two most basic viewpoints in the algorithms of fluid mechanics. In Lagrangian description, the computational mesh moves with the associated material particle. In Eulerian, the computational mesh is treated as a fixed reference frame and the fluid moves with respect to the grid. The ALE description was developed to represent the generalisation of the two classical concepts. The ALE description does not depend on the assumption of fluid particles and consider the mesh as a reference 2.3 Mesh Independence Study 19 frame travelling with an arbitrary velocity w in the space. Donea et al. (1982) summarised that, those descriptions can be classified by w as follows: 1. w = 0, the reference frame is static, corresponding to the Eulerian description, where motion is represented by velocity distribution on fixed nodes on the mesh. 2. w = v, where v is the actual particle velocity. The reference frame moves with the particle, corresponding to the Lagrangian decription. 3. 0< w < v, the reference frame is not static and travels at a velocity w different from the particle velocity v, corresponding to the arbitrary Lagrangian-Eulerian description. The ALE algorithm allows the computational mesh inside the domains to move arbitrarily to optimize the shapes of elements, while the mesh on the boundaries and interfaces of the domains can move along with materials to precisely track the boundaries and interfaces of a multi-material system. Also, ALE formulations can be reduced to either Lagrangian formulations by equating mesh motion to material motion or Eulerian formulations by fixing mesh in space. As a result, the versatility of ALE method allows it to perform comprehensive engineering simulations, including heat transfer, fluid flow, fluid-structure interactions and metal-manufacturing. In the present code, the location of each finite element node is updated at every time step according to the motion of the cylinder. The FEM node displacements are governed by (Zhao, 2013): — · (g—Sy) = 0, (2.19) where Sy is the displacement of the nodal points in the y-direction and g is a constant that controls the mesh deformation. To avoid excessive distortion of the near-wall elements, in each finite element, it is configured that g = 1/Aelement , with Aelement being the area of the element. A Galerkin FEM is applied to solve Equation (2.19) by specifying the displacements at all boundaries. On the cylinder surfaces, the nodal displacement Sy is identical to the displacement of the cylinder whereas Sy = 0 on all other non-cylinder boundaries. 2.3 Mesh Independence Study Mesh independence study is conducted to ensure that the mesh is sufficiently dense to obtain converged results. A close-up of the typical computational mesh around the cylinders is illustrated in Fig. 2.2. It has 16,407 nodes in total. The complete computational domain is of a square shape with a sufficiently large side length of 30D to eliminate the boundary effects 20 Methodology (Zhao and Cheng, 2014). The mesh is a combination of structured (around the cylinders) and unstructured (far from the cylinders) types. The number of elements along each cylinder circumference Nc is 152, and the minimum non-dimensional mesh size in the radial direction Dr⇤ is 1.30⇥103. The non-dimensional time step Dt⇤ in the simulation is chosen based on the Courant condition Dt⇤= 0.00025/U⇤m = 0.00025Dfn/(2pA1 f1), i.e., the Courant number is Dt⇤U⇤m/Dr⇤ = 0.19< 1, whereU⇤m is the non-dimensional maximum vibration velocity of the active cylinder. Fig. 2.2 Computational meshes for interaction between two cylinders with G/D= 0.2 and Dr  1.38⇥103 and Nc 152. The test case in G/D= 0.2, A1/D= 0.1, f1/ fn = 0.825, z = 0 and m⇤ = 2.5 is chosen for the mesh independence study. This is a challenging case, because of the large vibration amplitude of the passive cylinder, which results in a very small distance between the two cylinder at certain instances. The temporarily small proximity of the two cylinders adversely affects the accuracy of the computation, as the computational cells in the gap are greatly 2.3 Mesh Independence Study 21 0.155 0.16 0.165 0.17 0.175 0.18 0.185 0.19 1.00E-04 1.00E-03 1.00E-02 1.00E-01 A 2 /D Δr (a) 0.1605 0.161 0.1615 0.162 0.1625 0.163 0.1635 50 100 150 200 250 300 A 2 /D Nc (b) Fig. 2.3 Effect of Dr on the responding amplitude A2 with Nc = 116 and Effect of Nc on the responding amplitude A2 with Dr = 1.25⇥ 103, for the case G/D = 0.2,A1/D = 0.1, f1/ fn = 0.825,m⇤ = 2.5. squeezed and stretched periodically. Simulations are conducted in two scenarios. Scenario (a) has a constant Nc = 116, with Dr = 2.65⇥ 104, 3.22⇥ 104, 4.49⇥ 104, 6.36⇥ 104, 9.21⇥104, 1.38⇥103, 2.21⇥103, 3.87⇥103 and 2.43⇥102 in a series of meshes with different resolutions. Scenario (b) has constant Dr = 9⇥104, while Nc = 62, 80, 98, 116, 134, 152, 170, 188, 206, 224, 242, 260 and 278 in a series of meshes with different resolutions. Figs. 2.3a and 2.3b show the variations of A2/D with Dr and Nc, respectively. A2 is calculated to be (YmaxYmin)/2, where Ymax and Ymin are the maximum and minimum, respectively, of the Y2 values in the last 50 periods of steady oscillations. It can be seen that the change in A2/D is negligible when Dr  1.38⇥ 103 and Nc 152. In conclusion, a mesh with Dr = 1.30⇥ 103 and Nc = 152 would be sufficiently fine to achieve converged results. Hence, all the simulations with A1/D 0.02 adopted a resolution no coarser than this. To ensure the mesh independence of cases with larger oscillation amplitude of the active cylinder. Mesh independence study has been conducted for the scenario of G/D= 0.9,A1/D = 0.477, f1/ fn = 3.2,m⇤ = 1.5,z = 0,Rem = 10 with a variety of mesh density as listed in Table 2.2. Based on these meshes, typical cases were tested as seen in Fig. 2.4. It can be seen that the mesh with normal density can already provide a decent accuracy. In order to be conservative, the dense mesh is chosen to run all the simulations with 0.02 < A1/D 0.477, which is illustrated in Fig. 2.5. An additional mesh independence study is carried out at KC = 9,G/D = 2.5,A1/D = 1.432, f1/ fn = 2.8,m⇤ = 2,z = 0,Rem = 315 where the active cylinder’s amplitude is larger. Three sets of meshes were tested and compared for the suitability of the mesh density, as seen in Table 2.3. The amplitude in the first 200 oscillation periods of the active cylinder is 22 Methodology Table 2.2 Comparison of the passive cylinder’s displacement with different meshes for KC = 3,G/D= 0.9,A1/D= 0.477, f1/ fn = 3.2,m⇤ = 1.5,z = 0,Rem = 10. Mesh Resolution Fine Normal Coarse Very Coarse Nc 152 134 86 50 Dr 0.00130 0.00144 0.00192 0.00597 Nnode 30280 24050 15971 9722 Y2,min/D -0.04893 -0.04804 -0.04684 -0.04463 Y2,max/D -0.1982 -0.1972 -0.1955 -0.1930 calculated, and considering the irregular nature in regime G, the convergence is quite good. In addition, as seen in Fig. 2.6, the vortex shedding pattern is quite coherent already for three sets of the meshes. As a result, the dense mesh is chosen for the cases with large KC. Table 2.3 Comparison of the passive cylinder’s displacement with different meshes for KC = 9,G/D= 2.5,A1/D= 1.432, f1/ fn = 2.8,m⇤ = 2,z = 0,Rem = 315. Mesh Resolution Fine Normal Coarse Nc 164 140 128 Dr 0.0030 0.0050 0.0058 Nnode 33874 27625 22389 Y2,min/D -0.6871 -0.6645 -1.1603 Y2,max/D 0.4568 0.4593 1.1915 (Y2,maxY2,min)/2D 0.5720 0.5619 1.1759 2.3 Mesh Independence Study 23 Fig. 2.4 Comparison of the passive cylinder’s displacement time histories from different meshes for G/D= 0.9,A1/D= 0.477,m⇤ = 1.5,z = 0 and (a) Rem = 10, f1/ fn = 3.2, where the passive cylinder drifts towards the near side (b) Rem = 150, f1/ fn = 3.2, where the passive cylinder drifts towards the far side (c) Rem = 150, f1/ fn = 0.725, where the vibration amplitude is large due to the resonance. 24 Methodology Fig. 2.5 Computational meshes for interaction between two cylinders with G/D= 0.9 with Dr  1.30⇥103 and Nc 152. 2.3 Mesh Independence Study 25 (a) Coarse mesh (b) Coarse mesh (c) Coarse mesh (d) Coarse mesh (e) Normal mesh (f) Normal mesh (g) Normal mesh (h) Normal mesh (i) Dense mesh (j) Dense mesh (k) Dense mesh (l) Dense mesh Fig. 2.6 Vorticity contours of the mesh independence case at KC = 9,G/D= 2.5,A1/D= 1.432, f1/ fn = 2.8,m⇤ = 2,z = 0,Rem = 315. (a-d) Coarse mesh (e-h) Normal mesh (i-l) Dense mesh. 26 Methodology 2.4 Validation The numerical methods, including the streamline upwind Petrov-Galerkin FEM, the solution of the vibration equation, treatment of the moving boundary and adaptation of the mesh close to the boundary, have all been validated in many previous studies (Zhao, 2013; Zhao and Yan, 2013; Zhao et al., 2013; Cui et al., 2014; Zhao and Cheng, 2014; Lin et al., 2016). With regard to this thesis, the problem of interest is a single cylinder oscillating harmonically in an otherwise still fluid at KC = 5,Rem = 100 to validate our numerical method. The simulation is conducted on a mesh shown in Fig. 2.8, and the simulation results are compared with the experimental data obtained by Dütsch et al. (1998), as shown in figure 2.7. The horizontal and vertical components (u & v) of the simulated and measured velocities are compared along four horizontal sections (i.e. y/D= 0.6,0,0.6 and 1.2) at three phases (i.e. f = 180,210 and 330). It can be seen that the simulation results match well with the experimental data. These experimental data from Dütsch et al. (1998) are commonly used to validate numerical models regarding flow past cylinder scenarios (Guilmineau and Queutey, 2002; Yang and Balaras, 2006; Kim and Choi, 2006; Choi et al., 2007; Kim and Choi, 2019). In addition to the harmonic oscillation of a single-cylinder, simulations were conducted upon a case study that involves the interaction between two cylinders to validate the present numerical method. In this case study, the active cylinder is forced to oscillate along the centre line of the two cylinders, whereas the passive cylinder is neutrally buoyant (m⇤ = 1) and is free to move in the fluid subject to unbalanced hydrodynamic force without any constraint of spring or damper. Initially, both cylinders are at rest, and the active cylinder impulsively starts the oscillation by moving towards the passive cylinder. The diameters of the two cylinders are identical, while the active cylinder’s vibration amplitude is fixed at A1/D = 1.0. The simulation results of the present numerical model are compared with that of Gazzola et al. (2012), who discovered the threshold Reynolds numbers, above which the passive cylinder is repelled away from the active cylinder, and below which the passive cylinder is attracted towards the active cylinder. The threshold Reynolds number is shown to be a function of the initial distance between the two cylinders, as seen in Fig. 2.9a. To illustrate how the threshold Reynolds number is obtained from the present simulations, the passive cylinder’s displacements are plotted at a series of Reynolds numbers in Fig. 2.9b. The threshold Reynolds numbers obtained from the present model agree well with the results in Gazzola et al. (2012). In addition, the current numerical model has also been validated extensively for simula- tions in regime A, A*, C, E, F and G by Zhao and Cheng (2014), and achieved very good agreement with the experimental results by Tatsuno and Bearman (1990). 2.4 Validation 27 -1.0 -0.5 0.0 0.5 1.0 -1.5 -1 -0.5 0 0.5 1 1.5 x/ D -v / Um y/D=0.6, Exp. y/D=0, Exp. y/D=-0.6, Exp y/D=-1.2, Exp. y/D=0.6, Sim. y/D=0, Sim. y/D=-0.6, Sim. y/D=-1.2, Sim. (c) v at ψ=210° -1.0 -0.5 0.0 0.5 1.0 -1.5 -1 -0.5 0 0.5 1 1.5 x/ D -v / Um (e) v at ψ=330° -1.0 -0.5 0.0 0.5 1.0 -1.5 -1 -0.5 0 0.5 1 1.5 x/ D u / Um (f) u at ψ=330° -1.0 -0.5 0.0 0.5 1.0 -1.5 -1 -0.5 0 0.5 1 1.5 x/ D -v / Um (c) v at ψ=210° -1.0 -0.5 0.0 0.5 1.0 -1.5 -1 -0.5 0 0.5 1 1.5 x/ D u / Um (d) u at ψ=210° -1.0 -0.5 0.0 0.5 1.0 -1.5 -1 -0.5 0 0.5 1 1.5 x/ D -v /Um (a) v at ψ=180° -1.0 -0.5 0.0 0.5 1.0 -1.5 -1 -0.5 0 0.5 1 1.5 x/ D u /Um (b) u at ψ=180° Fig. 2.7 Comparison of fluid velocity distribution between the numerical simulation results and the experimental data by Dütsch et al. (1998). The horizontal & vertical velocity components (u & v) of the simulation results (continuous lines) and measured data (discrete symbols) are compared along four horizontal lines (i.e. y/D= 0.6,0,0.6 and1.2) at three different phases (i.e. f = 180,210 and 330). 28 Methodology Fig. 2.8 Computational meshes for validation case of a single cylinder vibrating in an otherwise still fluid with Nc = 134 and Dr = 1.83⇥103. 2.5 Data Processing Techniques This section explains the techniques to process and make sense of the data generated by a large number of simulations. Many scripts are created to automatically generate the input files, to efficiently exploit the high performance computing facilities and to aggregate the simulation results of 23,400 cases. Example data processing scripts are listed in Appendix A. Chapters 3 to 5 only discuss a few representative cases to illustrate the physical pattern. However, the generality of the conclusions is ensured by scrutinisation on all 23,400 cases that has been simulated. The simulation results were automatically summarised into figures by small programmes and then manually examined. This research deliberately implements the serial computing rather than parallel computing to eliminate the inefficiency due to additional data transfer cost. The simulations were all executed on HPC facilities. So the strategy was to run a large amount of serial cases at the 2.5 Data Processing Techniques 29 15 20 25 30 35 40 2 2.5 3 3.5 4 4.5 5 Th re sh ol d Re d/D Gazzola et al. (2012) Present (a) (b) Fig. 2.9 (a) Threshold Reynolds numbers as function of the normalised centre-to-centre distance d/D, wherem⇤= 1, A1/D= 1, and the active and the passive cylinders have identical diameters. The present simulation output agree well with the results from Gazzola et al. (2012). (b) Passive cylinder’s non-dimensional displacementY2/D versus the time normalized by the active cylinder vibration frequency t f1, for a series of Reynolds numbers (from top to bottom curves Rem= 100,60,50,40,30,27,25,20,19,18,15,10), with A1/D= 1,d/D= 3.5. Here, the threshold Reynolds number is 20. same time, typically hundreds or thousands of cases for each batch. Many bash scripts were created to execute the simulation on HPC and to transfer the large amount of data back to local nodes efficiently by compressing the files. Many figures in the results chapters Chapters 3 to 5 illustrate the variation of the passive cylinder’s responding amplitude. In this thesis, for the cases with periodical interaction, the amplitude of passive cylinder’s vibration is calculated as A2 = (Y2,maxY2,min)/2, where Y2,max and Y2,min are the maximum and minimum displacement in the last 50 periods of steady vibration. The vibration centre drift of the passive cylinder in Chapter 5 is calculated as DY 2 = (Ymax+Ymin)/2. The displacement/force spectra of the passive cylinder’s vibration were auto-generated by programmes. The displacement spectra are produced by conducting Fast Fourier Transform (FFT) to the original displacement time history. FFT is an algorithm of signal processing to quickly execute the discrete Fourier transform. It can be used to investigate the time histories on the frequency domain and thus to identify patterns that is usually undistinguishable in the time domain. It can be understood as using many sinusoidal components like y = Asin(wt+f) to fit the time histories. The results of FFT can be demonstrated by two figures. The first figure’s x-axis is the frequency w for all the components with its y-axis being the amplitude A of each corresponding components. The second figure has the same x-axis with y-axis being the phase f of each corresponding components. A series of FFT results can be combined together by adding a third axis. In Chapters 3 to 5, the third axis will be the 30 Methodology active cylinder’s oscillation frequency f1/ fn and thus a three-dimensional figure is plotted to demonstrate the physical pattern. Since FFT can also extract f of the frequency components, the phase difference between the passive cylinder and the active cylinder is also extracted by FFT. To ensure the accuracy of the FFT analysis, in this thesis, at least 50 periods of steady-state periodical vibrations time history are used in the transform. Zero padding is also applied during the transform, which, although adds some noise to the low amplitude regime in the frequency domain, greatly increases the accuracy while extracting the peak amplitude. In Chapter 6, for cases with relatively high KC and thus with irregular time history, the vibration amplitude and the vibration centre shift may both be functions of time, and we calculate them by carrying out fast Fourier transform to converting the displacement time history from the time domain to the frequency domain. In the frequency domain, the low frequency part represents the vibration centre shift, while the high frequency part stands for the vibration relative to the vibration centre, i.e. harmonics. We extract the high frequency parts, and convert the high frequency part back to time domain, and then calculate its amplitude as A2 = (Y2,maxY2,min)/2, since the extracted time history is very regular. 2.6 Chapter Summary This chapter describes the methodology of this study. The problems setup is illustrated and the corresponding dimensional analysis is carried out to determine the 6 non-dimensional groups. The governing equation is presented and the computational methods are discussed. Detailed mesh independence study is conducted to choose the appropriate mesh density and validation is achieved by comparing to both experimental and numerical studies with similar case setups. Chapter 3 Effects of Parameters Associated With the Active Cylinder in Periodic Regimes This chapter examines the effects of the active cylinder’s frequency f1/ fn and its oscillation amplitude A1/D on the passive cylinder’s oscillation amplitude A2/D. Here, the natural frequency fn is defined as fn = p k/m/2p . k is the stiffness of the spring mounted to the passive cylinder. m is mass of the passive cylinder. f1/ fn is considered to be the most influential parameter for A2/D. The resonance of the passive cylinder occurs at f1/ fn ⇡ fw/ fn = p m⇤/(m⇤+CA), where CA is the added mass coefficient of the passive cylinder, i.e. the added mass divided by the displaced fluid mass by the cylinder, and fw is the immersed natural frequency of the passive cylinder in water. The effects of A1/D is presented together with f1/ fn for a clear comparison between their effects and also to demonstrate their coupled effect. To better demonstrate the meaning of the active cylinder’s frequency f1/ fn and its oscilla- tion amplitude A1/D, the forced oscillation equation of the active cylinder, i.e. Equation (2.4), is restated here: Y1 = A1 sin(2p f1t) (3.1) 3.1 Resonance of Passive Cylinder In the periodic regimes, the vibration of the passive cylinder will converge after a few initial periods, as seen in Figure 3.1, where the time history of the passive cylinder is illustrated. So the converged amplitude A2 is a representative value to describe the vibration of the passive cylinder. In this thesis, for the cases with periodical interaction, the amplitude of 32 Effects of Parameters Associated With the Active Cylinder in Periodic Regimes Fig. 3.1 Convergence time history for a case in periodic regime A/A⇤. After a few initial steps, the vibration amplitude of the passive cylinder converges. passive cylinder’s vibration is calculated as A2 = (Y2,maxY2,min)/2, where Y2,max and Y2,min are the maximum and minimum displacement in the last 50 periods of steady vibration. The amplification factor A2/A1 is often used to compare amplitude of the passive cylinder with that of the active cylinder. If A2/A1 > 1, it means that the vibration of the passive cylinder is even larger than that of the active cylinder. Figure 3.2 shows the variation of the amplification factor A2/A1 with f1/ fn. The major resonance peak occurs at the immersed natural frequency f1/ fn = 0.775⇡ fw/ fn = p m⇤/(m⇤+CA) (assumingCA = 1), whereas another minor peak of A2/D takes place at half the immersed natural frequency f1/ fn = 0.375⇡ 0.5 fw/ fn. The reason for the existence of a minor peak will be discussed later. The oscillation amplitude of the active cylinder and the passive cylinder is in general linearly correlated. Figure 3.2 demonstrates the relationship between the active cylinder vibration amplitude A1/D and the passive cylinder vibration amplitude A2/D at z = 0 and 0.2. At z = 0, it can be seen in Fig. 3.2a that A2/D generally follows a linear relationship with A1/D, except near the two resonance peaks. Other non-dimensional parameters have a much smaller influence on this A1A2 relationship than the f1/ fn. At z 0.2, the linear correlation of vibration amplitude is strengthened between the passive cylinder A2/D and the active cylinder A1/D, as demonstrated in Fig. 3.2b. Here, this linear relationship is examined by the variation of amplification factor A2/A1 with the amplitude of the active cylinder A1/D, as seen in Fig. 3.2b. The A2/A1 f1/ fn curves overlap each other regardless of the varying A1/D, which indicates a strong linear correlation, even at resonance. Harmonics are discovered in the response vibration of the passive cylinder, and sometimes, the secondary harmonic can exhibit amplitude larger than the fundamental harmonic, causing secondary resonance at half the immersed natural frequency. In Fig. 3.3, the FFT-generated 3.1 Resonance of Passive Cylinder 33 (a) (b) Fig. 3.2 Variation in A2/A1 with f1/ fn at G/D = 0.4,A1/D = 0.025 0.1,m⇤ = 1.5, and (a) undamped z = 0 (b)damped z = 0.2. A2 and A1 are linearly correlated unless when the resonance occurs or the damping ratio z >= 0.2. displacement amplitude spectra show that the vibration history of the passive cylinder in each case typically contains three distinct frequency components. The corresponding frequency for each frequency components is equal to one, two, or three times of f1/ fn as f/ fn = n f1/ fn,n 2 {1,2,3}, i.e. at least three harmonics exist. As a result, when the active cylinder vibrates at a frequency with f1/ fn = ( fw/ fn)/n (n2 {1,2,3}), one of the harmonics thus reaches the immersed natural frequency, as f/ fn = n f1/ fn ⇡ n[( fw/ fn)/n] = fw/ fn (n 2 {1,2,3}). Since f/ fn = fw/ fn, resonance then occurs at that particular harmonic and yields a significant increase in its amplitude. For example, Fig. 3.3a, 3.3b, and 3.3c show the close-up of the amplitude spectra at f1/ fn ⇡ fw/3 fn, f1/ fn ⇡ fw/2 fn, and f1/ fn ⇡ fw/ fn, respectively. At f1/ fn = 0.25⇡ fw/3 fn, three rows of noticeable frequency components are observed, as seen in Fig. 3.3a. These rows, from the furthest to the nearest row, correspond to the harmonics with f/ fn = f1/ fn, f/ fn = 2 f1/ fn and f/ fn = 3 f1/ fn. Overall, the furthest row of f/ fn = f1/ fn has the largest amplitude. The middle row of f/ fn = 2 f1/ fn has the second largest values of the amplitude. The nearest row of f/ fn = 3 f1/ fn has the smallest values of the amplitude. In Fig. 3.3a, the amplitude values of the third harmonic with f/ fn = fw/3 fn reach their maximum at f1/ fn = 0.25 ⇡ fw/3 fn due to resonance, as seen in the red dashed rectangle of Fig. 3.3a. Similarly, in Fig. 3.3b, two sets of harmonics are observed, i.e. f/ fn = f1/ fn, f/ fn = 2 f1/ fn. The amplitude values of the harmonics with f/ fn = 2 f1/ fn peak at f1/ fn = 0.375⇡ fw/2 fn, as seen in the red dashed rectangle of Fig. 3.3b. Here, the amplitude values of the components on the middle row also exceeds that of the furthest row, meaning that the second harmonic has an amplitude larger than the first harmonic. This results in the minor peak found in Fig. 3.2a. In Fig. 3.3c, only one 34 Effects of Parameters Associated With the Active Cylinder in Periodic Regimes 0.30.250.20.15 0 0.1 0 2 10-4 0.5 4 1 0.420.40.38 0 1 0.36 2 0 3 4 10-3 5 0.5 1 0.34 0.80.780.760.74 0 0.72 0.02 0.7 0.04 0 0.06 0.08 0.1 0.12 0.5 1 1.5 (d) Fig. 3.3 Amplitude spectra showing the response of the passive cylinder with various f1/ fn at G/D= 0.4, A1/D= 0.075, m⇤ = 1.5. (a) FFT spectra at f1 = fw/3⇡ 0.25 (b) FFT spectra at f1 = fw/2⇡ 0.375 (c) FFT spectra at f1 = fw ⇡ 0.75 (d) An overview of FFT spectra. It can be seen that, at f1 = fw/2, there are two distinct frequencies while at f1 = fw there is only one. The red dashed thin line tracks the dominant frequencies. The red dashed thick squares highlight the resonating frequency component. 3.1 Resonance of Passive Cylinder 35 1.1 1.05 1 0.95 0 1 1.5 2 0.005 0.01 0.015 0.02 Fig. 3.4 Amplitude spectra showing (a) responding displacement of the passive cylinder and (b) force coefficient upon the passive cylinder at 0.95 f1/ fn  1.1, G/D= 0.4, A1/D= 0.075, m⇤ = 1.5, z = 0, Rem = 100. The dominant frequencies for displacement and force are different. row of the frequency components ( f/ fn = f1/ fn) is identified, whose values of amplitude reaches maximum at the immersed natural frequency f1/ fn = 0.75. This constitutes the major resonance peak in Fig. 3.2. Fig. 3.3d provides an overview of the FFT spectra. The red dashed line denotes the dominant frequency in each case. It is clear that the dominant frequency is equal to f1/ fn except at f1/ fn = 0.375⇡ fw/2 fn, where the minor resonance is observed. Although the above analysis is for a specific case with f1/ fn being the only variable, all other cases simulated with various A1/D,G/D, f1/ fn,m⇤,Rem,z follow almost the same pattern. The multiple frequency components observed in the passive cylinder is as if the active cylinder is play a sound of a single tone and the passive cylinder responds with another sound of harmonics. The variation of force acting on the passive cylinder CY2 with f1/ fn generally follows the same pattern as the displacement, except at 0.95 < f1/ fn < 1.1 with z = 0. In this range, the displacement amplitude spectra only have one row of noticeable components for f/ fn = f1/ fn, as shown in Fig. 3.4a, and their amplitude decreases steadily with the active cylinder frequency f1/ fn. In contrast, at the same range of 0.95 < f1/ fn < 1.1, two rows of distinct frequency components, i.e. f/ fn = f1/ fn and f/ fn = 2 f1/ fn, are found in the force amplitude spectra shown in Fig. 3.4b. At f1/ fn = 1, the amplitude of the further row with f/ fn = f1/ fn reaches its minimum and becomes much smaller than the nearer row. Meanwhile, the amplitude of nearest row f = 2 f1 increases slightly with f1/ fn. It can be seen that a discrepancy between the dominant frequencies of Y2 and CY2 is discovered at 0.99 f1/ fn  1.01, which is demonstrated by the red dashed lines tracking the dominant frequencies at 0.99 f1/ fn  1.01. 36 Effects of Parameters Associated With the Active Cylinder in Periodic Regimes (a) (b) Fig. 3.5 Variation of the oscillation phase difference Df21 between the passive and the active cylinder (for the fundamental frequency components) with f1/ fn at A1/D= 0.025 0.100,m⇤ = 1.5,G/D= 0.4,Rem = 100 and (a) z = 0 (b) z = 0.2. The increase of the active cylinder oscillation frequency causes the phase difference to shift from anti-phase to in-phase at the immersed natural frequency. Df21 decreases with A1/D. At z = 0, A1/D affects more the phase difference at low frequencies compared with that at z = 0.2. 3.2 Phase Lag Shift from Anti-phase to In-phase The phase difference between the vibration of the passive cylinder and the active cylinder changes for 180 when the active cylinder’s oscillation frequency increases beyond the passive cylinder’s resonance frequency. The phase difference is defined as the vibration phase of the passive cylinder minus that of the active cylinder, i.e. Df21 = f2f1, where the phases of the passive cylinder and the active cylinder are obtained through FFT. The phase difference is dominantly controlled by the active cylinder vibration frequency f1/ fn. As seen in Figure 3.5, a 180 phase jump for Df21 occurs at the immersed natural frequency, below which the two cylinders vibrate in anti-phase, i.e. Df21 ⇡ 180, and above which they vibrate in phase, i.e. Df21 ⇡ 0. The phase jump frequency is equal to the resonance frequency, and the resonance occurs when Df21 ⇡ 90. The phase difference is not sensitive to other parameters except damping ratio z . Df21 decreases slightly with active cylinder vibration amplitude ratio A1/D at low active cylinder frequency f1/ fn. At z = 0, this impact of A1/D diminishes with f1/ fn, demonstrated as the overlapped curves in Fig. 3.5a, whereas, at z = 0.2, the impact of A1/D remains constant, as seen in Fig. 3.5b. 3.3 Effects on Flow Field In regimes A and A⇤, no vortex shedding is identified and the flow is laminar and symmetrical, which reflects the feature of the regime A⇤ for single cylinder vibration. There are two types 3.3 Effects on Flow Field 37 (a) (b) Fig. 3.6 Velocity vectors on the left of the passive cylinder atG/D= 0.4,A1/D= 0.075,m⇤= 1.5,z = 0,Rem = 100, and f1 = 45 with (a) f1/ fn = 0.74, Df21 = 90.7 and (b) f1/ fn = 1.00, Df21 = 0.58. The vortex is always attached to the surface of the cylinders on their two sides, but it is dissipated before they can detach from the surface. Df21 is found to have significant influence on the flow field. The velocity vector is drawn on every grid point and the vector scale factors are (a) 0.06 grid units/magnitude and (b) 0.6 grid units/magnitude. 38 Effects of Parameters Associated With the Active Cylinder in Periodic Regimes Fig. 3.7 Velocity vectors (u⇤i = ui/ fnD) at f1/ fn = 1.00, f1 = 0. A pair of small vortices show up in the wake of the passive cylinder when the passive cylinder reaches its highest or lowest position. These tiny vortices only exist for a short period of time in each cy- cle. The velocity vector is drawn on every grid point. The vector scale factor is 1.6 grid units/magnitude. Fig. 3.8 Velocity vectors at f1/ fn= 0.74 and f1= 135 withG/D= 0.4,A1/D= 0.075,m⇤= 1.5,z = 0,Rem = 100. At exactly 90 < f1 < 180, the zero-velocity point travels from the bottom of the passive cylinder (at f1 = 90) to the top of the active cylinder (at f1 = 180). The velocity vector is drawn on every grid point. 3.3 Effects on Flow Field 39 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) p*: Fig. 3.9 Non-dimensional pressure p⇤ contours at f1 = (a & e & i) 0, (b & f & j) 90, (c & g & k) 180, (d & h & l) 270. (a-d) f1/ fn = 0.55, Df21 = 168.71; (e-h) f1/ fn = 0.74, Df21 = 90.673; (i-l) f1/ fn = 1.0, Df21 = 0.582 with G/D = 0.4,A1/D = 0.075,m⇤ = 1.5,z = 0,Rem = 100. The motion of the passive cylinder is found to have great influence on the distribution of pressure. Streamlines are plotted to show the status of the fluid. 40 Effects of Parameters Associated With the Active Cylinder in Periodic Regimes of hydrodynamic forces upon the cylinder surface, i.e. shear and pressure forces. In the present study, the pressure force always has an amplitude larger than shear force, so the pressure force is the dominant contributor to the motion of the passive cylinder. Therefore, pressure contours are used to visualise the pressure distribution around the cylinders. For comparison purposes in different configurations, corresponding figures are with the same phase of the active cylinder’s vibration, f1. f1 = 0 is defined as the moment when the active cylinder is at its highest position. In the following discussion, the default condition is G/D = 0.4, A1/D = 0.075, and m⇤ = 1.5. The flow fields at different values of f1/ fn are examined because f1/ fn has a dominant effect upon the flow field compared with other non-dimensional groups. In regimes A and A⇤, for both the active and the passive cylinders, vortices are generated on two sides of the cylinder at every f1/ fn, but the vortices are dissipated before they can detach from the surface. This vortex generation-dissipation process occurs twice in each cycle of the vibration. Figure 3.6 shows two typical flow patterns observed at f1/ fn = 0.74 and f1/ fn = 1.0 respectively. At f1/ fn = 0.74, the first type of the flow pattern is observed, as seen in figure 3.6a. When the passive cylinder moves downwards, the fluid is still moving upwards due to the previous upward motion of passive cylinder. This creates a vortex that moves outwards as well as upwards relative to the passive cylinder. The vortex grows larger as it moves out, but eventually, the streamlines can no longer be closed, and thus the vortex is dissipated. This process can be affected by f1/ fn. At f1/ fn = 0.74, the resonance occurs and the amplification factor is large, i.e. A2/A1 ⇡ 1.7. This means that the passive cylinder vibration amplitude is much larger than that for the active cylinder, and the flow field close to the passive cylinder is dominantly controlled by the passive cylinder itself. Consequently, the velocity vector field near the passive cylinder looks very similar to a single cylinder vibrating in still fluid. In contrast, at f1/ fn = 1.00, we see the second type of the flow pattern, as demonstrated in figure 3.6b. Since the passive cylinder amplitude is much smaller than that for the active cylinder, as A2/A1 ⇡ 0.25, the flow around the passive cylinder is greatly influenced by the active cylinder, which results in a unique flow pattern. A zero-velocity point can be seen close to the vortex at the coordinate of about (0.58,0.57). The fluid below that point has been dragged down due to the downward motion of the active cylinder while the fluid above is still moving up due to inertia, forming a vortex smaller and weaker than that at f1/ fn = 0.74. It is also interesting that the position of the zero-velocity point is relatively stable throughout the entire cycle. If higher f1/ fn is applied, the zero-velocity point appears upwards. At f1/ fn = 1.00, when the passive cylinder has either reached its highest or lowest position, i.e. f1 ⇡ 0 or 180, a pair of vortices appears below the passive cylinder, as seen 3.3 Effects on Flow Field 41 in figure 3.7. This is because the active and the passive cylinders move in phase and the upward motion of the passive cylinder causes the fluid to flow down along its surface and to collide with the fluid pushed up by the active cylinder. The two flows with opposite directions collide, and therefore a pair of vortices are generated. Conversely, if the active and the passive cylinders are both moving downward, reaching their lowest positions, a pair of vortices with opposite vorticity is produced. At f1/ fn = 0.74, a moving zero-velocity point is found in the gap and on the boundary dividing the fluids flowing upwards and the fluids going downwards, as seen in Fig. 3.8. This point is found to travel downward from the bottom of the passive cylinder to the top of the active cylinder only, which occurs twice in each cycle. At f1/ fn = 0.74, the oscillation phase difference is Df21 = 90. Therefore, for every 90, the passive cylinder and the active cylinders switch between moving in the same direction and in the opposite direction. For example, in figure 3.8, the zero-velocity point appears when the passive cylinder and the active cylinders are moving in opposite directions, i.e. towards each other or away from each other, and that point travels only from the passive cylinder to the active cylinder. This is because A2/D is larger than A1/D, i.e. A2/A1 ⇡ 1.7, and thus the passive cylinder has a larger maximum velocity. At 90 < f1 < 180, the passive cylinder is accelerating to its maximum upwards velocity, while the active cylinder is decelerating to zero velocity and moving downwards to its lowest position. The passive cylinder thus drives its surrounding fluid upwards, reversing the flow direction from the previous downwards movement. Here, a zero-velocity point appears. As this process carries on, more and more fluids are dragged upwards by the passive cylinder, and the boundary thus continues to move downwards. Consequently, the zero-velocity point moves downwards as well, which eventually touches the near-side surface of the active cylinder. At that moment, the active cylinder velocity reaches zero, and the zero-velocity point disappears. Conversely, when the passive cylinder and the active cylinders move towards each other, the same process repeats, although the fluid in the gap changes from flowing upwards to going downwards, as the zero-velocity point travels. The existence of zero-velocity point can be affected the structural damping factor z , which will be discussed later in Chapter 4. Figure 3.9 depicts contours of non-dimensional pressure and streamlines of the flow around the cylinders. At f1/ fn = 0.55, i.e. figure 3.9a-3.9d, since A2/D and f1/ fn are both low, the fluid is less disturbed and thus its pressure contours conduct much less fluctuation compared with that at f1/ fn = 0.74 & 1.00. At f1/ fn = 0.55, since the amplification factor is A2/A1 ⇡ 0.25, the active cylinder has more control over the pressure distribution compared with the passive cylinder. The active and the passive cylinders vibrate in anti-phase and thus always move in opposite directions. As a result, the streamlines do not connect the 42 Effects of Parameters Associated With the Active Cylinder in Periodic Regimes active cylinder and the passive cylinder directly through the gap. In other words, the flow direction is reversed in the gap. At f1/ fn = 0.74, f1 = 90, as depicted in figure 3.9f, when two cylinders move close to each other, the fluid is pushed out of the gap and continues to flow due to inertia, and the pressure in the gap gradually becomes negative. When two cylinders move away from each other, the fluid flows back to fill the gap, where the pressure is increased. Since Df21 = 90.673, the two cylinders switch periodically from moving in the same direction to moving in the opposite direction, which causes a unique pressure distribution, as demonstrated in figure 3.9e-3.9h. Correspondingly, the distribution of streamlines switches periodically from connecting the cylinders directly, e.g. figure 3.9f, to not being so, e.g. figure 3.9g. At f1/ fn = 1.00, i.e. figure 3.9i-l, the two cylinders vibrate in phase and A2/A1 ⇡ 0.25. As a result, the pressure distribution is mainly controlled by the active cylinder while the passive cylinder only has a minor contribution. There are always streamlines directly connecting the two cylinders because they vibrate in phase. Despite the identical phase f1 and amplitude A1/D of the active cylinder, the pressure contours at f1/ fn = 0.74 and that at f1/ fn = 1.0, as seen in figures 3.9f and 3.9j, are drastically different, because the phase of the passive cylinder at f1/ fn = 1.0 is 90 lagged behind that at f1/ fn = 0.74. In other words, the pressure distribution on the surface of the passive cylinder is greatly influenced by the motion of the passive cylinder itself. The pressure contours caused by the active and the passive cylinders interfere each other. For example, at f1/ fn = 1.00 and f1 = 0, i.e. figures 3.9i, the two cylinders vibrate in phase and both reach their highest positions with their velocities decelerating to zero. Since the active cylinder has a much larger amplitude than the salve and they vibrate in phase, the pressure distribution is dominantly controlled by the active cylinder. For the same reason, the motion of the passive cylinder and the friction on its surface only slightly affects the distribution of the negative pressure and do not affect on the distribution at the far side of the active cylinder. Some low pressure areas are attached to the left and right surfaces of the active cylinder or the passive cylinders. The vortices can be identified by the closed streamlines on the two sides of the cylinders. For example, for f1/ fn = 1.00, at f1 = 90 and f1 = 270, i.e. figures 3.9j and 3.9l, the negative pressure (dashed lines) attached to the shoulders of the active cylinder corresponds to the vortices in figure 3.6. These vortices can also be identified by the streamlines in figure 3.9j and 3.9l. In summary, although the passive cylinder is driven by the disturbed fluid around it, it also feeds back to the flow and changes the surrounding pressure distribution. When the passive cylinder is free to move, as in the past studies, its existence does not influence the disturbed flow much, and the flow pattern is similar to that of a single cylinder oscillating in a still fluid. When the passive cylinder has a different density from the fluid and is elastically-mounted, 3.4 Chapter Summary 43 its influence on the disturbed flow can be significant; therefore the coupling between the active and passive cylinders is more complex. 3.4 Chapter Summary Detailed numerical simulations are conducted to investigate the effects of active cylinder’s am- plitude A1/D and frequency f1/ fn upon the mechanical interactions between two submerged cylinders. With considered cases in this chapter, all the interactions reach a periodic steady state after a few cycles. This chapter discusses only the characteristics of the interaction during the steady state. Active cylinder’s oscillation frequency f1/ fn is found to play a major role in governing A2/D and Df21. The integer multiples of the active cylinder’s vibration frequency, i.e. harmonics, are distinguished by FFT in the responding vibration of the passive cylinder, and the magnitude of these harmonics is related to the hydrodynamic force acting on the passive cylinder. When n f1/ fn = fw/ fn, n 2 {1,2,3}, resonance occurs at the corresponding harmonic. This explains the secondary resonance occurring at f1/ fn = ( fw/ fn)/2, which takes place at the second harmonic f/ fn = 2 f1/ fn. At structural damping z = 0, A1 is linearly correlated with A2 except when the resonance occurred. At z  0.2, the linear correlation can be assumed even at resonance. At z = 0, the pattern of the force acting on the passive cylinder is very similar to the pattern of the resulting displacement. However, at 0.99  f1/ fn  1.01, the dominant frequencies of the external force and the displacement are distinctively different.As f1/ fn increases from 0.05 to 2.4, the phase difference Df21 between the passive cylinder and the active cylinders observes a 180 change, from being anti-phase to being in-phase, and the phase shift occurs abruptly at around f1/ fn = fw/ fn. At Df21 ⇡ 90, the passive cylinder’s vibration amplitude A2/D reaches its maximum. Small vortices are generated on the left and right shoulders of the two cylinders, but no vortex shedding takes place, which is in line with the flow features in regime A⇤ (Tatsuno and Bearman, 1990). The variations in Df21 and A2/A1 are also reflected in the changes in the pressure distributions. Chapter 4 Effects of Parameters Associated With the Passive Cylinder in Periodic Regimes This chapter focuses on the effects of parameters relevant to the passive cylinder, i.e. the damping ratio z and the mass ratio m⇤ of the passive cylinder. Each parameter affects the cylinder-to-cylinder flow-mediated interaction in its unique way. To better demonstrate the meaning of the damping ratio z and the mass ratio m⇤, the mass-spring-damper system of the passive cylinder, i.e. Equation (2.5), is restated here: ∂ 2Y2 ∂ t2 +4p fnz ∂Y2 ∂ t +4p2 f 2nY2 = 2 p U2m m⇤D CY2 (4.1) Although this chapter focus on the effects of parameters related to the passive cylinder, the analysis on data are often presented in variation of the active cylinder’s oscillation frequency f1/ fn, because f1/ fn is the most influential non-dimensional group due to resonance as mentioned in Chapter 3, so the display of results can be less meaningful if not shown with a variety of f1/ fn. Such manner of demonstration will allow clear comparison between effects of different non-dimensional groups. 4.1 Structural Damping Ratio The structural damping ratio z of the passive cylinder is an important non-dimensional group that it not only affects vibration amplitude of the salve but also alters the phase difference between the passive cylinder and the active cylinder. The increase in z can strengthen the linear relationship between the amplitude of the passive cylinder A2 and that of the active cylinder A1. A critical damping ratio is found that beyond which the resonance amplitude increases with the active cylinder’s oscillation amplitude A1/D, and 46 Effects of Parameters Associated With the Passive Cylinder in Periodic Regimes (a) X: 2.4 Y: 174.3 f 0 (d eg re es ) (b) Fig. 4.1 (a) Variation in magnification factor of the passive cylinder A2/(F0/k)with f1/ fn, and (b) variation in force-displacement phase difference for the passive cylinder (for frequency components with f/ fn = f1/ fn) with f1/ fn at G/D = 0.2, A1/D = 0.1, m⇤ = 2.0, and z = 01.4. The red dashed line is the locus of maxima by assuming harmonic force input. below which the resonance amplitude decreases with A1/D. It is interesting to find that the flow characteristics are subtly influenced by z . Being counter-intuitive, the damping ratio of the passive cylinder affects the pressure distribution at the far side of the active cylinder. z also alters the timespan for the flow in the gap to reverse its direction. 4.1.1 Approximating the Passive Cylinder’s Motion to be Harmonic in the Context of Varied Structural Damping This section discusses the appropriateness to approximate the passive cylinder’s vibration as sinusoidal with a frequency identical to that of the active cylinder in the context of varied structural damping factors by examining the magnification factor and the displacement-force phase difference. With such an approximation, the response of the passive cylinder can be described as Eqs. (4.2) and (4.3): FY2(t) = F0 sin(2p f1t) (4.2) Y2 = A2 sin(2p f1t+f0) (4.3) where the amplitude of the force acting on the passive cylinder is calculated as F0=(FY2,max FY2,min)/2. FY2,max and FY2,min are, respectively, the maximum and minimum force values in final 50 cycles of steady vibration. f0 is the phase difference between the sinusoidal force and the displacement. 4.1 Structural Damping Ratio 47 In Fig. 4.1a, the magnification factor A2/(F0/k) of the passive cylinder’s displacement is calculated from the simulation results as: A2 F0/k = ( A2 D )/( F0/(D/2) 2p3m⇤ f 2nD2r ) (4.4) which shows the same pattern as that of a second-order linear system subject to harmonic forcing. The analytic maxima locus, Eqs. (4.5) and (4.6), can be deduced from substituting FY2(t) = F0 sin(2p f1t) into Eq. (2.5), the motion equation of the passive cylinder. ( f1/ fn)peak = q 12z 2 (4.5) (A2/(F0/k))peak = 1 2z p 1z 2 (4.6) The peaks of the simulated magnification factor match the above analytic maxima loci very well as indicated by the red dashed line in Fig. 4.1a. This corresponds to the approximation of the harmonic response of the passive cylinder. Moreover, as seen in Fig. 4.1b, the simulated displacement-force phase difference f0 agrees exactly with the analytic phase difference described in Eq. (4.7): f0 = arctan 2z ( f1/ fn) 1 ( f1/ fn)2 (4.7) which is deduced from Eqs. (2.5) and (4.2). Due to the almost perfect matching between the numerical and analytical results, the curves are nearly indistinguishable. Therefore, only the simulated curves are demonstrated in Fig. 4.1b. The harmonic assumption of Eq. (4.2) can capture the major dynamic features of the passive cylinder regardless of the varying damping factor. The harmonic assumption of Eq. (4.2) leads to minor inaccuracies, since the vibration of the passive cylinder is predominantly but not entirely harmonic. For example, It leads to a discrepancy at f1/ fn = 0.4, where secondary resonance exists. However, the secondary resonance can be explained by multiple superimposed harmonic input forces (discussed further in Section 4.1.2). Another difference from the analytic solution is that A2/(F0/k) does not reach infinity at f1/ fn = 1 and z = 0, which means that the input force is not purely harmonic in that situation. Despite these minor discrepancies, the harmonic assumption is capable of capturing the general features of the passive cylinder vibration for all the cases. 48 Effects of Parameters Associated With the Passive Cylinder in Periodic Regimes 4.1.2 Effects of Structural Damping on the Vibration of the Passive Cylinder The structural damping can significantly affect passive cylinder’s vibration amplitude and phase lag. It can also alter the relationship between other non-dimensional groups and the passive cylinder’s vibration pattern. The amplification factor A2/A1 decreases with the damping factor z , as seen in Fig. 4.2a. This decrease is most dramatic when the response is close to resonance. For example, the resonance amplitude at z = 0.4 is only a seventh of that at z = 0. The resonance amplitude is highly sensitive to z when damping is small, but it becomes less sensitive to z , with the increase of the damping factor. For example, the resonance amplitude decreases by half with the increase of z from 0 to 0.05, whereas the resonance amplitude decreases by only a third from z = 0.05 to z = 0.10. At z 0.8, a resonance peak can no longer be observed. Instead, the responding amplitude increases with the frequency monotonically. At z  0.10, the secondary resonance increases with the vibration amplitude of the active cylinder. However, at z 0.20, A1/D can no longer affect the pattern of the minor resonance. As the active cylinder vibrates at a very low frequency, the responding amplitude of the passive cylinder, regardless of z , converges to zero. If the active cylinder vibrates at a very high frequency, the responding amplitude converges to a certain value above zero, i.e. A2/A1 = 0.2. This means that at either a very low or a very high vibrating frequency, the effect of damping on the responding amplitude of the passive cylinder becomes very small, which is in contrast with that at resonance. These patterns apply to all the cases. As demonstrated in Section 4.1.1, the passive cylinder response can be approximated as harmonic with a frequency equal to the active cylinder frequency, with its frequency identical to that of the active cylinder. The passive cylinder actually vibrates with more than one frequency components according to the FFT results, although the dominant frequency component is, in most cases, the first harmonic with f/ fn = f1/ fn, i.e. the fundamental frequency component. The amplitude of the frequency component with f/ fn = f1/ fn is usually much larger than other frequency components. This is why the passive cylinder response can be approximated as harmonic with a frequency of the active cylinder. For example, in the amplitude spectra plotted in Fig. 4.3, two harmonics are typically observed in the response of the passive cylinder. The one having the same frequency as the active cylinder f/ fn = f1/ fn is the dominant component, whereas the amplitude of the second harmonic, i.e., component with a double frequency f/ fn = 2 f1/ fn, is much lower. The amplitude of the second harmonic is generally small, but it becomes comparable with the amplitude of the first harmonic as the active cylinder’s frequency is about f1/ fn = 0.375 as shown in Fig. 4.3. Such large amplitude occurs because the frequency of the second 4.1 Structural Damping Ratio 49 (a) (b) Fig. 4.2 (a) Variation of A2/A1 and (b) variation of phase difference between the two cylinders Df21 with f1/ fn (for the frequency components with f/ fn = f1/ fn) with f1/ fn at G/D = 0.2 ,A1/D = 0.1, m⇤ = 2.0, and z = 0 1.4. A2/A1 is negatively correlated with z , particularly within the regime of resonance. The phase difference curve converges at Df21 ⇡ 80, f1/ fn ⇡ 0.775. The increase in damping slows down the 180 phase shift of the passive cylinder as f1/ fn reaches its immersed natural frequency. harmonic f = 2 f1/ fn coincides with the immersed natural frequency. As mentioned before, the displacement amplitude of the passive cylinder decreases with z . At half the submerged natural frequency, a threshold z is found, under which the amplitude of the second harmonic is greater than that of the first harmonic, and beyond which it is lower than that of the first harmonic. Fig. 4.3b illustrates a case with the damping coefficient above the threshold. From 50 Effects of Parameters Associated With the Passive Cylinder in Periodic Regimes the curves demonstrated in Fig. 4.2a, the threshold damping coefficient for such a setup is found to be around z = 0.1. Dominant influence of the second harmonic can be observed in Fig. 4.2a and Fig. 3.3 at half the immersed natural frequency with z < 0.1, but they are not observed with z > 0.1. In other words, the secondary resonance can be suppressed by adding value to the structural damping factor. The existence of this threshold damping coefficient signifies that the two harmonics are not equally influenced by damping. Because the displacement is more sensitive to damping at resonance than at other situations. When the active cylinder vibrates at 1/2 the immersed natural frequency of the passive cylinder, the second harmonic of the passive cylinder’s displacement with f/ fn = 2 f1/ fn coincides with the immersed natural frequency, causing resonance. The increase in damping causes more amplitude reduction for the second harmonic than that for the first harmonic, whose frequency does not coincide with the immersed natural frequency of the passive cylinder. In the phase analysis, the vibration phase of the passive cylinder is represented by the phase of its first harmonic with f/ fn = f1/ fn. This representation captures the predominant characteristics of the motion, except at the the secondary resonance, where the displacement of the passive cylinder has two harmonics with comparable magnitude. Such a representation enables us to compare the phases Df21 of the passive cylinder and the active cylinders, because the fundamental frequency of the passive cylinder synchronizes with the active cylinder vibration at the steady state. Fig. 4.2b illustrates the vibration phase difference Df21 = f2 f1 between the passive cylinder f2 and the active cylinder f1, at various z . Regardless of z , it is seen that, at a very low and very high f1/ fn, the values of Df21 are approximately 180 and 0, respectively. The two cylinders’ vibration is in anti-phase at low frequencies and in phase at high frequencies. The phase of the passive cylinder changes abruptly at resonance when structural damping is low, but the phase change with the frequency becomes increasingly gradual with increasing damping. For example, at z = 0, the phase change of the passive cylinder occurs with an almost vertical gradient at the immersed natural frequency, whereas at z = 1.4, the phase difference reduces gradually with the increment of f1/ fn. Although damping ratio affects the rate of the phase change, the phase difference remain constant at different z , as all the curves pass through Df21 ⇡ 80 and f1/ fn ⇡ 0.775. It means that the phase difference is independent of damping at resonance. Later in Section 4.1.4, the effects of damping on flow features at resonance are investigated. In addition, it is seen that, at a low damping of z  0.2, the relative resonance amplitude decreases with the input amplitude A1/D, e.g. Fig. 4.4a. Conversely, given a relatively high damping of z 0.4, the resonance amplitude increases with A1/D, e.g. Fig. 4.4b. This reversal of the correlations indicates the existence of a threshold damping factor, under which A2/A1 decreases with A1/D, and beyond which A2/A1 increases with A1/D. 4.1 Structural Damping Ratio 51 (a) (b) Fig. 4.3 Amplitude spectra showing the responding displacement of the passive cylinder with various f1/ fn at G/D= 0.2, A1/D= 0.1, and m⇤ = 2.0. (a) An overview of the FFT spectra (b) the FFT spectra at f1 = fw/2 ⇡ 0.375 . The red dashed thin line tracks the dominant frequencies. It can be seen that, at f1 = fw/2, the high frequency component is always smaller than the fundamental component, whereas for the undamped cases, a high-frequency component becomes larger than the fundamental frequency, thus causing a minor peak of resonance. 52 Effects of Parameters Associated With the Passive Cylinder in Periodic Regimes (a) (b) Fig. 4.4 (a) Zoom-in at peaks for variation of A2/A1 at z = 0.2, and (b) zoom-in at peaks for variation of A2/A1 at z = 0.4 for G/D= 0.4, A1/D= 0.0250.1, m⇤ = 1.5. A threshold damping ratio of the passive cylinder is discovered, beyond which its peak amplification factor decreases with the active cylinder vibration amplitude, and under which the peak increases with the amplitude. In this scenario, the threshold lies in z = 0.20.4. In summary, the vibration amplitude of the passive cylinder decreases with z , and the magnitude of this decrease drops with damping as well. The increase in damping reduces and can eventually eliminate the secondary resonance at half the immersed natural frequency. From Fig. 3.2b in Chapter 3, it is also known that the increase in damping strengthens the linear relationship between the active cylinder and the passive cylinder vibration amplitude. A threshold damping factor is found that, below which its peak amplification factor A2/A1 decreases with the active cylinder amplitude A1/D, and beyond which the peak increases with A1/D. The above results show that the relative amplitude can be affected by the damping factor in both a direct and an indirect way. 4.1.3 Force Acting on the Passive Cylinder For the mass-spring system of the passive cylinder, the input force determines the time history of its displacement. The input force acting on the passive cylinder is controlled by the status of the surrounding fluid in the process of this flow-mediated interaction. In this section, the fluctuation of the pressure and the flow distributions in one cycle of the passive cylinder vibration is examined, and how the force acting on the passive cylinder changes correspondingly. The examined scenario is only one typical resonating case at G/D= 0.4, A1/D= 0.075, m⇤= 1.5, f1/ fn = 0.74,z = 0,Rem = 100, and Df21 = 90.673, but it is capable to represent the most common situation encountered and to reveal the underlying principles in this process. It will be discussed later that the decrease of Rem can enhance the contribution from shear force, but the pressure force is always significant. 4.1 Structural Damping Ratio 53 (a) y/ D 2 1 0 -1 -2 x/D 10-1 (b) x/D 10-1 (c) x/D 10-1 (d) x/D 10-1 p*: (e) Fig. 4.5 (a–d) Non-dimensional pressure contours p⇤ = p/r f 2nD2 at G/D = 0.4,A1/D = 0.075,m⇤ = 1.5, f1/ fn = 0.74,z = 0,Rem = 100, and Df21 = 90.673. The dashed lines indicate negative values of non-dimensional pressure. The colour map shows corresponding colour for each value of non-dimensional pressure. The velocity vectors are plotted every 20 points. The vector scale factor is 0.3 grid units/magnitude. (e) The y-direction force coefficient upon the passive cylinder and its shear and pressure components were examined. Pressure rather than viscosity is the main contributor to the force acting on the passive cylinder. 54 Effects of Parameters Associated With the Passive Cylinder in Periodic Regimes As discussed in Sections 4.1.1 and 4.1.2, the force acting on the passive cylinder can be treated as harmonic, especially at resonance. By examining the time history of the force coefficientCY2 at resonance, i.e. see Fig. 4.5e, the pressure and the shear force components is predominantly sinusoidal. The pressure distributions at the maxima, minima, and transient equilibrium of the force are demonstrated in Figs. 4.5a-4.5d. The force acting on the passive cylinder is most significantly induced by pressure rather than viscosity, as seen in Fig. 4.5e. The pressure force is induced by the pressure distribution surrounding the passive cylinder, which is illustrated in Figs. 4.5a-4.5d. It is seen that the maximum attractive and repulsive forces occur when the pressure field within the gap is at its highest and lowest positions, as shown in figures 4.5b and 4.5d respectively. At these positions, the pressure gradients are at the steepest, which poses the greatest acceleration on the fluid surrounding the passive cylinder. In Figs. 4.5b and 4.5d, the pressure patterns are almost identical. If you reverse the sign of the pressure contours in Fig. 4.5b and adjust for the pressure reduction due to vortices, you can then obtain Fig. 4.5d. The passive cylinder is at transient equilibrium in both Figs. 4.5a and 4.5c, where distinctively different patterns are observed. In Fig. 4.5a, the pressure magnitude is very low at the far side of the passive cylinder as well as within the gap, while in Fig. 4.5c, a higher magnitude of pressure is found at the same regions. The passive cylinder reaches its maximum velocity at both Figs. 4.5a and 4.5c, creating a pair of vortices much stronger than that in Figs. 4.5b and 4.5d. However, the direction of the velocity is opposite in Figs. 4.5a and 4.5c, which reverses the sign of the surrounding pressure contours. Nevertheless, the strong vortices can only decrease the pressure, which strengthens the negative pressure and weakens the positive pressure. The downward momentum of the passive cylinder in Fig. 4.5a results in a lower pressure at the far side of the passive cylinder, while in Fig. 4.5c, the upward momentum increases the pressure below the passive cylinder. On the circumference of the passive cylinder, the pressure distribution is negative in Fig. 4.5a and is positive in Fig. 4.5b except in the shoulder regions characterized by strong vortices. Nevertheless, the circumferential pressure distributions are symmetrical regarding the x-axis in both Figs. 4.5a and 4.5c. As a result, the passive cylinder reaches transient equilibrium in Figs. 4.5a and 4.5c. Compared with the pressure force, the shear force is also sinusoidal but has a very limited contribution to the force acting on the passive cylinder, since the shear force is much smaller than the pressure force. The shear force maintains a constant phase difference from the pressure force regardless of f1/ fn and z . At resonance, the shear force slightly increases the amplitude of the force acting on the passive cylinder. In addition, the fluid flows in and out of the gap periodically in response to the sinusoidal vibration of the active cylinder and 4.1 Structural Damping Ratio 55 the passive cylinder, which causes the horizontal and vertical components of the fluid velocity in the gap to change sinusoidally as well, as demonstrated in Figs.4.5a-4.5d. 4.1.4 Effect of Structural Damping on Flow Field Section 4.1.3 has demonstrated that pressure is the main contributor to the passive cylinder vibration. This section further discusses the effects of z on the pressure and flow fields surrounding the cylinders at resonance, as seen in Fig. 4.6. As previously discussed in Section 4.1.2 with Fig. 4.2b, when resonance takes place, the variation of the damping does not affect the phase difference between the two cylinders, which facilitates the flow field comparison among the cases with different damping ratios. The effect of z on the pressure and the streamline distributions is demonstrated in Fig. 4.6 by varying the value of z from 0, 0.1 to 0.8 while keeping other parameters unchanged. For comparison purposes in different configurations, corresponding figures on the same column are with the same phase of the active cylinder’s vibration, f1. f1 = 0 is defined as the moment when the active cylinder at its highest position. When z = 0, the pressure fluctuates most severely on the far side of the passive cylinder, as seen in Figs. 4.6a and 4.6c. With the increase in damping, the fluctuation at the far side of the passive cylinder becomes less severe, whereas the pressure fluctuation within the gap becomes significant as seen in Figs. 4.6a, 4.6e and 4.6i. This phenomenon indicates that, with the increase in structural damping, the main contributor of the passive cylinder’s motion changes from the pressure fluctuation at the far side regime of the passive cylinder to that within the gap. This is because the increase in damping severely reduces the vibration amplitude of the passive cylinder, as demonstrated in Fig. 4.2a, and thus the pressure accumulated in the gap cannot be alleviated in time due to the inadequate displacement of the passive cylinder, resulting in a stronger pressure fluctuation in the gap. The restricted motion of the damped passive cylinder also prevents the generation of a strong vortex, corresponding to a shorter lifespan of the vortices attached on the left and right of the passive cylinder, as shown in Fig. 4.7. The weakened vortices increase the pressure on the left and right sides of the passive cylinder. Being counter-intuitive, although the structural damping ratio is entirely associated with the passive cylinder, the pressure fluctuation intensity below the active cylinder slightly decreases with structural damping, as seen in Figs. 4.6a, 4.6e and 4.6i. This is because a damped passive cylinder displaces less amount of the fluid, and thus smaller volumes of fluid are pushed down to flow past the active cylinder and to strengthen the pressure below it. In the gap, a zero-velocity point travels downward from the passive cylinder to the active cylinder, as seen in the red dashed circle of Fig. 4.8b. That point appears when the passive 56 Effects of Parameters Associated With the Passive Cylinder in Periodic Regimes (a) y/ D 2 1 0 -1 -2 (b) (c) (d) (e) y/ D 2 1 0 -1 -2 (f) (g) (h) (i) y/ D 2 1 0 -1 -2 x/D 10-1 (j) x/D 10-1 (k) x/D 10-1 (l) x/D 10-1 p*: Fig. 4.6 Non-dimensional pressure contours p⇤= p/r f 2nD2 atG/D= 0.2; A1/D= 0.1; m⇤= 1.5; f1/ fn = 0.78; and f1 = 0 (a, e, and i), 90 (b, f, and j), 180 (c, g, and k) and 270 (d, h, and l). (a–d) z = 0, Df21 = 46.6; (e–h) z = 0.1, Df21 = 50.0928; (i–l) z = 0.8, Df21 = 68.309. The cylinder on the top is the passive passive cylinder. The cylinder on the bottom is the active active cylinder. The dashed lines indicate negative values of non-dimensional pressure. The color map shows corresponding color for each value of non-dimensional pressure. With the increase in damping, the main driver of the passive cylinder changes from the pressure fluctuation at the far side of the passive cylinder to that within the gap. The black sticks and the dots indicate the displacement of the cylinder in one cycle. 4.1 Structural Damping Ratio 57 (a1) y/ D (b1) (c1) (a2) y/ D x/D (b2) x/D (c2) x/D Fig. 4.7 Velocity vectors on the left of the passive cylinder at G/D= 0.2, A1/D= 0.1, m⇤ = 1.5, and f1/ fn = 0.75, with (a) z = 0, Df21 = 108.4; (b) z = 0.1, Df21 = 95.62; (c) z = 0.8, Df21 = 68.309 and (1) f1 = 90; (2) f1 = 180. The velocity vector is drawn on every grid point and the vector scale factors are 0.15 grid units/magnitude. The lifespan and the strength of vortices both decrease with damping. The black sticks and the dots indicate the displacement of the cylinder in one cycle. 58 Effects of Parameters Associated With the Passive Cylinder in Periodic Regimes (a1) (b1) (a2) (b2) Fig. 4.8 Velocity vectors in the gap at G/D= 0.2, A1/D= 0.1, m⇤ = 1.5, and f1/ fn = 0.75, with (a) z = 0, Df21 = 108.4; (b) z = 0.1, Df21 = 95.62; and (1) f1 = 158; (2) f1 = 189. The velocity vector is drawn on every grid point and the vector scale factors are 0.011 grid units/magnitude. The zero-velocity point (see dashed circle in b1) travels from the bottom of the passive cylinder to the top of the active cylinder. For the damped case with z = 0.1, the stagnant flow point appears earlier than the undamped case and has a longer lifespan. For both the damped and undamped cases, the stagnant flow point disappears at f1 = 189. The black sticks and the dots indicate the displacement of the cylinder in one cycle. 4.2 Mass Ratio of the Passive Cylinder 59 cylinder and the active cylinder move either closer to or farther away from each other and the fluid flow in the gap is changing direction. The increase in damping lengthens the lifespan of such a point. For example, at z = 0, that point exists in the time frame f1 = 135 184, as shown in Figs.4.8a1 and 4.8a2. By comparison, at an increased damping z = 0.1, that point exists in the time frame f1 = 161 184, as illustrated in Figs. 4.8b1 and 4.8b2. At the moment of f1 = 161, the zero-velocity point of the undamped case has just been produced, as demonstrated in Fig. 4.8a1, whereas that of the damped case is already on the way downward, as seen in Figs.4.8b1. At the moment of f1 = 184, for both the damped and the undamped cases, the zero-velocity point disappears, as seen in Figs. 4.8a2 and 4.8b2. It is clear that the lifespan of the zero-velocity point is lengthened due to the damping effect, indicating a sustained period of opposite flows within the gap. The longer lifespan of the zero-velocity point allows the pressure to accumulate in the gap for a longer period and, consequently, to generate a more violent pressure fluctuation, as shown in Figs. 4.6c, 4.6g and 4.6k. A larger damping ratio also results in a greater fluctuation of the transient gap distance between the two cylinders, which means the fluid is squeezed out of the gap rather than vibrating together with the passive cylinder and the active cylinder. This corresponds to the lower flow velocity in the gap and the increased lifespan of the zero-velocity point. 4.2 Mass Ratio of the Passive Cylinder Another important parameter is the mass ratio m⇤, which is the mass of the passive cylinder divided by that of the displaced water, i.e. m⇤ = mc/mdisp = mc/(rD2p/4). As seen in Fig. 4.9, the mass ratio influences the natural frequency of the passive cylinder in fluid as fw/ fn = p m⇤/(m⇤+CA). This submerged natural frequency is approximately equal to the active cylinder frequency that causes the resonance of the passive cylinder. The resonance frequency increases with m⇤, but m⇤ appears to have no effect on the resonance amplitude. As a result, the resonance frequency in Fig. 4.9 increases with m⇤. It is also interesting to note that the curves with various m⇤ converges at the point (A2/A1, f1/ fn) = (1,0.36), as seen in the dashed rectangle in Fig. 4.9. This convergence point is independent of A1/D because A2/D is linearly correlated with A1/D at f1/ fn = 1, as seen in Fig. 3.2. This convergence point is only affected by G/D. In other words, if f1/ fn = 1, A2/A1 is solely dependent on G/D. If we assume CA = 1, the calculated p m⇤/(m⇤+CA) does not exactly match the res- onance frequency fw/ fn found in the simulations. There is always a discrepancy of the magnitude of 102. Conversely, if we calculate the added mass coefficient as CA = m⇤[1 ( fpeak/ fn)2], where fpeak is the resonance frequency obtained directly from the simulations, 60 Effects of Parameters Associated With the Passive Cylinder in Periodic Regimes Fig. 4.9 Variation of A2/A1 with f1/ fn and m⇤ at G/D= 0.2,A1/D= 0.1,Rem = 100,z = 0. m⇤ affects the resonance frequency. The red dashed rectangle highlights the convergence point. the CA can be found to be in the range of 0.700.85 for m⇤ = 1.52.5. The calculatedCA increases with m⇤, G/D as well as with A1/D. In terms of the mass ratio’s effect on phase difference, since phase jump frequency is equal to the resonance frequency, m⇤ determines at which f1/ fn the vibration of the two cylinders abruptly changes from being in phase to being anti-phase, as seen in Fig. 4.10. The structural damping factor affects the mass ratio’s influence on the frequency-amplitude relationship, as seen in Fig. 4.11a. Given a damping factor larger than zero, the resonance amplitude of the passive cylinder decreases with its mass ratio, e.g. the dashed red lines. However, for undamped cases, the mass ratio does not affect the resonance amplitude at all, as demonstrated by the black solid lines. Despite the negative contribution to the peak amplitude, the increase in damping does not allow the mass ratio to affect the immersed resonance frequency. With z = 0, the mass ratio is seen to be unable to affect the vibration amplitude at f1/ fn = 1.0, because the black solid lines cross the same point at f1/ fn = 1.0 as highlighted in the red dashed box in Fig. 4.9. However, with z > 0, the lines representing different mass ratios give different amplitude at f1/ fn = 1.0, and the separation of these lines increases with damping, as highlighted in the red dashed box in Fig. 4.11a. Hence, at z > 0, the responding amplitude becomes dependent on the mass ratio. It is also observed that the frequency disparity between the force and the displacement at f1/ fn = 1.0. For all cases, the 4.3 Chapter Summary 61 Fig. 4.10 Variation of the oscillation phase difference between the two cylinders with f1/ fn at A1/D= 0.1,m⇤ = 1.52.5,G/D= 0.2,Rem = 100,z = 0 the phase change occurs. G/D does not affect much the phase difference mass ratio does not affect the phase lag Df21 at a very high or very low f1/ fn, as shown by the lines converging at the far left and far right in Fig. 4.11b. However, the increase in mass ratio causes the phase change as well as the resonance to occur at a larger f1/ fn, corresponding to the increase of the immersed natural frequency. This pattern is not influenced by the variation of the damping ratio. Additionally, at z = 0.2, despite the variation of resonance frequency with m⇤, the phase lag between the two cylinders at the resonance frequency remains constant regardless of m⇤, e.g. Df21 ⇡ 70 in Fig. 4.11. At z = 0, the phase difference at resonance is observed to be lower as Df21 ⇡ 90. This indicates that the phase difference at resonance decreases with damping ratio. In a nutshell, a non-zero damping ratio leads to a negative correlation between the mass ratio and the relative resonance amplitude, whereas at zero damping ratio, mass ratio does not affect the resonance amplitude but only the resonance frequency of the passive cylinder. 4.3 Chapter Summary This chapter carries out parametric study on the effects of the passive cylinder’s structural damping and its mass ratio on its responding vibration. It is found that, with a variety of damping ratios, the motion of the passive cylinder can generally be regarded to be sinusoidal with a frequency identical to that of the active cylinder oscillation. Although multiple frequency components were discovered in some configurations, the dominant and 62 Effects of Parameters Associated With the Passive Cylinder in Periodic Regimes (a) (b) Fig. 4.11 (a) Variation in A2/A1 with f1/ fn, and (b) variation in displacement phase difference Df21 between the passive cylinder and the active cylinder (for the frequency components with f/ fn = f1/ fn) with f1/ fn at G/D = 0.2,A1/D = 0.1,m⇤ = 1.5 2.5,Rem = 100 and z = 00.2. The color and line type denote z , whereas the marker type denotes m⇤. With a non-zero damping, the peak A2/A1 decreases with m⇤. The curve of Df21 disperses at resonance, which follows the same pattern as the undamped cases. 4.3 Chapter Summary 63 fundamental frequencies are equal to the frequency of the active cylinder in most cases, so, in regimes A and A⇤, it is usually accurate enough to simplify the vibration of the passive cylinder as harmonic. The increase in damping reduces the passive cylinder vibration amplitude and particularly weakens the resonance. With the presence of damping, the resonance amplitude decreases with the mass ratio. The increase in damping strengthens the linear correlation between the active cylinder’s amplitude and the passive cylinder’s amplitude. It is discovered that a threshold damping ratio of the passive cylinder, beyond which the amplification factor A2/A1 decreases with the active cylinder’s vibration amplitude A1/D, and under which the factor increases with the amplitude. Damping also slows down the switch of the vibration phase of the passive cylinder near resonance, but the phase difference Df21 at resonance remains constant regardless of the damping ratio. The damping ratio does not significantly influence the dependence of Df21 on other parameters. The force acting upon the passive cylinder mainly arises from the pressure gradient rather than viscous shearing. Hence, the pressure difference on the two sides of the passive cylinder determines its driving force. It is seen that the maximum attractive and repulsive forces occur when the pressure in the gap is at highest and lowest, respectively. With the increase in damping, the reduction in the vibration amplitude of the passive cylinder is accompanied by the increased variation of the gap size between the two cylinders. As a result, the magnitude of the pressure oscillation within the gap increases but the pressure change on the far side of the passive cylinder decreases. Correspondingly, the increase in damping gives rise to a sustained period of opposite flows within the gap, leading to an increased lifespan of a zero-velocity point within flow in the gap. Mass ratio m⇤ determines the values of fw/ fn and thus controls the the exact frequency for the resonance f1/ fn. At z = 0, mass ratio does not affect the resonance amplitude of the passive cylinder. However, at z > 0, resonance amplitude decreases with mass ratio. Mass ratio affects the phase difference between the two cylinders only at resonance frequency regardless of the damping ratio. In other words, mass ratio does not affect the phase difference at very low or very high frequency f1/ fn. Chapter 5 Effects of Intermediate Fluid and Gap in Periodic Regimes This chapter focuses on the effects of Reynolds numbers Rem and gap ratio G/D on the interactions between the two cylinders. It is reminded here that the Reynolds number in this thesis is defined by maximum velocity of the active cylinder, diameter of the cylinder and the viscosity of the fluid, i.e. Rem =UmD/n = 2pA1 f1D/n . The gap distance G is the clear distance between the active cylinder and the passive cylinder. In this chapter, Reynolds number ranges from 10 to 150. The Reynolds number is found to affect the vibration centre drift of the passive cylinder which can be repelled or attracted by the active cylinder with different Reynolds number. The correlation between the Reynolds number and the resonance amplitude can be altered by the change of structural damping factor. Gap ratio can in general reduce the vibration amplitude of the passive cylinder but it almost has no influence on the phase-lag, which is counter-intuitive. 5.1 Reynolds Number Reynolds number significantly affects the interactions between the two cylinders, and thus the displacement of the passive cylinder. In general, with the increase of the Reynolds number, both the resonance vibration amplitude and the resonance frequency is increased, while the phase difference between the active cylinder and the passive cylinder is reduced. A threshold Reynolds number is discovered, beyond which the passive cylinder vibrates steadily about a position that is closer to the active cylinder than the initial stationary position, and below which the passive cylinder is repelled away from the active cylinder. This is similar to the critical Reynolds number discussed by Gazzola et al. (2012). The difference is that in the 66 Effects of Intermediate Fluid and Gap in Periodic Regimes 0 10 20 30 40 50 60 70 80 90 -0.02 -0.01 0 0.01 0.02 0.03 0.04 Fig. 5.1 Convergence time history for a case with Re = 140. After a few initial steps, the vibration amplitude and the centre drift of the passive cylinder converges to constant values. current scenario, the passive cylinder is constrained by a spring rather than being free and it is not neutrally buoyant, so it is not carried away by the flowing fluid but vibrates about a constant position away from the initial position. The influence of the damping factor and the mass ratio on the relationship between the Reynolds number and the passive cylinder’s displacement is also examined. We then further discuss the influence of the Reynolds number on the flow field adjacent to the two cylinders. 5.1.1 Effects of Reynolds Number on the Passive Cylinder’s Vibration Centre Drift Similar to Case B in Gazzola et al. (2012), the passive cylinder in the presented cases is also observed to be repelled away or attracted towards the active cylinder, depending on the value of Rem. Nevertheless, it should be noted that the repelling and attraction discussed is referred to the centre of the passive cylinder’s vibration relative to its stationary position at time zero - either closer or further away from the active cylinder. For this reason, in the following discussion, this phenomenon is addressed as "vibration centre drift". In the periodic regimes, the vibration amplitude and the centre drift of the passive cylinder will converge after a few initial periods, as seen in Figure 5.1. So the converged amplitude A2 and the vibration centre drift DY 2 are useful to describe the general vibration patterns of the passive cylinder. In this thesis, for the cases with periodical interaction, the amplitude of the passive cylinder’s vibration is calculated as A2 = (Y2,maxY2,min)/2, where Y2,max and Y2,min are the maximum and minimum displacement in the last 50 periods of steady vibration. The vibration centre drift of the passive cylinder is calculated as DY 2 = (Ymax+Ymin)/2. At a very low Keulegan-Carpenter number of the active cylinder KC = 0.157 0.628 and a 5.1 Reynolds Number 67 (a) (b) (c) (d) Fig. 5.2 (a) Variation in vibration centre drift DY 2 with f1/ fn at G/D = 0.9,A1/D = 0.477,m⇤ = 1.5,z = 0 and Rem = 10 150. (b) Zoom-in at the secondary and the pri- mary resonance regime (c) Further zoom-in at the primary resonance regime (d) Variation in vibration centre drift DY 2 with Rem at G/D = 0.9,A1/D = 0.477,m⇤ = 1.5,z = 0 and f1/ fn = 13.2. fixed Reynolds number of Rem = 100, the phenomenon of vibration centre drift was hardly observed in the previous studies (Lin et al., 2018a,b). However, when the KC is increased to 3 and a variety of Rem from 10 to 150 is tested, the vibration centre drift of the passive cylinder is clearly observed. A critical Reynolds number is discovered that, beyond which the passive cylinder is repelled away from the active cylinder, and below which the the passive cylinder is attracted towards the active cylinder. For example, as seen in Fig. 5.2a, the critical Reynolds number is at about Rem = 40, when the frequencies are relatively high at f1/ fn > 1. The variation of the vibration centre drift demonstrates a unique pattern at both the secondary and the primary resonance regimes at 0.25< f1/ fn < 0.85, as seen in Fig. 5.2b. The two drops in the curves at f1/ fn = 0.35 and at f1/ fn = 0.7 correspond to the secondary and primary resonance regimes, 68 Effects of Intermediate Fluid and Gap in Periodic Regimes (a) (b) Fig. 5.3 (a) Variation in DY 2/A1 with f1/ fn and (b) Zoom-in for its resonance regime at G/D= 0.9,A1/D= 0.477,m⇤= 1.5,Rem = 10,70 and z = 00.2. The marker type denotes z , whereas the line type denotes Rem. respectively. This indicates that, when resonance occurs, the passive cylinder tends to be attracted towards the active cylinder. On the contrary, when the active cylinder oscillates at a relatively high frequency at f1/ fn > 1, the resonance-induced drop in the curve is no longer observed. An increased frequency is found to amplify the effects of Rem on the repelling or the attraction of the passive cylinder. Figure 5.2c is a close-up at the primary resonance at f1/ fn = 0.640.8, where the attraction effect due to the resonance is clearly identified. In addition, the curves for Rem 50 converge at the point of f1/ fn = 0.78,A2/A1 = 0.01. This indicates that, at Rem 50, when the active cylinder vibrates at the frequency of f1/ fn = 0.78, the distance of the vibration centre drift is constantly DY 2 = 0.01A1, being independent of the Reynolds number. Nevertheless, at Rem = 10, this pattern of convergence is not observed. Moreover, the information in Fig. 5.2c can be presented in another form. DY 2/A1 can be plotted against Rem rather than f1/ fn, as shown in Fig. 5.2. At a high frequency f1/ fn 1, the passive cylinder is increasingly repelled from the active cylinder with the rise of the Reynolds number, especially at 10 100, the vibration centre drift DY 2 becomes insensitive to the Reynolds number. A convergence point is observed at Rem = 50 and DY 2/A1 = 0.01, which means that the vibration centre drift of the passive cylinder is independent of the active cylinder’s vibration frequency at Rem = 50 and f1/ fn > 1. Also, it is notable that the critical Reynolds number, i.e. the Rem corresponding to zero vibration centre drift, slightly increases with the active cylinder’s vibration frequency. The influence of the damping factor upon the passive cylinder’s vibration centre drift is most significant at the resonance regime at f1/ fn = 0.30.8, but is much less influential at low and high frequencies at f1/ fn < 0.25 and f1/ fn > 1, as seen in Fig. 5.3a. In other words, 5.1 Reynolds Number 69 (a) (b) Fig. 5.4 (a) Variation in DY 2/A1 with f1/ fn and (b) Zoom-in for its resonance regime at G/D = 0.9,A1/D = 0.477,z = 0.2,Rem = 10,70 and m⇤ = 1.5 2.5. The marker type denotes m⇤, whereas the line type denotes Rem. at resonance, the passive cylinder tend to be attracted towards the active cylinder with a high damping factor. This pattern corresponds well to the damping factor’s effect on the passive cylinder’s responding amplitude, where the damping effect also becomes most significant at the resonance regimes and diminishes at both very high and very low frequency regimes, as demonstrated in Lin et al. (2018a). As seen in Fig. 5.3b, it is interesting that a higher damping factor can actually push the passive cylinder further away from the initial vibration centre, which is counter-intuitive. The Reynolds number affects the damping factor’s influence on the frequency-amplitude relationship, as seen in Fig. 5.3a. In general, at the resonance regime, the vibration centre drift is more sensitive to damping factor with a higher Reynolds number. For example, as seen in Fig. 5.3b, with the damping factor z rising from 0 to 0.2, the vibration centre drift DY 2 increased by 0.003 from -0.023 to -0.020 at Rem = 10, whereas it increases by 0.012 from -0.008 to 0.004 at Rem = 70. The increment at Rem = 10 is only 25% of that at Rem = 70. Mass ratio m⇤ also plays an important role on the vibration centre drift of the passive cylinder. In contrast to the damping factor, the mass ratio becomes more influential over the vibration centre drift when the frequency is beyond the resonance regime, as seen in Fig. 5.4a. With a further examination, it is found that the correlation between the mass ratio and the vibration centre drift reverses twice at Rem = 10, but it reverses for only once at Rem = 70, as seen in Fig. 5.4b. For cases with low Rem = 10, vibration centre drift is positively correlated with m⇤ at low frequencies. As f1/ fn goes beyond 0.5, the correlation is reversed that the vibration centre drift decreases with m⇤. When f1/ fn is further increased beyond 0.8, the correlation is reversed back to be positive again, and remains positive for larger frequencies. 70 Effects of Intermediate Fluid and Gap in Periodic Regimes Here, for Rem = 10, the correlation is reversed twice at f1/ fn = 0.5 and f1/ fn = 0.8, and it ends up in positive correlation. For cases with high Rem = 70, vibration centre drift is positively correlated with m⇤ at low frequencies, similar to low Rem cases. When f1/ fn goes above 0.7, the correlation becomes negative, and it remains negative for all frequencies at f1/ fn > 0.7. Here, the correlation is reversed once, and it ends up in negative correlation. In summary, at a low frequency f1/ fn < 0.5 or a high frequency f1/ fn > 0.8, with the increase of the mass ratio, the passive cylinder is dragged towards its initial position, where DY 2/A1 = 0. However, at the primary resonance regime 0.5< f1/ fn < 0.8, the increase of mass ratio may cause the passive cylinder to be attracted towards the active cylinder, which is counter-intuitive. 5.1.2 Effects of Reynolds Number on Passive Cylinder’s Vibration Am- plitude The amplitude spectra of passive cylinder’s displacement offer an comprehensive overview to the dynamic characteristics of the passive cylinder, as depicted in Fig. 5.5. It is seen that the amplitude of the dominant frequency components at resonance increases exponentially with Rem. As seen in Fig. 5.5a, a series of frequency components are observed with almost zero component frequencies f/ fn ⇡ 0. This indicates that the vibration centre of the passive cylinder at the periodic state shifts from its initial position, giving rise to a component with an extremely low f/ fn. It is also seen that the component amplitude at zero frequency increases with active cylinder’s oscillation frequency f1/ fn, which means when the active cylinder vibrates at a higher frequency, the passive cylinder drifts further away from the initial position. The influence of the Reynolds number on the component amplitude at zero frequency is more complicated. It initially decreases with Rem at low Reynolds numbers and, when Rem goes beyond a critical value, e.g. Rem = 50 in Fig. 5.5c, the amplitude increases with Rem again. The relationship between the Reynolds number and the vibration centre drift of the passive cylinder will be further discussed in Section 5.1.1. A critical damping factor is discovered at z = 0.1, below which the resonance amplifica- tion factor increases with the Reynolds number, and beyond which the factor decreases with the Reynolds number, as seen in Fig. 5.6a. It is also seen that the increase of the Reynolds number amplifies the effect of the damping factor. In other words, the peak amplification factor becomes more sensitive to the damping factor at a high Reynolds number. The same pattern applies to the secondary resonance regime as shown in the inset in Fig. 5.6b, although the secondary peak amplification factor has a critical damping factor of z = 0.025, rather than z = 0.1 for the primary resonance peak in this case. The effects of the damping factor 5.1 Reynolds Number 71 (a) (b) (c) (d) Fig. 5.5 Amplitude spectra showing the responding displacement of the passive cylinder with f1/ fn = 0.053.2 at G/D= 0.9,A1/D= 0.477,m⇤ = 1.5,z = 0 with (a) Rem = 10 (b) Rem = 30 (c) Rem = 50 (d) Rem = 110. The dashed thin line tracks the dominant frequencies. (a) (b) Fig. 5.6 (a) Variation of passive cylinder’s amplification factor A2/A1 with active cylinder’s oscillation frequency f1/ fn at G/D = 0.9,A1/D = 0.477,m⇤ = 2.0,Rem = 10 110 and z = 00.2. (b) Zoom-in at resonance regimes. The marker type denotes damping factor z , whereas the line type denotes the Reynolds number Rem. 72 Effects of Intermediate Fluid and Gap in Periodic Regimes are weakened at very low or very high frequencies, which is consistent with the conclusion made by Lin et al. (2018a). The mechanism of the critical damping factor will be discussed in detail later. With a damping ratio less than 0.1, the passive cylinder’s resonance amplitude and the resonance frequency both increase with the Reynolds number, as seen in Fig. 5.7, which matches the pattern shown in Fig. 5.5. At f1/ fn < 1, the passive cylinder’s vibration amplitude generally increases with Reynolds number, whereas at f1/ fn > 1 the amplitude decreases with Reynolds number, as demonstrated in Fig. 5.7a. As seen in Fig. 5.7b, when Rem increases from 10 to 150, the resonance amplitude doubles from 0.3 to 0.6. Meanwhile, the resonance frequency increases from 0.625 to 0.74. In terms of secondary resonance at 0.34< f1/ fn < 0.4, as Rem rises from 30 to 150, the amplitude of the secondary resonance doubles from 0.075 to 0.14 and the secondary resonance frequency rises from 0.36 to 0.375. The situation at Rem = 10 is special that the secondary resonance is not observed, and, here, the amplitude at Rem = 10 is higher than those at Rem = 30 50. The vibration phase difference between the two cylinders is demonstrated in Fig. 5.7c. Overall, the increase of Rem causes the phase difference curves to shift upwards. The steepness of the phase change at resonance slightly increases with Rem as well. Reynolds number does not qualitatively affect the relationship between the mass ratio and the peak amplification factor, as seen in Fig. 5.8. The primary and secondary resonance frequencies increase with the mass ratio, regardless of the Reynolds number. 5.2 Flow Fields Around the Two Cylinders This section examines the effect of the Reynolds number upon the flow fields surrounding the two cylinders. The typical cases with parameter combinations of G/D = 0.9,A1/D = 0.477,m⇤ = 1.5, f1/ fn = 2.8 and various Rem are examined in detail. The major flow feature presented below is representative to other cases examined in this study. The flow resulting from the flow-mediated interaction between the two cylinders moving along the vertical (y) axis has the following symmetry properties: u1(x,y, t) =u1(x,y, t) (5.1) (u1,u2)(x,y, t) = (u1,u2)(x,y, t+T ) (5.2) where Eq. 5.1 represents reflection symmetry about y-axis, while Eq. 5.2 stands for the periodic nature of the flow-mediated interaction in the current range of parametric space. 5.2 Flow Fields Around the Two Cylinders 73 (a) 0.5 0.6 0.7 0.8 0.2 0.25 0.3 0.35 0.4 0.45 0.5 (b) (c) Fig. 5.7 (a) Variation of amplification factor A2/A1 with active cylinder’s oscillation frequency f1/ fn (b) Zoom-in at primary and secondary resonance regimes and (c) Variation of phase difference between the two cylinders Df21 with f1/ fn (for the frequency components with f/ fn = f1/ fn) with f1/ fn at G/D= 0.9 ,A1/D= 0.477, m⇤ = 1.5,z = 0,Rem = 10150, i.e. KC = 3,b = 3.3 50. A2/A1 is positively correlated with Rem, particularly within the regime of resonance, and the resonance frequency increases with Rem. The resonance amplitude at Rem = 150 is as large as 1.6 times of that at Rem = 10. The phase difference is shifted towards the positive side with the increase of Rem. 74 Effects of Intermediate Fluid and Gap in Periodic Regimes (a) (b) Fig. 5.8 (a) Variation of passive cylinder’s amplification factor A2/A1 with active cylinder’s oscillation frequency f1/ fn at G/D= 0.9,A1/D= 0.477,z = 0,m⇤ = 1.52.5,Rem = 10 110 and (b) Zoom-in at resonance regimes. The marker type denotes mass ratio m⇤, whereas the line type denotes the Reynolds number Rem. Since the scenario is symmetric regarding y-axis, Figs. 5.9 and 5.10 present only the left half of the region around the two cylinders. With the increase of the Reynolds number, more vortices are generated. For example, pressure coefficient and velocity vectors in the gap at a high frequency f1/ fn = 2.8 with a constant phase of f1 are plotted in Fig. 5.9, where the pressure coefficient is calculated as Cp = p⇤/[2p(A1/D)( f1/ fn)]2 = p/(rU2m). At Rem = 10, no vortex is generated, as seen in Fig. 5.9a, whereas at Rem = 50, a pair of vortices are generated in the gap, as shown in Fig. 5.9b. At a relatively high Reynolds number Rem = 150, 2 pairs of vortices are generated in the gap, and another pair of vortices is observed at the far side of the active cylinder, as demonstrated in Fig. 5.9c. Although vortices are observed, vortex shedding does not occur at Rem = 10150, and the vortices are dissipated before they can be shed away from the two cylinders. The pressure decreases at the near side of the active cylinder with the increase in the Reynolds number, because the generated vortices help reduce pressure. The increase of Reynolds number can also be interpreted as the relative reduction of viscosity, causing the fluid to flow out of the gap more easily and thus more rapidly reducing the pressure accumulated in the gap. Also, the vibration amplitude of the passive cylinder decreases with Reynolds number, corresponding to the reduced pressure fluctuation surrounding the passive cylinder. The fluid around the passive cylinder tends to flow upwards with the progression of Reynolds number at f1 = 180, as seen in Fig. 5.9a1, 5.9b1 and 5.9c1. The increase of the 5.2 Flow Fields Around the Two Cylinders 75 (a1) y/ D (b1) (c1) (a2) y/ D x/D (b2) x/D (c2) x/D Fig. 5.9 Contours of pressure coefficientCp and velocity vectors (1) in the gap at f1 = 180 and (2) in the far side of the active cylinder at f1 = 0, givenG/D= 0.9,A1/D= 0.477,m⇤= 1.5, f1/ fn = 2.8, with (a) Rem = 10, Df21 = 30.6; (b) Rem = 50, Df21 = 16.8; (c) Rem = 110, Df21 =12.22. The velocity vector is drawn on every two grid points and the vector scale factors are 0.1 grid units/magnitude. 76 Effects of Intermediate Fluid and Gap in Periodic Regimes (a) y/ D x/D (b) x/D (c) x/D Fig. 5.10 Pressure coefficient contours and velocity vectors in the gap at G/D= 0.9,A1/D= 0.477,m⇤ = 1.5,z = 0, and f1 = 180 with (a) Rem = 10, f1/ fn = 0.65,Df21 = 27.05, (b) Rem = 50, f1/ fn = 0.7,Df21 = 64.66, (c) Rem = 110, f1/ fn = 0.72,Df21 = 71.42, The velocity vector is drawn on every two grid points and the vector scale factors are 0.1 grid units/magnitude. Reynolds number, i.e. the decrease of viscous influence, contributes to the slow dissipation of the secondary vortices at the near side of the passive cylinder in Fig. 5.9c1, which causes the fluid to flow upwards, favouring the repelling of the passive cylinder. This explains the increase of the vibration centre drift with the Reynolds number as seen in Fig. 5.2d. This mechanism is very similar to the cases where the passive cylinder is not constrained by a spring (Gazzola et al., 2012, p. 14). This difference of flow structure will be further discussed later by vorticity plots and streamlines. At the resonance regime, as seen in Fig. 5.10, less vortices are generated in the gap between the two cylinders, compared with the high frequency situation demonstrated in Fig. 5.9. At Rem = 150, only one pair of vortices are generated in the gap at the resonance frequency, as shown in Fig. 5.9c1, while two pairs are observed at the high frequency as demonstrated in Fig. 5.10c. By comparing the velocity fields in Figs. 5.9 and 5.10, it is seen that the fluid flows more violently at a higher frequency, because the maximum velocity of the vibrating active cylinderUm is increased, contributing more dynamic energy imparted to the surrounding fluid and thus amplifying attraction and repelling of the passive cylinder. The secondary vortices at the near side of the passive cylinder is observed at the high frequency as seen in Fig. 5.9c1 but it is not seen at the resonance frequency Fig. 5.10c. The 5.2 Flow Fields Around the Two Cylinders 77 reduction on the active cylinder’s oscillation frequency weakens the secondary vortices in the gap, and thus reducing the repelling effect on the passive cylinder. This is consistent with the pattern shown in Fig. 5.2a. Pressure coefficient and streamlines at a high frequency f1/ fn = 2.8 are demonstrated in Fig. 5.11. With the increase of Reynolds number, the intensity of pressure fluctuation in the gap and at the far side of the passive cylinder both decreases, as seen in Fig. 5.11a1, 5.11a2 and 5.11a3. Since the pressure difference on the circumference of the passive cylinder is the main driver of its motion (Lin et al., 2018a), the vibration amplitude of the passive cylinder is reduced due to the increase in the Reynolds number. This confirms the results shown in Fig. 5.7, where the amplification factor decreases with the Reynolds number at a high frequency. Also, with the increase of Rem, the influence of vortices on the pressure is strengthened. This effect is indicated in Fig. 5.11a3 as the two small local low pressure spots located symmetrical beneath the active cylinder and in Fig. 5.11c3 as the two small local low pressure spots located symmetrical above the active cylinder. The vorticity contours, Figs. 5.12 and 5.13, demonstrate the flow pattern difference between the case with Rem = 10, which is lower than the critical Reynolds number and the case with Rem = 150, which is higher than the critical Reynolds number. The change of flow field is presented in Fig. 5.12 when Rem is increased from 10 to 150, which means the viscous effect is reduced. At Rem = 10, the vortices are in general much larger than those at Rem = 150 due to high viscosity. As demonstrated in Fig. 5.13a, at f = 180, the large vortices drive the surrounding fluid to flow downwards. The passive cylinder is immersed in the downward flow, and it is driven towards the active cylinder. As mentioned before, at Rem = 10, an extra pair of vortices is never observed during the periodical vibration of the passive cylinder.At Rem = 150, low viscosity causes the vortices in the gap harder to dissipated, and the size of vortex also becomes smaller. Taking Fig. 5.12c2 for example, the pair of vortices in the gap can be seen throughout the whole cycle, see Fig. 5.12a2-5.12d2, of the flow-mediated interaction. This pair of vortices has opposite signs. The positive vortex is always on the left, whereas the negative vortex is always on the right. Together they propel a fluid flow to push away the passive cylinder. The flow structure is also demonstrated by the streamlines in Fig. 5.13b. The existence of these vortices throughout the periodical vibration of the salve cylinder causes its vibration centre to drift towards the far side. The vortices at Rem = 10 are in general much larger than those at Rem = 150 due to high viscosity. As demonstrated in Fig. 5.13a, at f = 180, the large vortices drive the surrounding fluid to flow downwards, producing a strong attraction towards the active cylinder. The passive cylinder is immersed in the downward flow, and it is dragged towards the active cylinder. The high viscosity also leads to quicker dissipation of the vortices. At 78 Effects of Intermediate Fluid and Gap in Periodic Regimes (a1) y/ D (b1) (c1) (d1) (a2) y/ D (b2) (c2) (d2) (a3) y/ D x/D (b3) x/D (c3) x/D (d3) x/D Fig. 5.11 Pressure coefficient contours and streamlines at G/D= 0.9,A1/D= 0.477,m⇤ = 1.5,z = 0, and (1) Rem = 10, f1/ fn = 2.8,Df21 = 30.64, (2) Rem = 50, f1/ fn = 2.8,Df21 =16.77, (3) Rem = 110, f1/ fn = 2.8,Df21 =12.22, (a) f1 = 0 (b) f1 = 90 (c) f1 = 180 (d) f1 = 270. The numbers [1-4] identify the case with different Re, whereas the alphabets [a-d] indicate the instances in one cycle of the active cylinder oscillation. The dots on the sticks indicate the position of the cylinders in one cycle of motion. 5.2 Flow Fields Around the Two Cylinders 79 (a1) y/ D (b1) (c1) (d1) (a2) y/ D x/D (b2) x/D (c2) x/D (d2) x/D Fig. 5.12 Evolution of non-dimensional vorticity contours atG/D= 0.9,A1/D= 0.477,m⇤= 1.5,z = 0, f1/ fn = 3.2, and (1) Rem = 10, (2) Rem = 150, (a) f1 = 0 (b) f1 = 90 (c) f1 = 180 (d) f1 = 270. The numbers [1-2] identify the case with different Re, whereas the alphabets [a-d] indicate the instances in one cycle of the active cylinder oscillation. 80 Effects of Intermediate Fluid and Gap in Periodic Regimes (a) y/ D x/D (b) x/D Fig. 5.13 Streamlines and non-dimensional vorticity contours at G/D = 0.9,A1/D = 0.477,m⇤ = 1.5,z = 0, f1/ fn = 3.2,f1 = 180, and (a) Rem = 10, (b) Rem = 150. 5.2 Flow Fields Around the Two Cylinders 81 Rem = 10 extra pair of vortices is never observed in the periodical vibration of the passive cylinder, favouring the attraction of the passive cylinder. In summary, at Rem = 10, the elastically-mounted passive cylinder’s vibration centre drifts towards the active cylinder, whereas at Rem = 150, the flow pattern favours the repelling of the passive cylinder. Also, at around Rem ⇡ 60, where the effects contributing to repulsion and attraction are almost equal and the vibration centre of the passive cylinder stays at the initial position. The time histories of the total hydraulic force coefficient show that the amplitude of the shear force decreases with Rem due to the decreasing viscosity. The pressure force amplitude rises with the Rem at z = 0. The increase in pressure force exceeds the decrease in the shear force, so the amplitude of the force acting on the cylinder increases as well. Therefore, vibration amplitude increases with Rem at z = 0. At z = 0.2, the shear force drops with Rem, whereas the pressure force remains almost constant with the increase of Rem. Consequently, the amplitude of the force acting on the cylinder reduces. As a result, the vibration amplitude decreases with Rem at z = 0. This applies for both the primary resonance, Fig. 5.14a-5.14f, and secondary resonance, Fig. 5.14g-5.14l. The primary resonance, the pressure coefficient and velocity vector plots, Figs. 5.15 to 5.18, demonstrate that, at z = 0, the intensity of pressure fluctuation around the passive cylinder increases with Rem, whereas, at z = 0.2, the intensity decreases with Rem. For example, as seen in Fig. 5.15b1-5.15b3, at z = 0, the positive pressure above the passive cylinder increases as Rem goes up from Rem = 10 to Rem = 110, whereas the negative pressure below the passive cylinder decreases only slightly. Similar patterns can be observed in other sub-figures in Fig. 5.15. So, at z = 0, the pressure fluctuation, in general, is strengthened with Reynolds number. This is consistent with the time histories of the pressure coefficient shown in Fig. 5.14a-5.14c, where the pressure force acting on the passive cylinder fluctuates with smaller amplitude while Rem goes up. At z = 0.2, as shown in Fig. 5.16b1-5.16b3, the positive pressure above the passive cylinder remains almost constant with the variation of Rem, while the negative pressure below the passive cylinder is weakened with the increase of Rem. Together with other sub-figures in Fig. 5.16, it is clear that the pressure fluctuation is weakened due to the increase of Rem. This corresponds to the decrease in the amplitude of the pressure force with Rem as seen in Fig. 5.14d-5.14f. Similar pattern can be found for the primary resonance pressure coefficient contours and velocity vectors shown in Figs. 5.17 and 5.18. 82 Effects of Intermediate Fluid and Gap in Periodic Regimes (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Fig. 5.14 Time histories of hydraulic force acting on the passive cylinder at primary or secondary resonance with G/D = 0.9,A1/D = 0.477,m⇤ = 2, and (a) z = 0, Rem = 10, f1/ fn = 0.7 (b) z = 0, Rem = 50, f1/ fn = 0.745 (c) z = 0, Rem = 110, f1/ fn = 0.765 (d) z = 0.2, Rem = 10, f1/ fn = 0.73 (e) z = 0.2, Rem = 50, f1/ fn = 0.775 (f) z = 0.2, Rem = 110, f1/ fn = 0.795 (g) z = 0, Rem = 10, f1/ fn = 0.39 (h) z = 0, Rem = 50, f1/ fn = 0.39 (i) z = 0, Rem = 110, f1/ fn = 0.39 (j) z = 0.2, Rem = 10, f1/ fn = 0.39 (k) z = 0.2, Rem = 50, f1/ fn = 0.39 (l) z = 0.2, Rem = 110, f1/ fn = 0.39. 5.2 Flow Fields Around the Two Cylinders 83 (a1) y/ D (b1) (c1) (d1) (a2) y/ D (b2) (c2) (d2) (a3) y/ D x/D (b3) x/D (c3) x/D (d3) x/D Fig. 5.15 Pressure coefficient contours and velocity vectors at G/D = 0.9,A1/D = 0.477,m⇤ = 2,z = 0 and (1) Rem = 10, f1/ fn = 0.7, (2) Rem = 50, f1/ fn = 0.745, (3) Rem = 110, f1/ fn = 0.765, (a) f1 = 0 (b) f1 = 90 (c) f1 = 180 (d) f1 = 270. Primary resonance occurs at around f1/ fn = 0.75 and the damping factor is zero. 84 Effects of Intermediate Fluid and Gap in Periodic Regimes (a1) y/ D (b1) (c1) (d1) (a2) y/ D (b2) (c2) (d2) (a3) y/ D x/D (b3) x/D (c3) x/D (d3) x/D Fig. 5.16 Pressure coefficient contours and velocity vectors at G/D = 0.9,A1/D = 0.477,m⇤ = 2,z = 0.2 and (1) Rem = 10, f1/ fn = 0.73, (2) Rem = 50, f1/ fn = 0.775, (3) Rem = 110, f1/ fn = 0.795, (a) f1 = 0 (b) f1 = 90 (c) f1 = 180 (d) f1 = 270. Primary resonance occurs at around f1/ fn = 0.75 and the damping factor is relatively high at z = 0.2. 5.2 Flow Fields Around the Two Cylinders 85 (a1) y/ D (b1) (c1) (d1) (a2) y/ D (b2) (c2) (d2) (a3) y/ D x/D (b3) x/D (c3) x/D (d3) x/D Fig. 5.17 Pressure coefficient contours and velocity vectors at G/D = 0.9,A1/D = 0.477,m⇤ = 2,z = 0, f1/ fn = 0.39 and (1) Rem = 10, (2) Rem = 50, (3) Rem = 110, (a) f1 = 0 (b) f1 = 90 (c) f1 = 180 (d) f1 = 270. Secondary resonance occurs at f1/ fn = 0.39 and the damping factor is zero. 86 Effects of Intermediate Fluid and Gap in Periodic Regimes (a1) y/ D (b1) (c1) (d1) (a2) y/ D (b2) (c2) (d2) (a3) y/ D x/D (b3) x/D (c3) x/D (d3) x/D Fig. 5.18 Pressure coefficient contours and velocity vectors at G/D = 0.9,A1/D = 0.477,m⇤ = 2,z = 0.2, f1/ fn = 0.39 and (1) Rem = 10, (2) Rem = 50, (3) Rem = 110, (a) f1 = 0 (b) f1 = 90 (c) f1 = 180 (d) f1 = 270. Secondary resonance occurs at f1/ fn = 0.39 and the damping factor is relatively high at z = 0.2. 5.3 Gap Distance 87 5.3 Gap Distance In terms of the initial clear gap distance G/D, the passive cylinder vibration amplitude A2/D decreases with the gap ratio G/D, as shown in Fig. 5.19. At a low frequency of the active cylinder oscillation f1/ fn < 0.6, G/D barely affects A2/D. At a high frequency f1/ fn > 1.0, the passive cylinder amplitude is more sensitive to G/D compared with that at a low active cylinder frequency frequency f1/ fn < 0.6. At the resonance frequency, G/D is most influential to A2/D. The gap ratio G/D does not affect much the phase difference, as shown in Fig. 5.20, which means the distance between the two cylinders can hardly influence the phase difference. The variation of z has no qualitative influence on the dependence of the amplification factor on initial gap distance G/D, as seen in Fig. 5.21a. Regardless of damping ratio z , G/D controls neither the resonance frequency nor Df21, as shown in Fig. 5.20 and Fig. 5.21b. 88 Effects of Intermediate Fluid and Gap in Periodic Regimes Fig. 5.19 Variation of A2/A1 with f1/ fn and G/D at m⇤ = 2.5,A1/D= 0.05,Rem = 100,z = 0. A2/D decreases with G/D. Fig. 5.20 Variation of the oscillation phase difference between the two cylinders with f1/ fn at A1/D = 0.05,m⇤ = 2.5,G/D = 0.21.0,Rem = 100,z = 0. G/D does not affect much the phase difference. 5.3 Gap Distance 89 (a) (b) Fig. 5.21 (a) Variation of A2/A1 with f1/ fn, and (b) variation of displacement phase difference Df21 between the passive cylinder and the active cylinder (for frequency components with f/ fn = f1/ fn) with f1/ fn at G/D= 0.30.9,A1/D= 0.05,m⇤ = 2.5, and z = 0.2. A2/A1 decreases with G/D, but the initial gap distance has little effect on Df21. This pattern is exactly the same as the undamped cases. 90 Effects of Intermediate Fluid and Gap in Periodic Regimes 5.4 Chapter Summary This chapter studies the effects of non-dimensional groups associated with the intermediate fluid, i.e. the Reynolds number Rem and the gap ratio G/D, upon flow-mediated interactions between two cylinders immersed in an otherwise still fluid. Compared with the gap ratio, the Reynolds number is found to convey a much more complicated and significant impact upon the flow-mediated interaction. For the vibration centre drift, a critical Reynolds number is discovered, beyond which the passive cylinder is repelled away from the active cylinder and below which the the passive cylinder is attracted towards the active cylinder. Here, the "attraction" and "repelling" indicate that the passive cylinder vibrates steadily with its vibration centre maintaining a certain distance away from its initial position and the vibration centre does not change with time. The existence of the critical Reynolds number is particularly obvious at high frequencies f1/ fn > 1, while at the resonance regime this pattern is not significant. At the secondary and the primary resonance regimes, the passive cylinder tends to be attracted towards the active cylinder. When the Reynolds number is greater than Rem = 100, the change of Rem can hardly cause the vibration centre drift of the passive cylinder. Both the passive cylinder’s resonance amplitude and resonance frequency are found to increase with the Reynolds number with a damping ratio less than 0.1. The phase difference between the passive cylinder and the active cylinder decreases with Reynolds number. A critical damping factor is discovered, below which the resonance amplitude increases with the Reynolds number and beyond which the amplitude decreases with the Reynolds number. The increase of the mass ratio shifts the vibration amplitude curves to high-frequency region, but the mass ratio does not affect the overall trend of the relationship between the Reynolds number and the passive cylinder’s vibration. At the resonance regime, the passive cylinder is increasingly attracted towards the active cylinder with the increase of the damping factor. This effect can be amplified at small Reynolds numbers. The damping factor does not significantly influence the vibration centre drift at high frequencies f1/ fn > 1. With the increase in the mass ratio, the passive cylinder keeps approaching to its initial position. Contrary to the influence of the damping factor, the effect of the mass ratio upon vibration centre shift grows with the increase of f1/ fn. The increase of the Reynolds number can result in more and stronger vortices to be generated in the gap. At high Reynolds numbers, the secondary vortices in the gap are generated, causing the fluid to flow towards the passive cylinder, contributing to the repelling force on the passive cylinder. The increase of the frequency causes the active cylinder to vibrate more violently, and also amplifying both the repelling and the attracting action on the passive cylinder. 5.4 Chapter Summary 91 At resonance, the increase of Reynolds number leads to a decreased amplitude for shear force and an increased amplitude for pressure force. At z = 0, the increase in pressure force amplitude can outweigh the drop in shear force amplitude, thus causing a resonance amplitude positively correlated to Rem. Whereas at z = 0.2, this is not the case, resulting in a negative correlation between resonance amplitude and Rem. This explains why the critical damping factor exists. The effects of gap ratio is less significant than the Reynolds number. The passive cylinder’s vibration amplitude on the whole decreases with G/D, especially at resonance. Being counter-intuitive, the gap ratio between the passive cylinder and the active cylinder turns out to not affecting the vibration phase difference between the two cylinders at all. The gap ratio does not affect the resonance frequency as well. This independence of the phase and resonance frequency upon gap ratio is not altered by the variation of any other non-dimensional group in all the cases. Gap ratio only affects the amplitude of the passive cylinder. Chapter 6 Effects of Active Cylinder’s Amplitude and Frequency in Less Regular Regimes In this chapter, the parametric exploration is extended into the regimes where the flow pattern is more complicated with vortex shedding that significantly impacts the vibration of the passive cylinder. An overview in the passive cylinder’s response is given on the effects of various Keulegan-Carpenter Number KC = 2pA1/D and the active cylinder’s oscillation frequency f1/ fn with the Stokes Number fixed at b = Rem/KC = 35. After that, the effects of the active cylinder’s oscillation frequency f1/ fn is discussed with the KC and b chosen to be in regime C, E, F, G. Both the vortex dynamics and the response of the passive cylinder is examined in detail. 2176 combinations of parameters are simulated and examined, but only representative cases are discussed. Here, the active cylinder’s oscillation amplitude is represented by KC rather than A1/D for the convenience of comparison with previous research. 6.1 Overview of KC Effects This section gives an overview of the effects caused by varying the KC on the responding vibration of the passive cylinder. In general, the passive cylinder vibrates increasingly complicated and aperiodic with KC, as seen in Fig. 6.1. In regime A, as discussed in the previous chapters regard periodic regimes, the flow- mediated interaction between the two cylinders reaches a steady state after several initial periods of transition, as shown in Fig. 6.1, and the vibration centre shift can hardly be identified. In regime A at KC ⇡ 4, although the interaction is still steady and repetitive after initial transition, the passive cylinder can be attracted towards or repelled away from the 94 Effects of Active Cylinder’s Amplitude and Frequency in Less Regular Regimes (a) KC = 1 0 10 20 30 40 50 60 70 80 90 -0.02 -0.01 0 0.01 0.02 0.03 0.04 (b) KC = 4 0 10 20 30 40 50 60 70 -0.02 0 0.02 0.04 0.06 0.08 (c) KC = 5 0 10 20 30 40 50 60 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 (d) KC = 6 0 5 10 15 20 25 30 35 40 45 50 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 (e) KC = 7 0 5 10 15 20 25 30 35 40 45 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 (f) KC = 8 Fig. 6.1 Typical time histories of passive cylinder’s vibration for G/D = 2.5,m⇤ = 2,z = 0,b = 35, f1/ fn = 2.8, (a) KC = 1,Rem = 35,A1/D = 0.159, (b) KC = 4,Rem = 140,A1/D= 0.637, (c) KC= 5,Rem = 175,A1/D= 0.796, (d) KC= 6,Rem = 210,A1/D= 0.955, (e) KC = 7,Rem = 245,A1/D= 1.114, (f) KC = 8,Rem = 280,A1/D= 1.273 active cylinder depending on the Rem, as discussed in Chapter 5. In regime C at KC = 5, even though the response of the passive cylinder is on the whole repetitive, irregularity starts to appear for the drift of the vibration centre. It is not entirely independent of time any more, and the vibration centre starts to drift up and down rather than standing still. In regime E at KC = 6, the passive cylinder vibrates steadily in some parts of the history, but irregular displacement can occur from time to time. The passive cylinder is slightly attracted towards the active cylinder rather than repelled away as in KC= 45. In regime F at KC= 78, the response has become very irregular without any periodic vibration identified. The irregularity of the passive cylinder’s response in regimes C, E, F, G is due to unstable vortex shedding from the active cylinder, which will be discussed later. The effects of KC can also be reviewed in the frequency domain, as seen in Fig. 6.2. With the increase in KC, the irregularity clearly increased, and frequency component associated with the irregular vibration concentrated in the low-frequency band f/ fn< 1, and tend to peak at the immersed natural frequency f/ fn ⇡ 0.77. The fundamental pattern of harmonics and resonance, as discussed in Chapter 3, always exists regardless of the superimposed irregular frequency components. The increase of KC and f1/ fn can both add to the irregularity. The frequency component with f/ fn ⇡ 0 represents the significance of the vibration centre drift, 6.1 Overview of KC Effects 95 which grows with the KC in regime A and C from KC = 3 to 5, as shown in Figs. 6.2a-6.2c. However, the magnitude of vibration centre shift drops significantly at KC = 6, because the flow structure in regime E is fundamentally different from regime A and C, favouring the attraction of the passive cylinder. In regime F at KC = 78, the vibration centre drift grows large again. Resonance amplitude increases with KC as expected. In short, the overall irregularity and vibration amplitude increase with KC. Despite the growth of absolute responding amplitude with KC, amplification factor and phase difference for the harmonic components is at large independent of KC, as seen in Figs. 6.3 and 6.4. At KC = 17, linear relationship persists between the oscillation amplitude of the passive cylinder A2 and the active cylinder A1. At the minor and major resonance peaks, the linearity is weakened, which is coherent with the pattern in the periodic regimes Chapter 3. The amplification at the major resonance experiences a relatively large decrease from KC = 6 to 7. The phase difference only slightly decreases across all the frequencies with KC. The magnitude of decrease is less than 10. Similar patterns can be observed for both damped and undamped cases demonstrated in Fig. 6.3 and Fig. 6.4 respectively. The vibration amplitude is extracted from the time history of the passive cylinder’s displacement. The data processing method is discussed in Section 2.5. In summary, the irregularity and the overall amplitude for the vibration of the passive cylinder increase with KC = 1 7 at b = 35. However, the amplification factor and the phase difference of the harmonic components barely changes with KC at both damped and undamped conditions. The following sections discuss the response patterns of the passive cylinder and the flow structure. The vortex shedding mechanism is examined in detail 96 Effects of Active Cylinder’s Amplitude and Frequency in Less Regular Regimes (a) (b) (c) (d) (e) (f) Fig. 6.2 Amplitude spectra for the displacement histories of the passive cylinder at G/D= 2.5,m⇤ = 2,z = 0,b = 35, (a) KC = 3 in regime A, (b) KC = 4 in regime A, (c) KC = 5 in regime C, (d) KC = 6 in regime E, (e) KC = 7 in regime F, (f) KC = 8 in regime F 6.1 Overview of KC Effects 97 0 0.5 1 1.5 2 2.5 3 3.5 0 0.02 0.04 0.06 0.08 0.1 0.12 (a) 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.094 0.096 0.098 0.1 0.102 0.104 0.106 0.108 0.11 (b) (c) Fig. 6.3 At structural damping z = 0, (a) Variation of amplification factor A2/A1 for only the harmonic components with f1/ fn (b) Zoom-in for resonance (c) Variation of oscillation phase difference between the two cylinders Df21 with f1/ fn (for the frequency components with f/ fn = f1/ fn) with f1/ fn atG/D= 3.0,A1/D= 0.1591.114,m⇤= 2.0,z = 0,Rem = 35245 and KC = 17,b = 35. The amplitude of the harmonic components is extracted by filter out the low-frequency components by FFT. The data processing method is discussed in Section 2.5. 98 Effects of Active Cylinder’s Amplitude and Frequency in Less Regular Regimes (a) (b) (c) Fig. 6.4 At structural damping z = 0.02, (a)Overview and (b)Zoom-in for variation of amplification factor A2/A1 with f1/ fn and KC. (c) Variation of oscillation phase difference between two cylinders Df21 (for the harmonic component with f/ fn = f1/ fn) with f1/ fn and KC atG/D= 3.0,A1/D= 0.1591.114,m⇤= 2.0,Rem = 35245 and KC= 17,b = 35. 6.2 Regime C 99 6.2 Regime C In regime C, KC = 5 is a threshold above which the irregularity of the displacement time history is significantly increased and can no longer be approximated as purely periodical. With the increase of frequency ratio, the vibration of the passive cylinder gradually changes from a relatively sinusoidal pattern to pulse beating and the irregularity becomes more and more apparent, as seen in Fig. 6.5. At f1/ fn = 0.05 to 1.0, the vibration of the passive cylinder is dominantly repetitive with only minor pulse beats. At f1/ fn = 1.2 to 2.0, the pulse beats are relatively strong compared with low frequency. At f1/ fn = 2.4 to 3.2, the pulse beating is significant with its amplitude becomes larger than the vibration at the active cylinder’s frequency. The passive cylinder’s vibration centre drift towards the far side significantly increases with the active cylinder’s oscillation frequency at f1/ fn > 1.2. Resonance of the passive cylinder still occurs at about the immersed natural frequency f1/ fn = 0.77. The increase of irregularity can also be shown in the amplitude spectra of the displacement histories in Fig. 6.6. 2 minor frequency components with their frequency being 10th and 20th of the active cylinder’s oscillation frequency are observed at f1/ fn 1, which will be discussed later. Harmonics are more obvious when the active cylinder’s oscillation frequency is very low. At KC = 5, vortices shed from the active cylinder becomes stronger with the increase of its frequency f1/ fn, as seen in Fig. 6.7. Although the switch of skewing pattern is discovered, the vortex shedding process is on the whole repetitive and stable. 2 pairs of vortices rotating in opposite directions are shed from each oscillation cycle of the active cylinder. The strength of the vortices increases with the active cylinder’s oscillation frequency, causing a stronger stream and thus a more significant pulse beating. The mechanism of pulse beating will be discussed later. The vorticity generated by the passive cylinder is the strongest at f1/ fn = 0.77 where resonance occurs. The skewing pattern of the vorticity contours switches left and right periodically, as seen in Fig. 6.8. The switch occurs after approximately 10 cycles of the active cylinder’s forced oscillation, as seen in Fig. 6.9a. This periodical switching matches the pulse beating of the passive cylinder’s vibration. The vorticity skewing switching can also take place for every 20 cycles of the active cylinder. The skewing switching occurs intermittently between 20 and 10 cycles of the active cylinder’s oscillation. The vibration of the passive cylinder thus has 2 minor frequency components at 10th and 20th of the active cylinder’s oscillation frequency, as highlighted in Fig. 6.9b. The switching vorticity pattern indicates the periodical change of the flow structure, which causes the low-frequency beating of the passive cylinder. The periodical switching of flow structure can be observed in all frequencies f1/ fn in RegimeC. With the increase of the active cylinder’s frequency, the beating frequency tends to become 100 Effects of Active Cylinder’s Amplitude and Frequency in Less Regular Regimes (a) f1/ fn = 0.05 (b) f1/ fn = 0.35 (c) f1/ fn = 0.77 (d) f1/ fn = 0.82 (e) f1/ fn = 1.0 (f) f1/ fn = 1.2 (g) f1/ fn = 1.4 (h) f1/ fn = 1.8 (i) f1/ fn = 2.0 (j) f1/ fn = 2.4 (k) f1/ fn = 2.8 (l) f1/ fn = 3.2 Fig. 6.5 Displacement time history of the passive cylinder in regime C with KC = 5,b = 35,G/D= 2.5,A1/D= 0.796,m⇤ = 2,z = 0,Rem = 175 with (a) f1/ fn = 0.05, (b) f1/ fn = 0.35, (c) f1/ fn = 0.77, (d) f1/ fn = 0.82, (e) f1/ fn = 1.0, (f) f1/ fn = 1.2, (g) f1/ fn = 1.4, (h) f1/ fn = 1.8, (i) f1/ fn = 2.0, (j) f1/ fn = 2.4, (k) f1/ fn = 2.8, (l) f1/ fn = 3.2. KC = 5 is a critical value that above which the irregularity of the displacement time history is significantly increased. With the increase of frequency ratio, the vibration of the passive cylinder gradually changes from a repetitive pattern to pulsed beating. 6.2 Regime C 101 (a) f1/ fn = 0.05 (b) f1/ fn = 0.35 (c) f1/ fn = 0.77 (d) f1/ fn = 0.82 (e) f1/ fn = 1.0 (f) f1/ fn = 1.2 (g) f1/ fn = 1.4 (h) f1/ fn = 1.8 (i) f1/ fn = 2.0 (j) f1/ fn = 2.4 (k) f1/ fn = 2.8 (l) f1/ fn = 3.2 Fig. 6.6 Amplitude spectra of the passive cylinder’s displacement In regime C with KC = 5,b = 35,G/D = 2.5,A1/D = 0.796,m⇤ = 2,z = 0,Rem = 175 with (a) f1/ fn = 0.05, (b) f1/ fn = 0.35, (c) f1/ fn = 0.77, (d) f1/ fn = 0.82, (e) f1/ fn = 1.0, (f) f1/ fn = 1.2, (g) f1/ fn= 1.4, (h) f1/ fn= 1.8, (i) f1/ fn= 2.0, (j) f1/ fn= 2.4, (k) f1/ fn= 2.8, (l) f1/ fn= 3.2. 102 Effects of Active Cylinder’s Amplitude and Frequency in Less Regular Regimes (a) f1/ fn = 0.05 (b) f1/ fn = 0.05 (c) f1/ fn = 0.05 (d) f1/ fn = 0.05 (e) f1/ fn = 0.77 (f) f1/ fn = 0.77 (g) f1/ fn = 0.77 (h) f1/ fn = 0.77 (i) f1/ fn = 2.8 (j) f1/ fn = 2.8 (k) f1/ fn = 2.8 (l) f1/ fn = 2.8 Fig. 6.7 Vorticity contours In regimeC with (a-d) f1/ fn = 0.05, (e-h) f1/ fn = 0.77, and (i-l) f1/ fn = 2.8 with (a & e & i) f = 0 (b & f & j) f = 90 (c & g & k) f = 180 (d & h & l) f = 270 at KC = 5,b = 35,G/D= 2.5,A1/D= 0.796,m⇤ = 2,z = 0,Rem = 175. 6.2 Regime C 103 lower than 10th and 20th of f1/ fn. This periodical switch of the streaming and vorticity contours skewing direction was not found when only the active cylinder exists neither in regime C nor in other identified regimes (Tatsuno and Bearman, 1990). The mechanism of the beating is examined in further details at f1/ fn = 3.2 as seen in Fig. 6.10 and Fig. 6.11. At relatively high Reynolds number, the flow structure causes the passive cylinder to drift away from the active cylinder, as described in Chapter 5. The vortex pairs shed from the active cylinder produce a stream that pushes the passive cylinder away. In regime C, the stream produced by the active cylinder periodically changes direction, swinging from slight left to slight right and back to left, which is unlike in regime A and A⇤, the direction of the stream is always along the centre line of cylinders. Typically a full beating can take 20-30 oscillations of the active cylinder. High f1/ fn cases can take more oscillations. At instant A Fig. 6.11a, the streaming direction points at the passive cylinder, so the vibration centre of the passive cylinder drifts away from the active cylinder, as seen in Fig. 6.11. At instants B-D, the vortex pairs are shed towards the left side of the active cylinder. Since the negative vortex, i.e. rotating clockwise, the passive cylinder is slightly attracted towards the active cylinder, reaching the lowest position at instant C. At instant E-F, the streaming direction switch to the right. Notably, left-to-right switching is faster than right-to-left. So the passive cylinder is pushed away again, reaching highest position at instant E, but not as high as the position at instant A. At instant F, the positive vortex at the right side of the passive cylinder creates a downward flow, attracting it towards the active cylinder, reaching the lowest position at instant F, but not as low as instant C. In a nutshell, the periodical switching of the streaming direction causes the passive cylinder to be repelled and attracted periodically, resulting in the beating pattern of in its time histories. This section has shown the flow-mediated interaction in regime C. On the whole, the vibration is regular with only minor irregularity appearing when the active cylinder is vibrating at a high frequency f1/ fn 1.2. The switch of the streaming direction leads to the pulse beating in the displacement time histories of the passive cylinder. The amplitude of the beating increases with the the active cylinder’s frequency f1/ fn. This gradual switch of the streaming direction was not observed when the passive cylinder does not exist (Tatsuno and Bearman, 1990). In each of the active cylinder’s oscillation, 2 vortex pairs were shed. The switch of the streaming direction corresponds with the switch of the vortex shedding direction. In a nutshell, the flow-mediated interaction in regime C is at large regular. The slight irregularity is added by the switch of streaming direction, since each switch does not always take the same number of the active cylinder’s oscillations. 104 Effects of Active Cylinder’s Amplitude and Frequency in Less Regular Regimes (a) Instant A, Skew Left (b) Instant B, Skew Right (c) Instant C, Skew Left (d) Instant D, Skew Right Fig. 6.8 Non-dimensional Vorticity Contours in Regime C with f1/ fn = 0.05 and KC = 5,b = 35,G/D = 2.5,A1/D = 0.796,m⇤ = 2,z = 0,Rem = 175, corresponding to instants (a)A (b)B (c)C (d)D in Fig. 6.9. The shape of vorticity contours switch the direction of skewing for approximately every 10 periods. 6.2 Regime C 105 A, Skew Left B, Skew Right C, Skew Left D, Skew Right (a)Displacement Beating Frequency (b)Frequency components Fig. 6.9 (a)Displacement time history and (b) amplitude spectra of the passive cylinder in Regime C with f1/ fn = 0.05 and KC = 5,b = 35,G/D = 2.5,A1/D = 0.796,m⇤ = 2,z = 0,Rem = 175. The marked locations in (a) corresponds to the instants ad in Fig. 6.8. 106 Effects of Active Cylinder’s Amplitude and Frequency in Less Regular Regimes A B E D F C Fig. 6.10 Displacement time history of the passive cylinder in Regime C with f1/ fn = 3.2 and KC = 5,b = 35,G/D = 2.5,A1/D = 0.796,m⇤ = 2,z = 0,Rem = 175. The marked locations AF corresponds to the Figs. 6.11a-6.11f. 6.2 Regime C 107 (a) Instant A (b) Instant B (c) Instant C (d) Instant D (e) Instant E (f) Instant F Fig. 6.11 Vorticity Contours for a typical cycle of beating in Regime C, corresponding to instants (a)A (b)B (c)C (d)D (e)E (f)F in Fig. 6.10 with f1/ fn = 3.2 and KC = 5,b = 35,G/D = 2.5,A1/D = 0.796,m⇤ = 2,z = 0,Rem = 175. The passive cylinder is pushed further away when the streaming direction is pointing at it. 108 Effects of Active Cylinder’s Amplitude and Frequency in Less Regular Regimes 6.3 Regime E At regime E with KC = 6, the displacement time histories become more irregular, i.e. less periodic with more random motions, compared with regime C KC = 5, especially at high frequency f1/ fn, as seen in Figs. 6.12 and 6.13. The relatively stable beating pattern can no longer be observed in regime E, as shown in Fig. 6.12. Intermittent large displacements of the passive cylinder are discovered. With the increase in active cylinder’s oscillation frequency f1/ fn, the abrupt large displacements occur more frequently, and the displacement scale relative to the steady small vibration grows as well. The distribution of frequency components is increasingly concentrated at typically f/ fn  1. The dominant frequency of the passive cylinder’s vibration shifts from equalling the active cylinder’s oscillation frequency f1/ fn or its harmonics to being very low at f1/ fn > 2.0. It is also interesting to note that the vibration centre shift in regime E is less significant than that in regime C, despite the increase of the Reynolds number. In regime E, the vortex pairs are shed from only one side of the active cylinder, as seen in Fig. 6.15, which is coherent with the scenario with only the active cylinder (Tatsuno and Bearman, 1990). In general, the vortex shedding is steady and asymmetric that it only takes place at one side of the cylinders. However, the flow pattern changes between one pattern and its mirror-image intermittently, as seen in Fig. 6.14. During the transition between the 2 patterns, the vortex shedding can make a direct impact on the passive cylinder. The vortices shed from consecutive cycles can sometimes merge to create a stronger vortex. As a result, the vibration of the passive cylinder is much less steady and become more irregular during the transition, e.g. corresponding to the irregular vibration shown in the time history in Fig. 6.12. Irregularity increases with the oscillation frequency of the active cylinder f1/ fn. Between each transition, the passive cylinder vibrates steadily for a few periods. The number of these continuous steady cycles is reduced by increasing the frequency of the active cylinder’s oscillation f1/ fn, corresponding to the more irregular displacement time history as seen in Fig. 6.12. The magnitude for the abrupt large vibration amplitude increases with f1/ fn as well. During the transition between one pattern and its mirror-image, vortices can merge to create a stronger vortex. When it comes close to the passive cylinder, larger vibration displacement is induced. This process is similar to the vortex shedding in regime G. The variation of the active cylinder’s oscillation frequency can alter the vortex shedding pattern during the steady-state of the oscillation, as seen in Fig. 6.15. With the increase in f1/ fn, the vortex shedding at the temporary steady state becomes more curved, and the streaming more circular. At f1/ fn = 0.05, vortices shed below the active cylinder travel along a straight line at an angle of 45 to the x-axis. At f1/ fn = 0.77, these vortices travel on a line that is slightly curved to the left, rather than along a straight line, which results in 6.3 Regime E 109 (a) f1/ fn = 0.05 (b) f1/ fn = 0.35 (c) f1/ fn = 0.77 (d) f1/ fn = 0.82 (e) f1/ fn = 1.0 (f) f1/ fn = 1.2 (g) f1/ fn = 1.4 (h) f1/ fn = 1.8 (i) f1/ fn = 2.0 (j) f1/ fn = 2.4 (k) f1/ fn = 2.8 (l) f1/ fn = 3.2 Fig. 6.12 Displacement time history of the passive cylinder in Regime E with KC = 6,b = 35,G/D= 2.5,A1/D= 0.955,m⇤ = 2,z = 0,Rem = 210 with (a) f1/ fn = 0.05, (b) f1/ fn = 0.35, (c) f1/ fn = 0.77, (d) f1/ fn = 0.82, (e) f1/ fn = 1.0, (f) f1/ fn = 1.2, (g) f1/ fn = 1.4, (h) f1/ fn = 1.8, (i) f1/ fn = 2.0, (j) f1/ fn = 2.4, (k) f1/ fn = 2.8, (l) f1/ fn = 3.2. At KC = 6, the time histories become more irregular compared with KC= 5, especially at high frequency f1/ fn. 110 Effects of Active Cylinder’s Amplitude and Frequency in Less Regular Regimes (a) f1/ fn = 0.05 (b) f1/ fn = 0.35 (c) f1/ fn = 0.77 (d) f1/ fn = 0.82 (e) f1/ fn = 1.0 (f) f1/ fn = 1.2 (g) f1/ fn = 1.4 (h) f1/ fn = 1.8 (i) f1/ fn = 2.0 (j) f1/ fn = 2.4 (k) f1/ fn = 2.8 (l) f1/ fn = 3.2 Fig. 6.13 Amplitude spectra of the passive cylinder’s displacement in Regime E with KC = 6,b = 35,G/D = 2.5,A1/D = 0.955,m⇤ = 2,z = 0,Rem = 210 with (a) f1/ fn = 0.05, (b) f1/ fn = 0.35, (c) f1/ fn = 0.77, (d) f1/ fn = 0.82, (e) f1/ fn = 1.0, (f) f1/ fn = 1.2, (g) f1/ fn= 1.4, (h) f1/ fn= 1.8, (i) f1/ fn= 2.0, (j) f1/ fn= 2.4, (k) f1/ fn= 2.8, (l) f1/ fn= 3.2. 6.3 Regime E 111 (a) t⇤ = 7219.47 (b) t⇤ = 10879.20 Fig. 6.14 Vorticity contours in Regime E with f1/ fn = 0.05 at KC = 6,b = 35,G/D = 2.5,A1/D= 0.955,m⇤ = 2,z = 0,Rem = 210. The flow pattern changes between one pattern and its mirror-image intermittently. With the increase of the f1/ fn, the interaction remains steady for less cycles of the active cylinder oscillation and the passive cylinder’s vibration becomes more irregular as well. 112 Effects of Active Cylinder’s Amplitude and Frequency in Less Regular Regimes a vortex accumulated on the left side of the active cylinder. At f1/ fn = 3.2, the curvature is more severe with strong vortex appear to the left of the active cylinder. The vortex can accumulate both above and below the active cylinder. The vortices shed on the outer skirt of the circular streaming keeps separated from each other, whereas the vortices in the inter side of the vortex shedding curve tend to be distorted and merged. Since the curvature for the vortex shedding path increases with f1/ fn, consecutive vortices merge more easily. The basic vortex shedding mechanism is however not altered by f1/ fn. In regime E, chaotic vibration of the passive cylinder is observed at KC= 6. In Fig. 6.16a, we see that the time histories with z = 0 and z = 0.02 almost exactly overlap each other despite the difference of damping factor, whereas, in Fig. 6.16b, we see that the minor difference of initial condition, i.e. damping factor, leads to dramatic deviation of time history after t fn = 18, which is an indication of chaos. In summary of this section, the flow-mediated interaction in regime E is more irregular than that in regime C. The one-side "2P" vortex shedding mechanism and intermittent switching of shedding direction are coherent with the single-cylinder cases. During the switching, the vortices can have a direct impact on the passive cylinder, disrupting its steady vibration with a frequency identical to the active cylinder. With the increase in the active cylinder’s oscillation frequency, the switching of the vortex shedding direction occurs more frequently, taking place after fewer cycles of the active cylinder’s oscillation. The streaming becomes more circular as well. At high f1/ fn, vortices shed from consecutive cycles merge more often. Accompanied with the frequent switch of the shedding direction, the passive cylinder’s vibration can become very irregular, i.e. dominantly influenced by the vortices rather than the oscillation of the active cylinder. Additionally, with the slight variation of the structural damping ratio from z = 0 to 0.02, the time history of the passive cylinder’s vibration indicates the emergence of chaos. 6.3 Regime E 113 (a) f1/ fn = 0.05 (b) f1/ fn = 0.05 (c) f1/ fn = 0.05 (d) f1/ fn = 0.05 (e) f1/ fn = 0.77 (f) f1/ fn = 0.77 (g) f1/ fn = 0.77 (h) f1/ fn = 0.77 (i) f1/ fn = 3.2 (j) f1/ fn = 3.2 (k) f1/ fn = 3.2 (l) f1/ fn = 3.2 Fig. 6.15 Vorticity contours in Regime E with (a-d) f1/ fn = 0.05, (e-h) f1/ fn = 0.77, and (i-l) f1/ fn = 3.2 at KC = 6,b = 35,G/D = 2.5,A1/D = 0.955,m⇤ = 2,z = 0,Rem = 210. Sub-figures in the same column are at the similar state of vortex shedding process. With the increase in f1/ fn, the vortex shedding at the temporary steady state becomes more curved, and the streaming more circular. 114 Effects of Active Cylinder’s Amplitude and Frequency in Less Regular Regimes (a)KC = 5 with z = 0 and z = 0.02 (b)KC = 6 with z = 0 and z = 0.02 Fig. 6.16 Typical time histories of passive cylinder vibration for G/D = 2.5,m⇤ = 2,z = 0,b = 35 (a)KC = 5 with z = 0 and z = 0.02, (b)KC = 6 with z = 0 and z = 0.02. In (a), we see that the time histories with z = 0 and z = 0.02 almost exactly overlap each other despite the difference of damping factor, whereas, in (b), we see that the minor difference of initial condition, i.e. damping factor, leads to dramatic deviation of time history after t fn = 18, which is an indication of chaos. 6.4 Regime F 115 6.4 Regime F In regime F, the irregularity becomes more severe compared with regime E. Even if the active cylinder oscillates at a low frequency, the irregularity can still be apparent, as seen in Fig. 6.17. The amplitude of the sudden disruptions in the vibration time history of the passive cylinder is enlarged with the increase in active cylinder’s oscillation frequency f1/ fn. The distribution of the passive cylinder’s vibration frequency is more and more concentrated around its immersed natural frequency, as seen in Fig. 6.17. The vibration centre of the passive cylinder is in general slightly attracted towards the active cylinder, which will be discussed later by viewing the vortex patterns. For single-cylinder cases, the flow in regime F is stable and repetitive. The typical flow pattern can be seen in Fig. 6.19 by Zhao and Cheng (2014). In each cycle, 2 vortex pairs are shed from the oscillating cylinder towards two opposite directions along a diagonal line. This vortex shedding pattern was also referred as "2P" by Williamson (1985) and "Diagonal" by Obasaju et al. (1988). For two-cylinder cases in this study, the flow pattern switches between steady patterns and irregular ones. The transition among different steady patterns is accompanied by very irregular flow-mediated interactions between the two cylinders. For steady states, 3 patterns are identified, as seen in Fig. 6.20. The first pattern is referred to as Pattern F1. A typical cycle can be seen in Figs. 6.20a-6.20d. The pattern looks similar to that of the single-cylinder case, since vortex pairs are also convected towards the diagonal direction. The vortices being shed downwards demonstrate a similar pattern to that of the single-cylinder cases, where a vortex pair is shed from half cycle of the forced oscillation and the vortex shedding occurs only on one side of the active cylinder. Nevertheless, vortex pairs convected upwards are shed with a different mechanism. The vortices are generated from both the left side and the right side of the active cylinder, which is more similar to the pattern in regime A, as seen in Fig. 6.20b. The second pattern is referred to as Pattern F2, as seen in Figs. 6.20e-6.20h. For vortex shedding pattern above the active cylinder, in each cycle, one vortex pair is shed on the left side of the active cylinder, and an additional minor negative vortex is generated and convected to merge with the negative one of the vortex pair. Vortex shedding pattern below the active cylinder is similar to that of the Pattern F1 and the single-cylinder cases with one vortex pair shed from the left side of the active cylinder. So for the Pattern F2, 2 vortex pairs together with an extra negative vortex are shed during each cycle. As for the Pattern F3 in Figs. 6.20i-6.20l, one major vortex pair together with an additional minor vortex is shed below the active cylinder, and the major vortices are shed from both sides of the active cylinder. The vortex pairs shed above the active cylinder is similar to the single-cylinder cases. The mirror-image of these patterns can also occur. For all 3 patterns of the steady state, 116 Effects of Active Cylinder’s Amplitude and Frequency in Less Regular Regimes (a) f1/ fn = 0.05 (b) f1/ fn = 0.35 (c) f1/ fn = 0.77 (d) f1/ fn = 0.82 (e) f1/ fn = 1.0 (f) f1/ fn = 1.2 (g) f1/ fn = 1.4 (h) f1/ fn = 1.8 (i) f1/ fn = 2.0 (j) f1/ fn = 2.4 (k) f1/ fn = 2.8 (l) f1/ fn = 3.2 Fig. 6.17 Displacement time history of the passive cylinder in Regime F with KC = 8,b = 35,G/D= 2.5,A1/D= 1.273,m⇤ = 2,z = 0,Rem = 280 with (a) f1/ fn = 0.05, (b) f1/ fn = 0.35, (c) f1/ fn = 0.77, (d) f1/ fn = 0.82, (e) f1/ fn = 1.0, (f) f1/ fn = 1.2, (g) f1/ fn = 1.4, (h) f1/ fn = 1.8, (i) f1/ fn = 2.0, (j) f1/ fn = 2.4, (k) f1/ fn = 2.8, (l) f1/ fn = 3.2. At KC = 8, irregularity is apparent even at very low frequency. 6.4 Regime F 117 (a) f1/ fn = 0.05 (b) f1/ fn = 0.35 (c) f1/ fn = 0.77 (d) f1/ fn = 0.82 (e) f1/ fn = 1.0 (f) f1/ fn = 1.2 (g) f1/ fn = 1.4 (h) f1/ fn = 1.8 (i) f1/ fn = 2.0 (j) f1/ fn = 2.4 (k) f1/ fn = 2.8 (l) f1/ fn = 3.2 Fig. 6.18 Amplitude spectra of the passive cylinder’s displacement in Regime F with KC = 8,b = 35,G/D = 2.5,A1/D = 1.273,m⇤ = 2,z = 0,Rem = 280 with (a) f1/ fn = 0.05, (b) f1/ fn = 0.35, (c) f1/ fn = 0.77, (d) f1/ fn = 0.82, (e) f1/ fn = 1.0, (f) f1/ fn = 1.2, (g) f1/ fn = 1.4, (h) f1/ fn = 1.8, (i) f1/ fn = 2.0, (j) f1/ fn = 2.4, (k) f1/ fn = 2.8, (l) f1/ fn = 3.2. At KC = 8, irregularity is apparent even at very low frequency. 118 Effects of Active Cylinder’s Amplitude and Frequency in Less Regular Regimes Fig. 6.19 Vorticity contours for a typical regime F flow with only one forced oscillating cylinder at Re= 150, KC = 12 and b = 12.5. (Zhao and Cheng, 2014). The flow oscillates on the horizontal direction. the vortices have a direct impact on the passive cylinder, making its displacement irregular. During the unsteady phase, the flow structure is unpredictable and several vortices shed from consecutive cycles can merge to more severely interfere the vibration of the passive cylinder, plucking the passive cylinder and enlarging the amplitude of its vibration frequency components at about its immersed natural frequency, which is also observed in regime G. The existence of the passive cylinder has disrupted the flow structure, making it switching among several steady patterns and unsteady transition phase, which is dissimilar to the pattern previously observed in the single-cylinder cases. In summary, the flow-mediated interaction in regime F can switch between being steady and being unstable. 3 steady patterns are identified. Typically, only the vortex shedding pattern on one side of the active cylinder is similar to that for the single-cylinder cases, whereas on the other side, the pattern can be different that vortex pairs can be shed from both the left and the right of the active cylinder. An additional minor vortex can also be observed in 2 of the 3 steady patterns. Irregular flow occurs during the transition from one steady pattern to another. Corresponding to the vortex dynamics, the vibration of the passive cylinder is steady and periodic when the steady vortex shedding patterns take place and the 6.4 Regime F 119 (a) Pattern F1 (b) Pattern F1 (c) Pattern F1 (d) Pattern F1 (e) Pattern F2 (f) Pattern F2 (g) Pattern F2 (h) Pattern F2 (i) Pattern F3 (j) Pattern F3 (k) Pattern F3 (l) Pattern F3 Fig. 6.20 Vorticity contours for vortex shedding patterns in Regime F with f1/ fn = 0.05,KC = 8,b = 35,G/D = 2.5,A1/D = 1.273,m⇤ = 2,z = 0,Rem = 280 (a-d) Pattern F1, one vortex pair shed from both sides convected upward and another pair shed from the left side convected downward (e-h) Pattern F2, one pair + one negative vortex shed upward and one pair shed downwards from the left side (i-l) Pattern F3, one pair shed upwards from the left side, and one pair + one negative vortex shed downward. The mirror-images of these patterns can also occur. 120 Effects of Active Cylinder’s Amplitude and Frequency in Less Regular Regimes abrupt surge of the amplitude occurs when the vortex shedding is irregular, causing several vortices to merge and significantly impact the dynamics of the passive cylinder. 6.5 Regime G 121 6.5 Regime G In regime G, the vibration of the passive cylinder is very irregular as seen in Fig. 6.21 and Fig. 6.22. Irregular pulse beating can be observed in the time histories, e.g. when the active cylinder oscillates at f1/ fn = 0.77 from t⇤ = 20 to t⇤ = 80. As the active cylinder oscillates at a higher frequency, the frequency components of the passive cylinder are more and more concentrated in the low-frequency band at f/ fn  1. In regime G, most commonly, 2 major vortex pairs are shed from each oscillation cycle of the active cylinder, and each pair is shed from opposite sides of the active cylinder, as seen in Fig. 6.23. An additional minor vortex can accompany each vortex pair. Due to the unsteady flow, a pair of major vortices can sometimes be shed from both sides of the active cylinder. Sometimes, two vortex pairs are shed from the same side of the active cylinder rather than two different sides. In regime G, the direction of the vortex shedding changes irregularly, and is not stably diagonal as in regime F of the single-cylinder cases in relative sinusoidal flow. Due to interactions with irregular vortex shedding of the active cylinder, the vibration of the passive cylinder becomes irregular as well, as seen in Fig. 6.24. The amplitude of the passive cylinder’s vibration can suddenly increase, for example, as seen in Fig. 6.24 between the instants A and P. The corresponding vorticity contours demonstrate that the sudden increase of the amplitude is caused by the interaction between the passive cylinder and the vortices shed from the previous cycles of the active cylinder as seen in Fig. 6.25. The streaming in regime G is very circular with vortices travelling on elliptical trajectories, and the vortices have a high chance to merge and become a very large vortex. Instants A and P in Fig. 6.25 corresponds to Figs. 6.25a and 6.25p, respectively. From instant A at time t⇤ = 79.69, 3 negative vortices shed from consecutive cycles of the active cylinder merges around the passive cylinder and carries the passive cylinder away, as seen in Fig. 6.25a-6.25h. This is because circulatory streaming allows the vortices to stay around the passive cylinder and merges together to create a powerful drive. For occasions without the significant amplitude, as exemplified in Figs. 6.25m-6.25p, the vortex pairs do not directly impact the passive cylinder and thus do not result in its significant displacement. It is also seen that the direction of vortex shedding from the active cylinder is unstable, causing the irregular displacement time history of the passive cylinder. It is worth noticing that through instants A-P in Figs. 6.25m-6.25p, at the bottom right corner, a positive vortex is growing. This is because the vortices shed from each oscillation cycle of the active cylinder stays there due to circular streaming and positive vortices shed from consecutive cycles merge and gradually forming a larger and larger vortex. 122 Effects of Active Cylinder’s Amplitude and Frequency in Less Regular Regimes (a) f1/ fn = 0.05 (a) f1/ fn = 0.35 (a) f1/ fn = 0.77 (a) f1/ fn = 0.82 (a) f1/ fn = 1.0 (a) f1/ fn = 1.2 (a) f1/ fn = 1.4 (a) f1/ fn = 1.6 (a) f1/ fn = 1.8 (a) f1/ fn = 2.0 (a) f1/ fn = 2.4 (a) f1/ fn = 2.8 Fig. 6.21 Displacement time history of the passive cylinder in Regime G with KC = 9,b = 35,G/D= 2.5,A1/D= 1.432,m⇤ = 2,z = 0,Rem = 315 with (a) f1/ fn = 0.05, (b) f1/ fn = 0.35, (c) f1/ fn = 0.77, (d) f1/ fn = 0.82, (e) f1/ fn = 1.0, (f) f1/ fn = 1.2, (g) f1/ fn = 1.4, (h) f1/ fn = 1.6, (i) f1/ fn = 1.8, (j) f1/ fn = 2.0, (k) f1/ fn = 2.4, (l) f1/ fn = 2.8. 6.5 Regime G 123 (a) f1/ fn = 0.05 (a) f1/ fn = 0.35 (a) f1/ fn = 0.77 (a) f1/ fn = 0.82 (a) f1/ fn = 1.0 (a) f1/ fn = 1.2 (a) f1/ fn = 1.4 (a) f1/ fn = 1.6 (a) f1/ fn = 1.8 (a) f1/ fn = 2.0 (a) f1/ fn = 2.4 (a) f1/ fn = 2.8 Fig. 6.22 Amplitude spectra of the passive cylinder’s displacement in Regime G with KC = 9,b = 35,G/D = 2.5,A1/D = 1.432,m⇤ = 2,z = 0,Rem = 315 with (a) f1/ fn = 0.05, (b) f1/ fn = 0.35, (c) f1/ fn = 0.77, (d) f1/ fn = 0.82, (e) f1/ fn = 1.0, (f) f1/ fn = 1.2, (g) f1/ fn = 1.4, (h) f1/ fn = 1.6, (i) f1/ fn = 1.8, (j) f1/ fn = 2.0, (k) f1/ fn = 2.4, (l) f1/ fn = 2.8. The dominant frequency of the passive cylinder’s vibration becomes equivalent to its im- mersed natural frequency at f1/ fn 1.0. 124 Effects of Active Cylinder’s Amplitude and Frequency in Less Regular Regimes (a) (b) (c) (d) Fig. 6.23 Non-dimensional vorticity contours for regime G with KC = 9,b = 35,G/D = 2.5,A1/D= 1.432,m⇤ = 2,z = 0,Rem = 315, f1/ fn = 2.0. A P Fig. 6.24 Passive cylinder’s vibration time history in Regime G with KC= 9,b = 35,G/D= 2.5,A1/D= 1.432,m⇤= 2,z = 0,Rem = 315, f1/ fn = 2.0. The instants A and P corresponds to Figs. 6.25a and 6.25p, respectively. 6.5 Regime G 125 (a) t⇤ = 79.69 (b) t⇤ = 79.80 (c) t⇤ = 79.91 (d) t⇤ = 80.02 (e) t⇤ = 80.13 (f) t⇤ = 80.24 (g) t⇤ = 80.36 (h) t⇤ = 80.47 (i) t⇤ = 80.58 (j) t⇤ = 80.69 (k) t⇤ = 80.80 (l) t⇤ = 80.91 (m) t⇤ = 81.02 (n) t⇤ = 81.13 (o) t⇤ = 81.25 (p) t⇤ = 81.36 Fig. 6.25 Non-dimensional vorticity contours in Regime G with KC = 9,b = 35,G/D = 2.5,A1/D= 1.432,m⇤ = 2,z = 0,Rem = 315, f1/ fn = 2.0. (a-h) Multiple negative vortices shed from the active cylinder merges around the passive cylinder and pushes it away. (i-p) The vortex pairs shed from the active cylinder goes around the passive cylinder without causing its significant displacement. Instants A and P in Fig. 6.25 corresponds to Figs. 6.25a and 6.25p, respectively. 126 Effects of Active Cylinder’s Amplitude and Frequency in Less Regular Regimes 6.6 Chapter Summary To summarise this chapter, the irregularity and the overall amplitude for the vibration of the passive cylinder increases with KC = 17 at b = 35. However, the amplification factor and the phase difference of the harmonic components barely changes with KC at both damped and undamped conditions. Regime C features pulse beating in the response of the passive cylinder due to the almost periodic gradual switch of the streaming direction, and the vortex shedding occurs for both sides of the active cylinder. In regime E, the passive cylinder responds steadily most of the time with irregularity occurs intermittently. This corresponds to the intermittent switching between one flow pattern and its mirror-image. During switching, the flow structure and vortex shedding are fairly irregular. In every cycle, 2 vortex pairs are shed from the active cylinder. In terms of regime F, the irregularity is of another level. Only a few periods of steady interactions can be observed. At least 3 steady patterns can be identified. In a single cycle, a vortex pair can be shed from one side of the active cylinder like in regime E and another vortex pair can emerge from both sides of the active cylinder like in regime C. During the steady state, the flow structure is diagonal in coherence with the single-cylinder cases. The transition between the steady patterns is very irregular with little pattern identifiable. In regime G, the flow is very irregular. Most of the time, 2 vortex pairs are shed from opposite sides of the active cylinder in one cycle. The flow is also very circular, enabling vortices shed from consecutive cycles to merge, becoming a much stronger vortex. Such large vortex can directly impact the passive cylinder, causing its significant displacement as shown in the time histories. Chapter 7 Conclusions This chapter provides a summary of the entire thesis and then draws conclusions on how the 6 non-dimensional groups influence the resonance phenomena and the vibration centre drift of the passive cylinder and the corresponding flow characteristics. At last, potential future work is suggested. 7.1 Overview This study focuses on the representative case of the flow-mediated interactions between two cylinders immersed in a quiescent fluid. The active cylinder undergoes forced vibration with a specified amplitude and frequency, whereas the passive cylinder is elastically-mounted with a damper. Both cylinders are only allowed to vibrate along the centreline connecting the two cylinders. For this representative case, 6 non-dimensional groups are identified by dimensional analysis. This study uses an in-house finite element code to simulate this problem of flow-mediated interaction. Simulations are conducted for a range of combinations of parameters. The active cylinder’s oscillation frequency f1/ fn ranges from 0.05 to 3.2; the amplitude of the active cylinder A1/D varies from 0.025 to 1.432; the mass ratio of the passive cylinder m⇤ takes the value of 1.5, 1.7, 2.0, 2.2 or 2.5; the structural damping factor of the passive cylinder ranges from 0 to 1.4; Reynolds number Rem varies from 10 to 310; the gap ratio G/D ranges from 0.2 to 3. The range of Keulegan-Carpenter number and the Stokes number of the active cylinder can then be calculated to be KC = 0.169 and b = Rem/KC = 3.3637, respectively. In total, 23,400 combinations of parameters are examined. This parametric space is chosen to reflect the values usually seen in engineering applications. 128 Conclusions 7.2 Parametric Study in Periodic Regimes A and A* Active cylinder’s forced oscillation frequency f1/ fn is the most significant parameter. The vibration frequency of the passive cylinder is usually synced to f1/ fn, thus adjusting f1/ fn can cause resonance of the passive cylinder, greatly amplifying its vibration amplitude. Oscillation of the active cylinder also causes harmonics of the passive cylinder, resulting in secondary resonance. Reynolds number Rem governs the drifting of the vibration centre. Its variation can alter the flow structure, causing the passive cylinder to be attracted towards or repelled away from the active cylinder. Structural damping factor z can significantly suppress the amplitude of the passive cylinder with a slight increase of value. It is also capable of affecting how other parameters influence the passive cylinder’s vibration. It can alter the relationship between other non- dimensional groups and the vibration pattern of the passive cylinder. It also can lead to less intensive the pressure oscillation at the far side of the active cylinder. The mass ratio of the passive cylinder m⇤ determines immersed natural frequency of the passive cylinder, and thus controlling its resonance frequency. At cases where the passive cylinder is structurally undamped, it can not influence the passive cylinder’s amplitude at resonance. However, at the damped case, it can reduce the overall responding amplitude, including at resonance. Active cylinder’s oscillation amplitude A1/D is generally linearly correlated with that of the passive cylinder, except at resonance and with very low structural damping. Gap distance G/D is negatively correlated the responding vibration amplitude of the passive cylinder. However, it turns out not to influence the phase difference between the passive cylinder and the active cylinder. 7.3 Interactions in Less Regular Regimes C, E, F and G The overall flow structures exhibited in less regular regimes are fundamentally different from that in the periodic regimes aforementioned. However, despite the irregular frequency components discovered in the amplitude spectra of the passive cylinder’s vibration, the harmonics previously discussed is persistent and always distinguishable. So the previous conclusions regarding harmonics and major and minor resonance still apply. The vibration centre shift pattern may, however, change due to the alteration of flow structure. Regime C features the passive cylinder’s pulse beating due to the switching between the skewed "L" shape flow pattern and its mirror-image, and the vortex shedding occurs for both 7.4 Future Work 129 sides of the active cylinder. In regime C, the switching of flow pattern is gradual, so the irregular transition in the following regimes does not occur. This pattern is fundamentally different from that of a single oscillating cylinder. In regime E, the passive cylinder vibrates steadily most of the time, even though irregu- larity may take place from time to time. This response pattern corresponds to the intermittent switch of the flow structure between one pattern and its mirror-image. During the transition, the flow structure and vortex shedding are fairly irregular. In every cycle, two vortex pairs are shed from the active cylinder. In terms of regime F, the irregularity further increases. Only a few periods of steady interactions can be observed. At least 3 steady patterns can be identified. In a single cycle, a vortex pair can be shed from one side of the active cylinder like in regime E with another vortex pair emerged from both sides of the active cylinder like in regime C. A minor extra vortex can be observed when the vortex pair is shed from both sides. During the steady-state, the flow structure is diagonal in coherence with the single-cylinder cases. The transition between the steady patterns is very irregular with little pattern identifiable. In regime G, the flow is highly irregular. Most of the time, two vortex pairs are shed from opposite sides of the active cylinder in one cycle. Sometimes, two vortex pairs can also be shed from a single side. The flow is also very circular, enabling vortices shed from consecutive cycles to merge becoming a much stronger vortex. Such a large vortex can directly impact the passive cylinder, causing its significant displacement as shown in the time histories. The existence of the passive cylinder disrupts the flow pattern, since the flow structure is more irregular compared with the single-cylinder cases and in some regimes, the flow pattern is very different from the single-cylinder cases. 7.4 Future Work A few areas are suggested to be further explored in the future, including 3D simulation, artificial intelligence, Floquet analysis, bio-engineering and vortex-induced vibration energy harvesting. 3D simulations at higher KC and b can be conducted to explore regimes with more complicated 3D flow features and to provide a more comprehensive understanding of the flow-mediated interaction phenomena. The configuration of the two cylinders that are not parallel to each other can be studied. Artificial intelligence can be useful in summarising a large number of CFD results. Machine learning or even deep learning may be applied to find patterns in the data, since 130 Conclusions a large amount of data are generated in such parametric studies. 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Appendix A Data Processing Scripts Thousands of lines of mini MATLAB programmes were created to execute the pre-processing and post-processing of the CFD simulations. The following sections present examples for many data processing scripts. A.1 Generation of Time History and Amplitude Spectra This is the most frequently used mini MATLAB programme for post-processing. 1 clear all 2 close all 3 fclose all; 4 global autocut_switch xlimax 5 global cutlength 6 autocut_switch =0;% 0 is to turn off autocut 7 cutlength =2;%first periods to cut off 8 syscount =0;%counter to go throught diff directories 9 xlimax =3.3;% maximum value on x-axis 10 for syscount =1:1 11 if syscount ==1 12 sysdir % data part_2 13 elseif syscount ==2 14 sysdir2 % data part_3 15 end 16 sys_pwd=pwd; 17 list0=dir(’g*a*m*d*e*r*’); %9326 -9340 18 n0=length(list0); 19 s=0; resc =0; 20 nstart =1; 21 for j=nstart:n0 %1:n0 138 Data Processing Scripts 22 cd(sys_pwd) 23 cd (list0(j).name) 24 listf1=dir(’0*’); 25 n=length(listf1); 26 cd (’../’) 27 str=list0(j).name; 28 gamr=s2gamr(str); 29 G=gamr (1); 30 A1=gamr (2); 31 m=gamr (3); 32 damp=gamr (4); 33 Re=gamr (5); 34 taskid=gamr (6); 35 cphs=zeros(1,n); 36 yphs=zeros(1,n); 37 f1s=zeros(1,n); 38 for i=1:n % i=1:n 39 f1=str2double(listf1(i).name); 40 f1s(i)=f1; 41 close all 42 cd(sys_pwd) 43 dirnamef =[ list0(j).name ,’/’,listf1(i).name ,’/’]; 44 cd(dirnamef) 45 disp(pwd); 46 fdir=pwd; 47 cd(’G:\My Drive\High_KC_Anime_Research_Figure_Plot ’); 48 tempo=[’plotting_both/’,list0(j).name]; 49 mkdir(tempo) 50 cd (tempo) 51 disp(tempo) 52 ttl=[’G=’,num2str(G),’ A_1=’,num2str(A1),’ m=’,num2str( m),’ f1=’,listf1(i).name ,’ Cylinder 2’]; 53 pdir=pwd; 54 pname=[ list0(j).name ,’_’,listf1(i).name]; 55 fdir=fdir;pname=pname;pdir=pdir; 56 zpf =[]; ttl_p =[]; 57 start =0; 58 f1=f1;A1=A1; 59 nT=4;%nT last periods will be displayed 60 ylimfftl =[]; ylimfftr =[]; 61 ylimthl =[]; ylimthr =[]; 62 mr=m; A.1 Generation of Time History and Amplitude Spectra 139 63 [err ,stperiods ,cphs(i),yphs(i)]= checkplot_CY_and_Y(fdir , zpf ,start ,pdir ,ttl_p ,pname ,ylimfftl ,f1 ,nT ,ylimthl ,A1,ylimfftr , ylimthr ,mr); 64 if err~=-1 65 saveas(gcf ,[ttl ,’.fig’]) 66 end 67 resc=resc +1; 68 result(resc ,:)=[gamr (1:3) f1 gamr (4) Re stperiods -1]; 69 cd (’../’) 70 cd (’../’) 71 csvwrite ([’gamfr_Re_stperiods.dat’],result) 72 end 73 figure 74 plot(f1s ,cphs)%plot phase of force coefficient (dominant frequency with largest amplitude) 75 plot(f1s ,yphs)%plot phase of displacement 76 plot(f1s ,cphs -yphs) % plot phase difference 77 end 78 end 79 function [err ,stperiods ,cphs ,yphs]= checkplot_CY_and_Y(fdir ,zpf ,start , pdir ,ttl_p ,pname ,ylimfftl ,f1,nT,ylimthl ,A1 ,ylimfftr ,ylimthr ,mr) 80 cphs =0; yphs =0; 81 global autocut_switch xlimax 82 setlatex 83 if ~exist(’pname’,’var’) || isempty(pname) 84 pname=’’; 85 end 86 if ~exist(’fdir’,’var’) || isempty(fdir) 87 fdir=pwd; 88 end 89 cd(fdir) 90 if ~exist(’start’,’var’) || isempty(start) 91 start =1; 92 end 93 [t1 ,y1]= textscan_ft(A1,f1);% 1 denotes force coefficient 94 dt1=t1(20)-t1(19); 95 [t2 ,y2 ,v2]= textscanty; % 2 denotes displacement 96 dt2=t2(20)-t2(19); 97 fdname=’Verification ’; 98 t1=t1 (4000:(end -4000)); 99 y1=y1 (4000:(end -4000)); 100 t2=t2 (4000:(end -4000)); 101 y2=y2 (4000:(end -4000)); 140 Data Processing Scripts 102 if autocut_switch ;[t1,y1,t2,y2]= autocut_for_irregular(f1 ,t1 ,y1 ,t2,y2) ;end 103 note1=’Cy02’; 104 note2=’Y’; 105 L = length(y2); %total points after cut 106 stperiods=floor(L/(1/f1/dt1));% number of steady periods 107 if L-ceil (1/f1 .*10/ dt1) <=0 108 disp(’force less than 10 steady periods ’) ;err=-1; 109 return; 110 else 111 err =1; 112 end 113 if L-ceil (1/f1 .*10/ dt2) <=0 114 disp(’disp less than 10 steady periods ’) ;err=-1; 115 return; 116 else 117 err =1; 118 end 119 hold off 120 figure (6) 121 h(2)=plot(t2,y2,’color’,’k’,’LineStyle ’,’-’); 122 ylabel(’$y/D$’,’Interpreter ’,’LaTex’,’FontSize ’ ,20) 123 xlabel(’$tf_{n}$’,’Interpreter ’,’LaTex’,’FontSize ’ ,20) 124 axr = gca;axr.YColor = ’k’;%set color of yaxis 125 hold off 126 if ~exist(’ylimthl ’,’var’) || isempty(ylimthl); else; yyaxis left; ylim(ylimthl); end 127 if ~exist(’ylimthr ’,’var’) || isempty(ylimthr); else; yyaxis right; ylim(ylimthr); end 128 if ~exist(’f1’,’var’) || isempty(f1) 129 else 130 temp=[’$f_1/f_n=’,num2str(f1),’$’]; 131 hl = legend ([’$Y_2/D$\ \ \ ’,temp]); 132 set(hl ,’Interpreter ’,’latex’,’FontSize ’ ,15) 133 end 134 if ~exist(’ttl’,’var’) || isempty(ttl); else; title(ttl); end 135 if ~exist(’pdir’,’var’) || isempty(pdir); else; cd(pdir); 136 printpdf(gcf ,[pname ,’_’,note2 ,’_Disp_Only_Time_history_all_used. eps’]); 137 saveas(gcf ,[pname ,’_’,note2 ,’_Disp_Only_Time_history_all_used.fig ’]) 138 end 139 figure (7) 140 cphs =0; A.1 Generation of Time History and Amplitude Spectra 141 141 [f, P1]= fftzp2(t1,y1); 142 plot(f,P1,’color’,’black ’) 143 xlim ([0 3.7]); 144 if ~exist(’ylimfft ’,’var’) || isempty(ylimfftl) 145 else 146 ylim(ylimfftl); 147 end 148 if ~exist(’ttl’,’var’) || isempty(ttl) 149 else 150 title(ttl); 151 end 152 xlabel(’$f_2/f_n$’,’Interpreter ’,’LaTex’,’FontSize ’ ,20) 153 ylabel(’$C_{AY2}$’,’Interpreter ’,’LaTex’,’FontSize ’ ,20); 154 hold on 155 yyaxis right 156 yphs =0; 157 [f, P1]= fftzp2(t2,y2);plot(f,P1,’color ’,’r’,’LineStyle ’,’--’); 158 xlim ([0 xlimax ]); 159 if ~exist(’ylimfftr ’,’var’) || isempty(ylimfftr) 160 else 161 ylim(ylimfftr); 162 end 163 axr = gca;axr.YColor = ’k’;%set color of yaxis 164 if ~exist(’f1’,’var’) || isempty(f1) 165 else 166 temp=[’$f_1/f_n=’,num2str(f1),’$’]; 167 hl = legend ([’$C_{Y2}$ ’,temp],[’$Y_2/D$\ \ \ ’,temp]); 168 set(hl ,’Interpreter ’,’latex’,’FontSize ’ ,15) 169 end 170 xlabel(’$f/f_n$’,’Interpreter ’,’LaTex’,’FontSize ’ ,20) 171 ylabel(’$A/D$’,’Interpreter ’,’LaTex’,’FontSize ’ ,20); 172 if ~exist(’pdir’,’var’) || isempty(pdir) 173 else 174 cd(pdir) 175 printpdf(gcf ,[pname ,’_’,note1 ,’_FFT_.eps’]); 176 end 177 end Listing A.1 Mini MATLAB programme used to process simulation data into figures for time histories and amplitude spectra 142 Data Processing Scripts A.2 Mesh Generation by Gmsh The following Gmsh script was used to generate the structured and unstructured mesh used for simulations. 1 Mesh.ColorCarousel =2; 2 3 //input 4 G=2.5;// clear gap ratio 5 length =30;// length *2= computational domain boundary length 6 rt2 =0.08;// mesh size around smaller square 7 ref80 =1.8;// mesh size on far square boundaries 3.0 8 9 Spc_sixth_cylinder =28;// Spc_sixth_cylinder *6= total transfinite around one cylinder 10 Pr =1.065;// radiate lines progression rate 11 Spr =54;// radiate lines transfinite num 12 13 scale=G*1.2; // enlarge factor for smaller square 14 15 Spc=Spc_sixth_cylinder *2; 16 Cy=G/2+0.5; 17 ms =0.0001;// reference element size 0.3 (useless as it is overwritten) 18 sq2d4 =0.35355339059327376220042218105242; 19 20 dx =0.25; 21 dy =0.43301270189221932338186158537647; 22 dx2=Cy /1.7320508075688772935274463415059; 23 dy3 =0.81649658092772603273242802490196* Cy; 24 25 26 //Point (36) = {-0.4, 0, 0, 0.001};// local refinement 27 28 // square boundary 29 Point (1) = {-length , length , 0, ref80}; 30 Point (2) = {length , length , 0, ref80}; 31 Point (3) = {length , -length , 0, ref80}; 32 Point (4) = {-length , -length , 0, ref80}; 33 Line (1) = {1, 2}; 34 Line (2) = {2, 3}; 35 Line (3) = {3, 4}; 36 Line (4) = {4, 1}; 37 38 // Cylinders A.2 Mesh Generation by Gmsh 143 39 Point (5) = {0, 0.5+G/2, 0, 1.0}; 40 Point (6) = {0, -0.5-G/2, 0, 1.0}; 41 42 Point (7) = {-dx, Cy+dy , 0, 1.0}; 43 Point (8) = {dx, Cy+dy , 0, 1.0}; 44 Point (9) = {dx, Cy -dy , 0, 1.0}; 45 Point (10) = {-dx, Cy -dy , 0, 1.0}; 46 47 Point (11) = {-dx, -Cy -dy , 0, 1.0}; 48 Point (12) = {dx, -Cy -dy , 0, 1.0}; 49 Point (13) = {dx, -Cy+dy , 0, 1.0}; 50 Point (14) = {-dx, -Cy+dy , 0, 1.0}; 51 52 Circle (15) = {7,5,8}; 53 Circle (16) = {8,5,9}; 54 Circle (17) = {9,5,10}; 55 Circle (18) = {10 ,5,7}; 56 Circle (19) = {14, 6, 13}; 57 Circle (20) = {13, 6, 12}; 58 59 Circle (21) = {12, 6, 11}; 60 Circle (22) = {11, 6, 14}; 61 62 //small block around cylinder 63 Point (23) = {dx2 , 2*Cy , 0, ms}; 64 Point (24) = {-dx2 , 2*Cy , 0, ms}; 65 Point (25) = {-dx2 , 0, 0, ms}; 66 Point (26) = {dx2 , 0, 0, ms}; 67 68 Point (27) = {dx2 , -2*Cy , 0, ms}; 69 Point (28) = {-dx2 , -2*Cy , 0, ms}; 70 //Point (29) = {-1.5, 0, 0, ms}; 71 //Point (30) = {1.5, 0, 0, ms}; 72 73 //Line (23) = {25, 24}; 74 Circle (23) = {25, 5, 24}; 75 //Line (24) = {24, 23}; 76 Circle (24) = {24, 5, 23}; 77 //Line (25) = {23, 26}; 78 Circle (25) = {23, 5, 26}; 79 Line (26) = {26, 25}; 80 //Line (28) = {25, 28}; 81 Circle (28) = {25, 6, 28}; 82 //Line (29) = {28, 27}; 144 Data Processing Scripts 83 Circle (29) = {28, 6, 27}; 84 //Line (30) = {27, 26}; 85 Circle (30) = {27, 6, 26}; 86 Line (31) = {7, 24}; 87 Line (32) = {10, 25}; 88 Line (33) = {9, 26}; 89 Line (34) = {8, 23}; 90 Line (35) = {14, 25}; 91 Line (36) = {13, 26}; 92 Line (37) = {12, 27}; 93 Line (38) = {11, 28}; 94 95 //block lines tran 96 Transfinite Line {23, 24, 25, 28, 29, 30} = Spc Using Progression 1; 97 // Cylinder lines 98 Transfinite Line {22, 20, 21, 18, 16, 15} = Spc Using Progression 1; 99 // radient lines tran 100 Transfinite Line {32, 31, 34, 33, 35, 36, 38, 37} = Spr Using Progression Pr; 101 //edit 102 Transfinite Line {21 ,29 ,15 ,24 ,19 ,26 ,17} = Spc_sixth_cylinder Using Progression 1; 103 104 Line Loop (39) = {38, 29, -37, 21}; 105 Plane Surface (40) = {39}; 106 Line Loop (41) = {20, 37, 30, -36}; 107 Plane Surface (42) = {41}; 108 Line Loop (43) = {19, 36, 26, -35}; 109 Plane Surface (44) = {43}; 110 Line Loop (45) = {22, 35, 28, -38}; 111 Plane Surface (46) = {45}; 112 Line Loop (47) = {17, 32, -26, -33}; 113 Plane Surface (48) = {47}; 114 Line Loop (49) = {16, 33, -25, -34}; 115 Plane Surface (50) = {49}; 116 Line Loop (51) = {15, 34, -24, -31}; 117 Plane Surface (52) = {51}; 118 Line Loop (53) = {18, 31, -23, -32}; 119 Plane Surface (54) = {53}; 120 121 Recombine Surface {54, 52, 50, 48, 44, 46, 42, 40}; 122 123 Transfinite Surface {48}; 124 Transfinite Surface {50}; A.2 Mesh Generation by Gmsh 145 125 Transfinite Surface {52}; 126 Transfinite Surface {54}; 127 128 Transfinite Surface {40}; 129 Transfinite Surface {42}; 130 Transfinite Surface {44}; 131 Transfinite Surface {46}; 132 133 Line Loop (79) = {55, 56, 57, 58}; 134 Line Loop (80) = {24, 25, -30, -29, -28, 23}; 135 Plane Surface (81) = {79, 80}; 136 Line Loop (82) = {1, 2, 3, 4}; 137 Plane Surface (83) = {80, 82}; 138 139 Recombine Surface {83}; 140 Coherence; 141 142 // refinement around the cylinders 143 Point (84) = {-scale , (Cy+scale), 0, rt2}; 144 Point (85) = {scale , (Cy+scale), 0, rt2}; 145 Point (86) = {-scale , -(Cy+scale), 0, rt2}; 146 Point (87) = {scale , -(Cy+scale), 0, rt2}; 147 148 //+ 149 Line (88) = {84, 85}; 150 Line (89) = {85, 87}; 151 Line (90) = {87, 86}; 152 Line (91) = {86, 84}; 153 154 Line {88} In Surface {83}; 155 Line {89} In Surface {83}; 156 Line {90} In Surface {83}; 157 Line {91} In Surface {83}; 158 159 Coherence; 160 161 // Boundary Condition 162 Physical Line(" Cylinder 1", 2) = {19 ,20 ,21 ,22} ; 163 Physical Line(" Cylinder 2", 3) = {15 ,16 ,17 ,18} ; 164 Physical Line("Outer Wall", 4) = {1,2,3,4} ; 165 166 //Mesh nodes 146 Data Processing Scripts 167 Physical Surface ("Main mesh", 1 ) = {83, 46, 40, 42, 44, 48, 50, 54, 52}; Listing A.2 Mini MATLAB programme used to process simulation data into figures for time histories and amplitude spectra