Investigation of Fluid Force Coefficients of a Towed Cylindrical Structure undergoing Controlled Oscillations

Transcription

1 Investigation of Fluid Force oefficients of a Towed ylindrical Structure undergoing ontrolled Oscillations by Muhamad H Kamarudin School of Mechanical Engineering The University of Western Australia Perth This thesis is submitted for the degree of Masters of Engineering Science of The University of Western Australia December 213

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3 The two great unsolved problems were reconciling quantum mechanics and general relativity, and turbulence. - Heisenberg

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5 Abstract This thesis presents the results on the experimental measurement of hydrodynamic forces on a circular cylinder undergoing prescribed harmonic motions in a steady uniform flow. The objective of the study is to quantify the hydrodynamic coefficients in both transverse and in-line directions under different combinations of frequency and amplitude of illation. A forced illation experimental rig has been developed utilizing the scotch yoke mechanism to cover the range of desired amplitudes and frequencies of the harmonic illations. The rig was mounted onto the towing carriage facilities of 4m long and towed at a constant speed. All experiments were performed at Reynolds number of 12 with aspect ratio of 1. Non-dimensional illating frequency ranging from.4 f / f 2. 2 were tested at different non-dimensional amplitudes of.2 A / D 1. in the transverse direction of the tow. Fluid force analysis and its breakdown into components in-phase with velocity and acceleration are presented to investigate the general response characteristics of varying amplitude and frequency of motion. 3-dimensional response characteristics are plotted in order to offer a better representation of the fluid force coefficients as a function of frequency and amplitude. The results reveal that the response at higher amplitudes i.e. A / D.9 and A/ D1. deviate from the behaviour observed from the lower amplitude cases. No change of energy transfer observed over the frequency range investigated for the higher amplitude cases. The lock-in envelope upon which the cylinder frequency locks-in with the natural vortex shedding frequency of the system producing resonance behavior is also plotted. Furthermore, the numerical simulation results are also presented as a basis of comparison with experimental results. The results from the numerical simulations are used qualitatively to discuss the characteristics of the fluid-structure interaction which is not obtainable from the experiments, complementing the quantitative experimental findings. The phase switch between vortex formation and cylinder motion and also the formation of limit cycle in the lock-in range from the numerical results are presented. Finally, the conclusions of the findings and the recommendations for future work are presented. i

6 Acknowledgements First of all, I thank God for giving me the ability to carry out this research and to complete it despite the extended period of time. Tremendous thanks to my supervisor, Prof. Krish Thiagarajan who has been my supervisor since my undergraduate years and has helped me a lot throughout my research despite this research was undertaken on a part-time basis and without any special funding. It would be almost impossible to complete this research without his help. Thanks again to Krish who had inspired me to pursue the area of offshore engineering and I think I made the right choice. Special thanks to Dr. Roshun Paurobally for reviewing, commenting and provided suggestions to improve this thesis. I thank Roshun for this willingness to accept the role of supervision despite the research was coming to an end at the time of him accepting the role. I would also like to thank Mr. Andrew Liew for his time and effort in the lab in developing the forced illation apparatus. His time spent from conceptual drawings, manufacturing, calibrating to the testing of the apparatus is greatly appreciated. I also thank all the personnel in the mechanical and electrical workshop involved in fabrication process of the experimental apparatus and to the hydraulics laboratory personnel in providing full support for the experiments. ii

7 Table of ontents Abstract... i Acknowledgements... ii List of Figures... vi List of Tables... ix 1. Introduction Motivation and Background Scope of Work Outline of Thesis Literature Review and Theory Stationary ircular ylinder The Navier-Stokes Equation Theoretical Description of Flow Past a Bluff Body Mechanism of Vortex Formation and Vortex Shedding Flow Regime Oscillating ircular ylinder Equation of Motion Oscillating ylinder Wake Structure Methods of VIV Investigation Free Vibration Experiments Forced Vibration Experiments Numerical Simulations Semi-Empirical Models Description of Experiment Towing arriage Facilities Developing a Forced Oscillations Experimental Rig Experimental ondition iii

8 3.3.1 alibration of Instruments onduct of the Experiment Quality ontrol End condition effect Free surface effect Blockage effect Residual waves Limitation in towing tank length Data Analysis Outline of Post Processor Definition of Direction Signal Processing Differentiation of Displacement Signal Hydrodynamic Force oefficients Fluid Forces on ylinder Fourier oefficient Analyses Definition of Hydrodynamic Force oefficients Hydrodynamic Force Decomposition Numerical Simulation Approach and Methodology Numerical Model Description Mesh onsideration for Turbulent Flows Numerical Simulations Parameters Model Validation Mesh onvergence Analysis Applicability and Limitations of 2-Dimensional Modeling Results Stationary ylinder Results Oscillating ylinder Results Total In-line and Transverse Force oefficients iv

9 6.2.2 hange of Energy Transfer between ylinder and Fluid Added Mass Effect on the Transverse Force Surface Plots of all Amplitude Ratios Response haracteristics of Force oefficients at Higher Amplitude Range Lock-in Range omparison with Numerical Results Phase Switch between Vortex Formation and ylinder Motion Limit ycle Predicting Free Vibration onclusions and Recommendations for Future Work onclusions Recommendations for Future Work References v

10 List of Figures Figure 1-1: Independence Hub Semi-Submersible... 2 Figure 1-2: ross-flow Parametric Response Model... 4 Figure 2-1: Ideal Flow Streamlines and Pressure Distribution around Bluff Body... 9 Figure 2-2: Development of Boundary Layer and Shear Layer. Figure from Sumer and Fredsøe [45]... 1 Figure 2-3: von Karman Vortex Street. Reproduced from 1 Figure 2-4: Gerrard s Vortex Formation Model [16] Figure 2-5: Strouhal Number, St. Figure by Norberg [32] Figure 2-6: RMS Transverse Force oefficient, _. Figure by Norberg [32] Figure 2-7: haracterization of the different Flow Regimes. Figure by Sumer and Fredsøe [45] Figure 2-8: Mass, Damper and Spring System of 1-DOF Motion Figure 2-9: Williamson-Roshko Map of Wake Structures for illating cylinder. Figure from Williamson and Roshko [51] Figure 2-1: Detailed Williamson-Roshko map for wake structures for illating cylinder. Figure from Williamson and Roshko [51] Figure 2-11: Free Vibration Experimental Set-up by Vikestad [48]... 2 Figure 2-12: Forced Vibration Experimental Set-up by Bishop and Hassan [8] Figure 2-13: Lift Force Response. From Bishop and Hassan [9] Figure 2-14: Excitation oefficient and Added Mass oefficient ontours by Gopalkrishnan [17] Figure 2-15: ontour Plots of Force oefficients and Boundaries of different Vortex Modes from Morse and Williamson [3] Figure 2-16: Instantaneous Vorticity ontour showing Vortex Switching [1] Figure 3-1: Schematic of Towing arriage Tank (Plan View) L rms vi

11 Figure 3-2: Scotch Yoke Mechanism Figure 3-3: Forced Oscillation Rig Model Schematic Figure 3-4: Experimental Rig Figure 3-5: Experimental Setup in Towing Tank Figure 3-6: Data ollection Flow hart Figure 4-1: Definition of Direction Figure 4-2: Measured Forces and Displacement Signals Figure 5-1: Description of Numerical Domain and Boundary ondition... 5 Figure 5-2: Wall y value around wall for / f 1. f with 32 x 1 domain Figure 5-3: ell ourant Number Figure 5-4: Map of Vortex Synchronization Regions for Model Validation Figure 5-5: Validation Results showing different Vortex Pattern Figure 5-6: Mesh Density onvergence Plot omparison Figure 6-1: Histogram of Strouhal Number, St for Stationary ylinder Figure 6-2: Histogram of RMS Transverse oefficient, L _ rms for Stationary ylinder Figure 6-3: Histogram of In-Line oefficient, Dm for Stationary ylinder Figure 6-4: Stationary Transverse Force Profile showing non-constant Force Magnitude Figure 6-5: Stationary In-Line Force Profile showing High Frequency Fluctuation Figure 6-6: Transverse Force Profile from Khalak and Williamson at Re 1, 6[24] 66 Figure 6-7: Forced Oscillation Transverse Force Profile Figure 6-8: Forced Oscillation In-line Force Profile Figure 6-9: Mean In-line oefficient for A / D. 3 as a function of f / f Figure 6-1: RMS Transverse oefficient for A / D. 3 as a function of f / f Figure 6-11: Phase Angle for A / D. 3 between transverse force and cylinder displacement as a function of f / f... 7 Figure 6-12: Mechanical Energy Transfer for A / D. 3 between cylinder and fluid as a function of f / f... 7 vii

12 Figure 6-13: Transverse Force oefficient In-Phase with Velocity for A / D. 3 as a function of f / f Figure 6-14: Transverse Force oefficient In-Phase with Acceleration for A / D. 3 as a function of f / f Figure 6-15: Added Mass oefficient for A / D. 3 as a function of f / f Figure 6-17: Surface Plot of Phase Angle, as a function of f and A/ D Figure 6-18: Surface Plot of Energy, E as a function of f and A/ D Figure 6-19: Surface Plot of Transverse Force oefficient In-Phase with Velocity, as a function of f and A/ D... 8 Figure 6-2: Surface Plot of Added Mass oefficient, and m as a function of f A/ D Figure 6-21: Surface Plot of Transverse Force oefficient In-Phase with Acceleration, lh as a function of f and A/ D Figure 6-22: Transverse Force oefficient In-Phase with Velocity, and In-Phase with Acceleration, lv lv lh Figure 6-23: Phase Angle and Energy Plots for A / D. 3 and A / D Figure 6-24: Motion and Transverse Force Spectra before Lock-in ( f. 262 Hz) Figure 6-25: Motion and Transverse Force Spectra inside Lock-in ( f. 574 Hz) Figure 6-26: Motion and Transverse Force Spectra above Lock-in ( f Hz) Figure 6-27: Transverse Force Profile before Lock-in ( f. 262 Hz) Figure 6-28: Transverse Force Profile inside Lock-in ( f. 574 Hz) Figure 6-29: Transverse Force Profile above Lock-in ( f Hz) Figure 6-3: Lock-in Region formed between f / f for different A/ D... 9 Figure 6-31: omparison between Numerical and Experimental Results for Figure 6-32: omparison between Numerical and Experimental Results for lv lh Figure 6-33: Instantaneous Vorticity and Streamline ontours viii

13 Figure 6-34: Limit ycle for A / D. 3 showing change in direction of motion (from clockwise to anti-clockwise) during phase switch List of Tables Table 3-1: Towing arriage Devices Table 3-2: Experimental Parameters Table 3-3: alibration Factors Table 5-1: Validation ase Matrix Table 5-2: Mesh Density onvergence Results Table 6-1: Summary of Stationary ylinder Results ix

14 Abbreviations FD FL N DAQ D DOF DNS DNV DPIV FFT LES LVDT PISO QUIK RANS RMS RP rpm UWA VIV omputational Fluid Dynamics ourant-friedrichs-lewy omputer Numerical ontrol Data Acquisition Direct urrent Degree of Freedom Direct Numerical Simulation Det Norske Veritas Digital Particle Image Velocimetry Fast Fourier Transform Large Eddy Simulation Linear Variable Displacement Transducer Pressure Implicit with Splitting of Operators Quadratic Upstream Interpolation for onvective Kinematics Reynolds-Averaged Navier-Stokes Root Mean Square Recommended Practice Revolution per minute University of Western Australia Vortex Induced Vibration x

15 Roman Symbols A amplitude of illation a, a1, a n Fourier coefficient B blockage ratio ( D / w) b 1,b n Fourier coefficient A D added mass in-line coefficient Dm mean in-line coefficient L L rms transverse coefficient _ root-mean-square of transverse force coefficient lh lv m D E F D force in-phase with acceleration coefficient force in-phase with velocity coefficient added mass coefficient cylinder outer diameter energy in-line force F magnitude of illating in-line force d F ds F L magnitude of illating in-line force (Strouhal component) transverse force F magnitude of illating transverse force l F ls f magnitude of illating transverse force (Strouhal component) vortex shedding frequency (stationary cylinder) f illation frequency f v F r g * vortex shedding frequency (illating cylinder) Froude number gap to diameter ratio g 2 acceleration of gravity (9.81m / s ) G k G generation of turbulence kinetic energy due to k generation of turbulence kinetic energy due to h minimum depth of submergence min k stiffness, turbulence kinetic energy L length of cylinder xi

16 m m b m w * m n p mass of cylinder mass of beam mass of cylinder enclosed water mass ratio integer number pressure Re Reynolds number ( UD / ) S G stability parameter St Strouhal number ( f D / U) S k, S w user defined source terms t time T U u r period of illation fluid velocity friction velocity near wall V reduced velocity ( U / f D ) w y y y width of towing tank specific dissipation rate cylinder displacement, distance of the first cell from wall cylinder velocity cylinder acceleration y non-dimensional wall distance ( u y / ) Y k Y z dissipation of k dissipation of elevation point above a reference plane xii

17 Greek Symbols t time step size x cell length of grid cell in numerical domain damping of system fluid density turbulence kinetic energy dissipation rate mean standard deviation ζ damping ratio wavelength ( UT / D), angle (cell in numerical domain) velocity potential phase angle (stationary cylinder), phase angle (illating cylinder) fluid kinematic viscosity k effective diffusivity of k effective diffusivity of non-dimensional time step turbulence kinetic energy dissipation rate angular illation frequency ( 2 / T) Mathematical Operators differential operator partial differential xiii

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19 HAPTER 1: INTRODUTION hapter 1 1. Introduction 1.1 Motivation and Background Vortex Induced Vibration (VIV) of ocean structures has been an active topic of research at least for the last four decades. The research in VIV covers a wide range of applications such as offshore structures, heat exchangers, bridge decks, chimney stacks, drilling and production risers and marine cables. In the offshore industry, circular cross section such as pipelines and risers used to transmit petroleum fluid are the most susceptible to VIV which can lead to excessive stress and diminishing fatigue life. The problem becomes even more prominent in recent days when the oil and gas industry moved into deep water exploration with high ocean currents. VIV is a major factor affecting all stages of development of offshore structures from conceptualization to design, analysis, construction and monitoring. As the industry strives to develop less accessible deep water offshore hydrocarbon reservoirs, an increasingly frequent problem encountered is component failure and reduced component fatigue life related to VIV which can potentially lead to significant cost for remediation. Under such deep water conditions, the demands for long circular cylinders as offshore structures are increasingly required and prediction of VIV response is increasingly crucial. The subject of VIV consists of a number of disciplines from fluid mechanics, structural mechanics, vibrations, omputational Fluid Mechanics (FD) to acoustics and complex demodulation analysis. VIV is an inherently nonlinear, self-governed, multi-degree-offreedom phenomenon. Despite research conducted for more than four decades, much has yet to be discovered to understand the nature and characteristics of the kinematics and dynamics of this complex non-linear problem. Ultimately with better understanding, the phenomenon can be better predicted and hence prevention measures can be taken. Industrial applications highlight the lack of today s knowledge to predict the dynamic response of this fluid-structure interaction problem thus leading to a relatively large Factor of Safety in design. Figure 1-1 shows the Independence Hub semi-submersible in Mississippi anyon in approximately 24m of water depth to appreciate the scale of the typical project involved in the oil and gas industry indicating the prediction of VIV is crucial to avoid any cost impact due to structural failure. The system consists of a 12 leg mooring system, 12 x 1 inch risers and 4 x 8 inch risers. Gas is exported by a 2 inch riser. 1

20 HAPTER 1: INTRODUTION (a) Schematic of Independence Hub semi-submersible [49] (b) Photograph of Independence Hub semi-submersible [4] Figure 1-1: Independence Hub Semi-Submersible The prediction of the VIV phenomenon requires a large number of parameters such as Reynolds number, mass ratio, damping ratio, aspect ratio, stiffness and seabed proximity in order to accurately quantify the response. Experiments performed by different research groups are often hard to compare due to the wealth of parameters involved in the typical free vibration experiments. The method of forced vibration has been widely employed as means to aid VIV prediction apart from free vibration experiments. It is often more convenient to conduct a forced vibration experiment where the cylinder is given a prescribed harmonic motion allowing one to have better control of the parameters hence minimizing uncertainties. The ultimate aim is to quantify the fluid forces acting on the cylinder under the prescribed motion. 2

21 HAPTER 1: INTRODUTION The motivation for the present work is that the industry still does not have a full understanding of VIV for cylindrical structures. It is much easier and more reliable to quantify the forces using forced vibration due to the constancy of the desired amplitude and frequency. It is a well-known fact that the force coefficients used in designs do not come from free VIV tests, but come from forced vibrations experiments [37]. Hence, it is of fundamental importance to accurately quantify the force coefficients based on forced vibration experiments in order to shed light into the free vibration phenomenon. One of the most common tool used to predict VIV in industry is that of DNV-RP-F15 [13] parametric response model. DNV-RP-F15 is a recommended practice to predict fatigue damage due to direct wave loading and VIV of free spanning pipelines. This parametric response model is based on the results from various free vibration experiments and plotted as a function of reduced velocity, V r producing an envelope curve. The curve of the envelope is defined by mathematical expressions describing the physical phenomenon. Figure 1-2 (a) and (b) show the cross-flow parametric response model from DNV-RP-F15 and the data from some of cross-flow experimental results respectively. As the DNV cross flow response model is a function of eigen frequency, in order for the response model to give conservative results for all conditions the data basis on which the model is built should cover all frequency ratios. Furthermore, as the eigen frequency is a function of added mass as pointed out by Vikestad et al. [48], and in addition the added mass is a function of illation parameters i.e. amplitude and frequency [37], one will find it very arduous to quantify the natural frequency of an ever changing amplitude and frequency of illation in free vibration experiments. All these factors contribute to the uncertainties in the prediction of VIV based on free vibration experiments. It is with this motivation that the scope of work of this research is formed to expand the effort made by previous researchers in the course to increase the understanding of VIV phenomenon using the forced vibration method. 3

22 HAPTER 1: INTRODUTION (a) ross-flow Response urve from DNV-RP-F15 [13] (b) ross-flow Response urve from Mørk et al. [28] Figure 1-2: ross-flow Parametric Response Model 1.2 Scope of Work The objective of this research is to carry out a detailed experimental investigation of a plain cylinder undergoing forced vibration covering a wide range of amplitudes and frequencies under a constant Reynolds number. The ultimate aim is to analyze the behaviour of the hydrodynamic coefficients associated with the fluid-structure interaction under varying amplitudes and frequencies. The scope of work consists of: 4

23 HAPTER 1: INTRODUTION An experimental apparatus to simulate forced vibration of a plain rigid cylinder is specifically designed and developed. onsideration is given to the dimension and robustness of the set up in the development of the apparatus to cover the range of amplitudes and frequencies required. The set-up is mounted onto a towing tank and towed in one direction at a constant speed to simulate steady current while prescribing the harmonic motion in 1-DOF i.e. transverse direction. Only transverse illation is investigated. Experiments are conducted covering a range of amplitudes and frequencies (bracketing the natural Strouhal frequency) and the signals measured via strain gauges and Linear Variable Displacement Transducer (LVDT) are obtained via a data acquisition system to be analyzed. The stationary cylinder runs are also conducted to benchmark the response under identical test conditions. 2-dimensional numerical simulation employing the method of omputational Fluid Dynamics (FD) has been carried out to serve as a comparison with the experimental results. The simulation is conducted only for one amplitude ratio. The qualitative results from the simulation are used to complement the experimental results. The fluid force coefficients are calculated from the experiments and simulation to analyze the response characteristics under varying amplitudes and frequencies. An attempt at understanding the behavior of the hydrodynamic coefficients at different illating conditions is discussed. 1.3 Outline of Thesis The outline of the thesis is as follows. hapter 2 gives a review of the literature and the theory behind the phenomenon of VIV. The theory covers the fundamental aspects of stationary cylinder before looking deeper into the premise of the pertinent aspects of an illating cylinder. A discussion on the various methods used to investigate VIV is also included and the final part of the section discusses some of the most important contribution from early to the recent research being made on forced vibration. 5

24 HAPTER 1: INTRODUTION hapter 3 presents the experimental set-up used to conduct the experiments. The development of the forced illation experimental rig is included to demonstrate the process involved within the course of this research. This chapter also gives the experimental parameters involved and the experimental matrix used for the experiment. A few issues have also been addressed to ensure the quality and reliability of the force measurements such as free surface effects and blockage ratio from the apparatus used. The various hydrodynamic coefficients used to present the results have been defined in this section. The method used for fluid force decomposition is also presented. hapter 4 describes the 2-dimensional numerical simulation methodology used to conduct the numerical analysis for comparison with experimental results. The numerical model and the mesh consideration are presented in this section. It also illustrates the validation process and the convergence analysis to ensure robustness of the numerical model used. The last part discusses the applicability and limitations of the 2-dimensional numerical model. hapter 5 presents the results from the experiments conducted. The results have been discussed in detail for one amplitude ratio and the 3-dimensional surface plots of the fluid force coefficients are presented for all test matrix. The experimental results have been compared with the numerical results and how the numerical results complement the scope of work of the thesis has also been presented. hapter 6 presents the conclusions from the work conducted and recommendations for future work. 6

25 HAPTER 2: LITERATURE REVIEW AND THEORY hapter 2 2. Literature Review and Theory The fluid-structure interaction between a cylindrical body and its ambient fluid flow has received great attention owing to its importance in the offshore industry. The main concern within the scope of this study is the effects of Vortex-Induced Vibration (VIV) to a bluff body that can compromise its structural integrity. The problem lies in the complexity of the viscous flow around the bluff body and the dynamic interaction with the structure. It is well-known that when the vortex shedding frequency brackets the natural frequency of a bluff body, for example a cylinder, lock-in or synchronization will occur [11]. Lock-in can occur over a range of flow speeds dependent on the amplitude of illation, A and Reynolds number, Re. Even though the phenomenon of VIV is inherently self-excited and a multi-degree of freedom phenomenon, early studies were oriented towards studying the effects of forcibly illating the cylinder at constant amplitude and frequency in the transverse direction i.e. 1-DOF. This approach isolates the problem by focusing on the mere effects of frequency and amplitude by creating a sinusoidal motion to study the wake state and forces on the system. The convenience and simplicity of the externally driven forced vibration where one can control the amplitude and frequency of the motion has stimulated enthusiasm to study how forced vibration can be used to predict the VIV phenomenon, which is self-regulated in nature. 2.1 Stationary ircular ylinder The following sections discuss the fundamentals of stationary circular cylinder which form the rudimentary element for the discussion of the phenomenon of VIV and cylinder under forced vibration. It provides the background to the basic of flow dynamics past a circular body and the mechanics of vortex formation and shedding which serve as the essential component in the following section of illating cylinder The Navier-Stokes Equation The elementary governing formulation to describe a viscous flow is given by the Navier- Stokes equation presented in Equation 2-1 [11]. 2 U t 1 ( U. ) U p U g Equation 2-1 7

26 HAPTER 2: LITERATURE REVIEW AND THEORY T where U is a vector representing the fluid velocity ([ u v w] ), is the differential operator, is the fluid density, p is the pressure, is the kinematic viscosity of the fluid and g is the acceleration due to gravity. In an inviscid, irrotational and incompressible fluid, the Navier-Stokes equation reduces to Equation 2-2 also known as the Bernoulli equation [11]. p t 1 2 U 2 gz constant Equation 2-2 where is velocity potential and z is the elevation point above a reference frame Theoretical Description of Flow Past a Bluff Body When a fluid flows past a stationary body, a region of disturbed flow is formed around the body [53]. This region of the disturbed flow is largely dependent on the geometrical parameters of the body namely the shape, size and its orientation and also the flow parameters namely the velocity, U and viscosity, ν of the fluid. Based on the parameters mentioned above, the flow of a cylindrical body can be described by the non-dimensional quantities called the Reynolds number given by Equation 2-3. DU Re Equation 2-3 where D is diameter of the cylinder, U is the flow velocity and ν is the fluid kinematic viscosity. In an ideal flow, the streamlines around a cylinder in uniform current can be described as shown in Figure 2-1, also known as the potential flow theory. In potential flow theory, the water particles maintain the same velocity in the downstream stagnation point as in the upstream stagnation point as shown in Figure 2-1(a). Figure 2-1(b) shows the pressure distribution comparison for the experimental and potential flow. It can be seen that the pressure distribution is symmetric for the ideal fluid. The potential flow theory disregard the contribution of the viscous layer around the body and hence leading to zero drag phenomenon. This is commonly known as d Alembert s paradox. The pressure distribution shown in Figure 2-1 (b) is from an experiment by Achenbach [1] where the distribution is asymmetric for a viscous flow. 8

27 HAPTER 2: LITERATURE REVIEW AND THEORY. stagnation. point (a) Streamlines of Flow for Ideal Flow (b) Pressure Distribution [1] Figure 2-1: Ideal Flow Streamlines and Pressure Distribution around Bluff Body As the fluid flows past the solid body, it is divided into two regions; a very thin layer in the neighbourhood of the body called the boundary layer (see Figure 2-2) and the region outside this layer. In the boundary layer region, friction plays an essential part and outside this region, the friction effect can be neglected. This strong concept of boundary layer was developed by L. Prandtl in 194 [41]. The viscous flow in the boundary layer has a peculiar property that under certain conditions, the flow in the immediate neighbourhood of the body becomes reversed. Thus, the flow will separate at one point of the body dependent on its geometry due to the existence of an adverse pressure gradient. In the case of a bluff body such as subsea pipeline, the body will experience a boundary layer separation. This separation is then accompanied by the formation of eddies in the wake of the body. The transition point between the boundary layer and the wake region is called the separation point. The separation occurrence is dependent on the Reynolds number. The Reynolds number also determines whether the separation will be laminar or turbulent. This important concept of boundary layer separation gives the foundation to the whole phenomenon of viscous flow past a body and its fundamental effects pertinent to the discussion in the subsequent sections Mechanism of Vortex Formation and Vortex Shedding The phenomenon of boundary layer separation is closely related with the uneven pressure distribution in the boundary layer itself. For Reynolds number greater than about 4, the boundary layer will separate due to the adverse pressure gradient [45]. The change in pressure distribution is the result of the divergent geometry of the flow environment behind the cylinder. The separation of boundary layer from the surface forms a shear layer and it is highly unstable. The shear layer contains a considerable amount of vorticity and it causes the shear layer to roll into a discrete vortex 9

28 HAPTER 2: LITERATURE REVIEW AND THEORY downstream of the body. Figure 2-2 shows the development of boundary layer and the shear layer to form the vortices [45]. The size of the vortex will increase appreciably and it becomes separated shortly afterwards and moves downstream of the fluid. onsequently, the pressure distribution and the field of flow in the wake suffer a radical change. The change in pressure field causes a considerable suction in the eddying region behind the cylinder. This suction will result in a larger pressure drag on the body. Figure 2-2: Development of Boundary Layer and Shear Layer. Figure from Sumer and Fredsøe [45] Further downstream of the body, the vortices will move in a particular discernible pattern in which they will move alternately clockwise and counter clockwise. The instability of the flow arises as the shear layer vortices shed from both the top and the bottom surfaces interact with one another, known as Karman vortex street. Figure 2-3 shows the von Karman vortex street from a cylindrical body using dye visualization. Figure 2-3: von Karman Vortex Street. Reproduced from 1

29 HAPTER 2: LITERATURE REVIEW AND THEORY A physical description of the mechanics of the vortex formation and vortex shedding has been given by Gerrard [16], described using Figure 2-4. When the vortex from the top surface that acts in the clockwise direction becomes stronger and larger, it will draw the opposing vortex from the bottom surface across the wake in the anti-clockwise direction. Some of this flow is drawn into the growing vortex via flow path a and some across the wake into the developing shear layer via flow path b. Flow path c remains in the near wake and is suppressed half a period later. The emergence of the opposing vortex will suppress the vorticity on top of the surface and thus the vortex on the top surface is shed and moves downstream of the wake. At this instant, a new vortex will form on the top surface and now the vortex on the bottom surface will grow stronger and larger and then draws the top vortex across the wake. This will lead to the shedding of the bottom surface vortex. The cycle continues and the shedding will take place alternately as long as there is flow across the body. Figure 2-4: Gerrard s Vortex Formation Model [16] The first evidence of vortex shedding can be traced back to the ancient Greeks times where the tensioned string vibrating in the wind was described as the Aeolian Tones. It was only in 1878 that Strouhal made the first scientific investigation of a vibrating wire in air and inferred that the frequency of the Aeolian tone varied with the diameter and velocity of the relative motion [35]. He also found that resonance will occur when the Aeolian tones coincided with one of the natural tones of the wire where the sound was greatly reinforced. Lord Rayleigh later in 1896 defined a non-dimensional quantity that is known as Strouhal number. The vortex shedding frequency for a stationary cylinder, f can be described by a relationship between current velocity, U and the cylinder diameter, D given by Equation 2-4. f D St Equation 2-4 U 11

30 HAPTER 2: LITERATURE REVIEW AND THEORY This is known as the Strouhal relationship. The Strouhal number as a function of Reynolds number for a stationary cylinder is given in Figure 2-5 [32]. Over a wide range of Reynolds number, the Strouhal number is approximately.2. Figure 2-5: Strouhal Number, St. Figure by Norberg [32] If the vortex shedding frequency matches the resonance frequency of the structure, the structure will begin to resonate and its movement can become self-sustaining eventually leading to structural failure. A classic example of catastrophe due to resonance is the collapse of Tacoma Narrows Bridge, a suspension bridge in Washington in 194 [44]. A stationary cylinder will also experience forces due to the fluid-structure interaction namely the in-line and the transverse forces. Throughout this thesis and the following convention, the in-line (drag) force is defined as the force in the direction of the flow path and the transverse (lift) force is the force perpendicular to the flow path. In most of the research involving VIV, the fluctuating transverse force is of interest. For a stationary cylinder, the time-averaged transverse coefficient is zero due to the symmetry of the body and the flow. Figure 2-6 shows the variation of RMS of fluctuating transverse coefficients with Reynolds number. 12

31 HAPTER 2: LITERATURE REVIEW AND THEORY Figure 2-6: RMS Transverse Force oefficient, _. Figure by Norberg [32] L rms 2.2 Flow Regime Examination of the flow field around a stationary cylinder in steady uniform flow reveals several distinct flow regimes as shown in Figure 2-7 [45]. These regimes are characterized by the flow character based on vortex creation downstream of the body. No flow separation is observed for Re 5. The first separation occurs when Reynolds number becomes 5. In the range of 5 Re 4, a symmetric pair of vortices is present and above this threshold the flow becomes unstable and shedding begins to occur. While in the Reynolds number range of 4 Re 2 the vortex street is still in the laminar regime, above this range the transition to turbulence is observed in the wake. At Re 3, the wake is completely turbulent. The understanding of the different flow regime with Reynolds number is crucial to appreciate the level of turbulence involved and hence to understand the complexity of the fluid-structure interaction. 13

32 HAPTER 2: LITERATURE REVIEW AND THEORY Figure 2-7: haracterization of the different Flow Regimes. Figure by Sumer and Fredsøe [45] 14

33 HAPTER 2: LITERATURE REVIEW AND THEORY 2.3 Oscillating ircular ylinder In the previous section an introduction to the fundamentals of a stationary cylinder was presented. The dynamics of an illating circular body was found to vary from a fixed body due to the interaction between the moving body and the flowing fluid. This section addresses the governing equation for an illating rigid circular cylinder and also the wake dynamics of an illating body in free stream to provide better understanding of the characteristics of an illating structure pertinent during the event of VIV Equation of Motion The canonical arrangement for the study of VIV has been the elastically-mounted rigid circular cylinder restricted to only 1-DOF motion i.e. transverse direction. In investigating the VIV phenomenon, experiment has been conducted with a freely illating cylinder with controlled flow velocity, mass, damping and stiffness while the displacement and the fluid forces acting on the body are measured. However, in the illating cylinder case, the body is prescribed with certain displacement of known amplitude and frequency and the fluid forces are measured. The system shown in Figure 2-8 is a classical forced vibration free-body diagram of mass, damper and spring system with fluid supplying the forcing term. Such a system can be described by Equation 2-5. my y k y Equation 2-5 where y, y, and y are the displacement, velocity, and acceleration respectively in the transverse direction, m is the mass including added mass, is the damping and k is the stiffness of the structure. The fluid forcing term, F L is the transverse component of the fluid force contributing to the motion. One can rewrite Equation 2-5 as: m 1 2 F L 2 y y k y L DLU Equation 2-6 Equation 2-6 presents the basic governing equation involved in forced vibration phenomenon of a cylindrical structure. 15

34 HAPTER 2: LITERATURE REVIEW AND THEORY Figure 2-8: Mass, Damper and Spring System of 1-DOF Motion Oscillating ylinder Wake Structure For a stationary cylinder, the wake structure downstream of the bluff body will assume the von Karman vortex street structure with Re 4 as described in Section The opposite signed vortices will be shed alternately from the top and bottom of the body into the wake and will be dissipated away. In an illating cylinder phenomenon, the wake structure will no longer conform to the von Karman type structure but with various other structures. As the illating cylinder imposes its frequency and amplitude of the wake during lock-in, there is an increase in transverse force compared to the stationary cylinder. Williamson and Roshko [51] carried out a detailed flow visualization study of a forced illation cylinder with various amplitudes (up to 5D) and frequencies. They have found several vortex synchronization regions with respect to the amplitudes and frequencies the cylinder illates at deduced from aluminium particle patterns. They presented a map of the amplitude and frequency space for the forced illation of a cylinder with respective vortex structure for a range of Reynolds number they investigated i.e. 3 Re 1. The wake structures are categorized based on number of vortices shed per cycle of illation denoted by S (single) and P (pair) type of vortices. In 2S mode, a single vortex is formed in each half-cycle and in 2P mode two pairs of vortices are shed per cycle. The P+S mode is an asymmetric form of 2P mode where the second pair in the cycle is replaced by a single vortex. The map of Williamson and Roshko is presented in Figure 2-9 and a detailed view of the 2S, 2P and P+S region is detailed in Figure

35 HAPTER 2: LITERATURE REVIEW AND THEORY Figure 2-9: Williamson-Roshko Map of Wake Structures for illating cylinder. Figure from Williamson and Roshko [51] 17

36 HAPTER 2: LITERATURE REVIEW AND THEORY Figure 2-1: Detailed Williamson-Roshko map for wake structures for illating cylinder. Figure from Williamson and Roshko [51] 2.4 Methods of VIV Investigation This section summarizes the current methods used to investigate the phenomenon of VIV. A detailed insight into the methods applied in studying VIV is beyond the scope of the present work. Bearman [7], Sarpkaya [37] and Williamson and Govardhan [5] among the researchers that reviewed the research carried out for the past decades over which several methods have been adopted. The sole purpose for the study of VIV is to increase the understanding of the phenomenon to enable better prediction of the response under various parameter spaces and then mitigate or control the effects. The methods of investigating VIV can be classified into four broad categories as follows: 18

37 HAPTER 2: LITERATURE REVIEW AND THEORY Free Vibration Experiments; Forced Vibration Experiments; Numerical Simulations; and Semi-Empirical Modeling. The review of the methods available will include the early work that has been conducted in this area. The literature is rich in various contributions from different researchers in the field of quantifying the response of a circular cylindrical structure due to VIV. A review of the literature discussed below will focus in the area of forced vibration experiments and numerical simulations pertinent to the study presented herein. Obviously there is a wealth of resources related to the above mentioned scope. However, the research materials presented herein are among the contribution that has made the highlight in the field of using forced vibration method to predict VIV Free Vibration Experiments Free vibration VIV experiments have been carried out extensively over the past decades. Sarpkaya [37] categorized the free vibration experiments into two groups based on how the eigen frequency of the system is generated as follows: Rigid cylinder tests. The test cylinder is supported by springs in the direction the cylinder illates and the eigen frequency is controlled by the spring stiffness and mass of the illating parts. The cylinder is allowed to illate either in transverse or in-line direction or may be free to illate in both directions. Flexible beam tests. The eigen frequencies are controlled by the mass, bending and axial stiffnesses and length of the beam. Figure 2-11 shows the experimental set-up used by Vikestad [48] for a freely illating cylinder in the transverse direction with a spring mounted set-up. The mass, stiffness and damping would be fixed while varying the current speed to obtain results over a range of reduced velocities. The results extracted from this type of experiment would typically be the response amplitude i.e. amplitude and frequency of illation, transverse and in-line forces and frequencies. Flexible beam tests are often carried out as scaled models of real slender marine structures. With its slender characteristic, the structure has more than one illation amplitudes over the length of the structure. This gives rise to a few different mode shapes. It is expected that the maximum response amplitude for a flexible beam is higher than the rigid cylinder tests. 19

38 HAPTER 2: LITERATURE REVIEW AND THEORY Figure 2-11: Free Vibration Experimental Set-up by Vikestad [48] Forced Vibration Experiments Forced vibration experiment has been adopted as one method to study VIV where the cylinder is given a prescribed motion often harmonically. ontradictory to free vibration experiment where the mass, stiffness and damping of the system govern the response of the structure, these three parameters are immaterial in forced vibration. In simple terms, forced vibration is a simplification of free vibration experiments where one can have better control of the parameters. The ultimate aim is to quantify the fluid forces acting on the cylinder under the prescribed motion. The majority of the forced vibrations experiments conducted aimed at finding the relationship between forced and free vibrations and hence uses the forced vibration to predict VIV. However, there are concerns as to the limitations of forced vibration in the prediction of free VIV problem, where a free VIV is driven internally by the wake at an average frequency and the former is governed externally at an exact frequency and given amplitude ratio ( A/ D ). In the words of Sarpkaya [37], It should be the ultimate objective of VIV research to predict to the extent possible, the kinematics and dynamics of self-excited vibrations from forced vibration (physical/numerical) experiments and equally important the dynamics of forced illations from the physical/numerical experiments with selfexcited illations. Forced illations can be performed in either a towing tank or current tank. A motion generation system is required to achieve the desired illation. This type of experiment is often conducted for a rigid cylinder to avoid any interference from the structural eigen frequencies. 2

39 HAPTER 2: LITERATURE REVIEW AND THEORY The combination of forced and free vibrations has also been conducted to a limited extent. Moe and Wu [27] used an experimental set-up where the motions were forced in transverse direction and free in in-line direction. Vikestad [48] in his experiment used an electric motor to excite additional frequencies in a free illations experiment in order to measure damping of the system. Early Work To the best of the author s knowledge, Bishop and Hassan were among the first to research into the response of a cylinder subjected to forced vibration in uniform flow. They produced a pair of papers published in 1964 investigating the behaviour of transverse and in-line forces of a stationary cylinder in a flowing fluid [8] and an illating cylinder in a flowing fluid [9]. Focus will be given to the latter paper which is more relevant to the current study. They used a 2.54cm diameter cylinder suspended horizontally in a water channel 22.86cm wide and 4.7cm deep. The cylinder was made to illate via a means of scotch yoke mechanism to attain the desired frequency range of the cylinder illation of.5hz to 9.Hz. The Reynolds number investigated was in the range of 36 Re 11. The forces acting were measured by resistance wirestrain gauges. Figure 2-12 shows the set-up of forced vibration used by Bishop and Hassan [8]. Figure 2-12: Forced Vibration Experimental Set-up by Bishop and Hassan [8] 21

40 HAPTER 2: LITERATURE REVIEW AND THEORY They reported that when the driving frequency of the cylinder approaches the Strouhal frequency, f the forces become synchronized where the system illates at the imposed frequency of the cylinder, f deviating from the Strouhal relationship. This synchronization occurs over a range of frequencies. Within this synchronization range, the phase and amplitude of the transverse and in-line forces change with variation of frequency. The results obtained suggested the existence of wake illator phenomenon where the forces acting on the circular cylinder was imposed by what they termed as non-linear self-excited fluid illator. They also observed hysteresis where the driving frequency jump occurs with change in frequency and also characteristics of frequency demultiplication by which the forces are synchronized when the driving frequency is in the region of the integral multiple of the Strouhal frequency. Their work gained vast interest from other researchers especially in later years in which their data was used as benchmark for other work. Figure 2-13 shows the response diagram of transverse forces for different Reynolds number at amplitude ratio of.3. The points fixed by the dotted lines represent conditions for stationary cylinder. The figure shows that with the change in Reynolds number, a new wake with different natural frequency is obtained. However, the data for the transverse and in-line forces are given in arbitrary units making it rather difficult for quantitative comparison. In the effort to correct the magnitudes of the transverse force coefficient from the inertia component due to the illation, they subtracted the inertial transverse force measured in still water from the total measured force. This method is questionable as it assumes that the variation of added mass does not depend on the flow velocity which the later researchers have proven otherwise. Figure 2-13: Lift Force Response. From Bishop and Hassan [9] 22

41 HAPTER 2: LITERATURE REVIEW AND THEORY Since the early work by Bishop and Hassan, several researchers have used forced vibration method to investigate VIV. In 1978, Sarpkaya [38] performed a forced vibration experiment in an open-return water tunnel to obtain the force-transfer coefficients for the component of the transverse force that is in-phase and out-of-phase with cylinder illation. The Reynolds number investigated ranged from 7 to 11,. The force coefficients were then used to predict the amplitude response of a freely vibrating cylinder in uniform flow via a differential equation solved numerically. The author found good agreement between the predicted values and experimental data of Griffin and Koopman [19]. The much referred literature that reports the vortex flow pattern for different amplitudes and frequencies in forced vibration experiment by Williamson and Roshko [51] has been described in Section Another highlight in the forced vibration finding was made by Gopalkrishnan [17]. The experiment conducted consists of 6 frequencies (.5 / f. 35 ) for 6 different amplitude ratios (.15 A / D 1. 2 ) for a cylinder forced to vibrate in transverse direction at Reynolds number of 1,. The author plotted the contour plots of the transverse force coefficient in-phase with velocity (excitation force) and also the added mass coefficients deduced from the force in-phase with acceleration as shown in Figure The excitation force coefficient gives information of the energy transfer between the fluid and the cylinder. The positive coefficient indicates excitation while a negative coefficient indicates damping. The author pointed out the role of the transverse force phase angle in causing the amplitude-limited nature of VIV. The phase angle changes abruptly as the cylinder takes control of the illating system when it reaches a particular critical frequency. The work has gained considerable attention and the data presented has been the benchmark of many studies that followed. f 23

42 HAPTER 2: LITERATURE REVIEW AND THEORY (a) Excitation oefficient contour (b) Added Mass oefficient contour Figure 2-14: Excitation oefficient and Added Mass oefficient ontours by Gopalkrishnan [17] 24

43 HAPTER 2: LITERATURE REVIEW AND THEORY Dahl [13] recently developed a large matrix of 2-DOF forced motions of a cylinder to identify the force coefficients associated with the different motions of the cylinder. The database consists of 234 test runs performed with varied A/ D in the in-line ( A / D. 75 ) and transverse (.25 A / D 1. 5 ) directions with 4.5 V 8. The author remarked that combined in-line and transverse motions enhance vortex induced vibrations, causing large third harmonic forces. The transverse force in particularly consists of large amplitude third harmonic components of lift in a 2-DOF experiment. The database shows that in the region for which the free vibrations may occur, third harmonic forces increase almost linearly as a function of in-line motion. Another recent research in forced vibration that is worthy of mention in this literature review is the work of Morse and Williamson [3]. The authors made extensive measurements of fluid forces on a cylinder controlled to illate in a transverse direction in uniform flow at Reynolds 4. The test matrix consists of 568 runs ranging from A/ D 1. 6 and 16 where is UT / D. This is by far the most complete work on forced vibration experiments. The authors mapped the vortex formation region and found remarkably similar boundaries separating different vortex shedding modes mapped by Williamson and Roshko [51]. The authors also found a new characteristic that is the existence of a region where two vortex formation regimes overlap, termed the 2P O mode where the wake can intermittently change between 2P and 2P O. This is made possible with their high resolution data. The authors later used these data to predict the response of free vibration i.e. elastically mounted cylinder [31]. They solved the combine amplitude and frequency equations proposed by Khalak and Williamson [25] and plotted the contour of the predicted free vibrations response (amplitude and frequency) for a particular value of combined mass damping, * ( m A ) ζ. By matching the experimental parameters (mass ratio, damping ratio, normalized velocity) the authors found the agreement between predicted and measured free vibrations responses by Govardhan and Williamson [18] is much closer than has been found in previous studies. This is only made possible because of the high resolution of their force data and the careful matching of the experimental arrangement between the controlled and free vibration cases. The authors then introduced the concept of energy portrait, a plot of energy transfer into the body and the energy dissipated by damping which enables one to identify the stability of the predicted amplitude response solution. Figure 2-15 shows the contours of the force coefficients in-phase with velocity, sin() and the force coefficients in-phase with acceleration cos( ) and the Y boundaries between the different vortex modes. Y r 25

44 HAPTER 2: LITERATURE REVIEW AND THEORY (a) Force oefficient In-Phase with Velocity (b) Force oefficient In-Phase with Acceleration Figure 2-15: ontour Plots of Force oefficients and Boundaries of different Vortex Modes from Morse and Williamson [3] 26

45 HAPTER 2: LITERATURE REVIEW AND THEORY Numerical Simulations With the advent of computing power, numerical simulations have received considerable attention from researchers for the study of VIV owing to the convenience of modeling and away from laboratory environment. Over the years, many numerical models have been developed which centred around solving the time-dependent Navier-Stokes equation to represent the physical phenomenon of the fluid-structure interaction. There have been primarily three different computational approaches in the open literature towards describing the flow field and forces for this problem, namely the Reynolds- Averaged Navier Stokes (RANS) such as k and k models, Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS) [37]. The RANS method uses a set of RANS equation to model the turbulent flows whereas in the LES, the large, energy-carrying eddies are directly computed or resolved and the remaining small eddies are modelled. The DNS method numerically solves the full Navier-Stokes equations and no modelling of the flow is required. DNS requires a vast computing power and time. RANS is the most widely used approach for calculating flows due to its reasonable computational resource requirement. Although DNS and LES provide a much deeper insight in the wake-boundary-layer interaction mechanism compared to RANS simulation, as was noted by Sarpkaya [37], RANS codes are more robust, less time consuming and have been applied as practical design analysis tools with reliable solutions. It is worth noting that Reynolds-averaging procedure erases the random disturbances in RANS calculations, so that the fluid forces and the body response tend to be quite repeatable in the lock-in region. Even though there are arguments that RANS simulation results do not agree with those from experiments, careful comparison between the numerical results and the experimental data may help in assessing the effects of random disturbance in the time series of forces and displacement in experiments and those in DNS/LES Vortex Switching Phenomenon The vortex switching phenomenon due to the dynamics of the wake has been observed on several numerical studies conducted. The following paragraphs cite a few of the research in numerical studies and the insight they provide into cylindrical body illations. Blackburn and Henderson [1] studied the flow dynamics of a 2-dimesional flow past a transversely illated cylinder at Re 5 and made the observation of vortex switching i.e. the abrupt alteration of the phase of vortex formation with respect to cylinder motion. The authors investigated the effects of varying the frequency ratio, f / f where f is the vibrating frequency and f is the vortex shedding frequency 27

46 HAPTER 2: LITERATURE REVIEW AND THEORY of a stationary cylinder. The authors suggested that the switching occurs due to the change in sign of mechanical energy transfer between the body and the fluid. The authors categorized the flow regimes into three aperiodic regimes namely the quasiperiodic ( f / f. 77 ), weakly chaotic (.95 / f. 95 ) and chaotic regimes ( / f 1. 15). Figure 2-16 shows the instantaneous vorticity contours over half a f motion cycle for a cylinder illating at f / f. 875 (Figure 2-16 (a)) and f / f.975 (Figure 2-16 (b)) for A / D. 25. At the same cylinder position the shedding of the vortices take place on the different side of the cylinder suggesting the phenomenon of vortex switching. f (a) / f. 875 f (b) / f. 975 f Figure 2-16: Instantaneous Vorticity ontour showing Vortex Switching [1] In a pair of papers ([3] & [4]), Anagnostopoulos presented a numerical study to investigate a flow past a transversely excited cylinder and also observed the vortex switching phenomenon. It was remarked that the switching is due to the change in phase angle between the transverse force and cylinder displacement which is strongly dependent on the parameters of the cylinder excitation. The switch in timing is accompanied by instability in the near-wake rendering the wake aperiodic. The author then determined the lock-in zone using the fluid velocity traces. Guilmineau and Queutey [21] investigated numerically using an O-type structured grid the forced illation of circular cylinder in transverse and also in in-line directions. Both studies have been conducted at low Reynolds number i.e. Re 185and Re 1 for the transverse and in-line respectively. For the transverse illation studies, they observed the phenomenon of vortex switching from one side of the cylinder to the other suggesting the development of a high degree of concentration of vorticity in the wake near the cylinder. 28

47 HAPTER 2: LITERATURE REVIEW AND THEORY Semi-Empirical Models Another contributing effort that has been made towards the understanding of VIV is the development of semi-empirical models. The semi-empirical models of structures undergoing vortex-induced vibration can be classified into three main branches being the wake-illator coupled model, the single degree of freedom (1-DOF) model and force decomposition model. The wake-illator model is self-exciting and self-limiting and has been used to describe both rigid and elastic cylinders. The model assumes that the flow is 2-dimensional and hence is only limited to moderate to large response amplitude. Often the objective is to obtain the equations of the cylinder illator and the fluid illator by independent means and then use them to predict the response of combined fluidstructure systems. Bishop and Hassan [8] are credited with first suggesting the idea of using van der Pol type illator to represent the time-varying forces on a cylinder due to vortex shedding. Hartlen and urrie [22] formulated the most noteworthy of the illator models and subsequently Skop and Griffin [42] developed a model to resolve the inadequacies of the Hartlen and urrie s model. 1-DOF model uses a single ordinary differential equation to describe the behavior of the cylinder illator. The effects of the flowing fluid are characterized as the fluid damping and stiffness. These parameters are in turn dependent on illation amplitude, reduced velocity and Reynolds number. Basu and Vickery [34] and Goswami et al. [23] used these models to predict the structural response. A force decomposition model was first introduced by Sarpkaya [38]. He decomposed the transverse force on an elastically mounted rigid cylinder into a fluid force in-phase and out-of-phase with the cylinder displacement. He showed through a parametric study that the maximum response of the cylinder is governed by the stability parameter, S G for values larger than unity. 29

48 HAPTER 3: DESRIPTION OF EXPERIMENT hapter 3 3. Description of Experiment This section gives a detailed description of the experimental setup and the apparatus used to conduct the forced vibration experiments. The purpose of the experiment was to quantify the fluid force coefficients associated with prescribed illation of the cylinder at certain amplitude and frequency combinations. The experiments were conducted using a purpose-built forced vibration rig mounted onto the towing carriage facilities. A cylinder was attached vertically to the rig while being towed by the carriage and was prescribed a fixed amplitude and frequency in the transverse direction of the tow. The motion of the cylinder was measured by the Linear Variable Displacement Transducer (LVDT) that was attached to the rig. Strain gauges were used to measure the transverse and in-line forces acting on the cylinder. 3.1 Towing arriage Facilities The experiments were conducted in the towing carriage facilities at the Hydraulic Laboratory, The University of Western Australia. The tank is 55m long, 1.25m wide and has a maximum depth of 1.5m, with a usable track length of 4m. The aim of the carriage is to tow the cylinder through still water to simulate steady, uniform current condition. The tank has a pre-installed towing carriage mechanism consisting of an electric motor (ASEA model, with 3.9kW and 152rpm) with continuous cable drive, and fitted with a nitrogen gas braking system to provide enough braking power for the carriage to stop. The velocity of the carriage is controlled through the use of a variable speed controller with maximum run speed of 1.5m/s. The speed of the carriage can either be controlled using a set speed dial or a variable speed dial. The set speed dial has been set with a fixed velocity for each dial sequencing from 2-6 which varies the velocity of the carriage from.2m/s,.5m/s,.8m/s, 1.1m/s and 1.5m/s respectively. Setting the set speed dial to 1 enables the user to use the variable speed dial to achieve a speed at a finer resolution that is not provided by the set speed dial. However, the use of the variable speed dial to achieve the desired speed is subjected to human error due to the difficulty to control the exact position of the dial to achieve the same target speed for every run. This is undesirable as inconsistency of the tow speed is inherent between different experimental runs. An external tachometer is pre-installed to indicate the speed of the carriage during its motion by measuring the rotational speed of the wheel 3

49 HAPTER 3: DESRIPTION OF EXPERIMENT mobilizing the carriage. A digital distance recorder is utilized to measure the distance travelled by the carriage. Table 3-1 lists the devices attached to the carriage. Device Functionality Units Electric Motor arriage driver - Digital Speed Translator Indicate towing speed mm/s Speed Dial Provide speed control over the carriage - Digital Distance Recorder Indicate the distance travelled by the carriage mm Table 3-1: Towing arriage Devices The experiments were conducted by using the set speed dial to ensure accuracy and consistency of the towing speed for all experimental runs. The towing carriage was mobilized by an electric motor with continuous cable drive. The electric motor is able to drive the carriage in both directions (i.e. forward and reverse motion) along the tank. For safety purpose, two stoppers were installed at each side of the tank to prevent the carriage from crashing onto the two ends of the tank. Figure 3-1 shows the plan view schematic of the towing carriage tank. 1.25m Figure 3-1: Schematic of Towing arriage Tank (Plan View) 3.2 Developing a Forced Oscillations Experimental Rig A substantial amount of time and effort has been spent to develop the forced illation setup to accommodate the towing facilities and also at the same time to satisfy the test matrix i.e. amplitude and frequency ranges. Several design iterations have been undertaken to arrive at the optimal design for the apparatus to conduct the forced illation experiments. Several ideas were put forward to find an illating mechanism 31

50 HAPTER 3: DESRIPTION OF EXPERIMENT that would output a sinusoidal motion and could be fitted within the towing carriage facilities. The initial design that was put forward adopted the concept of a crank-slider mechanism that consisted of two arms moving in one plane which converts the linear motion to rotational motion. alculations showed that in order to achieve the desired amplitude of illation, the mechanism required a long lever arm otherwise the motion would not be sinusoidal. This idea was later abandoned as it posed a problem as the set-up needed to fit within the tank and the length of the lever arm required exceeded the tank dimension. The current setup utilizes the scotch yoke mechanism as shown in Figure 3-2. The scotch yoke mechanism is a linear actuator that converts rotational motions into simple harmonic motions. The mechanism would yield a sinusoidal motion amplitude output independent of the rotating arm. The kinematics of scotch yoke can be expressed by the equation of motion given by Equation 3-1. y t) Asin ( ) Equation 3-1 ( t where y is the linear displacement, A is the amplitude of illation, is the angular illation frequency, t is time and is the initial phase of the illation. The linear displacement, y follows the sinusoidal pattern given by constant rotation,. Differentiating Equation 3-1 gives: y t) A cos ( t ) Equation 3-2 ( 2 y ( t) A sin ( t ) Equation 3-3 Equation 3-2 and Equation 3-3 are essential to estimate the drag, friction and inertial forces involved during the illation in order to determine the motor sizing to drive the mechanism. 32

51 HAPTER 3: DESRIPTION OF EXPERIMENT y y Figure 3-2: Scotch Yoke Mechanism The setup was constructed on a steel plate which was then mounted onto the towing carriage facilities. The motion was achieved by the rotating shaft which then transfers the motion to the T-slider as an illatory motion. The cylinder was attached vertically to the T-slider via a support beam (with a plate attached to it) that illates together with the slider. In order to minimize friction during sliding to ensure smooth motion, the slide pack with ball bearing has been adopted. The shape of the motion of the slider is a pure sine wave over time given a constant rotational speed of the shaft. Figure 3-3 presents the schematic of the set-up. Rotating shaft Motor T-slider Support beam Test cylinder Figure 3-3: Forced Oscillation Rig Model Schematic 33

52 HAPTER 3: DESRIPTION OF EXPERIMENT The rotation of the shaft was driven by a motor with constant current input to ensure constant rotational speed. The frequency of the motion is controlled by changing the input voltage of the D motor through the implementation of a control box. The speed control has a resolution of up to.1rpm enabling a very fine control of the speed of the motor and hence the control of the frequency of illation. Two motors were used rated at 5N.m each. In addition, an externally mounted tachometer was equipped to the experimental set-up to record the frequency of the illation for comparison with the LVDT signal which recorded the motion. The desired amplitude of the motion was achieved by deciding the length of the rotating shafts. Multiple shafts with calibrated holes were utilized to define the desired amplitude of motion. These shafts were fabricated using a N machine with dimensions precise to less than.1mm. This gives very accurate output of the desired amplitude of the motion. Figure 3-4 shows the actual experimental rig and the associated components involved. Rotating Shaft LVDT Strain gauges D Motor Test cylinder Figure 3-4: Experimental Rig 34

53 HAPTER 3: DESRIPTION OF EXPERIMENT 3.3 Experimental ondition A smooth cylinder was used in this experiment and Table 3-2 presents the relevant experimental parameters. Parameter Symbol Value ylinder diameter D.6m ylinder length L.6m Aspect ratio L / D 1 ylinder mass m.593kg ylinder mass (incl. enclosed fluid) Support beam mass (with plate attached) m m 2.29kg w m b Table 3-2: Experimental Parameters 1.168kg To ensure good quality of the output data, detailed attention to boundary conditions have been given for the set-up of the apparatus. The cylinder was located at the centre of the tank in order to minimize any wall effects. The bottom of the cylinder was kept 3mm away from the bottom of the tank to encourage 2-dimensional flow. No end plate was used, such that the bottom of the tank acts as the finite boundary for end condition. The top part of the cylinder was kept about 3.5D away from the water surface to eliminate free surface effects. Figure 3-5 shows the setup of the experimental rig in the towing tank. 3.5D Test cylinder 6mm x 6mm.625m 3 mm Figure 3-5: Experimental Setup in Towing Tank 35

54 HAPTER 3: DESRIPTION OF EXPERIMENT The cylinder system was equipped with bi-axial strain gauges to measure both the transverse and in-line forces acting on the cylinder. The strain gauges were attached to the aluminium support beam connecting the cylinder and the sliding block as shown in Figure 3-4. The motion of the cylinder driven by the scotch yoke mechanism was measured by the LVDT with one end connected to a fix support and the other end connected to the T-slider which illates at the same motion as the cylinder. The output data from the strain gauges and the LVDT were passed through an amplifier before being collected using Agilent Data Acquisition system via a junction box. The amplifier gain used was 5 for the transverse and in-line force channels and 1 for the displacement i.e. LVDT channel. ollection of the experimental data was accomplished by digital sampling at 1Hz of the amplified analogue signal across three data acquisition channels. Figure 3-6 shows the data collection sequence. The gathered data will be post-processed as explained later in Section 4.3. Figure 3-6: Data ollection Flow hart alibration of Instruments alibration of the strain gauges and LVDT were performed prior to the experiments to obtain a calibration factor for results processing. The strain gauges (attached to the supporting beam) were calibrated by fixing one end of the beam and applying a known weight to the other end of the rod. The weights were first measured using a digital scale with.1g of accuracy. The voltage change for the different weight applied were recorded and plotted to obtain the calibration factor. The weights were increased and then decreased for both the transverse and in-line strain gauges to confirm consistency and accuracy of the calibration process. The LVDT was calibrated by connecting it to the apparatus and activating the illations. As the rotating shafts were fabricated to high precision, by running the apparatus with different amplitude, the voltage recorded during the illation was used for calibration where the maximum voltage captured represents the amplitude of illation. 36

55 HAPTER 3: DESRIPTION OF EXPERIMENT Table 3-3 shows the description of the instrument and their respective calibration factors. Instrument Description alibration Factor Transverse strain gauge In-line strain gauge Four gauge Wheatstone bridge bending beam configuration Four gauge Wheatstone bridge bending beam configuration.567 V/N.122 V/N LVDT Electrical stroke of 5mm to 15mm 25.5 V/N Table 3-3: alibration Factors 3.4 onduct of the Experiment The conduct of the experiment has been planned in such a way that it is timely, consistent, repeatable and the quality of the output can be ensured. The following steps represent the procedures followed during the conduct of the experiments. 1. heck and record settings of all instruments. 2. Ensure the output file for each output channels is correctly named. 3. Perform zero setting for all channels. 4. Begin zero data logging for 1s prior to start of the carriage. 5. Start the towing carriage and log data for 5s to allow carriage to accelerate to the specified tow velocity. 6. Start the motor for cylinder illation and log data for the period of constant velocity, illation frequency and amplitude to the end of the towing tank. 7. Turn off the motor before the carriage reaches the end of the working distance of the tank. Data logging for 5s before the tow carriage slows down to a complete halt. 8. Data logging for additional 1s. Terminate data acquisition system. 9. Return carriage to starting position for subsequent run. The run was only conducted in one direction of the towing i.e. forward direction to maintain consistency of the results collected even though the carriage is capable of forward and reverse motion. 37

56 HAPTER 3: DESRIPTION OF EXPERIMENT Approximately 33 values of non-dimensional illating frequency ranging from.4 f / f 2.2 where f is the illation frequency and f is the Strouhal frequency for a stationary cylinder were selected to cover the natural Strouhal number. The frequency was controlled by the speed of the rotating shaft based on the input voltage controlled by the control box. In addition, a tachometer was also used to record the rpm of the rotating shaft for comparison purposes. Each of the 33 illating frequency was tested at 9 non-dimensional amplitudes ranging from.2 A / D 1. yielding the actual illation amplitude from 1.2cm to 6.cm. All runs were investigated at a constant carriage speed of.2m/s corresponding to a Reynolds number of 12,. This towing speed was selected due to the readily set speed dial in the towing tank hence avoiding the use of a variable speed dial which is highly susceptible to inconsistencies due to human error. As rightly pointed out by Gopalkrishnan [17], the speed was also selected to give the best compromise between force measurement requirements and experimental accuracy. A larger velocity (or Reynolds number) will result in a larger force hence can be easily detected and measured by the instruments. A small velocity will lead to longer experimental run times. Before and after each run for each amplitude ratio was conducted, the fixed cylinder case was investigated as a control experiment to benchmark against other experimental data. This is important to provide confidence in the experimental results obtained. 3.5 Quality ontrol The following few issues have been addressed to ensure the quality and reliability of the force measurements for the calculations of hydrodynamic coefficients. The subsequent discussions stem from the fact that the results produced should be valid for an infinite length cylindrical structure in an infinite volume of water End condition effect In order to avoid 3-dimensional effects due to the finite length of the cylinder, the end of the cylinder would need to be sufficiently close to a boundary. Often, end plate is used in experiments to encourage 2-dimensional flows. Morse et al. [29] stated that a sufficiently small gap to diameter ratio i.e. g*<15% is required to correctly capture the response of the body. In the experiment, the cylinder was mounted vertically with the bottom of the tank acting as a finite end plate. The gap between the cylinder and the bottom of the tank was kept at 3mm (at initial position) giving g* of 5% which is deemed sufficient to suppress 3-dimensional effects. However it should be noted that the floor of the tank is not perfectly flat but with variations of approximately ±2mm. Nonetheless, this still satisfies 38

57 HAPTER 3: DESRIPTION OF EXPERIMENT the g*<15% condition specified by Morse et al. [29]. Furthermore, as shall be seen later the Strouhal number obtained for stationary cylinder is in the vicinity of the classical value of.2 in the Reynolds number regime investigated, proving the adequacy of the end condition Free surface effect are has to be taken in order to avoid any free surface effects as the result of cylinder interaction with the free surface. Bishop and Hassan [8] have used the Froude number criterion given in Equation 3-4 below to evaluate the effect of the free surface to the experiment. They stated that for the free surface effects to be negligible, much less than unity. gh min F r has to be U F r Equation 3-4 where hmin is the minimum depth of submergence. In their experiments, Fr was calculated to be.375 which was considered to be sufficiently low. Experiments conducted by Gopalkrishnan [17] and Vikestad [48] have reported Fr of.181 and.29 respectively which was found to be sufficiently low to neglect the effects of free surface interactions. It should be noted that the cylinder and hmin experiments, hmin F r recorded in the above experiments were for a horizontal is defined from the water surface to the cylinder level. In the current is taken to be the distance between the water level and the top cylinder level i.e. 3.5D to account for the worst case scenario. This yields F of.139 and hence the free surface effect is deemed negligible. The support beam connected the cylinder to the illating mechanism penetrating the water and its interface with the water surface creates transverse waves as it illates. The resultant surface waves produced were minimal in the order of 3mm. Therefore, no correction was implemented for the hydrodynamic forces Blockage effect Blockage refers to the effect of force measurement in a finite stream that is different from force measured in an infinite stream. The blockage effect is caused by the presence of the walls alongside the towing tank. Blockage ratio, B is defined as the ratio of the diameter of the cylinder and the width of the tank ( D / w). r 39

58 HAPTER 3: DESRIPTION OF EXPERIMENT Should the effect of the blockage found to be significant, correction factors should be applied to the measured forces. Zdravkovich [52] stated that correction factors are not required for B. 1 for Re 3. In the current experiment, the blockage ratio is.48 and hence no correction factors were applied Residual waves Residual small scale waves were generated as the result of towing and the illation between the cylinder and the fluid. This effect is more prominent especially with higher amplitude motion where the waves created were of relatively larger scale. The fluid needed to be at rest before the subsequent run of the experiment could be conducted in order to maintain the consistency and to avoid interference of the data logged for all runs. Therefore, certain interval of time was allowed ranging from 4 minutes (after lower amplitude run) to 1 minutes (after higher amplitude run) before the subsequent run was conducted. By observation, this could be confirmed when the waves subsided (no more fluctuations in water level) and the data logged from the force channels for the initial 1s zero run recorded a flat signal (no signal fluctuations of any scale) Limitation in towing tank length The towing velocity and the length of the towing tank determine the duration of the data collected. For robust data processing, the number of cycles collected has to be sufficiently large to obtain an accurate time-averaged force coefficient. As transient effects were present at the start and the end of each run, it is important to ensure that the steady state time series contain a sufficient number of cycles to calculate the timeaverage value of the coefficient especially for the low illation frequency run. The effective towing tank length is 4m. With the lowest frequency used being.25hz, this will give approximately 2 number of cycles. This was found to be sufficient for calculating the hydrodynamic coefficients. 4

59 HAPTER 4: DATA ANALYSIS hapter 4 4. Data Analysis The intent of this chapter is to detail the procedures for data processing from the experimental output. The total number of runs conducted is approximately 3. To ensure consistency and maintain effectiveness of the post processing, a MATLAB program was developed to batch-process the data. Signal processing, interpretation of data, force decomposition and quality assurance in the measured data will be described in this chapter. 4.1 Outline of Post Processor The post processor i.e. MATLAB code was developed to aid in the data processing of the large amount of experimental data. It was developed to process the data in a manner such that it processes from the raw data to the output of the desired hydrodynamic coefficient. As such, it is programmed to have the following characteristics: Reads the raw input files for each test case and display the time history of the raw data for the forces and displacement signals to visually analyze the quality of the data; Enable the user to select steady state section of the signals for data processing in a way that the selection of the initial and end time for the post-processing is consistent for the transverse force, in-line force and displacement signals; Filter the signal by using a pre-defined low pass filter criteria; and Results can be presented as an individual case file or a plot of a series of run to observe the varying pattern of coefficients with a fixed parameter e.g. frequency. 4.2 Definition of Direction This section presents the definition of the coordinate system used to define the direction of the towing, illation and force measurements. Figure 4-1 shows a schematic of the definition of the direction of coordinate system. The cylinder is illated perpendicular to the direction of the towing. 41

60 HAPTER 4: DATA ANALYSIS The in-line force measured refers to the force measured in the direction of towing i.e. z- axis and the transverse force measured refers to the force measured in the direction of illation i.e. y-axis. urrent z y x Test cylinder Towing direction Oscillation direction Figure 4-1: Definition of Direction 4.3 Signal Processing This section presents the procedures taken to process the raw data from the output channels. These raw data are then used to derive the hydrodynamic coefficients described in further detail in Section 4.4. There are three channels which give the following output data: Transverse force (measured from transverse strain gauge); In-line force (measured from in-line strain gauge); and Motion of the illation (measured from LVDT). To obtain the time series for the derivation of the hydrodynamic coefficients, the forces and motion signals are processed as per the following procedures: Select the steady state region with the same initial and end time for the motion and forces time series; Apply calibration factor for each signals; 42

61 Volts (V) HAPTER 4: DATA ANALYSIS Zero the signals and perform low pas filtering with cut off frequency of 1. Hz; Differentiate the displacement signal in time domain to obtain velocity time series of the illation; Differentiate the velocity time series (from above) to obtain acceleration time series of the illation; Subtract the inertia force due to acceleration of the cylinder mass from the total transverse force; Perform Fast Fourier Transform (FFT) of the displacement signal to obtain dominant frequency and compare with the reading from the tachometer; alculate the hydrodynamic coefficients and phase difference for each test case; and Plot the results for a set of runs. Figure 4-2 shows the raw data of the displacement, in-line and transverse forces from the output channels. The data shows initial zero period of 15s as described in the conduct of experiment in Section 3.4 before entering into the steady state region where the data was used for post-processing. The offset of the y-axis for the data collected is arbitrary before the zero of the signal being applied Displacement In-line Force Transverse Force Steady state section time (s) Figure 4-2: Measured Forces and Displacement Signals 43

62 HAPTER 4: DATA ANALYSIS Differentiation of Displacement Signal The differentiation in the time domain of the displacement signal to obtain the velocity, y and acceleration, y time series are given below. Velocity time series, yi yi y 1 Equation 4-1 t yi 1 2yi yi 1 Acceleration time series, y Equation t 4.4 Hydrodynamic Force oefficients The purpose of this section is to give the formulations and definitions of the hydrodynamic coefficients used in this thesis. The governing equations of the forces exerted on an illating cylinder are presented and the hydrodynamic force coefficients pertinent to the current study are defined Fluid Forces on ylinder A stationary cylinder subjected to fluid flow will experience illating forces due to vortex shedding (described in detail in Section 2). The transverse force, illate at the frequency of the vortex shedding and the in-line force, at twice the frequency of the shedding. where Fls and Fds F L will F D will illate FL Fls sin(2 ft ) Equation 4-3 FD Fds sin(2 (2 f) t ) Equation 4-4 are the magnitude of the illating Strouhal transverse and in-line forces respectively, f is the vortex shedding frequency i.e. Strouhal frequency and and are the phase angles. When a cylinder is subjected to an external illating force, an additional frequency component exists alongside the Strouhal component. The body will experience forces from both the Strouhal and illating body components. For a body illated harmonically, the transverse and in-line forces may be represented by Equation 3-9 and Equation 4-6 respectively. F L Fls sin(2 ft ) Fl sin(2 ft ) Equation 4-5 FD Fds sin(2 (2 f) t ) Fd sin(2 (2 f) t ) Equation

63 HAPTER 4: DATA ANALYSIS where F l and F d are the magnitude of the illating transverse and in-line forces at the illation frequency of f and 2 f respectively and and are the phase angles Fourier oefficient Analyses The illating force coefficients and phase angle can be determined via Fourier coefficient analyses. A waveform y (t) can be represented by a basic Fourier series given by Equation y ( t) a n1 a n 2nt cos T bn 2nt sin T Equation 4-7 where a, a n and bn are defined as below. a b n n a 1 T T y( t) dt 2 T 2 nt y( t)cos dt T T 2 T 2 nt y( t)sin dt T T Equation 4-8 Equation 4-9 Equation 4-1 Ignoring the Strouhal component as the primary focus is the lock-in response, expanding Equation 4-5 gives: F L F sin ( )cos(2 f t ) F cos( )sin(2 f t ) Equation 4-11 l l and F l sin ( ) can be directly related with a1 and Fl cos ( ) with b 1. The force magnitude, F l and phase angle can be described as: F l Equation a 1 b1 a arctan 1 Equation 4-13 b 1 45

64 HAPTER 4: DATA ANALYSIS Definition of Hydrodynamic Force oefficients The following gives the definition of the hydrodynamic coefficients used in the current scope of study. The following coefficients are defined in order to examine their behaviour to understand the fluid structure interaction involved at different amplitudes and frequencies relevant to a vibrating cylinder. Mean In-line oefficient, Dm The mean in-line coefficient is given by Equation Dm 1 T tt t FD ( t) 1 DLU 2 2 dt Equation 4-14 The mean in-line coefficient is calculated by averaging over a finite number of cycles of the normalized in-line force in the steady state section. While the focus of the research is mainly on the transverse force component, the in-line force provides insight into the behaviour of the mean in-line coefficient with varying illation amplitudes and frequencies, beneficial for the similar behaviour expected for the transverse force components. RMS Transverse Force oefficient, L _ rms A Root Mean Square (RMS) of the transverse force coefficient is given by Equation 3-19 below. L _ rms FL is the mean value of the transverse force and 1 n 2 F 1, F i n L i L Equation DLU 2 F L, i is the transverse force at time step i. The RMS determination is calculated in the steady state region of the time series for n number of time steps. The RMS value gives the statistical measure of the magnitude of the varying transverse force coefficient i.e. in sinusoidal form when the motion varies between positive and negative values. It is linked to the magnitude of the illating force but does not provide any information on illation frequency or phase angle. 46

65 HAPTER 4: DATA ANALYSIS Force In-Phase with Velocity oefficient, lv The force in-phase with cylinder velocity can be obtained by decomposing the total transverse force described in Section 4.5. Equation 3-2 gives the formulation for the force coefficient. lv FL sin ( ) Equation DLU 2 This coefficient suggests the direction of energy transfer between the body and the fluid. A positive value of the coefficient suggests energy transfer from fluid to body (excitation to the cylinder) while a negative value suggests energy transfer from body to fluid (damping). Force In-Phase with Acceleration oefficient, The force in-phase with acceleration coefficient is given below. This coefficient is orthogonal to the lh lv above and determines the inertial added mass force. Positive denoting negative added mass and vice versa. lh lh FL cos ( ) Equation DLU 2 Added Mass oefficient, m The added mass coefficient is defined as: m FL cos ( ) Equation D 2 L A 4 2 where is the angular illation frequency and A is the amplitude of illation. The added mass coefficient is based on the force in-phase with acceleration normalized with respect to the mass of displaced water and acceleration amplitude of the harmonic illation. It represents the mass of the fluid around the body which is accelerated by the movement of the body. In this study, the cycle-averaged added mass coefficient is presented. The characteristics of added mass coefficient is important to appreciate that for a separated time-dependent flows e.g. VIV, the frequency at lock-in cannot remain 47

66 HAPTER 4: DATA ANALYSIS constant throughout the synchronization range as the added mass is a function of time, motion and orientation of the body [37]. Energy, E The mechanical energy transferred between the flowing fluid and the illating cylinder per motion cycle can be written in dimensionless form as below. 2 E 2 U D 2 T yf L dt Equation 4-19 The indication of the sign of the energy transferred provides the same information as lv described previously. The quantity E is positive when work is done on the cylinder and negative when work is done on the fluid. The excitation or damping of the system can be obtained from this quantity by observing the change in sign of the energy transfer. 4.5 Hydrodynamic Force Decomposition This section describes the method used to decompose the hydrodynamic force into a component that is in-phase with velocity and a component that is in-phase with acceleration. When a cylinder is illated or experiences an illatory load, there exists a force components that are in-phase with velocity and acceleration of the motion in the total transverse fluid force measured. This force needs to be decomposed from the total force to study its behavior with varying parameters. In this thesis, Fourier-averaged method has been adopted to decompose the transverse force. Under the assumption of single frequency harmonic motion, the total transverse force can be represented by Equation The motion of the illating cylinder is given by: 2 t y( t) Asin Equation 4-2 T 48

67 HAPTER 4: DATA ANALYSIS The force in-phase with velocity and in-phase with acceleration can be formulated by multiplying the fluid force by y and y respectively and integrating over a whole number of cycles. This gives: T T FL y 1 dt AFl sin T 2 F L y dt T A Fl cos Equation 4-21 Equation 4-22 The derivation of the time-averaged fluid force coefficients in-phase with velocity, and in-phase with acceleration, lh are given below. lv lv Flsin 1 2 U DL 2 T F y L 2 dt T 1 2 A U DL 2 Equation 4-23 lh Flcos 1 2 U DL 2 T F L y 2 dt T A U DL 2 Equation 4-24 The phase angle, of interest is the phase between the body motion and the force. The initial point of integration is taken at zero up-crossing of the displacement signal. oefficients are calculated for all zero up-crossing in the steady state region and averaged over the number of cycles. 49

68 HAPTER 5: NUMERIAL SIMULATION APPROAH AND METHODOLOGY hapter 5 5. Numerical Simulation Approach and Methodology Apart from experimental runs conducted to investigate the fluid-structure interaction of an illating cylinder, numerical approach has also been used to provide further insight into the phenomenon. This chapter provides the approach taken and the methodology used to investigate the flow dynamics associated with uniform flow past an illating cylinder using a 2-dimensional omputational Fluid Dynamics (FD) approach. 5.1 Numerical Model Description The simulations were carried out using the state-of-the-art FD tool, ANSYS FLUENT. The unsteady Reynolds-Averaged Navier-Stokes (RANS) equation was used with pressure based implicit formulation. A 2-dimensional numerical domain with an O-type grid as shown in Figure 5-1 was employed to represent the model and its vicinity. The domain consisted of velocity inlet, outflow and a non-slip wall representing the cylinder. The domain was set to be twenty times the diameter of the cylinder (2D) in the radial direction. This distance would allow the upstream flow to be fully developed by the time it reaches the cylinder and also allows sufficient length for the vortices to be fully captured downstream of the cylinder. The illatory motion of the cylinder was achieved with the use of a moving dynamic mesh with smoothing and local remeshing methods. Velocity Inlet Outflow Non-slip wall Quadrilateral mesh 2D Non-slip wall Triangular mesh Figure 5-1: Description of Numerical Domain and Boundary ondition 5

69 HAPTER 5: NUMERIAL SIMULATION APPROAH AND METHODOLOGY A standard k model was chosen as the turbulence model for the simulation. The standard k model is an empirical model based on model transport equations for the turbulence kinetic energy ( k ) and the specific dissipation rate ( ) that can be described by the following transport equations respectively [5]. k k ku ( ) ( i) k Gk Yk Sk Equation 5-1 t xi x j x j u ( ) ( i ) G Y S Equation 5-2 t xi x j x j where gradients, G k represents the generation of turbulence kinetic energy due to mean velocity G represents the generation of, diffusivity of k and, respectively, due to turbulence, k and represent the effective Y k and Y represent the dissipation of k and S k and S are user-defined source terms. A first-order implicit scheme was used for the unsteady term and second order upwind scheme for turbulence kinetic energy ( k ) and the specific dissipation rate ( ) equations. Quadratic Upstream Interpolation for onvective Kinematics (QUIK) scheme was applied for the momentum equations and Pressure Implicit with Splitting of Operators (PISO) algorithm for velocity-pressure coupling equation in the k solver. A mesh with 32 points in the angular direction and 1 points in the radial direction (32 x 1) was selected in this study which discretized the domain into mesh elements. 5.2 Mesh onsideration for Turbulent Flows Accurate computation of forces for turbulent flows highly depends on the size and quality of mesh around the wall. The numerical results for turbulent flows are more grid dependent than those of laminar flows. Sufficiently fine and high quality mesh enables the complex turbulent flows to be properly resolved especially at the boundary layer to yield high accuracy results. A fine grid was created near the cylinder and gradually coarser in the wake and in the far-field. 51

70 HAPTER 5: NUMERIAL SIMULATION APPROAH AND METHODOLOGY In simulations, the hybrid mesh approach was adopted for this numerical domain where quadrilateral mesh was used at the near wall while triangular mesh was used in the rest of the domain. Quadrilateral mesh was selected at the near wall as it allows better control of the mesh with lower skewness that improves the quality and convergence as well as avoiding numerical errors. Triangular mesh was employed for the rest of the domain due to its conformal nature that can be easily deformed without resulting in numerical error when using moving dynamic mesh. The quality of the mesh also depends on the cell skewness factor. Equation 4-3 gives the definition of skewness factor. Skewness factor e max e min max, Equation e e where max is the largest angle in the cell, min is the smallest angle in the cell and e is the angle for equiangular cell i.e. 6 o for triangular mesh and 9 o for quadrilateral mesh. The skewness factor ranges from (best) to 1 (worst). The worst cell skewness in the model is.586 associated with triangular mesh located approximately 1D distance away from the centre of the cylinder and is deemed acceptable. The worst skewness factor near the wall associated with the quadrilateral mesh is only.7. The use of this hybrid mesh approach allows the optimization of the dynamic mesh method as well as allowing accurate computation of forces near the wall with high quality mesh. The determination of the size of the mesh near the wall depends on flow velocity and hence the Reynolds number. A sufficiently fine mesh is required in order to accurately compute the forces around the cylinder. A finer mesh near the wall is required with higher flow velocity or Reynolds number. As a rule of thumb, the first point of the quadrilateral mesh from the wall was located at 1/Re away from the wall. The adequacy of the size of the mesh can be verified by checking the wall y values where its values should be less than unity around the circumference of the wall for the boundary layer to be sufficiently resolved. Wall y is a non-dimensional value measuring the average velocity of turbulent flows near the wall proportional to the logarithm of the distance of the first cell located near the wall. Equation 4-4 gives the definition of wall y [5]. 52

71 Wall y + HAPTER 5: NUMERIAL SIMULATION APPROAH AND METHODOLOGY u y y Equation 5-4 where u is the friction velocity near wall, y is the distance of the first cell from wall and is the kinematic viscosity of the fluid. Figure 5-2 shows the wall y value around the circumference of the wall for the case of f / f 1. with 32 x 1 domain where m position is the centre of the cylinder. The value of wall y is below 1 indicating sufficient mesh size near the wall for the Reynolds number simulated. Similar observation was made for other cases as well Position (m) Figure 5-2: Wall y value around wall for / f 1. f with 32 x 1 domain 53

72 HAPTER 5: NUMERIAL SIMULATION APPROAH AND METHODOLOGY 5.3 Numerical Simulations Parameters The simulations have been performed at Reynolds number of 12,. To the author s knowledge, most of the computational research in this scope of study has been concentrated at low Reynolds number even though not discounting the facts that few attempts have been made to quantify the flow dynamics at higher flow regime. The simulations were carried out for a range of frequency ratios in the vicinity of lock-in region i.e..7 f / f 1. 4 where f is the frequency at which the cylinder is illated at and f is the frequency of vortex shedding for a stationary cylinder. The amplitude ratio, A/ D has been fixed to A / D. 3. A non-dimensional time step, was set to be.2 where t U / D. The time step used in the numerical simulation was t. 15 to satisfy the former relationship. The adequacy of the time step size used can be confirmed by checking the cell ourant number in the numerical domain. The ourant number is the ratio of a time step size to a cell residence time defined in Equation 5-5 [5]. t FL Equation 5-5 x U cell / where t is the time step size, xcell is the length of the grid cell and U is the fluid velocity. To ensure a sufficient time step size is used, the cell ourant number should be less than unity. Figure 5-3 shows the cell ourant number in the numerical domain for the case of f / f 1. with 32 x 12 domain. Similar observation made for the rest of the cases simulated. It is observed that the cell ourant number in the numerical domain is less than unity and the ourant number at the area of interest i.e. near the wall is well below unity indicating that the time step size used is sufficient for the cases investigated. 54

73 ell ourant Number HAPTER 5: NUMERIAL SIMULATION APPROAH AND METHODOLOGY Far-field elements Outflow Velocity Inlet Wall Position (m) Figure 5-3: ell ourant Number The illating motion of the cylinder has been set to a harmonic motion defined by Equation 5-6. y Asin( t) Equation 5-6 where A is the amplitude of illation, is the angular velocity and t is time. 5.4 Model Validation To ensure robustness of the numerical domain and accuracy of results, a validation process has been undertaken. The numerical model was validated against published literature of Williamson and Roshko [51] for the case of a cylinder illating transversely in a free stream where the authors have proposed the map of vortex synchronization regions. The map of vortex synchronization regions categorized by the different modes of vortices shed proposed by the authors has been described in detail in Section

74 HAPTER 5: NUMERIAL SIMULATION APPROAH AND METHODOLOGY Figure 5-4 illustrates the different regions of vortex modes corresponding to different combinations of amplitude ratio ( A/ D ) and wavelength ratio ( / D ) where UT / D, U is the flow velocity, T is the period of illation and D is cylinder diameter. As part of the validation process, four cases have been selected from the map of vortex synchronization region to reproduce the vortex patterns found by Williamson and Roshko [51] from the numerical model used. Table 5-1 shows the different cases simulated to validate the illating cylinder model against the experimental data mentioned above. The cases simulated have been marked in Figure 5-4 for the relevant regions. ase Amplitude Ratio, A/ D Wavelength Ratio / D Vortex Pattern S P P + 2S P + S Table 5-1: Validation ase Matrix Figure 5-4: Map of Vortex Synchronization Regions for Model Validation 56

75 HAPTER 5: NUMERIAL SIMULATION APPROAH AND METHODOLOGY Figure 5-5: Validation Results showing different Vortex Pattern Figure 5-5 shows the results of the cases simulated to validate the illating model. The vortex patterns produced by the simulation agree well with the patterns produced experimentally as suggested in Figure 5-4 for a given amplitude and frequency. From the four cases validated, it can be concluded that the numerical model qualitatively agrees well with experimental results. 5.5 Mesh onvergence Analysis Mesh convergence analysis has been performed on the forced illation cylinder model. The study was used to verify that the mesh density used was sufficient to give a convergent result. The analysis has been performed with a similar O-type grid numerical domain with 4 points in the angular direction and 18 points in the radial direction (4x18). This discretized the domain into 461 mesh elements. The force coefficients were compared with the 32x1 domain to determine convergence of the mesh density. The time step size of t. 15 and Reynolds number of 12 were used for both models. The case of / f 1. was chosen for the convergence studies. f 57

76 HAPTER 5: NUMERIAL SIMULATION APPROAH AND METHODOLOGY L L x12 domain 4x18 domain time (s) (a) Transverse Force oefficient Profiles for different numerical domain D D x12 domain 4x18 domain time (s) (b) In-line Force oefficient Profiles for different numerical domain Figure 5-6: Mesh Density onvergence Plot omparison 58

77 HAPTER 5: NUMERIAL SIMULATION APPROAH AND METHODOLOGY Figure 5-6 shows the comparison plots for the two mesh densities mentioned above for the transverse, L and in-line, D force coefficients profiles. The plots show excellent convergence for the transverse coefficient and only minimal difference in the profile of the in-line force coefficient given the scale shown. Table 5-2 presents the results obtained for the comparison of the difference in mesh density with the chosen numerical model. The comparison of the results for the two different cases investigated indicates that mesh density convergence has been achieved. Only a maximum of approximately 4% difference was recorded in the variations of the vortex shedding frequency, f v. It suggests that 32 x 1 domain is adequate to investigate the current scope of work within the parameter space used. Mesh Density Number of Elements Dm L _ rms f v 32 x x Table 5-2: Mesh Density onvergence Results 5.6 Applicability and Limitations of 2-Dimensional Modeling It is noted that at Re 12, the wake of the cylinder is 3-dimensional and fully turbulent. The limitation of 2-dimensional calculation is that the influence of the spanwise wake turbulence is omitted. However, when the cylinder is illating, the vortex structure is altered and the flow between adjacent axial segments of the cylinder becomes coupled and the correlation length for vortex shedding increases [2]. This produces wake that is more 2-dimensional in nature. Transverse illation results in the correlation length of the wake vortices to increase diminishing the 3-dimensional influence of vortex shedding. A similar argument was put forward by Blackburn and Henderson [1] that suggested that the harmonic motion of a long circular cylinder seems to suppress three dimensionality and produces flows resembling more 2-dimensional in nature. This is due to the increase in spanwise correlation of forces with increasing cylinder motion amplitude. The three dimensionality effects are not regarded as an important driving mechanism in the fluid-structure interactions as the fundamental driving mechanisms are primarily 2-dimensional. 2-dimensional simulation at Re 4 is sufficient to avoid the suppression of phase switch due to the strong dissipation of flows [1]. 59

78 HAPTER 5: NUMERIAL SIMULATION APPROAH AND METHODOLOGY The simulations in the study were conducted at Re 12, with A / D. 3 and are not expected to provide noticeable difference from actual 3-dimensional results as the physical fluid motion becomes essentially 2-dimensional. The results presented here may provide some insights into understanding the wake dynamics and the physics of fluid-structure interactions at moderate Reynolds number and may also be used as a guide for high Reynolds number. 6

79 HAPTER 6: RESULTS hapter 6 6. Results This chapter presents the results for the stationary and illating cylinders. Stationary cylinder s transverse and in-line coefficients together with the Strouhal number are shown and compared with established data in the range of Reynolds number investigated. The thorough analysis of the results from the stationary cylinder provides a basis of confidence for the apparatus, data acquisition system and hence the output of the forced sinusoidal experiments. Fluid force analysis from strain gauges data from the forced illation experiments and its breakdown into components in-phase with velocity and acceleration are presented in order to examine the general response characteristics of varying amplitude and frequency of motion. In the sinusoidal illation tests, 3-dimensional response characteristics were plotted in order to offer a better representation of the fluid force coefficients as a function of frequency and amplitude. Furthermore, the numerical simulation results are also presented as a basis of comparison with experimental results for both stationary and illating cylinder cases. Finally, how the numerical simulation complements the experimental findings is discussed. 6.1 Stationary ylinder Results Tests with stationary cylinder in uniform flow were performed to determine transverse coefficients, in-line coefficients and the Strouhal number for the test cylinder. This was achieved by positioning the cylinder in the middle of the towing tank without activating the motor for the motion of the rotating shaft. The towing speed was kept constant at.2m/s yielding a Reynolds number of 12,. A number of runs were performed where 17 repetitions were made. The results from these stationary runs serve as a valuable statistical basis for the determination of force coefficients to follow in the forced illation experiments. It provides confidence in the experimental rig used and improved assurance on the forced vibration results obtained. The average Strouhal number calculated is.184 with a standard deviation of.35 or less than 4%. The duration of the steady state condition for the stationary cylinder run was approximately 9s resulting in a spectral resolution of.11hz, yielding a Strouhal resolution accurate to.3. Figure 6-1 presents the histogram of the Strouhal number obtained. omparing the average Strouhal number obtained with the data compiled by Norberg [32] as shown in Figure 2-5, it lies below the bandwidth of the compiled data 61

80 Number of Realizations HAPTER 6: RESULTS for the Reynolds number investigated. In the past, experimental data of Strouhal number and vortex shedding frequency has been obtained with the use of very large aspect ratio ( L / D1 ) hence eliminating the influence of boundary conditions. A number of studies have shown a decrease in vortex shedding frequency with decreasing aspect ratio. Gerich and Eckelman [15] stated that for low aspect ratio ( L / D15 ) cylinder, the shedding frequency would be influenced by the aspect ratio and the boundary condition used. Norberg [33] observed that L / D 25 is required for 1 4 Re 4x1 4 to acquire independent conditions i.e. 2-dimensionality case. This could explain the slight reduction in the vortex shedding frequency from the regular Strouhal frequency i.e. ~.2. However, the Strouhal number obtained is still within the practical and realistic limit. Aronsen [6] in his experiment reported a Strouhal number of.188 in the subcritical Reynolds number range which is very close to the present investigation Strouhal Number, St Figure 6-1: Histogram of Strouhal Number, St for Stationary ylinder The average of the root-mean-square (RMS) of the transverse coefficients, the mean in-line coefficients, Figure 6-2 and Figure 6-3 show the histogram of L _ rms and Dm calculated is.39 and respectively. L _ rms and Dm respectively. The transverse coefficient is presented in the form of an RMS value of the fluctuating transverse force to permit sensible comparison of the non-periodic nature of the force. omparing the L _ rms with the recent data compiled by Norberg [32] as shown in Figure 2-6, it can be seen that the present experiment lies in the lower bound region for 62

81 Number of Realizations Number of Realizations HAPTER 6: RESULTS the Reynolds number investigated. It is also worthwhile to observe in Figure 2-6 the wide scatter of L _ rms 8 7 data from the compilation of the existing experimental data L_rms Figure 6-2: Histogram of RMS Transverse oefficient, 7 L _ rms L _ rms for Stationary ylinder Dm Dm Figure 6-3: Histogram of In-Line oefficient, Dm for Stationary ylinder 63

82 HAPTER 6: RESULTS Table 6-1 summarizes the results for the stationary cylinder runs. It is observed that there is large scatter for the mean in-line force coefficients, the transverse force coefficients, L _ rms Dm and the RMS values of. The scatter can be explained from the fact that these illating forces are sensitive to three dimensionality of the flow. Vortex shedding from a stationary cylinder at the Reynolds number range investigated does not occur at a single distinct frequency, but rather it wanders over a narrow band of frequencies with a range of amplitudes and is not constant along the cylinder length. The lack of spanwise correlation and the irregular appearance of the vortex shedding cycles lead to the vortex shedding behavior from being a steady, harmonic and twodimensional process. The correlation length over which the stationary cylinder could be considered to experience 2-dimensional flow is of the order of 3 5 cylinder diameter [11]. The aspect ratio used in the current experiment was 1, and hence the flow past the stationary cylinder was subjected to three dimensionality effects as it was not being fully correlated over the entire length of the cylinder. Figure 6-4 and Figure 6-5 show force profiles for both the transverse and in-line forces respectively. The transverse force profile exhibits a non-constant force magnitude across the time series mimicking a random beating phenomenon. This would have contributed to the scatter in the RMS of the transverse force coefficient, L _ rms with repetition of the experimental runs. It is evident that there exists a high frequency fluctuation superimposed on the in-line force profile from Figure 6-5. More importantly to observe is the irregular low frequency modulations observed for the in-line force profile as the results of three dimensionality effects which contribute to the scatter in the mean in-line force coefficient, Dm. However, it will be observed in the subsequent section when the cylinder is subjected to an external force, the vortex-induced forces will assume a more consistent behavior minimizing the random fluctuations. Strouhal Number, St Dm L _ rms Mean, Standard Deviation, Table 6-1: Summary of Stationary ylinder Results It is also observed the rather large difference from the experimental data as presented in Table 6-1 and the numerical simulations results presented in Table 5-2. In comparing this, one has to bear in mind that in the compilation of different experimental data itself there exists a rather large scatter for example, L _ rms from Figure 2-6. While the schemes used in the numerical simulations aim to represent the actual flow conditions, it 64

83 Force (N) Force (N) HAPTER 6: RESULTS is inevitable that some discrete features of the flow may not be replicated hence affecting the forces computed to a certain extent. Nonetheless, the L _ rms of.47 from the numerical simulations of stationary cylinder is still within the range in Figure 2-6. The Strouhal calculated from the numerical simulation ( St. 234 ) even though is higher than the data in Figure 2-5 but is within the sensible range. Branković [12] conducted experiments for3 Re 21 and reported the same Strouhal value of.23. The data also closely matches the established data by Roshko [36] time (s) Figure 6-4: Stationary Transverse Force Profile showing non-constant Force Magnitude time (s) Figure 6-5: Stationary In-Line Force Profile showing High Frequency Fluctuation 65

84 HAPTER 6: RESULTS Figure 6-6 shows the transverse force profile from Khalak and Williamson [24] at Re 1,6for comparison with the transverse force profile from the present experiment at Re 12,. Khalak and Williamson present only 1s of their force trace whereas here in Figure 6-4, 55s of such force trace is presented. The magnitude of the transverse force profile from the present experiment as shown in Figure 6-4 is comparable with Figure 6-6 given the Reynolds numbers difference in the two plots shown. This provides improved confidence in the experimental apparatus used and the output from the forced illation experiments. Figure 6-6: Transverse Force Profile from Khalak and Williamson at Re 1, 6[24] 6.2 Oscillating ylinder Results As mentioned in Section 3.4, experiments involving the cylinder subject to illating external force were conducted for a large number of frequency and amplitude combinations. Approximately 33 values of non-dimensional illating frequency ranging from.4 f / f 2. 2 where f is the illation frequency and f is the Strouhal frequency for a stationary cylinder were selected to cover the natural Strouhal number. Each of the 33 illating frequency was tested at 9 non-dimensional amplitudes ranging from.2 A / D 1. yielding the actual illation amplitude from 1.2cm to 6.cm. Similar to the stationary cylinder run, all runs were investigated at a constant carriage speed of.2m/s corresponding to a Reynolds number of 12,. For ease of observation, the results will be discussed in detail for one amplitude of illation over a range of frequencies and the collation of results covering all amplitudes and frequencies investigated will be presented as 3-dimensional surface plots in Section To demonstrate the primary response characteristics of the forced sinusoidal illation, results for A / D. 3 are presented. Figure 6-7 and Figure 6-8 show the forced 66

85 Force (N) Force (N) HAPTER 6: RESULTS illation transverse and in-line profile respectively for amplitude ratio of A / D. 3. It can be seen that the force profiles assume a more consistent behaviour suppressing the random fluctuations as compared to the stationary force profiles shown previously in Figure 6-4 and Figure 6-5. The external force exerted on the cylinder to produce the illation manages to shield the small disturbances to the system thus eliminating the random variation in the force profile. The controlled illations also curtail the 3- dimensional flow producing wake that is more 2-dimensional in nature. As mentioned in Section 5.6, the spanwise correlation of forces will increase with increasing cylinder motion amplitude permitting a more regular force profile time (s) Figure 6-7: Forced Oscillation Transverse Force Profile time (s) Figure 6-8: Forced Oscillation In-line Force Profile 67

86 HAPTER 6: RESULTS Total In-line and Transverse Force oefficients The behaviour of the mean in-line force coefficients for A / D. 3 with variation of the non-dimensional frequency is presented in Figure 6-9. The mean in-line coefficients resemble the stationary cylinder case in Table 6-1 at low frequency but with general trend of increasing in magnitude as the illation frequency increases. Note that the mean in-line coefficient for the stationary cylinder in this experiment is The sharp amplification peak is observed at approximately / f. 9, somewhat below the natural Strouhal frequency i.e. f / f 1 after which the magnitude of the mean in-line coefficient drops but to a level higher than the stationary cylinder value. The steep rise in the coefficient in the vicinity of the natural Strouhal frequency is due to the resonance between the illation frequency and the natural Strouhal frequency. Another amplification peak is also observed at / f 1. 6 denoting augmentation due to higher f order harmonic of the Strouhal component. However, it is also noticed that the second peak begins deviating away from the second harmonic of the natural Strouhal relationship and with less pronounced increase in magnitude. This suggests that at frequency higher than the natural Strouhal frequency (recall that Strouhal relationship is only valid for stationary cylinder), the effects of the Strouhal weaken. 2.4 f Dm Dm f /f f / f A / D. as a function of f / f Figure 6-9: Mean In-line oefficient for 3 68

87 HAPTER 6: RESULTS A similar explanation can be attributed to the variation in the root-mean-square (RMS) of the transverse force coefficient, _ for A / D. 3 and is presented in L Figure 6-1. At low frequency the RMS transverse force coefficients are very small resembling the stationary cylinder case in Table 6-1. However, it dramatically increases and peaks at / f. 9, just below the Strouhal natural shedding frequency due to the f effects of resonance. However, the second peak of the amplification is not as pronounced as the mean in-line force coefficients in Figure 6-9 due to the weakening of Strouhal effects. 8. rms L _ rms 3. L_rms f /f f / f Figure 6-1: RMS Transverse oefficient for A / D. 3 as a function of f / f hange of Energy Transfer between ylinder and Fluid The variation of the transverse force phase angle, and the mechanical energy associated with it, E are presented in Figure 6-11 and Figure 6-12 respectively. Both the phase angle and mechanical energy provide the information on the power transfer between the cylinder and the fluid. This is important towards understanding the fluidstructure interaction in characterizing the response of the system with variations of external excitations. The transverse force phase angle, is defined as the phase difference between the transverse force and the displacement of the cylinder. The mechanical energy transferred from the flowing fluid to the illating cylinder per 69

88 E Φ (deg) HAPTER 6: RESULTS f /f f / f Figure 6-11: Phase Angle for A / D. 3 between transverse force and cylinder displacement as a function of f / f E f /f Figure 6-12: Mechanical Energy Transfer for A / D. 3 between cylinder and fluid as a function of f / f f / f 7

89 HAPTER 6: RESULTS motion, E was defined previously in Equation The quantity E is positive when work is done on the cylinder and negative when work is done on the fluid. Positive values of E correspond to phase angle, between and 18. The illation of an elastically mounted cylinder requires positive energy transfer. However, a cylinder which is forced to illate is not subjected to this constraint and all values of are physically possible. It is evident from the phase angle plot that the phase angle is between and 18 for frequency ranges of f / f. 65 and / f. 91. At lower frequencies, the phase angle has a value of 18 f indicating that the energy is being transferred from the fluid to the cylinder. As the frequency increases and approaches the Strouhal frequency region, there is a dramatic drop in the phase angle into the negative range. The negative phase angle suggests that phase change occurs where the cylinder now takes control of the system due to the resonance as it is in the vicinity of the natural Strouhal component. In this range, energy is being transferred from the cylinder to the fluids. With an increase in frequency and moving away from the Strouhal frequency, it is noticed that the phase angle jumps to be in the range 18 where the fluid takes charge of the system again i.e. work done on the cylinder. This can be confirmed by analyzing the plot of energy transfer, E. As defined earlier, positive value of E indicates that energy is being transferred from the fluid to the cylinder. In the same ranges of the phase angle as stated above, the energy plot indicates positive values. For.65 / f. 91, the change in signs of the phase f angle and energy marked the change in the phase relationship between the transverse force and cylinder displacement. It is interesting to note that both phase angle and energy remain within 18 and positive in sign respectively after an abrupt jump away from the Strouhal frequency. In other words, the fluid dominates the system where work is done on the cylinder. No further phase change is noted approaching the higher harmonic in the Strouhal component. This can be attributed to the weakening in the Strouhal component higher than the primary component as the frequency increases. The resonance that occurs is rather weak in the higher harmonic Strouhal to enable the cylinder to dominate the system. A similar finding is noted in the discussion of the weak second amplification peak of and L _ rms previously in Section Dm In addition, the change in phase relationship is also accompanied by a change in the coefficient of force in-phase with velocity component of the transverse force, lv which provides the information on the energy transfer between the cylinder and the fluid. The variation of lv against the non-dimensional frequency is presented in Figure 6-13 for A / D. 3. A positive value of the coefficient suggests energy transfer from fluid to 71

90 HAPTER 6: RESULTS body (excitation to the cylinder) while a negative value suggests energy transfer from body to fluid (damping). In the frequency ranges of f / f. 65 and f / f. 91, lv has a positive value indicating energy transfer from the fluid to the cylinder and the value changes to be in the negative range for.65 f / f. 91 indicating energy transfer from the cylinder to the fluid. This agrees well with the findings of the phase angle, and energy, E above. lv lv f f /f / f Figure 6-13: Transverse Force oefficient In-Phase with Velocity for A / D. 3 as a function of f / f Added Mass Effect on the Transverse Force Another important response characteristic that has to be looked at is the added mass effect on the transverse force. Added mass can be defined as the mass of the fluid around the body which is accelerated with the movement of the body due to the action of pressure [45]. In the words of Sarpkaya [37], Added mass is one of the best known, least understood and most confused characteristics of fluid dynamics. 72

91 HAPTER 6: RESULTS lh lh f /f f Figure 6-14: Transverse Force oefficient In-Phase with Acceleration for A / D. 3 as a function of f / f / f m m f /f Figure 6-15: Added Mass oefficient for A / D. 3 as a function of f / f f / f 73

92 HAPTER 6: RESULTS One often encounters the general force expressions for viscous flows using a velocitysquare in-line force plus an ideal inertial force with added mass coefficient, m taken to be 1 for a cylinder (e.g. Morison s equation). Bishop and Hassan [9] reported their transverse and in-line forces and assumed the added mass coefficient to be equal to its ideal (potential flow) value of 1. This invalidated their force measurements as the added mass is a function of time, type of motion, orientation of body through the fluid, illation frequency and amplitude. The quantification of the value of added mass can be obtained by determining the force in-phase with acceleration component of the transverse force. The added mass coefficient is then calculated as the force in-phase with acceleration normalized with the mass of the displaced water and the acceleration amplitude of the harmonic motion as given in Equation Figure 6-14 presents the transverse force coefficient in-phase with the acceleration, for A / D. 3. It shows a sudden dip at approximately f / f. 9 (just below the Strouhal frequency) and is amplified steadily henceforth. The cycle-averaged added mass coefficients, m is presented in Figure Similarly, the coefficients illustrate a dramatic variation of the fluid force just before the Strouhal frequency. The added mass value varies notably in the vicinity of the resonant region. It is observed that the classical assumption of 1 is only valid at frequencies higher than the resonant or Strouhal frequency. m lh At low frequency, the added mass tends to be in the negative region. The comprehension of negative mass may appear paradoxical. Nevertheless, few other researchers have reported the value of negative mass e.g. Gopalkrishnan [17], Sarpkaya [39] and Vikestad et al. [48] among others. Sarpkaya [37] pointed out that m is the cycleaveraged sum of the masses transported during the periods of acceleration (where cylinder amplitude toward the mean position i.e. y ) and deceleration (cylinder moving towards its maximum position i.e. y A ). Thus, cycle-averaged negative added mass means that the drift mass during the deceleration periods is larger than that during the acceleration periods. From the aforementioned, it could be surmised that the added mass does not assume a constant value and the classical assumption of 1 for cylindrical object is not valid over a wide range of frequencies. In the present experiment, 1 is only valid for frequency ratio higher than f / f 2.. m m 74

93 HAPTER 6: RESULTS Surface Plots of all Amplitude Ratios The hydrodynamic force coefficients for all amplitude and frequency ratios investigated are presented in this section. The results are presented in the form of 3-dimensional surface plots to illustrate the change in the response characteristics with variation of amplitude together with the variation in frequency. Each surface plot contains data for 33 discrete frequencies (.4 f / f 2. 2 ) for each 9 amplitude ratios (.2 A / D 1. ). No smoothing of the data points has been applied and only linear interpolation between points is used. The change in magnitude of the force coefficients is manifested by different colours in the surface plots to aid better visualization of the variations in the response. The complete results for the hydrodynamic coefficients are presented in Figure 6-16 to Figure Note that the results presented are for the illation frequency, f not the non-dimensional frequency f / f as the previous plots. It was found that for 3-dimensional surface plot the presentation based on gives a better visualization than f / f due to the spread of the frequency axis. Hence, the results are presented for.25 f Mean In-line oefficients. It is evident from the surface plot of Figure 6-16 that amplification of the coefficient is observed in the vicinity of the resonant region i.e..49 f.613 (corresponding to.8 / f 1. ) for all cases investigated indicating the strengthening of the vortex forces. The second amplification is not readily discernible for the higher harmonic of the Strouhal frequency. The mean in-line coefficient, f Dm shows general increase as the amplitude of the illation increases. This could be explained as follows; as the amplitude of the illation increases, the apparent area of the cylinder exposed to the free stream increases hence contributing to larger in-line force. Phase Angle. The behaviour of the transverse force phase angle with the displacement amplitude is presented in Figure The response characteristic of the transverse force phase angle is consistent for almost all amplitude ratios across the frequency range analyzed except for higher amplitude ratio i.e. A/ D. 9 and A / D 1.. It is noticed that at these amplitude ratios the transverse force phase angle remains in the positive range ( 18 ) without changing sign near the resonant range. The discussion on the nature of higher amplitude ratio will be detailed in a later section. f 75

94 HAPTER 6: RESULTS Energy. The surface plot of the energy transfer for all amplitude ratios is shown in Figure The preceding discussion on energy transfer response for A / D. 3 is also applicable for the other amplitude ratios. However, as observed for the phase angle, the higher amplitude cases illustrate that the energy does not change sign in the vicinity of the resonant point i.e. energy being transferred from the fluid to the cylinder observed for all frequencies analyzed at higher amplitude ratios. Transverse Force oefficient In-Phase with Velocity. The transverse force coefficient in-phase with velocity, provides the information on the energy transfer between the lv fluid and the cylinder accompanied by a change in phase angle and energy signs. The response for the lower amplitude ratios does not vary to a large extent and the changes sign in the region of the Strouhal frequency signifying the transfer of energy from cylinder to the surrounding fluid. However, at higher amplitude the change in sign is not observed as seen from Figure lv Added Mass. The added mass coefficient, m shown in Figure 6-2 peaks at around the Strouhal frequency, f and exhibits a steady value henceforth at high frequencies, higher than the Strouhal frequency at all amplitude ratios where the coefficients approach unity. This is consistent with the simplified classical assumption of added mass value previously made. The value does not exhibit any further behaviour of variation with increasing frequency upon arriving at the value of unity. At lower frequency and lower amplitude, the added mass value tends to progress towards the negative range as discussed previously but this is not observed for high amplitude ratios. The change in added mass value is a direct relationship with the transverse force coefficient in-phase with acceleration, shown previously. The magnitude of lh based on the derivation of added mass as lh revolves around at lower frequencies but varies to a large extent at higher frequencies for all amplitude ratios analyzed as shown in Figure It could be surmised that the behaviour of all hydrodynamic coefficients at low frequencies are very weakly dependent on amplitude except for the added mass coefficient, m as noted above. The response begins to diverge at higher frequency for different amplitude ratios. What is also interesting to observe is the rather non-variant nature of the added mass coefficient, frequencies as it reaches the value of unity. m for almost all amplitude ratios at higher 76

95 HAPTER 6: RESULTS Dm Dm f.9 f.95 (Hz) A/D D Figure 6-16: Surface Plot of Mean In-line oefficient, Dm as a function of f and A/ D 77

96 HAPTER 6: RESULTS Ø f f (Hz) Figure 6-17: Surface Plot of Phase Angle, as a function of f and A/ D A/D.9.8 A/ D 1 78

97 HAPTER 6: RESULTS E E f f.8 (Hz) Figure 6-18: Surface Plot of Energy, E as a function of f and A/ D A/D A/ D 1 79

98 HAPTER 6: RESULTS lv lv f f.75 (Hz) A/D A/ D 1 Figure 6-19: Surface Plot of Transverse Force oefficient In-Phase with Velocity, lv as a function of f and A/ D 8

99 HAPTER 6: RESULTS A/D A/ D f f.65 (Hz) m m Figure 6-2: Surface Plot of Added Mass oefficient, m as a function of f and A/ D 81

100 HAPTER 6: RESULTS lh lh A/D A/ D f f.7.65 (Hz) Figure 6-21: Surface Plot of Transverse Force oefficient In-Phase with Acceleration, lh as a function of f and A/ D 82

101 HAPTER 6: RESULTS Response haracteristics of Force oefficients at Higher Amplitude Range omparison of the response characteristics for fluid force coefficients shows a distinct disparity between low amplitudes and high amplitudes response. In the current experiments, the difference is noted for A / D. 9 and A / D1.. It is noticed that the component of the transverse force in-phase with velocity coefficient, lv that provides the information of energy transfer does not exhibit any change in sign at these higher amplitude ratios. This implies that no change of energy transfer takes place over the frequency range at higher amplitude of illation. Figure 6-22 (a) and (b) show the response comparison of the transverse force coefficients for A / D. 3 and A / D. 9 respectively. For the higher amplitude ratio ( A / D. 9 ), the value of lv remains in the positive range designating energy transfer from the fluid to the cylinder throughout the frequency range investigated. This suggests that even in the Strouhal frequency range, the effects of resonance are rather weak to allow the cylinder to facilitate the system i.e. work done on the fluid. Energy transfer is observed for A / D. 3 with the change in sign of lv as discussed earlier and also depicted in Figure 6-22 (a). To give a better representation of the phenomenon, the comparison plots for the phase angle, and energy transfer, E are presented in Figure 6-23 (a) and (b) respectively for A / D. 3 and A / D. 9. The same trend is observed where the phase angle for A / D.9 remains in the positive range ( 18 ) without change in sign as per A / D.3 shown in the same plot. Similarly, the plot of energy transfer shows that there is no change in sign for A / D. 9 where the energy is always positive i.e. work done on the cylinder throughout the range of frequencies investigated. Intuitively, this resembles the illation of an elastically mounted cylinder where positive energy transfer is required to produce the illations. 83

102 Transverse Force oefficient Transverse Force oefficient HAPTER 6: RESULTS lv lh f / f f /f (a) A / D. 3 (low amplitude) lv lh f / f f /f (b) A / D. 9 (high amplitude) Figure 6-22: Transverse Force oefficient In-Phase with Velocity, and In-Phase with Acceleration, lh lv 84

103 E HAPTER 6: RESULTS A / D.9 Φ (deg) A / D f / f f /f 1 (a) Phase Angle 8 6 A / D.9 E 4 2 A / D f /f (b) Energy Figure 6-23: Phase Angle and Energy Plots for A / D. 3 and A / D. 9 f / f 85

104 HAPTER 6: RESULTS The phase angle determines if the transverse force acts as the excitation or damping to the system. If the phase angle is 18, the component of the transverse force in-phase with cylinder velocity is positive suggesting energy transfer from fluid to the cylinder. This implies that the vortex shedding acts as an excitation to the cylinder motion. However, if the phase angle is 18, the component of the transverse force in-phase with the cylinder velocity is in the negative range suggesting energy transfer from the cylinder to the fluid implying that the vortex shedding acts as a damping to the system. At low frequencies, the lower amplitude case exhibits phase angles approaching 18 with increase in frequency. However, with further increase in frequency approaching Strouhal component, it dramatically switches sign to be in the negative region (indicating change in energy transfer). Moving away from the Strouhal frequency with further increase in frequency shows that the phase angle for the small amplitude case settles at approximately 16. While the low frequency range behaviour is similar for both the small and high amplitude cases, the rapid change in sign as the frequency approaches the natural Strouhal component is not observed for the high amplitude case. Instead of becoming negative in the phase angle value, the phase angle for the high amplitude case becomes even more positive. Further increase in frequency illustrates that the high amplitude case settles at a phase angle value slightly less than the low amplitude case i.e The fact that the high amplitude case does not exhibit any change in phase angle sign i.e. remains positive throughout the frequency range investigated suggests that the vortex shedding acts as an excitation to the cylinder motion without any damping effects. 6.3 Lock-in Range In the study of VIV, one would be interested to find out the lock-in range upon which the cylinder frequency locks-in with the natural vortex shedding frequency of the system i.e. f / f 1. producing resonance behavior. In forced vibration experiments, the lock-in phenomenon occurs when the vortex shedding is entrained by the cylinder motion, hence the vortex shedding frequency changes to match the cylinder illation frequency. The lock-in envelope has been made available by previous researchers such as Bishop and Hassan [9], Blackburn and Henderson [1], Gopalkrishnan [17] and Stansby [43]. To determine the lock-in boundaries for the current experiments, a Fast Fourier Transform (FFT) has been used to analyze the transverse force data. Lock-in behaviour 86

105 HAPTER 6: RESULTS is characterized when the transverse force data constitutes only the component at which the cylinder is illated at. Non lock-in behaviour is characterized when the transverse force data contains other components of frequency such as Strouhal frequencies apart from the cylinder illation frequency. Figure 6-24 shows the motion and transverse force spectra outside the lock-in region i.e. before lock-in for f. 262 Hz where several dominant frequencies apart from the illating frequency are observed. Figure 6-25 shows the spectra for f. 574 Hz that is inside the lock-in range and it exhibits only one dominant frequency coincident with the illating frequency. With further increase in frequency, lock-out occurs where another component re-appeared in the transverse force spectrum in the vicinity of the natural Strouhal component shown in Figure 6-26 for f 1. 86Hz. This is the example of non lock-in above the lock-in range. Note that the natural Strouhal frequency is.613hz (corresponding to Strouhal number of.184). The force profiles before, inside and above the of lock-in region are presented in Figure 6-27, Figure 6-28 and Figure 6-29 respectively corresponding to the motion and transverse force spectra in Figure 6-24, Figure 6-25 and Figure It can be observed that the transverse force profile outside of lock-in region consists of several frequency fluctuations due to different components of transverse force acting on the cylinder. The force profile in the lock-in region demonstrates a more consistent manner dominated by one single frequency of illation i.e. the illating frequency. 87

106 Lift Spectrum Motion Spectrum Lift Spectrum Motion Spectrum Lift Spectrum Motion Spectrum HAPTER 6: RESULTS 8 x Frequency (Hz) Figure 6-24: Motion and Transverse Force Spectra before Lock-in ( f. 262 Hz) 8 x Frequency (Hz) Figure 6-25: Motion and Transverse Force Spectra inside Lock-in ( f. 574 Hz) 8 x Frequency (Hz) Figure 6-26: Motion and Transverse Force Spectra above Lock-in ( f 1. 86Hz) 88

107 Force (N) Force (N) Force (N) HAPTER 6: RESULTS time (s) Figure 6-27: Transverse Force Profile before Lock-in ( f. 262 Hz) time (s) Figure 6-28: Transverse Force Profile inside Lock-in ( f. 574 Hz) time (s) Figure 6-29: Transverse Force Profile above Lock-in ( f Hz) 89

108 A/D HAPTER 6: RESULTS By performing FFT for every experimental runs conducted and determining the frequencies over which the lock-in occurs for each set of amplitude ratio, a figure showing the complete picture of the entire lock-in region can be plotted. For each of the amplitude ratio, two points are plotted being the changeover from non lock-in to lock-in state and from lock-in to non lock-in state (or lock-out) as shown in Figure 6-3. The region bounded by the two points for each amplitude ratio is the lock-in region. It should be noted that the determination of lock-in region becomes progressively more complicated with increase in illation amplitude. As such, results for A / D. 9 and A / D1. have been omitted due to high probability of uncertainties in determining the lock-in envelope. The boundary of lock-in range shows a general widening with increasing amplitude ratio. The results have also been compared with previous work being conducted available from published data i.e. Gopalkrishnan [17], Koopman [26] and Stansby [43] and found to be agreeable within the lock-in range from other researchers given the difference in Reynolds number. Based on the observation made from Figure 6-3, the present experiment conducted at higher Reynolds number generally shows a smaller lock-in envelope compared to the one conducted earlier at lower Reynolds number. However, results from Gopalkrishnan [17] shows an even tighter lock-in envelope compared to the current experiment even though the Reynolds number is slightly lower than the current study Lock-in.4.2 Non Lock-in Non Lock-in f /f Gopalkrishnan (Re=1) Koopman (Re=2) Stansby (Re=36) Present Experiment (Re=12) Koopman (Re=1) Koopman (Re=3) Stansby (Re=92) Figure 6-3: Lock-in Region formed between f / f for different A/ D 9

109 HAPTER 6: RESULTS The line of best fit is shown in the figure to represent the lock-in boundary. The lower bound lock-in boundary from the experiments can be described by Equation 5-1. ln ( A/ D) f / f.672 Equation and the upper bound boundary can be described by Equation 5-2. ln ( A / D) f / f Equation Note that this is only valid within the parameter space investigated. 6.4 omparison with Numerical Results This section will discuss the comparison of the experimental and the numerical simulation results, their similarities, differences and how the results from the numerical simulations complement this research. As stated in Section 5.3, numerical simulations for A / D. 3 and.7 f / f 1. 4 at Reynolds number of 12 were conducted for comparison purposes using the Reynolds-Average Navier-Stokes (RANS) standard k model. As such, the numerical and experimental results will be compared for the same amplitude ratio. Figure 6-31 and Figure 6-32 show the coefficients of transverse force in-phase with velocity, the numerical and experimental results. lv and acceleration, lh components respectively for both omparison of the experimental and numerical results shows divergence to a certain extent in the magnitudes of the coefficients. This is especially true in the higher frequency ratio region. Even though it was endeavoured to simulate the experimental condition in the numerical environment, it is unlikely that all the wake behaviours that could occur in physical experiments have been revealed in the numerical simulations. The difference in the turbulence levels, uniformity of flow and aspect ratio of the cylinder may also contribute to the observed differences between experimental and numerical results. To date, most of the numerical studies relating to forces induced by fluid-structure interaction are usually restricted to the lower end of the Reynolds number spectrum i.e. Re 1, mainly owing to the complex intrinsic feature of turbulence at 91

110 HAPTER 6: RESULTS lv lv Experiment f /f Numerical Simulation Figure 6-31: omparison between Numerical and Experimental Results for.5 f / f lv. -.5 Experiment Numerical Simulation -1. lh lh f /f f Figure 6-32: omparison between Numerical and Experimental Results for / f lh 92

111 HAPTER 6: RESULTS moderate to high Reynolds number. An intrinsic feature of turbulence at high Reynolds number is the huge value of the ratio of the largest to the smallest relevant turbulent scales. In space, the largest scales are of the order of the geometry concerned or the largest eddies of the flow field. The smallest scales can be taken as the sizes of the eddies where most of the viscous dissipation occurs [46]. In the subcritical Reynolds number range where Re 12 lies, transition waves will appear in the shear layers springing from the separation point resulting even in the vortices closest to the cylinder to be of turbulent in nature. This nature of complex turbulence is increasingly difficult to be captured with increasing Reynolds number. Due to the larger scales being highly dependent on the particular geometry and physics of the flow of interest, it is therefore very difficult, if not impossible, for the turbulence model to account for the effects of all scales [46]. The argument presented herein highlights the limitations of the RANS model especially to accurately model the higher turbulence regime hence contributing to the difference in force prediction between the numerical and experimental results. An effort to simulate an externally forced cylinder at moderate Reynolds number as a comparison to the experimental results shall pave the way to explore the suitability of using a numerical scheme to predict fluid induced forces of vibrating structures at moderate to high Reynolds number. It is also worth noting that a very small difference in frequency would lead to large disparity in the calculated coefficients. Even though the comparison in results are presented in terms of non-dimensional frequency i.e. f / f, one has to bear in mind that the Strouhal number from the experiments ( St. 184 ) differs from the Strouhal number from simulation ( St. 234 ). Albeit that these numbers are still within the range of other published work as discussed earlier, however as far as the comparison of Strouhal frequency is concerned, it differs to a great extent being f. 234 Hz and f.613 Hz for the numerical and experimental respectively. This would certainly augment the difference between the numerical and experimental results. Nevertheless, a closer inspection of the numerical results would reveal that even though the magnitude differs from the experiments, the trend of the plots agrees reasonably well within the vicinity of the natural Strouhal frequency range. In Figure 6-31, the plot shows a depression until f / f. 95 and then increases dramatically henceforth for the numerical results. This is not dissimilar from the experimental results which show dramatic amplification after / f While the numerical results show negative lv f values for.7 f / f 1. indicating energy transfer from the cylinder to the fluid, the experimental results show negative range for.65 / f. 91. The f 93

112 HAPTER 6: RESULTS numerical results show that the values of lv changes to negative sign thereafter which is not observed in the experimental results. In Figure 6-32, the values of lh illustrate a steady increase after f / f 1. and / f. 945 for the numerical and experimental results respectively which also indicate close agreement, before both curves gradually dip at higher frequency ratio. The difference in the frequency ratio ranges above could due to the resolution of the frequency ratio investigated in the numerical simulation. However, the two plots signify good agreement as far as the trend of the results is concerned in the region of interest i.e. the natural Strouhal frequency range ( / f 1.). Hence with this observation, the numerical results will be used to f qualitatively discuss the characteristics of the fluid-structure interaction in the natural Strouhal frequency range which is not obtainable from the experiments. This will complement the quantitative discussion and enhance the overall picture of the vibrating cylinder in free stream in this scope of work. f 6.5 Phase Switch between Vortex Formation and ylinder Motion As mentioned in the preceding section, the results from the numerical analyses will be used to qualitatively describe the characteristics of the fluid-structure interaction of the vibrating cylinder. In this section the dynamics of the vortex shedding with variation of illating frequency which lead to the phenomenon of phase switch will be discussed. For the range of frequency ratios investigated, the phase switch between the vortex formation and cylinder motion is observed in the lock-in region. The mechanism can be explained such as the cylinder illates at a higher frequency, the concentration of vorticity moves closer to the cylinder until it reaches a limiting position. With a further increase in the frequency of illation, the vorticity formed switches to the opposite side of the cylinder. The switching mechanism marks the change in the phase relationship between the vortex formation and cylinder displacement. The switch also produces a change in the phase of vortex-induced forces on the cylinder, and can affect the sign of mechanical energy transfer between the moving cylinder and the flow. Figure 6-33 (a) shows the instantaneous vorticity contours of the cylinder at the upper most position during illation for.8 / f When comparing the vorticity f contours, a distinct difference is observed after f / f. 975 where apparent tightening of the vortices near the cylinder is observed. An abrupt change in the vortex formation taking place on the opposite side of the cylinder is clearly observed at f / f 1.. The transition after f / f. 975 marks the occurrence of the switch in phase relationship between the vortex formation and cylinder motion. 94

113 HAPTER 6: RESULTS + + f / f f / f s f / f f / f f / f 1.1 (a) Instantaneous Vorticity ontours (b) Instantaneous Streamline ontours Figure 6-33: Instantaneous Vorticity and Streamline ontours (cylinder at upper most position) 95