Citation |

- Permanent Link:
- http://ufdc.ufl.edu/UF00075488/00001
## Material Information- Title:
- Vortex-induced transverse loading on an articulated tower
- Series Title:
- UFLCOEL-TR
- Alternate title:
- Vortex induced transverse loading on an articulated tower
- Creator:
- Omar, Ahmed Fahmy, 1959- (
*Dissertant*) University of Florida -- Coastal and Oceanographic Engineering Dept Sheppard, D. Max (*Thesis advisor*) Hoit, Marc I. (*Reviewer*) Wang, Hsiang (*Reviewer*) Zimmerman, David C. (*Reviewer*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- Coastal & Oceanographic Engineering Dept., University of Florida
- Publication Date:
- 1992
- Copyright Date:
- 1992
- Language:
- English
- Physical Description:
- xviii, 153 p. : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Acoustic spectra ( jstor )
Aerodynamic coefficients ( jstor ) Amplitude ( jstor ) Cylinders ( jstor ) Kinetics ( jstor ) Natural frequencies ( jstor ) Reynolds number ( jstor ) Signals ( jstor ) Thermistors ( jstor ) Velocity ( jstor ) Coastal and Oceanographic Engineering thesis Ph. D Dissertations, Academic -- Coastal and Oceanographic Engineering -- UF Hydrodynamics ( lcsh ) Vortex-motion -- Mathematical models ( lcsh ) Water waves -- Mathematical models ( lcsh ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Abstract:
- This research is an experimental investigation of vortex-induced transverse loading on fixed and compliant structures in nonplanar oscillatory flow. The effects of flow nonuniformity and transverse motion on the transverse force, vortex-shedding frequency, and vortex span-wise coherence length have been investigated. To quantify such effects, two types of experiments were performed. Each type of experiment consisted of a series of tests. All tess were performed by oscillating an instrumented 10 ft long, 6 inch diameter alumninum, articulated cylinder in a 30 ft diameter cylindrical tank with a water depth of 8.5 ft. In the first type of experiment, transverse motion was not allowed, while in the second type, the cylinder was allowed to move freely in the transverse direction. Reynolds numbers, Re, between 0.61-10^4 and 1.3x10^5 and Keulegan-Carpenter numbers, KC, between 2.4 and 9.35 were obtained. A new method, involving the use of miniature, quick response thermistors was employed for measuring frequency of vortex-shedding. For the range of Re and KC tested, transverse force amplitudes as large as 70% of the in-line force were measured. Transverse forces occurred at frequencies that were multiples of the driving frequency. The dominant vortex-shedding frequencies clustered around one fo the harmonics of the driving frequency. The specific harmonic depended on the value of KC. A mathematical model for computing transverse forces taking into account the dependency of the lift coefficient on both Re and KC as well as the fact that these forces have multiple frequency components has been also proposed. Using this model, the magnitude of the transverse force was found to be deterministic and repeatable. However, the phase of this force was random in nature. The constrained tower results for in-line and transverse force coefficients were found to agree well with data obtained by others for waves acting on vertical cylinders. Transverse motion was found to have a significant effect on vortex-induced loading. The effect of transverse motion was also found to increase the vortex correlation length, increase the lift coefficient (by at least two and one-half times) and alter the nature of the vortex-shedding. for the range of KC tested, the results showed that, the larger the value of KC, the larger the amplitude of maximum transverse motion and the larger the value of reduced velocity, Vr, at which this maximum occurs.
- Thesis:
- Thesis (Ph. D.)--University of Florida, 1992.
- Bibliography:
- Includes bibliographical references (p. 148-153).
- Funding:
- This publication is being made available as part of the report series written by the faculty, staff, and students of the Coastal and Oceanographic Program of the Department of Civil and Coastal Engineering.
- Statement of Responsibility:
- by Ahmed Fahmy Omar.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright Ahmed Fahmy Omar. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 28172461 ( OCLC )
## UFDC Membership |

Full Text |

UFL/COEL-TR/089
VORTEX-INDUCED TRANSVERSE LOADING ON AN ARTICULATED TOWER by Ahmed Fahmy Omar Dissertation 1992 COASTAL & OCEANOQRAPHIC EN*IiEERIN DEPARTMENT University of Floida o GainesvMe, FRodda 32611 VORTEX-INDUCED TRANSVERSE LOADING ON AN ARTICULATED TOWER By AHMED FAHMY OMAR A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1992 ACKNOWLED GEMENTS The author wishes to express his sincere appreciation and gratitude to the supervisory committee chairman, Prof. D. Max Sheppard, for his continuous support, guidance and friendship in and out of the academic framework. It was a real joy working under his patient leadership. His input and encouragement in this long endeavor proved invaluable. A special debt of gratitude is owed to Profs. Marc 1. Hoit, Hsiang Wang and David C. Zimmerman for serving as members of his Ph.D. supervisory committee. In particular Prof. David C. Zimmerman was most helpful during the early stages of this work. Appreciation is also extended to Prof. Robert G. Dean for many helpful discussions during the course of this study. Many thanks go to Sidney Schofield, Subarana Malakar, Vernon Sparkman, Chuck Broward and the other members of the Coastal and Oceanographic Engineering Department and Laboratory for their help, friendship and cooperation. The author would also like to take this opportunity to express gratitude to all his past teachers who contributed in one way or another to his achievement of this educational goal. This work could not have been accomplished without the support of the University of Florida and the US Army Corps of Engineers Coastal Engineering Research Center. The contribution of the water tank and other facilities by the Crom Corporation is gratefully acknowledged. Finally, the author is very grateful to his parents and brothers, for their patience, love and sacrifice during the course of his life. TABLE OF CONTENTS ACKNOWLEDGEMENTS................................... iii LIST OF FIGURES....................................... vi LIST OF TABLES........................................ xi LIST OF SYMBOLS.......................................xii ABSTRACT........................................... xvii CHAPTERS 1 INTRODUCTION.......................................1 1.1 Problem Statement................................... 1 1.2 Research Objectives...................................3 2 LITERATURE REVIEW..................................5 2.1 Overview..........................................5 2.2 Steady, Planar Flow.......................... 10 2.2.1 Transversely Constrained Cylinder in Steady, Planar Flow 10 2.2.2 Transversely Unconstrained Cylinder in Steady, Planar Flow 15 2.3 Oscillatory, Planar Flow................................17 2.3.1 Transversely Constrained Cylinder in Oscillatory, Planar Flow 17 2.3.2 Transversely Unconstrained Cylinder in Oscillatory, Planar Flow 22 2.4 Oscillatory, Nonpianar Flow .. .. .. ... ... ... ... ... ....24 2.5 Wave Flows .. .. .. .. ... .... ... ... ... ... ... ....26 3 EXPERIMENTAL INVESTIGATIONS. .. .. ... ... ... ... ....31 3.1 Scaling Parameters and Model Selection. .. .. .. ... .... ....31 3.2 Experimental Set-up .. ... ... ... ... ... ... ... .....34 3.3 Instrumentation and Calibration. .. .. .. .. ... ... ... .....37 3.3.1 Frequency Generator .. .. .. .. .... ... .. .... ....37 3.3.2 Force Transducers .. .. .. .. ... ... ... ... ... ....41 3.3.3 Linear Displacement Transducers .. .. .. ... ... ... ..42 3.3.4 Thermistors. .. .. .. ... ... ... ... ... ... .....42 3.3.5 Lowpass Filters .. .. .. .. .... ... ... ... ... ....43 3.4 Data Reduction. .. .. .. .. .... ... ... ... ... ... ....44 4 MATHEMATICAL MODELS ........................ 4.1 In-line Force .............. 4.2 Transverse Force .......... 4.2.1 Steady Flow Model .... 4.2.2 Quasi-Steady Model . 4.2.3 Series Model ......... 4.2.4 Proposed Model ....... 4.2.5 Fixed Tower ......... 4.2.6 Complaint Tower ...... 5 EXPERIMENTAL DATA ANALYSIS ............. 5.1 In-line Forces ............... 5.2 Transverse Forces ............. 5.2.1 Constrained Transverse Motion . 5.2.2 Unconstrained Transverse Motion 5.3 Sources of Uncertainty and Inaccuracy . 6 SUMMARY AND CONCLUSIONS ..... 6.1 Summary of the Results .......... 6.2 Conclusions ................ 6.3 Recommendations for Further Work. . APPENDICES A ANALYSIS TECHNIQUES .......................... B INSTRUMENTATIONS AND CALIBRATION DATA ...... C FLOW CHARTS OF THE VARIOUS COMPUTER PROGRAMS.... BIOGRAPHICAL SKETCH ........................... . .o .. o o. o. o. o o o .o o o o. o o o. o. o o . LIST OF FIGURES 2.1 Flow chart for cylinder-loading response .. .. .. .. ... .....7 2.2 Lift coefficient versus Reynolds number for steady, planar flow around a smooth, fixed cylinder .. .. .. .. ... ... ... ....12 2.3 Regimes of steady, planar flow across a smooth, fixed circular cylinder, (Ref. 14) .. .. .. .. .. ... ... ... ... ... ...13 2.4 Strouhal-Reynolds numbers relationship with transverse force spectra for steady, planar flows around a smooth, fixed cylinder (Refs. 14, 67). .. .. .. .. ... ... ... .... ... ... ... ....14 2.5 Span-wise coherence length versus Reynolds number for steady, planar flow around a smooth, fixed cylinder (Ref. 68). .. .. ....15 2.6 Schematic transverse force and corresponding response power spectra for steady, planar flow around a smooth, transversely unconstrained cylinder (Ref. 74) .. .. .. .. ... ... ... ... ....16 2.7 Lift coefficient versus Keulegan- Carpenter number for oscillatory, planar flow around a smooth, transversely constrained cylinder (for constant Reynolds number, Re). .. .. .. .... .... ...18 2.8 Lift coefficient versus Keulegan- Carpenter number for oscillatory, planar flow around a smooth, transversely constrained cylinder (for constant frequency parameter, P3).. .. .. ... .... .....19 2.9 Schematic transverse force power spectra for oscillatory, planar flow around a smooth, transversely constrained cylinder for various Keulegan-Carpenter number, KC (Refs. 2, 32, 47). .. .. .. ....21 2.10 Span-wise coherence length for transversely constrained cylinder in oscillatory, planar flow (Ref. 50). .. .. .. .. .. ... ... ..21 2.11 Schematic transverse force and corresponding response power spectra for oscillatory, planar flow around a smooth, transversely unconstrained cylinder for various reduced velocities, V, (Ref. 47). .23 2.12 Span-wise coherence length for transversely unconstrained cylinder near a wall in oscillatory, planar flow (Ref. 39). .. .. .. ....24 2.13 Lift coefficient harmonics versus Keulegan- Carpenter number, KG, - curve fit of data from harmonically oscillated articulated cylinder in still water, e data from waves impinging on a smooth, fixed vertical cylinder (Refs. 12, 13). .. .. .. .. .. .... ... ....26 2.14 Lift coefficient versus Keulegan- Carpenter number for regular waves around a smooth, fixed vertical cylinder .. .. .. .. .. ... ....28 -2.15 Schematic transverse force power spectra for regular waves around a smooth, fixed vertical cylinder for various Keulegan- Carpenter numbers, KG (Refs. 12, 73). .. .. .. .. ... ... .... ....30 2.16 Maximum transverse response for various flow configurations (0 steady, planar flow), (o oscillatory, planar flow), (e wave flows). 30 3.1 Schematic diagram of transversely constrained experiment set-up. 38 3.2 Schematic diagram of transversely unconstrained experiment set-up. 39 3.3 Photographs of experimental set-up. .. .. .. .. .. ... .....40 3.4 Block diagram of measurement system. .. .. .. .. ... .....41 4.1 Definition sketch for the articulated tower showing in-line motion. 48 4.2 Contour lines defining Error surfaces for the in-line force (Ref. 15). 51 4.3 Definition sketch showing tower elements and idealized flow. ...58 4.4 Definition sketch for the articulated tower showing transverse motion. .. .. .. .. .. ... ... ... ... .... ... ... ...62 5.1 Cm, versus R, for harmonically oscillated articulated tower (transversely unconstrained). .. .. .. .. .. ... ... ... ... ...64 5.2 CD versus R, for harmonically oscillated articulated tower (transversely unconstrained). .. .. .. .. .. ... ... ... ... ...65 5.3 Contour lines defining error surfaces of the in-line force (KG = 8.4 and R, = 8.4 x 10') .. .. .. .. ... .. .... ... ......66 5.4 Phase comparison between measured and predicted in-line forces a KCC = 6.6 and R, = 9.4 x 104 (b) KC = 8.4 and R, = 8.4 x 104 -measured,..predicted) .. .. .. .. .. ... .... ... ..67 5.5 Comparison of Cm. for different flow types .. .. .. .. .. .. ...68 5.6 Comparison of CD for different flow types .. .. .. .. .. .. ....68 5.7 Lift coefficient harmonics for harmonically oscillated articulated tower (transversely constrained), CL(I) to CL(4) .. .. .. .... ...71 5.8 Lift coefficient harmonics for harmonically oscillated articulated tower (transversely constrained), CL(S) to CL(8) .............. .72 5.9 Lift coefficient harmonics versus beta, CL(j) and CL(2) .......... 73 5.10 Lift coefficient harmonics versus beta, CL(3) and CL(4) ....... .....74 5.11 Phase angle associated with the lift coefficients harmonics, 0(1) to 0(4) ................................ 75 5.12 Phase angle associated with the lift coefficients harmonics, O(s) to 0(8) . . . . 76 5.13 Comparison between measured and predicted transverse forces (Re = 104 and KC = 5.45) ...... ...................... 77 5.14 Comparison between measured and predicted transverse forces (R, = 2 x 104 and KC = 5.9) .......................... 78 5.15 RMS lift coefficient versus KC ......................... 79 5.16 Comparison of CLrms data with those from waves ......... ... 80 5.17 Range of data from wave forces hydrodynamic experiments (Ref. 27) ........ ................................... 81 5.18 Transverse force power spectra for constant Re(R = 1.15 x 104). 83 5.19 Transverse force power spectra for constant /fl( = 2,800) ..... 83 5.20 Thermistor power spectra for #3 = 1,834 ................... 84 5.21 Thermistor power spectra for P3 = 2, 161 .................. .84 5.22 Schematic illustration of thermistors locations ............... 85 5.23 Schematic illustration of the visualization of the vortex-shedding process ....... ................................. 86 5.24 Definition sketch of span-wise coherence length ............. 87 5.25 Span-wise coherence length for transversely constrained articulated tower ...................................... 88 5.26 Span-wise coherence length for transversely constrained articulated tower ...................................... 88 5.27 Comparison between measured and predicted transverse responses a) Re = 2.7 x 104 and KC = 5.0 (b) R, = 5.8 x 10' and KgC = 5.9 - measured, .... predicted) .......................... 92 5.28 Comparison of measured and predicted transverse responses (R, = 8.3 x 104 and KC = 6.7). ...................... 93 5.29 Comparison of measured and predicted spectra of transverse responses (Re = 8.3 x 104 and KC = 6.7) . 93 5.30 Comparison of measured and predicted transverse responses (R, = 10s and K C = 8.4) ........................ 94 5.31 Comparison of measured and predicted spectra of transverse responses (Re = 10' and KC = 8.4) . . 94 5.32 Thermistor power spectra at / = 14,265 and yrm.,s/D = 29.5%. 96 5.33 Thermistor power spectra at P = 14,265 and yrms/D = 20.9%. 96 5.34 Thermistor power spectra at 0 = 14,265 and y,,m/D = 8.2%. 97 5.35 Thermistor power spectra at P = 16,260 and yrm,,/D = 16.8%. 97 5.36 Thermistor power spectra at 0 = 16,260 and yrm,s/D = 8.8%. 98 5.37 Schematic of the effect of transverse motion on the behavior of vortex-shedding frequency . . 99 5.38 Thermistor power spectra at # = 5,400 and ym,/D = 56%. 101 5.39 Thermistor power spectra at P = 6,500 and yrms/D = 50%. 101 5.40 Thermistor power spectra at # = 9,900 and yrm,/D = 16%. 102 5.41 Thermistor power spectra at /3 = 9,900 and yrm./D = 28%. 102 5.42 Thermistor power spectra at # = 9,900 and yrm,/D = 35%. 103 5.43 Thermistor power spectra at P = 9,900 and yrm,/D = 40%. 103 5.44 Thermistor power spectra at / = 12,500 and yrm,/D = 33.5%. 104 5.45 Schematic of vortex-shedding frequency behavior (/ = 9,900). 104 5.46 Span-wise coherence length for transversely unconstrained articulated tower. .. .. .. .. .. . .. .. 105 5.47 Transverse motion power spectra for 8 = 5,400 . 107 5.48 Transverse motion power spectra for / = 6,500 . 107 5.49 Transverse motion power spectra for /3 = 9,900 . 108 5.50 Transverse motion power spectra for /3 = 12,500 . 108 5.51 Transverse motion power spectra for # = 14,265 .......... ..109 5.52 Tower measured free oscillation in still water ............... 110 5.53 RMS transverse motion versus frequency ratio .............. 113 5.54 RMS transverse response versus reduced velocity .......... ..114 5.55 Average highest 1/3 transverse response versus reduced velocity. 115 5.56 RMS transverse response versus Kuelegan-Carpenter number. 115 B.1 X-Y force transducer and wheatstone bridge circuit ........ ..130 B.2 Schematic of strain gauge amplifier/signal conditioning module. 131 B.3 Calibration curve for in-line force (for test runs at 0.4 > fd < 0.8 Hz.) ....... ................................ 132 B.4 Calibration curve for in-fine force (for test runs at 0.15 > fd < 0.3 Hz.) ....... ................................ 132 B.5 Calibration curve for transverse force .................... 133 B.6 Standard rigid linear displacement transducer .............. 133 B.7 Calibration curve of in-line linear displacement transducer. ... 135 B.8 Calibration curve of transverse linear displacement transducer. 135 B.9 Schematic of thermistor signal processing circuit .......... ..136 B.10 Schematic of thermistor testing set-up using flow visualization table. 137 B.11 Thermistor signal power spectra at various Reynolds numbers. 138 B.12 Schematic diagram and frequency response for second order, passive, Butterworth lowpass filter circuit (designed for the feed back control signal) ................................... 139 B.13 Samples of transverse force signals and their corresponding power spectra (a) before filtering (b) after filtering ................ 140 C.1 Computer code flow chart for CDCM ..................... 143 C.2 Computer code flow chart for CDCMN ................... 144 C.3 Computer code flow chart for CLEF ..................... 145 C.4 Computer code flow chart for CLEUF .................... 146 C.5 Computer code flow chart for ATVSR .................... 147 LIST OF TABLES 2.1 Categorization of flow and structural parameters influencing vortexinduced transverse loadings and response (numbers refer to references) 8 3.1 Test conditions for experiments I and II . 46 5.1 Samples of test results on in-line force data . 69 5.2 Test conditions for cases where large transverse motion were measured 90 5.3 Results of lift coefficients when transverse motion existed. 91 B.1 Specifications of strain gage amplifier/signal conditioning module. 131 B.2 Linear displacement transducer specifications . 134 B.3 Thermistor specifications . . 136 BA Linear drive motor specifications . . 141 LIST OF SYMBOLS A amplitude of transverse motion AR transverse amplitude to cylinder diameter ratio a in-line motion amplitude at the driving mechanism CD drag coefficient CD' drag coefficient associated with velocity raised to power n C., added mass coefficient CM inertia coefficient C8 structural damping coefficient CL lift coefficient CLn nth harmonic lift coefficient CLmi, nth harmonic lift coefficient of element "in" CLma maximum lift coefficient CL.rm root mean square lift coefficient d water depth d, water depth form the bottom hinge dr element length D cylinder diameter E[C] expected value of the square of the lift coefficient f frequency ( Hz) fd driving frequency ( Hz) - system natural frequency ( Hz) fT cylinder response frequency ( Hz) - vortex-shedding frequency ( Hz) - fluid frequency of oscillation ( Hz) FB buoyancy force FD drag force FL transverse (lift) force Fms root mean square force Fmax maximum transverse force Fv, root mean square of transverse force g acceleration of gravity h water depth H wave height la added mass moment of inertia Io cylinder mass moment of inertia about bottom hinge Im total cylinder mass moment of inertia about bottom hinge k, restoring moment due to weight and buoyancy of the cylinder k wave number KC Keulegan-Carpenter number If, stability parameter f cylinder length ts cylinder submerged length L vertical distance form bottom hinge to top hinge above the cylinder m added mass per unit length m, cylinder mass per unit length M ratio of frlfd MD moment about bottom hinge due to drag force Mfx moment about bottom hinge due to reaction "R" at the top hinge MfY moment about bottom hinge due to total transverse force Mg moment about bottom hinge due to weight and buoyancy of the cylinder ML measured moment due to total transverse force Mm measured moment due to total in-line force Mtotal total moment about bottom hinge N ratio of f/fd r. distance of zth tower weight component from the bottom hinge r,. distance of element n from the bottom hinge r distance of a general tower element from the bottom hinge Re Reynolds number R reaction of the in-line driving force at the top hinge S(nf) spectral density at frequency nf St Strouhal number S(f) auto-spectral density of signal x(t) at frequency f Ssu(f) auto-spectral density of signal y(t) at frequency f S (f) cross-spectral density of signals x(t) and y(t) at frequency f xiv T period of flow oscillation TV external generated turbulence U uniform flow velocity U.x maximum flow velocity U.MS rms of flow velocity V reduced velocity w, weight of zth component of tower Xm measured in-line linear motion of the tower ym measured transverse linear motion of the tower Y measured transverse linear velocity of the tower Y'm, measured transverse linear acceleration of the tower y, predicted transverse linear motion of the tower Yrms rms of transverse linear motion of the tower Y1/3 average of the largest 1/3 linear transverse motion of the tower y amplitude of transverse motion Yr.m maximum of rms amplitudes of transverse linear motion of the tower z span-wise coherence length of vortices flh hydrodynamic damping coefficient # frequency parameter (= Re/KC) 1.y(f) correlation coefficient between signals x(t) and y(t) at frequency f 5,2 average of least square errors S logarithmic decrement of free oscillation A, length of an element of the tower Art distance from in-line force reaction at top hinge to the center of in-line force transducer At distance from transverse force reaction at top hinge to the center of transverse force transducer 0 in-line angular deflection of the tower - in-line angular velocity of the tower - in-line angular acceleration of the tower v fluid kinematic viscosity - critical damping factor p, mass density of water ',- variance of transverse force o- circular frequency (rad/sec) 0(m) mth harmonic of the phase angle associated with lift coefficient Okm mth harmonic of the phase angle - transverse angular deflection of the tower - transverse angular velocity of the tower - transverse angular acceleration of the tower w cylinder frequency (radsec) Wd cylinder driving frequency (rad/sec) Wf fluid frequency of oscillation (rad/sec) o, vortex-shedding frequency (rad/sec) Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy VORTEX-INDUCED TRANSVERSE LOADING ON AN ARTICULATED TOWER By AHMED FAHMY OMAR August 1992 Chairman: Prof. D. Max Sheppard Major Department: Coastal and Oceanographic Engineering This research is an experimental investigation of vortex-induced transverse loading on fixed and compliant structures in nonplanar oscillatory flow. The effects of flow nonuniformity and transverse motion on the transverse force, vortex-shedding frequency, and vortex span-wise coherence length have been investigated. To quantify such effects, two types of experiments were performed. Each type of experiment consisted of a series of tests. All tests were performed by oscillating an instrumented 10 ft long, 6 inch diameter aluminum, articulated cylinder in a 30 ft diameter cylindrical tank with a water depth of 8.5 ft. In the first type of experiment, transverse motion was not allowed, while in the second type, the cylinder was allowed to move freely in the transverse direction. Reynolds numbers, R,, between 0.61 X 104 and 1.3 x 105 and Keulegan- Carpenter numbers, KC, between 2.4 and 9.35 were obtained. A new method, involving the use of miniature, quick response thermistors was employed for measuring frequency of vortex- shedding. For the range of Re and KG tested, transverse force amplitudes as large as 70% of the in-line force were measured. Transverse forces occurred at frequencies that were multiples of the driving frequency. The dominant vortex-shedding frequencies xvii clustered around one of the harmonics of the driving frequency. The specific harmonic depended on the value of KC. A mathematical model for computing transverse forces taking into account the dependency of the lift coefficient on both R, and KC as well as the fact that these forces have multiple frequency components has been also proposed. Using this model, the magnitude of the transverse force was found to be deterministic and repeatable. However, the phase of this force was random in nature. The constrained tower results for in-line and transverse force coefficients were found to agree well with data obtained by others for waves acting on vertical cylinders. Transverse motion was found to have a significant effect on vortex-induced loading. The effect of transverse motion was also found to increase the vortex correlation length, increase the lift coefficient (by at least two and one-half times) and alter the nature of the vortex-shedding. For the range of KC tested, the results showed that, the larger the value of KC, the larger the amplitude of maximum transverse motion and the larger the value of reduced velocity, V,., at which this maximum occurs. xviii CHAPTER INTRODUCTION 1.1 Problem Statement With the depletion of near shore oil reservoirs, exploration and production is moving to deeper water and more remote locations. In most cases, this means more severe environmental conditions and more stringent design and safety regulations. At the same time, there is a great need to reduce the cost of producing hydrocarbons. One way to cut the cost is to optimize structural designs. An essential component of optimization is being able to predict accurately the loadings to be experienced by the structure. This trend has recently led to a massive research effort into the design and assessment of the short and long term reliability of offshore structures. The design usually involves three major steps, first, the long term prediction of environmental conditions, second, the estimation of the forces associated with these environmental conditions and third, the determination of the effects of these forces on an intended structure and its ability to survive the expected extreme environment. It is interesting to note that by 1970, over 10% out of 200 drilling rigs had collapsed and a further 20% had suffered severe fatigue failure of structural members due to flow-induced vibrations (King, 1974). This is in addition to frequent shut-down of operation for days due to the large oscillations induced by fluid-induced forces; see Lewis et al. (1991) and Koch et al. (1991). In the past two decades, a significant amount of research on fluid-structure interaction problems has been conducted. This research can be divided into two categories, those concerned with i) the fluid-induced forces and ii) the fluid-induced response and/or vibration. The latter category has received the most attention. 2 The fluid-induced forces are comprised of in-line forces and vortex-induced transverse (lift) forces. Many studies have been done to predict the fluid in-line forces acting on a cylinder in steady and oscillating flow. The majority of those studies have been based on the Morison equation (Morison et al., 1950), where the two infine force components of drag and inertia are identified. The equation expresses these forces in terms of the velocity and acceleration of the fluid particles at the location of the cylinder. In comparison to in-line forces, vortex-induced transverse forces have received little attention. This is most likely due to the complexity of the vortexshedding phenomena and the lack of a clear cut methodology for analyzing the data. Structural elements are constantly subjected to loading due to wind and/or ocean currents and waves. Most flow situations encountered in nature are turbulent, nonplanar (nonuniform), and unsteady. To further complicate matters, the structural element of interest is often in close proximity to other members, compliant, and perhaps partially covered with biofouling. To predict the vortex-induced transverse loading and response of such a structural element under such complex flow conditions is truly a challenging task, but nevertheless, one that is routinely faced by design engineers in the offshore industry. The problems of in-line and transverse loading and the corresponding structural response are all related and interdependent. However, the quantities involved are difficult to isolate and to measure; and, therefore some of the reported data have been affected by the techniques, apparatus and/or instrumentation used. In addition, the diversity of the investigators' backgrounds, ranging from fluid to structural mechanics, resulted in differences in approaches taken and differences in the manner in which results were presented. This diversity of backgrounds has its advantages, but the lack of uniformity in data reporting makes it difficult to compare and extend data sets. The focus of this research was on the investigation of some of the fluid-structure interaction problems, in particular on quantifying the effects of flow nonuniformity 3 and transverse motion on vortex-induced transverse loading and on developing a mathematical model for predicting transverse forces. It is extremely difficult to obtain a theoretical solution for vortex-induced transverse force s especially at high Reynolds numbers. The difficulty is partly due to incomplete knowledge of the flow field around the structure, and to problems associated with the coupling of structural motion and fluid flow. Consequently, the approach for obtaining the solution to this problem has been the same as taken here, i.e. experimental. 1.2 Research Objectives The overall objectives of this study were 1. To design and conduct experiments to establish the dependence of the vortexshedding process, vortex span-wise coherence length, magnitude of the vortexinduced transverse forces and transverse motion on the in-line flow and structure parameters. The physical model must be sufficiently large to cover meaningful ranges of Reynolds (R,) and Keulegan- Carpenter (KC) numbers and 2. To develop an improved mathematical model for predicting vortex-induced transverse forces. To achieve these objectives, the following tasks were performed: 1. Design, construct and instrument an articulated tower system. Two designs were necessary, one with and one without transverse motion. 2. Perform two types of experiments by sinusoidally oscillating the articulated tower in an otherwise still-water tank. In the first type the tower transverse motion was constrained. During the series of tests performed for this type, the following quantities were measured 4 " the in-line and transverse forces. " the frequencies of vortex shedding along the physical model. " the in-line motion. During the series of tests for the second type, transverse motion was allowed and the following quantities were measured " the in-line forces. " the in-line and transverse motion. " the frequencies of vortex shedding along the physical model. 3. Develop a mathematical model for predicting the vortex-induced transverse forces. 4. Quantify the effects of flow nonuniformity and structure's transverse motion on the lift coefficients, frequency of vortex-shedding and spanwise coherence length of vortices. CHAPTER LITERATURE REVIEW In this chapter, a literature review and discussion of topics relevant to the work undertaken are presented. The review is limited to the vortex-shedding process and its induced transverse forces and concentrates on smooth, rigid cylinders in 1) steady, planar, 2) oscillatory, planar, 3) oscillatory, nonplanar, and 4) wave flows. The effects of transverse motion on the vortex-shedding process, span-wise coherence length and transverse loading are also discussed for those cases where sufficient data exist. It is the author's opinion that the above classification of quantities affecting vortex-induced loading and response will be beneficial in analyzing data from various investigators and in understanding the processes involved. Analytical and computational approaches to the transverse force problem have been hampered by the complexity of the processes. Thus, most of the work to date has been experimental, guided by dimensional analysis techniques. For this reason, only experimental work will be discussed in this review. 2.1 Overview When a viscous fluid such as water or air flows past a bluff body with sufficient velocity, flow separation occurs and a wake region is formed. Over a wide range of flow and structure parameters of interest, vortices are observed to form near the points of flow separation. For symmetric structure shapes, void of sharp edges, such as right circular cylinders, vortices are formed on both sides of the body. Under certain conditions these vortices remain attached to the body while under other conditions they are shed from the body in or out of phase with each other. The net effect of this phenomenon is a fluctuation in the points of flow separation, which in turn causes 6 a time varying distribution of normal and tangential stresses over the body. This results in time dependent in-line and transverse loads on the structure, even when the flow is steady and planar (uniform). The processes associated with flow separation are complex and difficult to predict. Yet minor changes in the separation point can result in relatively large changes in both the in-line and the transverse forces on the structure. This flow instability problem is sensitive to perturbations such as those introduced by surface roughness, motion of the body, free stream turbulence, flow orientation relative to the structure, flow around the ends of the structure, etc. In an attempt to understand and model this phenomenon, researchers have isolated various aspects of the problem starting with the (seemingly) simplest case of uniform, steady flow and moving toward the more complex flow and structure situations. The processes are of course nonlinear and thus their individual effects cannot be simply superimposed to obtain the combined effect. However, much can be learned about the mechanisms involved and some guidance for the design engineer can be achieved by such a process. Vortex-induced loads are of interest in a number of engineering disciplines and of particular importance in the design of offshore structures. Structural elements of interest are often compliant and subjected to complex flows (turbulent, oscillatory and nonuniform). For such a situation, the main danger from the vortex-induced loading arises from the possibility of resonance created by the vortex-shedding frequency being close to the system natural frequency (or one of its multiples). Large and damaging amplitudes of oscillation can result from the complicated and pernicious mechanism of resonance which can occur over a considerable range of conditions. The interaction between the flow and the structure's motion causes the frequency of vortex-shedding to be controlled by the response. This can result in what is known as "lock-in". Although the fluctuating pressure that causes the transverse force is predominantly at right angles to the direction of the approaching flow, it can also produce dynamic 7 forces in-line with the flow. The induced transverse motion can also increase the time-average drag significantly. The problems of in-line and transverse loading and the corresponding structural response are all related and interdependent as pointed out in several excellent reviews on-this subject (e.g. Bearman, 1984; Sarpkaya, 1979; Chen, 1987; Griffin, 1984). On the other hand, transverse forces depend on the nature of the flow, the structure's geometric parameters, and (when it exists) the structure motion. This is illustrated in the diagram in Fig. 2.1. As stated above, the flows are very complex. Perhaps the most surprising thing about this phenomenon is that it displays some degree of order and repeatability and thus predictability. Figure 2.1: Flow chart for cylinder-loading response. The quantities involved are difficult to isolate or to measure and therefore some of the reported data have been affected by the techniques, instrumentation and/or 8 apparatus used. In spite of these difficulties, researchers have managed to acquire at least a qualitative understanding of most of the processes and how the various geometric and flow parameters affect the transverse force. The matrix presented in Table 2.1 is an attempt to classify the existing work from the point of view of the more important geometric and flow parameters affecting the vortex-shedding process and the resulting transverse forces. Reference numbers for some of the more recent and historically important papers on the various subjects are given in the matrix. For a more complete list of references on these subjects the reader is referred to the review articles by Bearman (1984), King (1977) and Sarpkaya (1979). As one can see from Table 2.1, some areas have received more attention than others (e.g. steady and oscillatory planar flows have attracted most of the interest while other flows like oscillatory nonplanar and waves have received less attention). The author is aware that important proprietary research has also been conducted in this field, but these results are, of course, not available for review. Table 2.1: Categorization of flow and structural parameters influencing vortex-induced transverse loadings and response (numbers refer to references). Structure Parameter _______ (Rigid Cylinders)__Inline Transverse End Flow Smooth Rough Fixed Motion Motion L/D BR Effcts 1, 18, 34,63, 1,18 1, 76, 77,55, 63,33 18,34, 63, 77 77 19 Steady, Planar 71, 76, 77, 55, 19, 75, 67, 17 71, 75, 33 ____________ 19, 75, 67, 33, 17 60, 61, 2, 40, 46, 62, 60, 61, 62, 60, 61, 76, 80, 2, 40, 46, 71, 74, 80 43 Oscillatory, 71, 74, 76, 80, 43, 71, 45, 56, 43, 50, 32, 3, 45, 74, 42, 56, 70 Planar 50, 32, 3, 45, 42, 70 42, 69, 47, 78, 17 ____________69, 47, 78, 56, 70, 17 _______ __Oscillatory, 13, 11 13, 11 Non-planar ______ _7, 30, 65, 12, 11, 11 7, 30, 12, 11, 65, 11, 7, 65, Regular Waves 44,76,9,66 44,76,9,66 44,76 44,76 73, 23 73. 23 Irregular NNaves 9 Some of the earliest studies were on steady, planar, relative flow, perpendicular to rigid, fixed, circular cylinders (see King, 1977; Fleischmann and Sallet, 1981; Blevins, 1990). The term relative flow is used since in some cases the cylinder was towed through still water while in other cases the fluid was forced to move around a fixed cylinder. This was followed by experiments with oscillatory, planar, relative flow around circular cylinders. More recent experiments with oscillatory, nonplanar, relative flows have been performed both by moving articulated cylinders in still-water (Chakrabarti et al., 1983) and by subjecting cylinders to surface waves (see Bearman, 1988a; Isaascson et al., 1976, 1977; Sawaragi et al., 1977; Chakrabarti et al., 1976). While the above work was proceeding, other aspects of the problem such as the effects of surface roughness (e.g. Achenbach and Heinecke, 1981; Bearman, 1988a; Sarpkaya, 1976a, 1976b, 1990b; Wolfram et al., 1989), in-line and/or transverse motion (e.g. Chakrabarti et al., 1984; Maull and Kaye, 1988; Bearman and Hall, 1987; Donazzi et al., 1981; King, 1974; Laird, 1962; McConnell and Park, 1982a; Sarpkaya, 1978; Sumer and Fredsoe, 1988; Torum and Anand, 1985; Verley, 1980), structure aspect ratio (i.e., cylinder length to diameter ratio) (e.g. West and Apelt, 1982), flow blockage (e.g. Kozakiewicz et al., 1991; Torum and Anand, 1985; Ramamurthy and Ng, 1973; Yamamoto and Nath, 1976), free stream turbulence (e.g. Torum and Anand, 1985), end effects (e.g. Torum and Anand, 1985; Matten et al., 1978), etc. were also being investigated. One of the factors affecting the overall lift coefficient is the spanwise coherence length of the vortices (i.e. the length along the axis of the cylinder where the vortices are being shed in unison). Even though this parameter length has been the subject of much discussion, very few measurements with circular cylinders in water have been reported in the literature (see King, 1977; Wolfram et al., 1989; Obasaju et al., 1988; Kozakiewicz et al., 1991). It is thought that this length depends on: cylinder aspect ratio, end effects, free stream turbulence, nonuniformity of the in-line flow, roughness and roughness gradients along the cylinder, two and three 10 dimensionality of the flow (such as would be generated by uni- and multi-directional surface waves, respectively), nonaligned currents and waves, etc. During the past few decades, much has been learned about the vortex-induced transverse force problem; yet it is safe to say that more work is needed before reliable information required by design engineers is available. In the following sections a review of the vortex-induced transverse forces (in particular, the lift coefficients and frequency of vortex-shedding) and span-wise coherence length of vortices on smooth, fixed and transversely unconstrained cylinders in 1) steady, planar, 2) oscillatory, planar, 3) oscillatory, nonplanar, and 4) wave flows is presented. Lock-in conditions and their relation to the vortex-shedding frequency and the system natural frequency are also presented where sufficient data exist. Flexible cylinders will not be discussed in this review. This is because it is difficult to devise experiments with flexible cylinders that provide quantitative results specifically on the effects of flow nonuniformity. Results obtained from the more easily controlled experiments discussed here could; however, be helpful in analyzing the loading and response of flexible cylinders. 2.2 Steady, Planar Flow 2.2.1 Transversely Constrained Cylinder in Steady, Planar Flow Vortex-induced transverse forces on a smooth, circular cylinders in steady, planar, flows have received considerable attention during this century (see e.g. King, 1977; Fleischmann and Sallet, 1981). For this type of flow, it was found that Reynolds number, R, (relative magnitudes of inertia and viscous forces), was the most important parameter to characterize the flow around a circular cylinder. For steady, turbulent flows around right circular cylinders, the in-line loading has been formulated in terms of a drag force that is proportional to the square of the relative speed, the mass density of the fluid, and a projected area. The constant of proportionality is one-half the drag coefficient (CD/2). Plots of CD versus R, for this flow situation can be found in numerous publications (e.g. Sarpkaya and Isaacson, 1981). In the 11 absence of a better formulation, a similar model for the transverse or lift force in terms of the in-line relative speed squared, fluid mass density, and projected area has also been used. In this case, the constant of proportionality is one-half the lift coefficient (CL/2). In both cases, the coefficient is a catch-all term whose value depends to some degree on the flow and most, if not all, of the quantities given in Table 2.1. Sarpkaya and Isaacson (1981) showed that there is considerably more scatter in CL data for this flow than say for CD. In an attempt to show both the consistency and the scatter in the available experimental data, CL(rms) (= 2Fyrs/pwDU2) versus R, (= UD/v) has been plotted in Fig. 2.2 for twelve different investigators. Most of these data were taken from Sarpkaya and Isaacson (1981), but the results of two more recent studies (Dronkers and Massie, 1978; van der Vegt and van Walree, 1987) have been added. However, it should be pointed out that many of these data are for experiments conducted some years ago in air. In addition, it is possible that at least some of the cylinders experienced transverse vibration during the tests. Dronkers and Massie (1978) had a fixed vertical cylinder in a circulating flume while van der Vegt and van Walree (1987) towed a horizontal cylinder in still-water. These data appear to only add to the scatter. This lack of agreement in the reported data has been attributed to differences in free stream turbulence, aspect ratios, flow uniformity, etc. In structural design, the frequency components of the transverse force are perhaps as important as the magnitude. The basic phenomenon of vortex-shedding is illustrated in Fig 2.3, where the major regimes of vortex-shedding from a right circular cylinder in steady, planar flow are sketched, based upon the observations of various investigators. Recently, a flow visualization by van der Vegt and Walree (1987) has confirmed some of the patterns by which the vortex-shedding process occurs as presented in Fig. 2.3. In all the investigations carried out in this category, the frequency with which the individual vortices are shed was found to be proportional to the ratio U/D, 1.50 1.00 Symbol 0 A A X o U 0 + V I I in iiitl I I I 111111 Ref. # 57 57 57 57 57 57 75 57 57 19 57 57 00 I I I 0 I I 11I1IIII 13 & A Ah A A 00O Illllll 102 103 104 105 106 Re Figure 2.2: Lift coefficient versus Reynolds number for steady, planar flow around a smooth, fixed cylinder. so that f,, = StU/D, where the constant of proportionality is called Strouhal number and f, is the frequency of vortex-shedding. Later St was presented versus Re in an envelope, within 10% accuracy over a large Reynolds number range, see Fig. 2.4. Transverse force spectra (in this case, spectral density versus Strouhal Number) for various values of R, are also shown in Fig. 2.4. These spectra were computed by Schewe (1983), using data from steady flow around a fixed, rigid cylinder in a pressurized wind tunnel. It is important to note the relative energy levels in the various flow regimes. Single spike spectra similar to Schewe's spectra for Re = 1.3 X 10' (see Fig. 2.4) were recently obtained for water flows by van der Vegt and van Walree (1987). Thus for all practical purposes, it appears that vortices are shed at a single frequency for 60 < R, < 2 x 10' and well defined by a Strouhal number of about 0.2 for smooth cylinders. 111ii1i I i Q) E a -J 0 0.50 0.00 L..._ o oJo I I fill I Z 65 Figure 2.3: (Ref. 14). 3X 0 SZ Re < 33 X 106 LAMINAR BOUNDARY LAYER HAS UNDERGONE TURBULENT TRANSITION AND WAKE IS NARROWER AND DISORGANIZED 3.5 X 106 4 Re RE-ESTABLISHMENT OF TURBULENT VORTEX STREET Regimes of steady, planar flow across a smooth, fixed circular cylinder, Patal (1989) and others presented the relationship of Fig. 2.4 in the form of empirical equations: f, = (StU/D) = St (1 + 19.7/R,)(U/D) R, < 60 60< R,< 2 x10" In the turbulent regime (2 x 105 < R, < 7 x 106), the Strouhal number varies between 0.15 and 0.4 depending on the intensity of the free stream turbulence. In this region, the spectrum is broad banded, reduced in magnitude and very sensitive to flow disturbance. In the supercritical regime (R, > 7 x 106), the spectrum becomes narrow banded once again. (2.1) (2.2) Re < 5 REGIME OF UNSEPARATED FLOW 5T0 15 C Re < 40 A FIXED PAIR OF VORTICES IN WAKE ~40 Re < 90 AND90 < Re < 150 TWO REGIMES IN WHICH VORTEX STREET IS LAMINAR 150 C Re < 300 TRANSITION RANGE TO TURBULENCE IN VORTEX 300 4 Re Z 3 X 105 VORTEX STREET IS FULLY TURBULENT Hie M Re :=4.1 x 106 Figure 2.4: St rouhal- Reynolds numbers relationship with transverse force spectra for steady, planar flows around a smooth, fixed cylinder (Refs. 14, 67). The vortices shed along a cylinder can be in or out of phase with each other. The total transverse force on the cylinder is very sensitive to these phase angles. It is important to understand what quantities control or influence the shedding process. A parameter known as the span-wise coherence length (a length over which the vortices are considered well correlated) has been used as a measure of the coherence between the vortices shed along a cylinder. In general, the greater the coherence length the larger the total transverse force. This correlation length has been observed to vary with Reynolds number, surface roughness and free stream turbulence (King, 1977). The only data found that demonstrate the parameters influencing the coherence length support the importance of the Reynolds number (R,~). These data (by Scruton, 1967; taken from Overvik, 1982) are shown in Fig. 2.5 and generally indicate a reduction in the span-wise coherence length with increased Reynolds number. z 0 F-J c 0 0 1 Z 0 LU Re Fiue25cpnwscoeec eghvessRyod ubrfrsedpaa flwaon0 moh ie yidr(e.6) 2.ge2. : aSersely e Ucostrine Clndersu n nod Sbr steady, PlananFlo When the structure is allowed to move in the transverse direction and its support is such that a restoring force exists (resulting in a system natural frequency), there can be strong interaction between the transverse response and the transverse loading as indicated in Fig. 2.1. Extreme caution, therefore, should be exercised when using lift coefficient data obtained for fixed cylinders when the structural element is compliant. When the vortex-shedding frequency is in the vicinity of the natural frequency of the cylinder-support system, large transverse excursions can occur which in turn result in further increases in the transverse force. This large amplitude motion at the natural frequency can change the frequency of vortex-shedding to that of the oscillation frequency and "lock-in" occurs. Even though a number of experiments have been performed with cylinders free to move in the transverse direction, only a few investigators (e.g. King et al., 1973; King, 1974; Griffin and Koopman, 1977; Torumn and Anand, 1985) have reported 16 information on the response of the cylinder. The spectra of the transverse force and corresponding response by Torum and Anand (1985) were the only data found that show the relationship between the vortex-shedding frequency (ff) and response frequency (f,.). Torum and Anand's investigation was primarily to study wall effects, however, their results for the largest cylinder gap to diameter ratio (i.e., G/D = 3) shown in Fig. 2.6 should be very similar to an unobstructed cylinder. For these data, the authors did not report the values of Reynolds numbers. A value of v- = 10' M2/sec was assumed in the computations of Re, shown in the figure. FORCE RESPONSE 1 2 3 4 f / Figure 2.6: Schematic transverse force and corresponding response power spectra for steady, planar flow around a smooth, transversely unconstrained cylinder (Ref. 74). The spectra given by the authors were only for the conditions after lock-in. For those conditions, Fig. 2.6 shows that the vortex-shedding frequency displays a dependency on Reynolds number as for the fixed cylinder. As the vortex-shedding frequency moves further from the system natural frequency the response at the natural frequency decreases while the response at the vortex-shedding frequency increases. In a plot of vortex-shedding frequency versus velocity their results show that the vortex-shedding frequency follows the St = 0.2 relationship except in the neighborhood of the natural frequency where lock-in occurs. 17 It is known that the transverse motion of the cylinder has a significant effect on the vortex correlation and consequently on the transverse force. It has been demonstrated by Koopman (1967) and Toebes (1969) that the vortex span-wise coherence length for a transversely unconstrained cylinder exposed to a steady, planar current increases drastically with increasing amplitudes of oscillations. However, the author has not been able to locate information on lift coefficients for this flow situation. 2.3 Oscillatory, Planar Flow 2.3.1 Transversely Constrained Cylinder in Oscillatory, Planar Flow In oscillatory, planar flow around rigid, stationary cylinders both the acceleration and relative velocity of the free stream are constantly changing with time, but are uniform along the cylinder. The dimensionless groups that characterize this flow situation are the Reynolds Number (R, = U,,,aD/v) and Keulegan-Carpenter Number (KC = UmaTID = 21rA/D). Combinations of these groups such as / = Re/KC have also been used to correlate and present experimental data with varying degrees of success. Several different lift coefficients have been used (CLma-, = 2Fvjmax/pDUmax CLrms = 2FrmsIpDUmnax, CLrms = 2Fyrms/pDUr;ms, etc.) to present results (often without specifying which was used). Since the transverse force had been found in nature to exhibit some degree of irregularity, many investigators have commonly used an rms lift coefficient (CLrm,) and so is this review. The investigations for this flow were conducted by oscillating the flow past a fixed cylinder or by oscillating a rigid cylinder in still-water. Some very good work has been done in this area resulting in the largest data set for CL of any of the categories outlined in Table 2.1. In an attempt to compare the results, the author compiled rms lift coefficient data for smooth cylinders from a number of these investigations and plotted them versus KC. Some of the data were taken holding R, constant while other investigators maintained 3 constant. These data are plotted separately in Figs. 2.7 and 2.8. Those data not reported in terms of CLrms = 2Fwrms/pDU:ax 18 were converted to this definition prior to plotting. The data for Chaplin (taken from Bearman, 1988b) and Justesen (1989) in Fig. 2.7 had to be converted. Other investigators' data (such as Bearman et al., 1984; Maull and Milliner, 1987) are not shown since they presented their data in a manner that would be difficult, if not impossible, to convert to the coordinates used in these figures. In Fig. 2.7 recent data from Longoria et al. (1991) are presented along with those for Sarpkaya (1990a) and Skomedal (1989). Both plots show that the lift coefficient is a maximum at KC between 10 and 12. There is surprisingly good agreement among the data within each of the two plots. The fact that both plots have the same shape and magnitude means that for oscillatory, planar flow the lift coefficient depends primarily on the Keulegan-Carpenter number. II I I II I I Ii I I I I IIII 2 0) E %..o 01 01AAR& MAA I I I 1 1 III 1 10 100 KC Figure 2.7: Lift coefficient versus Keulegan-Carpenter number for oscillatory, planar flow around a smooth, transversely constrained cylinder (for constant Reynolds number, Re). The basic nature of the flow in this category depends on the period of time for which the flow continues in one direction before it reverses. If the period is very short, Symbol Ref # Re 0 25 (0.3-1) x 104 A 60 (2-20) x 104 V 49 7 x 104 105 * 61 (1-2) x 105 0 42 2.5 x 105 A 42 5x 105 * 42 7.5 x 105 a 42 1.0 x 106 J1~ 0vI* SA 59 4720 _E 59 6555 1-f ,3 o 79 >18600 GE11 -0 0 10 100 KC Figure 2.8: Lift coefficient versus Keulegan-Carpenter number for oscillatory, planar flow around a smooth, transversely constrained cylinder (for constant frequency parameter, /3). there will not be sufficient time for the vortices to form before the flow reverses. If the period is very long, the flow will be quasi-steady and will have the character of streaming flow, first in one direction and then in the other. For flow periods between these extremes some downstream vortex effects will occur. In nondimensional terms, it was found that, the parameter that best determines the general character of vortexshedding from a circular cylinder in oscillatory flows is the ratio of how far a fluid particle moves in one half-cycle to the characteristic cylinder dimension that the particle flows past. This nondimensional term is what is known by the KeuleganCarpenter number "KC = UmaTiD". Since the early seventies many vortex-shedding flow visualization studies have been conducted for this flow situation. Most of these investigations were carried out to further the understanding of the mechanisms and thus help in the development 20 of numerical models for predicting transverse force. Detailed descriptions of these processes have been given by Bearman (1988b), Sarpkaya (1976a), Skomedal et al. (1989) and Williamson (1985). Even though there is general agreement on the nature of the vortex shedding processes there are differences among researchers regarding the details. The subjectivity in the flow visualization techniques used to measure the conditions under which vortices are shed, the number shed, etc. probably accounts for many of the differences. Experimentally, the frequency of the vortex-shedding is computed from the spectral analysis of the transverse force. It is surprising that more investigators have not reported transverse force spectra. The spectra that have been reported (Justesen, 1989, KG = 1.7 to 15.6; Bearman and Hall, 1987, KG = 36.13; McConnell and Park, 1982b, KG = 37.7), are shown, schematically, in Fig. 2.9. The intent of this plot is to illustrate the behavior of the frequency and not the magnitude of the spectral density of the transverse force. The results are by no means conclusive, but the vortex energy for this flow appears to cluster around one of the harmonics of the in-line driving (or flow oscillation) frequency with moderate energy in the surrounding harmonics, depending, primarily, on the value of KCC. As KG increases the frequency of vortex-shedding increases. Correlation measurements have also been made for a fixed cylinder in oscillatory, planar flow, but unfortunately, only those results for correlation length measured by Obasaju et al. (1988) have been reported. They measured the span-wise correlation of vortex-shedding for a range of KG from 4 to 55. Their results show that the correlation length does not decrease monotonically with increasing KG. The highest correlation length was obtained at KG = 10 (see Fig 2.10) and at this value it was approximated by 4.5D. They also found that for KG > 30 the correlation is no longer sensitive to KG. _4,KC = 37.70 KC s30.13 A IIJVYK7 KC a 15.6 KC : 13.6 . 1 A --IN I" I I 1 I 4 KC2 12.5 KC.1O.6 S I I KC 8.8 KCa 6.8 / 'K S.0 I I'I KC 2.8 KC = 1.7 0 2 4 6 a 1B -- 1 IT Figure 2.9: Schematic transverse force power spectra for oscillatory, planar flow around a smooth, transversely constrained cylinder for various Keulegan-Carpenter number, KC (Refs. 2, 32, 47). 1 0.9 KC= 10 0.8 0.7 " 0.6 C, o 0.5 0 C .2 0.46 L 0.3 o 0.2 0.1 KC =22 I 0 1 2 3 4 5 6 7 8 9 10 correlation length (z/D) Figure 2.10: Span-wise coherence length for transversely constrained cylinder in oscillatory, planar flow (Ref. 50). -I- i I L / I I SI I I K 11 1 I I 1 1 I I I I I I I I I I I a I I I I I I l "" "W i "I- 1 II 1 1 1 % I K! 1 1. II I I I I I I I I l A = - KC a 12.5 . Kq= 106. a a . . . I K m 4.9 22 2.3.2 Transversely Unconstrained Cylinder in Oscillatory, Planar Flow The work conducted for this case can be put into two categories; 1) that which is concerned with the effect of the transverse motion on the transverse force (McConnell and Park, 1982a, b; Sarpkaya and Rajabi, 1979) and 2) that which is concerned with determining the conditions under which transverse motion can be excited (Bearman and Hall, 1987; Sarpkaya and Rajabi, 1979; Sumer and Fredsoe, 1988, 1989). In general, when the cylinder is allowed to have transverse motion, the vortex-shedding frequency as well as the strength of the vortices is modified. For this situation KC alone is no longer adequate to characterize the transverse loading. This was clearly demonstrated by Sumer and Fredsoe (1988, 1989) in their experiments that covered a wide range of KC (KC = 10 to 100) and a large range of reduced velocity, V = Um/Df, = 2r(A/D)(f./f,) = KC(f/f,). Their results demonstrated that, at least for the range of KC tested, both V and KC are necessary to describe the behavior of the transverse response. Their plots of transverse motion due to vortexshedding, i.e., Yrms versus V for constant values of KC show that for lower KC (up to 10) a single spike exists. As KC increases, the number of spikes increases until at KC 100 the response versus V is flat for 6 < V < 11. The reader is referred to their paper for a detailed interpretation of this behavior. To the author's knowledge, McConnell and Park (1982a) are the only investigators reporting lift coefficients for this situation. Their results showed that the lift coefficient increased up to twice that for a fixed cylinder. Others presented only structure response information such as yrms/D or ymax/D versus V or f/f,, ratio. As shown in Fig. 2.1, when transverse motion exists there can be an interaction between the response and the transverse force. The level of this interaction is very much dependent on how close the stationary vortex-shedding frequency is to the system's natural frequency. In order to study this phenomenon the transverse force and response should be measured simultaneously. Of the literature reviewed, 23 only McConnell and Park (1982b) reported spectra for both transverse force and the corresponding response. Schematized versions of these spectra are presented in Fig. 2.11. The frequency for the force and response spectra has been normalized by the cylinder natural frequency in still-water. Their results showed for 4.4 < V < 6.6 and f, /fd = 6.22 lock-in occurred, i.e., f/f,, = f,, = 1. This show that it is not necessarily for f/fd to be an integer to have a lock-in. Their results also show (at least qualitatively) the influence of the transverse motion on the frequency of vortex-shedding. FORCE RESPONSE Vr = 7.11 Vr = 6.06 Vr = 4.24 > f/f 1-f/f n 0 2 4 n 0 2 4 Figure 2.11: Schematic transverse force and corresponding response power spectra for oscillatory, planar flow around a smooth, transversely unconstrained cylinder for various reduced velocities, V, (Ref. 47). Data or information on the vortex span-wise coherence length for this category was not available until recently when Kozakiewicz et al. (1991) measured and reported the correlation length for a vibrating cylinder near a wall. The study is not directly related to the work undertaken here as it was mainly concerned with the wall effect on the vortex correlation; however, some insight can be gained from the results. Their results for the largest gap/diameter ratio (that is GID = 2.3) are reproduced 24 in Fig. 2.12. As shown in this figure the correlation between vortices is largest at KG = 6 (as compared to KG = 10 for a fixed cylinder [Obasaju et al., 1988]) and like the case of fixed cylinder is mainly dependent on KG. The results of their study also showed that for a fixed KG the larger the amplitude of transverse motion the larger the correlation. 0.9 0.8 oX -0.7 "- 0.6 .) " 0.5 CD 0 o 0.40.3 0 0.2 0.1 0 0 2 46 8 10 12 correlation length (z/D) Figure 2.12: Span-wise coherence length for transversely unconstrained cylinder near a wall in oscillatory, planar flow (Ref. 39). 2.4 Oscillatory, Nonplanar Flow The final flow configuration to be considered in this review is that of nonplanar, oscillatory relative flow around rigid, circular cylinders. Very little work has been done with these flow conditions as can be seen from Table 2.1. The need for information on the effects of flow nonuniformity and transverse motion on transverse force and vortex span-wise coherence length was the motivation for the work of this dissertation. Both transversely constrained and unconstrained cylinders will be discussed in this section. Surface waves acting on vertical cylinders are also included in this 25 category, but will be treated separately due to the effects of the vertical component of water particle velocity on the transverse force. The question is, what effect does flow nonuniformity have on the transverse force? More specifically, what effect does it have on the span-wise coherence of the vortices? Does the coherence length of the vortices diminish with nonuniformity as a result of KG gradients along the cylinder or do the higher energy vortices, associated with the regions of higher velocity and vorticity, dominate the shedding process? Also, what effect does the transverse motion of the structure have on the vortex-shedding process and transverse force? Chakrabarti et al. (1983) and Chakrabarti and Cotter (1984) conducted experiments on an articulated cylinder where the top was oscillated sinusoidally while constraining the transverse motion. They measured local and total forces on the cylinder and, among other things, investigated the frequency and magnitude of the transverse force for a range of KC values. They used a five term Fourier series (similar to Mercier, 1973) to represent the measured transverse force, defining a different lift coefficient for each term in the series. These lift coefficients are reproduced in Fig. 2.13 but will be discussed in the following section on waves. They did not calculate the phase angles associated with the different harmonics. However, they advised the use of a random phase angle to calculate the transverse force. Their results also show that the dominant frequencies of vortex-shedding are dependent on KG and cluster around the driving frequency for the lower values of KC. As KG increases, the frequencies cluster around a value twice the driving frequency but became less organized at higher KG. The trend continued but the basic assumption of clustering to one of the multiples of the driving frequency seemed to break down. The author was unable to find any published data on vortex span-wise coherence length for this flow category. Likewise, no information was found on the effect of transverse motion on transverse force and vortex span-wise coherence length. O.Oo0Lt 1. L -' -'' '.j L t~ L-' 00 00 0 5 10 15 20 25 5 10 15 20 25 0 5 10 15 20 25 (a) KC (b) KC (c) KC Figure 2.13: Lift coefficient harmonics versus Keulegan-Carpenter number, KC, - curve fit of data from harmonically oscillated articulated cylinder in still water, * data from waves impinging on a smooth, fixed vertical cylinder (Refs. 12, 13). 2.5 Wave Flows Flows generated by regular, unidirectional surface waves impinging on fixed vertical cylinders are discussed separately due to the three dimensionality of the particle motion and its potential effects on the transverse force. Several researchers (Bearman et al., 1985; Bearman, 1988b; Bidde, 1971; Chakrabarti et al., 1976, 1983; Chakrabarti and Cotter, 1984; Isaacson and Maull, 1976, 1977; Sawaragi et al., 1976, 1977; T0rum and Reed, 1982) have conducted experiments with regular waves and have reported lift coefficient data as a function of KC. Others (e.g. Maull and Kaye, 1988) have observed the effect of transverse motion on the in-line motion and transverse forces in regular waves. Random waves have also been investigated (see Bearman et al., 1985; Graham, 1987). Chakrabarti et al. (1976) measured vortex-induced transverse forces on local sections of a fixed vertical cylinder (same articulated cylinder they oscillated in still water) in regular waves. They used the same series technique as outlined in the previous section to obtain the lift coefficients. Data reported for the first three harmonics of lift coefficient are plotted in Fig. 2.13 along with curves representing (i.e. polynomial curve fits) data for the articulated tower discussed above under oscillatory, 27 nonplanar flows. Note that CL(1) and CL(3) from the articulated tower experiments are lower than the corresponding values for waves while CL(2), which is the larger of the coefficients, is about the same for waves and the oscillated articulated tower. More data and comparisons are needed before conclusions can be drawn for this flow situation. The experiments performed in this category (wave flows) covered a wide range of apparatus sizes ranging from the 25 mm (1 in) diameter, 51 mm (0.167 ft) long cylinder (used by Sawaragi et al., 1976) to the 0.5 m (19.7 in) diameter, 10 m (32.8 ft) long cylinder (Bearman, 1988b). Since the transverse force was found to exhibit some degree of irregularity, many of the investigators presented their data in terms of an rms lift coefficient. As in the case of planar flows discussed above there are differences in CLr,, used by the different investigators. Once again the author compiled available data for CLrms and plotted them versus KC (see Fig. 2.14). Those for Bidde (1971) were not included since they are defined in terms of CLm. In this figure, Bearman's data are based on an rms velocity from measurements near the location of his instrumented segment. It is not clear what definition was used by Isaacson and Maull (1976) but the author assumed it to be based on the maximum value of the velocity at the instrumented segment as computed using linear wave theory. Sawaragi et al. (1976) did not define their CLa, but since they compared their results with Isaacson and Maull the same assumption made for Isaacson and Maull was applied to them. As would be expected there is more scatter in these data than for the less complex flows. This is particularly true for the data of Sawaragi et al. for larger values of KC and kh between 1.2 and 2.4. Possible reasons for such scatter include the large differences in model sizes, differences in H/D, and differences in ellipticity of the particle motion (Bearman et al., 1985 found this was not a factor for the conditions of their experiments). However, it is interesting to note that the lift coefficient is a maximum at about the same KC values as for oscillatory, planar flow (KC between 10 and 12). 1.8 1 0 [ SI 1.6 [ 3 Symboll Ref #I Kh _ 1.4 1.2 nm3] 0 0 1.0- N E 0 0 0 0.8- 0 0 F0 0 0.6 Oo 0 0.4- 0 a) o o * ,0 o o 1.0 0 0.2- 8 * 0 cb 0 -? I I I II 5 10 15 20 25 30 KC Figure 2.14: Lift coefficient versus Keulegan-Carpenter number for regular waves around a smooth, fixed vertical cylinder. Some flow visualization tests have also been conducted in this flow category to study the vortex-shedding process and its frequency. Bearman (1988b), Isaacson and Maull (1976) and Sawargi et al. (1976) carried out visualization tests to characterize the frequency of the vortex-shedding process. However, due to the subjectivity in the visualization techniques the behavior of vortex-shedding frequency is examined from the transverse force power spectra. The spectra obtained by Chakrabarti et al. (1976) and by Torum and Reed (1982) were thus compiled and plotted in Fig 2.15. From these spectra, one can see the existence of multiple frequencies at all values of KC, but the dominant frequencies have the same trend as in oscillatory, planar flows. That is, at lower values of KC the dominant frequencies cluster around the wave frequency. As KC increases to 9 the frequencies cluster at twice the wave frequency, then at three times the wave frequency as KC increases further, etc. N 7 0.46-2.28 0 30 0.76-0.79 0 66 0.6-1.0 a 66 1.2-2.4 29 The author was unable to find reported data on span-wise coherence length or transverse motion for this flow category (waves). However, as mentioned before, some investigators allowed transverse motion in their studies. These investigators were mainly concerned with determining the conditions under which transverse motion can be excited since this motion can cause shut-down of operations and fatigue of structural members (such as marine risers). In an attempt to quantify such conditions, the maximum values of yms/D (defined as Yrms ,ID) from several investigators' data were plotted versus several of the pertinent parameters. A plot of YrmsID versus V, (see Fig. 2.16) was found to be a meaningful. This plot shows that the maximum values of the response (Y,,ms/D) for different flow conditions fell in a relatively narrow range of reduced velocity, V,. In summary, the existing data on the vortex-induced transverse loading and response for fixed and transversely unconstrained, smooth, cylinders were reviewed for the different flow situations. As a result of this review one can say there have been significant advances in the understanding of vortex-induced forces on bluff bodies for some types of flow, especially steady and oscillatory, planar flows. In spite of these advances, however, more work is needed before the effects of many of the flow and geometric parameters on the transverse force and response can be quantified for the design engineer. Lack of consistency in data reporting has hindered progress in some areas while the difficulty of measuring some of the important quantities (such as, span-wise coherence length of vortices) has limited the data available for analysis. With an increasing awareness of the problems associated with inconsistent data reporting and with rapid advances in instrumentation technology, hopefully, these problems can be eliminated. This study focuses on the effects of flow nonuniformity and transverse motion on vortex-induced transverse forces (in particular, lift coefficients and frequency of vortex-shedding) and on the vortex span-wise correlation length. KCz 16 KC=9 Figure 2.15: Schematic transverse force power spectra for regular waves around a smooth, fixed vertical cylinder for various Keulegan-Carpenter numbers, KC (Refs. 12, 73). 1.41 1 * 0 0 0 0 0 E0 0 0 0 0 0 0.4 0.21I Symbol 03 0 Ref # 18, 21,52 40, 46, 71 74, 75, 70 00 0 0 I I I I I I I 2 4 6 8 Vr 10 12 14 16 Figure 2.16: Maximum transverse response for various flow configurations planar flow), (o oscillatory, planar flow), wavee flows). (0 steady, 0 65, 2 CHAPTER 3 EXPERIMENTAL INVESTIGATIONS 3.1 Scaling Parameters and Model Selection The modelling of fluid-induced dynamic forces exerted on bodies immersed in a viscous fluid has always presented difficulties with regard to similarity and scale. This is especially true when the structure is such that it is free to move. If dimensional analysis is applied to the particular problem of a transversely unconstrained bottom-hinged cylinder in nonuniform oscillatory fluid flow, the amplitude of transverse motion can be expressed in terms of the lift coefficient, CL, reduced velocity, V, stability parameter, Is, and added mass to cylinder mass ratio, m/ms, (see Appendix A for details), i.e., y/D = Y(CL, V,K,mams) (3.1) where CL = CL(Re, KC, Ty). (3.2) If the cylinder is smooth and the external generated turbulence "Tv" in the flow is small, Eqn. 3.2 reduces to CL = CL(R,,KC). (3.3) Consequently, Eqn. 3.1 reduces to y/D = Y(R., KC, V, K8, mram,). (3.4) For complete similarity the values of all the parameters should be the same for both model and prototype, i.e., R UD UD V 1' KC = (-)m = (UT) Vn T = (1 M36 (D)P and ~m8 where m and p refer to model and prototype, respectively. Of these similarity groups the most important parameters are the Reynolds number (R,), Keulegan-Carpenter number (KC) and reduced velocity (V,). Hydrodynamic force coefficients for large values of Re are needed in full-sized offshore structures applications. These conditions are difficult if not impossible to produce in laboratory experiments. Wave tank testing generally produces Reynolds numbers (based on structure diameter) up to approximately 5 x 104, whereas most prototype structures experience Reynolds numbers well beyond 10', i.e., in the upper subcritical, critical and supercritical regions (according to steady flow principal Reynolds number flow regimes around smooth cylinders). Wave tank testing has also the additional problem of not being able to precisely control Reynolds number. Ideally both Reynolds number and reduced velocity must be scaled. It is practically impossible to achieve both a desired Reynolds number and reduced velocity at the same time unless a full scale structure is tested. For an average prototype articulated tower, such as the "ELF" loading tower operating in the North Sea, (diameter = 4.5 m, length = 150 m and water depth = 135 m), the reduced velocity, V varies between 0 and 12 for a current velocity varying between 0 and 2.5 knots. It can also be as high as 20 for a 100 year design wave (having a 17 sec. period and 30 m height according to Kirk and Jain, 1977). 33 In laboratory testing it is important to cover the range mentioned above for both prototype Reynolds number and reduced velocity. Using a constant diameter cylinder this can be done by adjusting either the natural frequency f,, or the velocity range or both. In the tests carried-out in this investigation the natural frequency was held approximately constant. The intent of this investigation was to study the effect of flow nonuniformity and transverse motion on the vortex-induced transverse loading. In order to make the results of this study as useful as possible to the design of offshore structures, the model was designed to produce ranges of similarity parameters as close as possible to those experienced by prototype structures where vortex-induced loading is important. Other geometric parameters such as the length/diameter ratio (f ID) should also fall within the prototype range. A survey of the available literature revealed that vortexshedding is considerable in the following ranges of similarity parameters: 3 x 104 < R, < 106 7 5.0 < KC < 12.07 1.0 < V, < 20.0 and flD between 20 and 30. The requirements for achieving similarity (i.e., equivalent values for model and prototype) for Reynolds number conflicts with the conditions required for similarity for reduced velocity and other parameters. For example, to produce high values of Reynolds number requires high values of flow velocity since V is fixed and the cylinder diameter is constrained by the ratio t1D. To satisfy reduced velocity similarity the higher flow velocities imposed by Reynolds number similarity require high system natural frequency which means a very light structure which is limited by required rigidity. Also higher flow velocities means higher amplitudes of oscillation or higher driving frequencies which are constrained by the available driving mechanisms. Due to these complications, the design of the cylindrical model used in these experiments was accomplished with the aid of a computer program that optimized the range of the parameters within the limitations imposed by the drive mechanism, tank, budget, etc. In other words, given the values of the facilities available such as maximum water 34 depth, maximum in-line velocity and others, the program selects the dimensions that cover ranges of the similarity parameters overlapping with those of prototypes. In this study, model dimensions of 0.1524 m diameter and 3.05 m length were selected, iLe, a ratio of f/D = 20. Such a model covered ranges of R, between 6.1 x 10' and 1.3- x 105, KC between 2.5 and 9.5 and V, between 3.15 and 46. For an elastically-mounted cylinder, experiments should be carried-out at constant K, to facilitate comparison of maximum response amplitudes for different conditions. In this investigation, despite the changes in temperature during the course of testing, I,( was approximately constant. The ratio of the fluid added mass to the structural mass (m,,/m,) should be modelled properly. In air m0,/m, = 0(10-') and thus is of little importance. In water, however, the ratio ma/m, = 0(l) and affects both the maximum amplitude of motion due to vortex-shedding and the velocity range over which lock-in occurs. 3.2 Experimental Set-up Very little information exists on the effect of flow nonuniformity and structure transverse motion on vortex-induced transverse loading and vortex span-wise coherence length. To investigate such effects two types of experiments were designed and performed. Both experiments were performed in a circular water tank with a rigid articulated tower model. In the first type of experiment the model was constrained from transverse motion while in the second type it was free to move in the transverse direction. As discussed before in Chapter 2, there are two different ways to produce a relative oscillatory motion between a cylindrical model and the surrounding fluid. One method is to oscillate the fluid past a stationary cylinder while the other is to oscillate the cylinder past the stationary fluid. Kinematically there is no difference between the two situations when viewed from the appropriate reference frame. Experimentally, there are differences in implementation between the two methods. 35 In the experiments of this investigation, an articulated cylinder was oscillated in an otherwise still fluid. This approach was taken because it was relatively easy to implement and because it allowed more precise control over the parameters. It also allowed the fluid-induced forces to be determined more directly. In other words one ran determine, after subtracting the inertial force due to the mass of the oscillating cylinder, the coefficient of added mass (C,) instead of the inertia coefficient (Cm, i.e., 1 + C,) since no horizontal pressure gradient exists in the flow field. On the other hand, this approach has certain disadvantages which must be overcome. These are: i) waves and free surface disturbances can be created by the oscillating cylinder, ii) the drive mechanism can transmit vibrations to the cylinder and surrounding fluid and iii) the inertia force due to the mass of the oscillating body has to be accounted for in the measured force signal. In this experiment, several ideas and designs were employed to overcome these difficulties. Due to the tank to cylinder diameter ratio (= 60) wall effects were negligible (in terms of a blockage ratio it was 1.6%). This is well below the 6.0% Emit given by West and Apilt (1982) for no influence from blockage on the Strouhal number. Soft and porous packing materials were also placed at water surface halfway between the cylinder and the tank wall to absorb surface disturbances and reduce wave reflections from the wall (see Fig. 3.3). Two heavy steel I-beams were placed across the tank to support the linear driving motor. The feed back control system for the linear drive motor was initially a source of high frequency vibration, but this was minimized by using a lowpass analog filter in the feed back circuit. The two types of experiments used in this study were designed to obtain the information needed to quantify the dependence of the vortex-shedding process, vortex span-wise coherence length and transverse force on the flow and structure parameters. All tests were performed in a deep circular water tank 9.15 m in diameter and 3.1 m high. The tower model was a right circular rigid aluminum cylinder with a diameter 36 of 0.1524 m, a thickness of 3.0 mm and a length of 3.05 m. It was instrumented with miniature, quick response, (2.15 mm diameter bead) thermistors embedded in its surface to measure the frequency of vortex shedding along the tower. These thermistors were located at the leading and trailing sides of the tower at 20 different locations (total of 10 thermistors along each side). All the wires of the thermistors were run through the inside of the tower. The signal processing circuits for the thermistors were mounted inside the cylinder near the top. The cylinder was attached to the bottom at the center of the tank through a low friction hinge designed to allow X and Y motion while constraining cylinder rotation about its axis. Its top was attached through a vertically sliding shaft to a linear drive electric motor mounted horizontally between the I-beams. The vertical sliding shaft was designed to account for the change in the vertical position of the top of the tower during its rotation about the bottom hinge. Ultra low friction linear ball bushings for the vertical sliding shaft were mounted in the upper end plate of the cylinder. An X Y force transducer was inserted between the top of the tower and the table of the linear drive motor. To negate the need to measure forces at the base, a pin joint was placed between the tower and the X Y force transducer at the top. Two Linear Displacement Transducers (LDT) were used to measure in-line and transverse displacements of the tower. The active strokes of the in-line and transverse LDT were 0.66 m and 0.46 m, respectively. All alignments were made prior to adding water to the tank. In the first type of experiments, tests were performed with the configuration shown in Fig. 3.1. An in-line simple harmonic motion was imposed at the top of the tower while constraining the transverse motion. Quantities measured during these tests were i) in-line position, ii) in-line force, iii) transverse force and iv) vortex-shedding frequencies from the 20 thermistors mounted along the tower. The second experiment type configuration shown in Fig. 3.2 used the same tower model with the exception of the mechanism that attaches the cylinder to the linear 37 drive motor. The tower was driven with the same in-line motion as in the first type experiment, but in this case it was allowed to respond freely in the transverse direction. Quantities measured in these tests included i) in-line position, ii) in-line force, iii) transverse motion and iv) vortex-shedding frequencies from the 20 different thermistors along the tower. The water depth for all the tests (type I and 1I) was 2.65 m. Photographs of the general setup and transverse motion mechanism are shown in Fig. 3.3. 3.3 Instrumentation and Calibration As discussed in the previous section, the following quantities were measured 1. In-line tower position. 2. In-line force. 3. Transverse force (type I experiment only). 4. Transverse response (type II experiment only). 5. Thermistor signals. Other signals such as the input driving frequency were measured and held constant during each test. Quantities such as water temperature and surface wave activity were also monitored. A block diagram of the measurement system used is shown in Fig. 3.4. The instrumentation used to measure these quantities are listed below. The calibration procedures used, where needed, are also discussed. 3.3.1 Frequency Generator The sinusoidal motion imposed at the top of the tower was generated using a Hewlett Packard (HP) frequency generator. A Hewlett Packard (HP) frequency counter in parallel with the frequency generator was also used to continuously monitor the input driving frequency (fd) during each run. 1 Tank 2 Articulated Tower (10' x 6"0) 3 Linear Motor Drive 4 Force Transducer 5 Trandsucer Coupling 6 Upper Cylinder Bearing Guide 7 Linear Bushing Bearing 8 Vertical Sliding Shaft 9 Universal Joint 10 Lumber Frame Cover Figure 3.1: Schematic diagram of transversely constrained experiment set-up. 1 Tank 2 Articulated Tower (10' x 6"0) 3 Linear Motor Drive 4 Force Transducer 5 Trandsucer Coupling 6 Upper Cylinder Bearing Guide 7 Linear Bushing Bearing 8 Vertical Sliding Shaft 9 Spacer 10 Universal Joint 11 Linear Bushing Bearing 12 Lumber Frame Cover Figure 3.2: Schematic diagram of transversely unconstrained experiment set-up. Figure 3.3: Photographs of experimental set-up. 41 ln-line Position -- i Analog/Digital Processing iSwitc1 Inline Position Signal Transverse In-11nPositiono Signal Sinalrs FFroquncy Position S Counter Y LPFSignal H P F u n c t i o n C n r l e Generator Linear Motor & Controller Figure 3.4: Block diagram of measurement system. 3.3.2 Force Transducers Due to the high cost of commercial X Y (horizontal components) load cells, the one used in these experiments was designed, constructed and tested in the department laboratory. The transducer was a beam-type load cell (see Appendix B, Fig. B.1) made of 304 stainless steel with overall dimensions 3.8 x 3.8 x 7.5 cm. It was constructed as if two I-beam were placed flange-to-flange so that their webs make a "90'" angle. The thickness of each web which was required to withstand the maximum loads anticipated in its direction was calculated using Hooke's law, with a safety factor of 1.5. With the scantlings selected, the transducer had a resolution of 0.2 N in each direction. Each web was instrumented with four (two on each side) active 350 ohm strain gages (with a gage factor of 2.03 1%). These gages formed the 42 elements in a standard four-arm wheatstone bridge circuit and provided both high sensitivity and temperature compensation. The strain gage bridge was connected to a strain gage amplifier, the Omega DMD 465 (see Appendix B, Fig. B.2 and Table B.1 for schematic and specifications). A pulley system was developed to calibrate each channel of the transducer. The calibration was carried out by varying the loads on one channel in both directions (i.e., in tension and compression). Loads were added in increments then removed in increments to test for hysterisis. At the same time a constant load was maintained on the other channel to examine the side load effect. This was repeated for different constant side loads. The calibration curves are shown in Appendix B, Figs. B.3, to B.5. Note that the two channels are very well isolated (i.e., loads in one direction have minimum influence on the transducer output in the other direction). 3.3.3 Linear Displacement Transducers The tower's in-line and transverse motions were measured using two, MTS Temposonics II, LDT (Linear Displacement Transducers) systems with AOM (Analog Output Modules), see Appendix B, Fig. B.6 for a schematic setup and Table B-2 for the specifications of the AOM. Both LDTs were calibrated by giving the tower known displacements measured with a rule while the output from the A/D data acquisition board was recorded. A statistical analysis (average and standard deviation) of the recorded calibration data at each position was then carried-out. This procedure was repeated for the different positions around the vertical position of the tower i.e., in the positive and negative directions. The resultant data gave straight line calibration curves for each transducer, as shown in Appendix B, Fig. B.7 and Fig. B.8. 3.3.4 Thermistors The vortex shedding frequencies were measured by using miniature, quick response, thermistors embedded in the surface of the tower. Dimensions, thermal and electrical properties of these thermistors are given in Appendix B, Table B-3. The 43 principal is similar to that of a hot film anemometer. That is, the overheated thermistor is cooled by the flow of the fluid past its 2.15 mm diameter bead which in turn reduces its electrical resistance. The circuit designed for processing the signals of these thermistors is given in Appendix B, Fig. B.9. The concept of using thermistors to measure frequency of vortex-shedding was tested using a thermistor embedded in a 0.065 m diameter and 0.61 m long PVC cylinder. The cylinder was then placed in a steady flow on a flow visualization table as shown in Appendix B, Fig. B.10. Three different flow velocities with Reynolds numbers of 4.6 x 10' 9.7 x 104 and 5 X 104 were used. The thermistor output signals were sampled at 50 Hz and a spectral analysis performed on each signal. The spectra of the signals (see Appendix B, Fig. B-1 1) showed a single large spike (relative to other frequency components in the spectrum). The Strouhal number, St at these spikes was found to equal 0.23, 0.188 and 0.123, respectively for the Reynolds numbers above. 3.3.5 Lowpass Filters Analog filters must always be used with caution since they can produce undesirable as well as desirable effects. The undesirable effects are in the form of signal phase shift and amplitude attenuation. To minimize these effects, a second order, Butterworth, lowpass filter was designed for the in-line position signal to reduce the noise in the signal to the servo controller. The cut-off frequency was carefully selected for the filter after considering the useful driving frequency range. Since the highest driving frequency was 1.0 Hz, a cut-off frequency of 3.0 Hz for the feed back loop filter was deemed sufficiently high. A schematic diagram of this filter is shown in Appendix B, Fig. B.12. Two other lowpass filters were used for the in-line and transverse forces. Their cut-off frequencies were selected after examination of the unfiltered signals from exploratory runs for all the proposed driving frequencies. Spectral analyses were then performed on the measured signals. The results of these spectra showed no strong frequency components (even for the highest driving frequency, 1.0 Hz) above 15.0 Hz 44 for the in-line and 20.0 Hz for the transverse force signals. Accordingly 17.0 Hz and 22.0 Hz cut-off frequencies for the in-line force and transverse force signals were selected. A sample output of the transverse force time series and its power spectrum before and after filtering is given in Appendix B, Fig. B.13. - Other more standard instruments used in these experiments included: A Normag DC-linear drive motor (specifications given in Appendix B, Table B.4); DC-servo controller (Moog, Model 82 300); glass bulb mercury thermometer and a 12-bit, multi-function, high speed A/D Metrabyte/Asyst/DAC expansion board with analog input accuracy of 0.01% and a variable input voltage range. This board was installed in an IBM compatible personal computer. 3.4 Data Reduction As mentioned before, two types of experiments were performed. Type I was with the transverse motion constrained and type II was with a cylinder free to move in the transverse direction. Tests were run for different values of in-line amplitude and driving frequency. In the first type experiment, the cylinder driving frequency (fd) was varied from 0.4 Hz to 1.0 Hz. At most frequencies the amplitude of oscillation varied from 0.1 m to 0.28 m. A total of 86 runs were made during this experiment. Of these 8 runs were performed to test the repeatability. In each run, in-line and transverse forces, in-line position and the signals from the 10 thermistors were recorded. During the test runs GLOBALLAB software was used to perform preliminary analysis of the measured data. The tower driving frequency (fd) and amplitude (a) were also monitored during each run using a frequency counter and oscilloscope. In the type II experiment, where transverse motion was allowed, the driving frequency (fd) was varied from 0.15 Hz to 0.8 Hz. At each frequency the amplitude was varied from 0.076 m to 0.28 m. A total of 190 runs, including 36 runs to test repeatability, were made during this experiment. In each run the in-line force, in-line 45 and transverse position and the thermistor signals were measured and recorded. To reduce the scatter in the measured data of in-line force and transverse motion some longer test runs, where only in-line force, transverse motion and the signal of vortex shedding from the top thermistor, were made. -Since the Direct Memory Access (DMA) page registers cannot be incremented by the controller, the maximum data area available (64K, a page, for 32, 767 conversions) was used to acquire data for each run. This resulted in a duration of 63 sec for each recorded signal in the first type experiment where 13 data channels were acquired and 91 sec (273) sec for the second type experiment when 9 or (3) channels were used. The maximum obtainable in-line amplitude was found to vary with the driving frequency (fd). The intent was to obtain as wide a range of Keulegan- Carpenter (KG) and Reynolds (R,) numbers as possible. The range covered by these parameters and others during both experiments together with a summary of the test conditions are shown in Table 3.1. The natural frequency (f,,) and damping ratio ( ) included in Table 3.1 were obtained by measuring the frequency of free oscillation in still water. The measured value of (f,,) compared well with the analytical value obtained from the free vibration response predicted by a computer program developed by Omar and Sheppard (1991) for predicting response of articulated towers under the action of wind, current and waves. Table 3.1: Test conditions for experiments I and II. Item Exp. I Exp. II minimum fd 0.5 Hz 0.15 Hz maximum fd 1.0 Hz 0.8 Hz minimum a 0.085 m 0.08 m maximum a 0.283 m 0.29 m temp. range 83 880 F 60 740 F minimum Re 6.1 x 103 8.4 x 103 maximum Re 2.15 x 104 1.3 x 10 minimum KC 2.6 2.4 maximum KC 8.65 9.35 minimum #3 1792 3151 maximum/ 3692 18336 minimum V 2.66 maximum V 36.10 f,. 0.15 Hz 0.12 water depth 2.625 m 2.625 m No. of channels 13 or 14 3 or 9 sampling frequency 40 Hz 40 Hz CHAPTER 4 MATHEMATICAL MODELS 4.1 In-line Force The following equation of motion was used to reduce the in-line force data: ImO() = Mtotal = M + MD+ MfX, (4.1) where Im = + = total mass moment of inertia about the bottom hinge, I = cylinder mass moment of inertia, I' = added mass moment of inertia, Mtotai = total moment about bottom hinge, M9 = moment due to tower weight and buoyancy, MD = moment due to drag, Mf = moment due to linear drive motor and 0(t) = in-line deflection angle. For the tower shown in Fig. 4.1 the moments Mg,MD and Mfy are given by Mg = ( nw, r,- p :=1 0d/ s MD = -Pw,,, D CD o 2= RL, My, = R L, , Ae g r dr) sin 0, (r 0)r 0| r dr and where A, = 7rD2/4 is the cross sectional area of the tower element. The added mass moment of inertia, I can be expressed as Idl/cosO fdl/cos0 D2 r2 S= I drpmD22d. Jo 0 4 C dr (4.5) (4.2) (4.3) (4.4) Figure 4.1: Definition sketch for the articulated tower showing in-line motion. The motion of the tower was imposed by a horizontal linear drive motor mounted directly above the tower as shown in Fig. 4.1. The position of the moving table of the motor and thus the upper end of the shaft connected to the tower was monitored by a LDT (Linear Displacement Transducer). Due to the relatively small angular movement of the tower, the arc LO(t) can be approximated by the measured horizontal linear displacement Xm(t). Substituting Eqns. 4.2 4.5 into Eqn. 4:1 and replacing LO by X,,, results in I. 7 r D2 dl,,X,,t -(+ pw D2 C-Xm m -L 1-P2 L cos3[Xm(t)/L] 1 d4 1 DC Xm(t) jXm(t) 8 P D L2 cos4[Xm(t)/L]CD n d2 +( w,1 p g D2 d ( )sin[X,,(t)/L] = RL. (4.6) S=1 8 cos2[Xm(t)/L] Since the measured in-line moment Mm,,(t) = RArt, R = Mm,(t)/Art. Substituting this expression for R into Eqn 4.6 and rearranging results in I. M,,(t) L/Art + (t) 7X D2 ( w,r, pg D 2 [X(t)/L])sin[Xm(t)/L] = s=x8 Cos2(Xm.(t)/L] 2 d p. D2L 1 O m 1 12 L cos[Xm(i)/L] C (t) p L2 CD [Xm(t)/L] CD XmM(t)IXm(t)I. (4.7) 8 L2 cos4 [X,,,(t) /L] The quantities on the left hand side in Eqn. 4.7, Mm,,(t), m(t) and Xm(t) are known from the measured values. The in-line motion Xm(t) was monitored by the LDT and found to conform with the input signal, a sin wdt, where,"a" is the amplitude of oscillation and wd the oscillation circular frequency. Thereupon, the velocity Xm(t) and acceleration Xm(t) were computed from the time derivatives of Xm(t) = a sinwdt. Equation 4.7 can be written as fm(t) = fI(t) Cm + fD(t) CD, (4.8) where M,.. (t)L I "(-' n r d2 fm(t) = + -Xm() (Zwr, -pwgD2 1 sin(X,,(t)/L], ArL L 8 cos2[Xm(t)/L])sin[Xm(t)/L] x d3 fi(t) = p. D2 () and 12 L cos3[X(t)/L]x(t) and fD(t) 1 d = --pDL2 cos4[Xm(t)/L] 1(t)X1,(t) CD and Cm are the unknown quantities in Eqn. 4.8 and can be obtained by minimizing the squares of the differences between the computed and measured values, i.e., by minimizing S where 1N 2= [fI(t,) Cm + fD(t) CD fm(t,2. (4.9) N is the number of data points in one cycle. From this equation the minimum CD and C,, were obtained by solving the two simultaneous equations OCD 0 and (4.10) - = 0- (4.11) 8C Equation 4.10 and Eqn. 4.11 can be written in a matrix form as SfI(t)fD(t ) i C = { = 1 z t,)fot,, (4.12) 1=l S I(t,) fDt = t =I fm(t)fI(t ) The solution of Eqns. 4.12 results in the following equations for CD and Cm CD = [f(t)fD(t)] ,) [f,.(t,) [fD(t)fI(t,)] and (4.13) -[f[(t,)]E[ff(t,)]- [E fD(t,)f t,)]2 C = Elfm(t,)f(t1)] f ) E[fm(t,)fD(t,)] E[fD(t,)fI(t,)] (4.14) m E[ff(t.)] E[ff,(t,)]- [E fD(t,)f (t,)]2 where the summations are evaluated from z = 1 to N for each cycle. To put more emphasis on the large data values and thus further reduce the differences between the measured and predicted forces in the neighborhood of maximum forces, the weighted least squares technique was also applied using f,(t,) as a weighing factor. The suitability of the in-line force data for determining CD and Cm was also evaluated using Dean's (1976) approach. In this approach, the mean square error 62 given by Eqn. 4.9 defines a quadratic "error surface" which is a minimum at the CD and Cm obtained by solving Eqn 4.12. The suitability of the data can then be evaluated from the lines of constant error values, ,2(CD, Cm), which are ellipses. For example, the steeper the slope of the error surface with CD, the better the data are suited for evaluating the drag coefficient, see Fig. 4.2. CD and Cm are time averages over several cycles. This study did not deal with the instantaneous values of these coefficients. However, since the measurement errors in the data must introduce some uncertainty, the variance and covariance of the C (C )min Lines of Equal Errors, N. c (CD min CD (CD) min D (a) Data Well-Conditioned for Deter- (b) Data Well-Conditioned for Determining Drag Coefficients mining Inertia Coefficients Figure 4.2: Contour lines defining Error surfaces for the in-line force (Ref. 15). estimates (CD and C,,) were computed. The variance, of, in the value of any function can be written as (4.15) For estimating the uncertainties i N f 2 01 = D( ) . i =1 D ay, in CD and C,,, Eqn. 4.12 can be rewritten as [aj]{a} = {#} 2 E aks a, = pk, 3=1 = 2 x 2matrix, or 1_=I X3(t,)Xk(t,), = a vector of length 2, oi = Ifm(t,)Xk(t,) k = 1,2 and = 1,2and k = 1,2. The inverse matrix Ck [a)1 is closely related to the probable (or more precisely, the standard) uncertainties of the estimated parameters a = (CD, Cm). The (4.16) where [] ak.1 {f M /k (4.17) 52 solution to Eqn. 4.12 (or equivalently Eqn. 4.16) is 2 2 N a = j[al'j Pk = Ck [ m f(t,) Xk(t,)]. (4.18) k=1 k=1 s=1 Note that a, corresponds to f and f,,(t,) to y, in Eqn. 4.15 thus Of a, 2 y (t) Ck Xk(t,). (4.19) ay' f"I t') k=1 Consequently, the variance associated with the estimate parameters a, is 22 N U2(a) = ZZ C pkCal [ Xk(t,) Xt(t,)]. (4.20) k=1=1 t=1 The final term in the square brackets is just the matrix [a]. Since this is the matrix inverse of [C], Eqn. 4.20 reduces to a 2(a,) = C,,. (4.21) In other words, the diagonal elements of [C] are the variances of CD and Cm, while the off-diagonal elements Cyk are the covariances between CD and C, (i.e., COV(CD, Cm)). An estimate of the goodness-of-fit of the data to the model is still needed. A simple measure of the goodness-of-fit defined as the average percentage error between the measured and predicted signals (fm(t,) and f,,(t,)) was used. This was defined as ERR = 100 E[fm(t) fP(t)]2 (4.22) \ E f'2i, f(t,) where the summation is evaluated for z = 1, 2, ..., N. Drag Force CD, for most situations, has been found to depend on Reynolds number, and thus on the velocity. In an attempt to minimize this dependence and maintain the drag coefficient constant, an investigation of the power to which the velocity must be raised in the drag force equation was made. The drag moment in Eqn. 4.3 was replaced by the following expression 1 dl/coso MD =-- pw DCD (rO) sgn(r0) r dr, (4.23) 53 where CD is the drag coefficient associated with this expression and "sgn" is a sign function equal to 1 depending on the sign of the argument, rO. Making the same assumption Xm(t) = LO(t) and carrying out the integration of Eqn. 4.23, the equation of motion (Eqn. 4.7) reduces to L I, Mm Wt' + Xm.t) Are L (w r, p g D1 )sin[Xm(t)/L] = 8= cos2[Xm(t)/L - rpw D 2 d Cm Xm (t)12P L cos3[Xm(t)/L] (t1 d CD Xm(t)I gn[X,(t)]. 2 P ncos[Xm(t)/L]' n + 2' L2 sgn (t)]. (4.24) In this equation, the quantities on the left hand side, M,.(t),,,,(t) and Xm(t) are known from the measurements. Using fn(t) and fi(t) defined before, and defining 1 fdr = 2 pwDsgn[Xm(t)], C(t) = [ di ] and cos[Xm(t)/L] Xm(t) D(t) = L Equation 4.24 reduces to fm(t) = fI(t) Cm + fd [C(t)]n+2 [D(t)]" dD/(n + 2). (4.25) The unknown quantities in this equation are Cm, COD and n which can also be obtained by using the least squares equation: 1 N S= N E[fh(t,) Cm + fdr[C(t)]+2 [D(t,)]" dD/(n + 2) f(t,)]2 (4.26) or the weighted least squares equation 1C2 6n, = g f (t.)[f;(t,) Cm + fI[Ct ,)Jn+2 [D(t,)] doD/(n + 2) fm(t,)]2. (4.27) N 54 The minimum Cm,, dD and n were then obtained by solving the coupled nonlinear system of equations given by ac"'= 0,(4.28) OS2 =0and (4.29) 8dD 8 = 0. (4.30) an The Newton-Raphson method for nonlinear systems of equations was applied to solve this set of equations (Eqns. 4.28 to 4.30). The method is given in many text books (e.g. Press et al., 1988). 4.2 Transverse Force For several reasons the transverse force has been the subject of greatest interest in this study. First, its amplitude can, under certain conditions, be as large as that of the in-line force. Second, the transverse force can give ris e to fluid-induced oscillations and to fatigue failure Third, even small transverse motions of the body regularize the wake motion, alter the span-wise correlation of the vortices, and drastically change the magnitude of both the in-line and transverse forces. With all the information available on vortex- shedding, there is no simple, explicit formula to predict the time variation of the vortex-induced transverse force. This is because of its dependency on the type of flow and structure motion. In most types of flows it also exhibits some degree of irregularity that is usually due to the alternating eddies behind the structure. Therefore, most formulations of transverse forces are semi-empirical and based heavily on experimental data. Of course, there are many numerical models available (e.g. Navier-Stokes based-models, discrete-vortex models, wake oscillator models) but due to their limitations (such as laminar, two-dimensional flows at low Reynolds number, sensitivity to the numerical technique used and their dependency on experimental data and flow visualization) they are not reviewed here. 55 A number of authors, including Maull and Milliner (1978), Sarpkaya and Shoaff (1979) and Graham (1980), have proposed a method that uses Blasius equation and the "discrete-vortex method". Such a method assumes that the induced-force consists of a component due to the attached unsteady irrotational flow past the body and a component generated by the vortices shed into the flow as a result of separation. The application of this model requires detailed information on vortex strengths and trajectories. Such information usually is difficult to obtain, especially for the cases involving wave flows. In the following sections, the existing semi-empirical models that predict transverse forces are briefly discussed. 4.2.1 Steady Flow Model This model is similar to the drag force in Morrison's equation and is written in terms of the lift coefficient, CL as FL = p,DiCLU2. (4.31) 2 The model was originally developed for steady flow where it has been confirmed by many investigators that the transverse force spectra has a single frequency, predicted by Strouhal number, St = 0.2 for 103 < R, < 2 x 101. Some researchers and designers use this steady flow model for predicting transverse forces for oscillatory and wave flows. In order to account for the time variation of flow velocity and transverse force, a maximum transverse force which yields a lift coefficient defined by maximum transverse force CLma2 (4.32) has been used. Others use an rms transverse force that gives a lift coefficient defined by rms value of the transverse force D 2 ULrms = k%.0,3) 56 Other methods based on semi-peak-to-peak values of the transverse force or different velocities, such as those corresponding to the maximum in-line force rather than to the maximum velocities, will not be discussed here in order to avoid confusion. In the author's opinion, the use of such methods for oscillatory flows, especially wave flows, is one of the reasons for the scatter in the existing CL data as illustrated in Fig. 2.14. This is because, unlike the in-line force in oscillatory and wave flows, the transverse force has multiple frequencies and exhibits some degree of irregularity which are a result of the different shedding mechanisms in these types of flow. 4.2.2 Quasi-Steady Model Recently the steady flow model has been extended to include the frequency of vortex-shedding. Verley (1980), followed by McConnell and Park (1982b), suggested the following simple quasi-steady model FL = 1 pW D i CL U2 sin wt. (4.34) The model is based on the instantaneous values of flow velocity and vortex-shedding frequency. It also assumes that at the beginning of each half cycle the flow starts from rest again (i.e., does not account for the flows previous history) and the phase is brought to either 00 or 1800. In an attempt to improve this model, Bearman et al. (1984) proposed the following quasi-steady model 1 FL = fp DeCLU2n cosqS sin2 Wft, (4.35) where is a function of time given by 0 = 0.2KC [1 cos wft] + V, (4.36) wf is the frequency of flow oscillation and 0 is a constant to be adjusted for every half cycle. This model is also based on the instantaneous flow velocity. It does not predict the sign of the force which depends upon the sense of the vortices shed. It is based on the assumption that the Strouhal number is constant and equal to 0.2 57 (the appropriate value for steady flow). With a suitable choice of the lift coefficient CL for each half cycle of the incident flow, the model seems to work reasonably well for KC > 20. However, the variation of force with time was found to be somewhat regular and the vortices tend to form and shed in a certain prescribed manner. In general, such a model may be helpful in understanding the flow phenomenon but may not be suitable for design purposes as it is sensitive to the phase angle and still needs an experimentally determined CL. 4.2.3 Series Model Since transverse forces exhibit some degree of irregularity, Mercier (1973) found it is appropriate to express the transverse force in the form of a series. Later Isaacson (1974) and Chakrabarti et al. (1976) used the same idea. Their results showed this model to be superior to previous models with regards to the force-time history. In general, the series form of the transverse force can be written as 1 N FL(t) D t U2 L CL(n) cos(2rnft + 0(,)), (4.37) where N is the number of harmonics, CL(,) is the lift coefficient at the nth harmonic and 0(n) is the phase angle associated with CL(n). In 1977 Sawaragi et al. used the same model, but introduced the assumption that the transverse force is a random variable. Moreover, the spectrum of the force in the region of the dominant harmonic can be treated as a narrow-band spectrum. Based on these assumptions, they reduced the series model (Eqn. 4.37) to FL(t) SaLE f cos(2rnft + 0(n)), (4.38) n=1 where OL is the variance of the transverse force defined by O2 = {lp U2 }2. 1E[C 2], (4.39) and S(nf)Af is the spectral energy of the transverse force at frequency nf. 4.2.4 Proposed Model Fixed and compliant structures in deep water under the action of nonuniform oscillatory or wave flows have varying values of KG and R, starting from zero at the bottom to a maximum at the surface. Therefore, using the maximum velocity in the series model discussed above does not yield a representative total transverse force. Furthermore, experimental investigations show that, in general, the vortex-shedding process for this type of flow depends on KG and R,. Thus, the lift coefficient is not a constant over the structure. In an attempt to improve the series model the following assumptions were made. The structure was divided into N number of finite elements with length At. Each element was then considered to be subjected to uniform flow (see Fig. 4.3). The total transverse force thus could be expressed by 1 M N FL(t) = ~p. D E CLmn, U,' mna, At cos(27rmnft + 0(mn)). (4.40) 2 m=~1= This representation allows CL to be dependent on KG, R, and the frequency of vortexshedding. Figure 4.3: Definition sketch showing tower elements and idealized flow. 4.2.5 Fixed Tower Transverse force data measured during the first series of tests were reduced using the proposed model presented in section 4.2.4. Since the measured data was in terms of the moment of total transverse force about the bottom hinge (ML(t) L), the moment form of Eqn. 4.40 was used, i.e. L 1 M N ML(t) Ar 2pD E CLmn U, r,, A cos(27rmft+4(m)), (4.41) Ar m=1 n=1 where rn is the distance of element n from the bottom hinge. Substituting Un = a, Wd sinWdt, where an (= ar/L) is the amplitude of the tower oscillation at the element n, and wd (= 27rfa) the driving frequency, into Eqn. 4.41 results in L 1 a2w M N ML(t) = p. D L Z E CL mn r Al cos(27rmft + 0(m)). (4.42) t m=1 n=1 Equation 4.42 is analogous to the Fourier series expression M M f(t) = E Cm e'mwdt = E Cm cos( mwdt + (m)) m=1 m=1 M = (am. cos mW dt + bm sin mwdt), (4.43) m=1 where M = is the number of harmonics, f(t) = ML(t)L/Art, am 2 jM am = "-=1 f(t) cos m Wdt, bm = M =1 f(t) sin mwdt, (m) = 0(m) = tan-' bm/am and Cm a2 + b = p, D (a2 w/L2) E CLn r3 AL. In Eqn. 4.43 the coefficients Cm were obtained by taking the FFT of the measured moment of transverse force ML(t)L/Art. Then knowing C m (Fourier components), the minimum lift coefficients at each element along the tower (CLn, n = 1,2, 3, ..., N.) were obtained by using the method of least squares: m 1 a2w C r AL) C,,]2, (4.44) m= 2 L [(n D 60 where the minimum CLm,,, are the solutions of ----n = 0, m = 1,2, ..., M and n= 1,2,...,N. (4.45) OCLmn In other words, for each harmonic the minimum lift coefficients along the tower (CLn) are the solutions of a62/OCLn = 0 where, n = 1, 2, ..., N. This results in the following N simultaneous equations AX=B, (4.46) where 36 33 33 33 1 1 2 1 r3 rnrn 33 r6 33 33 r2 r1 1.. 23 2rn A 33 33 33 6 Srr1 nT2 rn3 n , CL1 CL2 X =and CLn 3 B={ } rn The solution of Eqn. 4.46 at each harmonic is then given by {X} = {CL.} = [A]-'{B}. (4.47) 4.2.6 Complaint Tower When the articulated tower was allowed to respond freely in the transverse direction, the transverse force was not measured. Instead the transverse motion was measured. Structure motion was found to significantly increase the transverse force (see McConnell and Park, 1982a). One way to quantify the effect of transverse motion is to compute the change in the transverse force due to motion. This force can be determined by computing the force needed to produce the measured response. This 61 method was used in this study. The measured data for the tower response in the transverse direction was reduced using the following equation of motion Imb(t) = E Mtota, = Mg + MD + Mfy, (4.48) where I,,, Mtotai, Mg and MD are defined in Eqn. 4.1, and 0(t) and Mfy are the transverse deflection angle and applied transverse moment, respectively. For the tower shown in Fig. 4.4 the moments Mg, MD and M~, were defined as k di / cos 0(t) Mg = (Z1w, r, p, A gr dr) sin (t) (4.49) s=1 0} 1 N MD 2= pD n CD rn 7 A 2(t) sgn[b(t)] (4.50) 1 M N Mfy = 2p,D E E CLn mar,,, ,,rn Alcos(2rmft + 0(,)) (4.51) m=1 n=1 where Ac = 7rD2/4 is the cross sectional area of the tower element. The added mass moment of inertia I' is given by N I = p D Cm A (4.52) n=1 Substituting these moments into Eqn. 4.48, making the assumption that the measured transverse response ym(t) Lk(t) and di/ cos #(t) = di gives N " (Io + E -p, D' Cmn rn Ai)y'4(t)+ n=1 4 1 N P,, D E CD n r' AeY (t) sgn[ym(t)] + 2L p D E CDn,,r S@U n=1 k (7 p g D2 d -w,r,)ym(t) = s=1 1 a2w M N3 -PWD L-E E CLm nr At cos(2rmft + (,(m)), (4.53) n=l n=1 where N is the number of elements comprising the tower's underwater portion, M the number of transverse force frequency harmonics and k the number of the different components making up the tower. Figure 4.4: Definition sketch for the articulated tower showing transverse motion. The left hand side of Eqn. 4.53 is known since it is composed of the measured quantities 5.m(t),m(t) and ym(t). The quantities !rm(t) and im(t) were obtained numerically by taking the first and second derivatives of the measured response ym(t). Equation 4.53 can then be written as 1 2 M DL 3 & CL rrcos(2rmft + 0(m)), (4.54) =1 n=1 where f(t) is the left hand side of Eqn. 4.53 but of opposite sign. Equation 4.54 is similar to Eqn. 4.40 thus the same procedures used for reducing data using Eqn. 4.40 were applied to obtain the lift coefficients (CL) along the tower for each harmonic. CHAPTER 5 EXPERIMENTAL DATA ANALYSIS This chapter contains the data, data analysis and results from 276 test runs for two types of experiments. The sources of uncertainty and inaccuracy associated with the data are also discussed. The results indicate significant effects of flow nonuniformity and structure transverse motion on the vortex-shedding process, loading and thus the structure response. 5.1 In-line Forces The in-line force data were analyzed using Eqn. 4.8 from which the inertia and drag coefficients (Cm, and CD) were obtained by the least squares curve fit technique. It was assumed that these coefficients are constant over one cycle. The mean values of Cm and CD were then computed for 13 to 73 cycles depending on the driving frequency, fd. The results show that Cm and CD do not vary significantly, from one cycle to the next. The average percentage error between the measured and predicted forces was also computed. For most of the results the error for Cm and CD was less than 5%. A flow chart of the computer program (CDCM), written to analyze these in-line force data, is given in Appendix C. For the range of Reynolds numbers, R,, and Keulegan- Carpenter numbers, KCC, tested, the Cm, and CD data show a clear dependency on R, when KC is held constant. This is consistent with the results in the lower range of KC for a mechanically oscillated cylinder in still water, see Chakrabarti et al. (1983). The results for Cm and CD versus Re for different values of KC are shown in Figs. 5.1 and 5.2. These figures show that, for the range of IKC and Re tested, Cm and CD are strongly dependent 64 on KG and R, and are decreasing as R, increases. This trend is also consistent with the trend reported by others for the range of the parameters tested (see Sarpkaya, 1976a). Figure 5.1 also suggests that the fluid added mass increases as the cylinder amplitude of oscillation increases, i.e., as KG increases. Figure 5. 1: Cm, versus unconstrained). 4 6 8 10 12 14 16 Re x 10-4 Re for harmonically oscillated articulated tower (transversely It was observed by many investigators (e.g. Sarpkaya, 1976a) that, for lower range of KG, the in-line force is usually inertia dominated. The data from these tests were also more suitable for extracting C, than CD, since the maximum KC attained was 9.35. This was examined by calculating the "error surfaces" which are a minimum at CDmin and Cmm.i.. The method used is discussed in detail by Dean (1976). Figure 5.3 shows an example of these error surface contours for one of the test runs. Most of the other runs show the same shape which means that the data are well- conditioned 65 2.0 Symbol KC 1.6 o 3.0 x 4.5 + 5.5 * 6.5 1.2- [ 7.5 1.2 8.5 S"M 9.5 0.8 * 0.4 + 0 I I I I I I 0 2 4 6 8 10 12 14 Re x 10-4 Figure 5.2: CD versus R, for harmonically oscillated articulated tower (transversely unconstrained). for determining Cm. The scatter in CD could be caused by the errors from various sources as will be discussed later in this chapter. In summary, the number of useful data points for establishing the CD Re relationship is small compared to the number used for Cm especially in the lower range of KC. The phase relationship between the measured forces and the corresponding calculated forces was also examined. The phase differences were generally found to be small. This is illustrated in Fig. 5.4, which shows a few examples of the force time history. In an effort to show the effect of flow nonuniformity on these data, a comparison of the present CD and C, data with those obtained by Sarpkaya (1976a) in uniform oscillatory flow and Bearman et al. (1985) in waves was made. The results are shown in Figs. 5.5 and 5.6. Although both Cm and CD show the same trend for the range 66 2.05 2.05 1.73 3E+003 2 1.6FE+003 1.41 1.2E+003 1.41 / ...._. 800 1.09 01-.0 1.090 8 0.77 0 C) 0 0.77 0.46 0.46 0.14 0 400 o.14 -0.18 B -0.18 -0.50 111 -0.50 -1.50 -1.20 -0.91 -0.61 -0.32 -0.02 0.27 0.57 0.87 1.16 1.46 1.75 2.05 CD Figure 5.3: Contour lines defining error surfaces of the in-line force (KC = 8.4 and Re = 8.4 x 104). of KC tested, C, does not exhibit a dependency on the type of flow. On the other hand, CD seems to be affected by the flow nonuniformity in that its magnitude is less for nonuniform and wave flows than for uniform oscillatory flow. The CD values in waves and nonuniform oscillatory flows were found to fall within the same range. The differences between CD values in uniform and nonuniform oscillatory flows could be attributed to the varying R, along the cylinder in the nonuniform oscillatory and wave flow cases since Re varies from a value close to zero at the bottom to a maximum value at the water surface. This in turn could cause the flow to change from subcritical to critical or even supercritical (depending on the water depth and the cylinder diameter) along the cylinder. It may also cause the vortices to separate from the surface at different moments in time along the cylinder leading to a phase difference between the vortices and consequently a smaller correlation length. 67 50 40 ,; S 30-I S20 ! I i z '- 10 U. 0 0 i!/ "' 10S-20 -30i -40 0 2 6 6 8 10 14 16 a time (see) 30 20 E 10 0 C -0 o. ... prdct. T-10 I . -20 -30 2 4 6 8 10 12 14 16 b time (sec) Figure 5.4: Phase comparison between measured and predicted in-line forces (a) KC =6.6 and Re= 9.4x104 (b) KC =8.4and R = 8.4 x104 (- measured, ...predicted). 6.0 10.0 KC 20.0 40.0 60.0 100.0 Figure 5.5: Comparison of C, for different flow types. I I I l l I I I I I I I I I II + 130+.%. , vq ** "WV v& % B *1*2v Yv IM H9VyV V *.: *%%. : oo0o 0000 A e ,+ ,o,*, v, Sarpkaya (uniform oscillatory) e Bearman (regular waves) 0, ,, author's investigation. I I 1 1 I I s I I I I I III 4.0 6.0 8.010.0 KC 20.0 40.0 60.0 100.0 Figure 5.6: Comparison of CD for different flow types. I I a I i I I I I II1 1 I I I I I I I II -~ +B a,+,a,o, Sarpkaya (uniform oscillatory) * Bearman (regular waves) *,w,*, .,,O,n author's investigation I I I I I I I I I I I I I I I I I 3.0 2.0 21.0 0* 0.5 2.0 1- 1.0 0 U 0.5 0.1L 1.0 69 Equation 4.25 was also used instead of Eqn. 4.8 to reduce the in-line force data and to examine the drag force-velocity relationship. A flow chart of the computer program (CDCMN) to compute Cm, CD and n is given in Appendix C. Examples were run to test the drag-velocity relationship. The results show that the power "n" to which the velocity is raised is close to 2 for most of the cases tested; see Table 5.1 for samples of the results. Table 5.1: Samples of test results on in-line force data RX KC] [nCD [CMJ 5.82 x 104 5.9 1.98 0.35 1.15 8.35 x 104 8.4 1.81 0.19 1.16 9.50 x 104 5.8 1.97 0.55 0.72 8.30 x 104 6.7 1.86 0.23 0.90 9.40 x 104 6.6 2.04 0.13 0.78 1.20 x 105 6.6 1.96 0.15 0.69 4.20 x i04 6.5 1.88 1.00 1.77 3.32 x 104 6.5 2.05 0.88 2.30 5.2 Transverse Forces In this section the analysis of the data pertaining to vortex-induced transverse forces is presented for both the transversely constrained and the unconstrained experiments. This includes the analysis of measured transverse forces, thermistor signals and transverse motion. 5.2.1 Constrained Transverse Motion When the tower was constrained from motion in the transverse direction, the transverse forces were analyzed using the proposed transverse force model, Eqn. 4.42. The lift coefficients CL(n) and their associated phase angles 0(,q for the first eight harmonics were evaluated using Eqns. 4.47. A flow chart of the computer program (CLEF) written to obtain CL(,) and 0(,, from the measured transverse force data is given in Appendix C. The results obtained were plotted versus different parameters, 70 such as, KC, R, and P3. The values of the lift coefficients, CL(n) were found to be dependent on both KC and R,. However, the CL(n) values were found to correlate better when plotted versus KC for constant /3, see Figs. 5.7 and 5.8. In general, CL (n) decreases with increasing KC as shown in Figs. 5.7 and 5.8. For KC > 7.0, CL(n) values start to increase slightly indicating the possibility of a multi-peak relationship with KC. This behavior has also been observed by Bearman et al. (1981) and Ikeda and Yamamoto (1981). The dependency of the CL (n) values on P3 and R, was also found to be strong, especially for the first four harmonics where the lift coefficients are the largest, see Fig. 5.7. CL(S) were also plotted versus /3 for constant KC as shown in Figs. 5.9 and 5.10. The trend exhibited in Fig. 5.9 suggests that there is a critical value of / or R, at which the lift coefficient is a maximum. Of all the different harmonics, the largest value of the lift coefficient occurs at the driving frequency. In general, the magnitudes of the different components of the lift coefficients were found to vary between 0.0 and 0.7 except at 83 = 2, 100 where, CL(1) reached a value of 1.4. On the other hand, the associated phase angles show no orderly dependency on R,, KC or 3. As can be seen in Figs. 5.11 and 5.12, the phase angle data associated with the lift coefficients (CL(,)) appear to be random. This random phase could be related to the shedding process that exhibits modes of behavior with vortex interaction between newly shed and earlier formed vorticies. The variability in the modes and thus phasing of the vorticies shed along the cylinder could also be a major factor. The phase relationship between the measured transverse forces and the corresponding predicted forces using the proposed model for computing transverse force show that while the maximum forces are reasonably correlated, the phase differences are often large. A few examples of the force profiles are shown in Figs. 5.13 and 5.14. I I I I I I Symbol E 1792 + 2160 2453 10 2780 X 3130 M 3690 0.41- 01 1 a I I I I I I I I 1.2 - Symbol p 0 1792 + 2160 2453 0 2780 X 3130 M 3690 I- 1 0 U ia Figure 5.7: Lift coefficient harmonics for harmonically oscillated articulated tower (transversely constrained), CL(1) to CL(4). 0.0 - 1.61 I I I I I I I I 1.2F 0.4 1 2 4 6 a 1C KC 1F I I i I I i I I Symbol i 1.2 179 + 2160 * 2453 10 2780 X 3130 0.8 0 3690 0.4 x I.01 1 1 1 1 1 1 I I 1.21 0.81 0.4 [ 2 4 6 i KC 1. I I I I I I I I Symbol 0 1.2 I 1792 + 2160 * 2453 D 2780 X 3130 0.8 IN 3690 0.4 0- Figure 5.8: Lift coefficient harmonics for harmonically oscillated articulated tower (transversely constrained), CL(5) to CI,(s). Symbol P 0 1792 + 2160 2453 0 2780 X 3130 00 3690 Symbol I0 N] 1792 + 2160 * 2453 1] 2780 X 3130 - M 3690 -4 ,x 0.8 0.4 04 0.8 0.4 Figure 5.9: Lift coefficient harmonics versus beta, CL(l) and CL(2). 0.8 0.4 LI II Symbol KC 2- 2.5 + 3.5 * 4.5 o 5.5 x 6.5 - a >7.0 -- 0 12 UU 16bUU0 2000 _ 0.8 0.4 2400 2800 Figure 5.10: Lift coefficient harmonics versus beta, CL(3) and CL(4). 1. I I I I II I I I I I I I I Symbol p I 1792 .+ 26n 180 10 182453 + 4 m 2780 x 4 n 3130 a x m 0 + L30 0 + + a N x 3690 o 3690 0 2* 8 1 0 2 68 1 x x Symbol on ab x -- W 1792 x 0 -180 m +1 2160 -180 * 2453 S2780 x 3130 30 3690 -360 I l I I360 II I 0 2 4 6 0 4 68 10 KC KC 360 I I I I I I I 3 1 1 1 1 1 1 SSymbol p W 1792 + 2160 180 2453 x N 180 _ 2780 x 0 3X 3130 a a 0 390 + o "x 9 m 0 3+X g X I * a am a x X" m N +-+ x a Symbol a + m l 1792 -180 -180 x + 2160 * 2453 O 2780 X 3130 M 3690 -360 1 I 1 1 I 1 .:360o 1 1 1 1 1 1 1 2 4 6 8 10 0 2 4 6 8 10 KC KC Figure 5.11: Phase angle associated with the lift coefficients harmonics, O(1) to 0(4). I I I I I I I I 360 1 1 1 1 1 1 I Symbol p - N 1792 + 2160 1 1 2453 180 0 2780 * X 3130 01 3690 1 a a 0 + - x * m x x + -180 3orn I I I I I I I I -360 I I I I I I I I I S , -I x * 0 x m 0- ~4 aonm x + N 1792 -180- + 2160 S2453 S2780 - X 3130 Il 3690 36 I I I I I I I KC KC Figure 5.12: Phase angle associated with the lift coefficients harmonics, (5) to (s). * U 8 1 4 KC 8 1 2 4 KC z 2.5 -2.5 5 S2.5 E 0 a 2. predicted transverse force time series 2time 3 4 sec] time (sec] frequency [Hz] 01 24 5 6 7 9 10 frequency (Hz] Figure 5.13: Comparison between measured and predicted transverse forces (Re = 104 and KC = 5.45). time [sec] 7 a 9 10 1.2, 0.8 0.e 0.4 0.2 m 4 5 8 10 time (sec] -20 21 " 20 - C a 6 4 2- 0 0 0 power specrumr ai measure. tranrse r iurc 1 2 3 4 5 6 7 8 9 11 frequency [Hz] power spectrum of predicted transverse force SA A . 1 2 3 4 5 frequency (Hz] 6 7 a 9 10 Figure 5.14: Comparison between measured and predicted transverse forces (Re = 2 x 10' and KC = 5.9). 5 time (sec] 4....... -' -- A *.-.....-........ # 79 For the purpose of comparison with other investigators' data the transverse force data were also analyzed in terms of CLrm, which is defined as CLrms = Frs (t) (5.1) 7 P, D t Ulm. where Fms(t) is the root-mean-square (rms) value of the measured transverse force. The results shown in Fig. 5.15 exhibit the same trend as the CL harmonics and are found to correlate well with KC for constant /3 with magnitudes varying between 0.12 and 0.6. In this figure the results show that, for KC < 5 (where almost no published data exist for nonuniform oscillatory flow) CLrms increases as KC decreases. However, as one would expect, according to other investigators data for 5 < KC < 11, CL,,,, increases as KC increases (see Fig. 2.14). 1.0 I I I I 0.8 Symbol f3 0 1792 + 2160 * 2453 0.6 2700 U)0 2780 E \ X 3130 Ml 3690 0.4 0.2 0 = =..,.0 I I I I I I II 0 2 4 6 8 10 KC Figure 5.15: RMS lift coefficient versus KC. 80 The present CLms data were also plotted with other investigators' data obtained for waves, see Fig. 5.16. As shown in this figure, where KC overlaps, the lift coefficients, CL,m,, for the oscillating tower in still water agrees very well with CL,,m, data for waves. For the range of R, and KC tested, Fig. 5.16 confirms the possibility of a secondary peak for KC between 2.0 and 3.0. This would give a KC spacing of z_ 8.5 from the main peak which is at KC z- 11.0. This is approximately the interval of KC found by Ikeda and Yamamoto as reported by Williamson (1985). Intervals between the CL peaks were found to be approximately 7.5 for KC < 70.0. 1.8 i I I 1 1.6 Symbol Reference 0 Bearman, 1988 0 0 Isaacson & Maull, 1976 1.4- 0 Sawaragl, et. al, 1976 o Sawaragi, et. at, 1976 1.2- 1 1 A Author's Inverstigation 00 0 1.2-0 o E 00 0 0.8- 0 0 0 .60- 00 MO M "0 *.O O - 0.4 o o 0.6 A 0 0 0.2-0 00 o,0 I 0 5 10 15 20 25 30 KC Figure 5.16: Comparison of CLrm, data with those from waves. Even though the data from proprietary studies are not available, the ranges of Re and KC covered by these data sets are generally known. To the author's knowledge CL data does not exist for most of the range of the parameters covered in these experiments. As indicated earlier this is an important range for many structural elements 81 and offshore structures. In a paper to be published soon by Horton et al. (1992) will show that there are no data available in this range, see Fig. 5.17. This figure (taken from Horton's to be published paper) is reproduced here with his permisson. f.0 < / ,/A,. /\ C , /y /O s-,-/.-/L (,/ W/\), ve z. **/2". :/. L.I.r Coo. i .. -'*/ //. 50 ~ v ' U -.3/0 9 jeA-.,/0. /07/ / ../D5 I I6 ]O j REYNOLDS NUMBER Figure 5.17: Range of data from wave forces hydrodynamic experiments (Ref. 27). There are several ways to obtain information about the frequency of vortexshedding. One common way is to perform a spectral analysis on measured transverse force data. In this study such a method was used and the assumption that the transverse force can be considered a stationary random process was made. Spectral analyses were carried out on the transverse force data to study the frequency content of the signals. This was done by using a commercial spectral analysis package called GLOBALLAB which uses an FFT algorithm. Fractional time series of 12.8 seconds 82 duration and 0.025 seconds time intervals (which gives a frequency resolution of approximately 0.078 Hz) were chosen. The final spectrum consisted of the average of 5 spectral estimates. Since the frequency of vortex-shedding in oscillatory flows is known (see literature review, Chapter 2) to be related to the in-line flow oscillation frequency, the frequencies in the power spectra were normalized by the driving frequency. Also to illustrate the dependency of the different harmonics in the transverse force on R, and KG, the power spectra were plotted, as groups, for a range of KG tested holding Re. constant. They were also plotted holding 03 constant. Figure 5.18 and Fig 5.19 show examples of these spectra for constant R, and /3, respectively. On investigating the frequency of the peaks of these force spectra, it was observed that the frequency of the dominant peak was always a multiple of the driving frequency (i.e., frequency ratio fl/fd = N, where N is an integer). This agrees with the findings of other investigators in oscillatory flows and waves (e.g. Chakrabarti et al., 1976, 1983; Bearman and Hall, 1987; Justesen, 1989; and others). The plots in Fig. 5.19, where 83 is the correlating parameter, exhibit the same trend as those of other investigators shown in Fig. 2.9. Because the frequency of vortex-shedding is known to be a very important parameter in this type of fluid-structure interaction problem, the measured signals from the thermistors mounted on the tower's surface were also analyzed using spectral analysis techniques. The spectra of the thermistor signals were found to have the same trend with KG, when P3 was held constant, as those for the transverse force data shown in Fig. 5.19. Examples of thermistor output spectra are shown in Figs. 5.20 and 5.21. Each figure comprises 5 spectra of signals from the top 5 thermistors (the only ones that worked for all of the tests) placed one diameter apart along the length of the cylinder, see Fig. 5.22. Similar results were found to exist for all values of /3 tested. This supports the finding that #3 rather than Re is a better correlating parameter for the vortex-shedding frequency. In general these spectra show that the frequency of the dominant peak is |

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