• TABLE OF CONTENTS
HIDE
 Front Cover
 Title Page
 Acknowledgement
 Table of Contents
 List of Figures
 List of Tables
 List of symbols
 Abstract
 Introduction
 Literature review
 Experimental investigations
 Mathematical models
 Experimental data analysis
 Summary and conclusions
 Appendix A: Analysis technique...
 Appendix B: Instrumentation and...
 Appendix C: Flow charts of the...
 Bibliography






Group Title: Technical report – University of Florida. Coastal and Oceanographic Engineering Program ; 89
Title: Vortex-induced transverse loading on an articulated tower
CITATION PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
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
Physical Description: xviii, 153 p. : ill. ; 28 cm.
Language: English
Creator: Omar, Ahmed Fahmy, 1959-
University of Florida -- Coastal and Oceanographic Engineering Dept
Publisher: Coastal & Oceanographic Engineering Dept., University of Florida
Place of Publication: Gainesville Fla
Publication Date: 1992
 Subjects
Subject: Vortex-motion -- Mathematical models   ( lcsh )
Water waves -- Mathematical models   ( lcsh )
Hydrodynamics   ( lcsh )
Coastal and Oceanographic Engineering thesis Ph. D
Dissertations, Academic -- Coastal and Oceanographic Engineering -- UF
Genre: government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (Ph. D.)--University of Florida, 1992.
Bibliography: Includes bibliographical references (p. 148-153).
Statement of Responsibility: by Ahmed Fahmy Omar.
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.
 Record Information
Bibliographic ID: UF00075488
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved, Board of Trustees of the University of Florida
Resource Identifier: oclc - 28172461

Table of Contents
    Front Cover
        Front Cover
    Title Page
        Title Page
    Acknowledgement
        Acknowledgement
    Table of Contents
        Table of Contents 1
        Table of Contents 2
    List of Figures
        List of Figures 1
        List of Figures 2
        List of Figures 3
        List of Figures 4
        List of Figures 5
    List of Tables
        List of Tables
    List of symbols
        Section 1
        Section 2
        Section 3
        Section 4
        Section 5
    Abstract
        Abstract 1
        Abstract 2
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
    Literature review
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
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        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
    Experimental investigations
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
    Mathematical models
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
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        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
    Experimental data analysis
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
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        Page 74
        Page 75
        Page 76
        Page 77
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        Page 79
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        Page 82
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        Page 115
        Page 116
        Page 117
    Summary and conclusions
        Page 118
        Page 119
        Page 120
        Page 121
        Page 122
        Page 123
        Page 124
    Appendix A: Analysis techniques
        Page 125
        Page 126
        Page 127
        Page 128
    Appendix B: Instrumentation and calibration data
        Page 129
        Page 130
        Page 131
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    Appendix C: Flow charts of the various computer programs
        Page 142
        Page 143
        Page 144
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        Page 146
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    Bibliography
        Page 148
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Full Text



UFL/COEL-TR/089


VORTEX-INDUCED TRANSVERSE LOADING ON AN
ARTICULATED TOWER


by



Ahmed Fahmy Omar


Dissertation



1992




COASTAL & OCENPo~IA HIC, ENGIEERIG DEPARTMENT
University of Florida Ganesvlle, Florida 32611
r





















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













ACKNOWLEDGEMENTS


The author wishes to express his sincere appreciation and gratitude to the su-

pervisory 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 work-

ing under his patient leadership. His input and encouragement in this long endeavor

proved invaluable.

A special debt of gratitude is owed to Profs. Marc I. 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 De-

partment 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, Nonplanar Flow . . . ..... ..... 24
2.5 W ave 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


_ II __ _~ ____ _r









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 ...........................


.........
. . . .
. . . .



. . . .













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 spec-
tra 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 spec-
tra for steady, planar flow around a smooth, transversely uncon-
strained 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, R). . . . ... 18

2.8 Lift coefficient versus Keulegan-Carpenter number for oscillatory,
planar flow around a smooth, transversely constrained cylinder
(for constant frequency parameter, ). . . . ... 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 spec-
tra for oscillatory, planar flow around a smooth, transversely un-
constrained cylinder for various reduced velocities, V, (Ref. 47). .23

2.12 Span-wise coherence length for transversely unconstrained cylin-
der near a wall in oscillatory, planar flow (Ref. 39). . ... 24


_____ ____~ _~_ _~_____ ____








2.13 Lift coefficient harmonics versus Keulegan-Carpenter number, KC,
curve fit of data from harmonically oscillated articulated cylin-
der in still water, 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, KC (Refs. 12, 73) .................... 30

2.16 Maximum transverse response for various flow configurations (0 ste-
ady, planar flow), (o oscillatory, planar flow), (* 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 mo-
tion . . . . . . . . .62

5.1 Cm versus R, for harmonically oscillated articulated tower (trans-
versely unconstrained) ......................... 64

5.2 CD versus R, for harmonically oscillated articulated tower (trans-
versely unconstrained). . . . .... ........ 65

5.3 Contour lines defining error surfaces of the in-line force (KC = 8.4
and R, = 8.4 x 104) ........................ 66

5.4 Phase comparison between measured and predicted in-line forces
Sa) KC = 6.6 and Re = 9.4 x 104 (b) KC = 8.4 and Re = 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


_~~~ _. ~~_________ _1








5.8 Lift coefficient harmonics for harmonically oscillated articulated
tower (transversely constrained), CL(s) to CL(s) . .... 72

5.9 Lift coefficient harmonics versus beta, CL(1) 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). . . . . . . . .. 776

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
(Re = 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(Re = 1.15 x 104). 83

5.19 Transverse force power spectra for constant /3(/ = 2,800). . 83

5.20 Thermistor power spectra for # = 1,834. . . ... 84

5.21 Thermistor power spectra for / = 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 articu-
lated tower.... ................ ........ 88

5.26 Span-wise coherence length for transversely constrained articu-
lated tower.... ................ ........ 88

5.27 Comparison between measured and predicted transverse responses
ea) R, = 2.7 x 104 and KC = 5.0 (b) R, = 5.8 x 104 and KC = 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 re-
sponses (Re = 8.3 x 104 and KC = 6.7). . . ... 93

5.30 Comparison of measured and predicted transverse responses (R, =
105 and K C = 8.4). ......................... 94

5.31 Comparison of measured and predicted spectra of transverse re-
sponses (Re = 105 and KC = 8.4). . . . .... 94

5.32 Thermistor power spectra at / = 14,265 and ym,,/D = 29.5%. .96

5.33 Thermistor power spectra at / = 14,265 and yrm,/D = 20.9%. .96

5.34 Thermistor power spectra at / = 14,265 and y,,m/D = 8.2%. .. 97

5.35 Thermistor power spectra at / = 16,260 and yrm,,/D = 16.8%. .97

5.36 Thermistor power spectra at 0 = 16,260 and ym,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 / = 6,500 and y,ms/D = 50%. 101

5.40 Thermistor power spectra at / = 9,900 and y,rm,/D = 16%. 102

5.41 Thermistor power spectra at / = 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 f = 9,900 and yrm,/D = 40%. 103

5.44 Thermistor power spectra at / = 12,500 and y,m,/D = 33.5%. .104

5.45 Schematic of vortex-shedding frequency behavior (/ = 9,900). .104

5.46 Span-wise coherence length for transversely unconstrained articu-
lated tower. .. .. .. .. .. . . .. .. .. 105

5.47 Transverse motion power spectra for / = 5,400 . .... 107

5.48 Transverse motion power spectra for / = 6,500 . .... 107

5.49 Transverse motion power spectra for / = 9,900 . ... 108

5.50 Transverse motion power spectra for 0 = 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 H z.). . . . . . . .. 132

B.4 Calibration curve for in-line force (for test runs at 0.15 > fd <
0.3 H z.) . . . . . . .. 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, pas-
sive, 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 vortex-
induced transverse loadings and response (numbers refer to refer-
ences). . . . . . . . . .. 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 mea-
sured. . . . . . . . ... .. 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

B.4 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

C' drag coefficient associated with velocity raised to power n

Cm added mass coefficient

CM inertia coefficient

C, structural damping coefficient

CL lift coefficient

Cn nth harmonic lift coefficient

CLmn ~ nth harmonic lift coefficient of element "m"

CLmax maximum lift coefficient

CLrms root mean square lift coefficient

d water depth

di water depth form the bottom hinge

dr element length

D cylinder diameter

E[C2] expected value of the square of the lift coefficient








f frequency ( Hz)

fd driving frequency ( Hz)

f system natural frequency ( Hz)

f, cylinder response frequency ( Hz)

f vortex-shedding frequency ( Hz)

f, fluid frequency of oscillation ( Hz)

FB buoyancy force

FD drag force

FL transverse (lift) force

Frms root mean square force

Fyma maximum transverse force

Fym root mean square of transverse force

g acceleration of gravity

h water depth

H wave height

aI 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

KI stability parameter

e cylinder length


~__ _








s, 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 fr/fd

MD moment about bottom hinge due to drag force

Mf, 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 fl/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

Sx,(f) auto-spectral density of signal x(t) at frequency f

S,,(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

Ty external generated turbulence

U uniform flow velocity

Umax maximum flow velocity

Urms rms of flow velocity

Vr 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

ym measured transverse linear velocity of the tower

ym measured transverse linear acceleration of the tower

y, predicted transverse linear motion of the tower

yrm, 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

Yrm maximum of rms amplitudes of transverse linear motion of the tower

z span-wise coherence length of vortices

Ph hydrodynamic damping coefficient

# frequency parameter (= Re/KC)

7-y(f) correlation coefficient between signals x(t) and y(t) at frequency f

8,2 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

Ar, distance from transverse force reaction at top hinge to the center
of transverse force transducer

0 in-line angular deflection of the tower

o in-line angular velocity of the tower

S- in-line angular acceleration of the tower

v fluid kinematic viscosity

critical damping factor

p, mass density of water

o' variance of transverse force

a circular frequency (rad/sec)

0(m) mth harmonic of the phase angle associated with lift coefficient

Om mth harmonic of the phase angle

S- transverse angular deflection of the tower

transverse angular velocity of the tower

transverse angular acceleration of the tower

w cylinder frequency (rad/sec)

Wd cylinder driving frequency (rad/sec)

wf fluid frequency of oscillation (rad/sec)

w, vortex-shedding frequency (rad/sec)


__~_~ ~I_ 1__ _IIC____ __~___~~_ _~~ _~_~II~













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 fre-

quency, and vortex span-wise coherence length have been investigated. To quantify

such effects, two types of experiments were performed. Each type of experiment con-

sisted 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, Re, 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 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


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 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 load-

ing. 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 1
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 interac-

tion 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 trans-

verse (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 in-

line 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 vortex-

shedding 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, non-

planar nonuniformm), 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 trans-

verse forces especially at high Reynolds numbers. The difficulty is partly due to

incomplete knowledge of the flow field around the structure, and to problems as-

sociated 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 vortex-

shedding process, vortex span-wise coherence length, magnitude of the vortex-

induced 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 (Re) 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 2
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 vor-
tex-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 Effects
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 Waves






9
Some of the earliest studies were on steady, planar, relative flow, perpendicu-

lar 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 Freds0e, 1988; T0rum and Anand, 1985; Verley, 1980), structure aspect

ratio (i.e., cylinder length to diameter ratio) (e.g. West and Apelt, 1982), flow block-

age (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 span-

wise 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 de-

pends 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 reli-

able 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, pla-

nar, 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 cylin-

ders 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, Re (relative magnitudes of inertia and viscous forces), was the most im-

portant parameter to characterize the flow around a circular cylinder. For steady,

turbulent flows around right circular cylinders, the in-line loading has been formu-

lated 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) (= 2Fyrmsl/pDU2) versus

Re (= 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 illus-

trated 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

O
A
A
X
o
0
x

+
V





I I I I 11iil


1 II 111111

Ref. #
57
57
57
57
57
57
75
57
57
19
57
57 00


I I I







0


I I I I IIIII









A
A


A

S


I 11111


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 Re 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 pres-

surized 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 105

(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 < Re < 2 x 105 and well defined by a Strouhal number of about 0.2

for smooth cylinders.


111111 I


Q)T
E
0-
0


0.50


0.00 ..


0o oo
I 14el 1


I I fill


I


"""'
















Z_ 65


Figure 2.3:
(Ref. 14).


^- 3 X 105s Re < 33 X 106
LAMINAR BOUNDARY LAYER HAS UNDERGONE
TURBULENT TRANSITION AND WAKE IS
NARROWER AND DISORGANIZED

3. X 106 < Re
d RE-ESTABLISHMENT OF TURBU-
LENT 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)

S= St(1 + 19.7/Re)(U/D)


Re < 60

60 < Re < 2 x 105


In the turbulent regime (2 x 105 < Re < 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



5 TO 5 < Re < 40 A FIXED PAIR OF
VORTICES IN WAKE

40 < Re < 90 AND 90 < Re < 150
TWO REGIMES IN WHICH VORTEX
STREET IS LAMINAR



150 < Re < 300 TRANSITION RANGE TO TURBU-
LENCE IN VORTEX
300 < Re Z 3 X 105 VORTEX STREET IS FULLY
TURBULENT






























le M Re =7.1 x 106
Figure 2.4: Strouhal-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 pro-

cess. 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 coher-
ence length support the importance of the Reynolds number (Re). 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.















0


5-J
Z
UI-



00

104 105 106
Re
Figure 2.5: Span-wise coherence length versus Reynolds number for steady, planar
flow around a smooth, fixed cylinder (Ref. 68).

2.2.2 Transversely Unconstrained Cylinder in Steady, Planar Flow

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; Torum and Anand, 1985) have reported






16
information on the response of the cylinder. The spectra of the transverse force
and corresponding response by T0rum and Anand (1985) were the only data found

that show the relationship between the vortex-shedding frequency (f,) and response

frequency (fr). 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 = 10-6

m2/sec was assumed in the computations of Re shown in the figure.

FORCE RESPONSE








Re= 3.35 x 10

Re= 3.1 x 104

Re = 2.94 x 104
1 2 3 4 f 1 2 3 4 --- ffn

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 depen-

dency 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 acceler-

ation 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 (Re = UmaD/v) and Keulegan-Carpenter

Number (KC = UmaxT/D = 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 (CLmax =

2Fjmax/pDUax, CLrm, = 2Fyms/pDUmax, CLm, = 2Fymms/pDUfms, 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 Re constant

while other investigators maintained # constant. These data are plotted separately

in Figs. 2.7 and 2.8. Those data not reported in terms of CLams = 2Fyrms/pDUma~






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.

I I I I I I I I I I I I I I II I


2



0
E

U 1-


0 0 1 A R I I I B1 1 I I I I '
1 10 100
KC

Figure 2.7: Lift coefficient versus Keulegan-Carpenter number for oscillatory, pla-
nar 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
S 49 7 x 104 105
61 (1-2) x 105
42 2.5 x 105
A 42 5x105
42 7.5 x 105
D 42 1.0 x 106


Ajr
*I
13^
















"- A 59 4720
E 59 6555
t 0a3 o 79 >18600


F130
o T I0



0 IV wVl IM 1 1 1 1 __ I 1l I I
1 10 100
KC

Figure 2.8: Lift coefficient versus Keulegan-Carpenter number for oscillatory, pla-
nar flow around a smooth, transversely constrained cylinder (for constant frequency
parameter, /).

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 vortex-
shedding 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 Keulegan-
Carpenter number "KC = UmaxT/D".

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 spec-

tral 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, KC = 1.7 to 15.6; Bearman and Hall, 1987, KC = 36.13; McConnell and Park,

1982b, KC = 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 vor-

tex 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 har-

monics, depending, primarily, on the value of KC. As KC 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 KC from 4 to 55. Their results show that the

correlation length does not decrease monotonically with increasing KC. The highest

correlation length was obtained at KC = 10 (see Fig 2.10) and at this value it was

approximated by 4.5D. They also found that for KC > 30 the correlation is no longer

sensitive to KC.






















_ 4 .,.KC = 3.137
KC 30.13


A11. IIJVYK


KC 15.6


KC s 13.6


. A .


1-i-- I- I I I I i-1


KC212.


KC.O.


I I I


KC 9.8


KC. 6.8


/ ^Ai 5J, K5


KC 2.8


KC 1.7


0 2 4 6 8 10


-- 111. 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).




0.9

KC= 10
0.8

0.7


S0.6-
0.6


o 0.5

C
.2 0.4-

L 0.3

KC = 42
0.2

0.1 KC = 22


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 os-
cillatory, planar flow (Ref. 50).


I


i


I L


/I I


/


/ r\ I .


I K


II I
11 I I I l


I I I I I I


I I I I I


I I


" "W


--


i


-L


In I Ar A-


I K!; a 8.k


II I I I I


A = -


KC a 12.5


. KC 10O.6


l | I I


I~


S KF 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 Freds0e, 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, =

Umax/Dfn = 27r(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 vortex-

shedding, 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 G_ 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 coeffi-

cient 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 interac-

tion 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, = ff, = 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
> I f/f 1 -f/f n
0 2 4 n 0 2 4 n

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 G/D = 2.3) are reproduced






24

in Fig. 2.12. As shown in this figure the correlation between vortices is largest at

KC = 6 (as compared to KC = 10 for a fixed cylinder [Obasaju et al., 1988]) and

like the case of fixed cylinder is mainly dependent on KC. The results of their study

also showed that for a fixed KC the larger the amplitude of transverse motion the

larger the correlation.

0.9 -

0.8

-0.7



0.6

0


a 0.2 5
KC = 65
0.5 -
0











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

KC 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 exper-

iments 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 (sim-

ilar 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 calcu-

late 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 KC increases,

the frequencies cluster around a value twice the driving frequency but became less

organized at higher KC. 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.



















o0 .0 'o0 1 L' ' i' I I I I I 1 0 .0 o I
0 5 10 15 20 25 0 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 ver-

tical 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 previ-

ous 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. poly-

nomial 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 CLrms 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 CLmax. In this figure, Bearman's

data are based on an rms velocity from measurements near the location of his in-

strumented 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 CLrms 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 differ-

ences 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
1.6 I I0 IS Imboll Ref # Kh
1.6 -] Symbol Ref # Kh _


1.4 -

1.2- *o *
0 0
1.0 0
o o
0 0.8 o o


r-IoP *m *
0 0 0
0.6 *0o I
8
0.4- 0 [o o O oo

0.2- 0

0 ?_ I I I I I
0 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 w 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
o 30 0.76-0.79
* 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 yrms/D (defined as Y,,,/D) from several investigators' data

were plotted versus several of the pertinent parameters. A plot of Y,,,mD versus V,

(see Fig. 2.16) was found to be a meaningful. This plot shows that the maximum
values of the response (Y,,,/D) for different flow conditions fell in a relatively narrow

range of reduced velocity, Vr.

In summary, the existing data on the vortex-induced transverse loading and re-

sponse 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 analy-

sis. 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 coef-

ficients and frequency of vortex-shedding) and on the vortex span-wise correlation

length.




















KC 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.
1.41 i ------


0



0 O


0.4 -


0.2 -


Symbol


0
o


Ref #
18, 21,52

40, 46, 71
74, 75, 70


o
*O


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), (e*wave flows).


(0 steady,


* 1 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 dimen-

sional analysis is applied to the particular problem of a transversely unconstrained

bottom-hinged cylinder in nonuniform oscillatory fluid flow, the amplitude of trans-

verse motion can be expressed in terms of the lift coefficient, CL, reduced velocity,

Vr, stability parameter, K,, and added mass to cylinder mass ratio, m,/m,, (see

Appendix A for details), i.e.,

y/D = Y(CL, V,,mJlm,) (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(Re, KC). (3.3)

Consequently, Eqn. 3.1 reduces to

y/D = Y(Re, KC, V,, K,, m,/m,). (3.4)

For complete similarity the values of all the parameters should be the same for

both model and prototype, i.e.,










UD UD
V 1'
UT UT
KC = ( )m = (-T)


Vl =( )M=( )
fnD fnD )'


pD2 pD2

( )m = ( )p
m, m,

where m and p refer to model and prototype, respectively.

Of these similarity groups the most important parameters are the Reynolds num-

ber (Re), Keulegan-Carpenter number (KC) and reduced velocity (V,). Hydrody-

namic force coefficients for large values of Re are needed in full-sized offshore struc-

tures 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 105, i.e., in the upper subcrit-

ical, critical and supercritical regions (according to steady flow principal Reynolds

number flow regimes around smooth cylinders). Wave tank testing has also the addi-

tional 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 (/D) should also fall

within the prototype range. A survey of the available literature revealed that vortex-

shedding is considerable in the following ranges of similarity parameters: 3 x 104 <

Re < 106, 5.0 < KC < 12.0, 1.0 < Vr < 20.0 and /D 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 /D. 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,

i.e, a ratio of /D = 20. Such a model covered ranges of Re between 6.1 x 103 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,

K, was approximately constant.

The ratio of the fluid added mass to the structural mass (ma/m,) should be

modelled properly. In air m,/m, = O(10-3) and thus is of little importance. In

water, however, the ratio ma/m, = 0(1) 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 coher-

ence 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

can 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%

limit 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 half-

way 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 in-

formation 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 sur-
face 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 II) 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.






40





























/ *.*... .. .-
jjgurfl.j. jogp : -"eim-.





















Figure 3.3: Photographs of experimental set-up.


A


ftml







41


i jn-line Position









Analog/Digital Processing




In-line Position
Signal Transverse
3.I Force Trnd












Frqu Swtcncy Position
Counter P Signal





HP Function owitn
Counter PController
Generator

Linear Motor & Controller

Figure 3.4: Block diagram of measurement system.



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 de-

partment 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 maxi-

mum 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 Tem-

posonics 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 re-

sponse, 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 ther-

mistor 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.11)

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 undesir-

able 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, Butter-

worth, 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 se-

lected. 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 (KC)
and Reynolds (Re) 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 (fn) 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 105
minimum KC 2.6 2.4
maximum KC 8.65 9.35
minimum # 1792 3151
maximum / 3692 18336
minimum Vr 2.66
maximum V, 36.10
fn 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:


Im(t) = Mtotal = M + MD + Mf,,


(4.1)


where
Im = +
= total mass moment of inertia about the bottom hinge,
Io = cylinder mass moment of inertia,
II' = added mass moment of inertia,
Mtota, = total moment about bottom hinge,
Mg = moment due to tower weight and buoyancy,
MD = moment due to drag,
MfX = moment due to linear drive motor and
O(t) = in-line deflection angle.

For the tower shown in Fig. 4.1 the moments Mg,MD and Mf are given by


Mg = (Z w, r,- p
2=1
MD = -1p, DCo dl/o
= RL,
My, = RL,


', A g r dr) sin 0,

r )|r |r dr and


where Ac = 7rD2/4 is the cross sectional area of the tower element. The added mass

moment of inertia, I, can be expressed as


I dl/cosO fdl/cos02 r
= I, dr= p Cm D2 2 dr.
Jo Jo 4


(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


(+ pw D c)2 -m
L 12 L cos3[Xm(t)/L]
1 d4
1P D CDXm(t) iXm(t)
8 D L2 cos4[X(t)/L]CD
n d2
+(E .1 r pg Dcs2 )sin[Xm,(t)/L] = RL. (4.6)
S=1 8 cos2[Xm(t)/L]

Since the measured in-line moment Mm(t) = RArt, R = M,(t)/Art. Substituting
this expression for R into Eqn 4.6 and rearranging results in










M,(t) L/Ar+t + t)
7X D2

( w, r, n- p g 2 [ )/)sin[Xm(t)/L] =
2 dl
pw D2L 3r Xm (1) -
12 L cos3[Xm(i)/L]C2 (t)

p D [Xt/LCD Xn(t)IXm (t)I. (4.7)
8 L2 cos [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) = f (t) Cm + fD(t)D, (4.8)

where

Mm. (t)L I "(t r d2
fm(t) = + Xm() (w,r, sin[Xn (t/L],
ArL L X=' 8 cos2[Xm(t)/L]
x ld
fi(t) = -rp. D2 X- () and
12 L cos3[X,(t)/L]
1 d4
fD(t) = dPw DX |
= L2 cos4[Xm(t)/L] 1(t)I X(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 2 where

1N
N [f=(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










CD =0 and (4.10)
CDn
S = 0. (4.11)
8Cm

Equation 4.10 and Eqn. 4.11 can be written in a matrix form as

E NE f2 (t") *I I CD -NJ=1 ( C= 1 E^/,)fz(t.)
SfI(t)f(t,) C = fm(t)f(t (4.12)
.E=1 /(tf.)D(t,) E=I fD(t1 ) C =1 f (t) fI(t,)


The solution of Eqns. 4.12 results in the following equations for CD and Cm

CD = E[fm(t)fD(t,)] f(t,) Z[f(t,)f(t,) E[fD(t)f(t)] and (4.13)
-[f[(t,)] [ff(t,)] [E fD(t,)fst,)]2

C = El(t,)fi(t)] E fD(t) E[fm(t,)fD(t,) E[fD(t,)fI(t)] (4.14)
SE[fl(t.)] E[f~(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 suit-
ability 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, S2(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











CH



(C )min


Lines of Equal
Errors, -
N.


(__ CD
(CD)min D (CDmin
(a) Data Well-Conditioned for Deter- (b) Data Well-Conditioned for Deter-
mining Drag Coefficients mining Inertia Coefficienta

Figure 4.2: Contour lines defining Error surfaces for the in-line force (Ref. 15).

estimates (CD and C,) were computed. The variance, or, in the value of any function

can be written as


(4.15)


For estimating the uncertainties


N af
01 f D( ) .
=1in CD and m, Eqn. 4.12 can be rewritten asy
in CD and C,, Eqn. 4.12 can be rewritten as


[a]{a} = {#}


2
E a a, = ,
3=1


=2 x 2matrix, or

1S,=l X3(t,)Xk(t,),

= a vector of length 2, oi

= EI fm(t,)Xk(t.)


k = 1,2 and j = 1,2and



k = 1,2.


The inverse matrix Cjk [a]~1 is closely related to the probable (or more pre-

cisely, the standard) uncertainties of the estimated parameters a, = (CD, C,). The


(4.16)


where
[I]




Okj


(4.17)






52
solution to Eqn. 4.12 (or equivalently Eqn. 4.16) is
2 N
= E [al-' k = C1 [E fm(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
S- ft) Ck Xk(t).- (4.19)
ay' af"I (t') k=l
Consequently, the variance associated with the estimate parameters a, is
22 N
U2(a) = EECk C [ Xk(t,) X(t,)]. (4.20)
k=1=1/= =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

a2(a,) = C,,. (4.21)

In other words, the diagonal elements of [C] are the variances of CD and Cm, while the

off-diagonal elements Ck 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 fp(t,)) was used. This was defined as

ERR = 100 [fm(t) fp(t)]2 (4.22)
E f'(2,)
.ft(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 fdl/cose
MD = --p DCD ((rO)n sgn(r0) r dr, (4.23)
i JO






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) 2 LO(t) and carrying out the integration of Eqn. 4.23, the equation
of motion (Eqn. 4.7) reduces to

L I,
Mm(t W + Xm(t)-
Are L
( -r, gD )sin[Xm(t)/L] =
(=1 8 cos2[Xm(t)/IL

12j L cos3[Xm(t)/L] m(t)-

1 D[ d l"2 CD X t) gn[X,(t).
2 cos[Xm(t)IL]' + 2 L2(4.24)
In this equation, the quantities on the left hand side, M,(t),X,,(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) = [ t ] and
cos[Xm(t)/L]
Xm (t)
D(t) = |I ,

Equation 4.24 reduces to

fm(t) = fi(t) C + fdr [C(t)]+2 [D(t)] CD/(n + 2). (4.25)

The unknown quantities in this equation are Cm, CD and n which can also be obtained
by using the least squares equation:

1N
2 = E[f(t,) Cm + fdr[C(t,)+2 [D(t,)]" dD/(n + 2) f,(t)]2 (4.26)

or the weighted least squares equation

1 N
6, = N f(t.)[f(t,) Cm + fd,[C(t ,)J+2 [D(t,)] CDI/(n + 2) fm(t,)]2. (4.27)
N






54

The minimum Cm, CD and n were then obtained by solving the coupled nonlinear
system of equations given by

S = 0, (4.28)
ac.

O= 0and (4.29)
OCD

= 0. (4.30)
On

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 rise 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 trans-

verse 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 105. 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
CLmaP = U (4.32)

has been used.

Others use an rms transverse force that gives a lift coefficient defined by

rms value of the transverse force
CLpms = U2 (4.33)
tpv" De U> max






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 = pW D 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 0 or 1800. In an attempt to improve this model, Bearman et al.

(1984) proposed the following quasi-steady model

1
FL = -p D CL U, cos q 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 I U l CL(n) cos(2Irnft + 0(,)), (4.37)
2 n=1

where N is the number of harmonics, CL(n) 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

g 2S(nf)Af
FL(t) = aLE cos(27rnft + 0(n)), (4.38)
n=l

where Oa is the variance of the transverse force defined by


a2 = {p D U}2. E[C2], (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 KC and Re 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 KC and Re. 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 AL. 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 CL mn Uma At cos(27rmft + 0(m)). (4.40)
m=1 n=l

This representation allows CL to be dependent on KC, R, and the frequency of vortex-

shedding.


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 us-

ing 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) ), the

moment form of Eqn. 4.40 was used, i.e.
L 1 M N
ML(t) pD p E CLtmnU, 2,marn ,A cos(2rmft+ (m)), (4.41)
Art 2 m=l n=l

where rn is the distance of element n from the bottom hinge. Substituting Un =

aWd sinwdt, where a, (= a r/L) is the amplitude of the tower oscillation at the

element n, and wd (= 27rfd) the driving frequency, into Eqn. 4.41 results in
L 1 a2w2 M N
ML(t)y- = -p. D E CLmn r Alcos(2mrmft + (m)). (4.42)
rt 2 L2m=ln=1
Equation 4.42 is analogous to the Fourier series expression
M M
f() = E C e'mwdt = E C cos( mwdt + (m))
m=1 m=l
M
= (a, cos mMwdt + bm sin mwdt), (4.43)
m=1
where
M = is the number of harmonics,

f(t) = ML(t)L/Art,
2 jM
am = E=1 f(t) cos m Wdt,
bm = E =1 f(t) sin mwdt,
(m) = 0(m) = tan-' bm/am and
Cm /a2 + b = p, D (a2 w/L2) E CL r 3 L.

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 Cm (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:
aMw 3 1 22 -Nc n]2
= 1 a2w C( r AL) C,]2, (4.44)
Cm= DL2 n







60
where the minimum CLmn are the solutions of


"- 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 (CL,)

are the solutions of O82/OCL, = 0 where, n = 1,2,..., N. This results in the following

N simultaneous equations

AX = B, (4.46)
where
16 r33 33 r 33
1 1r2 1 r3 1. rn
r33 r6 3 r3 33 rr3
r2 r1 1.. 23 2rn
A=
33 33 33 6...
nr r1 rnr2 rn 3 ,n
CL1
CL2
X = and

CLn


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 di-

rection, 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


Im.b(t) = Mtoat = Mg + MD + Mfy, (4.48)

where I,, Mtota, 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 Mfy were defined as

k di / cos 0(t)
Mg = (Zw, r, pw A, gr dr) sin 0(t) (4.49)
s=l 0
1 N
MD = pD n CDn rn At 2(t) sgn[Ob(t)] (4.50)
n=l
1 M N
Mfy = 2p,,D EE CLmn Un~marn,, Alcos(2rrmft + 0(,)) (4.51)
m=1 n=l

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' = pw D2 Cm r A. (4.52)
n=l

Substituting these moments into Eqn. 4.48, making the assumption that the measured

transverse response ym(t) Li(t) and dl/ cos (t) = di gives

N 21"t ; t)
(Io + -p, D' Cmn rn A)ym "(t) +
n=1 4
1 22 M
2L D E CD,. r, A (t) sn[ym(t) +
n=1 n
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)
2n=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 im(t),m,(t) and ym(t). The quantities Im(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 N
f(t) = p w Da CLmn r3 r cos(27rmft + (m)), (4.54)
m=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 nonuni-

formity 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 C, and CD were then computed for 13 to 73 cycles depending on the driving

frequency, fd. The results show that C, 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, KC,

tested, the C, and CD data show a clear dependency on Re when KC is held con-

stant. 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 KC and Re tested, C, and CD are strongly dependent






64
on KC and Re and are decreasing as Re 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 KC 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 KC, 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 Cmmin. 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 I I


Symbol KC
1.6 3.0
x 4.5
+ 5.5
6.5
12 I] 7.5
1. 8.5
0M 1 9.5


0.8



0.4 +o



0 I I I I I I
0 2 4 6 8 10 12 14
Re x 10-4
Figure 5.2: CD versus Re 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 Cm 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 0.2 4E+005
1.6E+003




1.41.2E+003 .4
800
1.09 0- 1.09

S0.770 0.







-0.18 B00 -0.18

-0.50 I -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, Cm 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 Re 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-
Ji"Ai A A A A :A
30 i
0 1 1
20- .
S10

: .




10 2 4 6 8 10 12 14 1-
S-20-

-30-
-0 I
-50

0 2 4 6 8 10 12 14 16
a time (sec)



30


20


E 10-










-30 ----- -- ----------------





.... predicted).
.- 1

T-10 I 8 .i ,









KC = 6.6 and Re = 9.4 x 104 (b) KC = 8.4 and R, = 8.4 x 104 (- 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.


S I I I I I I I I I I I I I I I I








v "* a :





000
A O



A,+ ,o ,, v, Sarpkaya (uniform oscillatory)
e Bearman (regular waves)
0o, A, H, author's investigation.
I I a 1 I I 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 I i I I I I 1 II I I I I I I I I II
*



W +A





o,+,a,o, Sarpkaya (uniform oscillatory)
o Bearman (regular waves)
*, ,*, m,o,O, ( author's investigation
I I I 1 I I I I I I I I I I~ I II


3.0

2.0



21.0
0


0.5


2.0 1-


1.0



0
0 0.5


0.1L
1.0


I






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 C,, 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

Re KC n CD CM
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 104 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 exper-

iments. 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 (,n) for the first eight

harmonics were evaluated using Eqns. 4.47. A flow chart of the computer program

(CLEF) written to obtain CL(n) 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, Re and /. The values of the lift coefficients, CL(n) were found to be

dependent on both KC and Re. However, the CL(n) values were found to correlate
better when plotted versus KC for constant /, 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 f and Re was also found to be strong,

especially for the first four harmonics where the lift coefficients are the largest, see

Fig. 5.7. CL(n) 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 P or Re

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 / = 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 /. As can be seen in Figs. 5.11 and 5.12, the phase angle data associated

with the lift coefficients (CL(n)) 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.





















CL3)


i-j
on












CD
0 0















9,


P1
9
CDO












CD







al


CL(1)
CaD









72
















n Co n 0










-g



-s












14 0 C') Q C0 0 4
I a




C. 04 4










E + Ox
0




S U;






o bj
c(0



o
[ j























iO )


5-4
8888 X


(Lo




















0.8




0.4


0.8




0.4


Figure 5.9: Lift coefficient harmonics versus beta, CL(1) and CL(2).





















0.8


0.4


Symbol KC
- 2.5
+ 3.5
4.5
o 5.5
x 6.5
a >7.0







--


0
12


:uu


16UU


2000


i 0.8





0.4


2400


2800


Figure 5.10: Lift coefficient harmonics versus beta, CL(3) and CL(4).


1.2




















II I =




+
-




+ +







U X 00
3 X
x. U


x 3 :!
X [2"







E *+ Ox
cno


10 0a a
-- -> -


0 0 0
( )
(L)o


I*


o o
(0


a s 0


a i


0


0



x +
a X 0
0-
*
.
x .a + _

g

F I 0c5 x0=



- E +I oxB

to ___


(c)~


F :-D B -





-v"


X 3
an x




m +X
+ +
4







a Dll
* + -


-- C
I I x
I~ x


a I S I a


I I I I I t i





S*E +*DxH ~


S3
O






xa
x x
D X



4-
i X



I A I -


I


-s






U
o








O
0






0







O
C
IU
-1-

























































a a
0 0 0
to ca


0 x


0


4m 00 U, co S)
C, Co,
cMcMCMCTC
"o,?oc(o

-E+-DXH
>.
w ___


S


x
0


0 0
ID cc


I I I I I

NOCTOO
-- enm- 00CD
m "Wo







B *
x 0



H a +
*+0D Xa _,
n X+ D


x 0

-IN




l I I l I l -


I I | I -




+




+ 0

0 1


oxo
a C











; ; W. t 0
0
.






I.tCMCM Q


"IE1+DxH

I I___
____ ____ S S i--


7 1? o -


a0


S.


CM


0


























0
ri


+










4.A
U-





x 1
.0













I I f I I I I
g IC
x








x 0






a -


-8
0

0





0




o
0






. "










fi







,


a




10
(O
{ij






.F?
Fr3


IY_


v


- J .. .


r,

-r


8 8
'7C


















z
2.5






a 5.











-2.5




- -5


predicted transverse force time series


S2 3 4

time [sec]


frequency [Hz]


S 2 3 4 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 ,

1 ,

0.8

S0.6

0.4

0.2


m




























4 5 6 7 9 10
time [sec]


1


0 -21











S21






















6

42
2-2C










8







2


0











0



0


power spec ru rn measureU transverse orce













0 1 2 3 4 5 6 7 8 9 1(
frequency [Hz]



power spectrum of predicted transverse force













A A A A ^ A A A .


1 2


3 4 5
frequency [Hz]


6 7 8 9 10


Figure 5.14: Comparison between measured and predicted transverse forces

(Re = 2 x 104 and KC = 5.9).


5
time [sec]


t.... f f- ........






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 = F (t) (5.1)
2 Pw D t Uma

where Frms(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 P 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, CLms

increases as KC increases (see Fig. 2.14).


1.0



0.8 Symbol -
1792
+ 2160
2453
0.6 2700
0 O 2780
\ \ X 3130
S3690
0.4 -



0.2 0r_.z=\= .-.,---



0 I I I I I I I
0 2 4 6 8 10
KC
Figure 5.15: RMS lift coefficient versus KC.







80
The present CLGms 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 coeffi-

cients, CLms, for the oscillating tower in still water agrees very well with CLrms data

for waves. For the range of Re 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

1.6 Symbol Reference
Bearman, 1988
S0 Isaacson & Maull, 1976
1.4 Sawaragl, et. al, 1976
o Sawaragi, et. at, 1976
1.2 n 3A Author's Inverstigation
0
1.2 ,o 0
o
O
S0 *0o o
0 0.8- o o
0ID *. *
0.6 A o M 5

0.4- 0^o 0 o

0.2

0 .2 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 exper-

iments. 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 permission.








o \- A .., \/o\
l
1. 0




/.. / \ y ./\ ve



Co /



/ \
/0 " ID1 I0j ]O
'iWr IXZ






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 vortex-
shedding. One common way is to perform a spectral analysis on measured trans-

verse 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 ap-

proximately 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 fre-

quency. Also to illustrate the dependency of the different harmonics in the transverse

force on Re and KC, the power spectra were plotted, as groups, for a range of KG

tested holding Re constant. They were also plotted holding 3 constant. Figure 5.18

and Fig 5.19 show examples of these spectra for constant Re and /, respectively.

On investigating the frequency of the peaks of these force spectra, it was ob-

served that the frequency of the dominant peak was always a multiple of the driving

frequency (i.e., frequency ratio f,/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 8 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 KC, when 3 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 / 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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