UFL/COELTR/089
VORTEXINDUCED 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
VORTEXINDUCED 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 Setup .... ....................... 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 Inline Force ............
4.2 Transverse Force .........
4.2.1 Steady Flow Model .
4.2.2 QuasiSteady 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 Inline 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 cylinderloading 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 StrouhalReynolds numbers relationship with transverse force spec
tra for steady, planar flows around a smooth, fixed cylinder (Refs.
14, 67) . . . . . . . .. 14
2.5 Spanwise 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 KeuleganCarpenter number for oscillatory,
planar flow around a smooth, transversely constrained cylinder
(for constant Reynolds number, R). . . . ... 18
2.8 Lift coefficient versus KeuleganCarpenter 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
KeuleganCarpenter number, KC (Refs. 2, 32, 47). . ... 21
2.10 Spanwise 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 Spanwise coherence length for transversely unconstrained cylin
der near a wall in oscillatory, planar flow (Ref. 39). . ... 24
_____ ____~ _~_ _~_____ ____
2.13 Lift coefficient harmonics versus KeuleganCarpenter 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 KeuleganCarpenter 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 KeuleganCarpenter
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 setup. 38
3.2 Schematic diagram of transversely unconstrained experiment setup. 39
3.3 Photographs of experimental setup. . . . ... 40
3.4 Block diagram of measurement system. . . ... 41
4.1 Definition sketch for the articulated tower showing inline motion. 48
4.2 Contour lines defining Error surfaces for the inline 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 inline force (KC = 8.4
and R, = 8.4 x 104) ........................ 66
5.4 Phase comparison between measured and predicted inline 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 vortexshedding
process. ................................. 86
5.24 Definition sketch of spanwise coherence length. . ... 87
5.25 Spanwise coherence length for transversely constrained articu
lated tower.... ................ ........ 88
5.26 Spanwise 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
vortexshedding 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 vortexshedding frequency behavior (/ = 9,900). .104
5.46 Spanwise 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 KueleganCarpenter number. 115
B.1 XY force transducer and wheatstone bridge circuit. ...... ..130
B.2 Schematic of strain gauge amplifier/signal conditioning module. 131
B.3 Calibration curve for inline force (for test runs at 0.4 > fd <
0.8 H z.). . . . . . . .. 132
B.4 Calibration curve for inline 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 inline 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 setup 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 inline 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 inline 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 vortexshedding 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 KeuleganCarpenter 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 inline 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 inline driving force at the top hinge
S(nf) spectral density at frequency nf
St Strouhal number
Sx,(f) autospectral density of signal x(t) at frequency f
S,,(f) autospectral density of signal y(t) at frequency f
S.(f) crossspectral 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 inline 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 spanwise coherence length of vortices
Ph hydrodynamic damping coefficient
# frequency parameter (= Re/KC)
7y(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 inline force reaction at top hinge to the center
of inline force transducer
Ar, distance from transverse force reaction at top hinge to the center
of transverse force transducer
0 inline angular deflection of the tower
o inline angular velocity of the tower
S inline 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, vortexshedding 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
VORTEXINDUCED 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 vortexinduced transverse loading
on fixed and compliant structures in nonplanar oscillatory flow. The effects of flow
nonuniformity and transverse motion on the transverse force, vortexshedding fre
quency, and vortex spanwise 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 KeuleganCarpenter 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 vortexshedding.
For the range of Re and KC tested, transverse force amplitudes as large as 70%
of the inline force were measured. Transverse forces occurred at frequencies that
were multiples of the driving frequency. The dominant vortexshedding 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 inline 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 vortexinduced 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 onehalf times) and alter the
nature of the vortexshedding. 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 flowinduced
vibrations (King, 1974). This is in addition to frequent shutdown of operation for
days due to the large oscillations induced by fluidinduced forces; see Lewis et al.
(1991) and Koch et al. (1991).
In the past two decades, a significant amount of research on fluidstructure interac
tion problems has been conducted. This research can be divided into two categories,
those concerned with i) the fluidinduced forces and ii) the fluidinduced response
and/or vibration. The latter category has received the most attention.
2
The fluidinduced forces are comprised of inline forces and vortexinduced trans
verse (lift) forces. Many studies have been done to predict the fluid inline 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 inline forces, vortexinduced 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 vortexinduced 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 inline 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 fluidstructure
interaction problems, in particular on quantifying the effects of flow nonuniformity
3
and transverse motion on vortexinduced transverse loading and on developing a
mathematical model for predicting transverse forces.
It is extremely difficult to obtain a theoretical solution for vortexinduced 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 spanwise coherence length, magnitude of the vortex
induced transverse forces and transverse motion on the inline flow and structure
parameters. The physical model must be sufficiently large to cover meaningful
ranges of Reynolds (Re) and KeuleganCarpenter (KC) numbers and
2. To develop an improved mathematical model for predicting vortexinduced
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 stillwater 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 inline and transverse forces.
the frequencies of vortex shedding along the physical model.
the inline motion.
During the series of tests for the second type, transverse motion was allowed
and the following quantities were measured
the inline forces.
the inline and transverse motion.
the frequencies of vortex shedding along the physical model.
3. Develop a mathematical model for predicting the vortexinduced transverse
forces.
4. Quantify the effects of flow nonuniformity and structure's transverse motion on
the lift coefficients, frequency of vortexshedding 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 vortexshedding 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 vortexshedding process, spanwise 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 vortexinduced
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 inline 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 inline 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.
Vortexinduced 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 vortexinduced loading
arises from the possibility of resonance created by the vortexshedding 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 vortexshedding
to be controlled by the response. This can result in what is known as "lockin".
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 inline with the flow. The induced transverse motion can also increase the
timeaverage drag significantly.
The problems of inline and transverse loading and the corresponding structural
response are all related and interdependent as pointed out in several excellent reviews
onthis 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 cylinderloading 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 vortexshedding 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
texinduced 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
Nonplanar
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 stillwater
(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), inline 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 inline flow, roughness and roughness gradients along the cylinder, two and three
10
dimensionality of the flow (such as would be generated by uni and multidirectional
surface waves, respectively), nonaligned currents and waves, etc.
During the past few decades, much has been learned about the vortexinduced
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 vortexinduced transverse forces (in particular, the lift coefficients and
frequency of vortexshedding) and spanwise 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. Lockin conditions and
their relation to the vortexshedding 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
Vortexinduced 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 inline 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 onehalf 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 inline relative speed squared, fluid mass density, and projected area has also
been used. In this case, the constant of proportionality is onehalf the lift coefficient
(CL/2). In both cases, the coefficient is a catchall 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 stillwater. 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 vortexshedding is illus
trated in Fig 2.3, where the major regimes of vortexshedding 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 vortexshedding 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 vortexshedding. 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 REESTABLISHMENT 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: StrouhalReynolds 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 spanwise 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 spanwise coherence length with increased Reynolds number.
0
5J
Z
UI
00
104 105 106
Re
Figure 2.5: Spanwise 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 vortexshedding frequency is in the vicinity of the natural frequency of
the cylindersupport 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 vortexshedding to that of the
oscillation frequency and "lockin" 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 vortexshedding 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 = 106
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 lockin. For
those conditions, Fig. 2.6 shows that the vortexshedding frequency displays a depen
dency on Reynolds number as for the fixed cylinder. As the vortexshedding frequency
moves further from the system natural frequency the response at the natural frequency
decreases while the response at the vortexshedding frequency increases. In a plot of
vortexshedding frequency versus velocity their results show that the vortexshedding
frequency follows the St = 0.2 relationship except in the neighborhood of the natural
frequency where lockin 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 spanwise 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 KeuleganCarpenter
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 stillwater. 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
KeuleganCarpenter 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 KeuleganCarpenter 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 (0.31) x 104
A 60 (220) x 104
S 49 7 x 104 105
61 (12) 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 KeuleganCarpenter 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 quasisteady 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 halfcycle 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 vortexshedding 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 vortexshedding 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 inline
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
vortexshedding 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 spanwise correlation
of vortexshedding 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 .
1i I I I I I i1
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 KeuleganCarpenter
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: Spanwise 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 vortexshedding
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 vortexshedding 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 stillwater. Their results showed for 4.4 < V, < 6.6
and f,/fd = 6.22 lockin occurred, i.e., f,/f, = ff, = 1. This show that it is
not necessarily for f,/fd to be an integer to have a lockin. Their results also show
(at least qualitatively) the influence of the transverse motion on the frequency of
vortexshedding.
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 spanwise 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: Spanwise 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
spanwise 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 spanwise 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 vortexshedding 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 vortexshedding 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 spanwise coherence length for this
flow category. Likewise, no information was found on the effect of transverse motion
on transverse force and vortex spanwise 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 KeuleganCarpenter 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 inline 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 vortexinduced 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
rIoP *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 KeuleganCarpenter 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 vortexshedding 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 vortexshedding process. However, due to the subjectivity in
the visualization techniques the behavior of vortexshedding 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 0.462.28
o 30 0.760.79
* 66 0.61.0
a 66 1.22.4
29
The author was unable to find reported data on spanwise 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 shutdown 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 vortexinduced 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 vortexinduced 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, spanwise 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 vortexinduced transverse forces (in particular, lift coef
ficients and frequency of vortexshedding) and on the vortex spanwise 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 KeuleganCarpenter 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 fluidinduced 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
bottomhinged 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), KeuleganCarpenter number (KC) and reduced velocity (V,). Hydrody
namic force coefficients for large values of Re are needed in fullsized 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 carriedout 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 vortexinduced 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 vortexinduced 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 inline 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 elasticallymounted cylinder, experiments should be carriedout 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(103) 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 vortexshedding and the velocity range over which lockin occurs.
3.2 Experimental Setup
Very little information exists on the effect of flow nonuniformity and structure
transverse motion on vortexinduced transverse loading and vortex spanwise 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 fluidinduced 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 Ibeams 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 vortexshedding process, vortex
spanwise 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 Ibeams. 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 inline and transverse displacements of the tower. The active strokes of
the inline 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 inline simple harmonic motion was imposed at the top of the tower
while constraining the transverse motion. Quantities measured during these tests
were i) inline position, ii) inline force, iii) transverse force and iv) vortexshedding
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 inline 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) inline position, ii) inline
force, iii) transverse motion and iv) vortexshedding 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. Inline tower position.
2. Inline 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 setup.
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 setup.
40
/ *.*... .. .
jjgurfl.j. jogp : "eim.
Figure 3.3: Photographs of experimental setup.
A
ftml
41
i jnline Position
Analog/Digital Processing
Inline 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 beamtype 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 Ibeam were placed flangetoflange 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 fourarm 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 inline 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 B2 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 carriedout. 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 B3. 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 vortexshedding 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 inline position signal to reduce the noise in
the signal to the servo controller. The cutoff frequency was carefully selected for the
filter after considering the useful driving frequency range. Since the highest driving
frequency was 1.0 Hz, a cutoff 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 inline and transverse forces.
Their cutoff 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 inline and 20.0 Hz for the transverse force signals. Accordingly 17.0 Hz and
22.0 Hz cutoff frequencies for the inline 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
DClinear drive motor (specifications given in Appendix B, Table B.4); DCservo
controller (Moog, Model 82 300); glass bulb mercury thermometer and a 12bit,
multifunction, 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 inline 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, inline and transverse forces, inline
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 inline force, inline
45
and transverse position and the thermistor signals were measured and recorded. To
reduce the scatter in the measured data of inline force and transverse motion some
longer test runs, where only inline 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 inline amplitude was found to vary with the driving
frequency (fd). The intent was to obtain as wide a range of KeuleganCarpenter (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 Inline Force
The following equation of motion was used to reduce the inline 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) = inline 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 inline 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 inline 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 inline 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 inline 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 WellConditioned for Deter (b) Data WellConditioned for Deter
mining Drag Coefficients mining Inertia Coefficienta
Figure 4.2: Contour lines defining Error surfaces for the inline 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
offdiagonal elements Ck are the covariances between CD and C, (i.e., COV(CD, Cm)).
An estimate of the goodnessoffit of the data to the model is still needed. A
simple measure of the goodnessoffit 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 NewtonRaphson 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 inline force. Second, the transverse force can give rise to fluidinduced oscillations
and to fatigue failure. Third, even small transverse motions of the body regularize the
wake motion, alter the spanwise correlation of the vortices, and drastically change
the magnitude of both the inline and transverse forces.
With all the information available on vortexshedding, there is no simple, explicit
formula to predict the time variation of the vortexinduced 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
semiempirical and based heavily on experimental data. Of course, there are many
numerical models available (e.g. NavierStokes basedmodels, discretevortex models,
wake oscillator models) but due to their limitations (such as laminar, twodimensional
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 "discretevortex method". Such a method assumes that the inducedforce 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 semiempirical 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 semipeaktopeak values of the transverse force or different
velocities, such as those corresponding to the maximum inline 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 inline 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 QuasiSteady Model
Recently the steady flow model has been extended to include the frequency of
vortexshedding. Verley (1980), followed by McConnell and Park (1982b), suggested
the following simple quasisteady model
FL = pW D CL U2 sin wt. (4.34)
The model is based on the instantaneous values of flow velocity and vortexshedding
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 quasisteady 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 forcetime 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 narrowband 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 vortexshedding
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 LE 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 vortexshedding process, loading and
thus the structure response.
5.1 Inline Forces
The inline 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
inline force data, is given in Appendix C.
For the range of Reynolds numbers, R,, and KeuleganCarpenter 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 104
Re for harmonically oscillated articulated tower (transversely
It was observed by many investigators (e.g. Sarpkaya, 1976a) that, for lower range
of KC, the inline 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 wellconditioned
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 104
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 inline 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
S20
30
0 I
50
0 2 4 6 8 10 12 14 16
a time (sec)
30
20
E 10
30   
.... predicted).
. 1
T10 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 inline force data
and to examine the drag forcevelocity 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 dragvelocity 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 inline 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 vortexinduced 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 multipeak 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)
ij
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 )
54
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
22C
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 rootmeansquare (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 vortexshedding in oscillatory flows is
known (see literature review, Chapter 2) to be related to the inline 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 vortexshedding is
known to be a very important parameter in this type of fluidstructure 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 vortexshedding
frequency. In general these spectra show that the frequency of the dominant peak is
