AN ABSTRACT OF THE THESIS OF
Ming-Kuang Hsu for the degree of Doctor of Philosophy in
Civil Engineering presented on October 16, 1986.
Title: Random Wave Forces on Cylinders
Redacted for Privacy Abstract approved: John H. Nath
The in-line force power spectrum is predicted by the frequency domain maximum force coefficient [Cun(z,fn)] vs. Keulegan-Carpenter number [Kn(z,fn)] relationship. The frequency domain Keulegan-
Carpenter number is defined as the velocity from the amplitude spectrum at fn divided by the respective frequency and diameter of the cylinder [Kn(z,fn) = un(z,fn)/(fn x D)].When Kn(z,fn) is small, the frequency domain maximum force coefficient closely follows the
2 (z,f ) = 21-/Kn(z,fn) for a smooth cylinder. The force relation Cpn n power spectrum can be estimated from a given wave spectrum by using linear wave theory and the above relation for Cnn(z,fn). The non- dimensionalized rms errors between measured and predicted spectra are used to evaluate the predictions.
The force time history can be predicted from the wave profile and an inverse Fast Fourier Transform method. This method is valid when Kn(z,fn) is less than 10 for smooth cylinder. Force time histories predicted by using this method compare reasonably well with measurements. RANDOM WAVE FORCES ON CYLINDERS
by
MingKuang Hsu
A THESIS
submitted to
Oregon State University
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
Completed October 16, 1986
Commencement June 1987 APPROVED:
Redacted for Privacy
Professor of Civil Engineering in Charge of Major
Redacted foriPrivacy Head of artment sf /Civil Engineerin
Redacted for Privacy
Dean of Graduate $1ool
Date thesis is presented October 16, 1986
Typed by Ming -Kuang Hsu ACKNOWLEDGEMENTS
The author wishes to express his deep appreciation to Professor
John H. Nath for his guidance and support during the course of this research.
During his graduate studies, the author received financial support from the American Petroleum Institute, the Chevron Oil Field
Research Company, the National Science Foundation, and computer funds from the OSU Milne Computer Center to all of whom he expresses his appreciation.
Finally, the author expresses his deep thanks to his parents and
Mr. and Mrs. Minore for their love. TABLE OF CONTENTS
Chapter 1Talt
1.0 INTRODUCTION 1 1.1 Predicting Ocean Wave Forces 1 1.2 Scope 2
2.0 THEORETICALCONSIDERATIONS FOR OCEAN WAVE FORCES 5 2.1 The Time Domain Force-Phase Method 9 2.1.1 The Time Domain Force-Phase-. Method 11 2.1.2 The Time Domain Force-Phase-S Method 16 2.2 The Frequency Domain Force-Phase Method 23 2.2.1 The Frequency Domain Force-Phase-. Method 27 2.2.2 The Frequency Domain Force-Phase-S Method 28 2.3 The Prediction of Force Power Spectra 28 2.4 The Prediction of Force Time History 31 2.5 Force-Phase Method Extension 31 2.5.1 Vertical Cylinders 32 2.5.2 Horizontal Cylinders 32
3.0 DESCRIPTION OF EXPERIMENTS 36 3.1 Laboratory Experiments 36 3.1.1 The SMC12 Cylinder 39 3.1.2 The SRC.02 Cylinder 39 3.1.3 The SP3-55D Cylinder 39 3.2 Field Experiments 43 3.2.1 NMI Second Christchurch Bay Tower 43 3.2.2 Wave Force Project II Tests 46
4.0 RESULTS AND DISCUSSION 49 4.1 The Cpn(Kn) Regression Lines 49 4.1.1 The Amplitude Filter 51 4.1.2 The Frequency Filter 60 4.1.3 Reynolds Number Effects 62 4.2 Force Coefficients 66 4.2.1 Laboratory Test Results 68 4.2.2 Field Test Results 74 4.2.2.1 NMI Second Christchurch Bay Tower 74 4.2.2.2Wave Force Project II Tests 78 4.3 The Prediction of Force Power Spectra 78 4.3.1 Evaluation Criteria for Predicted Force Power Spectra 80 4.3.2 Laboratory Tests 80 4.3.3 Field Tests 89 4.3.3.1 NMI Second Christchurch Bay Tower 89 4.3.3.2 Wave Force Project II Tests 94 4.4 The Prediction of Force Time Histories 101 TABLE OF CONTENTS (continued)
Chapter page
5.0 CONCLUSIONS 107
6.0 REFERENCES 108
APPENDIX A Constructing the Cu(K) Hyperbola 112
APPENDIX B The Derivative of C(K) with Respect to a 115 u APPENDIX C Comparison of Force Coefficients Obtained through Different Methods 123 LIST OF FIGURES
Figure page
1.1-1 Wave induced hydrodynamic loading regimes as determined by individual size (from Shield and Hudspeth, 1985) 3
2.0-1 Definition sketch of wave force on a vertical cylinder 6
2.0-2 Definition sketch for the phases and magnitudes of the wave profile, velocity and local force measurements on a vertical cylinder in periodic waves (from Nath, 1985b) 7
2.0-3 The Wave Force Project II wave record 10
2.1-1 Maximum force coefficient for smooth cylinders as a function of K (from Sarpkaya, 1976) 15
2.1-2 Maximum force coefficients for the VSMC12 and VSMC8 Cylinders (from Nath, 1985a) 17
2.1-3 Non-dimensionalized phase shift for the VSMC12 and VSMC8 cylinders (from Nath, 1985a) 17
2.1-4 Values of C and C from Eqs.(2.1-9) and (2.1-10) c4 m(1) using the cutves shown in Figs. 2.1-2 and 2.1-3. The and C data points are the least squares values of Cd for the VSMC12 and VSMC8 cylinders (from Nath,1985a51 18
2.1-5 The maximum force coefficients data points and their hyperbola for the VSMC12 and VSMC8 cylinders 21
2.2-6 Non-dimensionalized phase data points and the curve of predicted nondimensionalized phase 21
2.1-7 Drag and inertia force coefficients Cdd, and Cmdi calculated by TFPS method then convert4d into Cdi, and The data points are CmLby using Eqs. (C-8) and (C-9). the least-squares values of Cd and Cm for the VSMC12 and VSMC8 cylinders 22
2.2-1 The amplitude spectrum of WPII record-6887 26
2.5.1 Definitions of forces on vertical and horizontal cylinders 33
3.1-1 The OSU wave flume (from Nath, 1985a) 38 LIST OF FIGURES (continued)
Figure Page
3.1-2 Sectional elevation of a VSMC12 test cylinder (from Hudspeth et al., 1984) 40
3.1-3 Sectional elevation of a vertical 8.625-in. test cylinder (from Nath, 1983) 41
3.1-4 Side view of a horizontal 8.625-in. test cylinder (from Nath, 1985a) 42
3.2-1 NMI Christchurch Bay Tower side view (from Bishop, 1984a) 44
3.2-2 NMI Christchurch Bay Tower top view (from Bishop, 1984a) 45
3.2-3 Platform layout of Wave Project II (from Blank, 1969) 47
3.2-4 Location of WPII dynamometers and wave staff (from Blank, 1969) 48
(K) hyperbola and their asymptotes 50 4.1-1 The Cpn n
4.1-2 The velocity-force cross-amplitude spectrum 52
4.1-3 The velocity-force cross-phase spectrum 53
4.1-4 The amplitude filter: (a) before filter, (b) after filter 55
4.1-5 The band pass frequency filter: (a) before filter, (b) after filter 56
The C ) data points and their regression line 4.1-6 pn(Kn (before filter) 57
4.1-7 The C (K) data points and their regression line, afteru passing through an amplitude filter (cut-off amplitude = 0.01 spectral peak) 58
) data points and their regression line, 4.1-8 The Cpn(Kn after passing through an amplitude filter (cut-off amplitude = 0.1 spectral peak) 59
The C 4.1-9 pn(Kn) data points and their regression line, after passing through an band pass (0.1 Hz to 1.0 Hz) filter 61 LIST OF FIGURES (continued)
Figure Page
4.1-10 In-line force coefficient vs. K (from Bearman et al., 1985) 63
4.1-11 Maximum force coefficient for smooth cylinders as a function of K (from Sarpkaya, 1986b) 64
4.1-12 The C (Kn) data points and their regression lines for NMI larpn ge and small cylinder (11 records at three locations) 65
4.1-13 Maximum force coefficient for rough cylinders (from Sarpkaya, 1986b) 67
4.2-1 In-line force coefficients and phases for the VSMC12 cylinder (Run 32) 69
4.2-2 Transverse force coefficients of the VSMC12 cylinder (Run 32) 70
4.2-3 In-line force coefficients and phases for the API84 VSRC.02 cylinder (Run 23) 71
4.2-4 Transverse force coefficients for the API84 VSRC.02 cylinder (Run 23) 72
4.2-5 Horizontal force coefficients for the API84 HSRC.02 cylinder (Run 55) 73
4.2-6 Vertical force coefficients for the API84 HSRC.02 cylinder (Run 55) 75
4.2-7 In-line force coefficients for the NMI large cylinder level-3 (Record 71) 76
4.2-8 In-line force coefficients for the NMI small cylinder level-3 (Record 71) 77
4.2-9 In-line force coefficients and phases for the WPII cylinder position-7 (Record 6887) 79
4.3-1 Raw in-line force power spectra for the VSMC12 cylinder (pseudo prediction) 81
4.3-2 Smoothed in-line force power spectra for the VSMC12 cylinder (pseudo prediction) 82 LIST OF FIGURES (continued)
Figure Page
4.3-3 Raw in-line force power spectra for the VSRC.02 cylinder (pseudo prediction) 83
4.3-4 Smoothed in-line force power spectra for the VSRC.02 cylinder (pseudo prediction) 84
4.3-5 Raw horizontal force power spectra for the HSRC.02 cylinder (pseudo prediction) 85
4.3-6 Smoothed horizontal force power spectra for the HSRC.02 cylinder (pseudo prediction) 86
4.3-7 Raw vertical force power spectra for the HSRC.02 cylinder (pseudo prediction) 87
4.3-8 Smoothed vertical force power spectra for the HSRC.02 cylinder (pseudo prediction) 88
4.3-9 Raw in-line force power spectra for the VSMC12 cylinder (true prediction) 90
4.3-10 Smoothed in-line force power spectra for the VSMC12 cylinder (true prediction) 91
4.3-11 In-line force power spectra for the NMI large cylinder level-3, record-71 (pseudo prediction) 92
4.3-12 In-line force power spectra for the NMI small cylinder level-3, record-71 (pseudo prediction) 93
4.3-13 In-line force power spectra for the NMI large cylinder level-3, Record-71 (true prediction) 95
4.3-14 In-line force power spectra for the NMI small cylinder level-3, Record-71 (true prediction) 96
4.3-15 Raw in-line force power spectra for the WPII cylinder position-7, record-6887 (pseudo prediction) 97
4.3-16 Smoothed in-line force power spectra for the WPII cylinder, position-7 record-6887 (pseudo prediction) 98
4.3-17 Raw in-line force power spectra for the WPII cylinder, position-7, record-6887 (true prediction) 99 LIST OF FIGURES (continued)
Figure Page
4.3-18 Smoothed in-line force power spectra for the WPII cylinder, position-7, record-6887 (true prediction) 100
4.4-1 The wave profile for the VSMC12 cylinder test 102
4.4-2 The measured and predicted in-line force time history for the VSMC12 cylinder (true prediction) 103
4.4-3 The wave profile record during Hurricane Carla (record- 6887) 104
4.4-4 The measured and predicted in-line force time history for the WPII cylinder at position-7 (true prediction) 105
A-1 The Cu hyperbola and the coordinate systems 113
versus K for $ = 4480 (from Sarpkaya, B-1 C CmF 1176) 117
B-2 CAP and CCmFversus K for a = 5260 (from Sarpkaya, 1176) 118
B-3 C and C versus K for a = 4480, converted from CdF c4 mob and C cutves (Fig B-1) by using Eqs. (C-12) and (C-135- 120
B-4 C and C versus K for B = 5260, converted from CdF c14 m46 and C cutves (Fig B-2) by using Eqs. (C-12) and (C-137- 121
C-1 Drag coefficients from different methods 127
C-2 Inertia coefficients from different methods 127 LIST OF TABLES
Table Page
2.5-1 Summary of frequency domain maximum force coefficients, Keulegan-Carpenter numbers and phases 34
3.0-1 Summary of the test conditions 37
B-1 The values of the derivative of Cu (K) 119
C-1 Force coefficients from different methods 128 NOTATION
C(z) Drag coefficient for Morison equation d Drag force coefficient of steady state flow Cds
C (z) Drag force coefficient determined by Fourier analysis dF Drag force coefficient determined by the least-square CdL(z) method
CdF(z) [= C d4(K)] Drag force coefficient determined by the force-phase method
Root mean square of the maximum force coefficient CFrms(z)
Cm(z) Inertia coefficient for Morison equation
CdF(z) Inertia force coefficient determined by Fourier analysis
CdL(z) Inertia force coefficient determined by the least- squares method
C (z) [= C (K)] Inertia force coefficient determined by the m, m force-phase method
Total force coefficient, see Eq. (2.1-1) CT(z)
C ) Coincident spectral density function xy n
C (z) [= C (K)] Maximum in-line force coefficient of vertical cylinder
[= C (K)] Frequency domain maximum force coefficent Cpn(z,fn) un n for frequency fn(Kn) H C (z) Maximum horizontal force coefficient of a horizontal pn cylinder T C (z) Maximum transverse force coefficient of a vertical pn cylinder, calculated by in-line horizontal velocities V C (z) Maximum vertical force coefficient of a horizontal pn cylinder
D Diameter of a smooth cylinder
2 Mean square error between measured and predicted forces E rms Non-dimensionalized rms errors F(z,t) Force per unit of length
F(z,t) Non-dimensionalized force
Fm(z) Measured force (Appendix C)
F(z,t) Non-dimensionalized measured force (Appendix C) m
F(z,f ) Amplitude of in-line force amplitude spectrum for n n frequency fn
F(z,t) Predicted force time history P F(z,t) Non-dimensionalized measured force (Appendix C) P F(z) Maximum force during a wave cycle U
FnH(z,fn) Force amplitude in the horizontal direction (horizontal cylinder)
FnV(z,fn) Force amplitude in the vertical direction (horizontal cylinder)
FnT(z,fn) Force amplitude in the transverse direction (vertical cylinder)
F(z) Nondimensionalized maximum force U Gx(fn) Power spectral density function for X(t)
Gxy(fn) Cross-amplitude spectrum for X(t) and Y(t)
Gy(fn) Power spectral density function for Y(t)
H Wave height
Hn(fn) Transfer function
K(z) Keulegan-Carpenter number
K K value of the intersection of two asymptotes L1 and i L 2
Kn(z,fn) Frequency domain Keulegan-Carpenter number
L Wave length
L0 Deep water wave length
L1 Slant asymptote L Horizontal asymptote 2
N Total number of components
P N
C2
Q [° glidl EC-min a d.
Qxy(fn) Quadrature spectral density function R Reynolds number
S (z f ) Water particle acceleration power spectrum AA ' n
S (zf ) Wave force power spectrum FF ' n Wave force power spectral peak [SFF(z'fn)m]ll
SFF(z'fn)m Measured force power spectrum Predicted force power spectrum SFF(z'fn)p
Suu(z'fn) Horizontal water particle velocity power spectrum Water particle velocity power spectrum SVV(z'fn)
SuF(fn) Velocity-force power spectrum
T Wave period
X(t) Time series
Y(t) Time series
X Axis (Appendix A)
Y Axis (Appendix A)
R Axis (Appendix A) i Axis (Appendix A)
X' Axis (Appendix A)
Y' Axis (Appendix A) a Wave amplitude [transverse axis (Appendix A)] an(fn) Amplitude of wave amplitude spectrum b Transverse axis (Appendix A) d Water depth f Frequency f Fundamental frequency 1 f n x f n 1 Imaginary number m(K) See Eq. (2.1-18)
2 dC3dth 2 K 1= -[1 sin(4)] (Appendix B) p d K 2 n
d C 3 K11 , q kAppendix B) L d K s Water particle orbit size t Time u(z,t) Water particle velocity in x direction
Water particle acceleration in x direction u (z) Maximum water particle velocity in x direction during a wave cycle un(z,fn) Water particle velocity spectrum in x direction v(z,t) Water particle velocity in y direction vn(z,fn) Water particle velocity spectrum in y direction w(z,t) Water particle velocity in z direction
Water particle velocity spectrum in z direction wn(z'fn) x Axis y Axis z Distance upward from the still water surface r(z) Phase, see Eq. (2.1-2) Af Frequency increment
tn(fn) Phase
1 4 Tir a(K) [= 2 K
a 1= R/K] Frequency parameter
6 Angle (Appendix A)
Relative roughness of the cylinder
Wave profile
wt]
en(fn) Phase of wave phase spectrum with frequency fn
exy(fn) Cross-phase spectrum between X(t) and Y(t)
Down-crossing wave period
v Kinematic viscosity
p Mass density of fluid
(1)(z) Phase angle between peak velocity and peakforce in a wave cycle
Sa(z) Phase angle between peak velocity and peak acceleration in a wave cycle
(z) [= /(-90°)] Non-dimensionalized phase
71(z) [= 0/0a] Non-dimensionalized phase
4'n(z'fn) Phase between Fn(z,fn) and un(z,fn)
Phase between F (zf) and un(z,fn) OnH(z'fn) n ' n
Phase betweenFT 4'nT (z f) and v(zf ) (z1fn) ' n n ' n V ()nV (z'fn) Phase between F(zf) and w(z,f ) n ' n n n w [= 2x/T] Angular frequency wn [= 27 fn] Superscripts
F Fourier analysis method
L Least-squares method d Drag force m Inertia force (or measured quantity) n nth component p Predicted (forces, velocities, etc.)
Maximum value (forces, velocities, etc.)
0 Force-phase method
Abbreviations
API84 Americam Petroleum Institute Project-84
ARMC Artificial Marine Roughened Cylinder
ASCE American Society of Civil Engineering
FFT Fast Fourier Transform
FFPS Frequency Domain Force-Phase-Slope method
FFPO Frequency domain Force-Phase-0 method
NCEL Navy Civil Engineering Laboratory
NMI National Maritime Institute
OSU Oregon State University
OTC Offshore Technology Conference
RMS Root Mean Square
TFPS Frequency domain Force-Phase-Slope method
TFP0 Frequency domain Force-Phase-0 method
VSMC8 Vertical 8.625-in. Diameter SMooth Cylinder
VSMC12 Vertical 12.75-in. Diameter SMooth Cylinder V(H)SP3-55D Vertical (Horizontal) Dried Cylinder from the South Pass region of Gulf of Mexico. It was placed -55 feet below the sea surface for 3 years.
V(H)SRC.02 Vertical (Horizontal) Sand Roughened Cylinder (c = 0.023)
WPII Wave Force Project II RANDOM WAVE FORCES ON CYLINDERS
1.0 INTRODUCTION
1.1 Predicting Ocean Wave Forces
Offshore drilling platforms were first used in the Gulf of
Mexico in the late 1940's. Since then a wide variety of platforms has been built which are located in increasing depths and subjected to severe environmental conditions. Forces acting on the platforms must be predicted accurately to design safe, economical platforms for these severe environments.
Wave forces on offshore structures are generally calculated by two different methods; (1) the Morison equation, and (2) the diffrac- tion theory (Sarpkaya and Isaacson, 1981; Dean and Dalrymple, 1984;
Shields and Hudspeth, 1985; Chakrabarti, 1985).
The Morison equation is used when the structure exhibits no sensible effect on the wave field, and waves passing the structure remain essentially unmodified by the presence of that structure. The
Morison equation assumes that the force on the structure is the sum of a drag force and an inertia force which are 90° out of phase for linear waves. Experimentally determined drag and inertia coeff i- cients are required.
When the size of the structure is relatively large compared with the wave length, the body will substantially alter the wave field in the vicinity of the structure, thus influencing the forces on the structure. In this case, the diffraction of the waves from the sur- face of the structure must be taken into account, and the diffraction 2
theory should be applied. The effects of vortex shedding are usually
neglected.
Shields and Hudspeth (1985) provided a useful plot Fig. 1.1-1 to
help engineers quickly assess the appropriate technique for determi-
nation of the wave-induced hydrodynamic loads.
Randomness of the sea state should be considered when predicting
wave loads on offshore structures. Methods using the Morison equa-
tion to predict random loads on cylinders were studied by Wiegel et
al. (1957), Reid (1958), Borgman (1965,1967a,1967b,1969),
Bretschneider (1965,1967), Dean and Aagaard (1970), Evans (1970),
Thrasher and Aagaard (1969), Wheeler (1969), Nath and Harleman
(1970), Hudspeth, et al. (1974), and Heidman et al. (1979). Excel-
lent summaries were made by Borgman (1972a) and Sarpkaya and Isaacson
(1981).
In this thesis only cylinders which belong to the Morison equa-
tion regime are studied (i.e. cylinders with a diameter to wave
length ratio of 1/5, or smaller). A new method which can be used to estimate the random wave forces is introduced. Using this method, the force spectra and force time series can be predicted for a given wave profile.
1.2 Scope
Three problems are addressed: 1) the estimation of force power spectra; 2) the prediction of force time history; and 3) the determi- nation of force coefficients for the frequency domain Morison equa- tion. Wave Length L
wi Wave 11.10111
100 = Cylinder Diameter Drag 1 Dominant
10 e0 " 90% Drag 4464's Steel? o Vin' Morison II/D V.0% 1 10% Drag
1 Inertia Dominant Diffraction
I I I 1 5 10 100 1000
Hydrodynamic Loading Regimes
Fig. 1.1-1 Wave induced hydrodynamic loading regimes as determined by individual size (from Shield and Hudspeth, 1985). 4
The time domain forcephase method described by Nath (1985a) is
extended to the frequency domain to solve these problems. The basic assumptions used in this method are: 1) the cylinder is rigid; 2) the
Morison equation regime; 3) linear wave theory; 4) linear superposi
tion for water particle velocities, accelerations, and forces; 5) the forces in one frequency were induced by the wave kinematics in that frequency only; 6) no current; and 7) the axes of vertical (horizon tal) cylinders are perpendicular (parallel) to the wave crest.
Laboratory and field data are used in this study. Laboratory data were obtained from wave tank experiments performed at Oregon
State University. Smooth and rough cylinders in both the vertical and horizontal positions were studied. Field data came from the Wave
Force Project II and Second Christchurch Bay Tower projects. The field cylinders were vertical, and they were all considered to be smooth.
The predicted force power spectra are compared to actual labora tory and field data, and they are also compared with predictions obtained by the Borgman (1967a) method. The predicted force time histories are compared to measured data. 5
2.0 THEORETICAL CONSIDERATIONS FOR OCEAN WAVE FORCES
A definition sketch of wave forces on a small diameter vertical
cylinder is shown in Fig. 2.0-1. The coordinate system is located at
the still water level. The y axis is horizontal and perpendicular to
both the x and the z axes. Water particle velocities in the x, y,
and z directions are denoted as u, v, and w. The cylinder is located at x a 0, and the wave is traveling to the right. The force shown in
the figure is the in-line force.
The in-line force per unit length at position z can be expressed approximately with the Morison equation (Morison et al., 1950):
pD2 F(z,t) = Cd(z) u(z,t)lu(z,t)I + Cm(z) u(z,t) (2.0-1) where D is the diameter of the cylinder, p is themass density of the fluid, u(z,t) is the horizontal water particle velocity, u(z,t) is time derivative of the water particle velocity, and Cd(z) and Cm(z) are drag and inertia force coefficients, respectively. Relationships between the wave profile, the horizontal water particle velocity, and the in-line force are shown in Fig. 2.0-2 for an idealized record.
One problem involved in using the Morison equation is the determination of the force coefficients C d(z) and Cm(z). The least- squares method is usually used with short data samples. Data samples can be the whole wave cycle or only a part of it, such as the wave trough or the wave crest. Coefficients are then determined by mini- mizing the mean square error between predicted and measured forces.
In regular waves, the force coefficients determined in thisway often Direction of Propagation
L
is V
Jr- Still Water Level
Trough FlZ,t unit length
D =.-d //
Fig. 2.0-1 Definition sketch of wave force on a vertical cylinder.
0 7
acceleration force 773U /
\\ / ...... _, ,/
Fig. 2.0-2 Definition sketch for the phases and magnitudes of the wave profile, velocity, and local force measurementson a vertical cylinder in periodic waves (from Nath, 1985b). 8 vary from wave cycle to wave cycle. They are often averaged over several wave cycles.
Force coefficients are sensitive to the variation of the phase,
0(z), which is the time lag between peak velocity and peak force. A small change in 0(z) induces a large change in the force coefficients
(see Hudspeth et al., 1986). Small phase variations from wave cycle to cycle cause variations in vortex shedding in regular laboratory wave tests. Nonlinear wave effects, the return current, and other extraneous effects also cause variations. Small phase variations are impossible to prevent, and so are the relatively large variations of force coefficients.
To improve force predictions, a phase-insensitive parameter, the maximum force coefficient, Cu(z), is introduced.' For a unit length of cylinder, it is defined as
F (z) C (z) = u (2.0-2) 1 pDuu2(z) where F (z) is the maximum force per unit length on a cylinder in a wave cycle and uu(z) is the maximum ambient undisturbed horizontal velocity in the wave cycle.
The Keulegan-Carpenter (1958) number K(z) is an important param- eter in this thesis. The time domain Keulegan-Carpenter number is defined as
u (z) T K(z) = u (2.0-3)
1The subscript, p, will be used to indicate maximum values throughout this thesis. 9
for regular waves, where T is the wave period. For random waves the
time domain Keulegan-Carpenter number is defined as
u (z) t K (z) = P (2.0-4)
where t is the down-crossing wave period of an individual wave
selected from a random wave record. It may also be defined as the
time interval between two up-crossings. The KT(z) values determined
by this definition usually yield large numbers. For example, the
largest individual wave recorded during Hurricane Carla for Wave
Force Project II (Blank, 1969) had a wave height of about 38 ft with
a period of about 13 sec, Fig. 2.0-3. The water depth is about 98
ft. For a 1-foot diameter riser, the KT of this individual wave is
approximately 160. For 3-ft and 8-ft cylinders, the KT of this individual wave is about 53 and 20, respectively.
2.1 The Time Domain Force-Phase Method
When Starsmore (1981) studied field data from the Christchurch
Bay Tower and laboratory data collected by Keulegan and Carpenter, he found that the peak force correlated well with the Reynolds number and the Keulegan-Carpenter number. In order to obtain good correla- tions for the phases, he had to average several cycles. He also found that the force coefficients of the Morison equation could be replaced by total force coefficient CT(z) and phase angle t(z). The
CT(z) was defined as
C(z)K(z) d C(z) =/C2(z)+ ( ) . T (2.1-1) 40.0D
30.00
20.00 -43 4- 'CI-J L2 10.00 ccCD a. w
§ .00
10.00
4 I- 20.00 4 4 I I .00 131.07 26114 39A.22 524.29 TIME (sec)
Fig. 2.0-3The Wave Force Project II wave record.
,-- O 11
The CT(z) is closely related to the F(z). The r(z) was defined as u
2 C(z)w 20(z) u (z) -1 m -1 r(z) = tan r tan [2 cosq (2.1-2) LC(z)K(Z)] sP )] d where s is the water particle orbit size. Thus, the effects of phase
0(z) can be examined separately from the effects of peak force
F(z). If the force coefficients C(z) and C(z) are needed, they U d m can be obtained by recombining the peak force with a statistically stationary phase angle r(z).
Nath (1984b,1985a) used Cu(z) and $(z) to develop the force- phase method. His method is more straightforward and it has been used to analyze regular wave forces on vertical cylinders with surfaces of various roughness. Nath's ideas are extended here and applied to the analysis of random wave forces in the frequency domain.
Two time domain force-phase methods for simple harmonic waves will be reviewed before extending them to the frequency domain.
2.1.1 The Time Domain Force-Phase-0 Method
From linear wave theory (Ippen, 1966), for the coordinate system shown in Fig. 2.1-1, the wave profile n(t) can be expressed as
n(t) = a cosO (2.1-3) where a is the amplitude, 0 = wt, and w = 2w/T. (Where t is time).
When the cylinder is absent, linear wave theory shows that horizontal water particle velocity is 12
u(z,t) = uu(z) cose . (2.1-4)
The horizontal water particle acceleration is
u(z,t) = - -1,2.:r up(z) sine (2.1-5)
Substituting Eqs. (2.1-4) and (2.1-5) into Eq. (2.0-1), and non -
dimensionalized by dividing both sides by -ipDup2(z), we obtained the
nondimensionalized force F(z,t) as
2
1 it F(z,t) = C(z) coselcosel ---7--t- C(z) sine (2.1-6) d Kz)m
The phase of 0 atwhich F(z,t) is maximum is designated as 0(z),
as shown in Fig. 2.0-2. It can be obtained by setting dF/d0 = 0 and
solving for the phase. Thus
2 Cm(z) -sin-l[ w (2.1-7) 0(z) ' 2K(z) C (z)J ' ds where C (z) and Cd$(z) are force coefficients determined by the m(t) force-phase method. When positive maximum forces are considered,
0(z) is between 0° and -90°. In this thesis, phases shown in the
figures have usually been nondimensionalized by divided by -90°.
They are designated as 0(z).
Substituting Eq. (2.1-7) into Eq. (2.1-6) and letting cos24)(z) =
1-sin20(z), the nondimensionalized maximum force can be expressed as
2 2 w Cu(z) = C (z)[1-sin0(z)] - C (z) sin +(z) (2.1-8) dcp K(z) m()
The force coefficients Cd,(z) and Cm.(z) can be obtained from Eqs.
(2.1-7) and (2.1-8) as follows: 13
C (z) C (z) = U (2.1-9) dc0 2 1 + sin0(z) and
2K(z) Cu (z) -sin0(z) C (z) = (2.1-10) mc0 2 2 , n 1 + sin 0(z)
Note that Cu(z), $(z), Cds(z), and Cms(z) are functions of K(z) and R(z). To emphasis their relationships with K(z), it is sometimes more suitable to express them in terms of K(z) as Cu(K), 0(K),
C (K) and C (K). dO P mO Now consider the extreme conditions. When K(z) is very small, e.g. less then 8, the drag term of Eq. (2.1-8) is small compared to the inertia term, and it can be neglected. The maximum force occurs when 0(K) approaches -90°, and Eq. (2.1-8) can be simplified to
2 C(K) = It C (K) (2.1-11) u K m0
Assume that when K is small the flow field of an oscillating flow is similar to a wavy flow. Then according to Sarpkaya (1986a), the
Stokes force on a fixed circular cylinder in a sinusoidally oscillat- ing flow can be expressed in terms of a linearized Morison equation.
That is, when K << 1 and $(= R/K) >> 1 the inertia force coefficient can be expressed as
-1/2 -3/2 C = 2 + 4(.03) + (w0) (2.1-12) m
When $ is large, (e.g. larger than 5000), Cm approaches 2. Herein it is assumed that when a is large and K is small, Cm,(K) is equal to 2.
By taking the log of both sides of Eq. (2.1-11), we obtain 14
log Cu(K) = log(2w2) log K . (2.1-13)
In a loglog plot, Cu(K) is a straight line with a slope of 1.
When K is very large, the inertia term of Eq. (2.1-8) is small
compared to the drag term. Therefore, it can be neglected. Equation
(2.1-8) can be simplified to
Cu(K) = Cd.(K) C (2.1-14) Cds '
where Cdsis the steadystate drag coefficient.By taking the log of both sides, it becomes
log Cu(K) = log Cds . (2.1-15)
In a loglog plot, Cu(K) is a straight line parallel to the log K axis.
To prevent a sharp transition Cu(K) in a loglog plot, a hyperbola with Eqs. (2.1-13) and (2.1-15) lines as asymptotes is used. Referring to Sarpkaya's (1976) U tube results2 (Fig. 2.1-1), this is a good approximation for large 6(R/K) when the cylinder is smooth and a is greater than 3500. It is assumed that the logCu(K) curve of a wavy flow is similar to that of oscillatory flow. The laboratory tests and field tests studied in this thesis had a values greater than 3500. Thus in this thesis, the measured logCu(K) data points are fitted by a hyperbola. (The procedures are detailed in
Section 4.1 and Appendix A.)
2The C and K values of U tube tests are not functions of z. u 7.0 1 I 6.0 2,F2 + points cp pass through_ 5.0 5260
4.0
3.0
2.0
...... 1.5 0 497 ......
1.0 0.9 0.8 0.7 0.6
0.5 I 3 4 5 6 7 89 10 15 20 30 40 506070 100 150 200 K
Fig. 2.1-1 Maximum force coefficient for smooth cylinders as a function of K (from Sarpkaya, 1976). 16
Peak velocities and peak forces can be measured from strip chart
recordings. The Cu(K) data points are then calculated from
Eq. (2.0-2). By using the procedures described in Appendix A, a
Cu(K) hyperbola can be determined. The phase 0(K) can be measured
directly from the strip chart or determined from a cross-correlation
function (Bendat and Piersol, 1971). The nondimensionalized phases
0(K) (= +(K) /( -90 °)] are then plotted on log-log paper, and a
smoothly fitting curve is drawn.
With measured values of Cu(K) and 0(K), the force coefficients
can be obtained from Eqs. (2.1-9) and (2.1-10). Data examples from
Math (1985a) with curves fitted by eye are shown in Figs. 2.1-2,
2.1-3, and 2.1-4. In this example velocities are obtained from the wave profile by using stream function theory (Dean, 1965). Because this method is related to the measured phase 0(z), it is designated as the Time Domain Force-Phase-0 Method (TFP4)).
2.1.2 The Time Domain Force-Phase-S Method
In this section it will be shown that under some assumptions only the Cu(K) relationship is required to determine Cdo(K), Cro(K), and 0(K).
Equation (2.1-8) can be rewritten as
2 C (K) Cu(K) = C (K) + a(K) (2.1-16) d0 C (K) dO where a(K) = w4/4K2. By taking the derivative of Eq. (2.1-16) with respect to a(K), we obtain 17
60 I I I
10
CJ
o VSMC12
a VSMC8 0
I I I, 11, 1 1 1 I 1 1 lt .3 / 1 1 .3 10 100 1000 K
Fig. 2.1-2Maximum force coefficients for the VSMC12 and VSMC8 cylinders (from Nath, 1985a).
1 1 I 1 1 1 1 I rill,
1.00 _ _ _ ..o_ o_ _o_m o e: o A es_ D Tit) . 0 0 0 a o VSN1C12 a VSMC3
1 I 1 1 1 1 I 1 t 1 1 1 1 1 1 1 j .3 10 1CO 1CCO K
Fig. 2.1-3Non-dimensionalized phase shift for the VSMC12 and VSMC8 cylinders (from Nath, 1985a). 18
30 I I I 1 i I I 1 j i I 1 1 1 1 1 1 1 I I i I ill!
VSMC8
0 VSMC 12 10
C
2=,90 _ - 616 0 0
0
qes, _o et C
o o o O 0 o O oi 00 0 0
o
I I I I I .1 Ili in I I !Till! I I I 1111 1 10 100 1000
Fig. 2.1-4 Values of C d.and Cm.from Eqs. (2.1-9) and (2.1-10) using the curves shown in Fig. 2.1-2 and 2.1-3. The data points are the least squares values of Cd and Cm forthe VSMC12 and VSMC8 cylinders (from Nath, 1985a). 19
dC (K) dC (K) C2 (K) C2112 u d. a(K) d r da(K) da(K) da(K)LC%(K)-1 Cd.(K)
= P(K) + Q(K) + m(K) (2.1-17)
Appendix B illustrates that P(K) + Q(K) << m(K) is acceptable when K
is less than 40 and when K is large.The data for 40 < K < 90,
(Rodenbusch and Gutierrez, 1983), were not available when P(K) + Q(K)
<< m(K) was tested. When K is small, Eq. (2.1-17) can be approx-
imated as
2 dC (K) [C (K)] m(K) = (2.1-18) da(K) C (K) d.
An explicit hyperbolic function for C4(K) can be determined from experimental data (see Section 4.1 and Appendix A). Equations
(2.1-16) and (2.1-18) can then be used to solve for Cd4)(K) and
Cm,(K):
Cd.(K) = C (K) - m(K) a(K) (2.1-19) and
2 C (K) =IM(K)C(K) - m(K) a(K) . m. (2.1-20)
The phase, 4(K), can be determined by substituting Eqs. (2.1-19) and
(2.1-20) into Eq. (2.1-7), to yield
1 / m(K) a(K) .(K) = -sin (2.1-21) C (K) - m(K) a(K) 20
Examples are shown in Figs. 2.1-5 through 2.1-7. All but the veloc-
ity data used in this and previous examplesare the same (Hudspeth et
al., 1984). The velocities used in this exampleare from measure-
ments. In Fig. 2.1-6 the phase 4(K) is nondimensionalizedby divid-
ing by phase a(K) and denoting it as "i(K). Because this method is
related to the slope of the Cu(K)-a(K) curve, it is designatedas the
Time Domain Force-Phase-Slope method (TFPS).
One of the reasons for the scatter in the force coefficientsis
the variation of the phase. In the ocean, however, waves come from
all directions, the spatial phase shift between theaxes of the ver-
tical cylinder and wave staff vary from time to time. This addition-
al phase shift also affects the scatter of force coefficientsif it
is not taken into account.
To minimize the influence of spatial separation,one can install
the wave staff (or current meter) as close to the cylinderas pos-
sible. When the instruments are too close to the cylinder, however,
the vortex-shedding effects from thepresence of the cylinder alters
the wave field and the measured wave profile, distorting thewave kinematics.
The location of the wave staff (or current meter) isno longer important when the TFPS method is used (except that itmust not be in the cylinder wake), because the force coefficientscan be determined without knowing the phase. Theoretically, if measuring instruments are installed outside of the region influenced by vortex shedding, more accurate wave kinematics can be recorded and more accurate force coefficients can be calculated. More studies are needed to test the 21
1 I I 1 1 f III
-3 VSMC8 0 VSMC12
10
7'
r
LF
I I 0 11 111 1 I I tild K
Fig. 2.1-5 The maximum force coefficients data points and their hyperbola for the VSMC12 and VSMC8 cylinders.
K Fig. 2.1-6 Nondimensionalized phase data points and the curve of predicted nondimensionalized phase. 22
1 1 1i 1 1 T T 1 1 1 1 1 10 I 1 1 1
VSMC8 O VSMC12 Cm
.4° -of.- _.- o Cm o C- _0 ds Cd 0 0 00 0 0 0.1
0.01 I l I I 1II 1 I I 1 t I II1 1 I I I I I I I 10 100 1000 K Fig. 2.1-7Drag and inertia force coefficients Cdo and Cmo calculated by TFPS method then converted into Cc, and Cmi, by using Eqs. (C-8) and (C-9). The data points are the least-squares values of Cd and Cm for the VSMC12 and VSMC8 cylinders. 23 validity of this approach and to determine the optimal location of wave staffs and current meters.
2.2 The Frequency Domain Force-Phase Method
In this thesis random waves are studied in the frequency domain.
At x = 0, the random wave profile can be expressed as:
N n(t) = 1 an(fn) eiEen(fn) Sn(fn)3 (2.2-1) n=1 where j is the imaginary number, subscript n denotes the nth frequen- cy component, fn = n fl (f1 is the fundamental frequency), an(fn) is the ordinate of the wave amplitude spectrum for frequency fn, On(fn)
= 2wfnt, On(fn) is the frequency-dependent phase relative to an arbitrary origin of time, and N is the number of data points. The amplitude an(fn) and phase On(fn) can be obtained from the amplitude and phase spectra by using the Fast Fourier Transform (FFT).
For random linear waves, the horizontal water particle velocity at elevation z can be written as
N JEe(fn) (5n(fn)3 u(z,t) = u (z,f ) e (2.2-2) n=1 n n where un(z,fn) is the amplitude of the horizontal water particle velocity and On(fn) is the the phase for frequency fn.
By FFT methods, the measured random wave force, F(z,t), can also be decomposed as 24
N ilen(fn) c'n(fn) Sn(fn)1 F(z,t) - Fn(z,fn) e (2.2-3) n=1
where Fn(z,fn) is the amplitude of the in-line force for frequency
fn, and n(fn) is the phase lag between in-line force and horizontal
water particle velocity for frequency fn. The phase 4n(Kn) is
obtained from velocity-force cross spectrum analysis. The cross
spectrum analysis is described as follows.
The cross-spectral density function Gxy(fn) of a pair of time
history records X(t) and Y(t) is the Fourier transform of the cross-
correlation function (Bendat and Piersol, 1971). The cross-spectral
density function is generally a complex number such that
Gxy(fn) = Cxy(fn) Nxy(fn) (2.2-4)
where the real part Cxy(fn) is called the coincident spectral density
function and the imaginary part Qxy(fn) is called the quadrature
spectral density function. It is convenient to express the cross-
spectral density function in complex polar notation such that
-J6 (f ) G ) = IG (f)1 e xY n (2.2-5) xy n xy n where the cross-amplitude spectrum IGxy(fn)1 and the cross-phase
spectrum exy(fn) are related to Cxy(fn) and Qxy(fn) by
IG (f )1 =liCx2 y(f ) + Qx2y(f ) (2.2-6) xy n n n and 25
-1 Qxy(fn' ) exy(f) = tan [c . (2.2-7) n xy' n
Assume that the power spectral density functions for X(t) and Y(t) can be expressed as Gx(fn) and Gy(fn). Then without smoothing
1Gxy(fn)12 Gx(fn)Gy(fn) (2.2-8)
In this thesis X(t) = u(z,t), Y(t) = F(z,t), IGxy(fn) I SuF(fn), exy(fn) = n(z,fn), and SuF(fn) is the velocity-force cross-amplitude spectrum.
The time domain Keulegan-Carpenter numbers have already been defined in Eqs. (2.0-3) and (2.0-4). The frequency domain Keulegan-
Carpenter number, Kn(z,fn) is defined in a different way. It is
un(z,fn) K(z,f) = (2.2-9) n n D f n
According to linear wave theory, Kn at the still water level (z = 0) can be expressed as Kn(0,fn) = 2nan/D where, in this case, an is the amplitude of the amplitude spectrum at the frequency fn. The ampli- tude spectrum for the Hurricane Carla wave profile shown in Fig.
2.0-3 is shown in Fig. 2.2-1. The largest Kn(0,fn) for 1-ft diameter riser is about 3.4. The largest Kn(0,fn) for 3 and 8-ft diameter cylinders are about 1.13 and 0.43, respectively. Thus, it is that the largest Kn(z,fn) calculated from a random wave amplitude spectrum is much smaller than the largest Kr calculated from the same wave record.
The frequency domain maximum force coefficient, Cpn(z,fn), is defined as 2.40 WPII record-6887
2.00 zan(0,fn)= 2.0 ft fn = 0.102 Hz Kn(0,fn) 1.60 = 3.4
1.20
.80
.40
.00 .00 .10 .20
FREQUENCY (Hz)
Fig. 2.2-1 The amplitude spectrum of WPII record-6887. 27
F (z,f ) n n C (z,f ) (2.2-10) un n 1 -24 Dun2(z,fn)
where Fn(z,fn) and un(z,fn) are taken from the respective amplitude
spectra. When Kn(z,fn) is small C100(Kn) = 2.0, Cun(Kn) may be
expressed as
2 , 2 C [K. (z f )1 C [K(z f)1 " (2.2-11)3 unn n K m0 n ' n Kn n
A set of Morison force relations can be obtained by using Eq.
(2.1-6) if one assumes that the force components at any one frequency
result from wave components at only that frequency. Then the time
domain formulas can be used in frequency domain to determine the
force coefficients.
2.2.1 The Frequency Domain Force-Phase-0 Method
When experimental Cun(Kn) and On(Kn) values are both available,
the frequency dependent Cdcp(Kn) and Cmo(Kn) can be solved as
Cun(Kn) Cdo(K) (2.2-12)4 2 1 + sinOn(Kn) and
2 Kn Cun(Kn) -sinOn(Kn) C (K) = (2.2-13)5 m0 n 2 2 IT 1 + sin0n(Kn)
This method is designated as Frequency Domain Force-
Phase-0 Method (FFP0).
3Similar to Eq. (2.1-11). 4Similar to Eq. (2.1-9). 5Similar to Eq. (2.1-10). 28
2.2.2 The Frequency Domain Force-Phase-S Method
When experimental Cun(Kn) values only are available, the force
coefficients can be estimated by
cd0(%) = Cun(Kn) m(Kn) a(Kn) (2.2-14)6
and
Cm0(Kn) =Vfm(Kn)Cpn(Kn) - m2(Kn) a(Kn) (2.2-15)7
The phase can be determined by using Eq. (2.2-11). That is,
m(Kn) a(Kn) -1 (1)(Kn) = - sin (K) - m(K) a(K) (2.2-16)8
This method is designated as the Frequency Domain Force-Phase-
Slope Method (FFPS).
2.3 The Prediction of Force Power Spectra
By using the Morison equation and regression analysis, a pair of
force coefficients can be obtained from measured wave kinematics and forces. Then, by substituting these force coefficients back into the
Morison equation, the predicted forces are obtained and compared to
the measured forces in order to check the method of determining the coefficients. Herein this process is designated as pseudo prediction. If Cun(Kn) is obtained from Eq. (2.2-11) and On(Kn) is
6Similar to Eq. (2.1-19). 7Similar to Eq. (2.1-20). 8Similar to Eq. (2.1-21). 29
n assumed to be -90° or C ) and n n) are taken from other test results, the prediction is designated as a true prediction.
Borgman (1965,1967a,1967b,1972a) proposed a method to predict
the force power spectrum from a given wave spectrum, which is
described as follows:
Let SFF(z,fn) be the power spectral density for measured forces
and Svv(z,fn) and SAA(z,fn) are the power spectral densities for water particle velocities and accelerations. The velocity and accel- eration power spectra are calculated from the wave amplitude spectrum
by using linear wave theory. Borgman found that force power spectral densities can be approximated by
Cd(z) p D wn Cm(z) p D
S (z f) = [ FF ' n 2 VV(z ' fn) +[ 4 1 SAA(z' f n )
(2.3-1) where wn = 21rfn. A least-square fitting of the theoretical force power spectrum against the actually measured force power spectrum yields Cd(z) and Cm(z). This method is designated as the Borgman method. The Borgman method assumes that the force coefficients,
Cd(z) and Cm(z), are constant for the whole spectrum frequency range and thus for all Kn(z,fn) values. Because the Borgman method needs only a pair of force coefficients, it is useful in true predictions.
Brown and Borgman (1965) obtained frequency-dependent force coefficients by using the cross-spectral density between the force and wave profile. This method is designated as the Brown-Borgman method. The details will not be described here. For one example studied, they found that force coefficients exhibited a distinct 30 tendency to be functions of frequency. But the difficulty of explaining the negative force coefficients in the highfrequency region, and the difficulty of estimating the frequency-dependent force coefficients makes it less useful for truepredictions.
The method introduced in this thesis is simple. Only the
) relationship is used (Hsu and Nath, 1986). By rearranging C (Kn Eq. (2.2-10), the force amplitude Fn(z,fn) can be obtainedwith
p..D Fn(z,fn) Cun[Ku(z,fn)] (2.3-2)
) values is available If the regression line of the measured C (Kn from another test, the force amplitude Fn(z,fn) canbe obtained by using Eq. (2.3-2) and the wave force power spectrum can bepredicted by
2 F(z,fn) S (zf) = nA (2.3-3) FF ' n where Af is the frequency increment.
If the surface of the cylinder is smooth and all of the decomposed Kn(z,fn) values are less than 8, then Cun(Kn) can be
replaced by Eq. (2.2-11). The force amplitude can be expressed as
2 w 112, Fn(z,fn) (2.3-4) ITp ITTDT n\z' ni n n
or
2 D2 if fn un (z,f ) . (2.3-5) = p n 31
2.4 The Prediction of Force Time History
The wave grouping effects are usually studied in the time
domain.
By comparing Eqs. (2.2-2) and (2.2-3), the relationship between
velocity and force for frequency fn can be expressed as
F(zfn ) = Hn (f) u(zf ) . (2.4-1) n ' n n ' n
Where H(f n n) is the transfer function, which can be expressed as
icOn(zfn)
Hn(fn) = IHn(fn)1 e , (2.4-2)
and IHn(fn)I can be derived from Eq. (2.3-5) to yield
1 pn2 D2fn= pnD2wn. (2.4-3) IHn(fn)I 2
For pseudo prediction n(z,fn) can be estimated from velocity-force
cross spectrum analysis. For smooth cylinders when Kn is small, the
true prediction can be achieved by using +n(z,fn) = -90° and
2 C (z,f) = 2w/K . un n n An example will be shown in section 4.4.
2.5 Force-Phase Method Extension
The force-phase method was only applied to the in-line forces on vertical cylinders in previous sections. The transverse forces on vertical cylinders are of about the same magnitude, however, and there are no reasons to ignore them. Horizontal cylinders play an important role in offshore structures, and there are no reasons to ignore them either. Application of the FFP4, and FFPS to the trans- verse forces on vertical cylinders, and the horizontal and vertical forces on horizontal cylinders will be described as follows. 32
The wave forces acting on a vertical cylinder consist of in-line
forces and transverse forces. Similarly, the forces acting on a
horizontal cylinder consist of horizontal in-line forces and vertical
forces (see Fig. 2.5-1).
The transverse, horizontal, and vertical forcepower spectra can
be predicted (pseudo) by substituting maximum respective force coef-
ficients and correlated phases into the formulas derived in previous
sections. Those formulas are tabulated in Table 2.5-1.
2.5.1 Vertical Cylinders
The symbol Cun(Kn) is reserved for the frequency domain maximum
in-line force coefficient. The frequency domain maximum transverse
force coefficient is described as follows:
Theoretically, there is no flow in the transverse direction ina unidirectional wave field when the cylinder is not there. Therefore,
the maximum transverse force coefficient is defined by using the undisturbed in-line horizontal velocities un(z,fn) as
FTI(z,fn) T C (K ) = (2.5-1) un n 1 2 P D D(z2fn) where F(z,f n n) is the amplitude of the transverse force for frequency fn. The correlated phase (0(z,f) is defined as the phase between n n T F(z,f) and un(z,fn). n
2.5.2Horizontal Cylinders
The maximum horizontal force coefficient in the frequency H domain, c (z,f), is defined as: Un n 33
. 0 " . ;9I : . 0"
in-line force horiz. V force A x Horizontal transverse Ill- Cylinder force . V . .
Top View
wave direction z z
in-line horiz.i force force x
vertical force
Vertical Cylinder
Side View
Fig. 2.5.1 Definitions of forces on vertical and horizontal cylinders. Table 2.5-1 Summary of frequency domain maximum force coefficients, Keulegan-Carpenter numbers and phases
Vertical Cylinder Horizontal Cylinder
Force In-line Transverse Vertical Horizontal
T V H Maximum-um- 2 F (z,f ) 2 F (z,f ) 2 F (z,f ) 2 F (z,f ) pn n T pn n V pn n H pn n Force C C C pn pn n ) P coefficient pDu 2(z,f pDu 2(z,f ) pDw 2(z,f ) pDu 2(z,f ) pn n pn n pn n pn n
Keulegan- u (z,f ) u (z,f ) w (z,f ) u (z,f ) Carpenter pn n pn n pn n pn n number D f D f D f D f n n n n
t (z,f ) t(z,f ) t (z,f ) tn(z'fn) n n n n n n phase lag phase lag phase lag phase lag Phase between between between between T V H F (z,f) and F (z,f ) (z,f) and F (z,f ) pn n pn n pn n pn n and
u ) (z,f u (z,f ) w (z,f ) u (z,f ) pn n pn n pn n pn n 35
FnH(z,fn) H C (K ) pn n 1 (2.5-2) -2- p Dun2(z,fn)
H where F(z,f n n) is the amplitude of the horizontalforce for frequency H f . The correlated phase 0(z,f) is defined as the phase between n n H F(z,f) and u(z,f n n n).
The maximum vertical force coefficientin the frequency domain is defined as:
V F(z,f ) n n CV (K) = pn n 1 (2.5-3) 2- p Dwn2(z,fn)
V where F(z,f ) n n is the amplitude of the vertical force forfrequency f and w(zf ) is the amplitude of the vertical n' n ' n water particle velocities for frequency fn. V The correlated phase 0(z,f ) is n n V defined as the phase between F(z,f) and w(z,f). n n n n 36
3.0 DESCRIPTION OF EXPERIMENTS
With some exceptions, full scale measurements on offshore structures offer little assistance in the systematic study of wave forces, because of difficulties in quantifying the real sea environ- ment. Laboratory tests allow us to measure fluid loading on a cylin- der under controlled conditions. However, in model testing it is difficult to extrapolate the fluid loading to full scale conditions.
The scaling factors such as R and K can not match the ocean situation at the same time. Nevertheless, laboratory tests and field tests are necessary to check the validity of the theory.
The experimental data from Oregon State University (OSU), Wave
Project II (WPII), and the National Maritime Institute Second Christ- church Bay Tower (NMI) were used to check the new method. Only a brief review will be given to summarize general aspects of the experiments.More information can be found from cited references.
The test conditions are summarized in Table 3.0-1.
3.1 Laboratory Experiments
The Laboratory experiments described in this thesis were con- ducted in the OSU wave tank (Fig. 3.1-1). The overall length of the wave flume is 340 ft. The width is 12 ft and the depth is 15 ft, of which 3.5 ft is freeboard. Each test cylinder was located 136 ft from the wave board (Hudspeth et al., 1984).
Three cylinders were tested. Table 3.0-1 Summary of the test conditions
Date Project Test Water Spectrum # Cylinder Orientation Roughness* (year) Site Depth Diameter (ft) (in.) 1933 NCEL JONSWAP 12.75 Vertical SMC12 Scott 8.625 Vertical SRC.02 Bretschneider Vertical SRC.02 OSH Scott Vertical SP3-55 1984 AP184 Wave 11.5 Bretschneider Vertical SP3-55 lank Scott 8.625 Horizontal SRC.02 Bretschneider Horizontal SRC.02 Scott Horizontal SP3-55 Bretschneider Horizontal SP3-55
1961 WPII Gulf of 97.5 44.0 Vertical Maxico 1982 English 28.5 18.9 Vertical NMI Channel 110.2 Vertical
*SMC12, Smooth aluminum cylinder 12.75 in. SRC.02 Sand roughened cylinder - e = .023 (D = 8.625 in.) SP3-55D South Pass platform, 3-year old, -55 ft.
#JONSWAPspectrum (Hasslemann et al, 1973) Scott spectrum (Scott, 1965) Bretschneider spectrum (Bretschneider, 1959) 4-0 04- u, 1
4- i 4- Lr> movable concrete slab beach
18 ft.44 ft. 128.6 ft. 121.4 ft. 30 ft.
Fig. 3.1-1The OSU wave flume (from Nath, 1985a). 39
3.1.1 The SMC12 Cylinder
The SMC12 cylinder was a 12.75-in. diameter SMooth Cylinder
(Hudspeth et al., 1984). It had a 12-in. long local force transducer. This cylinder is detailed in Fig. 3.1-2. It was tested as a Vertical cylinder and is designated the VSMC12 cylinder.
3.1.2 The SRC.02 Cylinder
The SRC.02 cylinder was a Sand-Roughened Cylinder with e =
0.023, where e is the sand grain size divided by smooth cylinder diameter. It is designated as the VSRC.02 and HSRC.02 for Vertical and Horizontal tests, respectively. The positions of the VSRC.02 and
HSRC.02 cylinders are shown in Figs. 3.1-3 and 3.1-4.
3.1.3 The SP3-55D Cylinder
The SP3-55D cylinder is from the South Pass region of the Gulf of Mexico. It was placed -55 ft below the sea surface for 3 years and became covered with rough natural marine organisms (Nath, 1985b).
Before it was used to do the tests, it was Dried. It is designated as the VSP3-55D and HSP3-55D for Vertical and Horizontal tests, respectively. The arrangement of the HSP3-55D cylinder is the same as for the HSRC.02 cylinder.
For all of the vertical and horizontal cylinders, the center of the force transducers was 3.7 ft below the still water surface.
All of the cylinders were tested in regular waves and random waves. For periodic wave tests, the wave periods ranged from 2.5 sec. to 6.0 sec. with maximum wave heights of about 5 feet. The maximum Reynolds number reached was 3.1x105. The maximum Keulegan- 40
V 12 steel toot ]9.1 .If 'I 1U,511Mi 5,111 tO wool, 10,11, 7c7S 01". ^ par ooltS 7/60) 1 3 coo and onto. wags ocoaal ti toe a.. bettfal rt .4d-a. 4/cants raual ;.,.'''..../. ' 3.5' beano, ; q 1I/2' a stalolest fed ;I IIwan low frIetlen N I itOfile,n0S 2 plates 4 - "0;R.I.:;R. , 'Papa I Z rh,,-;.: , a Car ../c.1.45. III
,
bearing
11 It
11 bolo for II loess es< vire Direction of ."; 44. L 1
11:I
Per lool fare. tron10.er y ... V / aortal transoms+. line -._- 4 1 t aorta I sew. sae seta 1 1 s, , :. :.' .-..-,...... Hi.i. V o 0 Y, ,' . --1.- /6 R. 7" I h 41'
II It Swett, aluoiroa. carer P I pins. tr aloe. Pr I re 11 Meanest.Sol It 0 lonoilinetnolly.fill b II tart bats to r flames. , II
r flanges 7J' ( traits I)
It
II n
N l' II S'cl/4'
I Il sogriby wee 1 (Stool) 1 1 S
:
t I
I 'id strain bar I 4 alumina.. te.aer 7075 15 will eutl ti-jelint It 11 1 tn tenon 1 oat ii filT 0 belt ant*. plate to floor , of flame texistang plate) / le,,, ,, 1 I /,.,* alumnum flange
4o: :
G c) a p .t3 ELEvArtot
Fig. 3.1-2 Sectional elevation of a VSMC12 test cylinder (from Hudspeth et al., 1984). 41
12 x 8 rectangular steeltube. 39.7 plf
.0 0 0 . u0*io o. a? 1.0 0 I6
. o. .7
same rouahened "skins" as for the horizontal .77, cylinder tests
Lcvrrenf mefer
:9 o A 12'
. 0.:
0 1 2 guys ea. connection
0 8 total
V O 0 ui
Cof o 0 .
g .0-m...0 . 0 0'; '. O. r0000 ct. v g I . 0 17 0 0, co, " . . ELEVATION
Fig. 3.1-3 Sectional elevation of a vertical 8.625-in.test cylinder (from Nath, 1983). 42
SONIC WAVE SENSOR
TOW CARRIAGE
7.2'
OM.ma.MO. CURRENT METER
1.7'
8 5/8" DIA. CYLINDER SPECIMEN. 8.7 ft. LONG
7.8' 11.5'
Cs : "c) 0 ." 0 czj .
Fig. 3.1-4 Side view of a horizontal 8.625-in.test cylinder (from Nath, 1985a). 43
Carpenter number was 27, and maximum values of 8 (= R/K) were about
4x104. More details can be found in Nath (1985b).
For the random wave tests, each cylinder test comprised one run
of a simulated Scott spectrum and one run of a Bretschneider spec-
trum. Each spectrum had a target significant wave height of 3.2 ft
and a modal period of 3.9 sec. The actual values ranged from 3.0 to
3.21 ft and 3.72 to 3.85 sec., respectively (Nath and Yeh, 1986).
3.2 Field Experiments
Two sets of field data were used in this study: the WPII
project and the NMI tests.
3.2.1 NMI Second Christchurch Bay Tower
The NMI Second Christchurch Bay Tower (Bishop, 1985) had a large
column (cylinder) with diameter D = 2.8 m and a small column (cylin-
der) with diameter D = .48 m. The side view and plan view of the
tower are shown in Figs. 3.2-1 and 3.2-2. The wave profile was
obtained by a capacitance instrument Fig. 3.2-1.
The water particle velocities, influenced by both waves and
current, were measured by perforated ball instruments (Bishop,
1984a). One perforated ball instrument measured the wave kinematics
in two othogonal directions. Near the small cylinder, perforated
ball instruments were installed at four depth stations, Fig. 3.2-1.
At each station, a pair of perforated ball instruments were used to
give three velocity components. The predominant wave direction was determined from velocity measurements. Force measuring elements were
located at 5 different depths (see Fig. 3.2-2, Bishop 1984a). 44
WAVE STAFF RAISING ANO GENERAL USE (II) DERRICK
ANEMOMETER WINO MINI
WAVI BUOY RECEIVER AERIAL
BAROMETER
NAVIGATION LIGHT cr(WHITE) DETECTOR
.1161111IN TELEMETRY AERIALS .1111. DATUM (.10.041 -att FOG HORN
INSTILMENT ROOMS
ENTRANCE PORCH
ISI OATUM WEE) NAM PLATFORM
GENERALPURPOSE ULTRASONIC°Illleji (GA) ROOM TIDE SAUCE LEVEL was:maxi Wart N.WIt&($4) Mon'GAUGE MAIN COLUMN -re. 1 SCINCRNO UWOM -: . ACCELEROMETERS (20 FF) LEVEL 2 MR: 0 11 DIA '"'" ," :- ALS.WL can i I PIREORATED illki GALL PROM ....---H . ...; PAL L.W.S.(741) _ ._::.. ... i FLUEN SEA TEMPERATURE PRESSURE SENSOR TAPOmOS 124 OM . Ili -ja..."1 4------2.80 DIA MAN TOWER SMALL COLUMN gi FORCE SLEEVES LS OFF) FORCE SLEEVES 1.1 OFEI.--....,-'-'tivtl 3 . UrfirioEYES (4 Om ill 4p. DATUM (.1411) b RESAINFSECMCITS0Wt) DIA C°N IASCRO ril
TOWING EYES TONER DATUM C_ ,.,...... rn S 1:? (a OFF) ,. NN f o.crum (-040)i; , EARTH PRESSURE \S KV POWERCABLE INLET DUCT CELLS (11 OFF) VERTICAL STEEL CIRCUMFERENTIAL SKIRT
Fig. 3.2-1 NMI Christchurch Bay Tower side view (from Bishop, 1984a). 45
LARGE COLUMN
LATERAL.AXIS
THIRD 'N .g_KVI AY
ROTARY CUR RENT METER
Fig. 3.2-2NMI Christchurch Bay Tower top view (from Bishop, 1984a). 46
3.2.2Wave Project II Tests
The platform layout for Wave Project II (Blank,1969) is shown in
Fig. 3.2-3.The force measurement device of WPII had 16 local pres- sure transducers9, with an active length of 6 inches (Blank, 1969).
The devices were clamped onto one of the platform support piles.The outer diameter of each device was about 44 in. During hurricane
10 season, the eight meters were spaced vertically along the pile as shown in Fig. 3.2-4. A wave staff was placed about 3 ft seaward from the instrumented pile (see Fig. 3.2-3).
During Hurricane Carla only the data at positions 7 and 8 were legible and were analyzed. The biggest individual wave recorded during Hurricane Carla was recorded in record-6887.
9The forces were obtained by integrating the pressure around the cylinder. The pressure force varied linearly in the interval between cxo successive pressure transducers. During the hurricane season, waves were expected to be higher than during other seasons. Thus, near the free surface, the spacing between meters should be wider in order to record all possible waves. 47
ORIENTATION POSITION Qf CE1Tr1. N2I° 51' 10' WEST
LATITUDE 28° 47' 34.299' NORTH LONGITUDE 90° 12' 08.079' WEST LAMBERT COOROINATES:
X 2 2 362 358.15 Y a 47883.08
N
NITROGEN SUPPLY
2 DIESEL ELECTRIC GENERATORS
CALCO WEATHER STATION
FOG mcnn INSTALLATION DIESEL FUEL TANK
WATER TANK CRANE
INSTRUMENT HOUSE
8 WAVE DYNAMOMETERS ON SW LEG OF PLATFORM
BATTERY ROOM
CRANE RECORDING A CONTROL WAVE STAFF EQUIPMENT
Fig. 3.2-3Platform layout of Wave Project II (from Blank, 1969). 48
foECx NOTE:ALL DIMENSIONS OF DYNAMOMETERS REFER TO CENTER OF SENSING ELEMENT OF TRANSDUCERS. 11'8' 014AMOUTEI VISS CO2TACT NO 12N M 4't
WAVE STAFF
FAGOVE COUTACT 57
60'0" 818"(8.61" 3180""E CONTACT 40 01 2" AM 132 Ty CVTACT 40 19'5- (19.71. 20'2' myr 14"AEOVE CO7ACT 1 p 7 149'0' 04 Hy
1061"
ST'6" 87'7" (9 '91_
TT'4" 18 2'
55'4"
//44\"'/''``4111`"4/'`.'"'l DigENSIONSVALIO MAY 2, -JULY 5,1961
Fig. 3.2-4 Location of WPII dynamometers and wave staff (from Blank, 1969). 49
4.0 RESULTS AND DISCUSSION
Determination of the C (K un n) relationship is the major function of the FFP+ and FFPS methods; they will be studied in Section 4.1.
Applications of the Cun(Kn) relationship, such as for force coeffi- cient determination, the force power spectrum, and force time history prediction, will then be studied in Sections 4.2, 4.3 and 4.4, respectively.
4.1 The C (K un n) Regression Lines
The in-line force on the vertical cylinder will be used as an example to describe the procedures for obtaining a maximum force coefficient regression line from measured velocities and forces.
Other forces, e.g. the transverse forces, vertical forces, and hori- zontal forces, can be analyzed by similar procedures.
The procedures for obtaining the Cun(Kn) regression line are described below.
(1) C (K un n) data points are obtained by substituting measured velocities and measured forces into Eq.
(2.2-10).
(2) (K) data points are then removed. Scattered Cun n These procedures are described in sections 4.1.1. and
4.1.2.
(3) The remaining C (K) data points with small K un n n values (decomposed Kn values less than about 8) are
then fitted to a straight line, L1. An ideal L1 line
is shown in Fig. 4.1-1. 50
100 1-
1.- 1 --1--
. _t- .....--.-
4 1- -- -. _
10
7,------T- - a-- -, - . I.Y. L___ ,-- ...... 1 1.---.---4-..--" ) Hyperbola ... . n =Ma. ---- 1--4-4-7`.------
t . .1 1 1 -- . ..a.-
.,----- "S" 4- -_=-,.= ---='-' -: ----- . s...L'...i.i...,_;1_7-7.--;;-_:.'.'...--;: \,,..-"_7 -....._-_,. - =."- '-7-.':.-,== F,...-277.- - ,:.=i_.,-71.- =777:7:7 -- -- ...... - -.....-.- - - . - - .....L. 7...- _ -..,-...,.....-..--- -...
.------1. - -,--- ,--
A I . . i ' 0.1 * . 1 10 100 Kn
Fig. 4.1-1 The C pn(Kn) hyperbola and their asymptotes. 51
(4) The line L2 shown in Fig. 4.1-1 is then determined by
setting Cnn(Kn) equal to Cds, the steady state drag
coefficient. In this thesis Cdsis set equal to 0.5 for smooth cylinders and 1.0 for rough cylinders.
These numbers were obtained from OSU towing test
results (Nath, 1986).
(5) A hyperbola", the Clin(Kn) curve, was constructed by
using LI and L2 as asymptotes. (See Fig. 4.1-1).
The details of procedures (5) are described in Appendix A. Data were plotted on 6 x 5 cycle log-log graph paper, which can accommo- date all possible Kn regions.
4.1.1 The Amplitude Filter
Figures 4.1-2 and 4.1-3 show a typical force-velocity cross- amplitude spectrum and cross-phase spectrum. From these two figures, it can be found that for those frequencies with large cross- amplitudes the correlated phases, .n(z,fn), are near -90°. When the amplitudes are small, however, the phases become random because noise is a large percent of the signal. Phase randomness may be caused by environmental noise and other unknown effects.
Line L1 is determined by the least-squares method. In the least-squares method, all data points are weighted equally. If unimportant data points occur far from the regression line, the slope and intercept of the regression line may be greatly changed.
11Note that the maximum transverse force coefficient, CTn(Kn), was only fitted by a straight regression line, LI. The L2 line did not exist. .90 VSMC12 cylinder
.70
0 a) V) 4) 4- .50
-.N. 4- .30
.10-
-.10 i I I I I I I : # I .00 .20 .40 .60 .80 1.0 FREQUENCY (Hz)
Fig. 4.1-2The velocity-force cross-amplitude spectrum. 140.00
100.00
60.00
cu
as 20.00 c 4--
-? 20.00 I
60.00
-100.00 I 4- i 1 I I 1 1 1 .00 .20 .40 .60 .80 1.0 FREQUENCY (Hz)
Fig. 4.1-3 The velocity-force cross-phase spectrum. 54
Unimportant data points may thereby lead to inaccurate interpreta- tion. Therefore, the unimportant data points which were calculated from small-amplitude components were removed before the regression analysis.
Not all of the velocity and force components are used in a regression analysis to determine L1. The velocity-force cross- amplitude spectrum is used to help to determine which frequency will be used. The velocity-force cross-amplitude spectrum is first passed through an amplitude filter (Fig. 4.1-4), and then a band pass fre- quency filter (Fig. 4.1-5). The through frequencies that remain after passing through these two filters are used. This does not mean that the removed frequency components will be discarded. They are not used to determine line Lwhich will be used to construct the 1
C (K) curve. pn n All frequency components are used to predict the force power spectrum and the force time history (by inverse FFT).
The cut-off amplitude of the amplitude filter was selected by specifying a cut-off amplitude to peak amplitude ratio. The ratios were obtained by trial and error. Each trial ratio began at 0 and was then increased until most of the remaining data points were evenly distributed on both sides of the regression line. The VSMC12 in-line force data points and the regression lines corresponding to different cut-off amplitudes are shown in Figs. 4.1-6, 4.1-7 and
4.1-8. The results in Fig. 4.1-6 include all of the plotted Cun(Kn) data points. When the frequency components that have cut-off ampli- tudes of less than 1% and 10% are deleted, the results are like Figs.
4.1-7 and 4.1-8, respectively. This example illustrates that an 55
.90- Force - Velocity Cross-AmplitudeCross - Amplitude Spectrum peak
.70
r-4 C4 4 .30-
CUT-OFF AMPLITUDE
.10 - 0.1 peak height
.10 1 4 1 1 .00 .20 .40 .60 .80 1.0 Frequency(Hz) (a)
.90 Force-Velocity Cross-Amplitude Spectrum peak .70-
.10 -
1
.00 .20 .40 .80 1.0 Frequency(Hz) (b)
Fig. 4.1-4The amplitude filter: (a) before filter, (b) after filter. 56
.90 Force-Velocity Cross-Amplitude Spectrum peak .70
loweir upper bound of bou9d band-pass filter .50
.30
.10
.10 .00 .20 .40 .60 .80 1.0 Frequency(Hz) (a)
.90 Force-Velocity Cross-Amplitude Spectrum
.70
.10
G .10 f I f I .00 .20 .40 .60 .80 1.0 Frequency(Hz)
(b) Fig. 4.1-5The band pass frequency filter: (a) before filter, (b) after filter. IN-LINE FORCE 10000 1 CYLINDER VSMC12 RUN NO. NCEL R32-CR rms .. 0.3560 (log10-1091.4) 1000 . REGRESSION .E0. (2.2-11) *4 : CUT-OFF RMP.mi 0.00 . FREQ. BAND 0.00-11.00 Hz .. : 100 :
Co(Kn)
10
1
0.1 I11141111 0 L.1.311111.....,_..1_,II 111111 I 1 I 1111111 I II 11111J1 II 11111 0.001 0.01 0.1 I 10 100 1000 K n
Fig. 4.1-6The Cpn(Kn) data points and their regression line (before filter). IN-LINE FORCE 10000
CYLINDER VSMC12 RUN NO. NCEL RS2CR rm. ... 0.0751 Clog10log111 1000 REGRESSION EQ. (2.2 -11)
.. CUTOFF RMP.me 0.01 s FREQ. BRNO 0.00-4.00 Hz
100
.fit. Con(Kn) 4
lo
1
0.1 I 1 1 111111 1 1 1 111111 1 1 1 1 111111 1 1111111 1 a 1 111111 1 a IWO 0.001 0.01 0.1 1 10 100 1000 K n
Fig. 4.1-7 The Cu (1(e) data points and their regression line, after passing throughan amplitude filter (cutoff amplitude .. 0.01 spectral peak). IN-LINE FORCE 10000
CYLINDER VSMC12 RUN NO. NOEL R92-CR rm. 0.0943tlog10-logliml) 1000 REGRESSION EQ. (2.2-11) CUT-OFF AMP. - 0.10 FREQ. BRNO 0.00-4.00 Hz
100
C (K) pn n
10
1
0.1 0.001 0.01 0.1 1 10 100 1000 K n
Fig. 4.1-8The Cvn(Kn) data points and their regression line, after passing throughan amplitude filter (cut-off amplitude .. 0.1 spectral peak). 60
increase in the cut-off amplitude of the amplitude filter decreases
the data scatter.
It was found that a small cut-off amplitude, such as 1% of the spectral peak, was sufficient for the VSMC12 (smooth cylinder). The
C (Kn) data points for the SRC.02 (rough cylinder) are more scat- tered, and a higher cut-off amplitude, such as 10% of the spectral peak, was necessary. In field tests the noise/signal ratio is high, and even for smooth cylinders a higher cut-off amplitude of at least
10% of the spectral peak may be necessary.
4.1.2 The Frequency Filter
Figure 4.1-9 shows the Cun(Kn) data points for the VSMC12 cylin- der after passing them through a band-pass (0.1 to 1.0 Hz) frequency filter. A comparison of Fig. 4.1-9 with Fig. 4.1-6 indicates that some of the scattered data points in Fig. 4.1-6 have disappeared.
The cut-off frequencies were determined from the bandwidth of the wave spectrum as follows. A lower bound of 0.1 Hz was used for the
OSU laboratory tests, because only a small amount of energy existed in the region where frequencies are less than 0.1 Hz, they may be caused by surge and other effects. The upper bound was set at 1.0 Hz because only noise energy existed above that level. Peak frequencies in the WPII and NMI tests were lower than those measured in the OSU laboratory tests. Lower and upper bounds for the NMI tests were set at 0.1 Hz and 0.5 Hz, respectively. Lower and upper bounds for the
WPII tests were set at 0.05 Hz and 0.5 Hz, respectively. IN -LINE FORCE 10000
CYLINDERVSMC12 RUN NO. NCEL RS2-CA rm. m 0.2487 (1og10-1op1 ag1) 1000 REGRESSION EQ. (2.2 -11) CUT-OFF AMP. 0.00 FREQ. BAND 0.10-1.00 Hz
100
Con(Ku)
10
1
0.1 1 IIIIIILII I 11111111 I 1111111111 I IIIIIISJI I\IIIIlN 0.001 0.01 0.1 1 10 100 1000 K n
Fig. 4.1-9 The C.n(Kn) data points and their regression line, after passing through an band pass (0.1 Hz to 1.0 Hz) filter. 62
4.1.3 Reynolds Number Effects
The Reynolds number discussed in this section is calculated from
uD/v. For regular waves and random waves, uis the maximum water 1.1
particle velocity during a wave cycle. For oscillatory flow, uu is
the maximum water particle velocity during a oscillatory flow cycle.
Bearman (1985) and his co-workers studied the forceson cylin-
ders of planar oscillatory flow generated in a U-tube in the small K
region (K < 10) with 2.56 to 7.48 cm smooth cylinders. They found
rms viT (4.1-1) CFrms 1 p Du2 2 rms
where CFrmsis the non-dimensionalized root-mean-square force. By
comparing Eq. (4.1-1) with Eq. (2.1-11), it is evident that
Crus = Cu//2 when Cm0 = 2. Thus, by substituting Cu /,/2 for CFrms,
Eq. (4.1-1) is exactly the same as Eq. (2.1-11) when Crom= 2.
Bearman's results are shown in Fig. 4.1-10, where therange of
Reynolds numbers is between 350 and 16,650. Figure 4.1-10 shows that
the test data fit the theoretical line very well. Sarpkaya's(1986a)
U tube results are shown in Fig. 4.1-11. When K 4 8, his results are
also in good agreement with the theoretical line. The Reynolds num-
bers of Sarpkaya's tests were between 2x103 and 1.2x104 (for K4 8).
An example of the OSU random wave test results is shown in Fig.
4.1-6. The highest Reynolds number in the OSU tests was 3.1x105.
For the large and the small cylinders of the NMI field tests, the
Cun(Kn) data points from eleven records (superimposed) at three different depths are shown in Fig. 4.1-12. The highest Reynolds 63
100 so X 8 = 1665 08 =1204 60 O8 = 483 a8= 196 40
20
CFrnIs
10
ZS
6
4
2
0.1 0.2 0.4 0.6 U. I 2 4 6 8 10 K
Fig. 4.1-10In-line force coefficient vs. K (From Berman, etc. 1985). 25 o, fl= 2300 20 x, ii=--3435 15
#,fil = 4720 10,- .7_ e 4', fl = 6555 7 6 5 +, fl = 11525 C p 4 3 H 2 t- 8 o 0 o 0 00 O O° 1 .8 27r2 . 6 K
I 1 1 t . 4 I ) 1 1 1 1 1 1 1 1 CO CO 1i (*) V (1)(1)IN- OD 61 eg eg eg egeg (53 ...... 0.! (V) d- In COIN- K
Fig. 4.1-11 Maximum force coefficient for smooth cylinders as a function of K (from Sarpkaya, 1986b). IN-LINE FORCE 10000
-.: NMI SCBT RECORD 50.51.52.53.511 63.67.68.69.70.71 1000 D LARGE COLUMN LEVEL-3.11.S 0 SMALL COLUMN LEVEL-3.11.5
100
C p n
10
21r2 K 1 n
0.1
0.001 0.01 0.1 1 10 100 1000 K n
Fig. 4.1-12 The Cnn(Kn) data points and their regression lines for NMI large and small cylinder (11 recordsat three locations). 66 number obtained in the NMI tests was about 4.2x106. It occurred on the large cylinder at level 3. Even though R was high, the Cnn(Kn) data points fit the Eq. (2.2-11) line quite well because C pnwas determined from the frequency domain amplitude spectrum.
The R values covered by these five different tests are between
6 350 and 4.2x10 . It is suggested for in-line forces that the
Reynolds number may not be such an important factor, whether in oscillatory flow, regular waves, or decomposed random waves when
K < 8 for a smooth cylinder.
For roughened cylinders, the effective diameter should be used to determine Kand C ). n (Kn The effective diameter should be determined in a reasonable way. According to Nath (1986), the Cu(K) regression lines are not only a function of K, but also a function of
R. Sarpkaya's (1986b) U tube results from a sand-roughened cylinder
(c = 0.02) are shown in Fig. 4.1-13. For Kn 4 7, the Cun(Kn) data points fit the Eq. (2.2-11) line very well. By comparing Figs.
4.1-11 and 4.1-13, it can be seen that there are no significant dif- ferences between the Cpn(Kn) regression line and the Eq. (2.2-11) line when c and K nare small.
4.2 Force Coefficients
The FFPS method is used here to determine force coefficients and phases. When the C (K pnn) hyperbola is obtained, curves of drag coef- ficients, Cdo(Kn), inertia coefficients, Cm0(Kn), and phase, (Kn), can be calculated from the C pn(Kn) curve by using Eqs. (2.2-14), (2.2-15) and (2.2-16). To compare the force coefficients with those obtained from regular waves by using the least-squares method, 25 20 15 o, fl= 2412 x,1.3 = 3598
10 II, ft= 4924 8 +, f? = 6836 7 6 *,fl = 9354 5 0, fl = 14200 C 4 3
2
r-
1 27r2 .8 K . 6 K . 4 1 1111111f yr (LI W - N (V) Ni" W 03 (S) m 6) Cif N UILON
Fig. 4.1-13Maximum force coefficient for rough cylinders (from Sarpkaya, 1986b). 68
Cd,(Kn) and Cm +(Kn) were converted into CdL(Kn) and CmL(Kn) by using
Eqs. (C-8) and (C-9) described in Appendix C. Measured Cun(Kn) and measured $n(Kn) were also plotted on the same figure.
4.2.1 Laboratory Test Results
The Cun(Kn) hyperbola, curves of predicted force coefficients, curve of predicted non-dimensionalized phase, measured Cun(Kn) data points, and measured non-dimensionalized phases for the VSMC12 cylinder are shown in Fig. 4.2-1. The Cun(Kn) data points were tightly distributed along the hyperbola and the predicted non- dimensionalized phases are in good agreement with the measured phases
(close to -90°). The Cun(Kn)(K) data points and their regression line n for the VSMC12 cylinder are shown in Fig. 4.2-2.
The Cun(Kn) hyperbola, curves of predicted force coefficients, curve of predicted non-dimensionalized phase, measured Cun(Kn) data points, and measured non-dimensionalized phases for the VSRC.02 cylinder are shown in Fig. 4.2-3. The CL(Kn) data points and their regression line for the VSRC.02 cylinder are shown in Fig. 4.2-4. H The C (K) hyperbola, curves of predicted force coefficients, pn n curve of predicted non-dimensionalized phase, measured Cun(Kn) data points, and measured non-dimensionalized phases for the HSRC.02 cylinder are shown in Fig. 4.2-5.TheCTun(Kn)data points were tightly distributed along the hyperbola but the predicted non- dimensionalized phases are in poor agreement with the measured non- dimensionalized phases. The CV (K ) hyperbola, curves of predicted pnn force coefficients, curve of predicted nondimensionalized phase, IN-LINE FORCE 10000
CYLINDER VSMC12 RUN NO. NCEL RS2 SPECTRUM JONSWAP 1000 FREQ. BAND 0.10- 1.00 Hz CUT-OFF AMP. -0.01 MEASURED Cur, I- rms 0.0618
100 I- tin CdL Cd Cmt. Cm 10
1 1imiropirmstwitirgrs- ik
0. 1 011111 I 1 II 111d ../Lll L-1-1-1-1-uL lilt 0.001 0.01 0.1 10 100 1000 Kr,
Fig. 4.2-1 In-line force coefficients and phases for the VSMC12 cylinder (Run 32). TRANSVERSE FORCE 10000
CYLINDER VSMC12 RUN NO. NCEL R92 rm. 0.2429 (1og10-loglm11 1000 REGRESSION \ E0. (2.2 -111 CUT-OFF AMP. - 0.05 FREQ. BRNO 0.10-1.00 Hz
100 4'?' T C un
10
liNte
1
0.1 i tm mind i II mind I itmai) wail I mamma._ U. 0.001 0.01 0.1 K1 10 100 1000
Fig. 4.2-2Transverse force coefficients of the VSMC12 cylinder (Run 32). IN-LINE FORCE 10000
CYLINDERVSRC.02 RUN NO. RPM R2S SPECTRUM SCOTT 1000 FREQ. BAND 0.10- 1.00 Hz CUT-OFF AMP. -0.01 MEASURED Cmn rm. 0.1092 Cmn 100 4n CdL -- Cd Cmt. Ca. 10
/.00
...... 1
Mt X
0.1 'clod 1111111 I 111111111 I 1. ad 0.001 0.01 0.1 1 10 100 1000 Kn
Fig. 4.2-3 In-line force coefficients and phases for the VSRC.02 cylinder (Run 23). TRANSVERSE FORCE 10000
CYLINDERVSRC.02 RUN NO. RPI64 R2S rm. la 0.1976 (1og10loglwill 1000 REGRESSION EQ.(2.2-111 CUTOFF RMP.I* 0.05 FREQ. BRNO 0.10-1.00 Hz
100
T dun
10
1
+4'
0.1 L_LAAJ-1111_ 1 1 (atm! t i 1111141---JLLJ\gd a t i1W\III 1 Ill 0.001 0.01 0.1 1 10 100 1000 Kn
Fig. 4.2-4Transverse force coefficients for the VSRC.02 cylinder (Run 23). HORIZONTAL FORCE 10000
CYLINDERHSRC.02 RUN N. WIWI ASS SPECTRUM SCOTT 1000 FREQ. BAND 0.10- 1.00 Hz CUT-OFF AMP. m0.03 MEASURED Con rms - 0.0920 Cun 100 CdL Cd CmL Cl". 10
Oe.
1
f:11411- W x 1177"kli
0. 1 I 1 111(111 _1 1 1 111111 1 11 111111 1 a I flail 1 Il tx11.1 1 I 111111 0.001 0.01 0.1 K1 10 100 1000
Fig. 4.2-5 Horizontal force coefficients for the HSRC.02 cylinder (Run 55). 74 measured C uV(K) data points, and measured nondimensionalized phases n n for the VSRC.02 cylinder are shown in Fig. 4.2-6.
4.2.2Field Test Results
The field data studied here were from the NMI and WPII tests.
The results of these tests are shown in sections 4.2.2.1 and 4.2.2.2, respectively.
4.2.2.1 NMI Second Christchurch Bay Tower
The wave profiles, velocities, and force time histories of the
NMI tests were not available. Fortunately, NMI report-177 (Bishop,
1984a) provided figures for the velocity power spectra Suu(z,fn) and force power spectra SFF(z,fn). Values for SFF(z,fn) and Suu(z,fn) were digitized from the figures. The force amplitude Fn(z,fn) can be obtained by rearranging Eq. (2.3-3) as
Fn(z (4.2-1) 'fn) SFF(z,fn) A f
Similarly, the velocity amplitude un(z,fn) can be calculated from
u(z,f ) S (z,f) A f . (4.2-2) n n uu n
The velocity power spectra calculated from the perforated ball measurements were used and in this thesis it was assumed that the measured velocities were pure wave kinematics.
Record 71 level-3 is an example. The Cun(Kn) hyperbola, curves of predicted force coefficients, curve of non-dimensionalized phase, and C data points for the NMI large and small columns are shown in Lin
Figs. 4.2-7 and 4.2-8. VERTICAL FORCE 10000
CYLINDERHSRC.02 RUN NO. RPM R55 SPECTRUM SCOTT 1000 FREQ. BAND 0.10- 1.00 Hz CUT-OFF AMP. 0.03 in MEASURED CA. rms 0.1570 Curl 100 n Cdl. . Cc' CmCmL 10 // // / // / // c/ 1
. I
II A _._ 0.1 I 1 11111 ...... J..... I.11111111' I 11 11111111 0.001 0.01 1 10 100 1000 Kn
Fig. 4.2-6Vertical force coefficients for the HSRC.02 cylinder (Run 55). IN-LINE FORCE 10000
NMI SCBT \\ RECORD 71 % \ LARGE COLUMN LEVEL-9 .. 1000r . \ FREQ. BAND 0.10- 0.50 Hz \ CUT-OFF AMP. 0.05 . \ s MEASURED C ja, . V rm. 0.01196 - ., CA, . 100 . , ..r. . . \ . \ CdL
- . \ . \ C,L .. \ Cl'. 10 7- . : ..N , \ .... \
...... , _...... ___\7 ...... " t - '. N '...... --...... "'''
g lid I a 0.1 a eat a wail tIII fill' I a at_usil a III.4s\.',.. 0.001 0.01 0.1 1 10 100 1000 K n
Fig. 4.2-7 In-line force coefficients for the NMI large cylinder level-3 (Record 71). IN-LINE FORCE 10000
NMI SCBT RECORD 71 SMALL COLUMN LEVEL-9 1000 FREQ. BAND 0.10 0.50 Hz CUTOFF Mr. - 0.05 a MEASURED C un rms se 0.0519 C A n 100 in CdL Cd Cmt. Cm. 10
1
0 . 1 i i i I,,,,,,1 i , ...".1 ii Mil 1 I L11111LJLJ1...a... 0.001 0.01 0.1 1 10 100 1000 Kn
Fig. 4.2-8 In-line force coefficients for the NMI small cylinder level-3 (Record 71). 78
4.2.2.2 Wave Force Project II Tests
WPII had only one wave staff (Blank, 1969) andno velocity
measurements. The velocities predicted from thewave profile were
treated as un(Kn). Those velocities were assumed to be in thesame
direction as the vector sum of the orthogonal forcemeasurements. The C (Kn) hyperbola, curves of predicted forcecoefficients, curve
of nondimensionalized phase, the Con(Kn) datapoints, and non-
dimensionalized measured phases of position-7 duringHurricane Carla
(record 6887) are shown in Fig. 4.2-9. The Cpn data points and
measured non-dimensionalized phasesare quite scattered.
4.3 The Prediction of Force Power Spectra
The results of force power spectra predicted bypseudo and true
predictions will be shown in the following sections. Both pseudo and
true predictions use Eqs. (2.3-2) and (2.3-3) to predict theforce
power spectra.For pseudo prediction the maximum force coefficient H vs. Kn hyperbolas [e.g. C (K), CV(K ) or C (K) hyperbolas] are pnn pn n pn n obtained from measured wave profiles (velocities)and measured forces
by using the method described in section 4.1. Only the in-line force
power spectrum for a smooth vertical cylinder cannow be predicted by
true prediction (when Kn is small). Equation (2.2-11) was used in
true prediction.
The NMI spectra were previously smoothed bya 14-point box car
moving average method. For OSU and WPII tests, a 9-point boxcar
moving average method was used to smooth both themeasured and the predicted raw spectra (Borgman, 1972b). IN-LINE FORCE 10000
CYLINDER NFII RUN.NO. 6887 F0S7 FREQ. BAND 0.05-0.50 Hz 1000 CUT-OFF AMP. 1E0.05 MEASURED C ras 1.1 0.1507 Cut,
100 N CdL Cd \) Cog. CM. 10 N N/ N 1
N .1':1",fil l K log K m: X K
0.1 I tI u u$1, I It till 0.001 0.01 0.1 1 10 100 1000
Fig. 4.2-9 In-line force coefficients and phases for the WPII cylinder position-7 (Record 6887). 80
4.3.1 Evaluation Criteria for Predicted Force Power Spectra
The non-dimensionalized root-mean-square-error E r of spectral amplitudes are used to evaluate the goodness of the pseudo prediction
and true prediction. It is defined as
N 1 (z,f ) )] - [S (z,f ) ]12 FF nm FF np " n=1 E = rms (4.3-1) [SFF(z'fn)m]ti
where SFF(z,fn)m is the measured spectral amplitude, SFF(z,fn)p is
the predicted spectral amplitude, N is the number of frequencycompo-
nents involved, and (SFF(z,fn)mill is the spectral peak of measured
spectrum.
4.3.2Laboratory Tests
In-line, vertical, and horizontal force power spectra predicted
by the pseudo prediction for cylinders of various roughness will be
described first. The pseudo predictions of transverse force power
spectra are poor and they are not shown here. Then the true pre-
dicted in-line force power spectra for vertical smooth cylinders will
be described.
Pseudo predictions of the in-line force raw spectra and smoothed
spectra of the VSMC12 and VSRC.02 cylinders are shown in Figs. 4.3-1
through 4.3-4. The predicted smooth cylinder in-line forcespower spectrum is in good agreement with the measured spectrum. The smoothed spectrum can be predicted even more precisely.
The raw and smoothed spectra for horizontal and vertical forces acting on HSRC.02 cylinders are shown in Figs. 4.3-5 through 4.3-8. IN-LINE FORCE POWER SPECTRA (RAW) 0
CYLINDER VSHC 12 MEASURED 0 RUN NO. NCEL R32 NEW METHOD 0 SPECTRUMJONSWRP (PSEUDO PREDICTION) 0 CO NEW METHOD Erm = 0.0259 -0 0 NE 0 11
O O CD
0O o Vs\e,ALL, 1 1 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 FREQUENCY (Hz)
Fig. 4.3-1 Raw inline force power spectra for the VSMC12 cylinder (pseudo prediction). 0 0 IN-LINE FORCE POWER SPECTRA (SMOOTHED) C3 CU CYLINDERVSMC 12 MEASURED 0 RUN NO. NCEL R52 NEW METHOD 0 SPECTRUM JONSWAP (PSEUDO PREDICTION) 0 CO NEW METHOD Errs. = 0.0186 .-.1
O O 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 FREQUENCY (Hz)
Fig. 4.3-2Smoothed in-line force power spectra for the VSMC12 cylinder (pseudo prediction). 1N-LINE FORCE POWER SPECTRA (RAW)
CYLINDER VSRC.02 MEASURED RUN NO. API84 R23 NEW METHOD 0 SPECTRUM SCOTT (PSEUDO PREDICTION) 0 NEW METHOD Erse - 0.0317
0 0
4. IL. cr)O
0 O j. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 FREQUENCY (Hz)
Fig. 4.3-3Raw inline force power spectra for the VSRC.02 cylinder (pseudo prediction). 0 IN-LINE FORCE POWER SPECTRA (SMOOTHED)
In CYLINDERVSRC.02 MEASURED RUN NO. AP1811 R23 NEW METHOD 0 SPECTRUMSCOTT (PSEUDO PREDICTION) 0 NEW METHOD Erma I.0.0180
0.00 0.10 0.20 0.30 0.110 0.50 0.60 0.70 FREQUENCY (Hz)
Fig. 4.3-4 Smoothed in-line force power spectra for the VSRC.02 cylinder (pseudo prediction). CI 0 HORIZONTAL FORCE POWER SPECTRA (RAW) 0 04 CYLINDERHSRC.02 MEASURED 0 RUN NO. SPUN R55 NEW METHOD 0 SPECTRUM SCOTT (PSEUDO PREDICTION/
CO NEW METHOD Eros . 0.0412 ..-41
..- ./.'_. - ,
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 FREQUENCY (Hz)
Fig. 4.3-5Raw horizontal force power spectra for the HSRC.02 cylinder (pseudo prediction). 0 HORIZONTAL FORCE POWER SPECTRA (SMOOTHED) CI CU CYLINDERHSRC.02 MEASURED 0 RUN NO. API84 R55 NEW METHOD CI SPECTRUM SCOTT IPSEUDO PREDICTION) NEW METHOD Erma as CO 0.0276 ...
.... C.:4 -40 ME 0 CNI .....
Ct CO 0 CO
0
0
0.00 0.10 0.20 0.90 0.40 0.50 0.60 0.70 FREQUENCY (Hz)
Fig. 4.3-6Smoothed horizontal force power spectra for the HSRC.02 cylinder (pseudo prediction). VERTICAL FORCE POWER SPECTRA MAW)
CYLINDER HSRC.02 MEASURED RUN NO. RPM R55 NEW METHOD SPECTRUM SCOTT (PSEUDO PREDICTION) NEW METHOD Eros m 0.1040 ""
".4O
ME
Iv)
aO Co CU
cn O° II g Il I
A
O of I I 0.00 0.10 0.20 0.90 0.40 0.50 0.60 0.70 FREQUENCY (Hz)
Fig. 4.3-7 Raw vertical force power spectra for the HSRC.02 cylinder (pseudo prediction). VERTICAL FORCE POWER SPECTRA (SMOOTHED) C3 U) CYLINDER HSRC.02 MEASURED RUN NO. API84 RSS NEW METHOD 0 SPECTRUM SCOTT (PSEUDO PREDICTION) 0 NEW METHOD Eros 1.0.0681 40
0 0 Al
0 .-.0
0 C3
I 1 r I I 0.00 0.10 0.20 0.30 0.40 o.s0 0.60 0.70 FREQUENCY (Hz)
Fig. 4.3-8Smoothed vertical force power spectra for the HSRC.02 cylinder (pseudo prediction). 89
Predicted horizontal force power spectra were in good agreement with
measured spectra. The predicted vertical force power spectra were in
poor agreement, however, especially in the low and high frequency
region (Fig.4.3-8).
The true prediction of in-line force power spectra and smoothed
spectra for the VSMC12 cylinder are shown in Figs. 4.3-9 and 4.3-10,
respectively. Equation (2.2-11) was used in this true prediction.
The predicted and measured spectra are in good agreement.
4.3.3 Field Tests
The field data used here are from the NMI and WPII measurements.
Only in-line force power spectra are shown here.
4.3.3.1 NMI Second Christchurch Bay Tower
Both pseudo and true predictions for the NMI cylinders are shown. The pseudo prediction are shown first, then the true predictions.
Pseudo predictions of the in-line force power spectra for small and large cylinders in record 71 are shown in Figs. 4.3-11 and
4.3-12. Measured and predicted spectra are shown in the same plot.
The spectra that Bishop(1984b) predicted by using Borgman's method are also plotted for comparison. Bishop applied long term averaged force coefficients to predict the force power spectra for a short record. The method may be considered as midway between pseudo and true prediction. Measured velocities were used in all of the pseudo predictions. For a large cylinder, the power spectra predicted by the new method are in better agreement with theory than those O C3 IN-LINE FORCE POWER SPECTRA (RAW)
CYLINDERVSMC 12 MEASURED RUN NO. NCEL R32 NEW METHOD O SPECTRUM JONSWRP (TRUE PREDICTION)
co NEW METHOD Erma m 0.0287
OCI
CV
O O CO
u-) OO
fj O ca I I I 0.00 0.10 0.20 0.30 0.110 0.50 0.60 0.70 FREQUENCY (Hz)
Fig. 4.3-9 Raw inline force power spectra for the VSMC12 cylinder (true prediction). 0 CI IN-LINE FORCE POWER SPECTRA (SMOOTHED) 0CI CU CYLINDER VSMC 12 MEASURED 0 RUN HO. NCEL R32 NEW METHOD 0 SPECTRUMJONSWRP (TRUE PREDICTION) 0 Co NEW METHOD Erms = 0.0912 .....
0 0 N ....
0 0 CO
aO 0 0.00 0.10 0.20 0.90 0.40 0.50 0.60 0.70 FREQUENCY (Hz)
Fig. 4.3-10 Smoothed in-line force power spectra for the VSMC12 cylinder (true prediction). 0 0 IN-LINE FORCE POWER PSPECTRA 0 CU NMI SCBT 0 RECORD 71 NEW METHOD Erma 0.0576 0 LARGE COLUMN LEVEL-3 BORGMAN METHOD Erma 0.0619 0 CO MEASURED .-. NEW METHOD (PSEUDO PREDICTION) BORGMAN METHOD 0 N0 ....
0 0 CO
0
I 0.00 0.05 0.10 0.15 0.20 0.25 0.90 0.35 FREQUENCY (Hz) Fig. 4.3-11 Inline force power spectra for the NMI large cylinder level-3, record-71 (pseudo prediction). 0 IN-LINE FORCE POWER SPECTRA CNI 0 NMI SCBT RECORD 71 NEW METHOD Erma) = 0.0855 SMALL COLUMN LEVEL-9 BORGMAN METHODErma = 0.0885 MEASURED NEW METHOD (PSEUDO PREDICTION) BORGMAN METHOD
0CO 0 ..._
. c; r I 1 1 r 1 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 FREQUENCY (Hz) Fig. 4.3-12 In-line force power spectra for the NMI small cylinderlevel-3, record-71 (pseudo prediction). 94
obtained by using Borgman's method. For small cylinder, the new
method was more precise than Borgman's method in the high frequency
regions, but Borgman's method was in better agreement with measured
spectra in the low frequency regions.
The true predictions of in-line force power spectra for both the
small and the large cylinders in record 71 are shown in Figs. 4.3-13
and 4.3-14. Measured velocities were used in true prediction. The
true predictions are not as good as those obtained by using Borgman's method.
4.3.3.2Wave Force Project II Tests
Both pseudo and true predictions for the WPII cylinder are shown here. The pseudo prediction will be shown first, then the true predictions.
Predicted raw and smoothed spectra, predicted by the pseudo prediction, for the WPII cylinder (record-6887) at position-7 are
shown in Figs. 4.3-15 and 4.3-16. These predictions did not agree well with measured values. One reason may be that the velocities used to predict force power spectra were predicted from the wave record, and the effects of current were ignored.
The true predicted raw and smooth spectra are shown in Figs.
4.3-17 and 4.3-18. By comparing the Erms values, it can be found that the true predictions are better than the pseudo predictions. IN-LINE FORCE POWER PSPECTRR
NMI SCBT 0 RECORD 71 NEW METHOD Erma 22 0.0963 0 LARGE COLUMN LEVEL-9 BORGMAN METHODErma i 0.0619 0 MEASURED CO .-6 NEW METHOD (TRUE PREDICTION/ BORGMAN METHOD 0O 0
O O
O O I I 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 FREQUENCY (Hz)
Fig. 4.3-13 In-line force power spectra for the NMI large cylinder level-3, record-71 (true prediction). 0 IN-LINE FORCE POWER SPECTRA CM CI NMI SCBT RECORD 71 NEW METHOD Erma m 0.1025 SHALL COLUMN LEVEL-9 BORGMAN METHODErma m 0.0885 MEASURED NEW METHOD (TRUE PREDICTION) BORGMAN METHOD
..-CV =N Z 2W Y0CO U. - U- (1)
0 CI
0 CI
i 1 I 1 1 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 FREQUENCY (Hz)
Fig. 4.3-14 In-line force power spectra for the NMI small cylinder level-3, record-71 (true prediction). 0 0 IN-LINE FORCE POWER SPECTRA (RAW) O
CYLINDER WPII MEASURED 0 RUN.NO. 6887 P037 NEW METHOD 0 (PSEUDO PREDICTION) 0 Ctl NEW METHOD Erma 0.0613 Cr)
cr) c ..0 NI 0 CU
1 . . It , t, _Afii ... I 1 r 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 FREQUENCY (Hz) Fig. 4.3-15 Raw in-line force power spectra for the WPII cylinder position-7, record-6887 (pseudo prediction). 0 0 INLINE FORCE POWER SPECTRA (SMOOTHED) 0
CYLINDER WPII MEASURED RUN.NO. 6887 POS7 NEW METHOD 0 (PSEUDO PREDICTION) N0 NEW METHOD Erma 0.0867 V) 01O 0 NI 0 =4_ Al
0 O CD- ..-.
.11
0 0 0.00 0.10 0.15 0.20 0.25 0.35 FREQUENCY (Hz) Fig. 4.3-16 Smoothed in-line power spectra for the WPII cylinder, position-7, record-6887 (pseudo prediction). 0 . IN-LINE FORCE POWER SPECTRA (RAW) 0
CYLINDERWPII MEASURED 0 RUN.NO. 6887 POS7 NEW METHOD C3 tTRUE PREDICTION) N NEW METHOD Erma - 0.0552 Cr)
t
%
0 C3 I
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 FREQUENCY (Hz) Fig. 4.3-17 Raw in-line force power spectra for the WPII cylinder, position-7, record-6887 (true prediction). 0 INLINE FORCE POWER SPECTRA (SMOOTHED) 0 =I CYLINDER WPII MEASURED 0 RUN.NO. 6887 POS7 NEW METHOD 0 (TRUE PREDICTION1 0 NEW METHOD Erms Am cr) O0 Cu O O A si ..1 .1/ I k O O O 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 FREQUENCY (Hz) Fig. 4.3-18 Smoothed in-line force power spectra for the WPII cylinder, position-7, record-6887 (true prediction). 101 4.4The Prediction of Force Time Histories Two true prediction examples will be shown here. The force time history for the VSMC12 cylinder (laboratory test) will be shown first, then the force time history for the WPII cylinder (field test) during Hurricane Carla. In-line forces for the VSMC12 cylinder were predicted by the true prediction. The measured wave profile is shown in Fig. 4.4-1. The wave amplitude and phase spectrum are obtained by using FFT. The velocity phase spectrum is the same as the wave phase spectrum. The velocity amplitude spectrum is obtained from the wave amplitude spec- trum by using linear wave theory.Equation (2.2-11) then was used to estimate Fn(z,fn). Phase shifts between the velocity phase spectrum and the force phase spectrum .n(z,fn) were set equal to -90° in this prediction. The inverse FFT is used to obtain the in-line force time history. The predicted force time history and the measured force time history are shown in Fig. 4.4-2.They are in good agreement. In-line forces for the WPII cylinder record-6887 at position 7 were predicted. The measured wave profile is shown in Fig. 4.4-3. The predicted force time history and the measured force time history are shown in Fig. 4.4-4.Most of the peak positions were shifted. However, the shape of the predicted force time history compares reasonably well with measurements. If the directional spectrum is considered the prediction may be improved. For design, one usually develops a design wave spectrum, or it is specified. The phases of the frequency components can be deter- mined from a random number generator and by assuming that the phases 4.00 4....--....-.--4 VSMC12 cylinder 3.n0-.- 2.00 -a- 1.00 - ... 4- A ---- .00 ...... 1.00 r f 1 4 2.00 f 1 r i 1 .00 1.29 24.58 36.86 49.15 61.' TIME( SEC) Fig. 4.4-1 The wave profile for the VSMC12 cylinder test. 2 5.00 I MEASURED PREDICTED =Mb OwlsIlm MEP IMO 15.00 ....-. V) 1:) --... 5.00-r - 5.00-* -r 15.00 25.00 i I i I 4 4 I 4 4 .00 12.29 24.58 36.86 49.15 61.' TIME( SEC) Fig. 4.4-2 The measured and predicted in-line force time history for the VSMC12 cylinder (true prediction). 35.00 WPII record-6887 .4. 25.00 4. H = 38 ft T= 13 sec. 11 15.00 .ID. 4.3 4.- 4. 5.00 4. 5.00 15.00 I i 4 I I 4 1 i I .00 32.77 65.54 98.30 13/.07 163.E TIME(SEC) Fig. 4.4-3 The wave profile record during Hurricane Carla (record-6887). 400.00 .P. MEASURED PREDICTED 300.00 4. (,) 200.00 ..0 A. I- "I- -Ir .1. 1 100.00I/ vs 200.00 I I I ; I I I I 1 .00 32.77 65.54 98.30 131.07 163.E TIME(SEC) Fig. 4.4-4The measured and predicted in-line force time hostory for the WPII cylinder at position-7 (true prediction). 106 are uniformly distributed between 0 and 2w. Thus, the amplitude and phase spectra of the forces can be determined with linear wave the- ory, Eq. (2.2-11), and the assumption at all the Sn s-90° (-w/2). Thence the time domain realization of the force can be determined with inverse FFT. 107 5.0 CONCLUSIONS 1. The in-line force power spectrum for smooth vertical cylinders can be estimated with surprising accuracy by using only the inertia term in the Morison equation. 2. By adding another assumption that the phase lag between the velocity spectrum and the force spectrum is -90°, the force time history can be predicted with surprising accuracy. 3. The new method is most suitable for preliminary design, because there is no need to worry about how to choose the usual force coefficients. 4. The new frequency domain Keulegan-Carpenter number was defined as K n(z,fn) = un(z,fn)/(fnx D), which turned out to be smaller than 4 even during Hurricane Carla in the Wave Force Project II measurements. 5. In small Kn(z,fn) regions, maximum force coefficients closely 2 follow the relation C (z,f) = 2.1r/K(z,f) for a smooth pn n n n cylinder. 6. For small Kn, the phase lag between un(z,fn) and Fn(z,fn) approaches -90° for smooth cylinders. 7. When the force power spectra pseudo prediction method was applied to forces on horizontal cylinders, the results were in good agreement with measured values. Smooth cylinder pre- dictions were more precise than the predictions for rough cylinders. 108 6.0 REFERENCES Bendat, J.S., and Piersol, A.G. (1971), Random Data: Analysis and Measurement Procedures, John Wiley & Sons, Inc. Bearman, P.W., Downie, M.J., Graham, M.R., and Obasaju, E.D. (1985), "Forces on Cylinders in Viscous Oscillatory Flow at Low Keulegan-Carpenter Number," J. Fluid Mech., Vol. 154, pp. 337-356. Bishop, J.R. (1984a), "Wave Force Investigations at The Second Christchurch Bay Tower," National Maritime Institute, Report 177. Bishop, J.R. (1984b), "A Spectral Analysis of Wave Forces and Particle Kinematics from The Second Christchurch Bay Tower,"National Maritime Institute, Report 179. Bishop, J.R. (1985), "Wave Force Data from The Second Christchurch Bay Tower,"OTC, Paper 4953. Blank, L.S. (1969), "Wave Project II User's Guide,"Chevron Oil Field Research Company. Borgman, L.E. (1965), "Wave Force on Piling for Narrow-Band Spectra," ASCE, Vol. 91, No. WW3, pp. 65-90. Borgman, L.E. (1967a), "Spectrum Analysis of Ocean Wave Force on Piling," ASCE, Vol. 93, No. WW2, pp. 129-156. Borgman, L.E. (1967b), "Random Hydrodynamic Force on Objects," Ann. Math Statist., Vol. 38, pp. 37-51. Borgman, L.E. (1969), "Ocean Wave Simulation for Engineering Design," ASCE, Vol. 95, No. WW4, pp. 557-583. Borgman, L.E. (1972a), "Statistical Models for Ocean Wave Forces," Advance in Hydroscience, Vol. 8, pp. 139-181. Borgman, L.E. (1972b), "Confidence Intervals for Ocean Wave Spectra," Proceedings, 13th Coastal Eng., Vol. 1, Chapter 10, pp. 237-250. Bretschneider, C.L. (1959), "Wave Variability and Wave Spectra for Wind-Generated Gravity Waves," U.S. Army Corps of Engineers, Beach Erosion Board, Tech. Memo. No. 118. Bretschneider, C.L. (1965), "On the Probability Distribution of Wave Force and An Introduction Drag Coefficient and Correlation Inertial Coefficient," Proc., 10th Coastal Eng., Chapter 9, Vol. 1, pp. 183-217. 109 Bretschneider, C.L. (1967), "Probability Distribution of Wave Force," ASCE, Vol. 93, No. WW, pp. 5-26 Brown, L.J., and Borgman, L.E. (1967), "Tables of the Statistical Distribution of Ocean Wave Forces and Methods of Estimating Drag and Mass Coefficients," Coastral Eng. Res. Center Tech. Memo No. 24. U.S. Army Corps of Engineers. Chakrabarti, S.K. (1985), "Recent Advances in High-Frequency Wave Forces on Fixed Structures," ASME, Journal of Energy Resource Technology, Vol. 107, pp.315-328. Dean, R.G. (1965), "Stream Function Representation of Non-Linear Ocean Waves," J. Geophys. Res., Vol. 70, No. 18, pp. 4561-4572. Dean, R.G, and Aagaard, P.M. (1970), "Wave Forces: Data Analysis and Engineering Calculation Method," Jour. Petrol. Tech., March, pp. 368-375. Dean, R.G., and Dalrymple, R.A. (1984), Water Wave Mechanics for Engineers and Scientists, Prentice-Hall, Inc., Englewood Cliffs, NJ. Evans, D.J. (1970), "Analysis of Wave Force Data," Jour. Petrol. Tech. Hasselmann, K., et al. (1973), "Measurements of Wind Wave Growth and Deacy During the Joint North Sea Wave Project, Deut. Hydrogr. Z. Reihe A, No. 12. Heideman, J.C., Olson, 0.A., and Johansson, P.I. (1979), "Local Wave Force Coefficients," Civil Engineering in the Ocean IV, ASCE Proceedings of Specialty Conference, San Franciso, CA. Hsu, M.K., and Nath, J.H. (1986), "Random Waves on Vertical Cylin- ders," Ocean Structural Dynamic Symposium'86, OSU. Hudspeth, R.T., Dalrymple, R.A., and Dean, R.G. (1974), "Comparison of Wave Forces Computed by Linear and Stream Function Methods," OTC, Paper 2037. Hudspeth, R.T., Jones, D.J, and Nath, J.H. (1984), "High Reynolds Number Wave Force Investigation in A Wave Flume, Vol. I," Final Report to NCEL and MMS., OSU. Hudspeth, R.T., Nath, J.H., and Khare, P. (1986), "Wave Phase/ Amplitude Effects on Force Coefficients," ASCE, (to be published). Ippen, A.T. (1966), Estuary and Coastline Hydrodynamics, McGraw-Hill Book Company, Inc. 110 Keulegan, G.H., and Carpenter, L.H. (1958), "Forces on Cylinders and Plates in An Oscillating Fluid," Journal of Research of National Bureau Standards, Vol. 160, No. 5, pp. 423-440. Morison, J.R., O'Brien, M.P., Johnson, J.W., and Schaaf, S.A. (1950) "The Force Exerted by Surface Waves on Piles," J. Pertol. Tech., Am. Inst. Ming Engng., Vol. 189, pp. 149-157. Nath, J.H., and Harleman, D.F. (1970), "Response of Vertical Cylinder to Random Waves," ASCE, WW 2, pp. 373-386. Nath, J.H., and Hudspeth, R.T. (1984a), "High Reynolds Number Wave Force Investigation in A Wave Flume, Vol.1," Final Report to NCEL, OSU. Nath, J.H., Hsu, M.K., Hudspeth, R.T., and Dummer, J. (1984b), "Laboratory Wave Force on Vertical Cylinders," Ocean Structural Dynamic Symposium'84, OSU, pp. 312-330. Nath, J.H. (1985a), "Wave Force-Phase Method for Vertical Cylinders," Final Report to NCEL, OSU. Nath, J.H. (1985b), "Hydrodynamic Coefficients for Marine Roughened Cylinders," Final Report to The American Petroleum Institute, OSU. Nath, J.H., and Yeh, R.C. (1986), "Breaking Waves - Their Influence on Wave Spectra," Progress Report to The Office of Naval Research, OSU. Nath, J.H. (1986), "Wave Force Coefficient Variability for Cylin- ders," Ocean Structural Dynamic Symposium'86, OSU, in press. Reid, R.O. (1958), "Correlation of Water Level Variation with Wave Forces on a Vertical Pile for Nonperiodic Waves," Proc. 6th Coastal Eng., Chapter 46, pp. 749-786. Rodenbusch, G., and Gutierrez, C.A. (1983), "Forces on Cylinders in Two-Dimensional Flows," Technical Progress Report Vol. 1, BRC 13-83, Bellaire Research Center (Shell Development Company), Houston, TX. Sarpkaya, T, (1976), "Vortex Shedding and Resistance in Harmonic Flow about Smooth and Rough Cylinders at High Reynolds Numbers," SL76021, U.S. Naval Post Graduate School, Monterey, CA. Sarpkaya, T., and Isaacson, M. (1981), Mechanics of Wave Forces on Offshore Structures, Van Nostrand Reinhold Company. Sarpkaya, T. (1986a), "Force on a Circular Cylinder in Viscous Oscil- latory Flow at Low Keulegan-Carpenter Numbers," J.Fluid Mech., Vol. 165, pp. 61-71. 111 Sarpkaya, T. (1986b), "In-line and Transverse Forces in Smooth and Rough Cylinders in Oscillatory Flow at High Reynolds Numbers," Naval Postgraduate School, NPS69-86-003 Scott, J.R. (1965), "A Sea Spectrum for Model Tests and Long-Term Ship Prediction," J. Ship Res., Vol. 9, pp. 145-152. Shields, D.R., and Hudspeth, R.T. (1985), "Hydrodynamic Force Model Research for Small Semisumersibles," NCEL tm no: M-44-85-06 Starmor, N. (1981), "Consistent Drag and Added Mass Coefficients from Full Scale Data," OTC, Paper 3990. Skjelbreia, L. (1959), "The Clamp-on Wave Force Meter", Research Report 611, California Research Corporation. Thrasher, L.W., and Aagaard, P.M., (1969), "Measured Wave Force Data on Offshore Platforms," OTC, Paper1006, Vol. I, pp. 71-82. Wheeler, J.D. (1969), "Method for Calculating Force Produced by Irregular Waves," OTC, Paper 1007, Vol. I, pp. 83-94. Wiegel, R.L., Beebe, K.E., and Moon, J. (1957), "Ocean Wave Forces on Circular Cylindrical Piles," ASCE, J. Hyd. Div., Vol. 83, pp. 1199-1 1119-36. APPENDICES 112 APPENDIX A CONSTRUCTING THE C(K) HYPERBOLA u The method of constructing a Cu(K) curve described here is valid for both periodic waves and decomposed random waves. In order to simplify notations, the log(K) and logCu(K) axes are replaced by X and Y axes. Procedures for constructing the hyperbola are shown below. The hyperbola with respect to X and Y axes (Fig. A-1) can be expressed as 2 -2 X Y (A-1) 2 a2 b 1 where a and b are transverse axes. The value of a is set equal to 0.1 in this thesis. The relationship between a and b can be expressed as a = tan (A-2) where d is the angle between asymptote L1 and the Y axis.The angles and 8 can be determined from asymptotes L1 and L2 (Fig. A-1). The asymptotes L1 and L2 are the same as those described in section 4.1. Relationships between the X - axes and the X' - Y' axes can be expressed as X = X'cosO + Y'sine (A-3) and Y = -X'sine + Yisine . (A-4) 113 Y Y elm Fig. A-1 The Cu hyperbola and thecoordinate systems. 114 Relationships between the X' - Y' axes and the X - Y axes can be expressed as X' = X - Xi (A-5) and Y' =Y - Y (A-6) i where (Xi, Yi) is the intersection of the two asymptotes L1 and L2. By substituting Eqs. (A-5) and (A-6) into Eqs. (A-3) and (A-4), then substituting Eqs. (A-3) and (A-4) back into Eq. (A-1), we can obtain a hyperbola with respect to the X - Y axes. The ordinates of the hyperbola are obtained by using a digital computer. 115 APPENDIX B THE DERIVATIVE OF C (K) WITH RESPECT TO a The C can be expressed as C2 C =C + a (B-1) u d. C d. where a = w4/4K2. The derivative of C(K) with respect to a can be u expressed as d Cu d Cm rc d a d a , d. 4 Cd, 2 dC C 2 d r_24.1 da d a Cd, Cd, =P + Q + m . (B-2) By substituting a = w4/4K2 into Eq. (B-2), after rearrangement Eq. (B-2) can be rewritten as d C sispl-q+ m (B_3)12 d a where 2K3 [1 -sin2(0)]d Cd. (B-4) P d K 2 and d Ciro 3 2K q (B-5) d K 2 w 12Note that P + Q = p + q, but P # p and Q * q. 116 The range of K is roughly divided into four K regions : (1) K 4 10, (2) 10 < K < 40, (3) 40 < K < 100, (4) 100 < K. (1) K < 10 In this region, the phase, 6, approaches -90°, Cd, = w2/2K and Cm0approaches 2, so d C 2 lim = (B-6) d K 2 k+0 K and d C lim =0 d K (B-7) k+0 Substituting Eqs. (B-6) and (B-7) into Eqs. (B-4) and (B-5), we find that when K approach 0, p = 0, q = 0, and (p+q)/m = 0. (2) 10 < K < 40 For 10 < K < 40, both inertia force and drag force are impor- tant, but no theoretical conclusions about p and q can be made. Experimental results are therefore used to evaluate the value of p + q. Unfortunately, no periodic wave results are available in this region of K with large 8. The only data available with large K and R are Sarpkaya's U tube results (Sarpkaya, 1976b). His tests are not as close to reality as real wave tests, but they can give us a rough idea about the value of p+q/m. Sarpkaya used the Fourier analysis method to estimate CdF and CmF Figs. B-1 and B-2. In order to maintain consistency, his ' results was converted into C and C by using the formulas in dO m0 Appendix C. The converted Cd, and Cm0 were then plotted on log-log thi :1 .11 1 11,1 " IMEEMBEIBINE EIRIMMINMEllth 111M611131111MEMEMEMBEHME111111 , wpm i-r10t NEM t -:iii .+T Pit11111 MAINE 3'1/MIMI INIRMRffiffli 111111fflmsallill Jig MIMI WWII= WIN MEE MEE ,AilElltiiiiillMilinffitE Emma=- 4' 111n IPiff MEN ONIIMENEMENNEIMMEMBIEMBIN Effl 1111 hit:BRIT: F;ii : MEMEMIE1110 ti I 11 ENHILIEMEENW! mom firdEgagfililiglliffg MOE pitzfil ;Ir. ffliiiimirmIgiffilfflillillillileffiffillEM NIEMEN EMIMillINEIMPLEAMEMEIN r1 :1 11 118 3.0 : IL: 1-r7-.Erj-17 ii-r. 11 t-rli C :I. ;i:r.::ri-.."...P. --fill111.1'Lir.: 2.5 !:::.:.:,...!:-::t . . . :. 1 4:: tt:: !hi11;). lifi I i. ;illti:114i Pi 1 ii-::,,i, tiit11: :. r ki t::: t:i! 1,1.14.4 rirti. ;4!Iii,ii I. f.i.!:::: :::: i!.: u, .1, t::::,.;:::....:_:1:: ilt 2.0 ..tti ::;" .:.;Ltt..:::: ,....t: [i4'''' ''''-1.': `-=';-::iiii t":4 iiii .-.U--..i.";-.:'Cr:r;!:44, iii, Iti; ,::; iii ''''' '' '' i ii:;;.::1": =JP.;II:Pt,,.:.:, .,.-...:ii.1;i-; LE:: .r,.". ,.. ,, 4_-i-.-r..,t-rti';'..4.,,'illt;'. r::.:11:: 0...1h'i: 1. -..., tl,, 1,1 1- 1-.1,111-r, t,,:tit ft.'ir .t t*.,,h.!,ut,...;::::::::m .,...,..tg1 4.,[:.1.4, t ,1:!;:tt: tt:1:l'.'r.:11t-',,.! 1.5 u:', :111 -rv 11-41.:4 1.0.1.11fli111111:7411'.'11:!:,':11;::::::::::WI .,-:-..,--f --.r,: 0 ,r,.,I.,:Int...... ::,; t.T.:.::7,-1 , ,4,41,, :t,,,.,,,.r.t1.Ili? 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UT/0 Ji: 1i: fi.::::---:F:IF11:::: tif..'-:7"--.1;--775.11 Itli4lfs: iiir.1 dr: ffilq7.....*:. ,.:',tt .7E-r-.--s-t r -r= - .-..--..-: .:; .:: IT? rTiii r ...... -c:.7 i=:::,--.....i..1.1. _-: :.-::::-:::: 1-- =----1-*T_-:-.:-- :..t ::: :::::-.. . 1r _. _-1,E-7.1FF-ct:-*=7-,-- - :::: 2.0 i.V:1... -1+::1L: :=1:::; -,t.-- :::r:..- . T 127-7.:7--ii-ft-7;r1,W7-:ii 51F:i'!, s ,...17:;?'.71.;&Ait41...7 -----r-: ; . ic;r215!!' '14Ai P--r---- 14t;eali....:.:tsi:::::i'.-.3 ;.... - 4 .-',"" :lii- -."', . .4'I-t'..44 "--1...-; i-,-.' 1. - ;;;.tv.;: ",, ;::: ....::,..fp r::,_;:! 41 7.7:::: rr.-..::'1',....p,i,:W.,...,,',..,- r:.:1 ..',..11!.-1--,-,.....'7.--'-'4-.:'..11.1rr,.r,v.r4 ,,,.,:i..1r:4::14. ' ..Lir.i.11,1i...tet4...tly,;Ir. --;.- r::: t--,-i-qtr;,..,,-;i--,t.:1,115ttt,iiiirrt444.+ .r --r-,. 4.t F...:::. *I 'r it g,;,', ."'"t1-.1 X IT.I. '44:"::It- r: i.r,r-174 ;1 i',,;' . ,;,,cri Wit.11111'ilia. 5260 ir r:".. :1;; ,,,ITT-li: 44-,....ii. ' i :r5,, $. jii4 -r,;,, -4. 'rri- 1. ttr.-'71,t tri41t4..:fti;4 I4 i!-'r-, t44.i+ 0. 0 '6-475"..!.11i.i.4 -...-p.,:p i F.'. Etcrfii 1'7:'-i:1Nil:rf..1::t7.;: 0 1 17.77E:f1.1:! : g4:: F.L. rj 0 fitf'.4--i:. -i! i li,F,i,-.4.`i.::::;;..;.: : I.:iili:-.1.= '..-- Tr;::i i.:.; - 0. tt..n:. ,"- f-7-ilii.:- iiillF1-1:,..41i -F?1,11:19.i.:"':' .1:::::'' %.;.1:1If.'::7.1.. ..' :-...,',': 0. 1-.':11.11:1--14:.1--1-.TETI;ITIPli:1-1 .. 1.-?^""; -Fi i:iii!1.111111:111::i 'ill:iiiiiiiii, El"TIF-iiii tFili111f...::"411:-F:f.:II7-:..ITE-i*Flitill.:::--71":::' K :1 U T/0 0. IIII-iiii4;11.11141i::1-i'illliE':i------EtCg:IR:111-7--:;;:i':illi :-11147:hilfil-.11i'iri;.:1;,.:::::.: !: 3 4 5 6 7 890 15 20 25 30 40 SO 60 70 100 Fig. B-2 CdF and CmF versus K fora 5260 (from Sarpkaya, 1976). 119 paper and smooth fitting curves were drawn, Figs. B-3 and B-4. From these curves the slopes of log(C4) and log(Cms) were estimated. Since d C C d log(C 11S) (B-8) d K K d log(K) and d C C d log(C ) ._121.....21_. ...._lni dK Kdlog(K) (B-9) Then p and q can be written as 2 2 C,4A K d log(C ) d(j) p - - [1 - sin(-40]2 .---=.7---- (B-10) 2 d log(K) w and C K d log(C ) m q-___at (B-11) 2 d log(K) it Finally p, q, and (p+q)/m are determined by Eqs. (B-10) and (B-11). The results are shown in Table B-1. Table 8-1 Values of the derivative of C(K) u a . 4480 a . 5260 K p+ q m p+q/m p+ q m p+q/m 10 -0.13 3.44 -3.8% 0.09 3.69 2.4% 20 0.082 7.00 -1.2% -0.18 7.35 -2.4% 40 0.59 12.60 4.7% -0.66 13.10 -5.0% It is found that 10 < K 4 40 the values of (p+q)/m were small. 120 10 "T. 1 _ - _ - _ -. --__. - _- = =7= = 4480 , c d4) o 11'4 0.1 -r 1 10 100 K Fig. B-3 Cd4, and Cuvi. versus K for B = 4480, converted from CAF and CmFcurves (Fig. B-1) by using Eqs. (C-12), and (C-T3). 121 10 '14 .i - - - ....-....- . - - -_ . - - - - - ^ _ -_ ---- = 5260 C -4 .. 0 C -=. t 0.1 4 ' 10 100 K Fig. B-4Cdo and Criph versus K for e - 5260, converted from Car and C curves mF (Fig. B-2) by using Eqs. (C-12), and (C-13). 122 (3) 40 < K 4 100 No data are available to examine the values of (p+q)/mat the time this study is carried out. (4) 100 < K When K is large it is assumed that CII(K) = Cds(K) = Cds, the maximum force is total drag force, so Co(K) is equal to 0. Since Cds is a constant, the values of the derivative of Cd.(K) and Cm +(K) with respect to K must be 0. Thus both p and q are equal to 0. After examining all four regions of K, it can be found that (P41) << m is acceptable for K < 40 and for when K is very large, if Cp(K) = 2w2/K is true when K is small and Cu(K)= Cds is true when K is large. 123 APPENDIX C COMPARISON OF FORCE COEFFICIENTS OBTAINED THROUGH DIFFERENT METHODS In the Morison regime, wavy flow data are often analyzed by the least-squares method, because least squares minimize the root mean square error between predictions and measurements. The Fourier anal- ysis method is often used for oscillating flow data, such as U tube tests (Sarpkaya, 1976 ) or cylinders oscillated in still water. Although the data base are the same, the values of the drag coeffi- cients and the inertia coefficients are different from method to method. In order to compare the force coefficients obtained from other publications, the relationships of force coefficients from different methods should be established. The relationships of force coefficients are very complex before the completed formula is derived. Simplified formulas will be used in this thesis, involving only the first harmonic component of force. When only the first harmonic component of the force is con- sidered, the measured force Fm, with phase lag 0 in respect to peak velocity, can be written as F = F cos(A + 0) (C-1) m p 1 2 where 0 = wt. Dividing by p Du will non-dimensionalize F so 2 m' that F = Cu cos(A + 0) (C-2) m 124 (a) Derivation for the Least-Squares Method The force coefficients determined by the simplified least- squares method shown here are designated as CdL and CmL. The mean square error between the measured and predictedforces is defined as 2 1 Iff " " 2 E = j (F F ) de (C-3) m p 0 where F is the measured force and Fp is the predicted force. m To determine C and C the mean square error is minimized by dL rwL, taking the derivative with respect to each coefficient: 2 d E = 0 (C-4) d CdL and -2 d E = 0 (C-5) dCmL By substituting Eq. (C-3) into Eq. (C-4) and Eq.(C-5), and rearrang- ing the terms, 2w 4 C = - f F coselcosel de (C-6) dL 3w m 0 and 2w C = f F sin° de (C-7) mL 2 m 1T 0 Equation (C-2) is now substituted in Eq. (C-6) and Eq. (C-7) to obtain 125 32 C cos. C (C-8) dL w and sin0 Cp (C-9) CmL = 2 (b) Derivation for the Fourier Analysis Method The force coefficients determined by Fourier analysis are desig- and CmF. The drag coefficient can be obtained by nated as CdF replacing F by Fm then multiplying both sides of Eq.(2.1-6) by cos8 and integrating over one wave cycle of 0 to 2w(Sarpkaya, 1976): 27r C = f F cose de (C-10) dF 4 m 0 The inertia coefficients can be obtained by multiplying Eq.(2.1-6) by sins and integrating from 0 to 2w: 2 -2K CmF = F sine de (C-11) 3 f m 11 0 By substituting Eq. (C-2) in Eq. (C-10) and Eq.(C-11) and rearrang- ing the terms, 3w C = cos, C (C-12) dF 8 and C = sin. C (C-13) mF 2 126 The relationships between these force coefficients andthe maximum in-line force coefficient for a verticalcylinder Cu are shown in Fig. C-1 and Fig. C-2. The phases and force coefficients obtained by differentmethods are tabulated in Table C -i. The results of extreme conditions are obtained by set 4)= -90°, when K is small, and set $ =0° when K is large. 127 1.2 C dF 1.0 cosh C 8 C \I / - dL 32 / . cosh 4/ ,j CuC 97 C ---, 0 1 = 0.5 1 + sni 0 OMR .0' sr 0.0 I I I 90° -60° 30° 0 (0 f K) (K .0) Fig. C-1 Drag coefficients from different methods. 1.0 .\ C C K mF mL N\ 0.5 sing) 2 C C Tr K 2 sin0 .\ Cu 72 l+sin0 I 0.0 1 90° -60° 30° 0° (0 F K) (K Fig. C-2 Inertia coefficients from different methods. Table C-1 Force coefficients from different methods NumberCarpenterKeulegan Phase sin(-0) CoefficientMaximumForce Force-Phase Method Least-Squares Method Fourier Analysis Method K (0) 0 Cu C (10 C MO CdL C mL C dF C mF Small -90 1 2 n2 C o 2 0 2 0 2 K 2 -.. -90 1 2n2 K 2 sin, C u art Median 05 0S Cds S 1 + sin20 Cu Kn2 1 + sing 9w32 cosy C 0 ,2K sink C 0 8 cos + C w Kn2 sink C 0 Large 0 0 C ds Cu 0 1.13 Cu 0 1.18 Cu 0