Citation
Heave reponse of a deep draft spar platform with a centerwell

Material Information

Title:
Heave reponse of a deep draft spar platform with a centerwell
Series Title:
Heave reponse of a deep draft spar platform with a centerwell
Creator:
Miller, William
Place of Publication:
Gainesville, Fla.
Publisher:
Coastal & Oceanographic Engineering Dept. of Civil & Coastal Engineering, University of Florida
Language:
English

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.

Full Text
UFL/COEL-97/017

HEAVE RESPONSE OF A DEEP DRAFT SPAR PLATFORM WITH A CENTERWELL by
William Miller, Jr. Thesis

1997




HEAVE RESPONSE OF A DEEP DRAFT SPAR PLATFORM
WITH A CENTERWELL
By
WILLIAM MILLER, JR.

A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
UNIVERSITY OF FLORIDA

1997




ACKNOWLEDGMENTS

I would like to thank Dr. Sheppard, my advisor and supervisory committee chairman, for his help and support during my research and Dr. Ahmed Omar for helping me to understand some of the processes involved. Thanks also go to the staff of the Coastal and Oceanographic Engineering Laboratory. Special thanks go to Chuck Broward who designed and built the instrumentation used in my experiments and Vernon Sparkman who built the model. None of this would be possible without their help. My gratitude also goes to Dr. Sheng for his assistance in helping me understand the intricacies of numerical solutions.
Many thanks go to my classmates and friends in the Coastal Engineering Department at the University of Florida, especially Becky, Al, Eric, Greg, Mike, Hugo, S anti, Mark, Chris and Wayne, all of whom acted as sounding boards and lent encouragement when necessary. Special thanks go to Dr. Dean and Dr. Thieke for their patience and understanding in scheduling my defense.
Finally, I'd like to thank my parents, Bill and Claudette, who have always supported me in my endeavors.
Partial funding for this study was provided by the Offshore Technology Research Center (OTRC) of Texas A&M University.




TABLE OF CONTENTS
page
ACKNOWLEDGMENTS.....................................................................1i
LIST OF TABLES.............................................................................v
LIST OF FIGURES ............................................................................ Vi
LIST OF SYMBOLS........................................................................... ix
ABSTRACT ................................................................................. xii
CH-APTER 1: INTODUCTION..............................................................1.
CHAPTER 2: GENERAL CONCEPTS AND PREVIOUS WORK ....................... 7
Spar Buoy/Platform Motion............................................................... 7
Effects of a Moonpool or Centerwell ................................................... 21
Large Spar Platforms..................................................................... 25
CHAPTER 3: CENTERWELL MATHEMATICAL MODEL DEVELOPMENT ....27
Momentum Equation for Heave ......................................................... 27
Inviscid Forces on the Fluid in the Control Volume .................................... 31
Viscous Damping ......................................................................... 36
Complete Equation for Heave Motion .................................................. 36
Analytic Approximation to the Complete Equation for Heave ....................... 40
CHAPTER 4: EXPERIM4ENTAL RESULTS ............................................... 47
Model Description and Scaling .......................................................... 47
Free Response Tests...................................................................... 49
Configuration for the Forced Motion Experiments..................................... 52
Experimental Results..................................................................... 56
iii




CHAPTER 5: MODEL CALIBRATION AND CENTERWELL HEAVE
D Y N A M IC S ............................................................................................. . 68
M odel C alibration .............................................................................................. 68
Sensitivity of the Centerwell Heave Response Dynamics to Secondary Effects ........ 80 Summary of Centerwell Heave Motion ............................................................... 86
CHAPTER 6: APPLICATION AND CONCLUSIONS ............................................. 91
Application of Centerwell Heave Dynamics to the Spar Platform ......................... 91
Affect of a Centerwell on the Spar Platform ........................................................ 94
Conclusions and Questions for Further Study ........................................................ 103
APPENDIX A: DERIVATION OF ANALYTIC SOLUTIONS TO THE
CENTERWELL AND SPAR GOVERNING EQUATIONS .......................... 105
Solution to the Centerwell Equation of Motion ..................................................... 105
Coupled Spar and Centerwell Equations of Motion ............................................... 112
APPENDIX B: FORTRAN AND MATLAB PROGRAMS ......................................... 114
Fortran 90 Numeric Solution to Centerwell Governing Equations ......................... 114
Fortran 90 Analytic Solution to Centerwell Governing Equations .......................... 120
MATLAB Analytic Solution to Centerwell Governing Equations .......................... 122
MATLAB Analytic Solution to Coupled Spar and Centerwell Governing
E qu ation s ................................................................................................ 12 5
L IST O F R E FE R E N C E S ............................................................................................. 131
BIO GRA PH ICAL SKETCH ....................................................................................... 136




LIST OF TABLES

Table page
Table 3. 1, Comparison of term in Aalbers' equation (Aalbers, 1984) and the
parameters in Figure 3.1............................................................. 38
Table 3.2, Order of Magnitude of Individual Terms in Heave Equation .................. 43
Table 3.3, Orders of Magnitude of Complete Terms ........................................ 44
Table 4. 1, Orifice Plate Diameters............................................................ 49
Table 4.2, Summary of Free Response Damped Natural Frequency (fd)...................... 51
Table 4.3, Estimated experimental error values .............................................. 61
Table 4.4, Synchronous motion frequencies (o),) by orifice size............................ 63
Table 5. 1, Added mass coefficient derived from the......................................... 69
Table 5.2, Maximum amplitude ratio comparison for non-linear damping only........... 77
Table 5.3, Maximum amplitude ratio comparison for combined linear and non-linear
damping .............................................................................. 80
Table 6. 1, Important dimensions for coupled equation solutions.... ..................... 94




LIST OF FIGURES

Figure P-age
Fig. 1.1, Drilling and Production Platforms (Clauss et al, 1992) ............................... 1
Fig. 1.2, Artist's impression of a Spar platform in the Gulf of Mexico
(Deep Oil Technology/ Rauma Repola) ........................................................ 5
Fig. 1.3, Chevron Spar Platform Mooring Arrangement (Glanville, 1991) ................... 5
Fig. 1.4, Chevron Spar Platform Elevation View (Glanville, 1991) ............................... 6
Fig. 2.1, Spar Buoy Arrangement ( Lewis, 1967, p. 628) ............................................ 8
Fig. 2.2, Added mass coefficients for heave motion of a vertical cylinder, G = gravity,
A = Diameter, T = draft (Sabuncu and Calisal, 1981, p. 41) ......................... 18
Fig. 2.3, Added masses of vertically oscillating components of offshore structures
(Clauss et al, 1992, p. 281) ......................................................................... 19
Fig. 2.4, Com posite Spar Platform ............................................................................ 20
Fig. 2.5, Examples of Semi-Submersible Platforms (Clauss, 1992) ........................... 21
Fig. 3.1, Spar and Centerwell Configuration and Definition of Terms. (not to scale) ....... 28 Fig. 3.2, Control Volume and Control Surface for deriving the Centerwell Equation
of M otion. (not to scale) .............................................................................. 28
Fig. 3.3, Spar diagram to scale .................................................................................. 33
Fig. 4.1, Scale drawing of model and orifice plates .................................................... 50
Fig. 4.2, Fram e and drive assembly ........................................................................... 55




Fig. 4.3, Plot of synchronous frequency with orifice diameter ratio ................................. 63
Fig. 4.4, Results for experiments conducted with an orifice area ratio of 0.52 ................. 64
Fig. 4.5, Results for experiments conducted with an orifice area ratio of 0.64 ................. 65
Fig. 4.6, Results for experiments conducted with an orifice area ratio of 0.74 ................. 66
Fig. 4.7, Results for orifice experiments conducted with an orifice area ratio of 1.0
(i.e. n o o rifi ce) ................................................................................................ 6 7
Fig. 5. 1, Added mass coefficient vs. the cube of the orifice diameter ratio ...................... 70
Fig. 5.2, Mathematical model solutions for area ratio 0.74, non-linear damping only
w ith ts = C.2 = 0 .............................................................................................. 7 5
Fig. 5.3, Numerically determined mean centerwell level variation with frequency ............ 76
Fig. 5.4, Amplitude ratio vs. frequency plot near the synchronous frequency for
orifi ce area ratio 0.74 ....................................................................................... 78
Fig. 5.5, Linear damping coefficient vs. orifice area ratio ................................................ 78
Fig. 5.6, Solution plots for an orifice area ratio of 0.52 ................................................. 81
Fig. 5.7, Solution plots for an orifice area ratio of 0.64 ................................................. 82
Fig. 5.8, Solution plots for an orifice area ratio of 0.74 ................................................. 83
Fig. 5.9, Solution plots for the no orifice case ............................................................... 84
Fig. 5. 10, Plots of effect of varied spar acceleration added mass term ( Q for an
orifi ce area ratio of 0.64 ................................................................................... 88
Fig. 5.11, Phase plot for linear varying spar acceleration coefficient for an orifice area
ratio o f 0 .6 4 ..................................................................................................... 8 8
Fig. 5.12, Results of solutions calculated with E = 0 and an orifice area ratio of 0.64 .. ... 89 Fig. 5.13, Linear damping only solution results for an orifice area ratio of 0.52 ............. 90
Fig. 6. 1, M odel response w ith no orifice ....................................................................... 97




Fig. 6.2, Comparison of the model spar and centerwell phase with no orifice............ 97
Fig. 6.3, Model response with orifice area ratio 0.6 ......................................... 98
Fig. 6.4, Comparison of the model spar and centerwell phase with an orifice
area ratio 0.6 ......................................................................... 98
Fig. 6.5, Chevron Spar response with no orifice based on scaled coefficients
discussed in Chapter 5............................................................... 99
Fig. 6.6, Comparison of the Chevron Spar and centerwell phase with no orifice ......... 99
Fig. 6.7, Chevron Spar response with an orifice area ratio 0.6 based on scaled
coefficients discussed in Chapter 5 ................................................ 100
Fig. 6.8, Comparison of the Chevron Spar and centerwell phase with an orifice
area ratio 0.6........................................................................ 100
Fig. 6.9, Neptune Spar response with no orifice based on scaled coefficients
discussed in Chapter 5.............................................................. 101
Fig. 6. 10, Comparison of the Neptune Spar and centerwell phase with no orifice....... 101 Fig. 6. 11, Neptune Spar response with an orifice area ratio 0. 6 based on
scaled coefficients discussed in Chapter 5......................................... 102
Fig. 6.12, Comparison of the Neptune Spar and centerwell phase with an
orifice area ratio 0.6................................................................ 102




LIST OF SYMBOLS

added mass
centerwell cross-sectional area spar cross-sectional area measured to the outer diameter orifice area
orifice area ratio AR/A centerwell added mass associated with waves and spar acceleration spar, wave and effective damping per unit mass viscous damping coefficient damping coefficients associated with wave and spar velocities effective damping coefficient per unit mass linear or potential damping coefficient per unit mass added mass coefficients non-linear or viscous damping coefficient per unit mass general diameter centerwell diameter spar outer diameter model and prototype outer diameter, respectively

a
A,
Ao AR AR aw, as Bs, Bw, Be bv b,, b, Ce
CL Cm, Cr2, cms CN
D Di Do
Dom, Dop




DR DR Fw
h, h, h
h0,h0i, holi1,

H(o) HAW
K
k
ke1 k.
L
nCW, ms
RAO s, s, "9
SAWP
so
SOM, Sop
J3

orifice diameter
orifice diameter ratio DR/DI wave force acting on a buoy or the centerwell height of water column within the centerwell measured from the base of the centerwell and associated time derivatives position of the centerwell water level measured from still water level and associated time derivatives first order solution to the centerwell governing equation and associated time derivatives second order solution to the centerwell governing equation and associated time derivatives frequency transfer function amplitude of centerwell water surface motion hydrostatic pressure coefficient per unit mass wave number or pipe resistance coefficient pipe entrance and exit resistance coefficients general length scale (usually referring to spar draft) total mass of centerwell and spar response amplitude operator spar displacement measured from still water level and associated time derivatives
amplitude of spar motion spar draft
model and prototype draft, respectively damping ratio




6 logarithmic decrement
4, phase of centerwell water surface motion
110 wave amplitude
r/, i7, y water surface displacement due to waves measured from still water
level and associated time derivatives A(w0) dynamic magnification factor
p t1, 2 added mass coefficient ratios
0 phase of spar motion
won natural frequency (radians/second)
(0o reference natural frequency used for comparison purposes
(radians/second)




Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science HEAVE RESPONSE OF A DEEP DRAFT SPAR PLATFORM WITH A CENTERWELL
By
William Miller, Jr.
December 1997
Chairman: D. Max Sheppard
Major Department: Coastal and Oceanographic Engineering
This thesis examines the heave response of a spar type platform consisting of a deep draft buoyant annular structure with a centerwell or moonpool which is open to the sea at the bottom. The governing equation for the heave response of this free flooding centerwell is derived using conservation of momentum and its coefficients are calibrated with experimental data. An analytical solution to this governing equation is obtained and compared to a numerical solution, giving satisfactory results. Sensitivity tests are conducted with both the numerical and analytical solutions to explore the effects of the different components of the centerwell heave motion.
Finally, the governing equation of the centerwell is coupled with that of the spar platform and numerical simulations are conducted to examine the effects of the centerwell on the heave motion of the spar platform. The results identify potential design problems due to the interaction.




CHAPTER 1
INTRODUCTION
In their quest to find and develop new sources of hydrocarbons to feed the world's
constantly growing hunger for fuel, the petroleum industry is pushing offshore into ever
deeper water. The current depth limitations of the various field production options are:
piled steel platforms, 0 to 500 meters, guyed towers, 200 to 600 meters, tension-leg
platforms, 150 to 900 meters, and semisubmersible platforms, 100 to 1500 meters (Clauss
et al., 1992, p. 130). Figure 1.1 shows various production platforms currently in use.
Fixed platforms
Bottom supported
compliant platforms
Floatirn compliant
platformm s
Concrete
platforms Pile supported
gravity foundation) jackets Guyed Compliant Articulated
towers piled towers towers Tension leg Catenary anchored
platforms floaters
(tankers. semisubmersibles)
Fig. 1.1, Drilling and Production Platforms (Clauss et. al., 1992)




"A series of new alternative platform concepts are being investigated by the offshore industry for deepwater development and production prospects. One concept which is receiving a considerable amount of attention is the Spar Platform" (Niedzwecki et al., 1995, p. 91). According to Halkyard et al. (1991, p. 41), the Spar Platform, also known as a Tension Buoyant Tower (TBT), has the potential to extend depth capabilities to as much as 3000 meters.
The Spar Platform is a large, slender, deep draft, cylindrical floating structure. It consists of a large, buoyant vertical cylindrical hull topped by a deck structure. The hull may be used for storage purposes and usually has a centerwell or moonpool. Drilling and production risers are deployed through this centerwell which also provides them protection. The deck, which consists of multiple levels, is located well above the waterline and provides the necessary space for production and drilling operations, pipe storage, drilling related equipment storage, process equipment, power plant, water treatment equipment and other facilities. "Its design is intended to eliminate first and second order heave forces due to the local seaway environment" (Niedzwecki, et al. 1995, p. 41).
Glanville et al. (1991) described a Spar Platform concept designed for Chevron by Deep Oil Technology and Rauma Repola. The design has a hull diameter on the order of 42.5 meters (140 feet) and a draft of 198 meters (650 feet). The transverse centerwell area was divided over the upper 50 meters of the hull and had a combined area of 32 percent of the waterplane area of the hull giving it an effective centerwell diameter of 24 meters. The deck structure was located 20 meters above the waterline. The design operating depth for this platform is 820 meters in the Gulf of Mexico. Figure 1.2 is an




artist's impression of a Spar platform in the Gulf of Mexico. Figures 1.3 and 1.4 are drawings of a proposed Chevron Spar.
Recently, Oryx installed the world's first production spar (the Neptune Spar) in the Gulf of Mexico approximately 135 miles southeast of New Orleans. Deep Oil Technology and Rauma Repola of Finland designed and built the unit. This platform has a 198 meter draft and an outer diameter of 22 meters and a square 10 by 10 meters centerwell. It is moored at a location with a depth of approximately 700 meters (http://www.offshoretechnology.com/projects/neptune/ index.html).
"The concept of a deepwater Spar Platform has its origins in the design of oceanographic spar buoys" (Niedzwecki et. al., 1995, p. 41). The FLIP (floating instrument platform) Ship was built in 1962 as a stable platform for oceanographic measurements. Rudnick (1967) documented its favorable motion properties. Newman (1977, p. 317 321) stated that "FLIP will be stable with practically no heave motion since its resonance frequency is beyond any predicted wave excitation." In the 1970s Shell constructed the Brent Spar for oil storage and offloading in the North Sea. Other uses of the design were discussed by van Santen and de Werk (1976), including a helicopter landing base, an offshore loading terminal and an oil storage terminal. However, additional uses of the Spar design have not developed to date.
In the late 1980s and early 1990s, the Spar design was revived and proposed as a low cost production facility for remote sites. The features which make the Spar attractive are described by Glanville et. al. (1991) and include: simplicity of design, favorable motion characteristics, insensitivity to water depth (i.e. the unit may operate in a wide range of




water depths with little variation in the design), ability to support a high deck load, the centerwell protects the risers from wave loads, the risers can be supported in tension by means of flotation, low cost of shipyard construction, oil storage availability in tanks located in the hull, and the structure may be relocated.
Though exhaustive studies have been performed on the basic spar design, the effect of the centerwell on the dynamics of such a deep draft facility as the Spar Platform have not been specifically addressed in the open literature.. Informal model tests have demonstrated a tendency for the water in this centerwell to move synchronously with the Spar platform at certain frequencies. A small lip located at the bottom of the centerwell seems to enhance this tendency, even though the flow restriction due to the orifice created was negligible.
The dynamics of this "synchronous centerwell motion phenomena" are the primary interest of this study. This paper will attempt to (1) experimentally verify and document the synchronous centerwell response and determine what effect a bottom orifice has on the phenomena; (2) examine experimentally and through numerical simulations the dynamics of the heave motion of the fluid in the centerwell; (3) numerically simulate the combined Spar-centerwell system in waves and determine potential design problems and/or advantages due to the presence of the centerwell.




Fig. 1. 3, Chevron Spar Platform Mooring Arrangement (Glanville, 199 1)

Fig. 1.2, Artist's impression of a Spar platform in the Gulf of Mexico (Randall, 1997, p.
58)




I
Fig. 1.4, Chevron Spar Platform Elevation View (Glanville, 1991)




CHAPTER 2
GENERAL CONCEPTS AND PREVIOUS WORK
Spar Buoy/Platform Motion
The heave motion of a spar buoy in waves has been investigated extensively. Nearly every fluid mechanics and engineering mathematics text uses the system to illustrate basic principles of mechanical oscillations, buoyancy, damping, resonance and the use of ordinary differential equations. For the most part, these texts treat the floating buoy as a mass-spring system and the result is a second order ordinary differential equation or a system of coupled second order equations, depending on the number of degrees of freedom. Non-linear terms for drag are introduced in several texts and papers specifically concerned with buoy design and dynamics.
The basic equation of motion for a floating structure is found by "equating the external forces on the structure (assumed to be prevented from moving) to the reactive forces acting on the structure (which is assumed to be moving in a calm fluid)" (Natvig and Pendered, 1980, p. 99). For a spar type buoy in heave the equation reduces to the second order ordinary differential equation below. This is the form resulting from the direct application of Newton's Second Law in an inviscid fluid.




Referring to figure 2.1, s is the buoy displacement, y is the water surface displacement due to the wave. In equation 2.1, a is the added mass or hydrodynamic mass, b is the potential damping coefficient and c is the hydrostatic restoring force coefficient, also termed the spring constant when associating the system with a mechanical oscillator.
z

Fig. 2.1, Spar Buoy Arrangement
Equation 2.1 assumes that the wave is long with respect to the buoy and therefore the buoy is "hydrodynamically transparent." In other words, "there is little change to the incident wave when it passes the structure" (Clauss et. al., 1992, p. 222). The assumption is also known as the "long-wave approximation" (Newman, 1977).
Clauss et al. (1992, p. 220) divides the forces experienced by the buoy into "(1) Froude-Krylov force: pressure effects due to undisturbed incident waves, (2) hydrodynamic added mass and potential damping force: pressure effects due to relative acceleration and velocity between water particles and structural components in an ideal




fluid, and (3) viscous drag force: pressure effects due to relative velocity between water particles and structural components."
Newman describes the Froude-Krylov hypothesis as "the assumption that the pressure field is not affected by the presence of the body and can be determined from the incident wave potential by itself The diffraction potential is neglected completely..." (1977, p.305). Newman goes on to state that, given the wavelength is large compared to the beam, the Froude-Krylov exciting force and moment are the leading-order contributions in the vertical plane.
Assuming a linearized form of the well known Bernoulli equation, Clauss et al. (1992, p. 265) describes the Froude-Krylov force in equation 2.2, where 0o, is the incident wave velocity potential. Newman (1977, p. 299-309) ), Hooft (1982, pp.144-153) and Sarpkaya and Isaacson (1981, pp. 441-443) develop similar equations and results.
FK = -p f pdS :p Eq. 2.2
For a spar buoy in heave due to harmonic wave motion, the Froude-Krylov (FFK) force is reduced to the wave portion of the hydrostatic restoring force which in deep water is FF = pgAekql= cek~q (Newman, 1977, p. 357-359; Hooft, 1982, p. 144-151; Clauss et al., 1992, p. 265-268).
Newman (1977, p. 289-296), Sarpkaya and Isaacson (1981, pp. 438-440), Hooft (1982, pp.112-114) and Clauss et al. (1992, pp. 288-291) demonstrate that near a free surface, the hydrodynamic added mass and potential damping force can be derived from the dynamic part of the unsteady Bernoulli equation. Applied to the velocity potential of




the waves generated by the oscillating body, this results in equation 2.3, where Se is the equilibrium body surface.
pJ- dS =-(a +b ) Eq. 2.3
s, Ot
Substitution of equations 2.2 and 2.3 into the equation for Newton's second law of motion results in an equation of the form of equation 2.1. This may be rearranged to give equation 2.4 in the common mass-spring system notation for a shallow drafted buoy.
(M+a)N +b&+cs = ai+ b + cq= f (t) Eq. 2.4
This is the simplest form of the equation of motion for a buoy in heave derived in basic design and dynamics texts (Korvin-Kroukovsky, 1955; Lewis, 1967; Berteaux, 1976; Newman, 1977; Hooft, 1982; Patel, 1989; Clauss et al., 1992; etc.). Many of the authors perform the derivation with the coefficients expressed as matrices and the variables as vectors to develop two and three dimensional coupled systems of equations and apply equation 2.4 to up to six degrees of freedom. However, most authors studying multiple degrees of freedom limit themselves to heave and pitch. The equation of concern in this paper is limited to a single degree of motion since "the heaving motion.., mode of oscillation is of prime interest with spar type structures" (Kokkinowrachos and Wilckens, 1974, p. 99). As will be noted later in this chapter, Aalbers (1984) in studying the motion of water in a moonpool also choose to limit his investigation to heave.
For deep drafted buoys and structures, a wave "attenuation factor" should be applied to the wave force terms in equations 2.1 and 2.4 to account for the variation of water particle motion with depth (Berteaux, 1977, p. 54, and Lewis, 1967, p. 629). This




attenuation factor is derived directly from linear wave theory and in deep water the factor reduces to ekz, where k is the wave number (for deep water k = O'1g, with w0 the wave radian frequency) and z the depth (negative downward) (Dean and Dalrymple, 1991). For very deep draft structures where the "draft (S) is large compared with z, the factor becomes approximately ekz = e-kSo.' (Lewis, 1967, p. 629). If it is assumed that most of the wave forces act on the bottom of the buoy, the application of the attenuation factor results in equation 2.5 below.
(m+a)N+b+cs=e-kSo (ail+bi +cekz'i) =f(t) Eq. 2.5
As previously noted, the "linear damping" term represents the "potential damping" due to the body moving through an inviscid fluid. Glanville et al. (1991, p. 61) refer to this damping term as "linear damping... due primarily to radiation," alluding to a dependence on wave generation by the buoy or platform.
The potential damping may be obtained theoretically by integrating the pressure distribution over the body based on the potential of the undisturbed wave (Vugts, 1968a, p. 16). This procedure is known as "the linear diffraction problem" and should be used when "the structure spans more than about a fifth of the incident wave length" (Sarpkaya and Isaacson, 1981, p. 382).
However, Newman (1962 and 1977, p. 303-307) favors a method due to Haskind (1957) known as the "Haskind Relations, which express the damping coefficients in terms of the exciting forces." Therefore, an "actual solution to the diffraction problem can be avoided" (Vugts, 1968, p. 16). This expression is endorsed by Garrett (1971, p. 129),




Vugts (1968, p. 16; 1971, p. 21), Kokkinowrachos and Wilckens (1974, p. 107-108), Sarpkaya and Isaacson (1981, p. 440-445) and Hooft (1982, p. 121-124) for motion in the presence of a free surface. Newman derives the relation and notes that it is not dependent on the long-wave approximation. The resulting equation is identical among the authors and is
bii k 211 gg (2)dO. Eq. 2.6
8 7rpgC,, o17,,
In equation 2.6, k is the wave number, Cg is the wave group velocity, 77o is the wave amplitude and X is the complex amplitude of the exciting force. Assuming a single frequency linear wave of the form 7 = 17, coscot) and returning to the long-wave approximation, X can be derived from equation 2.5. The result in complex form is X, = ekz '1o(-aco2 +c-ibco) Eq. 2.7
Hooft (1982, p. 121) indicates that the equation should be evaluated for each wave frequency in the incident wave spectrum. Integrating the equation for an axial symmetric structure with a vertical axis, and assuming deep water linear waves, leads to the following equation for the linear damping coefficient in heave: b X C3 Eq. 2.8
2pg3 I
However, the problem of the damping term does not end there. The linear system of equation 2.5 has neglected the viscous effects. For large manned platforms this may be sufficient. "Due to the large diameter of such structures the potential flow linear solution




is normally of primary importance while in this case viscous effects are negligible" (Kokkinowrachos and Wilckens, 1974, p. 99).
However, many authors include these viscous effects and when doing so "it is advisable to stick to the simple structure of the Morison-formula" (Kokkinowrachos and Wilckens, 1974, p. 105). The Morison equation assumes that the total wave force is "the sum of the inertial force and the viscous force" (Newman, 1977, p. 42) where the viscous force is proportional to the square of the velocity. Including such a "viscous damping" term results in equation 2.9.
(m + a) + (blinea, )s+ (bnoninear ) s +cs = f (t). Eq. 2.9 This is the equation given by Hooft (1970, p.35), Ochi and Malakar (1984, p. 5) and Glanville et al. (1991, p.61). Snyder and Darms (1967, p. 151), Gudmestad and Connor (1983, p. 184), Patel (1989, p. 316) and Clauss et al. (1992, p. 278) express the viscous damping in terms of the relative velocities between the spar and the wave particles as was done for the potential damping in equation 2.1 resulting in equation 2.10.
(m +a) + (bnea,) + (bnonunear ) -s + cs = ai + (bnear) + (bnoniear )I s i + c7.
Eq. 2.10
To develop a linear approximation to equation 2.10, Gudmestad and Connor (1983, pp. 185-186) suggest that for s >> the (bnon,)il-(il-s) term may be replaced by (bnonlinear ) li l-(b nonlinear )MfPatel (1989, p. 316) adds (bnonliear)I to both sides of the equation to obtain a (bnonlinear )I - s (i- ) + bnonlinear 1ssI term in the wave force expression. He then states that




the "relatively large magnitude of the inertia term in the wave force and the decay with depth of wave particle velocities.., makes the" (bnonlinear) s(l- s)+ bnoniInear s "term small for members with significant submergence below still water level." Thus, for s >> rf,
(bnoniear) 7 s|( ) +bnoninea, s -(bonlinear )s + bnoninear Is = 0
Then, for small il, the (bnonlinear) I s(il- s) + bnonline, term may be replaced by (bnoniner)li il and equation 2.10 can become equation 2.9 which is somewhat more manageable. Performing this expansion results in equation 2.11.
(m + a)+ ( b ) + ( bnonlnea, )| + cs = [ai + (bin,r)il+(bnoninear,) J l + c ]e.
Eq. 2.11
Snyder (1967, p. 152) and Hooft (1970, p. 35) describe the classical determination of the drag force on a structure where the non-linear damping coefficient can be defined in terms of the "area of projection of the body in the direction of velocity" (Ap) and a drag coefficient (CD) as given in equation 2.12.
bnon-inear = PCDAp. Eq. 2.12
Snyder and Hooft refer to the experimental work of Hoerner (1965) for values of the drag coefficient. Hoerner explains that "the drag on a smooth cylinder in supercritical (flow) condition (Reynolds number greater than 106) is approximately constant" (1965, p. 3-11 to 3-13). Hoerner plots experimental results for "cylindrical bodies in axial flow, with blunt shape" and gives a value of total CD for viscous pressure drag at approximately 0.85 for length (draft) to diameter ratios greater than 2. "The drag of these shapes




essentially consists of that of the forebody and base drag." It should be noted that these drag coefficient values are not those for a body in oscillatory motion.
Snyder (1967, p. 153), corrects for oscillatory motion by dividing the non-linear drag coefficient itself into forebody and base drag components, forebody drag defined as the form or pressure drag due to the body's shape and the base drag essentially as the skin friction component. He states that the drag coefficient (CD) "for surface piercing buoys... changes with the direction of relative flow. When the buoy is moving downward into the water the forebody drag coefficient must be used. When the buoy is moving upward with respect to the water, the base drag coefficient must be used." From Hoerner (1965), the base drag coefficient alone is approximately 0.2 for length to diameter ratios greater than
4 and up to 12.
Many designers and researchers linearize equation 2.10 or 2.11 by developing an "equivalent linearized damping coefficient" in accordance with a method developed by Kryloff and Bogoliuboff(Berteaux, 1976, App. I; Dao and Penzien, 1982; Gudmestad and Connor, 1983, p. 189; Tung, 1986; Patel, 1989, pp. 290-291 ; Ochi, 1990, pp. 389-409; Clauss et al., 1992, pp. 278-279). The principle of the Kryloff-Bogoliuboff method is to "consider an equivalent balance of energy during one cycle such that work per cycle of the two systems (non-linear and equivalent linear system) is the same" (Ochi, 1990, p. 389). Thus, the procedure is to express the energy dissipation for linear and non-linear damping over a cycle in terms of velocity, assume harmonic motion and equate the results and solve for the linear damping coefficient in terms of the non-linear coefficient. By integrating the




s term over a cycle the equation may be linearized using the approximation sin(cot) sin(cot) = (8/3K) sin(at).
Berteaux, Dao and Penzien, Gudmestad and Connor, Ochi and Tung derive their equivalent term as a frequency dependent term. Patel develops his expression in terms of the maximum energy dissipated and thus it is only dependent on the natural frequency of the system. Clauss et al. express the equation with an implied frequency dependence with the result being a function of the amplitude of the relative vertical velocity. Additionally, Ochi and Gudmestad and Connor explore several other linearization methods with special emphasis on the application of the resulting response transfer functions to random wave spectra. Several of the expressions for linearizing the damping term in accordance with the Kryloff-Bogoliuboff method are shown below.
Berteaux: btinear = (bnon-linear) a = 8 (bon-inea,)Rw Eq. 2.13
8(n .iea) 8 ~(b,o,,_inar)Ro)o E.2.13
Patel: blinear = (bnon-linear)m = (bon-inear)Roo Eq. 2.14
37c 31r
Clauss: blinear = bnon-linear) Eq. 2.15
37c
21
Ochi: binear 2 = (bnon-linear )O- Eq. 2.16
In equations 2.13 through 2.15, s = Rco is the amplitude of the structure's velocity for frequency co, and m = Rao is the amplitude of the maximum response velocity of the structure, which occurs at the natural frequency, il is "the vertical orbital




velocity" of the water particle "at the hull center and is the vertical velocity of the heaving semi-submersible" and Ii- "is the associated amplitude of relative vertical velocity" (Clauss et al., 1992, p.278). In equation 2.16, o-, is the standard deviation of the response velocity (Ochi, 1990, p. 391; Gudmestad and Connor, 1983, p. 189).
Clauss et al. expand this amplitude term with respect to the complex forms of the harmonic wave and spar motion resulting in the approximation of equation 2.17 (zhc is the hull center where the water particle velocity is taken).
blinear = bnon-iinear 8- .. (bnoninear )WT, e kz& Eq. 2.17 The dependence on response amplitude require that the linearized equation of motion must be solved iteratively to determine a response amplitude value that satisfies the corresponding equation. Additionally, equation 2.17 requires that the phase also be satisfied.
The hydrodynamic mass or added mass of the system is another frequency dependent term. Sarpkaya and Isaacson (1981, p. 440) note that "it should be emphasized ... that both the added-mass and the damping coefficients are frequency dependent." Kokkinowrachos and Wilckens (1974, p. 108) use a similar derivation to the Haskind Relations and assert a dependency of the added mass on the excitation force. Hooft (1970, p. 24-26) states "in general these coefficients ... depend on frequency (and) when the body has a three dimensional form ... the added mass is not only influenced by the projected area in the direction of oscillation but also by the length of the body in the direction of oscillation."




18
Sabuncu and Calisal (1981) performed a theoretical analysis of the added mass and damping coefficients for the motion of a vertical cylinder which included varied draft to diameter ratios. Figure 2.2 shows the deep water results for the added mass. In this figure T is the draft of the cylinder and A is its radius. The added mass coefficient corresponds to that in equation 2.4. Sabuncu and Calisal (1981, p. 60) state that "the results are compared to some available experimental numerical results. The agreement with numerical results are observed to be satisfactory." The figure shows that as the draft to radius ratio increases, the added mass coefficient becomes smaller and less dependent on frequency.
Hydrodynamic coefficients for vertical circular cylinders at finite depth Vertical cylinder heave deep water
A T/A-O.1
T T/A-0.3
4- 9. ThC3 T/A-O.5
--+A 4- T/A-3
7.2 A
S4;9
2.4
0 1.2 2.4 3.6 4.8 6
Omega Omega* A/G
Fig. 2.2, Added mass coefficients for heave motion of a vertical cylinder, G = gravity, A =
Diameter, T = draft (Sabuncu and Calisal, 1981, p. 41)
Hooft (1982, p. 150), Patel (1989, p. 286) and Clauss et al. (1992, p. 279-283) recommend values of constant added mass coefficients in their discussions of buoy and platform design. Clauss et al. incorporates a form coefficient to account for the draft to




radius ratio. Figure 2.3 shows the values recommended by Clauss et al. (ms3 is the added mass for heave motion). Patel recommends the added mass of a half hemisphere of the same radius as the cylinder and Hooft recommends using the added mass of a circular disc with this radius.
added masses mkj (direction of force k direction of motion j) a hulls with rectangular cross-section
2.0 j 4aQu
m33R = Co(UR) 9nbl (length I)
1. caissons columns
s mscs=C (qg)j 9R4S matc =Calac) Rg
Oa$
for comparison: horizontal circular cylinder
a
et3 al. 192 p. 281)C, =1 ....... .............. r...c .......... ...............
E
-o 1.2 _
.0
0
0 1 2 3 t, 5 6 7 8 9 ,,- 10
a -b .= R .ac tRc
a 2hcs' C 4he
Fig. 2.3, Added masses of vertically oscillating components of offshore structures (Clauss et al., 1992, p. 281) The effect of added mass may be used to minimize the heave of a buoy attaching a plate to the bottom of the buoy (known as a "damping plate"). The idea is to "minimize the heave by adding to the virtual mass of the buoy and also to provide some damping through the drag of the plate" (Cavaleri and Mollo-Christensen, 1981, p. 24). Cavaleri and Mollo-Christensen found through numerical simulations that for higher frequencies the damping plate was effective, but for lower frequencies it may actually lead to increased heave.




The hydrostatic portion of the pressure term is called the hydrostatic stiffness force, the buoyancy force, the restoration force or even the spring force when relating the system to a mechanical oscillator. This force results from "the total change in buoyancy from the initial calm-water condition as a result of both the change in water level" due to the wave "and vertical displacement" of the buoy (Lewis, 1967, p. 628). The hydrostatic force coefficient or spring constant is proportional to the waterplane area (Aw) of the buoy: c =pgA, (where p is the fluid density and g is gravitational acceleration).
Figure 2.4 illustrates a composite spar platform of the type used for storage and loading in the Brent Field in the North Sea. By proportioning the structure so "the length of the (surface piercing) column is such that the superstructure is high enough out of the water and that the main body is out of the wave zone" the design minimizes the wave forces experienced by the platform (van Santen and de Werk, 1976, p. 1106).
Fig. 2.4, Composite Spar Platform
Structures such as the composite spar are classified as semi-submersibles. Figure 2.5 shows examples of semi-submersible platforms classified as "multi-hull column stabilized structures" consisting of "an array of vertical columns of minimal waterplane




area, which are connected to caissons or longitudinal hulls with interconnecting truss-like structural members below the water surface" (Clauss et al., 1992, p. 76-77). The centerwell designed into the Spar Platform described in Chapter 1 would appear to have the same advantage of minimizing the waterplane area of the platform while providing the deep draft required to minimize wave forces.

PACESETTER(1973) AKER H-3(1974) BINGO 3000 (1982)
GVA 4000 (1983) TRENOSETTER (1987) BALDER (1978)
Fig. 2.5, Examples of Semi-Submersible Platforms (Clauss, 1992) Effects of a Moonpool or Centerwell
Garrett (1970) theoretically investigated the excitation of waves inside a partially immersed open circular cylinder. The cylinder examined was thin walled and floating in water of finite depth. Garrett developed his solution using the linear diffraction method

PENTAGONE (1969) SEDCO 700 (1973)

SEDCO 135 (1965)




and solving the continuity equation inside and directly below the cylinder. He concluded that (1) the phase of the response was independent of depth, (2) resonances occurred at wave numbers close to those of free oscillations in a cylinder extending to the bottom and
(3) away from resonance, the response amplitude was less then 10% of the incoming wave amplitude if the draft of the cylinder is greater than 4of the wave length of the incoming wave (p. 433). Garrett also noted that the sharpness of a resonance depended on the value of the ratio of the cylinder draft to incoming wave length, with the cylinder of greater draft producing the sharper resonance.
Mavrakos (1985) continued Garrett's work by theoretically examining a stationary, floating, bottomless cylinder with a finite wall thickness in a finite depth of water. By dividing the fluid into three regions (outside the cylinder, below the cylinder walls and inside and below the centerwell) and then solving the continuity equations in these areas, Mavrakos conducted numerical experiments to predict the motion of the fluid inside the cylinder. He concluded that the ratio of the outer cylinder radius to the centerwell radius had "practically no influence" on the amplitude of the response of the fluid in the centerwell. He also noted that increasing the outer radius to inner radius ratio for a constant depth to inner radius ratio tends to narrow the width of the resonance region.
Mavrakos (1988) went on to examine the forced motion of a floating, bottomless cylinder with finite wall thickness in finite water depth. Again, starting from the continuity equations and using a linearized Bernoulli equation to determine pressure, he derived equations for the hydrodynamic coefficients in finite water depth. As these equations are




taken into deep water, the damping coefficient vanishes while the added mass blows up. However, Mavrakos did note that resonance situations arise in the centerwell at frequencies corresponding to the wave lengths of free fluid motions in a cylindrical container with a bottom.
All of the above studies were theoretical and all used linearized approximations. Knott and Flower (1980) performed experiments in which energy losses were measured for oscillatory flow through a pipe exit with various shapes and curvatures and used photographic visualization to examine eddy formation near the exit. Dividing the losses into linear damping and non-linear "eddy damping," they found that the linear losses were accounted for by the theoretical viscous friction of the pipe walls and the eddy losses were dependent on the radius of curvature of the exit, the radius of the tube and the oscillation amplitude and frequency. From the flow visualizations it was found that "eddies formed even at low amplitudes (of oscillation), creating a local turbulent field below the mouth," however "eddy losses only begin to act once a certain amplitude of motion has been exceeded, and that critical amplitude is strongly dependent of the radius of the lip." (p. 163). "Photographic evidence... suggests that in some cases where high levels of turbulence occur on the outflow the induction of eddies back into the mouth appear to cause a blockage with necking of the flow and high shear stress gradients near the walls" (p. 160). Knott and Flower also concluded that "geometrical similarity was the most important consideration... since this ensured similarity of the dominant parameters" and that "larger scale operation involving higher Reynolds numbers would, if anything, suffer less eddy losses owing to the delayed separation of boundary layers" (p. 164).




Aalbers (1984) analyzed the water motions measured in the moonpool of a model ship. Based on the theory of Van Oortmerssen (1979), which describes a system of two floating structures in waves, Aalbers assumed that "the moonpool water column oscillates in such a way that it may be replaced by a frictionless piston inside the vessel" (Aalbers, 1984, p. 578) and developed the equations of motion accordingly. His equations are shown below.
For the moonpool:
{pA(T +h) +ah } h +bh + b2hh1 + pgAh
+ ({d + pA(T+h)+ah} +(eh, +bh) + pgAs Eq. 2.18
+higher order terms= F .
For the vessel:
f(M + a, IN + b, + c~s +dsh sN+ +eh +E.21 {M~a}~+S~+~s~~h(~h)+~h(+h)Eq. 2.19 +higher order terms= Fh .
In equations 2.18 and 2.19, s is the vessels vertical displacement (heave), h is the relative displacement of the surface of the moonpool (absolute motion is the sum of the ship's heave and the relative motion), T is the vessel's draft, ah and as are the added mass of the moonpool and vessel (from potential theory), bh and b, are the damping coefficients of the moonpool and vessel (also from potential theory) and b2 is the quadratic or vortex damping coefficient.
Van Oortmerssen investigated the interference effect of neighboring vertical cylinders and developed the "hydrodynamic interaction coefficients" (eh, eh, dh,, dsh) to describe the forces experienced by one body due to the motions of the other body (1979, p. 342). The coefficients are symmetric, therefore eh,=eh and dhs= dsh.




Finally, Fwh is the exciting wave force. The higher order terms are due to Bernoulli's expression for fluid pressure and are neglected by Aalbers (1984, p. 558). Aalbers also noted that the moonpool can be excited at frequencies equal to one half the natural frequency due to the non-linearities.
Aalbers experimented with three moonpools with diameters of 0.1, 0.2 and 0.3 meters. The model's draft ranged from 0.15 to 0.3 meters with a maximum draft to moonpool diameter ratio of 2. Tests were conducted in water depths equal to the model draft plus 0.6 meters. The results showed a "dependency of the added mass and both damping coefficients on the size of the moonpool, especially on the diameter." It was determined that the added mass was proportional to the diameter cubed (ah oc dia3), the linear damping was proportional to the diameter to the five halves power (bh ocdia2), which agrees with Froude scaling, and the quadratic damping was proportional to the diameter squared (b2 oc dia2 ) (Aalbers, 1984, p. 561).
Large Spar Platforms
The FLIP (Floating Instrument Platform) manned spar buoy was a large spar platform used for oceanographic research in the 1960s. FLIP had a draft of approximately 91.5 meters and a tapered hull with a diameter ranging from 6 meters at the keel to 3.8 meters at the surface. Rudnick collected and analyzed motion information onboard FLIP at sea and found that the maximum response amplitude to wave amplitude ratio was 0.177 at a frequency of 0.059 Hz (Rudnick, 1967, p. 264).




Glanville et al. did a preliminary motion analysis study on the Chevron Spar design. Their analysis was conducted using the JONSWAP spectrum and found a low frequency response with a signal to wave response of 0.24 at a frequency of 0.043 Hz (Glanville et al., 1991, p. 67). They noted that "in general, the computed response agreed with the results of the model experiments, including the low frequency effect" (p. 62).
Finally, Niedzwecki et al. (1995) conducted experiments on scale models of the Chevron Spar Platform described in Chapter 1 to investigate the surge motion of a tethered structure. One of their results was that "Morison's equation can produce satisfactory results for inertia dominated structures with diameter to wave length ratios less than 0.2 if it is accompanied by the appropriate modifications (for non-linearities)" (Niedzwecki et al., 1995, p. 105).




CHAPTER 3
CENTERWELL MATHEMATICAL MODEL DEVELOPMENT
Momentum Equation for Heave
Figure 3.1 shows the spar and centerwell configuration for which the equations of motion will be derived. It will be assumed that the centerwell air water surface has a uniform, level elevation. Two centerwell levels are of concern: the relative centerwell level or depth (h) measured from the bottom of the spar (and always a positive value) and the absolute centerwell level (ha) which is the displacement of the centerwell water surface measured from a fixed horizontal datum (the still water plane).
The centerwell is bounded by the spar inner diameter, D1, and has a cross-sectional area A, 1rDI The spar outer diameter, Do, also has a corresponding cross-sectional area, A0 2 Finally, an orifice plate may be placed at the bottom of the centerwell with an orifice diameter, DR, and area, AR = 4 A useful non-dimensional parameter is the ratio of the orifice area to the centerwell area referred to here as the "area ratio," AR= AR/A,.
The control volume used in the derivation of the governing equation is shown in Figure 3.2. The control volume (CV) is the fluid in the spar (i.e. the centerwell). It is




attached to the spar at the bottom and its upper surface moves with the centerwell surface.
Thus the control volume varies with time and has a volume V = hAI.
~Do
y
spar still .. ..... .
waterline ... V .............
S0 = still water dr aft
s = SPAR displacement centerwell from reference
h = centerwell depth S0 s So from bottom of spar
h
ha= centerwell surface displacement from reference
orifice lip
of = wave displacement
orifice
DR on Fig. 3. 1, Spar and Centerwell Configuration and Definition of Terms. (not to scale)

CV
Center Well
Orifice
Fig. 3.2, Control Volume and Control Surface for deriving the Centerwell Equation of Motion. (not to scale)




The integral form of the conservation of momentum equation for the control volume is:
F= f : JpdV + f p(Vr ii)dA Eq. 3.1
Ot or as
In equation 3.1 pi is the linear momentum in the differential volume, dV, at any time and pv(v, i) is the flux of momentum through the differential surface area, dA. Aalbers modeled the moonpool motion by assuming it was a "frictionless piston." The draft to centerwell diameter ratio for the Chevron Spar and thus the experimental model is approximately 8, while that of the vessel examined by Aalbers was 2. Thus, based on Aalbers success and the large draft to diameter ratio it is assumed that the fluid motion in the centerwell is restricted to the direction of the spar axis and that it has a uniform velocity equal to ha.
With the above assumption and noting from Figure 3.1 that h = SO -s+ha, the control volume term of equation 3.1 becomes the expression in equation 3.2.
d (JfdV = p_4_( fr kadV) = d f ( aAh) =Pd [ aA (S s+ha)]
di t~ v Pyjtj v -dtaJ dt
Eq. 3.2
Taking the derivative and recognizing that both s and ha vary with time while So is constant results in equation 3.3.
pd V pA [ha +ha (a = phAa +pA1h pAh.a
Eq. 3.3




Next consider the flux term of equation 3.1. Since the relative velocity of the upper surface of the centerwell control volume is zero, there is no flux of momentum through this surface. The only surface through which fluid and momentum may flow is the open surface at the bottom which is bounded by the orifice. The unit outward normal vector is downward on this surface. Therefore, if it is assumed that the flow is uniform and in the direction of the spar axis, vr V = 'vR, where VR is the average relative velocity through the orifice.
Applying the continuity equation to this control volume and this control surface gives the following:
Vr "fl -Vr -- AR
AR AR
This equation expresses the relative velocity over this portion of the control surface. To find the flux of momentum of the fluid, the absolute velocity must be determined. Under the same assumptions (i.e. uniform velocity in the direction of the spar axis) the momentum per unit volume of the fluid in the orifice is given by: pvAT_9+4)= P( + AA,+ ( AR AR AR
Thus the flux term in the momentum equation becomes equation 3.4, CS AR R -ARI" ARI"- l ):' .
Integrating over AR (the area of the orifice) yields equation 3.5 below:
cSI V(V -- PA 2 + ~A Ak _A'-I_ 2 -_ k ---IR]2 +AI &/a)k f V(V,-fi d( = A A A A a A .
CAR AR AR AR




= p r2 h -i- h +2 I2 Eq. 3.5
AR a AR AR a AR
Combining equations 3.3 and 3.5 results in equation 3.6,
F = pA hk a +h a) + ipA a a +2 a I
AR AR AR
PAIhha + PAIaA R -2ha + 2 AR (h2 2ha +s2]. Eq. 3.6
Finally, the conservation of linear momentum for the centerwell control volume reduces to equation 3.7.
YF=pAha +PA[l (haS) 2= pAihha +pA I(1- Alh2. Eq.3.7
Inviscid Forces on the Fluid in the Control Volume
Next consider the inviscid forces acting on the fluid in the control volume. The forces can be divided into body forces (FB) and normal surface forces (Fp) as shown in equation 3.8 below:
F=F + Fp pf dV- JpfdS, Eq.3.8
cV S
where fB is the body force per unit mass, p is the pressure and S is the surface area of the centerwell control volume.




If the only body force considered is gravity, then equation 3.8 becomes:
F = FB + Fp = -pgA~h + f pdS. Eq. 3.9
S
Due to symmetry the pressure forces on the vertical sides of the centerwell have a net value of zero. The pressure forces on the upper and lower surfaces of the centerwell are considered separately. On the upper surface, the pressure is atmospheric thus the integral over this surface is:
f pfidS = patinA1. Eq. 3.10
upper surface
At the bottom surface the pressure has both hydrostatic and dynamic components. The hydrostatic component is time dependent due to (1) the position of the spar base varying with time and (2) the time variation of the water surface due to waves. The variable dynamic pressure is the result of the relative motion between the base of the spar and the surrounding fluid. As noted in Chapter 2, Sarpkaya and Isaacson state that a structure must span more than approximately one fifth of the incident wave length before the system enters the diffraction regime where the long wave approximation does not apply (Sarpkaya and Isaacson, 1981, p. 382). For a structure such as the Chevron Spar with a 43 meter beam as described by Glanville et al. (1991), this corresponds to a minimum deep water wave period of approximately 12 seconds. Energy spectra for deep ocean waves show that the majority of deep ocean wave periods fall within a range of 2.8 to 21 seconds (Ochi, 1997) and the 100 year storm modal period used in the JONSWAP spectrum by Glanville et al. (1991, p. 65) for their analysis of the Chevron Spar was 14




seconds. Therefore, though near the limits of applicability, the Froude-Krylov hypothesis and the long wave approximation should be adequate for the present study.

_-r
Orifice
Fig. 3.3, Spar diagram to scale.

Consider the spar illustrated in Figure 3.3. This figure is a scale diagram of the spar model used in this thesis which is geometrically similar to the Chevron Spar. The draft to centerwell diameter ratio for this spar is approximately 8 and the orifice area is one half of the total centerwell cross-sectional area. This has been arbitrarily set as the minimum orifice area for this analysis.
Given the relatively small size of the orifice lip in comparison to the centerwell depth, it was assumed that the orifice lip imparts no significant acceleration to the

Centerwell
-Fr




centerwell fluid, but rather serves to accentuate the turbulent eddies which Knott and Flower found to form at the mouth of a pipe under oscillating flow (Knott and Flower, 1980). Therefore, corrections for these flow variations due to the presence of the orifice lip will be included in the non-linear viscous damping coefficient. The validity of this approximation can be determined by how well the derived equations predict the experimental results.
Assuming deep water waves and using the above approximations, the pressure on the lower surface of the control volume is given by the expression in equation 3.11. The hydrostatic portion is assumed to be due to the water level outside of the spar.
Power srface- 1 2h" -g(S -s iekz)]+Patm" Eq. 3.11
Integrating equation 3.11 over the centerwell cross-sectional area (A1) at the bottom of the control volume and recognizing that the unit normal to this surface is downward results in equation 3.12.
f pldS = f 2 (S s + Re Pkzj ( S
A,1 ij 2 a + Pat} I
=p f-!dS+phkA, +pg(So -s+ iek)A, -ptmA,. Eq. 3.12
Using the relation described in equation 2.3, the unsteady term may be reduced to the in-phase and out-of-phase terms below (note the potential term includes both the centerwell velocity potential, the spar velocity and the incident wave potential):




p fdS = -a,, ie") -bh -l a bh-,Eq.3.13
where a is the added mass and b is the potential damping coefficient. The results are separated into terms involving the relative velocity and acceleration between the centerwell fluid and the wave and between the centerwell fluid and the spar, thus leading to hydrodynamic coefficients for the wave (aw, bw) and the spar (as, bh,).
Incorporating equation 3.13 into equation 3.12 and combining the result with equation 3.10 results in the pressure force given in equation 3.14.
f pndS = patmZA, -a. (ha-ie)-bw ha-flek')-a, ( -)-b, (haS
1 .2
+ 2ph Ai +pgSo -s+ ek )A -PatmAl
=-(a as)a -(b, +bs)ha +aN +bs, +ph A, + pg(So s)AI
+ekz a(a +b, + pgA9 -).
Eq. 3.14
Substituting equation 3.14 into equation 3.9 and recalling that h=So-s+ha yields the final form of the inviscid force term in equation 3.15, where aT = a, +a, and bT =bw +b .
F = -pgA, (So s +ha) arTa -bTha +a,5 +b ,s + Iph A, +pg(So -s)AI
+ekz (a.i +b + pgAi 7)
= -(aTha +bk,,h +pgAha pha A,) +a,,+b +e K(ai+bw4+pgAz71).
Eq. 3.15




In equation 3.15, the added mass and potential damping coefficients should be dependent on frequency. However, as shown in Figure 2.2, the added mass is nearly constant with frequency for the draft to diameter dimensions of this centerwell (Sabuncu and Calisal, 1984).
Viscous Damping
Equation 3.15 does not include the damping effects due to skin friction and vortex generation at the bottom of the centerwell. Aalbers (1984, p. 578) and Knott and Flower (1980) found that a non-linear viscous damping term was required to account for these effects. Therefore, a viscous damping force dependent on the relative velocities between the spar and fluid is introduced as given in equation 3.16.
Fv = -b, ha -- ha-s- l). Eq. 3.16
Adding this force term to equation 3.15 results in equation 3.17.
SF = alha +bT +b ha -s-ek--Aa +pgA iha Iphf AJ +a,
+ b+b ha -ek' & +e' [a,+ (b, +bha -&-ekz 1)i +pgA, 7]
Eq. 3.17
Complete Equation for Heave Motion
Combining equations 3.7 and 3.17 and rearranging terms and dividing by the total mass (pAzh+aT = pA, h(1 +Cm +Cm2) = pAih(1+Cmr)) results in equation 3.18, where Cmr




is the total added mass coefficient, Cm is the wave added mass coefficient (a, = CnpAih) and cm2 is the added mass coefficient for the spar (a, = Cm2pAPh).
PAihha +pAj 1- AI h2 [aTha + (bT +bha -s-e )a +jpgAzh. P- a A]
+a,3+9 b, +bv a e e [a.+(b. +b ha -A-ek?) +pgAz97
~bT +b~ a -s'-e g11 Fi,2 (,A1 t2
ha + 'ha + g ha h2a- 1 2=
pAh(1+cmr) h(1+Cmr) h(1+CmT) 2 AR
cm2 (b, +bIha --ekj)
+Cmr pAzh(l +Cmr )
c b. +b,lha -A- ekz
+e kz (b+ b4+ek.i)
1+c r pA, h(l+c,,r) h(l+Cmr)
Eq. 3.18
Recall Aalbers equation for the moonpool fluid motion in Chapter 2 (equation 2.18). The correspondence between the terms in Aalbers' equation and the parameters shown in Figure 3.1 are given in table 3.1.
Converting Aalbers' terms in equation 2.18, and rearranging the terms and dividing by the total mass of the moonpool results in equation 3.19.
+ h b + b a g h
pAz h(1+c)a pAh(1+cr) h(l+cTr) Eq. 3.19
Eq. 3.19
b c ra
+ih ore tr-,1 Cm2
+higher order terms = s- + +FWh




Table 3.1, Comparison of term in Aalbers' equation (Aalbers, 1984) and the parameters in Figure 3.1. Aalbers term Figure 3.1 term
h ha-s
(moonpool surface displacement with
respect to vessel still water line relative
to the vessel)
T+h h
(moonpool depth)
bh b
(potential damping coefficient)
b2 by
(viscous damping coefficient)
A A1
(moonpool cross-sectional area)
dhs, ehs as, bs (possible equivalent terms)
(hydrodynamic interaction coefficients)
Converting Aalbers' terms in equation 2.18, and rearranging the terms and dividing
by the total mass of the moonpool results in equation 3.19.
ha b ha + -- b, + g h
+ pAh(1+ pAh(1+cmr) h + Eq. 3.20
Eq. 3.20
b c
+ higher order terms Ah(1 s CmT) CmT Fh
pA h(1 + cer 1+cmTr
Aalbers modeled the moonpool as a separate structure, a "frictionless piston," and
used the method of Van Oortmerssen (1979) to develop "hydrodynamic interaction
coefficients" to account for the interaction between the moonpool and the vessel.
Equation 3.18 does not use Van Oortmerssen's method. Even though the centerwell fluid
is assumed to move in a uniform manner, it is not treated as a separate structure in this
thesis. Rather, the centerwell is modeled as a fluid coupled to the spar through viscous
and normal forces, which both excite the motion and restrict it. Equation 3.18 contains




terms which are similar to (and are here assumed to be equivalent to) the hydrodynamic interaction terms of Aalbers.
The other major difference between equation 3.18 and equation 3.19 is the formulation of the viscous damping term. Aalbers does not include a wave induced water particle velocity term in his development of the relative velocity. This should be a minor component for the deep draft spar but is included in equation 3.18 for completeness. Aalbers was not considering a deep draft vessel in his derivation and so the lack of this component is puzzling.
Equation 3.18 is simplified to equation 3.20 by using the coefficients defined below.

Eq. 3.21

total linear (potential) damping coefficient: spar linear (potential) damping coefficient: wave linear (potential) damping coefficient: non-linear (viscous) damping coefficient: hydrostatic restoring force coefficient:

CL= -bT BW B
pAIh(l+c )=
BS = b,
B =pAzh(l + c.T)
CN= b
CN=pA, h(1 + c,Tr
g
Kh(1 +cmT)




non-linear term coefficient: E I
h(1 + Cm)
spar added mass ratio: Cm2
- I+ CmT
wave added mass ratio: = Cm
1+ CmT
The experimental testing was conducted without waves and with forced motion of the spar. Removing the wave forcing from equation 3.20 results in equation 3.21 which is the equation to be used to analyze the experimental data.
ha + (CL -CN" a S) a l+Kha _EK1- h212 s +CN ha -k2)S"
Eq. 3.22
Analytic Approximation to the Complete Equation for Heave
Equations 3.20 and 3.21 lend themselves quite nicely to numerical solution since they may readily be separated into a system of first order coupled differential equations. They may also be coupled with similar equations for the motion of the spar itself
However, it is desirable both from a computational efficiency standpoint and from an application standpoint to obtain an analytic solution to the equations if possible. Such a solution would allow the direct application of the resulting spar Response Amplitude Operator function to a wave spectrum in order to obtain the response spectrum of the system in random waves.




When the spar displacement (s) is small compared to the spar draft (So), as is undoubtedly the case for the deep draft Chevron Spar, the term h = h" -s + SO may be approximated by h = S. Several methods of converting the non-linear damping to an effective linearized damping were discussed in Chapter 2. These methods are usually applied to less complex flows than those in this problem. However, an approximation to the viscous damping in the form of that discussed by Clauss et al. may be appropriate. Using the approximation discussed by Clauss et al. resulted in the effective damping coefficient given in equation 3.22.
beff =b, 1 k- I = b,, 8-o H cos6- S cosO-ek Eq. 3.23
where HA q5, S~mp and 0 are the amplitudes and phases of the motion of the absolute centerwell level and the spar, respectively, and 7o is the amplitude of the wave which is assumed to have a cosine form. This approximation is discussed in greater detail in Appendix A.
In their discussion of the linearization of the viscous damping, Clauss et al. noted that, when linearized, this force "consists of a linear component with the same frequency (as the wave), and an additional term with triple the frequency" (Clauss et al., 1992, p.237). Equation 3.22 does not account for this triple frequency effect and thus a small discrepancy is expected near one third of the synchronous frequency in the comparison between a numerical solution of the equation of motion and this analytic approximation.




Utilizing the further approximation that z --S when the draft (So) is large compared to the structure displacement (s) and the wave amplitude (17o), the damping terms can be combined into a single effective linear term as shown in equation 3.23 below.
+~ +Ceha + Kha + EL 2 2 jh] ls+Beg+ekz(pwii+Bu+K77)
Eq. 3.24
8 HS
C' =CL+CM q 8 O L AWcoo)~ SAWcos(9)-ekz
C e = BS +C 8 --cor 77HA 40 ) _rSA CO40) -- e I
Be= s C31 r 77o 71o OO)- k
8 co, )BWe = BW + CN 8 coro H co) cos(O)-ekz
Note that the 8c9, /37c parameter is that of Clauss, Patel and Berteaux, however an equally valid parameter of 2o-,. 2/fr from Ochi could be used.
Equation 3.23 still contains the non-linear terms associated with the E coefficient. Examination of the relative magnitude of the terms in equation 3.23 may afford some insight into the method to be used. Specifically, recognizing that E is inversely proportional to the spar draft suggests that the associated terms may be neglected or approximated by a perturbation technique with E as the perturbation parameter.
The first step is to non-dimensionalize equation 3.23 by recognizing that all terms have dimensions of acceleration and then dividing by the gravitational acceleration (g). Assuming that the solution is sinusoidal, Table 3.2 shows the orders of values (where appropriate) and the orders of magnitude of the components of equation 3.23.




Table 3.2, Order of Magnitude of Individual Terms in Heave Equation
Term (s) Approx. Value Order of Magnitude
ha, ha,ha,s, s, N, q, i, -1-10 (m, m/s, m/s2) 1 10
g 9.8 m/s2 10
h, So -200 m 100
AR1 1 0 to -0.5 0.1
R
A, -500 m2 100
Cm, c cM, ~0.1 1 0.1 1
p 1025 kg/mn3 1000
e-kSo 0.002 (for T = 12 sec) 0.001
Ce, Bse, Bwe (see below) -0.1 1
Assume that the damping coefficient takes the form of b= pCf As, where Cf is the coefficient of friction and As is the surface area. Consider As to be the surface of the interior wall of the centerwell so As = rDSo (recall D is the centerwell cross-sectional diameter) and Az = iDJ From Hoerner (1965, p. 3-11 to 3-13), Cf can be estimated to be of order 0.1 to 1. The result is the following equation for the damping coefficients of equation 3.23.
b pCfrD, h Cf 0(0.1)
pAh(+C)p D h(+c)2 D ((0.0+c1m))(10)0(1)1)
pAzh(1+ cm) p~r1Dh(1+ cm D (1+ cm) O(10)1




Table 3.3, Orders of Magnitude of Complete Terms
ha sq_ o(1)
g 'g'g 0(10) = 0(0.1)
g g g O(10)
CL BS B C B Bw 0(0.01) or 0(1) ( t
a s w e so we _lO10' to O01
g' g' g 'g' g g O(10)
K 1 o ()
K I- O (0.01)
g h(1+ cm) 0(102)O(1)
E 1 0(i) = 0(10g h(1 + cm) 0(10)0(102)0(1)
(1-)2 2
AR 2 a+I O(0.o1)0(1) o(0.1)
(- -R)h 2(1- )ha +(1- )2)
Ps w c,. 0(1) -0(0.1)
g g g(1 +c C) 0(10)0(1)
F e" Oz(10-3)
F- e (= (. + B,,i + K3) (O(0.1) + O(0.1) + O(0.1)) 0(10-1)
g g 0(10)
If the relationships of Aalbers (1984) are used, the order of magnitude of the prototype size damping is ~0(0.01)0(D)= 0(0.01)0(100)= 0(1). Applying the
estimates for individual terms in Table 3.2 to the non-dimensionalized terms in equation 3.23, results in the orders of magnitude for the complete terms in Table 3.3. The wave force terms have been combined into a single term Fw which has units of force per unit mass.
Thus, the order of magnitude of the non-dimensionalized terms of equation 3.23 may be summarized below.




h Ch Kh a E 1 1 p Bs F
a +ea a +- 1 2 2 s e + w
g g g g AR g g g
O(0.1) O(0.1) O(0.01) O(10-4) O(0.1) O(0.1) 0(10-5)
Therefore, it is readily apparent that both of the terms associated with the E coefficient and the wave forcing are small compared to the rest of the terms. The wave forcing terms are linear and may be incorporated in the first order equation. The other small term may be treated by the perturbation technique described by Ochi (1990, p. 398), with the perturbation parameter s= E= 1/S o(1 + c). Letting ho be the first order solution and h, be the second order solution, equation 3.23 may be separated into the equations below.
(a) h o +C,ho +Kho =/ s+BeS+Fw and
(b) hi +Ceh1 +Kh, = 1- 0 2 Eq. 3.25AR) 2 0
If Fw, s, ho, and h, are assumed to be sinusoidal, equation 3.24 (a) will result in a solution of the form ho = f, (Ao cos t + Bo sin cot) where fa is the forcing function amplitude. Substituting this into equation 3.24 (b) and manipulating the squared terms will result in a solution of the form h1 = fa(A1 cos2cot +B1 sin2cot +Y). The final form of the solution is given in equation 3.25. Note that this equation predicts a mean centerwell surface level displacement (fas Y) which results from a reduction in pressure at the bottom of the centerwell due to flow through the orifice.
ha = f [(A0 cos cot + Bo sin cot)+ e(A1 cos2cot + B1 sin2cot) + 6Y] Eq. 3.26




A response amplitude operator (RAO), a frequency response function (H(o)) and the lead angle (0b) can then be determined. The complete analytic solution to equation
3.23 and the equations for, A0, Bo A1I, and B1 is given in Appendix A.
To summarize, the approximations made in obtaining an analytic solution to equation 3.20 are: (1) the wave amplitude and the amplitude for the spar and centerwell water surface motion are small compared to the spar draft so that h =S', and the position of the bottom of the spar is approximately -So, (2) the non-linear damping term may be approximated in the manner of equation 3.23 to yield an effective linear damping term, (3) the terms associated with the E coefficient of equation 3.20 may be assumed to be small for large spar drafts and treated by the perturbation technique with the perturbation parameter 6 E. The determining assumption in these approximations is that the spar in question is deep drafted. Thus, the solution developed should not apply to the moonpool experiments of Aalbers, however even Aalbers ultimately linearized his equation by simply neglecting the non-linear terms associated with the E term of equation 3.20. The final test will, of course, be the ability of the solution to correctly reproduce the experimental results to be described in the next chapter.




CHAPTER 4
EXPERIMENTAL RESULTS
Model Description and Scaling
Experiments were conducted in a wave tank at the University of Florida Coastal and Oceanographic Engineering Laboratory. The water portion of the tank is 1.2 meters (4 feet) deep, 0.6 meters (2 feet) wide and 28 meters (92 feet) long. The sides are constructed of 1.2 meter (4 foot) square glass panels.
The model was constructed of concentric transparent Plexiglas tubes capped at the end except for the centerwell area. The transparent model in conjunction with the transparent side panels of the wave tank allowed visual observation of the centerwell surface level displacement. Such an observation was vital to locating and examining the synchronous motion noted in Chapter 1.
Following the lead of Knott and Flower (1980, p. 164), it was decided to maintain geometric similarity as much as possible between the model and the Chevron Spar prototype. The size of the available tanks for conducting the experiment and the desire to minimize bottom influence restricted the model draft to approximately 0.6 meters (2 feet). To maintain geometrical similarity, the outer diameter was approximately 13 centimeters (5 inches) and the centerwell diameter 7.6 centimeters (3 inches). These dimensions yield




diameter-to-draft ratios of 0.22 and 0.13, respectively, which are those of the Chevron Spar design. The resulting length scale is 1/325 or 0.003
The standard procedure in ship modeling is to maintain Froude scaling between the model and the prototype. The total resistance is determined based on model tests and the skin friction is calculated mathematically based on Reynolds number. The difference between the skin friction and the total resistance is called residual resistance and is assumed to be a function of the Froude number only. Thus the residual resistance may be scaled directly from the model tests and the total prototype resistance is found by adding the scaled residual resistance and the skin friction calculated for the prototype Reynolds number (Todd, 1967, pp. 291-293; Randall, 1997, p. 159).
However, this leads to a problem for the oscillating flow of the spar platform in waves since the Reynolds number varies, making the skin friction coefficient variable with time. For the spar itself, many authors neglect the effects of viscous (Reynolds number dependent) forces all together, thus eliminating the problem. (Kokkinowrachos and Wilckens, 1974, p. 99). Unlike the solid spar, the motion of the fluid in the centerwell may be influenced by these viscous friction forces. Through physical experiments, Aalbers found that quadratic damping is proportional to the diameter of the moonpool squared (Aalbers, 1984, p. 561). Thus, Froude scaling may be used in the experiments and the scaling of friction forces to the prototype size should be adequately approximated by Aalbers relationship.
As constructed, the model's total length (draft plus freeboard) was 76 centimeters, the outer diameter, Do, was 14 centimeters and the inner (centerwell) diameter, DI, was 8




centimeters giving a wall thickness of 3 centimeters. The mass of the model was augmented with lead weights at the bottom and inside the model wall. This gave the model an overall mass of 6.6 kilograms (14.5 pounds) and a still water draft of 64.5 centimeters. The model-to-prototype scaling factors thus obtained were 1/310 for the length scale and L./LL = a1-/310 = 1/ 17.6 for the time scale (Froude scaling).
Three orifice plates were constructed to install on the bottom of the spar and partially obstruct flow through the centerwell. Orifice size was selected to provide as significant an obstruction as possible without exceeding the arbitrary limit of 50 percent area reduction discussed in Chapter 3. Table 4.1 summarizes the orifice diameter (DR) and orifice area ratio (AR/A,) information and Figure 4.4 illustrates the model and orifices to scale.
Table 4.1, Orifice Plate Diameters
Orifice Plate No. Orifice Diameter (DR) DRID AIA = AR
1 6.9 centimeters 0.86 0.74
2 6.4 centimeters 0.8 0.64
3 5.8 centimeters 0.72 0.52
Free Response Tests
Free response experiments were conducted by displacing and releasing the model in still water. The tests were conducted first with the centerwell fully open (referred to here as the "open bottom" configuration) and then with the centerwell completely closed (the "closed bottom" configuration). In the closed bottom configuration, the centerwell was filled with water to maintain the same draft as the open bottom configuration.




Do= 14 cm DI= 8 cm DRl= 6.9 cm DR2= 6.4 cm DR3= 5.8 cm
Centerwell
Orifice

-4DR-

Orifice 1 DR/ D = 0.86 AR/ A, = 0.74
Orifice 2 DR/DI = 0.8 AR/A, = 0.64
Orifice 3
- DR/D, = 0.72 AR/ A, = 0.S2

76 cm

Fig. 4.1, Scale drawing of model and orifice plates. A video recording was made of the model's motion against a grid placed on the tank side panel with a line on the model to mark the still waterline. Following the experiments, the video recording was reviewed frame by frame and the waterline mark noted as it passed each grid line. Special note was made when the spar crossed the zero level and at each peak and trough of its motion.
The system was assumed to be linear and the damped natural frequency (fd in Hz, COd in radians/sec) and the logarithmic decrement (b) were calculated from the video data. Once these values were obtained, equation 4.1 and the definitions below were used to

D, Do




calculate the added mass coefficient c. = a/m and the linear damping ratio
= b/2 (m+a)k. Table 4.2 shows the results of these calculations for both model configurations.
m(l + cm )N + bs + ks = 0
6 k k Eq. 4.1
P_8 = a=---(m+a 2
Table 4.2, Summary of Free Response Damped Natural Frequency (fd),
Added Mass Coefficient (c,) and Damping Ratio 6
Run Configuration fd (Hz) Cm 8)
1 open bottom 0.614 0.023 0.035
2 open bottom 0.610 0.037 0.032
3 open bottom 0.606 0.050 0.039
16 closed bottom 0.600 0.073 0.021
17 closed bottom 0.598 0.080 0.022
A noticeable difference exists between the added mass and damping ratio for the two configurations. The added mass coefficient of the model increased when the bottom is closed off indicting that the more bluff the body the greater the added mass. The damping ratio increases when the bottom is opened, presumably due to the increase in surface area subject to friction and the vortex generation by the bottom edge of the centerwell. Both of these results were expected.
The value of the damping ratio was of the order of magnitude (-0(0.01)) expected. The added mass coefficient is comparable to that of an ellipsoid with a large major axis to minor axis ratio and flow parallel to its major axis (Liggett, 1994, p. 114). A worthwhile comparison would be between the added mass obtained in the experiments and that




computed using the method of Clauss et al. (1992, p.28 1 and shown in Figure 2.3). This calculation results in added mass coefficients of 0.082 and 0.098 for the open bottom and closed bottom configurations, respectively. Using the two computed added mass coefficients, damped natural frequencies (fd) of 0.596 Hz and 0.592 Hz, respectively, were calculated. The coefficients are relatively close to the values calculated from the free response test data and the damped natural frequencies are certainly within the error of the experimental data.
Only a few of the experiments produced useable data. The rest were adversely affected by horizontal model motion disrupting the video focus, poor visibility of the waterline mark, the model pitch motion, and other problems associated with the unrestricted motion of the model. Despite the problems encountered with the free oscillation experiments and the measurement techniques, the experiments did provide useful information about the behavior of the spar with a centerwell. The possibility of using the added mass coefficients of Clauss et al. was confirmed. Also, the damping ratio was found to be small in both configurations.
Configuration for the Forced Motion Experiments
The bulk of the experimentation was conducted by directly forcing spar motion and measuring the response of the centerwell water level with electronic instrumentation. The instrument output was recorded using an analog-to-digital conversion card and Global Lab data collection software. The sampling frequency was 60 Hz with the run times varying from 2 to 4 minutes. The data were analyzed using programs written in MATLAB




which allowed for interactive graphical examination of the data and the use of built-in signal processing functions to analyze the time varying experimental results.
To conduct the experiments, a frame was constructed to house the model, provide a platform for the drive mechanism and instrumentation and to restrict the motion of the spar to heave alone. The frame consisted of a square upper and lower plate 46 centimeters (18 inches) on a side. The plates were separated by 122 centimeter (48 inch) rods at each comer. The frame stood on the lower plate and the model was located inside at an elevation which provided approximately 36 centimeters of separation between the bottom of the model and the lower plate at the lowest point in the model's stroke. Attached to each plate and running through special tubes in the wall of the model were four guide wires designed to minimize twisting and horizontal movement of the model.
Attached to the top of the model was a collar. Through the collar, on either side of the model, were two drive rods which penetrated the upper frame plate through slide bearings. Above the upper plate, the drive rods went through, but were not attached to a lower yoke. The drive rods were then connected to an upper yoke. A strain gage was located between the upper and lower yokes. A third drive rod was attached to the bottom of the lower yoke, directly below the strain gage. This drive rod was connected to a fly wheel on its lower end. The fly wheel was attached to a variable speed, rotary AC motor which provided the driving force for the tests. The apparatus is illustrated in Figure 4.2.
The instrumentation consisted of the strain gage mentioned above; a position indicator attached to the upper yoke to measure the spar displacement; a capacitance type




wave gage inside the centerwell for measuring the centerwell water level; and a capacitance wave gage located outside of the system.
The design of the position indicator was based on the capacitance wave gage. This device consisted of a stationary metal tube covered with rubber inside a second metal tube which was free to move over the stationary tube. The stationary tube was mounted to the top of the upper plate of the model frame and the moving tube was attached to the upper yoke of the drive assembly. As the yoke moved up and down, the moving tube exposed varying amounts of the stationary tube causing the capacitance to change. This changing capacitance was measured to give the position of the spar. This instrument, which was designed and constructed at the Coastal Engineering Laboratory, proved to be a reliable and stable position indicator with a linear response.
The motor initially used to drive the spar could not provide periods longer than approximately 1.5 seconds and motion in vicinity of this period was uneven and strained. The ideal natural period of the centerwell (i.e. the natural period without accounting for the damping or added mass) is 1.6 seconds (1T, = 21rf! g/IS ), hence the period of interest for investigating the synchronous motion of the centerwell was longer than the upper limit of the driving mechanism. The result was that the synchronous motion was not witnessed during the initial testing due to the under-powered motor.
In addition to the problem of an under-powered motor, the drive linkage proved unstable. The circular motion of the motor shaft was not well translated to the desired linear heave motion and significant lateral motion of the drive rods was produced. While the upper plate drive rod bearings and the guide wires limited the motion of the model to




55
the heave direction, the oscillation of the drive mechanism resulted in the entire frame vibrating. Thus, the resulting centerwell response was adversely influenced by secondary vibrations of the frame inducing seiching in the centerwell.

Fig. 4.2, Frame and drive assembly.
The first motor was replaced by a larger motor capable of driving the model at longer periods. To minimize the lateral oscillations of the drive rods, the upper plate bearings were reinforced and the frame was weighted. These adjustments solved most of the initial difficulties, however the reinforced bearings produced excessive friction forces. The magnitude of the friction forces was such that the damping and added mass could not be extracted from the force data.

model position indicator

upper yoke 0

lower yoke 0 position indicator Drive Assembly
lowriye rd support assembly
dtori drive rod bearings
motor
upper frame plat10 c position
collar indicator
upper yoke
lower yoke
motor
drive
rod
122 cm
modelflwhe
drive rod
- bearing "
lower frame plate 3c
18cm -

strain gage
fly wheel




However, as previously noted, the free response tests indicated the method of Clauss et al. provided an adequate added mass estimate for the Spar and numerical means of estimating the damping coefficient were available (see Chapter 2). Thus the loss of these data was not considered significant to the overall accomplishment of this study.
Experimental Results
A total of 88 separate experiments were conducted. The experiments covered oscillation periods from 0.6 seconds to 6 seconds which correspond to prototype wave periods from 10.6 to 105.6 seconds. Shorter periods were not tested due to continued vibrations experienced by the frame at higher frequencies. In addition, as noted in Chapter 3, the long wave approximation becomes suspect at periods shorter than 12 seconds (prototype scale). The majority of the experiments were performed with a constant amplitude of approximately 2 centimeters, although some early tests used a 4 centimeter stroke. Analysis of the data showed that the stroke difference did not affect the results.
Notable was the absence of radiated waves during the experiments. Radiated waves where not observed or measured by the wave gage located outside of the system (except within the first centimeter of the model). Even the small response in the immediate vicinity of the model was minor, on the order of millimeters or less.
As previously mentioned, the model position and centerwell level displacement data was analyzed using MIATLAB software. Since the centerwell wave gage was mounted to the model itself, the resulting water level information was relative to the model position. The information of interest was the position of the centerwell level with




respect to a fixed coordinate system. This was obtained by the simple formula ha = h + s S,,; where ha is the centerwell level with respect to a fixed coordinate system (absolute centerwell level), h is the height of the water column inside the centerwell with respect to the base of the model and s is the model displacement (see Figure 3.1). Thus the data to be analyzed was itself derived data and not directly measured.
The data were analyzed using two methods. After deriving the absolute centerwell level (ha) a Fourier analysis was conducted using the Fast Fourier Transform in IATLAB to obtain frequency, amplitude and phase information for the relative and absolute centerwell levels and the model position. Phase was measured as a lead angle with respect to the model displacement. The average frequency bin width and therefore the accuracy of the frequency was 0.03 Hz. For the most part, the spectrum of the relative centerwell level and the model displacement gave solid single frequency responses. In all cases where multiple frequencies existed, the secondary frequencies were within the neighboring bin or the magnitude of the secondary response was several orders of magnitude less than the primary response.
This was not the case for the absolute centerwell levels. Though the majority of the tests showed the same response as above, five cases gave very different results. In all of these cases the frequency was high and the amplitude response was small indicating a decoupling of the centerwell motion and the model motion. This could be expected since four of the five cases were tests with no orifice where the coupling is weak. In the case where this occurred with an orifice (AR = 0.64), the frequency was near the top of the




experiment band (1.39 Hz, (A). = 2.2) and the response was only 2 percent of the model displacement.
The Fourier analysis yielded good response amplitude results, but the phase results left much to be desired. When the absolute centerwell displacement was simulated using the frequency, amplitude and phase information obtained by the Fourier analysis and the time series was compared to the data time series, the results were good for amplitude and frequency matching, but inconsistent for the phase. Also discernible from examination of plots of the data with time were variations in the speed of the motor itself Even the model displacement times series, which was the most consistent of the three sets, exhibited a small frequency variations due to instabilities in the motor speed.
The hope for correcting this situation lay in the singularity of the spectra. The spectra showed a concentrated spike exclusively at one frequency. This suggested that the data could be analyzed by locating the local maxima and minima for each time series. With the time and magnitude of the local maxima and minima determined, mean values for the frequency, amplitude and phase of the time series were calculated. This method is referred to from this point on as the "maxima-minima analysis."
The maxima-n-finima method produced excellent results. The information developed for frequency, amplitude and phase were evaluated by comparing a generated time series with the experimental data and qualitatively ranking the results as good, fair, poor or unsatisfactory. A rank of good was given to 46 of 88 of the data set comparisons (52 percent), 24 ranked fair (27 percent), 13 were poor but usable (15 percent) and only 5 were deemed unsatisfactory and unusable (6 percent).




A product of the maxima-minima method was the variance of the means of frequency, amplitude and phase which could be used as an indication of the experimental error. The frequency variance yielded a maximum standard deviation of 0.03 Hz which was consistent with the frequency bin width of the Fourier analysis. Drift in the position and level instrumentation gave maximum amplitude errors of 0.06 centimeters and 0.15 centimeters for the model displacement and relative centerwell level, respectively. This results in an error of 0.21 centimeters for the derived absolute centerwell level. The variance of the mean amplitude from the maxima-minima analysis showed average standard deviations of 0.02 centimeters, 0.14 centimeters and 0.12 centimeters for the model displacement, relative and absolute centerwell level, respectively. Therefore, error for the model displacement, relative and absolute centerwell levels was set at the average of these; 0.04 centimeters, 0.15 centimeters and 0.17, respectively. To examine the ratio of the amplitude of the absolute centerwell level and the model displacement amplitude, the errors above were converted to percentages of their respective measurements. This gave the maximum errors for the model displacement and absolute centerwell level as 1 percent and 9 percent, respectively, thus value for the error in the amplitude ratio was approximately 10 percent.
The phase error estimate was more difficult to calculate. The phase of the centerwell level motion was calculated with respect to the phase of the spar. However, since both the period of the centerwell motion and the spar were used to calculate the phase, their respective errors must be included in the phase error estimate. With a frequency error of 0.03 Hz (for each frequency in the calculation) and a maximum period




of 2.5 seconds, the phase error may be estimated by AO = 2(1/Af / Tm,) = 27 degrees. Table 4.3 contains a summary of the experimental error values.
Figures 4.4 through 4.7 present the experimental results as plots of the absolute centerwell level to model displacement amplitude ratio and phase angle difference between the absolute centerwell level and the model displacement as functions of the frequency ratio. The phase angle is a lead angle such that if the motion of the model is of the form cos(x), the absolute centerwell level response is cos(x + 4)) with 4) the lead angle represented in the plots. The natural frequency of the centerwell was expected to be dependent on the orifice size. Therefore, when comparing the orifice configurations, the frequency ratio for the plots is defined as the response frequency (o)) divided by the ideal natural frequency (co) given by co,, = -g/So From this point on in this thesis, the natural frequency of the given system will be referred to as o, or -K (see equation 3.21). Note that this frequency is also the "synchronous frequency."
Both the results of the maxima-minima and the Fourier analysis are presented. As previously noted, the amplitude ratio was consistent between the two methods, but the phase angle shows a decided shift for the data analyzed with Fourier analysis.
Reflected in the plots is the distinct peak in the amplitude ratio corresponding to the point where the centerwell level moves in sync with the model. In other words, the water in the centerwell moved as though the bottom of the centerwell was closed at the bottom. The synchronous response phenomena occurred in all cases except the no orifice case shown in Figure 4.7.




Table 4.3, Estimated experimental error values.
Data Estimated Error
frequency 0.03 Hz
model displacement (S) 0.04 centimeters
relative centerwell level (h) 0. 15 centimeters absolute centerwell level (ha) 0. 17 centimeters
amplitude ratio (hals) 10 percent
phase angle 27 degrees
For the data developed using the maxima-minima analysis, the phase plots show a zero phase angle at the frequency corresponding to the synchronous motion. However, the data developed from the Fourier analysis show a 30 to 40 degree lead angle for this point in all of the plots. Since observation confirmed the zero phase angle, the Fourier analysis seemed to have introduced some bias in the data. The maxima-minima analysis results indicated the observed zero phase angle. All comparisons made beyond this point are based on this data extracted by the maxima-minima analysis.
It was noted that the no orifice condition did not produce synchronous centerwell motion. Observation showed that the synchronous motion could be initiated without the orifice, but could not be maintained for any significant length of time. When the time series were processed, the short periods of synchronous motion were averaged out by both analysis methods.
The no orifice configuration has a significant response over an extremely narrow frequency range. The amplitude ratio for this configuration drops very rapidly as frequency moves away from the natural frequency of the system. As the orifice size decreases, the slope of this drop grows shallower and the frequency range increases. As




expected, this suggests a weak coupling between the centerwell and the model over most of the frequency domain when the orifice is not installed. This is also reflected in the great variability of the phase data as compared to the configurations with the orifice installed.
Examining the data, and especially the phase data, it is apparent that the overall centerwell response becomes more consistent as the orifice size decreases. In other words, the coupling between the centerwell fluid and the model itself is increasing with decreasing orifice diameter. This is expected since it is primarily the orifice which maintains the coupling between the centerwell and the model. This coupling may be attributed to the model acceleration term (,ujQN of equation 3.22 and the damping and induced vortex formation due to the sharp edge at the centerwell exit and at the orifice lip. Also, the non-linear terms associated with the E coefficient have a dependency on the orifice area and so have an influence on the coupling.
The lack of success in maintaining the synchronous motion in the no orifice case does not necessarily imply that such motion is not possible for this configuration. The absence of such a response in these tests may be due to inadequate control of the model oscillation frequency combined with a narrow frequency band for attaining synchronous motion. The very narrow frequency band over which synchronous motion occurs in the no orifice case makes this condition difficult to achieve. Slight shifts in the motor frequency as noted in the data analysis could easily cause the response to move into the regions where little or no response exists.
The frequencies at which synchronous motion occurred were estimated from the data plots and are presented in Table 4.4. The table lists the orifice diameter ratio (DR)




and orifice area ratio (AR) and the corresponding frequency, radian frequency and frequency ratio. Figure 4.3 displays the frequency ratio versus the orifice diameter ratio and includes a linear least squares fit line to demonstrate the excellent consistency of the data. In Chapter 4 this information will be used to evaluate the centerwell added mass coefficient.
Finally, referring to only the data developed by the maxima-minima analysis, there is a distinct dip in the phase angle for frequency ratios of less than 0.5. This may be attributed to the 2ow term in the analytic solution described by equation 3.26.
Table 4.4, Synchronous motion fre uencies (wn) by orifice size.
DR AR frequency (Hz) (o. (radians/sec) (.lwo
0.72 0.52 0.5974 3.7536 0.9629
0.80 0.64 0.5940 3.7322 0.9574
0.86 0.74 0.5922 3.7209 0.9545
1.0 1.0 0.5844 3.6719 0.9420
Synchronous Frequency Variation with Orifice Dia. Ratio
0.97
0.96
0.96
3
0.58 y -0.0749x+ 1.0175
0.95 R = 0.9877 F
0.95
0.94
0.70 0.75 0.80 0.85 0.90 0.95 1.00
DR/DIJ
Fig. 4.3, Plot of synchronous frequency with orifice diameter ratio.




Amplitude Ratio Plot for Orifice Area Ratio= 0.52 + maxima-minima x fourier
1.2
1.0
X
' 0.6 +
+
.- +_X
X + X +
0.2+
0.0- +
0.0 0.5 1.0 1.5 2.0 2.5
m0/)o
Phase Plot for Orifice Area Ratio= 0.52 + maxima-minima x fourier 90
xx x
x
+ + + X
6 0 _X +
x
~X
+30 +
0 4 x
S~r +
-30 x
" + Xx
-60
+ + + + +
-90 + x
0.0 0.5 1.0 1.5 2.0 2.5
w0/ o
Fig. 4.4, Results for experiments conducted with an orifice area ratio of 0.52




Amplitude Ratio Plot for Orifice Area Ratio = 0.64 + maxima-minima x fourier
1.2
1.0
o x
0.8
0.6
S0.4
x *
0.2 +
0.0 0
0.0 0.5 1.0 1.5 2.0 2.5
(0 0 o
Phase Plot for Orifice Area Ratio 0.64 + maxima-minima X fourier 90
+
+
60 + +X
x 30
x
~x
S0
-30
x
-o x
-60 +
-90
0.0 0.5 1.0 1.5 2.0 2.5
Fig. 4.5, Results for experiments conducted with an orifice area ratio of 0.64.




Amplitude Ratio Plot for Orifice Area Ratio = 0.74 + maxima-minima x fourier
1.2
1.0
" 0.8S0.6
+
~0.4
a x
X
0.2 --.
S++ +
0.0
0.0 0.5 1.0 1.5 2.0 2.5
0)/0o
Phase Plot for Orifice Area Ratio= 0.74 + maxima-minima X fourier 90
+ +
60+ + + X-4
+ X +
C 30 +
XX
-30
~x X.
-60 x
+ + +x
-90
-120
0.0 0.5 1.0 1.5 2.0 2.5
010 o
Fig. 4.6, Results for experiments conducted with an orifice area ratio of 0.74.




Amplitude Ratio Plot for the No Orifice Case + maxima-minima X fourier
1.2
1.0
0.8 X +
S0.6
~0.4
0.2 --- +
x+
+ xXX4 ____ +x__ +x0.0 3
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0)/0o
Phase Plot for the No Orifice Case + maxima-minima x fourier 90 x
60 x
+ + ++
30
+
0
x x x
SXX X x
C x x
S-30
-60
-904- + +
++
-120 +
-150-
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0/Oo
Fig. 4.7, Results for orifice experiments conducted with an orifice area ratio of 1.0 (i.e. no orifice).




CHAPTER 5
MODEL CALIBRATION AND CENTERWELL HEAVE DYNAMICS
Model Calibration
The equation of motion for the centerwell driven by induced spar motion without waves was expressed in equation 3.22 and is repeated below.
ha, + (CL +CAa -])Aa + Kha, +EL 1 --[- j =s +(Bs +E[( CR 2I)4
With the spar undergoing forced motion, the first assumption is that the linear damping terms are equal, that is CL = Bs. This seems reasonable since both terms describe the same interaction between the spar and centerwell. This results in four unknown parameters to be determined from the data described in Chapter 4; the linear and non-linear damping coefficients, CL and CN, the added mass coefficient associated with the spar acceleration and orifice lip, Cm2 and the added mass coefficient associated with the centerwell motion, Cm. If Cmr2 is assumed to be small compared to Cm than the terms of equation 3.22 involving the added mass may be approximated as follows:
PS = Cm2/1+ Cm, K= g/[So(1 + cm)] and E= 11[S,(I + c)]).
Note also from Chapter 3 that the added mass coefficient (cm) is embedded in the damping coefficients as i/(I + Cm)




When the centerwell is experiencing synchronous motion, there is no relative velocity between the centerwell and the spar. Thus, for this case (and with CL = B,), equation 3.22 becomes equation 5.1 below, ha + Kha Eil usS. Eq. 5.1
In the synchronous motion condition the centerwell acceleration must also match the spar acceleration and ha -usg = (1- Jus)ia. The E term was shown to be small in Chapter 3 and is smaller still without its relative velocity component (/h2). Thus, equation
5.1 can be further simplified to equation 5.2 below.
h, + Kha = 0. Eq. 5.2
Equation 5.2 is the equation of motion for a mass-spring system undergoing a free, undamped oscillation. The fact that the centerwell in synchronous motion is equivalent to such a system is an interesting result in itself and allows the added mass coefficient to be calculated from the synchronous motion frequency, aO, by C. So O/gco, 1. Table 5.1 lists the added mass coefficient thus determined by orifice diameter ratio and area ratio.
Table 5.1, Added mass coefficient derived from the synchronous frequency of the centerwell DR AR o, (radians/sec) Cm
0.72 0.52 3.754 0.078
0.80 0.64 3.732 0.091
0.86 0.74 3.721 0.097
1.0 1.0 3.672 0.127




Aalbers found that the added mass of the of the moonpool was proportional to the radius of the moonpool to the third power. Using this relationship, the following can be derived:
if D3 D
a= Cnm = c f4D So oc D - cm OC D --m m D2So SO
Since the synchronous frequency varied with orifice diameter, it was concluded that the diameter of interest is the orifice diameter. Plotting the added mass coefficients of Table 5.1 against the orifice diameter to spar draft ratio results in Figure 5.1. The plot shows a linear variation of the added mass coefficient with the orifice diameter to spar draft ratio. Equation 5.3 expresses this relationship.
1.4 s-1 -0.05 Eq. 5.3
Plot of Added Mass Coefficient vs. DR/So
0.13
*~0.120.11
0.10
00.09
0.08 y 1.3993x- 0.0483
-6 0.07 R = 0.9859
0.06 I I
0.08 0.09 0.10 0.11 0.12 0.13
DR/So
Fig. 5. 1, Added mass coefficient vs. the cube of the orifice diameter ratio




Next consider the non-linear damping coefficient. The model is essentially a pipe with fluid flowing through it, so the methods used to calculate flow through pipes should be applicable. The Darcy equation, which is used to calculate head loss in a pipe, is given in equation 5.4:
V 2
hL=k- Eq. 5.4
2g
where hL is the head loss, v is the velocity of the fluid and k is the resistance coefficient (Crane, 1976, p. 2-8).
The non-linear damping coefficient can be divided into components derived from pipe flow processes and represented by different resistance coefficients. These coefficients when summed should yield the non-linear damping coefficient, C.
The resistance coefficient for the pipe itself is k = fL/D where f is the friction factor and LID is the length of pipe compared to its diameter. Multiplying the head loss by the hydrostatic coefficient (K = g/So(1I+ c.)) will convert the form of equation 5.4 to that of equation 3.22, i.e.
g -+ hL =(f L f L lI IvI v. Eq. 5.5
So(l+Cm) D)2,(+ .D 2S" (I+ Cm)
The model also has an abrupt entrance and exit. The resistance coefficients for an abrupt entrance and exit are 0.5 and 1, respectively (Crane, 1976, p. 2-11). Since the flow is oscillatory, the bottom of the centerwell will act as an entrance over half the cycle and an exit over the other half Thus, averaging these resistance coefficients and applying the conversion of equation 5.5 gives the component of the non-linear damping associated with




the entrance and exit losses. The resistance coefficient for this component is defined by equation 5.6.
entrance ___t_ +I IvIv. Eq. 5.6
__er___ Ie~ ) vv 2(.5 1 Ivv 0.375
2 2So(1 +cm) 2 2 2So(1 +C.) So(1+ Cn)
Finally, the last component of the non-linear damping to be considered is the effect of the orifice itself From Crane (1976, p. 3-5) the volumetric flow rate, q, through an orifice in a pipe can be described by equation 5.7: CAR 2 ghq CA Eq. 5.7
[ 1- DR4
where AR is the orifice area, DR is the diameter of the orifice divided by the diameter of the pipe and C is a flow coefficient.
Converting the volumetric flow rate in equation 5.7 to the flow rate in the centerwell calculated from the absolute centerwell level velocity (ha) and squaring both sides leads to equation 5.8, with C' = 1/C2 and DR4 =AR2.
A, CAR 2hghoA h _C_haAi= ala 4h
l-DR4 AR AR 1-AR2
i2 C22ghz AR2) 2
a I->h =
AR2 1-AR2 L 2gC 2AR2
g h= C (1 =1 haa Eq. 5.8
So(1+cm) 2So(1+c) AR2
Summing the components in equations 5.5, 5.6 and 5.8 gives equation 5.9 which expresses the complete non-linear damping term.




CN L.75 +Y(f +C' A2 -1)iS(+m Eq. 5.9
Note that the C' term disappears for the no orifice case when AR = 1. Neglecting the linear damping term (CL) and ,us, a least squares fit of the amplitude ratio data for the no orifice configuration resulted in a value for fLID of 2.8.
Using this value, another least squares fit was performed on the data collected for the configurations with the orifice installed. The C' coefficient was found to have a frequency dependence which resulted in different values above and below the synchronous frequency ("o,). The equations below summarize the least squares fit results for the nonlinear damping (with linear damping neglected).
(f L D = 2.8
C' =2.65 for D <.8 and co D1
( D DR
C' = 6.5(1-- for all D and co > co,
Still neglecting the ,Us and the linear damping terms, the above relationships were applied and the equation of motion was solved both numerically and analytically. The numerical solution was achieved using an adaptive step controlled fifth order Runge-Kutta algorithm with an embedded fourth order algorithm written in Fortran 90. A time series was obtained and analyzed using the maxima-minima method. The analytical solution of Chapter 3 and Appendix A was likewise computed using a program written in Fortran 90 and later rewritten in MATLAB.




Figures 5.2 and 5.3 display the results of these solutions for the configuration with an area ratio of 0.74. Figure 5.2 depicts the ratio of the centerwell and spar motion amplitudes and the phase difference between the centerwell and spar motion as functions of frequency normalized to the ideal natural frequency of the system. The experimental data for this configuration is also depicted. Figure 5.3 shows the mean centerwell. level offset noted in the analytical solution of the equations of motion found in Chapter 4. This phenomena was not anticipated prior to performing the experiments and the effect cannot be extracted from the data collected.
The numerical and analytical solutions show excellent agreement between one another for the amplitude ratio and the mean centerwell level. Only the slightest of disagreements occur at very low frequencies and near the synchronous frequency. Both solutions show a small peak in amplitude at half the synchronous frequency due to the 2wo term found in the analytical solution. The numerical solution gives a 3D peak as discussed by Clauss et al. resulting from the non-linear damping term which was not accounted for the linearization process for the equation solved analytically.
A large difference in the phase results of the solutions at low frequencies corresponding to these 2o) and 3w effects and oscillations in the numerical solution phase at higher frequencies illustrates the extreme sensitivity of the phase to secondary effects. This explains the difficulty in extracting reliable phase information from the data discussed in Chapter 4.




Non-Linear Damping Amp Ratio Plot for AR = 0.74 + maxima-minima analytic numeric
1.2 1.0
"0.8 "0.6
E 0.4
0.2+
+ + +
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
whI o
Non-Linear Damping Phase Plot for AR = 0.74
+ maxima-minima analytic numeric 120 90
- 60
+
30 +
0
-30
S-60
-90
-120
0.0 0.5 1.0 1.5 2.0 2.5 3.0
()/0oo
Fig. 5.2, Mathematical model solutions for area ratio 0.74, non-linear damping only with Its = Cm2 = 0.




Non-Linear Damping Mean CW Level Plot forAR 0.74
- analytic numeric
0.E+00
-2.&-04
* -4.E-04
-8.E-04
-1.E-03
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0) /0)D
Fig. 5.3, Mean centerwell level variation with frequency determined from numerical and analytical solution of the governing equation.
On the whole, agreement with the experimental data is good. Similar results were obtained for the experimental data with other orifices. The solution amplitude ratios do not agree as well with the experimental data at lower frequencies, but the experimental data itself is somewhat questionable at these frequencies suggesting scatter due to additional secondary effects unrelated to the heave motion. Possible causes of such secondary effects are centerwell seiching and drive force frequency variations, both observed during the experiments. The maximum amplitude ratio for each solution with its corresponding frequency is tabulated in Table 5.2.
As expected, the frequencies correspond closely with the values used to calibrate the added mass term. The numerical solution does not reach the synchronous condition observed in the experiments and its peak is reduced as the orifice size approached the




centerwell diameter. However, the analytical solution does reach this condition in all orifice configurations. Figure 5.4 examines the results in the vicinity of the synchronous frequency.
Table 5.2, Maximum amplitude ratio comparison for non-linear damping only.
numerical solution analytical solution
AR o) (rad/sec) Amp Ratio o (rad/sec) Amp Ratio
0.52 3.742 0.97 3.750 1.00
0.64 3.723 0.95 3.730 1.00
0.74 3.703 0.92 3.715 1.00,
1.0 3.672 0.88 3.668 1.00
That the analytical solution is essentially a linear solution and it does reach the synchronous condition suggests the linear damping term of the equation of motion cannot be neglected and has a significant effect near the synchronous frequency.
A new least squares fit of the amplitude ratio data near the synchronous peak was performed and the non-linear damping term (CL) of equation 3.22 was evaluated. Figure 5.5 shows the values obtained by this fit plotted against the orifice area ratio. An excellent inverse square relationship was found to exist as illustrated in the figure. Equation 5. 10 describes the resulting expression for the linear damping coefficient.
CL 0.06') s 1 ) Eq. 5.10
The linear damping coefficient has units of one over time in equation 3.22. Therefore scaling of this term should involve the natural frequency of the model and prototype. In addition, equation 5.10 shows a further dependency on the draft of the model. Such a scaling is described by equation 5.11 (M subscripts indicate the model value and P the prototype).




Non-Linear Damping Amp Ratio Plot for AR =0.74 + maxima-minima analytic -numeric
1.2
1.0
S0.8
S0.6
0 *2 0.0
0.90 0.92 0.94 0.96 0.98 1.00
Fig. 5.4, Amplitude ratio vs. frequency plot near the synchronous frequency for orifice area ratio 0.74.
Plot of Linear Damping Coefficient vs. AR
0.26
0.1o y 0 .0596x-1-119
R2 0.9974 20.16 .~0.11 .20.06
0.01
0.5 0.6 0.7 0.8 0.9 1.0
AR

Fig. 5.5, Linear damping coefficient vs. orifice area ratio




S__ 06__r__S,=O L~ MOM Eq. 5.11
soM( +Cm) "'Sop p
Using this relationship, the non-linear damping terms were reevaluated. The no orifice case gave a value of 1.1 for the f LID coefficient and the C' coefficient had a value of 0.4 for all orifice ratios and no longer exhibited a frequency dependence.
The non-linear damping term in equation 3.22 has units of one over length. This can be seen in the formulation of equations 5.5, 5.6 and 5.8 where the non-dimensional resistance coefficients are divided by the model draft. Scaling by the draft should be sufficient for the C' and the entrance and exit loss term, however thefLID term contains a draft to diameter ratio. Thus a more complex scaling involving both the diameter and draft is required. Such a scaling is described below (M subscripts indicate the model value and P the prototype).
f SoM) Djp Somt DspSoM
2SOCJDDM[2SOM SO 5P SOP N 5OP
Equation 5.12 summarizes the least squares fit result for the non-linear term, including scaling factors.
CN LV DI o .75+CQ I)s SO) Eq. 5.12
(f =1.1, C' =0.4
Recalculating the numerical and analytical solutions with the above coefficients results in the plots found in Figures 5.6 through 5.9. Table 5.3 shows the maximum amplitude ratio comparison for these solutions. Again the agreement between analytical




and numerical amplitude ratio solutions is excellent. The phase plots show a sensitivity to
non-linear damping effects which the numerical solution reproduces well in the lower
frequencies. Overall, agreement with experimental data is excellent and neglecting us
appears to be justified.
Table 5.3, Maximum amplitude ratio comparison for combined linear and non-linear dam ping.
numerical solution analytical solution
AR (o (rad/sec) Amp Ratio o (rad/sec) Amp Ratio
0.52 3.750 1.00 3.750 1.00
0.64 3.731 1.00 3.731 1.00
0.74 3.719 1.00 3.715 1.00
1.0 3.691 0.98 3.691 1.00

Sensitivity of the Centerwell Heave Response Dynamics to Secondary Effects
As noted, there is good agreement of both the solutions with the experimental amplitude ratio data over most of the frequency range. This does not indicate a contribution of the ps added mass term in equation 3.22, which results from the acceleration imparted by the orifice lip on the centerwell. The dimensions of the orifice lip would seem to justify this conclusion and lend validity to the original assumption that the effect of the orifice is only to increase the damping coefficient and therefore the coupling of the spar and centerwell through the relative velocity. Indeed, the amplitude ratio plots show a significant widening of the response curves with respect to frequency as the orifice area decreases (i.e. as the bottom is closed off). In the extreme case of a closed bottom, the amplitude ratio would necessarily be 1.0 for all frequencies and this is the limit suggested by such widening of the amplitude ratio plots.




Full Eqn Amp Ratio Plot for Area Ratio = 0.52 + maxima-minima analytic numeric
1.2 1.0
0
0.8
* 0.6 5 0.4
0.2 0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0) /0) o
Full Eqn Phase Plot for Area Ratio = 0.52 + maxima-minima analytic numeric 120 90
G~60- +

-30 +
-30
C~-6060 _+ +
-90 4-120
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0/) o Fig. 5.6, Solution plots for an orifice area ratio of 0.52.




Full Eqn Amp Ratio Plot for Area Ratio = 0.64
+ maxima-minima analytic numeric
1.2
1.0
" 0.8 0.6
0.4
0.2 +
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0)/oo
Full Eqn Phase Plot for Area Ratio = 0.64
+ maxima-minima analytic numeric
120
90- ++
~6030
0
-30
-90
-120
0.0 0.5 1.0 1.5 2.0 2.5 3.0
a)/Co o Fig. 5.7, Solution plots for an orifice area ratio of 0.64.




Full Eqn Amp Ratio Plot for Area Ratio = 0.74
+ maxima-minima analytic numeric
1.2
1.0
o
S0.8
0.6
0.4
0.2 +
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
(D /(D o
Full Eqn Phase Plot for Area Ratio = 0.74
+ maxima-minima analytic numeric
120
90 i
'n 60+ 30 +
0
-30
-60
+ +
-90
-120
0.0 0.5 1.0 1.5 2.0 2.5 3.0
(0/0 o
Fig. 5.8, Solution plots for an orifice area ratio of 0.74.




Full Eqn Amplitude Ratio Plot, No Orifice
+ maxima-minima analytic numeric
1.2
1.0
o
S0.8
S0.6
S0.4
0.2
+ + +
++ +
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Full Eqn Phase Plot, No Orifice + maxima-minima analytic numeric 120 90
'60
+ +
-30 \
+
S0
-30
.-60 +
+u +
-90 +
-120
-150
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Fig. 5.9, Solution plots for the no orifice case.




However, an examination of the effect of a term involving the spar acceleration is necessary for completeness. Figure 5.10 shows the results of such a sensitivity test conducted using the analytical solution. Recall that Jul = C12 AI+ Cm ). The effect of increasing the value of Cm,2from 0. 01 to 0. 1 on the amplitude ratio and the phase difference is enormous. This term causes the centerwell amplitude to exceed that of the spar motion and shifts the entire phase plot up approximately 30 degrees.
Little difference exists between the plots for cm2 values of 0.01 and 0. This lack of sensitivity indicates that the assumption of a negligible contribution by the spar acceleration term appears to be justified. A further check was performed by reducing the value of cm,2 by an additional order of magnitude and allowing it to vary linearly with frequency (i.e. Cm2 = 0.001(w). The result is a negligible difference with the amplitude ratio plot for the cm,2= 0 case, but an interesting development in the phase plot. Figure 5. 11 shows the phase plot for this case. Note the tendency of the phase to return to zero degrees as the frequency increases which is observed in the experimental phase data.
Given the difficulty experienced in extracting the phase data and the previously noted sensitivity of the phase, the result illustrated in Figure 5.11 cannot be considered conclusive. The experimental variation at higher frequencies may very well be due to experimental error or contamination of the data by secondary effects.
Another effect warranting investigation is that of the second order term associated with the E coefficient of equation 3.22. Both numerical and analytical calculations were performed by setting E = 0. The results are illustrated in Figure 5.12. As expected, the absence of this term had little overall effect. The small peak in amplitude ratio at one half




the synchronous frequency and the decrease in the phase at the same frequency due to the 2(o solution of Appendix A disappeared, indicating it was caused by the second order term. The continued presence of the one third synchronous frequency phase decrease indicates that this perturbation is not due to this term.
Finally, a least squares fit of the data over the entire frequency domain was performed to obtain a solution for the case of linear damping only. The numerical and analytical results of this solution are shown in Figure 5.13. Again, the results are as expected. The solution has little effect on the amplitude ratio except for the removal of the peak at one third synchronous frequency resulting from the 3w0 component of the nonlinear damping term. The same is true for the effect on the phase plot. Thus, the perturbation at one third synchronous frequency is shown to be due to the non-linear damping term.
Summary of Centerwell Heave Motion
The motion of the fluid in the centerwell is a result of the coupling between the centerwell and the spar. This coupling is primarily the result of the surface forces which derive from the relative velocities of the centerwell fluid and the spar itself There is little influence from the relative acceleration of the centerwell and spar.
When the centerwell moves in sync with the spar, no relative motion exists between the centerwell fluid and the spar and the centerwell exhibits the behavior of an undamped, freely oscillating mass-spring system. Though no actual coupling exists in this condition, the centerwell amplitude ratio curves are very steep. Thus, with the centerwell




initially in the synchronous motion state, a slight drift of its motion away from this condition will result in a rapid increase in relative velocity which will quickly restore the centerwell motion to synchronous motion with respect to the spar.
Also, the centerwell amplitude ratio curves widen with respect to frequency if an orifice is placed at the bottom of the centerwell. The amplitude ratio is only significant over a very narrow frequency band near the synchronous frequency when no orifice is present. This band gradually expands as the size of the opening at the bottom of the centerwell decreases. The equations which describe the non-linear or viscous damping coefficient would suggest that as the orifice lip becomes larger, the turbulence created at the bottom of the centerwell increases and vortex formation similar to that described by Knott and Flower increases, effectively closing off the bottom of the centerwell. That the linear damping or inviscid pressure force term also plays an important role was demonstrated by the fact that the numerical solution, which most accurately simulated the non-linear effects, did not reach the synchronous condition without a linear damping term.
Thus, a combination of linear pressure forces and non-linear viscous or vortex forming forces resulting from the relative velocity of the spar and centerwell fluid can drive the centerwell fluid into a state where these forces disappear and the centerwell oscillates as an undamped system. This is the basis of the synchronous motion observed in the experiments.
The effect of these forces on a spar with a centerwell system moving under the influence of waves will be examined in the next chapter.