IIYDRODYNAEC AS PECES OF MOORED SEMI - SUBMERSIBLES
A Literature Study
Arun Kr. Dey, BScEngg, MSc, CEng Report No. 926
January 1992
Deift University of Technology
Ship Hydromechanics Laboratory Mekeiweg 2
HYDRODYNAMIC ASPECTS OF
MOORED SEMI-SUBMERSIBLES
(A Literature Study)
By
ARUN KR. DEV, BSEngg, MSc, CEng
Shiphydromechanics Laboratory
Technical Univeûsity Deift
January, 1992
CONTENTS
Chapter Page
ABSTRACT I:
i INTRODUCTION 2
SEMI-SUBMERSIBLES' BASIC HYDRODYNAMICS 4
2.1 - General 4
2.2 - Wave Induced Forces (small bodies) 4
2.3 - Wave Induced Forces (large bodies) 5
2.4 - Second Order Forces 5
- References 7
MEAN AND LOW FREQUENCY SECOND ORDER
WAVE EXCITATION FORCES 8
3.1 -General 8
3.2 - Review of ;Past Works 8
3.3 - Review of Recent Works 17
- References 23
NON-LINEAR WAVE EFFECTS AND LOW FREQUENCY
HYDRODYNAMIC COEFFICIENTS 31
4.1 - General 3l
4.2 - Review of Past and Recent Works 3l
4.2.:1 - Non-linear Wave Effects 31
4.2.2 - Hydrodynamic COefficients 33
4.2.3 - Statistical Estimate 39
- References 42
ROLE OF THE ENVIRONMENT 48
5.1 - General 48
5.2 Review of Past and Recent Works 48
5.2.,1 - Waves, Wind and Cürrent 48
5.2.2 - Mooring System 52
- References 54
ABSTRACT
This treatisebased on a thorough literature search has been carried in the subject of mean and low frequency second order wave forces alOng with its related interests like viscous effects, low frequency hydrodynamic coefficients, wave directionality and grouping, mooring parameters and
statistical estimate for a moored offshore structure like semi-submersibles.It has been noticed that the present state-of-the -art of predicting the second order mean and low frequency wave force has more or less achieved its success though new problems are still being addressed towards an exact solutiòn
of such a complex problem.
It will not be too much too say that almost all the numerical solutions were accomplished through the use of linear potential theory (either 2-D or 3-D) with some kind of approximations in their formulations in order to ease the solutiOn technique.initially, experimental procedures were also adopted to understand the. root of the second order force.With the space of time, more and more
related interests are being initiated by the researchers in this field. Some of them like viscOus effects, low frequency hydrodynamic coefficients likeadded mass, bothpotential andviscous (linear and
non-linear) damping factors need profound knowledge for further treating the problems.For a moored structure, the linear wave radiatiOn damping is small andtherefore non-linear potential effects and viscous effects are important.Great uncertainty lies in choosing the appropriàte values of drag
coefficients.,
Theenvironmentlike wave-current interactiOns, 'low frequency turbulent wind spectra along with the
dynamic effects of the mooring system including their damping contributions have also significant
effects on the low frequency response amplitude
of a
moored structure like semi-submersibles.Especially the modification of the current and wave velocity fields and forces resultingfrom interaction of the velocity fields. 4
For greater accuracy, it is often suggested that the effect of wave directionality be accounted for
while predicting:the. motion characteristics of a moored structure.Similar is the case with the wave grouping which is important for moored structures where the second order' wave force produced by grouping may excite large resonant motions.
1. INTRODUCTION
Fixed or floating bodies in regular waves are subject to small second order mean wave forces and
in irregular waves an additional low frequency slowly varying wave forces due to the continuously
varying wave height and frequency besides the large linear wave forces in a seaway.Such second
order wave forces are found to be proportional to the square of the wave height while the first order wave forces are proportional the wave height in a linear way.These mean and slowly varying waves
forces, a common terminology in literature as wave drift forces (steady and oscillating), have
frequencies equal to the difference frequencies of two regular waves present in a regular wave group.
Especially when the relationship between the elongation and the tension of a moormg line is progressively non-linear, the resulting horizontal motion can indüce a rather high peak force in the
mooring lines.
The slowly varying component of drift forces is of particular importance for compliant structures
possessing low restoring forces in horizontal modes.Though the magnitude of the wave drift forces
are quite low compared to the first order wave forces, their presence 'while close to -the natural frequency of the system signify the ability to excite resonant motions of the moored structures
resulting large amplitude of motion especially when the system damping is very low.The slòw resonant motion is often a very significant part among all motions.For a moored-semi-submersible in a moderate sea-state, 70-80% of surge, sway, roll and pitch can be attributed to such resonant responses.For the same reason the vertical second order force can of importance for large
semi-submerged structures with large natural roll, pitch and heave periods.In case of a small initial statical stability, the mean' vertical moment can cause a semi-submersible to undergo sufficient tilt.
Several attempts have been made for the last three decades for the prediction of the second order
-wave forces with acceptable 'liinit.Detailed review of those methods would be given in Chapter 2but
basically two different methods exist so far - one is Pinkster's [3.39]1 "near field method" by which the second order forces are calculated by integrating the second order pressure over the
instantaneous wetted surface of the body and the other is' Maruo's [3-41 "far field method" basedon
the momentum principle later applied by Newman [36] also.The "far field method" beingopaque,
the "near field method" has become more clear-cut in understanding such a complex problem ina
more realistic way.
In a complete formulation of the slowly varying wave drift force ¡n a random sea, the problem of complete solution of the second order diffraction appears .The wave directional ity and the wave 'groupiness enhances the depth of the problem to be handled. Further a very 'low relative damping
is' present in the second order oscillatory systems, so its quantitative prediction is' of vital importance
in the determination of the motion responses. There 'are many ways in which a moored
semi-submersible can dissipate its energy of motion.The variOus sources of damping need to be dealt with differently.Finally, the information sometimes necessary is thestatistics in terms of the most probable
maximum value of motions and peak mooring forces in a given sea conditions.
Slowly varying resonant motions can be excited by any one of the environmental factors like wind, waves or current or by the mechanism of interactions among themselves.However, wind and current
are sometimes the prime source of dominant steady and low frequency forces acting on a moored
floating platform.Wind induced forces can be particularly important for any mooring system where
the peak energy of unsteady wind can occur at the system natural frequency.
The equation of motion used to predict the motion of moored floating body in waves is theone for
free floating bodies with the added restoring force due to mooring.To fit the restoring force due to
mooring in the equations of motions, it is necessary to know not only the static and dynamic
characteristics of the moormg system but also the dynamic interference of the mooring line and the floating body.In addition, mooring linedainping can be a significant part of the total surge damping of a moored semi-snl,mersible
Due to relative merits and demerits of various mooring systems, the offshore industiy sometimes
experiences major difficulties in mooring under storm conditions resulting operational downtime.Though various studies have contributed significantly towards understanding many:factórs
2. SEMI-SUBMERSIBLES' BASIC HYDRODYNAMICS
2.1 General
It is well known that semi-submersibles have better motion characteristics than
mono-hulls.Sometinies, they are referred to as "ColumnStabilized" vessels which imply that pitch and roll
stability is obtained primarily from the second moment of water plane area of the surface-piercing
vertical co1umns.Al1semisubmersibles share a very common objective of continued operation having minimum wave indúced mOtions and uninterrupted station keeping, even under relatively severe sea conditions.So, the primary design considération for any semi-submersible design is its "transparency" to environmental action'.
In practice, semisubmersibles' design is
a compromise among many factors
such aseconomics,motioncharacteristics, stationkeeping, deck load capacity, productionmethods, structtial requirements, stability, etc However as mentioned above, two major operating considerations are for
the vessel to have minimum motions and good station keeping capability in prevailing harsh
environmental conditions to provide a stable platform.
2.2 Wave Induced Forces (small bodies)
When a floating body undergoes oscillatory motions in waves, the magnitude of the bùdy motion depends in a complicated manner on the height, slope1 frequency and direction of the incoming. waves.For wave/height length ratios no greater than 1/30, linear wave theory is applicableThis has been the advantage that all the boundary conditions of the mathematical problem can be applied at
the average position of the oscillating body.In particulai, the hydrodynamic forces and moments can be calculated by integrating the pressureover the mean immersed huIl.The correlation between linear theory and full scale trials and laboratory experiments has indeedshown that (away from resonance) the displacement of the body is linearly proportional to wave height.
The presence of a free surface is an inherent problem.Thus even for a constant wave slope, the
motion changes significantly due to the frequency-dependence of the hydrodynarnic forces.
The procedure adopted in analyzing the motion characteristics of a floating bOdy such as a
semi-submersible is to assume that it behaves as a rigid body having six degrees of freedom.TIe external forces which excite the platform are associated with the fluid motion relative to thestructure.Starting
from the use of Morison equation, several "first generation" computational procedures have been
developed over the years for the' prediction of wave induced forces and motion responses.They are
described in detail in papers by Burke [2-1'], Paulliiig [2-2], Tasai, et al. [2-3], Hoóft [Z.. 4] .These procedures all share several basic assumptions concerning the computation of the hydrodynamic forces exerted by the fluid on the structure, the principal ones are as follows:
motion amplitudes are small.
the structure issubdivided into several slender cylindrical elements or small concentrated volumes. the distance between the elements are large compared to their cross-sectional dimensions and as such the interference effects can be neglected.
the typical cross-sectional dimensions of each element is small compared to the wave length and
thus the fluid pressures, velócities and accelerations at the surface of the element may be
approximated by the average values computed along the center line.
the principal parts of all members are deeply submerged below the free surface and therefore
infinite fluid values may be used for damping and added mass coefficients.Under these assumptions, drag forces due to wave dissipation are effectively neglected.
the forces associated with the incident wave motion are computed independently of the forces
associated with the motion of the structure.
This results in a system of external forces acting upon the structure, containing terms dependent upon the incident wave system and upon the motion of the body itself.This total system of external forces is then equated to the product of the body's mass and acceleration, yielding a system of differential
equations.These equations are then solved to obtain the frequency dependent motion of the structure.The linearity assumption ceases to be valid where the non-linear drag term in Morison
equation plays a crucial role in controlling the magnitude of the response.
Generally the prediction of the motions of semi-submersibles gives results which are considered acceptable comparedwith the results of model tests or in some cases with field measurements.
2.3 Wave Induced Forces flarge bodies
With the rapid advancement in offshore operations, certaintypesof sernisubmersiblesdò have larger
horizontal dimensions say the diameter or x-section of the columns of a semi-submersible.In such case, the ratio OIL (D=Diameter and L=Wave Length) is no more small. In this category, the
incident waves generally undergo an scattering/diffraction and wave forces calculations should then take account of such diffraction/scattering phenomenon.Furthermore, when the body consists of sharp
corners, flow separation and viscous effects can no longer be omitted.
Diffraction problems can still be solved with linear potential theory.The problem reduces to the determination of the velocity potential 4 which satisfies the Laplace equation within the fluid region.Additional assumptions are made to the linear boundary conditions where appropriate.The
total velocity potential frcomprises of "incident wave" ((W)) and "scattered wave" ()) potentials
for a fixed body and a
"radiated wave"((') potential
is added to them for a floating
body.Comparisons have been made between experiments and the solution for a vertical
cyl:inder.Reasonable agreement has been found which are acceptable for design purposes.
In order to meet the demands such as variety and complexity ofdifferent design configurations, such novel concept needs wide range of applications. Accordingly, improved analysishas been made based
on numerical approach which necessitates a surface integral equation which is then solved by a discretization procedure.Such method is now firmly established in modern day designpractice. It is worth mentioning here that the diffraction problem as discussed above is based on small amplitude wave theory within linear assumption.In practice importance be given to deal with non-linearities such as steeper waves.Such non-non-linearities arise due to the free surface boundary conditions otherwise the basic problems still lies within linear approach.Velocity potential is now extended up to the second order through perturbation technique via a power series.Similar to the linear diffraction problem, 4' is now expressed' as the sum of incident wave and scattered wave potentials.Difficulty arises in getting a suitable solution of 4 especially.At present considerable
efforts are being made towards understanding such phenomenon in a more satisfactory manner.
Further details about the diffraction problem is available in ref. [2-5] where relevant references are
given.
So far the mechanics of wave forces discussed is mostly offirstorder which is obviously responsible
for the first order motion responses associated with the frequencies of the waves.Apart from this
oscillatory natueof the forces and responses, steady force of second order also exist5both in regular
and irregular waves. Second order wave forces of varying nattire is found in a regular wave group in irregular waves which is associated with difference and sum frequency Difference frequency is
often termed as low frequency and sum frequency as higher frequency.Furthermore, such mean and
varying forces in horizontal plane are known as "Drift Forces" and in the vertical plane as "LiftíSuctiin Forces' In this literature study, most emphasis Would be given on this steady (mean)
and low frequency second order wave excitation forces m connection with different hydrodynamic aspects of a moored semi-submersible in a seaway.
REFERENCES IChapter 21
12.1] Burke, B.G.: "The Analysis of Motions of Semi-submersible Drilling Vessel", OTC' 1024, Offshore Technology Conference, Houston, Texas,, 1969.
[22]
Paulling, J.R.: "Wave Induced Forces and Motions of Tubular Structures", Proceedings ofthe ONR 8th Symposiumon Naval Hydrodynamics, Rome, Italy, 1:970,
[2.3] Tasal, F., et al.: "A Study on the Motión of Semisubmersible Catamaran Hull in Regular Waves", Report of Research Institute of Applied Mechanics of Kyushu University, University, VoLXffl, No60, 197
12-41 Hòoft, J.P: "A Mathematical Method of Determining Hydrodynamically Induced Forces
on a Semi-submersible", Transactions SNAME, VoL79, 1971
12-5] Sarpkaya, T. and Isaacson, M: "Mechanicsof. Wave Forces on OffshoreStructures", Van Nostrand Reinhold Company Inc., New York, 1981.
3. MEAN AND LOW FREOUENCY SECOND ORDER WAVE
EXCITATION FORCES
3.1 General
It has already been described that in regular waves, the drift force is steady.Its magñitude is also
constant.In irregular waves, the drift force has additional component which is oscillating with either low or high frequency.Though this kind of force is in general small compared to the first order wave
excitation force, the period of the oscillating drift force is found to be close to the natural period of
a floating structure or to its floating system resulting a highly tuned resonance if the static restoring force is small e.g. the horizontal oscillations of moored vessel or vertical' oscillations of small
water-plane area vessels.
In this chapter, 'a technical review regarding the study of the above would be presented in details.
3.2
Review of Past Works
One of the most early works was by Suyehiro [3-1] who understood the mechanism of the drift force as a result of the constant action of the wave pressure.He even did some simple experiments to measure the drift on a ship model subjected to beam waves.
Following Suyehiro, Watanabe [3-21 analyzed the drift force from the k*nematic theory by which the hydrostatic pressure of the undinturbed wave and a phase difference between the rolling oscillation' and the excitation moment of the wave was considered as the prime source of the: drift
force.
One of the pioneering works. was done by Havelock [3-3] on, the mechanism of the drift force.He
introduced a rather approximate expression for the Steady drift force which is as follows:
=! ?
(N a
+ N0. )where w = wave frequency, a, a0 = heave and pitéh amplitudes' and N, N9 = corresponding
damping coefficients.
Then one of the outstanding and quite leading works was by Maruo [3-41.He started the problem
for 2-Dimensional cases and thus also tried to give solution for 3-Dimensional' cases for a couple of
symmetric bodies.His solütion was based on the momentum theory.However his work does not 'include numerical results for oblique waves. The expression given by him for a 2-D' body is as
follows:
where r. is' the amplitude of the reflected waves due to sway, heave and roll in beam seas.
Ogawa [3-5] used Maruo's theory to calculate the drift force and moment on a ship in oblique seas.His expression was similar to Maruo's except multiplied by an additional term Sin where
is the angle of the incident wave with the ship's axis.
Newman [3-6] also applied the momentum principle to calculate the drift force and moment on an arbitrary geometry.This work was also a follow-up to Maruo's basic work but with a different
approach.His numerical results were based on slender-body theory associated with long wave length assumption.The expression given by Newman is in complex form:
2ir
+
p - 7r
IH(0)12 + Ii) dOwhere H(0) is the Kochin functiòn.
Grritsma and Beukelman [3-7] expressed the drift force in terms of the damping force of the
radiated waves:
1'X
J'b33vodx
in which b3 = sectional damping coefficient and = relative velocity term including heave and
pitch.
Hermans and 'Remery [3-81 discussed the resonance of a moored object in a regular wave trains built up by two regular waves with different amplitudes and with only a small difference ¿w in frequency.They demonstrated the results obtained from model tests of a barge.This work can be
considered as one of the first attempt to understand the slcwly varying drift force in a regular wave group.They used the expression for the drift force per unit length for a regulár wave in the following way:
-
1 2F =
p g a
where a = ra.R and R = reflection coefficient depending on kT where k = wave number and T is the draft of the' barge.
Ilsu and Blenkarn [3-9] showed the importance of the drift force in relation to the peak mooring force of a moored vessel.They showed that when a wave is propagated towards a moored vessel, a part of the 'incident wave is reflected and the remainder ¡s transmitted beyond the vessel.This
conservationof wave momentum results in a net force applied to the vessel for each waveThe result is a steady drift force in regular waves and ;iìi irregular waves a varying drift force occurs depending
on the changes in wave hèight and wave frequency (see Figure 3.1).For irregular waves, they even gave a step by step solution for the problems.The general validity of the method was supported by model tests.The drift force in regular waves is the main link to their analysis of the slowly varying
force
Verhagen [340] discussed the drift fOrce, both steady and slowly varying, through observations revealed through extensive test programs on the behavior of moored bodies in a seaway.His
expressions for the drift force in head seas and beam seas in irregular waves are as follows: (See also
Figures 3.2 and 3.3)
B2 "
F(co)= 4 pg -
Saw) do!Figure 3.1:Example of Drift Force Obtained From Wave Record Ref. [3-13]
= 2 p g L ¡
o o F(c1) 1/2 p g L W In rad.sc dwhere L, B = length änd beam of the ship; F(c) and F(c,.) = wave excitation force in regular
waves; S.(cz,) = irregular wave spectra.
0.8
05
o
o
Figure 3.2: Drift Force on a Floating Body in Figure 3.3: Drift Force on a Floating Body in
Head Seas Ref. [3-10] Beam Seas Ref. [3-10]
Verhagen and Van Sluijs [3-11] further analyzed the problem in a much more sophisticated way by mtroducmg the non-linear terms m the wave potential They also calculated the drift force on a cylindrical obstacle by integrating the pressure about the surface.Bernoulli's equation containing
Second order terms was also used.Such approach was one of the important works in the computation
of the drift force by the pressure integration mthod.The drift fòrce Spectrum was also discussed. Van Oortmerssen [3-12], in describing the interaction between a vertical cylinder and regular
waves, defined the drift force as a constant resistance force as proportional to the square ofthe wave
height.He then menti'oned the constant resistance force hi irregular waves as the slowly oscillating drift force.He further atthbuted the constant resistance force in terms of the non-linear part of the
Bernoulli's equation.The basic expression given by him is as follows:
0.5 -1.0 1.5
W In rad.nec'
F=JJi
{(.)2 + ()2
+ )2} Cos(n,x) dSôx ay
Remery and Hermans [3-13] discussed the mean drift force in irregular waves.According to them,
it appeared to be possible to determine the slowly oscillating character of a moored vessel in a given irregular wave trains when the reflection coefficients 'R' are known as a function of wave frequency w (See Figures 3.4 and 3.5) .It was clearly shown that resonance would occur in presence of regular
wave groups which encounter the ship in the vicinity of a frequency close to the natural frequency Kim and Chou [3-14] presented a new procedure based on strip theory for calculating the lateral drift force on a floating body in oblique waves.The disturbance of the incident wave in presence of a body was expressed by the sum of the diffracted and radiated waves which were then solved by
satisfying the kinematic boundary conditions on the body surface.Numerical results were compared with the experimental results and satisfactory agreement was found.Some of their arguments are quite
noticeable like the lateral drift force is significant in high frequencies while negligible in low frequencies and also the lateral drift force on a free body is generally less than a fixed body. Remery and Van Oortmerssen [3-15], in connection with the design of mooring system, gave a description of calculating the mean force induced by the complete environmental forces i.e. waves,
wind and current for tanker shaped bodies.The basic expression for the drift force was similar to the
one given in [3-12].An empirical expression for the drift force for ship shaped bodies in irregular seas of narrow spectrum as suggested by them is as follows:
=
-
'
g R2(ct,) L sin2cww 1/3
in which 1/3 =ignificant wave height (crest to trough) and R() = drift force coefficient for
flat plate for w = w = mean wave frequency.
spring cortstant C
-- T
wave direction sca'e 1:80 water depth 7320 rr mass M length L 182.40 m. breadth B. 48.96 m. draft T. 12.80 m. depth H. 19.20 m. displacement in sea-water . 109.683 tons cOunter mass M 840 tonsFigure 3.4: Test Set-Up and Main Particulars Figure 3.5: Non-dimensional Amplitude of
of Barge Ref. [3-13] Reflected and Scattered Wave Ref. [3-13]
Newman [3-16] made an analysis of the slowly varying second order force.Approximate results were
derived which only depend on the steady time averaged force in regular waves and does not necessitate to determine the second order force in presence of two simultaneous regular waves.
Pinkster [3-17] showed that the spectral density of the drift force can be calculated directly from the
2
R F d measured if.L..trOk.trk. IOflVfl20,,.,.
3PQt!B .:.
.R.
ri U!
. 0.75 o 850 w rad. sec 325 1.0 R o.spectral density of irregular waves using the mean drift force coefficients in regular waves.This obviates the need for an irregular wave model as suggested by Ilsu and Blenkarn [3-9].The
expressions for the spectral density of the low frequency part and mean part of the drift force are as follows:
00
S.(j) = 2 p2 g2
S.(w) S. (w +) R4 (w i/2) dw00
F = p g
S(w) R2(w) dwwhere R(w) = drift force coefficient based on the mean force in regular waves and S.() = spectral density of the regular waves. It was also indicated that the influence of the dimensions of the vessel and the influence of the higher order effects in the Iów frequency drift förce would be accounted in
order to further develop the prescribed method.
Salvesen [3-18] derived relations for the second order steady state fòrce in regular waves.It was
shown that the force can be expressed as products of the motion responses, the oscillatory potential
and the incident wave potential all of which are first order quantities.Strip theory was applied
towards the solution of the oscillatory potential in
terms of two dimensional
sectionalpotentials.Comparison between the computed results and the experimental data showed good
agreement for head waves but not encouraging in oblique waves.
Faltinsen and Michelsen [3-19] evaluated also the drift force and the moment.Based on the expressions given by Newman [3-6] and dealing with only the second order terms in the incident
wave, amplitude via velocity potential and asymptótic. expansion of Green's fünction, they obtained expressions for the drift force and the moment.The method is applicable for any wave direction and
finite water depth.For box shaped bodies, numerical results were compared with the experimental
values.Asymptotic values agreed quite well with the calculatiòns.
Wahab [3-20] studied three different methods of calculating the drift force and the moment acting
upon a ship in waves.Calculatiòn methods were simply extension of calculation procedures of ship motions in six degrees of freedom.However, the results of the different prediction methods showed deviations among themselves as well as from the limited available experimental data thus giving no
firm conclusion as to the suitability to any particular kind of the calculation method.
DaIzell [3-21] presented an 'investigation as to how to estimate the mean added resistance operator
from irregular waves' data.His mathematical model had a linear as well as a quadratic term which
were identifiable through namely cross-spectra and cross-bispectra analysis of random data.Such mathematical tool has a great potential in dealing with the slowly varying drift force in irregular
waves.
Rye, Rynnng and Moshagen [3-22] studied the slow drift oscillations of a moored structure in a
wave flume.They alsó showed the sensitiveness of the reflection coefficients (abbreviated normally
as R).Their method of analysis was based on the theory outlined by Remery and' Hermans [3-13]
and Pinkster [3-17]. A series of test programs were conducted.In case of irregular waves, the results
using the reflection coefficients from regular wave tests showed very poor agreement whereas the. same comparison using 'R' from regular wave group tests improved the results of comparison considerably (See Figures3.6 and 3.7).For the evaluationofextreme slow drift mooring forces, the slow drift spectrum would lead to an underestimatedvaluebecausaoLtheskewLnature..of the-wave
envelope.
Bowers [3-23] discussed the long period oscillations of a moored ship subject to random seas.An attempt was made to establish a relationship between the long period oscillation and second order
wave effects without taking any account of reflections from the vessel.The second order fOrce was
made up of two components - one from the integral of the first order pressure over the first order wave elevatiOn and the cther from the 'integration over the ship's draught of the second order
momentum flux The long period surging spectrum of a moored ship was found to agree with a model
inferring that the second order wave forces are more effèctive than the first order wave force
resniting ship's movements.
Figure 3.6:Reflection Coefficient Determined
from 'Regular Wave Group Tests Ref. [3-22]
Kirn and Breslin [3-24] evaluated analytically the complete quadratic responsefunction of the drift force in the bifrequency domain in order to isolate and evalùate the' effect on a. moored' ship of the
lowest frequency non-linear surging force.They carried out numerical calculations for the time historie of irregular waves, slow drift force 'and slow drift oscillation of a moored ship in head seas.It was also shown that,, given the quadratic frequency response, it is' possible to synthesize at least the non-linear low frequency resistance component in the time domain Verification through
experiments was not done for the analytical procedure develòped.However they recommended that
the procedure is reliable in' the early 'design stages of the mooring system of a ship 'or an ocean
structure.
Kaplan and Sargent [3-25] discussed the importance of the drift force in relation to the simúlation
of the motiOns of a semi-submersible pipe laying barge.The basic equation was given by:
Xb
R=-i- IN" V2d
A z zX3
where V is the relative vertical velocity along the craft and N' is the local two dimensional value of the vertical damping force which .showsthedependency only on the vertical motiónsThe formula
is simple and can be a quick practical tool for a rough estimate of drift forces and moments in oblique sea waves The longitudinal drift force m the ship axis system is then given by Xd =-RA Cosy and the lateral drift force is given by
=
R'A Sin'y wheteyJs the_anglof the wave0.1 -REFLECTION COEFFICIENT DETERMINED FROM REGULAR WAVE GROUP TESTS - REGULAR WAVE TESTS 0, -O - 0.2 ß.L 0.5 0.5 1.0 1.2 l. f (Hz) -k
Figure 3.7A Modified' Reflection Coefficient
(broken line) Obtained by Including the'
Results from the Regular Wave Group Tests
Ref. [3-22']' 0.6
Rtf)'
REFLECTION COEFFICIENT DETERMINED FROM . ' 0.6-0-s.i
e.:. Ru) 03-0.2-propagation relative to theship's longitudinal axis. The time history of the drift was determined from
the square of the envelópe which was the weighted" relative velocity term in the above equation.
Pinkster [3-26] presented detailed derivations for the calculation of the low frequency second order
force in irregular waves based on integration of the pressures over the wetted surface of the hull.ln order to enhance the knowledge of the distribution of the second order force over the length ofa
moored vessel, a series of model tests with a simplified hull form were carried out in head waves.An approximate calculated result was presented for the second order longitudinal force based on the strip
theory.Numerical results were presented albeit they disperse due to the rather simplified version of the formulae for the second order force.The method outlinedalso allowed the calculation of the quadratic transfer functions which give the in phase, out of phase and amplitude relationship
respectively.
Wichers [3-27] showed the effects of the steady current and wind while solving the general SPM
(single point mooring) problem of the slowly varying drift behavior of a moored object.}Ie outlined
a procedure to obtain practical values of added mass and damping to calculatethe nature of the
stability and also the natural frequency of the system.Finally, he concluded from both computations and model tests that a moored ship to a SPM can undergo slowly varying motions under the action of wind and current only even without presence of any external force like the slowly oscillating drift
force due to waves.
Pinkster and Van Oortmerssen [3-28] have described a method to obtain first order wave forces as well as mean second order wave forces on floating bodies in regular waves by linear potential
theory.The mean second order force was obtained by means of direct integration of the pressures on the hull based on the solution of the first order potential.Comparison of numerical results for a pipe laying barge with the experimental results showed good agreement.
Pijfers and BrInk [3-29] gave a calculation method to determine the mean and the slowly varying drift force on semi-submersibles due to the different environmental forces like waves, current and wind.Also shown a time domain method of calculating the slowly varying drift force in irregular
waves. Some of the important conclusions drawn out of the results of the calculations are significant
contribution of the effects of current and wind to the mean and slowly varying drift force (see Figures 3.8, 3.9, 3.10 and 3.1 1).They further suggested model experiments for accurate values of
the drag coefficients in a mean and harmonic flow superimposed on it.
Salvesen [3-30] presented a new method for computing the added resistance ofa ship with a constant
forward speed in regular head waves.It was concluded that added resistance being a second order
quantity can bewell expressed in terms of first order motion responses .Detailed comparison between the proposed theory with other existing theories as well as experiments were put forward in regular
head waves.Discrepancies in the experimental results can be as large as the differences between
different theories.It was concluded that the proposed theory has wide range of applicability for varied
ship hull forms than any other existing theory.It was also shown that the mean added resistance
(mean longitudinal drift force) in short-crested seaway can easily be obtained from the regular wave results.
Faltinsen and Loken [3-31] presented an approach to calculate the slow drift oscillation of a ship in irregular beam seas.The drift force and moment in oblique seas were also discussed.The
hydrodynamic boundary value problem was formulated and solved correctly to second order in wave
amplitude.The second order difference frequency force due to the presence of two simultaneous waves with frequencies co and was also calculated.The results of numerical computations were
compared with those of other methods for different hull geometry.The prediction methods were found quite satisfactory.
C
Pinkster and Hooft [3-32] discussed a method to calculate the mean and the low frequency second order forces which is applicable to all six degrees of freedom based on the direct integration of all contributions of the second order force over the instantaneous wetted surface of the hUll of the
structure.Attempts were made to make ail approximate solution of the second order velocity potential incorporating it into 3-D potential theory.Approximate resUlts were compared to some 2-D cases for which exact solutions are available.Further, the results of the computations of the total méan and the
low freqUency longitudinal force on a barge in head seas were presented.
Faltrnsen and Loken [3-33] presented a procedure to calculate the horizontal slow drift excitation force on an infinitely long horizontal cylinder in irregular beam sea waves.The hydrodynaniic boundary value problem was also solved correctly to the second order in wave amplitude.All non-linear terms were included in the theory.The results were presented in the form second order transfer functions.Numerical results were presented for in-phase and out of phase quadratic transfer functions for a rectangular cylinder of different dimensions in beam seas Finally, the method was
applied to calculate the mean and the slowly varying response of a moored ship subjected to waves, wind and current.
¿0 E e o 20 120
Figure 3.8:Longitudinal Drift Force Coefficient
Cas a Function of c,e for Different Current
Velocities V0 Ref. [3-29]
Pinkster [3-34] further discussed the mean drift force in regular waves by evaluating it to include the low frequency oscillating component for a floating Structure in a regular wave group consisting of two regular waves with small difference frequency.The results of compUtations were presented fbr a rectangular barge and also for a semi-submersible.The computed mean surging force was compared with the results of model tests.For the semi-submersible, the computed low frequency second order surging force in head seas was compared with the results obtained from a test in
irregular head waves using cross-bispectral analysis methods.
Molin [3-35]' showed the suitability of the expression for the drift force given by Maruo [3-4] and
Pinkster and Van Oortmerssen [3-28].
After comparing the computation results with the experimental results for different geometry hull shapes, he found that finite elements yield a betterapproximation of 4 inthe vicinity of the body than far away and as such Maruo's expression on the mean wetted surface of the body was selected and incorporated in his 3-D fluid finite element radiation-diffraction model in order to compute wave drift forces.
e s' o 20
t
20 05 lO 1.5 - wetjFigure 3.9:Transverse Drift Force Coefficient
as a Function of
ú for Different CurrentVelocities V0 Ref
29f
Iypel Ç1m IL o .0(.VcI ILClEOIVCI Vc.2n-
AVc..2e/5 Vclm/S :Tve i i e J¼ 97I'VI V ZmIL Vc I CIL4V0
V -2mLØmis
0.5 I.0 15 'e20
O I- 2
Current velocity V,, (misi
Figure 3.10: Longitudinal Drift Force in
Irregular Waves as a Function of Current
Velocity Ref. [3-28]
-Io
Figure 3.11:Transverse Drift Force in Irregular Waves as a Function of Current Velocity Ref.
[3-29]
Similar to the. above, Molin [3-36] tested different formulations for the drift force on different bodies such as a sphere, a rectangular bóx and a 200 kdwt tznIcer The resültsfröm these formulations were found to agree well with each other as well as with the experimental data in most cases except in the viclmty of the roll natural period where viscous effects are appreciable Recommendations were made
to carry out fUrther experiments in the conditiOn where the wave steepness is near to the breaking
point.
Withers [3-37] formulated a mathematical model in order to correlate, via simulation, thelòw frequency large amplitude motions and the resulting mooring forces for a tanker moored to a SPM
system and operating in uthdirectional current and wind.Encouraging results were obtained between
the presented mathematical model and the results of model tests.No significant differences were noticed between the two mathematical models - the constant frequency domain hydrodynarnic
coefficients model and the Cnmminc time domain hydrodynamic coefficients model when the system
is stable.In case of abrupt changes in motions, the Cumnuns mOdel was recommended for more
suitable applications.
Sidiropoulos and Muga [3-38] evaluated the drift force computation methods.Broadly speaking, two
categories were mentioned - the far field approach and thenear field approach.They also evaluated the validity of the "weak scattering" assumption (often used in conjunction with 2D strip theory in order to simplify computations) by performing a parametric study for a series of 6(six) rectangular
barges .They showed that the "weak scattering" is a function of the angle of incidence, the dimensions
of the structure and the frequency of the on-öoming waves.
Pinkster [3-39] has presented a complete works based on the 3-D potential linear theory iii relation to the mean and low frequency second order forces on floatmg bodies Detailed expressions for the second order forces based on the pressure integration method were shown.Extensive computational results for a tanker,, a barge, a semi-submersible and a submerged cylinder were presented and compared with experimental results for regular waves.Limited comparison was also done for wave groups and irregular waves for the low frequency force by using bi-spectral analysis. (See Figures 3.12, 3.13 and 3.14)
Chakrabarti [3-40] gave an excellent elaborate review of previous works both analytical and
experimental in connection with the mean and oscillating drift force on a moOréd ship withOut any
- Pierson-MoskorniIz V,,. s, Type I lype 2 21 rn/s O. .lonswQp P,erso. MOSkO,vil/
'
Jonsw. Type 1 Iype 2 jc 9OI.VJ .t'c 90C-Vc) Pie-112SEEIZ -2 0 1 2Cur ent velocity Vc(rnd's
20 E S-z 'I
S-t
io
00 TANKER
Q--2 o -COMPUTEDO.A. MEASURED (ASCENDING WAVE AMPLITUDE o..
4 -2
BARGE
4
Figure 3.12:Mean Longitudinal Drift Force in Head Wäves ief.
T-T
forward speed in head and beamseas with no considerations on the yaw mornent.Data from different sources were processed to show variations which were attributed to the differences in the numerical
results due to the implied assumptions on the structural form, theoretical approach and simplified
assumptions.
3.3
Review of Recent Works
Kaplan [3-41] has given an important review of some of the early works in connection with the mean and low frequency drift forces pointing out different aspects of the problntFinally he described an application to the analysis of a moOred or dynamically positioned vessel giving particular emphasis on the use of some method of restraint on vessel motion Both theoretical and experimental results are available.Results are as follows:
Pinkster [3-42] has compared the mean hotizontal and vertical drift force on a semi-submersible structure in both regular and irregular waves with those obtained from model tests.In absence of current, computations based on potential theory showed satisfactory compatibility even though viscous effects do exist in waves alone.He recommended further research in case of presence of
current including its effects of interaction with 'waves.
Kim and DaIzell [3-43] have given results of calculations for the aterái drift force for a cylinder and a series 60 hull by mtroducing a new technique mvolving application of the quadratic frequency
response function (QFRF) and the close fit method for flows induced by hull sections in the near field.The near field solutions were necessary in order to obtain the second order non-linear
SEMI-SUBMERSIBLE
COMPUTED
¿360 FACETS
Beam Seas; Significant Height=2 meters
Mean-ft St.Dev.-ft Max.Dis.-ft
ModeiScale
2450
19.80 67.70Simulation 26.30 20.30 74.80
50 'u1 a 2.5 TANKER 5.0 2.5 00 - COMPUTED
MEASURED (ASCENDING WAVE AMPLITUDE o.)
SEMI-SUBMERSIBLE COMPUTED 360 FACETS 2 2.5 BARGE o ea 00 4
Figure 3.13:Mean Transverse Drift Forces in Beani Waves Ref
t3-3i
interaction effects in the presence of a dual wave system. Such method has resulted the mean drift force showmg that the wave elevation term is dominant whereby the Bernoulli's quadratic term is secondaryThe computational results were presented for the mean drift force in the bi-frequency domain for a series 60, Cb=O 60 ship at 60 degrees headmg The authors further recommended experiments with the objectives of enablmg comparisons of experiment and theory m the most
important areas of bi-frequency domain and also for verifying the analytical works for the lateral drift
force in irregular seas.
Kokkinowrachos, Bardis and Mavrakos [3-44] have evaluated different computation methods
-momenturn method and dfrèct integration method - for the drift force due to regular waves in finite
water depth for a single body or in presence of another Computational results of both methods were
compared with experimental results and they found reasonably satisfactory For single bodies, the momthtum method is less expensive with regard to the computer time afl alsO numerically less
sensitive than the direct pressure integration method.For multi-body system, the methód of direct
Cintegration was found more tiseftul and as suth an effective tooL Such calculation showed that
dependmg on the frequency of encounter and the distance between the barges, the direction of the drift force becomes opposite to the wave propagation
Lundgren, Sand and Kirkegaard [3-45], out of scrutiny ofsome recent literatures, have shown that further classification be still needed towards predicting drift forces more realistically like accuracy for wave direction and grouping In addition to the contribution due to potential flow, due care be
given for viscous flowsuch as wave drag drift, current wave drag and current wave friction He also stressed on different aspects of hydrodynamic damping forces, other than potential one, arising from wave related (viscousflow),,current related (viscotisflow)and otherviscOus flows.Through Mathieu
instability problem, he has further concluded that wave directionality and grouping are of great
importance.
Clauss, Sul an and Schellin [3-46] have discussed a calculation method tO determine the second order drift force considering the effect of water depth The method was based on perturbation
techmque and pressure integration over the instantaneous wetted surface of the body otherwise called
the near field approachJn a critical way they have shown that the following relationship between
FX(2) and ra as follows:
FX)
where 2 < n < 3
around pitch resonance period.Mooring. forces and displacements were also found to be higher in shallower waters and also for higher waves.
Powers [3-47] has described the digital cross-bispectral analysis mthód to obtain the quadratic
transfer functions (QTF) which relate the high frequency wave input and the low frequency response
of a moored vessel i.e. the input-output relationship for a quadratically non-linear system with the
aid of imear and quadratic transfer functions He also pointed out that cross-bispectral techniques can
be utilized to determine quadratic transfer functionsThe cross-bispectrum can be directly computed from the FoUrier transforms of the sea wave input and the moored vessel response.
Salvesen, et al. [3-48] has presented a new computational method for predicting the surge motion of a tension leg platform ÇIIP) subjected to the combined environmental forces of wave, wind and
current.Both second order wave forces includinghydrodynamic coefficients were computed by a 3-D
hybrid-finite element method.In addition to the conventional Morison's drag terni, a new formüla
for predicting the viscoUs drag force was proposed.It was shown that the new drag formula produces
smaller values of steady surge displacements and larger values of slowly varying amplitudes than
Morison's drag formula showing that the latter over-predicts the additional viscous forces due to the
interaction between current, wave frequency motions and slowly varying motions.Furthermore, the results indicated that accurate predictions are necessary for the steady surge due to wave drift and the slowly varying surge due to unsteady wind excitation.
Withers [3-49] has presented the results out of a investigation into the low frequency surge motion
of a large storage tanloer moored in a linear system in deep water subjected to irregular head seas in
doing so, he found the amplitude of surge motions to be significantly influenced by an additional
damping (known as wave drift damping) superimposed on the low frequency
still waterdamping Taking into account of the quadratic transfer function of the wave damping in the equations
of motions, the computational results agreed well with the measured data.He has however stressed further investigations to get more insight into the "wave drift damping" phenomenon.
Pinkster and Huijsmans [3-50] have presented both theoretical and experimental results of the low
frequency drift force for a semi-submersible moored in irregular head waves.They showed that time
records of the low frequency drift force using impulse response function techmque lead to a much
more accurate results than the direct summation method though the difference between the measured
and the predicted drift force record is noteworthy.Similar trend was observed for the time domain
simulations of the low frequency horizontal motions.Detailed discussions were given regarding such differences.
Faltinsen, et al. [3-51] have developed a comprehensive theoretical model based on extension of existing different theories for the behavior of a tension leg platform ÇFLP).Forces due to wind and current were also incorporated in the hydrodynamic model .An extensive series of model tests was conducted in order to obtain a systematic evaluation of the theory and its effectiveness.
Chakrabarti [3-52] has carried out experiments on tanker and barge models in regular waves, wave groups and irregular waves in head seas for measuring the mean and the oscUlating drift force.The results showed that both steady and oscillating drift forces are influenced by the hull shape and the steady drift force is higher than the oscillating one.The oscillating drift force was found to be governed by the system damping.
Chakrabarti [3-53] has studied particularly the drift force due to viscous flow in addition to potential
effects for a fixed cylinder which is also applicable for a floating cylinder incorporating the motion of the cylinder.Appropriate expressions were derived for the contributing factors.Tests were
C
t. D. C' j" 00 TANKER 00 2.5 o -2.5 TANKERj
o o TANKER o 2 . 4 -2 00 2 SEMI - SUBMERSIBLE o COMPUTED O 360FACETS o 2 4 00 2 00 - COMPUTEDcoo MEASURED (ASCENDING WAVE AMPLITUDE °..°
BARGE -2 BARGE o 3-2 4
Figure 3.14:Mean Longitudinal and Transverse Drift Forces and Yawing Moment now uartering Waves Ref. [3-39] SEMI-SUBMERSIBLE JCOM350 PU T E O FACETS o 19 L
Io
2 4 2 4 2 4 2 4conducted in presence of waves and current on a small diameter cylinder as well as a large diameter cylinder.
Marthinsen [3-54] has presented a new method of calculating the slowly varying drift force in irregular waves by using the constant drift force in regular waves.The method was based on the
assumption that the wave energy spectrum is narrow banded and the forces can be calculated directly from the time history of incoming waves without calculating the energy spectrum.Computations with such method in long crested seas showed quite satisfactory agreement with Newman's [3-16] method. Marthinsen [3-55] has reviewed different ways of calculating the different forces in connection with the dynamic analysis of a mooring system.The basic theory of calculating the slOw drift oscillation
was similar to the previous reference. Viscous effects due to waves and current were described in addition to force due to wind and mooring.The results were presented for short crested seas for
limited comparison.
Chakrabarti and Cotter [3-561 have analyzed both first. and second order motions of a moored
semi-submersible analytically as well as experimentaliy.Viscouseffects wereconsidered as non-linear
damping terms in equations of motiOns as well as for calculating the drift force' due to the viscous effects employing simplified formulae.Predictiöns for the second order drift force due to regular wave groups and irregular waves were done based on the results of moored semi-submersibles subjected to regular waves.Correlations between experiments and analytical results appeared to be
quite satisfactory except some scattering of data in case of irregular waves.
Takagi, Saito and Nakamura [3-57] have made a comparative study on simulation methods for
motions of a moored body in waves having a non-linear system.Comparison was done between two
methods, convolution integral (C.!.) method and constant coefficient (C.C.) method and between
them and experiments.The results demonstrated better accuracy for the C.!. method with respect to
C.C. method.Similar works were also reported in Takagi and Saito [3-58].
Kagemoto and Vue [3-59] have developed a new method based on linear potential theory for the calculation of wave forces on multiple leg platforms taking into account the full hydrodynamic
interactions among the legs.The drift force can be obtained using diffraction characteristics of single
legs. only thus eliminating the vigorous calculation time by the complete 3-D diffraction-radiation
theory.The authors also presented some approximate fòrmulae fôr preliminary and quick estimation of interaction effects.
Morch and Moan [3-60] have described an advanced computational model for analysis of mooring systems and as. such validated the accuracy of the model against the full scale measurement of a semi-submersible.The computer model assumed both wind, current and waves to be stationary.Damping
forces were taken as the viscous drag force between the structure and the water masses using Morison equation.However increase of damping coefficients due to presence of. waves was not
considered and still water values were used which is rather contrary to many other recent findings.
Kishev, Ivanov and Pomeranetz [3-61] have summarized results out of experimental investigation
of the hydrodynamic coefficients for a series of semi-submersibles giving particular attention to their horizontal motions in the low frequency range.They suggested that the damping factor predicted by potential theory would lead to incorrect results particularly while predicting forces in the horizontal
plane where the percentage of viscous forces components are high and also they are subjected to a
significant scale effect.
Kim and Bao [3-62] have studied the usefulness of 2-D strip method to predict the lateral drift force on a semi-submersible in oblique waves.Theproposed method used.the well known Marno's formula
better comparison with the available experimental results than the 3-D theory.
Taylor and Hung [3-63] and [3-64] showed the importance of interference effects betweén multi
body structures like columns and pontoonsof semi-submersibles and tension leg platforms especially for the drift force and hydrodynamic coefficients like radiation damping and added mass.It was found
for example that for a group of N symmetrical members, the drift force and damping approach a
value of N2 times the value of single member in isolation over a range of low frequencies. However,
in case of lift force (vertical second order force) and added mass, such value is N times the single
body value.It was also found that as a first approximation, the contributiOn from underwater hull of both TLPs and semi-submersibles can be neglected though the authors suggested further investigation for such approximation.
Qi and Feng I365] have presented a method of determining the quadratic transfer function of a tension leg platform (FLP) through model tests using Hilbert transform and a low ultra -low pass filter (LPF).Besides, first order motion responses calculated with 3-D potential theory agreed sufficiently with experimental results.Wave envelopes were calculated from time records of the irregular waves using Hilbert transform.Timehistoriesof surge motionsof themodel were recorded in lOng crested irregular head seas.The low frequency response was then separated from the time history using a low pass fllter.The quadratic transform function was then obtained from the ratio of
the cross-spectrum and the auto-spectrum of the square of the wave envelope.
Qi, Feng and Zhang [3-66] have studied the low frequency surge motion of a tension leg platform
ÇFLP) in deep water in the severesea state (storm condition) performing a series of model tests.They also indicated about the low frequency still water damping and the wave drift damping in regular and
irregular waves via experimental results.They concluded that still water the low frequency surge damping coefficient of the UP was fôund to be proportional to the surge amplitude for the amplitudes tested.Further to the above, low frequency wave damping coefficient was found to be influenced by the geometrical configuration of the underwater hull and the low frequency s Urge damping in irregular waves was obtainedby using a random decrement technique which was then used in order to obtain the root mean square valUes of the low frequency surge motion of the UP, Hong 13-671 has applied a three dimensional method to compute the drift force on a
semi-submersible.Numerical results were compared with experimental results and other computed results
given by Pinkster [3-39].Comparison was satisfactory except for the small wave length/structure length ratios.The author attributed this discrepancy due to non-inclusion of viscous effects, finite
water depth effects and also the underwater horizontal bracing connecting the two hulls.
Matsui [3-68] has given the formulation of an exact second order theory for predicting the slowly
varying drift force on a compliant offshore structure in irregular seas.The wave diffraction has been
solved correctly to the second order in wave amplitude by application of Lighthill's [3-69] method.The theory was applied to calculate the slow drift overturning moment on an articulated
cylindrical column.His conclusion out of numerical results were as follows:
a.the slowly varying drift force may be predicted very well with the knowledge of the mean drift force in regular waves in short to moderate seas (high wave frequencies) when the drift force is dominated by the force components arising from products of first order quantities.
b.In extreme seas (low wave frequencies), diffraction effects due to the first order waves are small
that the contribution of the second order wave may be calculated quite accurately by neglecting such effects.
c.Newman's [3-16] approximate method is a practical way of calculating the slowly varying drift
force in short to moderate seas but underestimates the same in extreme seas associated with a longer mean wave period.
Langley [3-70] has examined the variability in computed results with the time domain simulations of the slow drift response of a moored vessel when the random sea-state is modelled
as a
superposition of regular wave components.Variations for the statistical moments under successive runs were investigated over an ensemble of program runs.Theoretical expressions for the likely accuracy of the statistical moments as predicted by a number of limited program runs were also
examined.
Langley [3-71] has described a frequency domain analysis of a moored vessel in a random sea including second order forces and converting non-linearities by a suitable equivalent linearisation
technique leading to an iterative solution forthe mean values and the covariances of the responseThe
method was found to be useful in the early design stage when parametric study is sometimes necessary because of reduced computer times than time domain analysis which of course can treat
the non-linear terms more accurately..
Kyozuka and' ¡noue 13-72] have calculated the second order hydrodynamic forces based on regular
perturbation theory while dealing with the steady tilt moment acting on a semi-submersible both
theoretically and experimentally.The results showed that for the steady' vertical force, hydrodynamic
interaction effects in addition to 3-D effects and column effects are small at least in thebeam sea
condition.
MaIsui [3-73], based on the theory outlined in the previous reference, has applied the same in order to compute the slowly varying drift force on a moored semi-submersibleSecond order hydrodynamic
forces were evaluated from numerical first order solutions based on hybrid finite element
technique.The results were presented in the form of time independent quadratic transfer functions for thedifference frequency as well as the mean frequency.They were compared with those of simplified
approaches ref. [3-16], ref. [3-23], ref. [3-39].Further to the above,, he has presented the results of the time domain simulation of the slow drift motion of the semi-submersible under typical sea
conditions.They were compared with those based on Newman's [3-161 approximation.Satisfactory
agreement was observed in comparison for the frequency range where the components due to the
products of first order effects dominate the total drift force.On the contrary, for the frequency range
where second order potential makes a significant contribution, Newman's approximate method
underestimates the drift force as well as the motions.
Shenyan and Maoxiang 13-74] have developed a 3-D hydrodynamic computation method for long crested waves.In addition to first order motions, second order difference frequency forces and
motions were calculated with/without the effect of set down and diffraction.Numerical 'results were compared with the tank test results of different structures, one of them being the semi-submersible,
with some observations for the wave damping coefficient.Satisfactory results were found between
computations and model test results.
Nielsen [3-75] has given a long wave length/small body approximation of the wave drift force on
fixed submerged bodies.The method was based on an approximation of the Kochin functión valid for
long waves i.e. an expression as 'ka'-> O.The method required only added mass and damping coefficient for estimation of the drift force and the reflection and transmission coefficients for
incident wave.Numerical results were given for a circular cylinder, an ellipse and a pair of ellipses. Mavrakos [3-76] has evaluated the second order steady vertical drift force and pitching moment due to regular waves on floating or submerged vertical axisymmetric bodies in finite water depth based
on momentum considerations.Some case studies were done for floating or submerged docks and vertical compound cylinders and results have been compared with other existing theories and
pertinent experimental results.
platforms with large draft. It assumed that the drift force is influenced by the scattering of the waves
from the vertical columns only and simplified the interactions between the columns thus avoiding the robust application of 3-D diffraction theory.The method was tested for two semi-submersible
platforms showing acceptable results compared to a conventional panel method approach.
Helvacioglu and Incecik [3-78] have calculated the second order mean force and slowly varying
force spectrum of a compliant structure (an articulated tower system) byusing analytical expressions available in different literatures. Finally they have produced the results forthe steady surge response
of the system due to environmental fôrces like second order wave force, wind and current.
Standing, Brendling and Jackson [3-79] have compared the predicted low frequency motions of
the semi-submersible support vessel "Uncle John" with the full scale measurements.The theoretical model often underestimated the low frequency motions both in the vertical and horizontal directions for certain factors like not including the viscous drag forces, inaccurate amount of system damping, etc.
Wu and Price [3-80] have evaluated wave drift forces on a vertical cylinder of arbitrary geometry with its application.to tension leg platforms in asimplifledmanner assumingnointeractionsbetween columns, no drift force contribution from the horizontal elements, negligiblé first ordèr responses and extending floating columns to the sea bed thus saving substantial computation time but at the
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