t
Mean Wave Drift
irces:
Theory and Experiment
by
R G Standing, N M C Dacunha
and R B Matten
Supported by Department of Energy
Offshore Energy Technology Board
OT-R-8175
NM' R 124
December 1981
National Maritime Institute
Feitham
Middlesex TW14 OLQ
Tel:0l-977 0933 Telex:263118
TECIINISCIEE UN1VERSITE1T
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Mekeiweg 2, 2628
D Deif t
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This report is Crown Copyright but may be freely reproduced for all purposes other than advertising providing the source is acknowledged.
DEPARTMENT OF ENERGY
OFFSHORE ENERGY TECHNOLOGY BOARD
NATIONAL MARITIME INSTITUTE
MEAN WAVE DRIFT FORCES: THEORY AND EXPERIMENT
by
structures. NNIWAVE has now been extended to compute mean and slowly-varying components of the second-order wave force. This and a companion report describe the underlying theory and experimental validation of the new computer model. Results are compared with published data and with other prediction formulae currently in use. Criteria for use of this computer model are also discussed.
Comparisons were made with two simple and widely-used approximate formulae. Havelock's formula proved to be a convenient means of extrapolating NMIWAVE results to high
frequencies, but is inadequate at lower frequencies. Newman's formula, based on long-wave slender-body theory, proved to have a very limited range of validity, and was nost unsatisfactory elsewhere.
The new programs were validated in three stages. First, they were checked agaínst each other. Two alternative sets of programs were developed, based on so-called near-field and far-field methods. Results from these two methods agreed well with each other. The near-field results, however, proved sensitive to the way in which the vessel's waterline was modelled.
Results from both sets of programs were then compared with existing published data, and found to agree well. Finally the programs were validated against experiments, undertaken specially for that purpose at NNI. Theoretical and experimental forces were found to agree excellently in regular wave conditions, but slightly less well in irregular seas.
The present report concentrates on mean drift forces. A comparison report discusses slowly-varying forces and low-frequency response motions.
Mean Wave Drift Forces
2.1 Background
2.2 The roles of wave diffraction and drag
Theoretical Methods for Diffracting Structures 8
3.1 Simple Analytic Formulae 8
3.1.1 Havelock's formula 9
3.1.2 Newman's formula -io
Concluding Comments 32
Acknowledgements 3'+
General Notation
35
References 37
3.2 Numerical Comparisons 12
Wave Diffraction Theory 13
1-f.i Mean drift forces: the far-field(wave momentum) method. 15 2-f.2 Mean forces: near-field method
i8
a Forces
i8
b Moment about a fixed point 20
c Moment about a point on the structure 21
d Comparisons with Pinkster's and momentum results 21
e Physical significance of the near-field terms 22
5. Comparisons with Published Data 2k
5.1 Calculations using the Far-Field Method 2k
5.2 Calculations using the Near-Field Method 26
6. Comparisons with Experimental Data 29
6.1 Regular Waves 29
6.2 Pairs of Beating Waves 29
6.3
Irregular Waves 306.+
Conclusions of Experimental Investigation 31 3 3 3Appendix 3: Derivation of mean second-order forces by the near-field method 5
Appendix k: Effect of the second-order potential on the mean force 62
Appendix 5: Subsidiary details of the theoretical calculations 6k
A5.1 Rectangular box 6k
A5.2 Vertical circular cylinder 61F
A5.3 The rectangular barge 65
A5.k
Submerged horizontal cylinder 66Appendix
6:
Experimental details and first-order results for the moored 67drill ship
A6.1 Experimental Conditions 67
A6.2 First-order analysis in regular waves 68
Tables Figures
A ship, moored at a jetty or offshore terminal in an exposed situation, will often undergo large-amplitude long-period horizontal motions, which cause severe loads in the mooring hawsers'' 2 Multi-point moorings will allow relatively
simple surge, sway and yaw motions, with typical periods in the range 30 seconds to
several minutes. A ship attached to a single-point mooring, however, may execute a highly-complex pendulum-type motion, with either 'fishtailing' or 'galloping'3. In this case the characteristic period may lie in the range 10-30 minutes.
There is unlikely to be sufficient energy in the wave spectrum at such low frequencies to excite a large 'linear' response, and various non-linear mechanisms appear to be involved. One such mechanism was discussed by Lean4. He found that a non-linear restoring force could give rise to subharmonic response. This effect is most marked when the ship is restrained by mooring lines and fenders, and the stiffness increases
instantaneously as the fenders come under compression.
A different mechanism is involved in the fishtailing motions of tankers at single-point moorings. Simulation studies5 have shown that these motions can occur even
in steady wind and current conditions, where they are associated with changes in the applied force as the ship's heading angle varies.
There is an additional source of low-frequency forcing in the presence of irregular waves. This third mechanism has recently attracted considerable research interest, and is the central topic of this and the accompanying (part 2) report. The mechanism is that of second-order wave drift forces.
There are a number of established procedures for predicting the behaviour of a ship8 in waves. These procedures are generally based on linear wave theory and linearised response equations. Linear theory implies a direct one-to-one relationship between each individual frequency component of the wave spectrum and corresponding components of the wave force and response spectra. These spectra therefore cover similar ranges of frequencies.
Second-order forces, however, can have both steady and low-frequency components. These components of the horizontal force are known as 'wave drift forces'. They tend to cause drifting of a free-floating vessel, and possibly a change of heading. They are,
therefore, of particular importance when estimating thecourseof a disabled ship, or that of a structure which has broken free from its moorings or during tow-out.
A ship in regular waves experiences a mean drift force, but no low-frequency component (as will be shown in the sequel to this report). Low-frequency forces are associated with wave grouping9' 10 The drift forces increase or decrease, depending on the local level of wave activity around the ship. These forces are usually small compared with the first-order wave-frequency components, but can excite large resonant response motions of a moored system if the damping is very low'1. The resulting low-frequency variations in the hawser load may be very much larger than either the mean or
wave-frequency components. Thus the low-frequency forces are of particular importance in the design of mooring systems,and are a major factor, for example, in the consideration
of storage systems or wave energy devices to be moored permanently in exposed locatIons.
Wave drift forces also affect the design of dynamic positioning systems. These systems generally aim to prevent both steady and low-frequency drifting, but not responses at wave frequencies.
Mean and low-frequency second-order wave forces have a number of
other important effects on ships and offshore structures. Second-order forces
are thought to be responsible for long-period heave motions of floating platforms with small waterplane areas'2, and y contribute to the heel of semi-submersibles13' 1 observed in some model tests. The added resistance of a ship advancing through waves is also associated with a second-order drift force, and seems to be sensitive to forward speed15.
These non-linear forces are less well-understood and less easy to predict than first-order forces. Great Gare must therefore be taken when planning and executing experiments in a wave tank9 , because the forces are small and can be etreinely
sensitive to the way in which the wave spectrum is constructed.16
This report examines various theoretical methods for predicting mean second-order wave forces in the absence of wind and current, and assuming that the vessel has no mean forward motion. It is arranged as follows. Section 2.2 discusses the relative
roles of wave diffraction and drag in the wave drift force process. Section 3
compares a number of different procedures for estimating drift forces associated with the wave diffraction process. Section 4 describes the basis of the NMIWAVE computer
program, which uses linear wave diffraction theory, and has now been extended to calculate wave drift forces. Two alternative procedures for predicting the mean force are
outlined. Section 5 then compares numerical results with both published data and experiment. A second report, to be issued shortly, will discuss low-frequency
components of the drift force, and resulting response motions.
2. Mean Wave Drift Forces
2.1 Background
The present report discusses the mechanism and prediction of mean wave drift forces and a follow-up reportwill examine low-frequency forces and responses. It is fairly
logical to proceed in this order for the following reasons.
(i) Mean loads are easier to measure in the laboratory, and easier to compute
theoretically. They can be calculated either directly, as second-order components of force acting on the structure, or indirectly, from momentum
changes in the far-field diffracted wave. The latter approach is often easier. In this respect it has been shown15 that the mean force depends only on the first-order díffracted wave, and not on second-order wave components. Slowly-varying forces, on the other hand, depend on both first and second-order wave fields, and have to be calculated by the direct approach.
(ji) Designers make use of various approximate formulae for estimating the
slowly-varying force (see for example reference 10). These formulae make use of data obtained for mean forces in regular waves.
Techniques for estimating the mean force range from simple analytic expressions17 , which
assume complete wave reflection, to sophisticated numerical methods. Several such techniques are described in the following sections.
2.2 The Roles of Wave Diffraction and Drag
There are two fundamentally different approaches to the calculation of first-order wave forces: one empirical and the other mathematical. The first method is based on Morison's equation'8, which represents the total force as the sum of separate drag
and inertial components with appropriate empirical coefficients. This method is widely used in the design of tubular frameworks for jacket structures. The second
technique is used in the design of ships and wide-body structures. It involves the mathematical solution19 of equations describing wave diffraction and radiation by
the ship, and based on ideal fluid assumptions.
The ranges of validity of these two methods can be defined very roughly in terms of the two ratios:
6/D, fluid particle orbit diameter to typical member diameter, DIX, member diameter to wavelength.
Note that the first parameter is related to the Keulegan-Carpenter number K = Umax TID, where T = wave period and Umax is the maximum fluid particle velocity. According
to linear deep-water wave theory, U1La = 1TÓ/T, so that K = Tr6/D.
The following criteria were proposed20. Wave loads on a fixed member should be calculated using:
Morison's equation (inertia + drag) when 6/D > 1,
either Morison's equation (inertia term only) or diffraction method when aID <1 and DIX <0.2,
wave diffraction method when D/X > 0.2.
Limitations on wave steepness ensure that the ratio 6/A is small, so that drag is usually unimportant when diffraction effects are signifícant, and vice versa. This
provides some justification for the use of a wave diffraction theory based on ideal flow assumptions.
Figure la shows these force regimes in a diagrammatic form. They are for a fixed vertical column which rises through the free-surface in deep-water waves, so that the typical orbit diameter 6 may be replaced by H.
Drag, inertia and diffraction play rather different roles in the wave drift force process. Drag and diffraction both affect the mean drift force directly, while inertial forces do not.
Wave diffraction alters the momentum of the wave train, and this change is associated with a mean force on the structure. Drift forces on ships tend to be associated with short-period waves, which are either diffracted or reflected. The ship is transparent to long-period waves, which therefore do not contribute to the drift forces,even though it
may simultaneously experience some very large first-order forces and response motions.
Drag affects the wave drift force in at least two distinct ways.
Near resonance, the amount of damping affects the phase difference between the first-order exciting force and response. One component of the wave drift force
involves the product of two such terms21'22 that enter as term IV on p.22. Huse23 has attributed negative drift forces on a semisubmersible to the phase difference between heave
force and pitch motion.
Drag forces, acting between the mean and instantaneous free surfaces, are in
phase with surface elevation (see figure lc))and therefore contribute to the mean drift force.
It is perhaps to be expected that if either drag or diffraction effects dominate the first-order problem. then the same is likely to be true of the second-order force. What is not clear, however, is what happens when the first-order forces are inertia-dominated, and both drag and diffraction effects are small.
The second-order 'set down.' wave can, in fact, give rise to an inertial type of drift force. This force i associated with the pressure gradient in the set-down wave, and has no mean component, for reasons discussed in appendix 4. The low-frequency set-down force may, however, be very significant, in conditions where the first-order loading is inertia-dominated24. This aspect of the problem will be discussed in the second report.
The following example illustrates some of the ways in which drag, inertial loads and wave diffraction affect the mean drift force. It shows how the wave loading regimes differ from those usually applied to first-order wave forces.
secondorder forces.
shown26 to be approximately
-pgDH2(TD/X)3 when DIX is small (1)
If wave diffraction effects are negligible, the mean force may be calculated by using Morison's equation18. The mean force due to conventional drag and inertial forces,
but integrated from mean water level to the instantaneous free surface (as shown in figure ic) is then
2iî
¶D2 2Tt2H m 4
-
_Cp____(T2) sin6
+CdPD(_f_)2 cos O cos O } cos Odo,
where the wave is described according to linear deepwater theory. The inertial contribution integrates to zero, as noted above. The drag contribution is
.- CdP
DH/T
(2)Comparing this expression with that obtained from wave diffraction theory (equation 2.2.1), the mean drift force is dominated by drag or diffraction depending on whether
HA2
D3 > or <60
The relationship between this single criterion and the 3 separate criterion for first-order forces may be seen as follows.
Drag dominates when
> 60 (.)2
(rearranging equation (3)). The limit on wave steepness for regular deep-water waves ensures that H/A <0.14. This means that drag dominates the drift force if H/D > 1. Thus if drag is significant for first-order forces, the same is generally true for mean wave drift forces.
Diffraction effects dominate when
(2.)3 >±Ji
X
60A
(rearranging equation (3) again). With the same limit on wave steepness, diffraction effects dominate the drift force if DIA > 0.13. This is in fact quite close to the criterion for significance of diffraction in first-order forces. Thus if diffraction is significant for first-order forces, the same is generally true for mean wave
drift forces.
The situation is less clear, however, when the first-order forces are inertia-dominated,with negligible drag and diffraction effects. The first-order forces on a semisubmersible in waves of small amplitude and long period, for example, are of this type. Wave drift forces on this vessel may be significant for the design of moorings or dynamic positioning systems, and may be associated with either drag23' 27
or diffraction22' 24, 26 or both effects, depending on the structure and wave
conditions. The above criterion (3) will give some guidance on the relative importance of these terms, though the situation is complicated by the first-order response of the
structure and possible significance of low-frequency set-down forces.
These force regimes are shown diagraiìuiiatically in figure lb. As before, these results apply to a vertical circular column which is fixed, rests on the sea-bed and pierces
the free surface in deep-water waves. Different criteria may apply to structures of a different shape, or ones which can respond to the waves.
pgH2R2.
3. Theoretical Methods for Diffracting Structures
Results from two simple analytic formulae will be compared with a more accurate, but very much more complicated, numerical solution. The theory underlying this numerical model will then be summarised.
These solutions are all based on classical linear wave theory and ideal flow assumption They are, therefore, applicable to structures of the wide-body type only, such as
ships, barges, floating caissons and (in some circumstances) semisubmersibles. The
drift forces are associated with wave diffraction and radiation, all viscous and drag effects being neglected. It is assumed that there is no wind or current, and that the structure has no mean forward motion.
The Cartesian coordinate system and sign convention used throughout this report are shown in figure 2.
3.1 Simple Analytic Formulae
It is convenient to consider first a two-dimensional model, in which the structure is very long, and its axis is horiontal and at right angles to the wave direction. The wave drift force is related to momentum in the incident, reflected and transmitted waves, and has been shown28 to be
- pgH2(l + R2 - T2) (1 + 2kd/sinh 2kd) (4)
for a regular wave of height R, wave number k 27r/A, d = water depth, R = ratio of reflected to incident wave height, T = ratio of transmitted to incident wave height.
For the special case of a non-absorbing structure, conservation of energy requires t ha t
R2 + T2 = 1,
According to Maruo's formula, above, the force is maximum when R = i (i.e. the wave is completely reflected). Havelock made this limiting expression the basis for estimating longitudinal drift force on a ship17. His formula is widely used in design because of its great simplicity, but is frequency-independent, and will seriously overestimate the drift force in long-period waves.
Each section of the ship is treated as part of an infinite reflecting wall. In an oblique sea, where a is the angle between the ship's waterline and the wave direction, Havelock showed that the longitudinal mean force is
çB/ 2
pgH sin2 a dy (5)
-B/2
where the integral is over the ship's beam B along the portion of waterline exposed to waves.
This formula can be generalised slightly for general angles of incidence a and structure shapes. The mean forces and turning moment are given by
x =-pgU2 j sin2 a n' dl La y =-pgH2 1 sin2 a n' dl iLa 2 z =-pgH2 sin2 a (x n'-yn') dl 2 1
where the integral is around segments of waterline La exposed to waves (i.e. for which sin a < O), and (n', n') is the unit outward normal to La.
1 2
These expressions will be described as 'Havelock's formulae' in the calculations described below, though are in fact slightly more general than Havelock's original
5 result
3.1.1 Havelock's formula
3.1.2 Newman's Formula
A smaller proportion of wave energy is reflected, and more transmitted, as the wavelength increases. Vessel response also starts to affect the drift force, and Havelock's simple formulae become unsatisfactory.
An analytic solution, due to Newman30, takes account of wave scattering and vessel response. Newman's formulae are quite easy to evaluate, but invoke the so-called slender-body assumptions. These are that the wavelength is of the same order of magnitude as the ship's length, and that both of these lengths are much greater than the ship's beam and draft. Newman derived the following expressions for the mean forces and moment:
= pgk3 1 B(x) ikx cos .
-[e
a+1r13ixns_!
B()
E cosa - in3 + in5*i
- k) cos i + iJ1 (kx - kE)
ddx
- ikxcos .
-= pgk3 sin B(x) Lae + in3 - ixn5j
L
B()
-ike cos * L a - 1fl3+ in5
e J0(kx - k) ddx
= -Pgk
sin 'L x B(x) En3 - xn5 -ikx cos i e dx (7)J fl3 = 1012_112 L íkx cos ) e dx (x)(I - I x)
o2
i JL i o B ikx cas 1) e dxand the constants I are the waterline moments n
I n
I = I B(x) x dx n
In the above equations B(x) is the beam of the ship at the waterline at longitudinal position x, i4i is the wave direction, * denotes the complex conjugate, J0 and J1 are
Bessel functions of the first kind. Physical quantities are understood to be the real parts of the complex mathematical expressions.
Newman's formulae have a number of peculiarities due to the slender-body assumptions.
The transverse force vanishes in beam seas. This occurs because the ship's beam is assumed small in comparison with the wavelength. The ship follows the orbital motions of a water particle, and is transparent to waves of all frequencies.
In very short waves the transverse force tends to infinity for all heading angles. The longitudinal force is unbounded in head and following seas. The turning moment, however, remains finite.
3.2 Numerical Comparisons
From the above comments it is clear that Havelock's formulae represent the asymptotic short-wave limit, while Newman's formulae are likely to be valid in long-wave
conditions. These two analytic solutions are compared with drift
forces calculated by a more complicated, but more accurate, numerical procedure based on the NMIWAVE computer program. This procedure is described in the following sections.
Figure 3 shows results of this comparison, and contains the longitudinal and transverse drift forces and turning moment on a drill-ship in oblique seas. Further details of the model and numerical procedure are given in section 6 . The NMIWAVE results are
reproduced from figure 11 , which also shows excellent agreement between the numerical
and experimental results. NMIWAVE nay therefore be regarded as a standard by which the other two more approximate methods may be judged.
Havelock's formula matches quite well with results from NNIWAVE in very short waves. NNIWAVE becomes increasingly expensive to run at high wave frequencies, because more facets are required in order to satisfy the criterion that the maximum facet diameter should be less than 1/7
of a wavelength. Havelock's formula therefore provides a convenient means of extrapolating NMIWAVE results to very high frequencies.
Newman's formula compares rather poorly with the NNIWAVE results over most of the frequency range. The two longitudinal force curves approach zero in a similar way at low frequencies, but behave very differently for W/L/g > 3 (or AIL c 0.7). The
transverse force, as predicted by Newman's formula, is several times too large at all frequencies. The shapes of the two moment curves agree rather better, but Newman's formula again gives numerical values which are 2 - 3 times too large.
From this very limited comparison of the three methods it is concluded that:
Havelock's formula provides a convenient meatis of extrapolating NMIWAVE results to high wave frequencies.
Newman's formulae are based on long-wave assumptions. Unfortunately the drift forces are small in long waves, and only become significant for wavelengths less than
the ship's length. Newman's method gives unsatisfactory results over this range of wave conditions.
c) Neither Havelock's nor Newman's method gives adequate results over a wide range of
wave conditions, and needs to be supplemented by a more accurate procedure.
Faltinsen and L$ken31 reached similar conclusions. They investigated several other procedures, including one based on Newman's theory, but solving the first-order
problem by strip-theory methods32. These methods are valid for wavelengths comparable with the ship's beam, and therefore over the range of wavelengths where drift forces are significant. Faltinsen and L$ken31 calculated drift forces on a tanker, using both this Newman-Helmholtz technique and the full 3-dimensional diffraction solution. They concluded that the Newman-Helmholtz technique should give satisfactory results
for ships of conventional form, provided the wave direction is not too close to the head or following sea condition. In such cases this technique may be regarded as a
possible alternative to the 3-dimensional model described below. The 3-dimensional solution should, however, be used for ships or structures of very full form, or for ships in head and following sea conditions.
4. Wave Diffraction Theory
John's'9 wave diffraction theory describes the scattering of small-amplitude waves by large objects placed in the sea. It provides a suitable basis for estimating wave loads on large fixed structures, such as storage tanks and gravity platforms, and the response motions of large free-floating or moored structures. John's theory was
formulated in terms of integral equations and Green's function. It has since provided
20, 33-42 . .
the basis for several computer programs , of which NMIWAVE is one. NNIWAVE s
development and validation, for the purpose of predicting first-order wave forces
and response motions, were described in a series of papers culminating in references33'34.
Linear wave diffraction theory is based on classical Airy wave theory and ideal flow assumptions, and predicts wave loads associated with both the local accelerating flow field (inertial loading) and with the wave scattering process. Results were found to agree well with experiment in the inertial and diffraction regimes of wave loading (i.e. when the Keulegan-Carpenter number, or ÓID ratio, is small). The basic assumptions of this method are:
of motion may also be linearised;
that the flow is inviscid, incompressible and irrotational,and therefore may be described in terms of a velocity potential 4>.
Appendix i describes the equations of motion of both fluid and structure, and the method of solution used within NMIWAVE.
The following points are noted.
The above solution represents the first term in a series of the form
(l)
+ E24>(2) + +
with associated forces
F =
CF(U
+ C2F(2) + +where c is a small parameter related to wave steepness.
In regular waves of height H the first-order solution is entirely linear, in the sense that the first-order velocity potential, pressures, wave forces and responses are all directly proportional to H.
The wave forces and responses all have sinusoidal time histories, with the same period as the (regular) incident wave. The mean force and response are zero.
Because the theory is linear, results for irregular seas can be obtained by linear superposition of regular wave components. Standard Fourier and spectral methods may be used8. First-order forces and responses cover the same band of frequencies as the wave spectrum, and their mean values are zero.
e) An explanation of mean and low-frequency drift forces is therefore sought in the
the second-order force
4.1 Mean Drift Forces: The Far-Field (Wave Momentum) Method
The mean first-order force is zero, as has been shown. Second-order forces, however, may have both mean and very low-frequency components. These wave drift forces are
the combined result of:
second-order pressures and interactions associated with the first-order wave field. pressures in the second-order wave (e.g. 'set-down').
Component (a) is the more easily computed of the two. It can be calculated directly from the linear diffraction solution. Component (b), however, requires a full
15
second-order model. Fortunately Salvesen has shown that the second-order wave gives rise to low-frequency forces, but no mean drift force (see appendix 4). For present purposes, therefore, the difficulties cf second-order wave diffraction theory can be ignored; the mean drift force depends on the first-order diffraction solution only.
It is convenient to consider first a train of regular waves. In this case there is a mean drift force, but no slowly-varying component. It will be shown later that the mean force in an irregular sea can be obtained directly from results in regular waves by a superposition formula similar to that used for first-order forces.
This train of regular waves impinges on a solid object, and the waves are scattered. Surface waves carry momentum, which changes when the waves are scattered. This
implies that there is a mean force (equal to the rate of change of momentum) acting on the structure. Newman3° showed that this mean force has longitudinal and transverse components
x' y'
and that there is a mean turning moment i associated with changes
in angular momentum, which are given by:
=
JJ
=
-
Ic
sin 8 + PUR(UR sin O +U0cos O) RdSdzM = - rr
URUS R2dOdZ z
JJ
where p is the hydrodynamic pressure; U is fluid velocity with radial and tangential components UR, U0; S is a large cylindrical control surface with radius R, and with its axis passing through the centre about which moments are taken. The bar denotes the mean over a complete wave cycle.
Faltinsen and Michelsen40 showed that the first-order pressures p and velocities U could be related to the source densities f() in first-order diffraction theory, giving
2i-/
Q() cos 8 cos O - k r Q2(0) cos d8= s(k)
1-sinhkd
Jo
21T
W a
= Ps(k) 1sinhkd
/
Q() sín 8 cos O -k I Q2(0) sin O do]Jo
= 1 PS(k){
2gk
2rr4k w cosh kd Q (8) cos O + Q' (8) sin _kJ O'(0)Q2(0)dO}
(8)
o
(9)
where s(k) = sinh 2kd + kd,
and O(e), Q(e) are defined in terms of the source densities
Q(0)e10(0) -2îi(2-k2)
/
k2d-v2d +6
J
L()
+f.()n
cosh k(y + d) e-ik( cos O + O sin8dS
j=l Jand = (ci, , y), y2 = = k tanh kd,
'(e) d6lde. Appendix 2 explains how these equations were derived.
The turning moment N, above, is about the origin of coordinates. NNIWAVE evaluates the moment about any fixed point, or about a point moving with the structure. The moment about a fixed point (xv, y) is simply
- xp + Yp Fx
The moment about a point moving with the structure requires an additional term
F
+ 2 l
where
l' 2 are first-order surge and sway motions at the relevant point, and F1, F2
represent total first-order fluid forces acting on the structure, including all hydrodynamic and buoyancy components.
Note that in the special case of a free-floating structure
F = M' =
If the moment and motions are relative to this free-floating structure's centre of gravity, then
- 2
-F1 = - wmn1, F2 =
where m is the structure's mass, so that
F2
+ 2 F1 = O
In this special case, therefore, the mean turning moment is the same, whether it is about the mean or instantaneous position of the structure's centre of gravity.
a) Forces
The far-field approach, described in section 4.1, requires little computational effort to evaluate the mean horizontal forces and turning moment. The vertical force, heel and trim moments, however, are associated with rates of change of vertical momentum, and involve conditions at the free surface and sea-bed. These components can be evaluated more straightforwardly by the near-field method, which also has to be used to compute the slowly-varying forces. This near-field method is more direct, and shows more clearly the effects of first-order waves, forces and response motions on the drift force. This method is more cumbersome to program on the computer, however, and more demanding in terms of computing time and storage.
Pinkster and van Oortmerssen22' 42 showed that the mean second-order force on a structure can be expressed as the sum of several terms:
= -pg
I 2 n' dl Lor
-+p
IV2n dS
JSo-PI
{----.n}n dS
JSoo
+ RF Iv V VI (10)where Lo is the mean waterline and So the mean underwater surface of the structure;
n is the normal to that surface,
o
4.2 Mean Forces: Near Field Method
I
II
i
(n1,
'2 "3
= I
i2 +
and (n1, n2, n3) is the normal to the mean hull surface at the waterline; n is the translational motion of the structure at the relevant point on its surface:
= n2 n3) + (rì4,
n5, n6)
x (x c)
The 3 x 3 matrix R contains
_o
6
0 114
-n
n, O_5
4 _jand represents a cross-product with the rotation vector (fl4 n
ne).
The force F is the total first-order fluid force, including all hydrodynamic and hydrostatic components. The buoyancy term= -PgzA
(n42 +n52)(O, 0, 1)
where A is the structure's waterplane area. The last term represents the effect of the second-order wave, and, as shown in appendix 4,
= P A (O, 0, 1)
where is the mean set-down pressure in the undisturbed incident wave.
The relative surface elevation
=
n3
--
14 + (x - x)n5
In the above expressions
r' ,
n, n,
F all represent first-order quantities; thusall second-order forces (except VI) involve products of two first-order quantities.
b) Moment about a Fixed Point
The mean second-order moment is expressed in a similar way. The derivation is
extremely lengthy, but is similar to that described in appendix 3 for the mean force. The moment about the (fixed) point Xg (at the structure's equilibrium centre of
gravity) has the same form as equation (10) except that n0, n' are replaced by vector cross-products
(x-x)xn, (x-x)xn'
- g o - g
-The total first-order force F is replaced by the total first-order moment about and the buoyancy term by
= pgV {n1 n6 +
- n6),
n2 n6 +
-
n4 n5 -
(xb - x )(ni52 -n62),
_T2 fl5 - fl1 n4
+ b -n4 n6
_(xb - x) n5 n6 }
+ n52)(A. Ax;
0)where A A are first-order moments of waterplane area
X, y A = (y-y )dA 'A g A = I (x-x )dA, X A g
b' zb) is the centre of buoyancy in equilibrium, and V the structure's mean
= pgV { (Yg - Ybn4 - n62), Xg -
xb)(nS
-
n6) + (Yg - n4 n5,(YgY) n
n- (x -x)
n } b46
g b56
+ 2)(A - A O) 5 y, X,d) Comparisons with Pinkster's and Momentum Results
The above expressions(1O)forF and M are identical to those of references ' , except
that the latter contain no buoyancy term WI. This term represents the effects of second-order motions on the hydrostatic restoring force,and is zero provided:
the moment is about the structure's actual (i.e. moving) centre of gravity rather than its mean (fixed) position;
= Xb Yg = This is always true for a free-floating structure, but may not be correct for a tethered platform;
z = O. This fixes the reference point for motions n at mean water level.
Pinkster, therefore, appears to have made these assumptions in developing his theory.
The significance of the last term in WI, involving the waterplane area, is clear. It represents the effect of the second-order heave motion -z (r42 + n52) on the
-A
O) y, X,c) Moment about a Point on the Structure
The corresponding expression for the mean moment about the structure's instantaneous centre of gravity (i.e. about a point moving with the structure) is identical except that the total first-order moment M is replaced by the first-order moment about the instantaneous centre of gravity, and the buoyancy term by
vertical force, heel and trim moments. It is analogous to the buoyancy stiffness force and moments due to heave in the first-order problem.
Transferring attention to the momentum method, it is clear that both this and the near-field technique make identical assumptions. They should therefore produce
identical results. Both methods have been programmed in conjunction with the NMIWAVE diffraction program. Some numerical comparisons between the two methods will be
shown in a later section.
e) Physical Significance of the Near-Field Terms
These expressions(IO) canbe interpreted in the following way, as shown in figure 4.
The first-order force represents pressures acting over the structure's mean wetted surface. Term I represents additional pressures acting between the structure's mean waterline and instantaneous free surface.
Bernoulli's equation contains a quadratic term pjU2. This second-order dynamic pressure is integrated over the structure's mean wetted surface.
The first-order force represents pressures acting on the structure as if it occupied its mean position in space throughout. By responding to the waves, the structure moves to a slightly different location. Term III represents the change in the force due to first-order motions through the pressure field.
This term has both hydrodynamic and hydrostatic components. Fluid pressures act at right angles to the structure's instantaneous surface, so that the force vector rotates as the structure rolls, pitches and yaws. Note that the hydrostatic
term has a horizontal component, because it includes hydrostatic pressures acting between mean water level and the structure's mean waterline (the former fixed in
space, and the latter moving with the structure). Term I includes hydrostatic pressure acting between the mean waterline and instantaneous free surface.
This term represents changes in the buoyancy force due to second-order motions. Second-order motions are of two types: second-order effects of first-order response motions, and second-order response to second-order forcing. This term includes
motions of the first type, but not of the second, because the present purpose is to calculate the second-order forcing only.
VI. This is the contribution from the second-order wave. As shown in appendix 4, the second-order wave is equivalent to a change in mean pressure throughout the fluid, due to set-down of the regular incident waves. It makes no contribution to the mean horizontal drift forces and turning moment, but may affect mean vertical forces,
trim and heel moments. The set-down term can make a significant contribution to slowly-varying drift forces in an irregular sea, however, as will be shown in part 2 of this report.
5. Comparisons with Published Data
In the previous section the two methods used to calculate mean drift forces and moments (i.e. the near field and far field methods) were described. The next
sections of the report describe the validation of the new computer programs, which were developed at NMI. These two programs calculate second order mean and slowly
varying wave drift forces, using the above two alternative methods. Results for mean forces are compared with:
simple analytic solutions
numerical results obtained elsewhere experiments performed at NNI.
Stages (i) and (ii) are described below, and stage (iii) in section 6.
The far field method was validated first by comparing results obtained at NMI with an analytical solution obtained by van Oortmerssen46 and one based on the MacCamy and Fuchs25 first order solution. In addition, this method was compared with results
40
obtained by Faltinsen and Michelsen who used a similar method to that used at NMI. These computations are described in part 5.1 below. The near field method has been developed by Pinkster22, who has published results for several simple cases. In part 5.2 below both NNI methods are compared with Pinkster's results.
5.1 Calculations Using the Far Field Method
Faltinsen and Michelsen4° have published results for the steady drift forces in regular waves on a floating rectangular box in deep water. The details of the box are contained in Appendix 5.
Their drift forces were calculated using equations (9). In order to test the MMI program that uses the same expression, Faltinsen and Michelsen's calculations were repeated. These new results are compared with Faltinsen and Míchelsen's values in Fig. 5(a) and Fig. 5(b). Fig. 5(a) shows the steady horizontal drift force in following waves for a fixed box and for a free floating box. Fig. 5(b) shows the results of calculations for quartering waves. In all cases the MMI results agree very well with the Faltinsen and Michelsen results.
The X-component of the mean drift force can be expressed as
F
=F
+F
X xl x2
where (taken from equations (9))
Ç a
xl = ps(k) sinh kd Q() cos cos S
21T
and = -ps(k) k F Q2(e) e e
x2 j
These two components were calculated separately, using NNIWAVE, and compared
individually with an analytic theory of NacCamy and Fuchs25 for a surface piercing vertical circular cylinder resting on the sea bed. Source densities derived from this solution were used to obtain Q() and Q(8) and then substituted into the expressions for F1 and
x2 Figure 6 shows the comparison between this analytic
solution and corresponding forces obtained using NMIWAVE. Agreement is excellent.
Figure 7 shows the total drift force, obtained by summing these two components. Also
sho are van Oortmerssen's46 results, based on Havelockrs17 deep water theory, and corrected for finite depth effects. Again there is excellent agreement, between the three sets of results.
These tests on the far field program developed at NNI show that:
it
produces results that agree well with Faltinsen and Michelsen, who use an identical expression for the drift forceit gives results that agree well with the exact expression for the drift force, published by van Oortmerssen, for a fixed surface-piercing vertical cylinder
It should be noted that Faltinsen and Michelsen and van Oortmerssen 6 also published experimental results and found reasonable agreement between theory and
experiment. Therefore, the tests on the far-field method confirm that the method used to calculate drift forces is valid and that the NNI program works correctly.
5.2 Calculations Using the Near Field Method
24 .
Pinkster has published results on the steady drift force on simple bodies using the near field method. One of these bodies is a simple barge, the details of which are in Appendix5. Mohn45 has also used the near field method to calculate mean drift forces on the same barge.
The NMI near field and far field programs were used to calculate steady surge drift forces in head waves and steady sway drift forces in beam waves for this barge. The results are shown in Fig. 8. Both NNI methods agree very well with each other and with Mohn's results. All these results also approach the high frequency limit for
the drift force given by Havelock's formula17 for head waves:
= - pgH2B
where B is the beam of the body and H is the wave height.
However, Pinkster's results are significantly greater than all the other results. It
is believed that most of this difference is due to the waterline integral term: i.e. term I in the near field formula for the drift force (expression (10)). In
order to show this, the contribution from each term ín the near field formula for the drift force (see expression (10)) on the barge in head waves was calculated
using the NMI program. These results are compared with those obtained by Pinkster in Fig. 9. There are two important points to note in this figure:
(i) NMI calculations for term I are significantly less in absolute value than
Pinkster's calculations. The other three terms agree very well.
ii) Term I is the largest and is opposite in sign to terms II, III and IV, whose sum is comparable in magnitude with term I. Therefore, small percentage errors
in term I give rise to larger percentage errors in the total drift force. In
this particular case terms II, III and IV are all marginally greater than Pinkster's, thus contributing even more to the differences in the total drift
force.
The differences in term I are almost certainly due to the differences in the way that the waterline was modelled.
Both NMIWAVE and Pinkster's program42 make use of pulsating point sources distributed over the body's surface. A point source is a good approximation to a spread source panel if the potential is evaluated some distance from the panel. Near the panel, however, the velocity potential is affected by the singularity at the point source. NMIWAVE2° uses a standard correction to replace this singularity when evaluating the potential due to a source exactly at the centre of the source panel. However,
this correction does not apply to other points near the singularity and so the usual practice at NMI is to avoid evaluating the velocity potential and terms dependent on it within a facet radius of the singularity47.
The waterline integral term contains the free surface elevation, which is calculated from the velocity potential at the mean waterline. It is important, therefore, when evaluating term I numerically to choose points on the mean waterline so that the
effects of the nearest source singularities are smoothed out. In the NNI calculations the barge was modelled with rectangular facets (see Appendix 5) and the mean
waterline was defined with twice as many points as the number of sources in the row nearest the waterline. Pinkster, in ref. 24, has shown that for his results on the
barge term I was evaluated at points nearest to the singularities. Fig. A3(b) shows these two different models of the waterline. Clearly Pinkster has evaluated term I at points where the nearest singularity causes a distortion in the velocity potential.
The NMI program was used to calculate the mean drift forces on the barge for 4 wave periods using these two different models for the mean waterline. The results are shown in Fig. A3(c) and clearly demonstrate that the NMI results agree very well with
Pinkster's results when the waterline is modelled in the same way.
It is important to point out that the II results agree well with Mohn45 who uses a finite element method to obtain the first order velocity potential. The problem of source singularities does not arise in his method, and therefore his results confirm the view that the source singularities account for Pinkster's results being different. It should be mentioned that Pinkster has also published experimental results for the barge, and these appear to agree with his theoretical results.
Pinkster has also published results for the drift force on a submerged horizontal circular cylinder24. The details of the cylinder are contained in Appendix 5. In this case the mean horizontal drift force is zero at all frequencies (see Ogilvie48). Pinkster has published results for the mean vertical drift force and trim moment24 on the cylinder. The NNI near field program was used to calculate the same quantities for this cylinder. (The far field method cannot be used for vertical forces). The
results are displayed in Fig. 10. The NNI results agree very well with Pinkster's results at all frequencies except the lowest. Since the cylinder is submerged there is no waterline contribution to the drift force. (In any case the waterline term only contributes to vertical forces if the surface of the structure is inclined at the waterline). These results confirm the víew that the differences observed above in forces on the barge are due to inadequacies in modelling the waterline term.
All the work above has shown that both methods for calculating the steady drift forces agree well with published data and with each other.
Experiments were performed on a moored drill ship subject to regular waves, beating waves and irregular waves in the No. 3 towing tank at NIMI. The basic experimental set-up and analysis are described in Appendix 6, together with comparisons of the measured first order motions and the NMIWAVE theoretical first order motions for this
ship. The results on mean drift forces are described below.
6.1 Regular Waves
As stated in Appendix 6 the mean experimental surge and sway forces and yaw moment were obtained for the ship in regular waves at model-scale frequencies ranging from 0.371Hz to 0.911HZ. The two NMI programs used the output from the first order NMIWAVE calculations (see Appendix 6) to calculate these steady forces. The
comparisons between NNIWAVE and experiment are shown in Fig. il(a) - 11(c). It can be seen that both theoretical methods agree well with each other and with the
experimental results. The near field method mostly produces slightly larger drift forces and yaw moment than the far field theory and is marginally closer to the
experimental results.
6.2 Pairs of Beating Waves
The mean forces in pairs of beating waves were also calculated. The waves were generated in the test tank by imposing a low frequency modulation on a regular wave of higher
frequency. The resulting wave should be equivalent to two superimposed regular waves of different frequencies but of equal height. However, problems in generating the higher frequency waves meant that in reality the spectrum in the tank contained two
frequency components of unequal height. This meant that there were difficulties in defining the central frequency, c, which in theory is the frequency of the regular wave on which the low frequency modulation is imposed.
The purpose of the analysis was to compare the drift forces in a regular wave of frequency c with the mean drift forces in a pair of beating waves with central frequency
. Therefore, was obtained for each pair of beating waves from the
power spectrum using:
S(f1)f1 + S(f2)f2 S(f1 + S(f2)
The experimental and theoretical results are compared in Fig 13. The theoretical
results are consistently larger than experimental values, hut generally by less than 30%. where S(f) is the power spectral density at frequency f, arid f1 and f2 are the two
frequency components referred to above. The beat frequency was
= li
-2
The mean drift forces in surge and sway were obtained from the means of the time histories of the cable tensions in the x and y directions. (The mean drift force
is equal in magnitude but of opposite sign to the mean cable tensions). These mean forces are plotted against f0, which has been non-dimensionalised toJL/g in Figs 12(a) and 12 (b). In these figures the experimental and theoretical results for mean drift forci in regular waves are also shown for comparison. The mean sway drift forces in pairs-of of beating waves with central frequency agree reasonably well with the mean forces in regular waves of the same frequency
f ig 12 (b)). The results for the surge force show
greater percentage differencesbetweenbeating waves and regular waves (fig 12(a)). Howeve
these forces are much smaller in magnitude and thereforeany measurement errors are magnified.
6.3 Irregular Waves
Finally the mean drift forces in irregular waves were obtained. The main objective of this work was to investigate the prediction of mean drift forces in irregular waves using the drift forces measured in regular waves. The mean drift force in irregular waves,
f(2),
can be obtained using an exact formula due to Pinkster49:(2)
2
sf
D)df
where S(f) is the wave power spectral density at frequency f and D(f) is the measured mean drift force in regular waves of frequency f and unit amplitude.
As stated earlier, D(f) was measured for sway and surge forces in 10 regular wave frequencies ranging from 0.371Hz to 0.911HZ. Intermediate values were obtained by linear interpolation. D(f) was extrapolated to the constant value given by
Havelockts formula17. The surge force was extrapolated for frequencies greater than 0.911Hz by using the theoretical values calculated by NMIWAVE for the mean surge drift force in regular waves. S(f) was taken from a spectral analysis of the
For both theory and experiment the mean drift force is much less in a JONSWAP spectrum than in the Pierson-Moskowitz spectrum. The JONSWAP spectrum has a much narrower band width than the Pierson-Moskowitz spectrum. The magnitude of the mean drift force is more sensitive to the position of the peak in a spectrum of narrow band width. In all four runs with a JONSWAP wave spectrum, the spectral peak occured at a relatively low frequency where D(f) was small. The Pierson-Moskowitz spectrum, however, had significant energy in the frequency band where the ship's drift force function D(f) is large. Therefore, the mean drift force would be larger in the Pierson-Moskowitz spectrum, as the results show.
Note that the results in Fig. 13 do not vary linearly with the square of the significant wave height for the JONSWAP spectra runs. This is because the mean zero crossing
periods of the 4 runs were also varied(see table I), thus moving the position of the peak of the spectrum. Since the drift force will be very sensitive to the position of
this peak, It is not surprIsing that there is such a variation in the drift force. For the Pierson-Moskowitz spectra the mean drift force does appear to be more nearly proportional to the square of the significant wave height.
6.4 Conclusions of experimental investigation
Comparisons between measured and computed mean drift forces showed the following results:
excellent agreement between the two sets of computed forces, based on two different formulae, and experimental values in regular wave conditions;
good agreement when measured force coefficients, obtained in regular wave groups (pairs of superimposed regular waves), were compared with the results for purely regular waves. As expected, agreement was best when the beat frequency (the difference between the two component frequencies) was low;
less good agreement in irregular waves, though the theoretical and measured force.s still differed by only about 30%.
It is not clear why the measured and computed results agreed less well in irregular than in regular waves. Possible explanations, however, include the following.
The superposition formula for mean drift forces in irregular waves involves the product of wave spectral density S(f) and che force coefficient D(f).
Conditions were such that S(f) peaked at a frequency where D(f) was very small, and became small as D(f) increased. The mean force was therefore sensitive to variations in both the force coefficient D(f) at low frequencies and the spectral density S(f) at high frequencies. Such variations might not be apparent when examining the coefficient or spectral density function as a whole, over the entire range of frequencies.
There were problems associated with both generating and measuring hígh-frequency wave components in the tank. There was, for example, some evidence that the
incident wave field was being contaminated by waves reflected from the ship model and tank walls. For this reason measurements obtained in high-frequency
regular waves were discarded. It was not possible, however, to remove high-frequency components from the irregular wave spectra.
The superposition formula is exact to second order, and assumes linear superpositic of the underlying wave components. Such assumptions appear reasonable for the greater part of the wave spectrum, but the shortest waves may have been rather
too steep, and therefore non-linear in their behaviour.
These problems warrant further investigation.
7. Concluding Comments
The NMIWAVE computer program is based on a first-order (linear) theory of wave
diffraction and structural response. It predicts first-order oscillatory wave loads, response motions of the structure, and tether loads. Since this program was completed in 1976, it has been applied to a wide range of ships and offshore structures,
including semisubmersibles, tethered buoyant and gravity platforms.
Several additional programs have now been developed. These extend NNIWAVE so that it can now predict mean and low frequency components of the second-order wave force. These forces affect drifting of disabled ships, low-frequency motions of moored vessels, and associated loads in mooring or dynamic positioning systems. They are
also considered responsible for low-frequency heave and steady heel motions cf semisubrnersible vessels.
The present report discusses the physical processes which can give rise to a mean wave
force. lt describes two alternative methods for calculating the mean force, based
on the NNIWAVE first-order solution, and compares the results with earlier published data, analytic solutions and experiment. Conclusions of this (the first) stage of
the present investigation are as follows.
The relative roles of drag and wave diffraction are examined. Wave drift forces on slender tubular structures are mainly associated with drag; forces on ships
and large-volume structures are dominated by the effects of wave diffraction and radiation. NumerIcal techniques, described in this report, apply to structures of the second type only.
Two different computer programs have been developed, corresponding to two
alternative ways in which to describe the problem. One program is based on the 'far-field' approach. Forces are derived from mathematical expressions for
momentum in the wave field far from the structure. This program is computationally very efficient but only calculates the mean horizontal force and turning moment.
The alternative 'near-field' program is more expensive to run on the computer, and requires considerably more disc and core storage. Second-order forces are calculated directly from pressures acting on the structure itself, and from its
response motions. This program can predict ali six components of the force and moment, and both the mean and low-frequency terms.
These new computer programs have been validated against simple analytic solutions, published data and experiment. There was generally very good agreement, but with some discrepancies as noted below.
(y) Results from the two alternative programs were compared with each other. These
programs maku use of the first-order NMIWAVE solution, but are otherwise quite distinct; they use different mathematical expressions and numerical procedures. The two suts of results were found to agree very well, thus giving confidence ín both methods of calculation.
(vi) One component was found to dominate the near-field force in nearly all the cases
examined. This component involved an integral of surface elevation around the structure's mean waterline. The drift force was found to be very sensitive to the spacing of mesh points around this waterline. It is recommended that these
points should be spaced at exactly half the distance separating adjacent sources in the NNIWAVE model. Some discrepancies in published results are attributed to inaccuracy in this waterline term.
Theoretical and experimental forces agreed excellently in regular wave conditions, but rather less well in irregular seas. The reason for this deterioration
is not clear, though a number of possible explanations are suggested.
Results from the new computer programs were compared with two simple analytic formulae, due to Havelock and Newman. Havelock's short-wave solution was found
to provide a useful means of extrapolating results from NNIWAVE to very high frequencies. Newman's long-wave theory, however, was found to have a very restricted range of validity.
A second report on low-frequency wave forces and response will follow.
Acknowledgements
This work was supported by the Department of Energy, through the Offshore Energy Technology Board, as part of an overall programme of research into fluid loading of offshore structures.
The authors also wish to thank their many colleagues at NNI, who assisted in carrying out the experiments, analysing the data, and running the computer models: in
particular to Miss J Masters-Thomas and members of the Offshore Group. Thanks are also due to U H Pinto, who developed a large part of the far-field computer program.
p water density
14) wave direction
Ç surface elevation wave amplitude = F172
Ç relative surface elevation
r
= (ct, ,y ): point on body surface S w angular frequency of regular wave
superscripts0' (2): equilibrium (zeroth order), first and second order quantities
subscript c central value
References
Stammers A J, Brockbank R and Wennink C J, 1977, 'Investigation of vessel mooring subject to wave action'; Dock and Harbour Authority, vol. 57 (no. 677), PP. 429-430.
Bowers E C, 1977, 'Model simulation of ship movements'; Dock and Harbour Authority, vol. 58 (no. 681), pp. 162-163.
Srheim H R, 1981, 'Analysis of motion in single-point mooring systems'; Norwegian Maritime Research, vol. 9, no. 1, pp. 2-13.
Lean G H, 1971, 'Subharmonic motions of moored ships subjected to wave action'; Trans. Roy. Instn. Nay. Archit., London, vol. 113, pp. 387-399.
Owen D C and Lightfoot B T, 1977, 'Theoretical Ship Techn. Res. Symp. (STAR), San Francisco, Soc. Nay. Atchit. Mar. Engrs., pp. 315-324.
Olsen O A, Braathen A, L$ken A E, Nyhus K A and Torset O P, 1978, 'Slow and high-frequency motions and loads of articulated single point mooring systems for large tankers'; Norwegian Maritime Res., vol. 6, no. 2, pp. 14-28.
Wichers J E W, 1979, 'Slowly oscillating mooring forces in single point mooring systems'; Proc. 2nd Conf. Behaviour Offsh. Struct. (BOSS '79), London,vol. 3.
Price W C and Bishop R E D., 1974, 'Probabilistic Theory of Ship Dynamics', Chapman and
Hall.
Spangenberg S and Jacobsen B K, 1980, 'The effect of wave grouping on slow drift oscillations of an offshore structure'; mt. Symp. Ocean Engrg. Ship Handling,
Co thenburg.
Remery G F M and Hermans A J, 1971, 'The slow drift oscillations of a moored object in random seas'; Offsh. Techn. Conf, paper no. OTC 1500, Houston.
Bowers E C, 1976, 'Long-period oscillations of moored ships subject to short-wave
Naess A and B$rresen R, 1978, 'on experimental prediction of low-frequency oscillations of moored structures'; Offsh. Techn. Conf, paper no. OTC 2879, Houston.
Numata E, Michel W H and Maclure A C, 1976, 'Assessment of stability requirements for semisubmersible units'; Trans. Soc. Nay. Archit. Mar. Engrs., vol. 84, pp.
56-Morrall A, 1978, 'The influence of the steady vertical component of wave force on the stability of semisubmersibles', Nat. Mar. Inst. rep. no. NMI R.46.
Salvesen N, 1974, 'Second-order steady-state forces and moments on surface ships in oblique regular waves'; Proc. Symp. Dyn. Mar. Vehicles Struct. in Waves, Inst. Mech. Engrs., London, pp. 212-226.
Dacunha N M C, Hogben N and Standing R G, 1981, 'Responses to slowly-varying drift forces and their sensitivity to methods of spectral modelling: a preliminary
assessment'; Nat. Mar. Inst. rep. no. 101.
Havelock T H, 1940, 'The pressure of water waves on a fixed obstacle'; Proc. Roy. Soc., London, vol. A175, pp. 409-421.
Morison J R, O'Brien M P, Johnson J W and Schaaf S A, 1950, 'The forces excited by surface waves on piles'; Petr. Trans. AIME, vol. 189, paper no. TP2846, pp. 149-15Z
John F, 1950, 'On the motion of floating bodies', parts I and II; Commun. Pure Appi. Math., vol. 2, pp. 13-57 and vol. 3, pp. 45-101.
Hogben N and Standing R G, 1974, 'Wave loads on large bodies'; Proc. Symp. Dyn. Mar. Vehicles Struct. in Waves, Inst. Mech. Engrs, London, pp. 258-277.
Havelock T H, 1942, 'The drifting force on a ship among waves'; Phil. Mag., vol. 33, pp. 467-475.
PinksterJA, 1979, 'Mean and low-frequency wave drifting forces on floating structures'; Ocean Engrg., vol. 6, pp. 593-615.
Huse E, 1977, 'Wave induced mean force on platforms in direction opposite to wave propagation'; Norwegían Mar. Res., vol. 5, no. 1, pp. 2-5.
Pinkster J A, 1980, 'Low frequency second order wave exciting forces on floating structures'; Netherlands Ship Model Basin rep. no. 650.
MacCamy R C and Fuchs R A, 1954, 'Wave forces on piles: a diffraction theory'; Beach Erosion Board, US Corps of Engrs., Washington DC, Tech. Memo, no. 69.
Karpinnen T, 1979, 'An approach to computing the second order steady forces on semisubmerged structures'; Helsinki Univ. of Technology, Ship Hydrodyn. Lab. rep. no. 16.
Pijfers J G L and Brink A W, 1977, 'Calculated drift force of two semisubmersible platform types in regular and irregular waves'; Offsh. Techn. Conf, paper no. OTO 2977, Houston.
Longuet-Higgins M S, 1977, 'The mean forces exerted by waves on floating or
submerged bodies, with applications to sand bars and wave power machines'; Proc.
Roy. Soc., London, vol. A352, pp. 463-480.
Maruo H, 1960, 'The drift of a body floating in waves'; J Ship Res. vol. 4,no. 3, pp. 1-10.
Newman J N, 1967, 'The drift force and moment on ships in waves'; J Ship Res., vol. 11, pp. 51-60.
Faltinsen O M and L$ken A E, 1978, 'Drift forces and slowly-varying forces on ships and offshore structures in waves'; Norwegian Mar. Res., vol. 6, no. 1, pp. 2-15.
Faltinsen O M and L$ken A E, 1978, 'Drift forces and slowly-varying horizontal forces on a ship in waves'; Symp. Appl. Math., Deift Univ. of Technology.
Standing R G, 1978, 'Applications of wave diffraction theory'; mt. J Num. Meth. in Engrg, vol. 13, pp. 49-72.
Standing R G, 1979, 'Use of wave diffraction theory with Morison's equation to compute wave loads and motions of offshore structures'; Nat. Mar. Inst. rep. no. NMI R. 74.
Standing R G, 1979, 'Use of potential flow theory in evaluating wave forces on offshore structures'; Proc. Conf. Power from Sea Waves, Inst. Math. Appls., Edinburgh.
Garrison C J, 1974, 'Dynamic response of floating bodies'; Offsh. Tech. Conf. paper no. OTC 2067, Houston.
Garrison C J, 1974, 'Hydrodynamics of large objects in the sea'; J 1-lydronautics, vol. 8, pp. 5-12.
Oortmerssen G van, 1972, 'Some aspects of very large offshore structures'; Proc. 9th ONR Symp. Naval Hydrodyn, Paris, pp. 975-1001.
Oortmerssen G van, 1976, 'The motions of a ship in shallow water'; Ocean Engrg., vol. 3, pp. 221-255.
Faltinsen O M and Michelsen F C, 1974, 'Motions of large structures in waves at zero Froude number'; Proc. Symp. Dyn. Mar. Vehicles Struct. in Waves, Inst. Mech. Engrs., London, pp. 91-106.
Lebreton J C and Margnac A, 1968, 'Calcul des Mouvements d'un navire ou d'une platforme amars dans la houle'; La Houille Blanche, vol. 23, pp. 379-389.
Pinkster J A and Oortmerssen G van, 1976, 'Computation of the first and second order wave forces on bodies oscillating in regular waves'; Proc. 2nd. Conf. Num. Ship Hydrodyn., Berkeley, pp. 136-156.
Erdelyi A, 1956, 'Asymptotic Expansions'; Dover, New York.
Longuet-Higgins M S and Stewart R W, 1962, 'Radiation stress and mass transport in gravity waves, with application to surf beats'; J Fluid Mech., vol. 13, pp. 481-504.
Mohn B, 1979, 'Second-order drift forces upon large bodies in regular waves', Proc. 2nd Conf. Behaviour Offsh. Struct. (BOSS '79), London, vol. 1, pp. 363-370.
Van Oortmerssen G, 1971, 'The interaction between a vertical circular cylinder and regular waves', Symp. Offsh. Hydr., Wageningen, NSMB pubi. no. 375, chap. Xl pp. 1-24.
Standing R G, 1978, User manual for the NMIWAVE program suite to compute wave loads and motions of offshore structures, XMl (1978).
Ogilvie J F, 1963, 'First and Second Order Forces on a Cyliner Submerged Under a Free Surface', J Fluid Mech., Vol. 16, pp. 451-472.
Pinkster J A, 1976, 'Low frequency second-order wave forces on vessels moored at sea', Proc. 11th ONR Symp. Naval Hydr., London, pp. 603-616.
Fryer D K, Gilbert G, Wilkie M J, 1973, 'A Wave Spectrum Synthesizer', J Hydr. Res., vol. 11 no. 3.
qf
Appendix 1. Solution of First-Order Problem
The incident wave is assumed to be regular, and the water assumed to be inviscid, irrotational and incompressible, so that its motions can be described in terms of a velocity potential. The waves are assumed to be small, so that the equations of motion may be linearised.
The velocity potential is written as a linear sum of components
6 -iWt
= (f +
d+ :: rì.)e
3=1
where the undisturbed regular incident wave is represented by
-gH cosh k(d + z) i(kx cos + ky sin ij - wt)
e
2w
cosh kd
and
d represents the wave diffracted by the body when held
stationary, q. represents the radiated wave associated with the jth component of body motion r. in calm water. A simple rigid body has 6 degrees of freedom, and
n, n2, n3, n4, n5, n6
representsurge, sway, heave, roll, pitch and yaw respectively relative to a point x. The angular wave frequency w = 2/wave period, and t is time. Cartesian coordinates are
chosen with z positive upwards, and the origin at mean water level, as shown in figure
The unknown potentials and
.
satisfy the equations:
the Laplace equation Vq O in the fluid
the linearised free surface condition g -w2q = O on z = O a radiation condition
hm R
( - ik) = OR
