### OF FlOATING

### Ofl SHORE

### STRUCTURES

Prof.dÈ.ir. J.A. Pinkster

Report No. 1050 December 1996

### FITDeift

Faculty of Mechanical Engineering audMarine TechnologyShip ijydromechanics Laboratory

Pinkster, J.A.

Hydrodynamic aspects of floating offshore structures by J.A. Pinkster - Deift:

Faculty of Mechanical Engmeermg and Marme Technology, Deift Umversity of Technology Ill - (MEMT, ISSN O9256555; 36)

With ref..

ISBN 90-370-0154-8

Subject headings: wave forces, offshore structures

Technische Universiteit Deift, Paculteit der Werktuigbouwkunde en Maritieme Techniek, Bibliotheek WBMT

MekelWeg 2,

2628 CD DeIft

Deze CIP-gegevens ongewijzigd 'afdrukken op de copyrightpagina van het bock, bici. het kopje 'CIP-CiEGEVENS 'KONINKUJKE BIBLIOTHEEK, DEN HAAG'.

Exploration and production of oil and gass offshore depends to a

con-siderabie extent on the behaviòur of floating equipment. There is a great

variety in the shapes and sizes of the floating units employed by the

indus-try . This variety is tö a large extent dictated by the particular function of

a unit. Not all floating structures are designed to float permanently, some

units may only be flöating for a short period of time, for instance during

transport from the construciton yard to the final destination where it may be placed on the sea-flöor. The environment in which floating units operate

is however, in many cases the same i.e. the open sea. The design of such

units must take into account the most important effects arising from such

an environment. Such effects include those due to wind, Waves and currents.

The response of a particular floating unit to these effects is dependent on the

type of.structure, the conditions to w.hich it is sub jécted and on the effects of

such aspects as the presence of a mooring system or whether a unit is under

tow. in this report a review is given of typical behaviour observed in floating

structure undèr various conditions. Much of the. data given in this part has

been derived from model tests carried out in a wind, wave and current

en-vironment. The characteristic behaviour has been confirmed from full scale observations however. After this introduction attention will be focussed en

tirely on wave-related phenomena concerning stationary floating structures. It is assumed that the reader i's familiar with the general' theory of ship

mo-tions in waves. Floating offshore structures can, as mentioned above, assume many forms. A numerical procedures to predict wave-induced loads and

mo-tions based on linear three-dimensional potential theory is discussed and the

validity of such methods, for which computer codes are generally available, s

demonstrated by comparison with results of model tests. Finally, an

impor-tant non-linear effect in the wave loads on floating structures .being the mean and low-frequency second order wave drift forces i's treated. Again results 'of

model tests are cOmpared with results of model tests.

This report was compiled as part of a PhD course given at the Deift University of Technology, 'held' at 20-24 May 1996 under auspices of the J.M. Burgerscentre' for Fluid Dynamics'.

i

### i

### Characteristic behaviour of floating structures.

51.1 Introduction 5

1.2 Thebehaviour of a bow-hawser moored vessel in wind and

current 5

1.3 The behaviour of a large concrete structure under tow 8

1.4 Lowfrequency horizontal motions of permanently moored tankers

in high seas. . 11

1.5 Motions and mooring forces of semisubmersibles 14

1.6 The vertical motions of ships in shallow water, long waves. . 17

1.7 Behaviour of a jetty-moored tanker 20

### Bibliography

252

### Prediction of loads and motions of structures in waves

272.1 Introduction 27

2.2 Theory 28

2.3 Solving for the potentials 32

2.3.1 A 'panel' method 32

2.3.2 Numerical Aspects 35

2.4 Some results of computations 38

2.4.1 The free floating hemisphere 38

2.4.2 Barge 43

### Bibliography

52### 3 Second order wave drift forces

5558

3.3.1 Second order waveforces on a body floating in waves 58

3.3.2 Co-ordinate systems 59

3.3.3 Motions of a point on the body . 60

3.3.4 Fluid motions . . _{6i.}

3.3.5 Body boundary conditions 62

3.3.6. Boundary conditions at infinity 64

3.3.7 Pressure in a point within the fluid 65

3.4 Second order wave forces 66

3.4.1 Second order wave'moment 69

3.5 Quadratic transfer functions for the mean and low frequency

wave drift force 70

3.5.1 Approximation for the contribution of the sec-ond

or-dr potential 71

3.5.2 Wave drift forces i:n regular wave groups 75

3.5.3 Frequency domain calculation of the mean and low

fre-quency drift force 77

3.6 Some results of calculations of wave drift forces . . .. 78

3.6.1 Mean drift forces iù regular waves 78

3.6.2 Components of the mean wave drift forces 81

3.7 Irregular waves 84

3.8 Simple frequency domain method to approximate the low fre-quency horizontal motions of a moored vessel in irregular head

or beam seas 90

3.9 Final remarks 93

3.10 Nomenclature 95

### Bibliography

_{97}

### A Wave drift forces in shallow water

101### A.i Abstract

### loi

A.2 Introduction 102

A.3 Theöry on second order drift fOrces 105

A.3.i The pressure integration method 105

A.4 The model basin 108

A.4.i The ship model 108

A.6 Waves 114

A.6.1 Wave set-down 114

A.7 Results of model tests in regular waves 125

A.71 Mean drift forces in head waves 125

A.7.2 Low frequency surge forces 125

A.8 Final remarks 129

### I3ibliography

130### Chapter 1

### Characteristic beL avio.ur of'

### floating structures.

1 ..1

### Introduction

Investigations into, the füidamenta1aspects of the behaviour of offshore struç-tures have often been guided by the behaviour as observed at sea and as

determined from model testing of such structures in realistically simulated

wind., wave and current environments.

In this section someexamp1es are given of the behaviour of these

struc-tures which have been shown to be of signi cant influence on the design Of the

structures. and: on operational aspects of work offhore., All of the examples covered here have motivated extensive fundamental researdii to determine

the governing factors of the observed behaviour.

### 1,2

### The: behaviour of a bow-hawser moored

### vessel in wind and current

At many locations offshore, huttl tankers,. which are either loading or

dis-charging crude oil offshore, are moored to SPM's using a bow-hawser.

It has been Observed that in conditions of steady wind and current, such vessels can carry out large amplitude hÖrionta11 moions (surge, sway and yawing) Which can lead to large peak loads in the bow-hawser. An example

showing the time record of the bow-hawser force of a 200 idwt tanker is

shown in Fig. (1.1).

This example shows that even in the absence of time variations in the

environmental forces, significant dynamic effects can occur In this particular

case it was shown that these effects are due to inherent instabilities in the horizontal motions of the tanker which are related to the lngth of the bow

hawser.

Fig. (1.2) shows the regions of unstable horizontal motions as a function of the angle between the wind and current on the one hand and the length of the bow-hawser on the other hand. See Ref. [1] for further details.

'o

11 _{o}

N

220 KOWT 1ANKER

Example of record of bow-hawaerforce

ASTERH' PROPULSION

E C UNSTABLE /

### /

200-150 V) 3### I

3 o 100 O -J z D 50220 kDWT TANKER - FULLY LOADED

COMPUTED

o i KNOTIJRRENT - 45 KNOTS WIND e 1 KNOT CURRENT - 22.5 KNOTS' WiND o I KNOT'CURRENT - O KNOTS WiND EXPERIMENT

A i KNOT CURRENT - 45 KNOTS WIND

(POl

UNSTABLE

STABLE

ASTERN PROPULSION P: -- - ASTERN PROPULSION'P: BUOY - HAWSER ELASTICITY

00 _{45}
o in degrees
STABLE
O TON
15 TONS
8.3 tim
go

Figure 1.2: Stability criterion fôr a 200 KDWT tanker in fully loaded

### 1.3

### The behaviour of a large concrete

### struc-ture under tow.

After completion of the platform in sheltered deep water, it is towed out to the field by 6 to 8 large ocean going tugs. On the way out to the open sea,,

narrow fjord passages have to negotiated which places demands on the swept

path and the draft of the platform.

Fig. (1.3 shows that whileunder tow, besides the constant motion in

the tow direction, this type of platform performs oscillatory motions in a11

six degrees of freedom Which have a bearing, among others, on such aspects

as the already-mentioned swept path and draft.

These oscillatory motions are related to a considerable extent to the os-cillatory components in the flow about the platform and the inherent towing stability of the platform on the other hand..

important aspects of the flow are the vortex-shedding phenomenon about the large diameter vertical columns. and flow, separation about the large base

structure. For the transportation of the platform, the towing force is of im-portance as this determines the, number and size of the tugs to be employed.

The force necessary to hold the platform on location prior to being ballasted down to. the sea-bed is determined, among others by the current force that may be exerted by a tidal current, see Fig. (L4).

Currents are usually assumed to be .constant in time. in some cases, such

as during the positioning of this type of platform current speed fluctuations may give rise to undesirablé horizontal platform motions.

Fig, (1.5) shóws an example of current velcity variations expressed in

terms of the measured mean, maximum and minimum current velocity and

the spectral density of the current velocity to a base of the frequency of velocity variations.

As is seen in the figure, velocity variations are low-frequency. 'Often

the natural frequency of the horizontal motions of a moored structure, r a structure being positioned is low due to the large mass on the one hand and

the relatively soft restraining system on the other hand. This can lead to

x,cOä m Y-COG rn. ZCOG m ROLL DEG. 'PrrcH DEG. YAW DEG T.FORCE:SB kN SPEED rnI 2:03 e (T\ LPORCEPS

### 'j

-0900-6I0 lOO l5OO 2CO 28)O 30 300 4000

o. io
n
E
C
'n_{z}
0.05
u
w
Q-'n

### i

I s Cune.? nito sFigure 1.4: Current force

MEAN CURRENT SPEED 0.86 mIs MAX. CURRENT SPEED 1.02 rn/s

MIN. CURRENT SPEED 0.33 mIs

-1..-:t.

0.0 0.5 1.0 LS

FREQUENCY in red/s

TIME ()

### 1.4

### Low frequency horizontal motions of

### per-manently moored tankers h. high seas.

Fig. (1.6) shows the wave elevation record and the motions of.a permanently moored tanker in high head sea conditions., It is seen that the héave and pitch

motions contain the same frequencies as the waves and have amplitudes in the ordr of the wave amplitudes and ware slope respectively.

The surge motions, besides containing (small) wave frequency motion components are dominated by low frequency motion components with

am-plitudes in excess of the wave amam-plitudes.

U4pon closer examinatiOn it appears that, the large amplitude surge

mo-tions contain frequencies centered around the natural surge period of the

moored vessel as dictated on the' one hand by the vessels mass and the other hand by the mooring stiffness The vessel is in fact carrying out resonant low

frequency surge motiöns in response to. small wave forces containing

frequen-cies corresponding to the natural surge frequency of the moored vessel. The response is further aggravated by the fact that the surge motion damping is

very small.

it will be clear that the mooring system of this vessel will be completely dominated by this effect and must, from the çnset, be designed to take this into account. This requires a carefull analysis of the phenomena involved. A

brief introduction is given here.

Wave loads on structures can be split into several components; Firstly there are wave loads which have the. same frequencies as the waves and are

linearly proportional jn amplitude, to. the wave amplitudes. These areJknown as 'first order' wave forces. Secondly, there are wave load components which

have frequencies. both higher and lower than the frequencies of the waves.

These forces are proportional to the square of the wave amplitudes and known as 'second order' wave forces.

Low frequency second order iave forces have fre4uencies corresponding to the fiequencies of wave groups present in irregular waves. These forces, which,, beside containing time-varying components, also contain a nonzero mean component, are known as 'wave drift' forces. This name is a conse-quence of the fact that a 'vessel floatiffg freely in waves will tend to drift in

the direction of propagation of the waves under influence of the mean second order fórces.

High frequency second order forces contain frequencies corresponding to

double the frequency of the waves (also known as "sum frequencies'). The reponse of the horizontal motions of moored structures to these forces is generally small.

The fact that iow frequency drift forces can cause lange amplitude

hori-zontal motion responses in moored vessels and that these motiousare related to the wave group phenomenon was demonstrated by Remery and Hermans [2].

One of the results of their investigations is shown in Fig. (1.7). In this figure. the low frequency surge motion amplitude of a barge moored in head seas consisting of regular wave groups (superposition of two regular waves

with small difference frequency corresponding, to the wave group frequency) is shown.

It is seen that when the period of the wave groups equals the natural

surge period of the moored vessel, large motión amplitudes are the result.

Fig. (1.8) taken' fröm Ref. [3], shows that the'low frequency surge motions

of a barge moored in head seas increase more or less' as a 'linear function of the square of the significant wave height óf the waves. This implies that the

4.0 20 E C Io O 0 05 lO, 1.5 20

NATURAL PERIOD'OF.SURGE MOTION

PERIOD OF WAVE GROURS

-Figure 1.7: Surge motions of a moored barge in regular head wave groups

0n,

Main oartluIársof bároe (full scale

Length, 230 m Breadth 3O5 rn Draught 7.5 m DIsPlacement 39263 m3

### /

IO_{}-'2 20 Cw. in m

Figure 1.8: Low frequency oscillatory surge motjòns of a barge i irregular head waves

### 1.5

_{Motions and mooring forces of sern!}

### submersiíbles.

That not only tankers and other monohuli vessels are subject to the effects of first and second order wave forces is shown by the results given in Fig. (1.9).

This figure shows that both the surge mctions of a large sèrni-subrnersible

crane vessel (SSCV) and the forces in the mooring lines: display, besides com-ponents with wave frequencies, also considerable low frequency comcom-ponents.

Fig. (1 .1O) shows a timerecord of the the low frequency second order wave

drift force on. a semi-submersible. The relationship between the occurrence

of groups of waves in the incident wave train and peaks in. the wave drift force are very clear.

Tension Leg Platforms, being semi-submersible type structures with

ver-tical mooring tethers which restrain the verver-tical motions of the platform,

are also subject to these wave loads. Fig. (1.11) shows the surge sway and yawing motions of a Tension Leg Platfòrm in irregular waves coming at an

angle to the longitudinal axis of the platform. The low frequency

compo-nents in all three motion componenta is clear. The correspónding forces in the mooring tethers of the TLP given in Fig. (1.12) are, howéver, dominated

by wave frequency components. Although not apparent from this figure, tether forces sometimes contain contributions from sum-frequency second order wave forces which can have frequencies co-inciding with the natural

frequency of the vertical motion of a TLP. These frequencies can be high due

oi L WAVE ('n) SURGE (m) loo MOORI '-IO liNE (I)

### IP

O_{Ql}

### IO

### O

O 100 -scordcFigure 1.9: Surge motions and mooring line force of a 55CV

-i

Wave(rn)

o - 25

Lortgi tudinl;-fcrc-e-( toni)

dr---

### -N

-' Measured - Cornputd L L__ t I 0 200 400 000 secondsseconde

o so ion

Figure 1.11: Computed and measured time traces of horizontal motions in

irregular seas approaching from 157.5 degrees

F, (ton) o Moos t, t CO Computed SURGE , fmf SWAY:

### N-:v1 \J /\'

(toi °T'W seconds . . 0 50 100Figure 1.12: Computed and measured time traces of tether forces in irregular seas approaching from 157.5 degrees

measured wavo
m
set-down - measured
2I _{0.5} --,--- - computed
Inrn
I L i
O lOO 2(x) 300
seconds

Figure 1.13: Wave and wave set-down records

### 1.6

### The vertical motions of ships in shallow

### water, long waves.

lt is weil known that one of the non-linear effects in irregular Waves is the

oçcurrence of Wave set-down.

Wave set-down is the phenomenon whereby the 'mean' waterlevel is

low-ered under groups of higher waves and increased under lower waves The

phenomenon has been extensively analysed among others, using potential

theory.

Based on potential theory, set-down appears as the low-frequency second order term in the power seriès expansion of the potential and the correspond-ing wave elevation.

An example of the set-down measured in an irregular wave trainis: shown in Fig. (1'.13). The set-down recòrd has been obtained by lòwpass filtering of

the measured wave elevation record. Bearing in mind the time scale shown in the figure, it is clear that the set-down wave components represent long

When the horizontal dimensions of a floating object are small compared to the wave length the vertical motions of such objects in the waves follow

closely the wave elevatiOn.

Fig. (1.14) shows that in set-down waves even relatively large vessels such as a 125000 m3LNG carrier have small dimensiòns relative to the set-down wave lengths and carry out vertical motions corresponding closely to

the set-down elevations.

O.25_ o -0.25

WATER DEPTH I DRAFT 1.2 O

MEASURED

---CALCULATED 2113r1.7m

LOW FREQUENCY HEAVE(rn.)

Figure 1.14: Second order vertical motions of a 125.000 m3 LNG carrier in

head seas

Fig. (1.15) finally, shows all components contributing to the total vertical

motions of the LNG carrier sailing in shallow water. The total vertical mo-tions consist of a sinkage contribution which is a static contribution related to square. of the forward speed. The wave frequency contribution is linearly related to the wave elevation and the 'group frequency' contribution is due

to the second 'order wave set-down effect in the incident waves.

'SPEED 10 KNOTS r WAVE FREQUENCY

### r

GROUP 'FREQUENCY (2 ,3) SINKAGE SPEED O KNOTS = 1.70 M; T1 = 16.5sFigure 1.15: Heave motion components of a 125.000 m LNG carrier in head

seas

SPEED 5 KNOTS

### 1.7

### Behaviour o a jetty-moored tanker.

At many locations througout the world vessels such as tan1çers are moored to jetties by means of mooring lines ad fenders. At a number of locations these jetties are situated in the open sea and the vessels moored! to them are' subject to the effects of Wind, waves' and' current which induce rnotiòns of the

vessel and forces in the mooring system. Sometimes also passing ships may

induce significant transient mooring loads.

In Fig. (1.16) through Fig. (1.21) some results are given of rnode tests with a 200 Kdwt tanker moored to a jetty in wind and waves. The data is

taken from van Oortmerssen [4].

Fig. (1.16) shows the spectral density of the irregular waves in which the model: tests were carried out. The following figures show time traces of the

motions,.mooring line forces and fender forces. Some figures give the spectral

density of the motions and the fender forces.

The results clearly shów that the. motions and mooring forces show two distinct frequency ranges, i.e. components with wave frequencies and coin-pnents with very löw frequencies.

From aforegoing discussions with respect to second order wave drift forces

¡t will be clear that low frequency components in the motions and forces

are related to the drift force phenomenon. However, in the case of jetty moorTngs, which bave highly nonlinear restoring characteristics (fenders are much stiffer than mooring lines)', even pure wave frequency 'excitation on the vessel can induce motions and mooring forces' which 'contain significant

sub-and superharmonics of the wave frequency. An example is shown in Fig.

(1.22). which has been taken from van Oor.tmerssen [5].

In some cases where the incident irregular waves are long, it has been

found that the corresponding set-down waves can contributie' significantly to the horizontal forces on the moored' vessel and the ensuing mooring forces.

E

¡

i

2.0

I-S

WAVE PEdiQUETICY U ed/U

Figure 1.16: Wave spectra used

- MODEL TEST MOORSIM 2.50 WAVE ml 200L SURGE 1ml WAVESPECTRUM Ho T I 104m 18.0* Z 158m 7.80 I 1.8801- 7.6 200 ¿00 600 SEC O IO S S

Figure 1.17: Deterministic comparison for beam waves (spectrum 3) and

wind (20 m-/s) 2.00 SWAY Im) -00D YAW Ide9.) - ..c>- - . .- .0 1.0 M UNE2 )TONF)0t" ... _.JV//_. -250.00_ 250.00 PENDER I lIONEl 0 -200.00

U LINE 2
- 150.00
- 100.00
FENDER 1rTON) _{0}
-100.00
0.00
-WAVE (m! O
2.00
-SURGE Im)
E-'
t.00
SWAY Im)

### E-

---030 YAW )de.)_{0. }-- 050L

### (TONFìo'

### J'\

Figure 1.18: Spectraofsway motion for beam waves (spectrum 1) and wind (20 m/s) MODEL TEST MOORS!M - ,/p5

### t\

0 000 400 600 SOC 01405Figure 1.19 Deterministic comparison for bow quatering waves and beam

0.5 00 MODEL TEST MOORSIM 00 0.5 1.0 FREQUENCY in rod/s

Figure 1.21: Spectral comparison of fendér fôrce, beam waves, spectr.um 3

MODEL TEST MOORSIM DIFFRAC t 3 i 1 t' I 00 05 '5 FREQUENCY in rad/s

MEASURED COMPUTED

-i N### -

_{\J'}

r'
i'
### _j \

### j\

___j_{'- FENDER i}FENDER 2 LINE 1 LINE 2 LINE 3 LINE 4 TIME WAVE SURGE SWAY HEAVE ROLL -

_{}

--PITCH
YAW
Figure 1.22: Computed and measured ship motions and mooring forces. Reg-ular waves from 90 degrees w = 0.2'12 rad/sec, wave height 0.32 m

### Bibliography

[i] Wichers,.J.E.W.'(19'79):'On the slow' motions of tankers moored to single point mooring systems', Proceedings Boss '79, London.

Remery,C.F.M..and.Hermans, J.A..(l971):'The' slow dift Oscillations of a moored object in random seas', Paper No 1500, OTC Houston, 1971.

Pinkster, J.A.(1976):'Low frequency second order wave forceson vessels 'moored at seal, Eleventh Symposium on Naval Hydrodynamics, Univer-sity College, London, 1976.

van Oortmerssen, G'.,van 'd'en Boom, H.J.J. and Pinkster, J.A.'(1984) "Mathematical Simulation of the behaviour of ships moored to offshore

jetties', ASCE, N:ov. 1984.

[51 van Oortrnerssen, G.'(1i976):'The motions of a mooréd ship in waves',

MARIN pu'b'Ikation Noi, 510,, 1'976.

### Chapte,r 2

### Prediction of loads

### motions of structures in

### waves

### .2.1

### Introduction

Methods.to evaluate the hydrodynamic loads and mötiöns of floating or fixed

structures in waves have ben developed based on linear three-dimensional

potential theory. See, for instance, van Oor,tmerssen [1:].

Experimental verification of results of computations has been carried out

for bodies with a large variety. of shapes. See van Oortmerssen [1] and

.Pinkster: [2]. Such comparisons have shown that 3-d diffraction methods generally can be applied to most body shapes and are therefore a good tool to inestigate such. effects.

in thjs part a briéf description is given of the 3-d diffraction program DELFRAC developed at the Deift University of Technology. By means of this. computational tool the wave-frequency hydrodynamic loads on free'-floating or fixed bodies and the wave-frequency motions of floating bodies as weil as

the second order wave drift forces can be computed.

The method is restricted to arbitrarily shaped bodies with zeio mean forward speed. For the majority of the fixed or floating structures in use

to-day in the Offshore industry this is an acceptable simplification. It should

be mentioned however, that nowadays attentiòn is being focussed on the development of such methods taking into account 'low forward speed such as

could be due, for instance to current effects. See Huijsmans '[3]. Treatment '27

of such methods is beyond the scope of this course.

### 2.2

### Theory

The theory is given for an arbitrarily shapedÇ fixed or free-'iloating body. Lse

is made of a right-handed, eathfiixed O- X1. - X2 - X3. system of axeswith origin at the mean water level and X3 axjs vertically upwards. The body

axis G - x.1 - - x3 has its origin in the center of gravity G of the body, x1 axis towards the bow, x2 axis to. port and x3 axis vertically upwards. In the

mean position the center of gravity of the body is located at X92,.X93

relative. .to the earth-bound system of fixed .axes. The wave elevation and all

potentials are referenced .to the fixed system of axes. Phase angles of body motions are referenced to the undisturbed wave elevation at the centre of

grav.ity of the body.

According to linear potential theory, the potential of a floating body is

a superposition f the potentials due to the undisturbed incoming wave

.the potential due to the diffraction of the undisturbed incoming wave on the

fixed body 4d and the potentials due to the six body motions

### j

= (2.1)

The potentials 'have to satisfy the following boundary conditions: In the

fluid domain,, the equation of continuity or Laplace equation:

a2

At the mean free-surface:

### a

At .the sea floor (X3 .= h):ax3.

an (2.5) in which:

= velocIty of a point on the hull of the bodr

il = normal vector of the hull, positive into thç fluid

The body motion and diffraction potentials have to satisfy a radiation

condition which states that at great distance from the body these potentials dissappear. This condition imposes a uniqueness which would not otherwise

be present. See ref[1].

The fre&surfáce condition fóllows from the assumptions that the pressure at the surface is constant and that water particles dó not pass through the free surface The condition on the wetted surface of the body assures the

no-leak condition of the (moving) hull. The condition at the seafloor is also

a no-leak condition.

Up to nOw no specific wave conditions have been applied.. The boundary conditions are general and apply to all possible realizations of wave con-ditions. In the following we will restrict ourselves to regular waves with constant amplitude, direction and frequency. Within linear potential theory,

wave loads, motions etc., in irregular long-crested or directionally spread seas

can be determined based on regular wave results by means of superposition.

In regular waves a linear potential which is a function of the

earth-fixed co-ordinates and of time t can be written as a product of a co-ordinate dependent part and a harmonic time-dependent part as fòllows:

(X1,X2,X3,t) = 4'(Xi,X2,X3).eT"t (2.6)

Following Tuck[4] a convenient formulation is obt3ined When writing: 7

### =iEqC

(2.7)j=O

in which represents the complex motion amplitudes associated with

the motion modes of the body (j = i, 6) and . the amplitude of the

undis-turbed incoming wave, ( represents the complex amplitude of the diffractiön potential

The aforementioned: boundary 'conditicns' for the potential similar condtions for the co-ordinate dependent part .

The velocity potential associated with the undisturbed

regular wave in water of constant depth h is given by:

.g cosh i(X3 * c) _{ei?c(X1 cosa-hXsina)}

O u coshKh by: in whkh: 4) result in long-crested (2.8,)

c =. the distance from the origin of the earth-fixed axes to the sea floor

h = waterdepth

The following relationship (dispersion' relationship) exists between the waterdepth h, the wave frequency w and' the wave number i:

w2 = u = KgtanhKh (2.9)

Inregular long-crested waves the undisturbed waveelevation is as follows

### lth/

g t

which results in:

<(X1,, X2,, t) _{= c0} e(X3 cosa+X2sina)iwt (2.11)

The motions: of the body in the j-'rnode relative' to 'its body axes are given

(2.10)

(2.12)

in which' the overline indicates the complex amplitude of the motion. In

the following, the overline Which also applies' to the complex 'potentials etc. is neglected.

The complex potential follows from the superposition of the undisturbed

wave potential o, the wave diffraction potential d and the potentials j

associated with the j-modes of mötion of the body (j = 1,6):

K = wave number

### a

= wave direction, zero' for wave travelling in the positive X1 directionw = wave frequency

in which:

S0= mean wetted surface of the body

nk= direction cosine of surface element dS for the k-mode

The .direction cosines are defined as follows:

nl cos(n,xi) = cos(n,x2) n3 = cos(n,x3) n4 = x2n3 - x3n2 n5 = x3n1 - x1n3 X1fl2 - X2fl1 (2.17)

The added mass and damping coupling coefficients are definçdas follows:

ai =

### _[pJJ.cjnk

dS]### [PJjnkds]

(2.18)so

### = -u {(+ d)'cO+

_{'i '4}

(2.13)
The fluid pressure follows from Bernoulli's law:

p(X1, X2,X3, t) _{=} _{= p(X.1, X2,}X3) -iwt (2.14)

with:

6

p(X1,X2,X3) (2.15)

The. wave force _{Xk is given by:}

For these coefficients it can be shown that the following symmetry

rela-tion ships apply:

alci

### = a

= (2.19)

### 2.3

### Solving for the potentials

### 2.3.1

### A 'panel' method

According to Lamb[5] the contributions of the body to the velocity potential

can be based on. a distributiön of sources over the'body surface. The. potential

at a point (X1, X2, X3) due to the bod.y in the mode j (j = 0, 7) is given by:

### q(X1,X2,X3) = Jfcr(Ai,A2,A3)G(Xi,X2,X3,Ai,A2,A3)dS (220.)

in which o(A1,A2,A3) is the source strength at a point on the mean

wetted part of the hull of the structure duè to the j-mode of the body. G(..)

is the Green function. or 'influence function". of a pulsating source located.in

(A1, A, A3) on the potential in a point, located at (X1,, X2, X3) and which

satisfies the equation of continuity, the linearised boundary conditions on the

free surface and on the sea-floor and the radiation condition at infinity.

The Green's function of a source 'located in (A1, A2, A3) satisfying these boundary conditions is given by Wehausen and Laitone [6]:

### =

_{+ }

### --72(

+ v))e cosh(A3 + c)'cosh(X3 ± c)### J0(R) d +

### J

¿ sinh ¿h - u cosh ¿h2( - u2)'cosh K(A3 + c) cosh (X3 + c)

### J0(R)

(2.21)### (iic2v2)'hv

8çb i

= fj=rj(X1,X2,X3)+

-JJoj(Ai,A2,A3)-G(Xi,X2,X3,Ai,A2,A3)d8

so

r = J(X1

### -

A1)### +(X2 -

A2)2 + (X3- A3)2 r1 = J(X1 -A1)2 +(X2 - A2)2 +(X3+2c+A3)2### R

sJ(Xi_Ai)2+(X2_A2)2John [7] has derived the following series for GO which is equivalent to

equation (2.21):

-G(X1.,X2,X3,A1,A2,A3) = 2ir

(ç2 - v2)h +"

coshic(A3 + c) cosh ic(X3 + c) {1o(icr) - iJo(icR)} +

cosj(X3 + c) cos¡(A3 + c)Ko(p1R) (2.22)

Where tj are the positive solutions of:

### ¡j tan(h) = 0

(2.23)Although these two representations are equivalent, one to the two may be

preferred with respect to numerical computations depending on the value of the variables. In general, equation (2.21) is the most convenient

representa-tion for calcularepresenta-tions, but when R = 0 the value of K0 becomes infinite, and therefore equation (2.22) must be used rhen R is small or zero.

In our case, use is made of the FI'NGREEN subroutine supplied by MIT for the purpose of computing the Green's function and its derivatives.

Com-putation of both formulations have been speeded up through the use of poly-nomial expressions. See Newman ([8]).

The unknown source strengths o(A1, A2, A3) are determined based on

the normal velocity boundary condition on the body, see equation (2.15)

combined with equation (2.20):

In the above equations, the operator signifies the gradient in the di

rection of the normal to the surface of the body. For the solution of the

motion potentials j, the righthand side of the above equation (2.24) is

given by the direction cosines defined by equation (2.17). For the solútion of

the diffraction potential d the right-hand side is given by:

### fl.=J=_

### a

Solving the integral equation (2.24) results in the unknown source strengths. Subtitution of these in equation (2.18) and in equation (2.16) yields the added

mass and damping coefficients and the wave forces respectively. Finally the motions (j are determined from the solution of the following coupled

equa-tions of motion:

6

### Mkj±akj)iwbkj+ckj}(j=Xk

### k=1,...,6

(2.26)with:

Xk, wave förce on the body in the k-mode

Mk = inertia matrix of the body for inertia coupling in the k-mode for acceleration in the j-mode

ak = added mass matrix for the force on the body in the k-mode

due to acceleration of the body in the j-mode

damping matrix for the force on the body in the k-mode dùe to velocity of the body in the j-mode

Ck, = spring matrix for the force on the body in the k-mode due to motion of the body in the j-mode

The bk and Ckj matrices may contain contributions arising from mechan-ical damping devices and springs.

The mean second order wave drift forces on the bod.y are computed by the

near-field pressure integration method, see ref[2] or by the far-field method,

see ref[9].

in which': Ann Anm i 2

### =

GmnLSn 4ir On### 2.3.2

### Numerical Aspects

By discretising the mean wetted surface of a body into N panels on which

the source strengths are homogeneously distributçd the following discretized form of equation (2.24) is obtained:

### 1N.

m

### l,..,N n4m (2.27)

The norrnaF velocity requirement can only be met at one point for each

panel into which the mean wetted' surface of the hull is divided. It is custom-' ary to .chose the centroid of the panel. Such points are known' as 'collocation' points.

Equation (2.27) is transformed into a set ofsimultanéous'linear equations in N uiknown comp1x source strengths:

(2.29)

This set of equations can be solved, by direct solution techniques or by iterative solvers. The computational effort involved with iterative solvers is quadratic in the number f unknowns while for direct solvers the effort is a

cubic function of t'he number of unknowns.

It should be noted that :in the eva1uation of the. coefficients' Anm 'due care

has to be tken when the field point (center of a target panel) is close 'to the source point. In this case, due to the' i/.r behaviour'of the' influence function

A11 Av1

### A22..

AIN ANN 01,j N,j fli,j flN,j (2.28)Gmn seen in equatiOn (2.21),, irîaccuracies will occtir. In this case the

I/r contribution is subtracted from the influence function and is analytically integrated over the finite sized source panel under the assumption that the

source strength is homogeneously distributed. In our case use has been made of the algorithm provided by Fang[1O].

Similar cases of inaccurate evaluations of the influence functions based

directly upon equation (2.21) and equation (2.22) are found when a source

panel is very close to the sea-floor and when close to the free surface. The

so-lution in both cases is to remove the relevant terms from the above-mentioned

equations and to insert relevant equivalent analytical integrations over the

source panel.

The above modifications concern simple contributions to the influence function such as the 1/i', 1/ri or logarithmic contributions.

Where more complicated' contributions are concerned such as expressed

in the Principle Value Integral part of equation (2.21), a more accurate

eval-uation of the contribution to the influence function can be made for smaller values of r by the application of Gauss quadrature methods. These simply

imply that for smaller values of r the source panel is split into 4or móre parts and the total influence íunctionis the sum of the inflúences from the

parts of the total panel under the assumption that thesource strength per

unit surface is the same fOr all parts of the panel. In this way the Influence function can be determined mre accurately without increasing the number

of unknowns to be solved.

A numerical problem associated. with the sOlution of equation (2.28) is the occurrence of so-called 'irregular frequencies'. For cçrtain, discrete wave

frequencies the determinant of the coefficients Amn will become zero and correspondingly the results of computations are unrealistIc. Due to the

dis-cretisation of the body, 'leakage' of the effects of these irregular frequencies to

a band around these frequencies occurs. Irregular frequencies are associated

with eigenfrequencies of the internal (non-physical) flow of the :body.

Methods to reduce the effects of irregular frequencies are, among others,

to increase the numberofpanels on the body or to 'close' the body by means of discretisation of the free surface. inside the body. Increasing the number of panels does not remove the irregular frequency but tends to restrict the effects to a narrower band around it. See, for instance Huijsmans [3].

It should be mentioned that irregular frequencies only occurr for free

char-A11 ANI A22 A1N ANN

### )=

### AB

### BA

acteristics.Solving equation (2.28) using direct solution methods (LU-decomposition):

are time consuming if the number of unknowns (N complex quantities o) is large. The time taken to solve the equations by a direct method is propor-tional to N3. Iterative solvers are proporpropor-tional to N2. However, due to other program overheads, such as the evaluation .of the coefficients Amn for each

wave frequency, iterative solvers only become faster for panel numbers larger than about 50 on body. Normally, for instance a tanker shaped vessél can

be desribed adequately by 300-600 panels. In day-to-day practice we tend to use direct method based on LU-decornposition.

A usefull method to reduce computation times lies in exploiting any ge-ometrical symmetry a body may have. A ship shaped vessel will generally have port-starboard symmetry while,, fcr instance a simple barge or a

semi-submersible will have port-starboard and fore-aft symmetry. In the latter case we need only describe one quadrant of the hull shape. The three

re-maining quadrants can be obtained by simply copying taking into account

sign changes in the co-ordinates.

The reduction in computation time lies in the fact that, for instance for a vessel with port-starboard symmetry, the matrix of influence coefficients

Amn also shows symmetry. In such cases the matrix of influence coefficients

has the following structure:

(2.30)

The right-hand-side of equation (2.30) expresses the fact that the self-influence of the port half of the vessel (upper left A) is the same as the self-influence of the starboard hair of the vessel (lower right A), The crossiniluence of soUrces

on the port half of the vessel on panels on the starboard half of the vessel is contained in the lower left matrix B. The corresponding cross-influence of sources on the starboard half of the vessel on panels on the port half are contained in the upper left matrix B.

For vessels which have double symmetry i.e. port-starboard and fore-aft

(A1,1 ... .. ..

### A22.:..

AN1 .. ANN### "A

### B C.D

### BA DC

### CDAB.

### DCBA

_{I}

(2.31)
We leave it as an exercise for the reader to deduce the effect that exploitation of such symmetries can have on the computational effort involved in solving equation (2.28).

### 2.4

### Some results of computations

In this section some results of computations based on 3-dimensional

diffrac-tion calculadiffrac-tions will be compared with analytical results 'for a free floatig hemisphere and with model test results for' a free floating barge.

### 2.4.1

### The free floating hemisphere

For the 'hemisphere floating in infinitely deep water analytical solutions 'for the' added mass,. damping,, first order wave forces and motions have been

given.by Kudou [11].

For the computations the mean submerged Ñurface of the body is

approx-imated by 206 panels or facets. See Fig. (2.1).

The amplitude of the surge and heave wave forces are given in Fig. (2.2)

and Fig. (2.3) respectively along with the corresponding phase angles. Phase anglés are given relative to the undisturbed incoming wave passing the center of gravity of the body. A phase angle is positive if the force in question reaches

its positive maximum value before the wave at the' center of gravity reaches

its positive maximum. value. Phase, angles are given in degrees. Wave force amplitudes have 'been made non-dimensional shown in the figures.

The. results of computations' of added mass and damping for the surge

### /

### /

.1 \_{/}

\ ### /

Q_{,-}

_{'S}

Figure 2.1: Distribution of facets on the sphere and thedistribution of line elrnents on the waterline

The amplitudes and phase angles of the surge and heave motions are shöwn in Fig. (2.6) and Fig. (2.7). Motions have been made non-dimensional

by dividing by the wave amplitude.

Comparison of the results of computations and analytical results show that the niimerical method can giv accurate results for this simply shaped

body.

A comparison of this type can 'be characterised as being -a 'verification' of the numerical method ie. that the computational method is providing the correct numbers when viewing the process as a discretisation of an analytical problem.

A second, at least as important a question is whether the calculations are giving results which match with results based on physical reality, in

other words, are the assumptions which have been made in order to be able

to obtain answers correct and are we predicting physical reality? When discussing such aspects we are referring to the 'validation' of the computional

method.

TOTAL NUMBER OF'WATERUÑE TOTAL NUMBER OF LINE ELEMENTS: 15 FACETS ON HULL: 206

2 2 (I) 1.5 F3 (1) P9A Ç8 2.5 2.0 1.0 0.5 0 05 1.0 1.5 2.

Figure 2.2: First order wave exciting force in surge

_.

J-500

250

o

Figure 2.3: First order wave exciting force in heave

ANALVICAL o COMPUTED

### D.ê----.

### NN

05 1.0 i:s 2:0 25 ka 500 25& F3 o1.0 0.75 0.50 0.25 100 0.75 0.50 0.25 o 00 05 05 1.0 15 ka hO 1.5 ka 20 25

Figure 2.4: Added mass and damping coefficients for surge

### /

b33/pvw

2.0

Figure 2.5: Added mass and damping coefficient for heave H ANALYTICAL

COMPUTED

:

2. 2. Ji) 3a rt 0. 2. 2 500 250 o 500 o ANALYTICAL e COMPUTED -...-.-_

___..-.---4----I -a---.. o 05 2.0 2.5Figure 2.6: First order surge motion transfer function

0 05 .l0 1.5 20 25

ka

Figure 2.7 First, ordèr heave motion transfer function

250 CX3 o

A body plan with the panel distribution is shown in Fig. (2.8). 3 LEnGTH 150METRES DREADIH SOMETRAS DRAFT jOHAnnES X

### 2.4.2

### Barge

For arbitrarily shaped' vessels it is not possible' 'to obtain results analytically.

In thiÑ case we can only compare results of computations with results of model tests.

For this part we have chosen to compare results of measurements of first

order wave forces on the fixed vesseland first order motions of the freé floating

vessel with results of computations.

Model tests and caiculatiöns were carried out 'for a barge shaped vessel

with the following dimensions:

Figure 2.8: Body plan of lay barge including facet and control point distri-bution

Length : 150 m.

Breadth : 50 m.

Draft : l'o rn.

Displacement : 73750 m3.

'C'.of G. above keel, KG' : 10 in.

Radius of gyration roll : 20 m.

Radius of gyration pitch : 39 m.

The model tests were carried out in the Shllôw Water Basin on MARIN.

This facility measures 21(1 m x 16 rn x 1.0 m. The facility is equiped Witha carriage spanning the basin to which the barge model was connected.

The model tests were carried out at a scale of 1:50. This means that the waterdèpth in full scale values amounted to 50 m,

Model tests were carried out for regular waves with frequencies ranging

from 0.3 rad:/s to 1.1 rad/s and wave directiòns of 900, 135° and 180° which

correspond to beam waves, bow quartering waves and head waves

respec-tively.

The. model test set-up for measuring the wave frequency first order

rno-tions consisted of a soft spring mooring system which restrained the model from floating away under the influence of wave drift forces but allòwed it to carry out wave frequency motions freely. The test set-up is shown in Fig.

(2.9).
BALL-JO INI
-FORCE TRANSDUCER
-SPRING
X2, F2 _{X6} 143
90
180

Figure 2.9: Set-up for tests to determine motions and mean drift forces For model tests in which the. first order wave loads were measured, the model was fixed rigidly to a space frame incorporating 6 force transducers. The set-up is shown in Fig (2.10). The space frame was connected to the

carriage which also carried all data. collecting and analysis equipment.

The results of first order wave force measurements are given in the form of non-dimensional amplitude transfer functions in Fig. (2A1) through Fig.

(2.22).

Results of motion amplitude transfer functions are shown in Fig. (223) through Fig. (2,34)

N

BARGE RIGID 6-COMPONENI FORCE TRANSDUCER UNIT ATTACHED TO CARRIAGE

FORCE TRANSDUCER

.

BARGE

Figure 2JO: Set-up for wave force tests

Comparison of the results:of measurenients and computations show that in general the computations correlate:well with the rneasurements Largest differences tend to occur in the wave frequency motions at the natural roll frequency of a vessel. In such cases the computations predict a large ampli-tude due to the fact that the vessel is 'being excited at a natural frequency and

at the same time, the roll damping tendä to be small thus leading to large

motion amplification. in such cases the model motions will be restricted.

due to viscous damping effects which have not been acçounted for in the computations.

1.8 1,. 2 Q. m X 0 o 0.6 COMPUTED MEASURED 4,5 0.18 O * 12

Figure2.11: Amplitude (f transverse Figure 2.13: Amplitude of roll mo-wave force, mo-wave direction 90° ment, wave direction 90°

0.9 0.6 o kfl D) o-X 0.3 s COMPUTED ME AS UR E D 30 4.5

Figure 2.12: Amplitude of vertical Figure 2f 14: Amplitude of

longitudi-wave force, longitudi-wave direction 90°' nal wave force, wave direction 135°

COMPUTED MEASURED . s O 1.5 3 0 q. 15 v-d-3,0 COMPUTED MEASURED . 3 4.5 i 3,0

0,1 0.12 1.8 1.2 O 0

Figure 2.15: Amplitude of transverse Figure 2.17: Amplitude of roll

mo-wave force, mo-wave direction 135° ment, wave direction 135°

11.5 COMPUTED MEASURED e COMPUTED MEASURED 0.6 e COMPUTED MEASURED 3 1 5 W\Jt7 3 0 LS 3.0 11.5

Figure 2.16: Amplitude of vertical Figure 2.18: Amplitude of pitch

mo-wave force, mo-wave direction Ï35° ment, wave direction 135°

1.8 1,2 s-no (1 X COMPUTED MEASURED 0.6 1.5 30 11.5 g 6 LA O. m X O

0.18

0.12

Figure 2.19: Amplitude of yaw mo- Figure 2.21: Amplitude of vertical

ment, wave direction 135° wave force,, wave, direction 180°

0.9 0.6 o o. O 15 3,0 wVt 4,5 0.3 . COMPUTED MEASURED O 30 II,5 6 o,

### a

X O 1.8 1.2 O### '5wVt

30 4.Figure 2.20: Amplitude öf 'longitudi- Figure 2.22: Amplitude of pitch

mo-wave force, mo-wave directiOn 180° ment, wave direction 180°

COMPUTED MEASURED -o 3 o o o "S-.. -e COMP UTED MEASURED s I 0.06 u . I, COMPUTED MEASURED 1 4,5 1.5 3 0

1.8 1,2 o LA ('J X 1.8 1.2 o LA X 00 0I O o s COMPUTED MEASURED 0.6 COMPUTED MEASURED

### 3y

COMPUTED MEASURED 0.6 COMPUTED MEASURED s o 0.6Figure 2.23: Amplitude' of sway Figure 2.25: Amplitude of roll

mo-tion, wave direction 90° tion, wave directión 900

1 5

w\/t7 3.0 11.5 1 5 3 0 4,5

Figure 2.24: Amplitude Of heaye Figure 2.26': Amplitude of surge

mo-tion, wave direction 90° tion, wave direction 135°

l,5 30 4,5 15 3 0 4,5 g 6 o LA 1.8 1.2 o LA X

Figure 2.27: Amplitude of sway Figure 2.29: Amplitude of roll mo-tion, wave direction 135° tion, wave direction 1350

o

30 45

Figure 2.28: Amplitude of heave Figure 2.30: Amplitude of pitch

mo-tion,, wave direction 135° tion,. wave direction 135°

1.8 1.2 15

### ,-30

(Ov Lig 4., COMPUTED L MEASURED### JL

COMPUTED MEASURED 0.6 I MEASURED COMPUTED 1.8 g COMPUTED MEASURED s 1.2 6 s o o### 1 5- 3.0

.5### ,-

_{3 0}

_{4,5}W LFg 0.6 e1 1.8 1.2 o Lfl m K o u, X

1.8 1.2 o X O 1.8 1.2 o ci X O o

Figure 2.32: Amplitude of surge Figure 2.34: Amplitude of pitch

mo-tion, wave direction 180° tion, wave direction 180°

COMPUTED MEASURED s 0.3 COMPUTED MEASURED e 0.6 COMPUTED MEASURED s o COMPUTED MEASURED 0.6

### \

3 0 4,5 15 3.0 4.5Figure 2.31: Amplitude of yaw Figure 23: Amplitude òf heave mo-tion, wave directiOn 135° tion, ware direction 180°

1.5 3 0 4,5 1:5 3 0 4,5 0. 0.6 o 0 1.8 1.2 o u, X

van Oortmerssen, G. (1!976): "The Motions of a Moored Ship in Waves", Ph.D. Thesis, Deift Univesity of Technology, 1976

Pinkster, J:.A. (1980) :"Low-freq.uency Second Order Wave Exciting Forces on Floating Structures", Publication No. 650, Netherlands Ship

Model Basin.

H:uijsmans, RH.M.. (1996):"Motions and drift forces on moored vessels

in current", Thesis TU-Deift, 1996 (to appear)

Tuck, E. (1970): "Ship Motions in Shallow Water", Journal of Ship Research, Vol. 14, No. 4, 1970, pp. 317-328.

Lamb., H. (1932): "Hydrodynamics", Sixth Edition (1932)

Wehausen, J.V.. and Laitone, EN. (1960):"Handbuch der Physik", Vol.

9, Springer Verlag, Berlin 1960.

John, F. (1949).: "On the Motion of Floating Bodies", Comm. on pure and applied mathematics, Part I: 2, 1949, pp. 13-57;. Part II: 3, 1950,

pp. 45-100.

Newman, J.N. (1985): "Algorithms for the Free-surface Green

Func-tion", Journal of Engineering Mathematics, 19,57-67.

Maruo, H. (1960): "The Drift of .a Body floating in Waves" Journal of

Ship Research, Vol., 4, No. 3.,. December 1960.

[l'o] Fang, ZogSherig "A new method 'for calculating the fundamental

po-tential functions induced by' a source/dipole polygon", Aplied

Mathe-niatics and 'Mechanics, '6(7), July 1985.

[11] Kudou, K.(1977): "The drifting force acting on a three- dimensional

### Second order

### wave

### drift forces

### 3 i

### Introduction

The impórtance of the mean and 'low frequency wave drjft forces from the

point of view of motion behaviour and mooring loads is generally recognized nowadays.

initiated by work by Hsu and Blenkarn [1] and Remery and Hermans

[2] who indicated that large amplitude low, frequency horizontal motions of moored vessels could be induced by slowly varying wave drift forces in

ir-regular waves and who stressed the importance of knowledge of the mean

second order forces in regulr waves in this problem, a considerable number of papers have been published which have increased insight in phenomena

nvolved. See for instance [3].,[4],[5] and [61.

Initially the solution to the problem of predicting môtions and mOoring forces was mainly sought in better quantitative data on the low frequency

wave drift forces. Accurate knowledge of these forces,, it was felt, would allow accurate estimates to he madé of low frequency horizontal motions and

moor-ing forces. Due to the resonant type motion response of the noored vessels to the wave drift forces, it, is nowadays' known that 'for accurate prediction

öf low frequency motions in irregular':seas attention must also be paid' to the hydrodynamic reaction forces. See for instance Wichers [7],i[8] and Prins. [9].

This chapter deals., almost entirely, with the wave drift exciting forces and moments On basis of experimental and theoretical results it is generally agreed 'that, for engineering purposes, the wave drift forces are second order

wave forces. This means that the forces' are quadratic functions of the height of the incident waves.

Based on the second order 'assumption it can be shown that, in order to

determine the tota1 second ordèr wave exciting forces in irregular waves, the mean second order force in reguilar waves' and 'the low frequency second order wave force in regular wave groups cOnsisting of two regular waves have to be known. See for instance Daizell [10].

In the following section a review is given of theories. developedi in the

course of time to détermine these forces.

### 3.2

### Historical development concerning

### sec-ond order wave forces

It is generally acknowledged that the existence of wave drift forces was first reported by Suyehiro tl 1;]. While experimenting with a model rolling in beam

seas he foirnd that the waves exerted a steady side force which he attributed

to the reflexion of the' incoming Waves by the model.

Watanabe [12] put forward a seçond order theory for the force based'on

the first order Froude-Kriloff force and the first order vessel mOtions. His

calculations showèd partial agreement with Suyehi'ro's 'experimental results;

Haverock [13] deduced a similar expression to that of Watanabe for 'the:'

mean longitudinal drift force on a vessel heaving and pitching in regular

head waves. Use was made of the Froude-Kriloff part of the first order wave

force and' pitching moment and the first order pitch and heave motions. The expression derived is simple and has been used many times to estimate the resistance increase due to waves of a ship travelling into head seas, see for instance Salvesen [14'].

These theories, however, give only a general indication of the wave drift

force since, for instance, diffraction effects' 'and' hydrodynamic effects following

from the body motions are nly partly taken into account.

M'aruo [15] gave a general theory for the determination of the mean second

order wave drift forces on 'floating bodies in regular waves. This t'heory is

exact within potential theory. The mean second ordér wave drift force is

derived using the equation of conservation of energy and momentum of the

from the body.

Similar theories based on energy and momentum considerations were

de-veloped, for instance, by Newman [16j and Faltinsen and Michelsen [18]. A

recent development by Mobï [1 9.] concerns the evaluation of Maruo's

expres-sion for the wave drift force at the mean position of the body instead of at a

control surface far from the body.

Gerritsma and Beukelman [20] determine. the increase in resistance due to waves of vessels travelling in head seas based on the assumption that the mean resistance increase (longitudinal drift force) can be. found by equating the damping energy radiated by the heaiing and pitching vessel to the work

done by the incoming waves. This theory has shown good results when

compared with the results of model tests. See for instance Dalzell [10].

.

In [21] and [221 Pinkster and Van Oortrnerssen introduced a method based

on direct integration of ali pressure contributions to the second order wave

forces on the wetted hull of the body. This method is also exact within

potential theory.

Faltinsen and Loken [23], [24] applied a similar method to the two

dimen-sional case of cylinders in beam waves and succeeded in giving a complete solution which includes the. mean Wave drifting force in regular waves and the low frequency oscillating wave drift force in regular wave groups.

In [25] and [26] Pin'icster and Hooft applied the method of direct

integra-tion to determine, for three-dimensional cases, the mean and low frequency

part of the wave drift forces using an approximation for the force contribution

due to the second order non-linear potential.

In [27] Faltinsen and Loken give a review of a number of the methods in use to calculate mean wave drift forces. In [19] Mohn discusses results obtained using Maruo's theory, Mohn's modification to Maruo's theory and the method of direct integration.

In general it has been found that for voluminous bodies, methods based

on potential theory give good results when compared with model test results. Discrepancies occur at wave frequencies near, for instance, roll resonance. In such cases viscous effects not accounted for by theory become important..

When regarding slender constructions such as some semi-submersibles, viscous effects may become important at all' frequencies. Wahab [28] and Pjjfers and Brink [29] and recently Dey [38] 'have developed approximative methods of calculation fór drift. forces which take these into account. Husé

drift forces on semisubmersibles. Dey [38 has carried out extensive sets of modl tests in ordèr to determine empirical coefficients necessary for deter-mining mean viscous drift forces on semi-submersible structures.

### 3.3

### Theory

In this part, expressions for the second order wave force and moment will be deduced' based on the direct integration, method. This method is straight forward and gives insight in the mechanism by which; waves and the body

interact to produce the force. Before thesection' dealing with thesecond order

wave force,, the boundary .cönditions for the various potentials are treated', special' attention being given, to the boundary conditions at the bödy.

The theory is developed in accordance with perturbation theory meth-ods. This means that ali quantities such as wave height,. motions, potentials,

pressures etc., are assumed to vary only very slightly relative to some initial,

static value and may all be written in the föllowing form:

### X = x(°) + x1

+### 2x2

(3.1)Where the affix (0) denotes the static'. value, (1) ndi'cates the flrs.t or-der oscillatory variation and (2) the second ordr variation. is some small

number with < 1.

First order in this case means, linear' with the wave height and second

order means that the quantity depends on the square of the wave height.

In the follówing, quantities are of second order if proceeded by 2 If, as in

many cases, the E2 is discarded, this' is due to 'the fact that 'the expressiòn contains 'only second order quantities. In suçh instances the second order quantities are recognized by the affix (2). or by the fact that a component is

the product of first order quantities with affix (1). Fôr instance, the pressure

### -

J{1)' 2

is recognized as a second' order quantity..

### 3.3.1

### 'Second' order wave 'forces on a body floating in

### wave s

For the derivation of. the second ord'er wave forces and moments it i's first

Figure 3.1: System of co-ordinate axis

The second system of co-ördinate axes is a fixed O - X1 X2 - X3 system with axes parallel to the G - x1 - x2 - x3 system of axes with the body in

the mean position and origin O in the' mean free' surface. The third system of co-ordinate axes is a G - Xi - X2 - X3 system of axes with origin in the centre of gravity G of the body and axes which are at ali times' parallel to the axes of the fixed O - X1 - X2 - X3 system.

assumed that the body is only allowed to move in' response to the first order

hydrodynamic forces with wave frequencies.. Other frequenciés or higher

orders of motion are not permitted which means that expressions obtained

for the second' order wave forces contain only the wave exciting' forces.

### 3.3.2

### Co-ordinate systems

Use is made fthree systems of co-ordinate axes (see Fig. (3.1);).

The first is a right-handed system of G

### - xi -

- x3 body axes withas origin the centre of gravity G and. with positive G' x3 axis vertically upwards in the mean position of the oscillating vessel.: The surface of the hull

is uniquely defined relative to this system of axes A point on the surface has as position the vector x. The orientation of a surface element in this system of axes is de- fined by the outward pointing normal vector .

If the body i carrying out smlil arn1itude motions under influence of' the first order oscillatory wave forces the resultant displacement Vector X cf a

point on 'the body, relativé to the fixed system of axes is a.small first ordér quantity also:

f)(1)

with

(1)

### »1)

+ R1

where the linearized' rotation transförmation R' is defined as:

with where: (1) (1) O) .(1) 5 54 0 (3.4)

Similarly, the velocity of a point on the body r lative,to the fixed system

of axes is also a first order quantity and follows from:

The orientation of surface ele ents of the body relative to the body axes

are dCnoted by the outward pointing normal vector . When the body is

carrying out small first order motions, the orientation of a surface element

relative to the fixed O- X1 - X2 - X3 or the .G - X1 - X2 - X3 'system f

axes becomes: = (1) .1 = cX (1)

### +

.(,i) Xfi O .(i) 55 (3.5) (3.6) (3.7)### -6

5 O (1)### s4 ]

(1) 54 0 (3.2) (3,3)Ñ = il +

### f(1)

(3.8)with

Ñ(1) _{= R1} _{(3.9)}

where R'1 is according to equation (3.4).

### 3.3.4

### Fluid motions

The fluid domàin is bounded by the free surface, the surface of the body

and the sea floor. Assuming that the fluid is inviscid, irrotational5 'homoge-neous and incompressible, the fluid motion may be described by means of the velocity potential :

)(l)

+ c24(2) _{+ ....} _{(3.10)}

The potentials are defined relative to the fixed system of O

### - Xi - X2 - X3

axes.

The first order potential (1) _{.consists of the sum .of potentials associated}

with the undisturbed incoming waves, the diffracted waves and wave due to

the first order body motions respectively,:

### ,I)

_{+}

### ,i)

_{+}

_{(3d 1)}

The first and second order potential must 'both satisfy the' equation of continuity within the fluid: domain, and the boundary condition at the fixed horizontal surface of the sea-floor. The boundary condition at the mean free surface becomes:

### gI

_{+}= 0 (3.12)

g -i- =

### 2(1).1)

### +

### 11):(x3

### +

(3.13)In order to derive equations (3.12) and (3.13) use is made of 'Bernoulli's equation and the assumption that fluidi elements in the free surface remain there at all times. By means, of 'aTayiór expansion 'the boundary conditions at the actual moving free surface are transformed into boundary conditions

### 3.3.5

### Body boundary conditions

In generali the boundary condition on thehulistates that the relative. velocity between the fluidi and the hull in the direction of the normal to the hull be

zero This means that no fluid passes through the hull. _{This boundary}

condition has to be satisfied at the instantaneous ]osition' of. the hull surface

and is as follows:

= . i (3.14)

Taking into accöunt equations (3.5) (3.8) and (3.10) and grouping powers

of results in t:he first and second order body boundary conditions. The
boundary conditions for the first order potential (1) _{on the body which}

states that,. to first order, there is no relative motion between the fluid and the body surface in the direction of the outward pointing normal vector is

as follows:

(3.15)

The boundary condition for the second order potentil 2 _{states that, to}
second order, the relative velocity in the direction of the ,outward pointing
normal N be zero or:

((1) (1)) _{(3.16)}

Equations (3.15) and (3.16) have to be satisfied at the instantaneous.

position of the surface: of the bod.y. Assuming that the motions are small, and applying a Taylor expansión, similar conditions may be posed on the

potentials at the mean position of the surface. The first order boundary condition becomes

= (1) . _{(3.17)}

The second order boundary condition becomes:

### (I(i)

### .) .

### jt')

_{. il +}(j(l') (1'))

_{(3.18)}

Where the additiOnal term. in equation (3.18) arises from the secoñd order correction to equation (3.15), when applying the Taylor expansion to the

### ve-((1).

_{+}1)

### +

(1))### =

It is customary to decompose this equation into two parts:

Diffraction problem:

Motion problem:

The methods in use to solve for the unknown potentials and 1) are

discussed at length in the literature. In the following it is assumed that the first order solution is known.

Substitution of the first order potential ) of equation (3.11) in the

non-homogeneous second order free surface boundary condition of equation (3.13). shows that the second order potential has, in general, the following

components:

(2)

_{=}

### +

### +

+### +

+### +

### +

+ + (3.22)Where the first nine components on the right-hand side are potentials

which are particular solutions to the following type of boundary condition,

e.g.:

### +

= +_{3X3}

_{gWtX3)}(3.23)

The last potential 2) is a potential which satisfies the homogeneous boundary condition:

### gI

+ = o (3.24)locity

### '(1)

Inequatión (3.17) and (3.18) the potentiais and their derivatives have to be taken at the mean .position. of the body.Substitution of equation (3.11) in boundary condition (3.17) gives the

following:

(3.19)

(3.20)

2)

is therefore an "ordinary" potential which satisfies the linearized free surface condition.

We will implify equation (3.21) by putting:

(2)

### =

### +

2) (3.25)In which ) represents the sum of the first nine components on the

right-hand side of equation (3.22).

may be regarded as the second ordèr equivalent of the first ordér

undisturbed incoming wave potential ).

Substitution of equation (3.25) in the second order boundary condition

(3.18) gives:

+

### _((1)

### )

(1)### il+

(il(i) (1)) -### Ñ1 (3.26)

or:

g

### _j(2) g

### ((1)

.

### +

### ()

jO) (3.27)If the right-hand sidè of this equation is known, may be solved using numerical methods.

In generaI, however, presents a problem due to the complexity of the

surface boundary conditiön of equation (3.13).

Faltinsen and Loken [24'] have found exact solutions for this problem for two-dimensional cases apjlicable to vessels in beam seas. Newman [17]. has

developed numerical methods to solve for the second order potential for the

three-dimensional case.

(2)

Pinkster in [26] and [39] has suggested an approximation for _{d} which is

applicable to three-dimensional problems. In a later section it will be shown
that the second order potential (2) _{is only responsible for part of the second}

order wave fOrces.

### 3,3.6

_{Boundary conditions at infinity}

For the potentials and a radiation conditiòn, which states that