Slowly-Varying Second-Order Wave Forces: Theory and Experiment

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Slowly -Varying Second

-

Order

Wave Forces: Theory and Experiment

by

R G Standing, N M C Dacunha and

R B Matten

Supported by the Department of Energy

Offshore Energy Technology Board

UT-R-8211

NMI R138

October 1981

National Maritime Institute

Feitham

Middlesex TW14 OLQ

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

SLOWLY-VARYING SECOND-ORDER WAVE FORCES: THEORY AND

EXPERIMENT

by

R G Standing, N M C Dacunha and R B Matten

calculates first-order wave forces and response motions of ships and offshore structures. NMIWAVE 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.

The companion report discusses various different procedures for estimating the mean force. The present report describes how one such procedure, based on the NMIWAVE program, has been extended to predict slowly-varying components of the second-order

force. Both horizontal drift and vertical components of the force are calculated. This report describes the basic theory, how slowly-varying forces arise and cause

a low-frequency response, and how forces in regular wave groups are related to forces in discrete and continuous irregular spectra.

Two contributions to this force are identified: one associated with wave diffraction and the other with gradients of second-order pressure. The gradient term includes a contribution from the second-order (set-down) wave. The relative sizes of the

diffraction, gradient and set-down terms are expressed in terms of ratios of three simple parameters. It is suggested that these three parameters may indicate the ranges of validity of approximate procedures due to Bowers and Newman, as well as of the NMIWAVE program.

Numerical comparisons confirmed that the Newman approximation is valid at low frequencies of second-order forcing. The more complete NMIWAVE solution should be used at higher frequencies.

The new computer programs were validated in two ways. Their predictions were compared both with published data and with results of experiments, undertaken specially for that purpose at NMI. There was sufficient agreement in general to establish confidence in the new NMI programs. Some discrepancies found in comparisor with published data are attributed to differences between numerical models. The experiments proved difficult to perform and to analyse. There was particular uncertainty about the amount of damping present.

1.2 Applications, guidelines and limitations 2

Basic Principles 4

2.1 Causes of low-frequency forcing 4

2.2 Low-frequency response 7

Theory

3.1 Second-order low-frequency forces 3.2 Second-order wave term

3.3 Symmetry and spectral relationships

3.4 Evaluation in terms of complex quantities 3.5 Response motions

Alternative Prediction Methods

41 The Bowers method

4.2 Newman/Pinkster approximation 4.3 Order-of-magnitude comparisons

Theoretical Calculations on a Moored Drill-Ship 5.1 The Bowers method

5.2 The complete calculation of the drift force spectrum 5.3 The Newman/Pinkster approximation

5.4 Results

5.5 Discussion of results 5.6 Conclusions

Comparisons with Published Data and Analytic Results 6.1 Calculations on a barge model

6.2 Horizontal cylinder of rectangular section 6.3 Rigid wall

6.4 Conclusions

Comparisons with Experimental Results

7.1 General experimental set-up and data analysis 7.2 Pairs of beating waves

7.2.1 Experimental and theoretical analysis 7.2.2 Discussion of results

7.3 Irregular waves

7.3.1 Experimental analysis

7.4 Conclusions of the experimental study Conclusions Acknowledgements Notation Reference s 10 11 12 14 17 lE 19 20 22 23 29 29 31 31 32 33 35 36 36 37 39 30 40 40 41 42 43 45 4 E 48 50 51 52 54

Appendix 3: slowly-varying drift forces on a rigid wall 64 Appendix 4: Estimates of the stiffness, damping and added mass 67

of the drill-ship.

Appendix 5: _{The magnitude of the Bowers force in deep water} 69

Tables 1 - 10 73

Tables Al-A3 85

1. Introduction

A companion report, reference 1, describes theoretical methods for calculating mean wave drift forces. The present report discusses the significance of slowly-varying components of the second-order wave force, methods for calculating these components and the associated low-frequency response motions of the structure. Such forces can cause drifting of disabled vessels, severe loads in mooring lines, or low-frequency heave and pitch motions of large semisubmersibles and similar structures.

Motions of ships and structures in waves are usually calculated using linear seakeeping theory. This theory supposes that the wave height H is small, and retains only first-order terms linear in H. A regular sinusoidal wave history causes a force and

response proportional to H, and at the same frequency. A superposition of waves of different frequencies (an irregular sea) gives rise to a similar superposition of forces and response motions, covering the same range of frequencies.

Any kind of non-linearity is likely to cause a response at a different frequency. One specific type of non-linearity is of interest here: the mean and low-frequency

components of the second-order wave force. In regular sinusoidal waves the second-order force has a mean component which acts roughly in a downwave direction. A ship or floating structure has to be moored or use thrusters in order to counteract this

so-called drift force.

In irregular sea waves the drift force varies continually. These low-frequency variations are generally small in magnitude, but can cause a large response motion

if the vessel's natural frequency happens to be excited, and the damping is low. Tensions in mooring lines may then become many times larger than the initial excitation force.

A conventionally-moored ship may thus experience low-frequency surge motions. A vessel at a single-point mooring may experience more complex 'fishtai].ing' or

'galloping' motions (2) . Structures with small waterplane areas, such as large semisubmersibles, may experience low-frequency heave, roll or pitch motions (3)

In order to estimate these motions for design purposes it is necessary to know both the low-frequency exciting force and the amount of damping present. Both affect the magnitude of any resonant response, and both quantities are poorly understood.

This report addresses the first problem: that of predicting the excitation force. In order to interpret the model test data, however, it proved necessary to measure the damping. Difficulties in measuring this damping will be discussed in a later section there are further difficulties in extrapolating these results to full scale. Further research is needed on this topic.

The first sections of this report describe theoretical methods for predicting the low-frequency forcing and response, and the origins of this force. Also discussed are the relationships between different prediction methods currently in use, and their general ranges of validity.

1.1 Summary of New Program Capabilities

Several new programs have been added to the NMIWAVE suite during the present investigation. These calculate second-order mean and slowly-varying wave forces and motion responses for either fixed, free-floating or moored diffracting

structures. The following components are calculated:

mean and slowly-varying forces in both regular waves and irregular wave spectra (II, §3.3),

mean forces in the horizontal plane by the 'far-field' method (I, §4.1),

mean forces (all 6 components) by the 'near-field' method (I, §4.2)

slowly-varying forces by the 'near-field' method, also by the earlier approximate Newman/Pinkster and Bowers formulae (II, §3, 4)

(y) mean offset and low-frequency response to the above forces (II, §2.2).

References above are to chapters in either this report (II) or the accompanying reference i (I).

1.2 Applications, Guidelines and Limitations

The main applications of these new programs will be in calculating:

mean forces acting on disabled vessels,

motion response of structures with very low natural frequencies (e.g. moored ships, tethered platforms, heave and pitch motions of semisubmersibles).

Approximate guidelines are given for:

neglect of drag forces (I, 3)

neglect of forces due to second-order waves (e.g. set-down) (II, §4.3)

use of the Newman/Pinkster and Bowers approximate formulae (II, §4.3).

The main limitations of these programs are as follows:

no current forces are included,

linear wave theory is assumed. Second-order wave effects are approximated,

2. _{Basic Principles}

2.1 Causes of Low-frequency Forcing

Low-frequency forces arise in many different ways, as described in reference 1. One important contribution, for a ship or semisubmersible at least (see reference 4) comes from pressures acting on the part-wetted area of hull surface around the

vessel's waterline. _{In conventional linear theory pressures are integrated up to} mean water level throughout the wave cycle, assuming a constant wetted _surface. Sinusoidally-varying pressures then give rise to a sinusoidal force. Additional forces acting between mean water level and the instantaneous free surface see figure 1) make a non-sinusoidal second-order contribution. _{It is these additional} forces that are of interest here. _{If the part-wetted members are slender tubular} components of a space-frame structure, then the additional forces are associated with drag. _{Conventional inertial forces (on a vertical member) are out of phase} with the surface elevation, and make no contribution (see reference 1) . Very large

structures such as ships, however, diffract the waves. _{The phases of both pressures} and surface elevation vary, and again combine to give a second-order force.

This report is concerned exclusively with forces which are associated with wave

diffraction, and which can be calculated under potential flow assumptions. Contributi from drag are not considered. The results can therefore be applied only to large-diameter structural members, such as those of ships, semisubmersibles, tethered buoyan

platforms, barges and caissons. Reference i shows that a rough criterion for use of

these results (at least for mean forces) is that the ratio HX2/D3 <60, where D is a

typical member diameter, H is wave height and X wavelength.

Several different components of the second order force may be identified.

Reference 1, following Pinkster (4) , shows that these include part-wetted surface effects described above, effects of first-order vessel motions on the instantaneous force, second-order pressures from the quadratic term in Bernoulli's equation, and forces due to second-order waves, such as set-down. All these terms involve products of first-order quantities. It will now be shown that forces of this type have both mean and low-frequency components. If the first-order incident wave is represented

as the sum of N superimposed sine waves, then the surface elevation may be written!

N

r,(x,y,t) = E a cos(k X COS _m + kmy sin _{m -} + £ ) (2.1)

m m m m

Any two first-order quantities X(t) , Y(t) may be written similarly: N X(t) = Z X a

_mm

cos _(emt - e m m m= 1 N

Y(t) =Z

Ya cos (et-c -y)

nfl n n n

n= i

where Pm' \- are phases of Xm, _n relative to the corresponding components of surface elevation.

Their product N N

XY = Z Z _{aman XmYn COS (emt -} E

-

ces (cnt - Cn -m=i n=l

N N

= Z Z a a_m X Y

_mn

{cos r(e

- e)t -

E

+ E

-

+

ml n=l

+ cos [(em + e )t -_n s_m - s i

-

1}

n m n

The second term represents a force at frequency (em + en) , which is too high to be of interest in the present context. The first term, however, represents a

low-frequency forcing with component frequencies (em - en) . This force can be re-expressed in the form:

N N

F(t) = Z Z a a

_{mn mn}

T cos E(e

_{L m}

- e )t - Em_n

+ C

-

6m rn=l n=l

where T,

are amplitude and phase functions, defined in chapter 3.

The simplest wave form which causes a low-frequency forcing consists of two superimposed sine waves (i.e. N 2) . The surface elevation is then a regular

sequence of wave groups, as shown in figure 2. The wave envelope has a characteristic period of 2iî

/

a1 - e2.

The low-frequency (and mean) force then has three

2

a1 T11 + a T22

+ a1 a2 {(T12 cos 612 + T21 cos 621) cos 612

+ (T12

Sfl

612 - T21 sin 621) sin °12

where 612 ₌ (01

--

El +

The first two components represent the mean force, which is equal to the sum of the mean forces acting in separate sequences of regular waves of amplitudes a1, a2 and frequencies 01, 02 respectively. The phases si,

2 of these individual wave

components have no effect on the mean force.

The third component represents a slowly-varying force whose magnitude is proportional to the product a1 a2, but is independent of the phases

2 The terms in T12

and T21 have been rearranged into components in-phase and out-of-phase with the wave envelope. The force varies with the same period as the wave envelope, but with a constant phase difference cz12, shown in figure 2, where

T12 sin 612 - T21

Sifl

621 tan a12

T12 COS 612 + T21 COS ₆₂₁

The slowly-varying force is therefore associated with the wave envelope. In an irregular sea, with many frequency components, the wave envelope depends on the group properties of the waves. Each pair of frequencies in the wave spectrum will give rise to a low-frequency force at their difference frequency. It is clear,

therefore, that the band widths of the wave and low-frequency force spectra must be similar. A narrow-band wave spectrum will give rise to a narrow-band force spectrum concentrated around zero frequency.

The amplitude and phase functions T12 and 612 depend only on the geometry of the system, and on the two frequencies 01,

0.

They are independent of wave amplitude and other frequencies present. This has an important consequence for the prediction of these forces. It means that force functions obtained by calculation or experiment

over a range of conditions with just pairs of beating waves can then be used to predict forces in any irregular sea state. Prediction methods described in this report are based on this premise.

2.2 Low-frequency Response

These second-order forces are often quite small compared with the linear components at wave frequencies. (In fact there is serious reason to doubt the validity of this kind of expansion procedure if the two terms are of comparable size) . The magnitude of the force is, however, often less important in the design and operation of a

structure than the system's response to that forcing. A freely-floating and disabled ship will tend to drift down-wave, possibly into danger(5). It may also turn, often beam-on to waves, so that it suffers severe roll motions.

If the ship is moored, then the ship and mooring act as a mass/spring system with a very long natural period. Low-frequency drift forces may excite this resonant motion, which can become severe if the damping is small. It is assumed here that

these motions can be described in terms of the linear equation

Ar + bn + Cn = F(t)

where F(t) is the low-frequency excitation force. The relationships between forcing and response are the same as for a conventional linear system. Thus if the forcing is periodic at frequency o it may be represented in complex terms as

F(t) = F exp (-iot)

and the response, also assumed periodic, is

n = n exp (-jot)

so that

n = F /{Ar

(2

_02)

-

2i }

o o - o o

-where o is the undamped natural frequency

= C/A

and the mooring force is

Cn =-F /2i

o o

The mooring force amplitude is l/23 times the original exciting force F. If the damping is low the mooring force can become many timesthe exciting force. For

example if = 0.05 (a reasonable value for a ship in surge, as found in model tests described later in this report) then the mooring force is 10 times the exciting force. Although the low-frequency forces are small, therefore, they can cause severe loads in mooring lines.

It is important to recognise here, however, that this kind of approach, in which the worst design condition is assumed to be regular resonant response, can lead to unduly pessimistic predictions of mooring loads. In an irregular sea the band of frequencie over which there is any significant dynamic amplification of mooring loads is very narrow: indeed the smaller the damping, and therefore the larger the amplification factor at resonance, the smaller is the width of the response peak. It is shown in appendix i that the root mean square mooring force therefore varies with instead of When is small the rms response may be less than the simple regular-excitation estimate would suggest. In such cases it is essential to use a full spectral procedure if estimates are not to beunduly conservative.

Figure 3 illustrates in spectral terms how a small second-order force can cause a

large low-frequency response. The wave spectrum (top left of diagram) causes a large force on the structure at wave frequencies and a smaller force at low frequencies

(top right) . The frequency response function (or response amplitude operator:

bottom left) is that of a lightly-damped resonant system. It allows a large response at the system's natural frequency, but filters out components at higher wave

Figure 4 shows motion response spectra for an actual ship, berthed at Acajutla, El Salvador. These results are reproduced from reference 6. The heave, roll and pitch spectra are predominantly at wave frequencies, while surge, sway and yaw

are at low frequencies, peaking around 0.01Hz. A note of caution is inserted here about regarding these low-frequency motions as entirely the result of second-order effects. This particular port is very exposed to Pacific swell. Bowers (7) found that the motions of the vessel were rather too large to be attributed entirely to second-order effects. In experiments designed to reproduce sea conditions at Acajutla he found it necessary to add a very long period wave. This

wave could not be detected by a conventional waverider buoy, but was measured by means of a bottom-mounted pressure sensor inside the port. In such conditions, therefore, low-frequency motions will be caused by a mixture of second-order effects and long-period swell.

Low-frequency motions of moored vessels may be excited in several other ways, a detailed discussion of which may be found in reference 8. It is shown, for example, that non-linearities in the mooring force, particularly if a combined mooring! fendering system is used, can give rise to subharmonics of the fundamental wave

frequencies, as well as so-called 'combination tones'. The latter represent response motions at various sum and difference frequencies. Such phenomena are distinguished

from second-order wave forces, because they arise through non-linearities in the response equations rather than in the wave forcing. They are therefore considered to be outside the scope of the present report.

3. Theory

Computer programs have been developed for use in conjunction with the NIWAVE

suite, in order to predict second-order slowly-varying wave forces and the associated response. The underlying theoretical assumptions are described in references

land 9. Briefly they are that:

the wave height is small compared with wavelength, water depth and body dimensions, so that the fluid motions may be described in terms of linear

(Airy) wave theory;

the response motions are also small, and of the same order of magnitude, so that the response equations may also be linearised,

the flow is inviscid, incompressible and irrotational, and so may be described in terms of a velocity potential .

The velocity potential is expressed in the form of a Stokes expansion:

= Ji) +

2(2)

+ +

as are the response motions of the structure:

= +

2(2)

+

3(3) +

and the wave forces:

= + +

2(2)

_{+ +}

where s is a small parameter related to wave steepness. The term F(o) represents the hydrostatic buoyancy force, which is in equilibrium with weight and tether loads.

The basic NMIWAVE program is described in reference 9. It calculates the first-order wave loads and responses of the structure (terms linear in s) by solving

hydrodynainic equations for linear wave diffraction and radiation, and linearised dynamic response equations for the structure. These equations represent coupled motions in surge, sway, heave, roll, pitch and yaw, and can include linearised tether loads.

The other program calculates directly the second-order forces acting on the structure. These two alternative techniques are known as the 'far-field' and 'near-field'

methods respectively. The far-field method requires less computer time and storage, but can only calculate mean components of the horizontal force and turning moment. The near-field method is more demanding of computer time and storage, but more versatile. It can calculate the mean vertical force, heel and trim moments. More important in the present context, it can be used to calculate the low-frequency forces.

3.1 Second-order low-frequency forces

Reference 1, following Pinkster (4) , shows that the second-order force can be expressed as the sum of six terms, represented pictorially in figure 5. These are as follows:

I

_-PgJ

r n'di o

representing changes in wetted surface area of the structure,

II

pJV2n

_{o -o}dS

arising from the quadratic term in Bernoulli's equation for pressure,

III

_J/t

n dS

representing the effects of first-order motions on the point of application of the pressure,

IV RF

representing the effects of first-order rotations on the direction of the force,

V a complex term representing second-order motions of the structure's centre of buoyancy and waterplane area, and which may be described in terms of products of first-order motions. Second-order motions, which are the result of second-order forcing, are omitted. They do not represent a forcing term in the second-order

equation of motion, and will appear eventually on the other side of that equation, VI J (2) - p

/tndS

S o

to be discussed below, and representing the effects of second-order waves. These include set-down, and reflections of the set-down wave from solid boundaries (see reference 10)

In the above expressions L is the mean waterline and S the mean underwater surface

o o

of the structure; n

o

is the unit normal to S and n' the unit normal to L in the

o

-

o

horizontal plane. First-order superscripts have been omitted in the following: r represents the relative surface elevation, the velocity potential, n the

structure's response at the relevant point on its surface, R a 3x3 matrix representinc rotational response, and the total fluid force, including both hydrodynamic and hydrostatic components. J2) is the second-order velocity potential, to be discussed below.

Details of term Vare given in reference 1, section 4.2 a), b) and c).

3.2 Second-order wave term

Set-down in a regular wave train causes at most a change in the mean pressure, and an associated vertical force, heel and trim moments (see reference 1) . The mean component of term VI is therefore relatively simple.

The low-frequency component is more complicated. Set-down in an irregular sea represents a long-period second-order wave which is tied to the underlying pattern of first-order waves, and travels with their group velocity. There are both vertical and horizontal pressure gradients in this wave, so that the resulting force has

components in both directions.

There are additional forces due to set-down in the first-order diffracted wave, interactions between the incident and diffracted waves, diffraction of the incident set-down wave, andreflectionof set-down from other fluid boundaries (e.g.at the wavemaker and beach in a wave tank). The last two components represent free waves, which have their own independent speed of travel.

Term VI depends on the second-order velocity potential 2), which is very difficult to compute. There are several different ways in which to approach this problem.

Lighthill (11) has shown that the second-order force can be expressed wholly in terms of first-order quantities by making use of certain reciprocal relationships. His expressions,however, require the evaluation of an integral of second-order pressures over the entire sea surface. The amount of numerical work required to achieve this end is likely to prove excessive, unless some approximation can be devised to represent asymptotic behaviour away from the structure. The development of such a solution may merit further study.

Pinkster (4) treats the set-down wave as a first-order free wave, but with g (acceleration due to gravity) reduced so as to achieve the correct phase speed. This procedure models only one of the diffraction components, perhaps not the major component. Pinkster showed that this procedure represents accurately the set-down in an undisturbed wave, and in the presence of a reflecting plane wall, but

seriously overestimates the set-down force on a cylinder.

Bowers (12) neglected diffraction effects altogether. Forces were calculated by using a Froude-Krylov type of approximation, from pressures in the undisturbed incident set-down wave. Bowers's expression is easy to evaluate, and gives an indication of the magnitude of set-down forces. These forces are in any case often very small, particularly in deep-water situations. Bowers's procedure was

adopted in the NMI program.

The purpose of evaluating this term, therefore, is not to obtain an accurate prediction of the set-down force, but rather to find out whether it is important. If it is significant there are a number of other issues that ought to be considered.

It may, for example, be necessary to take account of other second-order effects, such as reflection of the set-down wave at the beach ('surf beats') or (in model testing) similar long-period waves caused by the wavemaker. These issues are discussed in reference 10.

wh er e k2 sech2k d k2 sech2k d C m m n n mn o G m n + 2(o - o ) k k m n

mn

-cos (i -i ) + tanh k d tanh k di} / {K g sinh K d

co

-

m n m mn mn

mn

- On)2 cosh K d} mn where

-(o

- o

)t

+E -E

m n m

n-K cos ij = k cos iji - k cos _P

mn

mn m m n n

K Sin 4 = k sin j - k sin 4i

mn mn _m m n n

The underlying first-order wave is expressed as in equation (2.1).

The force is now obtained by integrating (2) over the structure's surface s, as in

equation VI of section 3.1.

3.3 Symmetry and spectral relationships

As shown in sections 3.1 and 3.2, the second-order force is a bilinear function of

the wave amplitudes a, a. _{If the surface elevation is of the form of equation}

(2.1), then the slowly-varying force may be written:

The set-down force (term VI) is therefore calculated by the NMI program as follows. The undisturbed set-down wave can be expressed in terms of the velocity potential:

N-1 N

(2) a a C cosh K (d+z)

= E E m n mn mn m=1 n=m+1

sin 1K

xcos)

+K

ysin

- mn

mn mn mn

-

L17

:

(3.1)

N N

F(t) = E E a_m a T

cos

[ (o -o )t_6m - +E

-n mn m n n m

n-m=1 n=1

There are two terms involving o and o which can be combined to give

m n

(T cos 6

+ T

cos 6 ) cos r

(o -o

) t -c +c

mn mn mn run - m n m

n-+ T

sin 6

-T

sin 6 ) sin

r

-o

)

t -E +:

mn mn orn orn - m n m

n-It is convenient to set p = (T cos

+ T

cos

mn mn mn nrc orn Q = i (T sin 6 - T sin 6 min mn mn rim mn

Then P and Q represent symmetric and antisyrmnetric matrices respectively:

P

=P 'Q

_=-Q

mn mn mn mn

Corresponding amplitudes and phases may be defined respectively by

F

='

₂ mn P mn tanc.

=Q /p

mn Tnn mn where F = F , = -ci mn nm mn orn

Then

N N F(t) = E E a a {p cos

r (o

- o

)t - E

+ E

11

_mn

_{sin [(0m_0n)tm}

+ E}

m n mn - m n m n m=1 n=1 or N N F(t) = E E a a F cos

r

(o - o )t - ci

- E + £

m=1 n=1 m n mn - rn n mn m n -2

¿

L

/Of this analysis is valid for either unidirectional or multidirectional (long

or short-crested) representations of the waves. The computer programs which evaluate Fmn, mn can in fact cope with pairs of wave trains from either the same or different directions. The spectral superposition formulae for multidirectional waves are

only slightly more complicated mathematically than those for unidirectional seas, but pose numerical problems. Very much larger quantities of data have to be handled

in order to provide coefficients at every pair of frequencies and directions. For this reason the present superposition programs are restricted to modelling unidirectic seas. In this case, if 5(0) is the first-order wave spectrum, then the second-order force spectrum may be written (see reference 4)

SF (o ) = 8 Jo s (o) S (o' + o) F2 (o, o+o' )do (3.2)

where F (o , o ) F in the earlier notation.

m n mn

The mean force in the same wave condition is

=2

_f

S(o)F(a, o) do

The NMI program evaluates these integrals numerically. In so doing it has to interpolate between discrete frequency pairs

0, 0

(m, n = i to N) at which the transfer function F has been calculated. Interpolation is by a standard four-point

3.4 Evaluation in terms of complex quantities

Terms I - V all involve products of first order quantities of the form XY. The

NMIWAVE program evaluates X and Y separately, using complex arithmetic for convenience. It is therefore convenient to continue to use complex notation, with the understanding

(as before) that physical quantities are represented as the real parts of complex variables.

If X and Y are written N -io t

X=

a X e m

m m

m=1 N

-lo

t

Y=

E a Y e n n n n= i

then X, Y

are complex quantities which contain information about both amplitude and phase.

The second-order force is now written (cf section 2.1): N N

-i(o -a )t

XY E E

X Y*a a

e m n

m n m n m=1 n=1

where * denotes the complex conjugate. _{Combining (o -o}

) and (o -o ) terms as before,

m n n m

and splitting the coefficients into symmetric and antisymmetric parts gives

-i a

mn

(X

* +

* y

F e

=P

±1

3.5 Response motions

The second-order equation of motion of the structure is assumed to be of tne same form as the first-order equation:

A

n(2)

+ bfl2 + cn(2)= F(t)

where is the second-order response, F(t) is the forcing, calculated as in previous sections. A is the mass (or moment of inertia) of the structure together with its added mass, b the hydrodynamic damping, and C the stiffness (hydrostatic and mooring). If F(t) is sinusoidal, then so too is (2), with the same frequency of variation, and canbe calculated straightforwardlyby standard techniques (see section 2

If the wave forcing is irregular, and represented in terms of a force spectrum SF(c) then the response has a spectrum:

SF

SR(0)

-A2r

_-

(02 - a22+4o2o22 i

O O

where = b/2v' and is the undamped natural frequency

= C/A

The new NMI programs also calculate the root mean square response and the mean respon: period. The mean square response is simply

1=

j0 SR(0)dO

and the mean period is defined as

effects on the mean wave drift force. This mean force is associated with changes in wave momentum due to diffraction, wave radiation and drag. Inertial forces make no direct contribution. By considering the mean force acting on a fixed vertical circular column, resting on the sea-bed and piercing the free surface in regular waves, it was shown that:

drag effects dominate roughly when HA2/D3>60,

diffraction effects dominate roughly when HA2/D3<60, where D column diameter, A = wavelength and H = wave height.

Mean forces occur in both regular and irregular wave conditions. Slowly-varying forces, however, can only be discussed in the context of irregular waves, and these

introduce considerable complications into the analysis. In order to eliminate one source of difficulty, drag forces will be considered no further here.

Attention will therefore be focussed on structures consisting mainly of large-diameter members. Examples include ships, barges and caissons, semisubmersibles and tethered buoyant platforms. In such cases wave loads are principally associated with inertial and diffraction effects.

As explained above, mean forces on such structures arise because of changes in wave momentum due to diffraction and radiation. Low-frequency forces arise in a similar way. There are, however, additional low-frequency forces associated with the

spatial gradient of second-order forcing. This forcing includes not only

variations in set-down and Bernoulli pressure, but also analogous variations in wetted surface and vessel motion contributions to the second-order force (in fact all of terms I - VI, described in section 3.1)

There are no such gradients in regular waves, because conditions are uniform

everywhere, but in an irregular sea conditions vary from one location to another. Resulting pressure differences cause second-order forces, particularly on long

An approximate method, due to Bowers (12), and applied by him to surge motions of a moored ship. Bowers neglected all wave diffraction and radiation effects, and therefore found zero mean force. He included gradients of forcing

associated with the undisturbed incident wave and surge response (terms I, II and III), and that of set-down (term VI of section 3.1).

An approximate method, due to Newman (14) and Pinkster (15) , in which the low-frequency forces are derived from mean forces in regular waves. These mean

forces are associated with wave diffraction and radiation. This method neglects all second-order pressure gradient effects, because these do not occur in

regular waves. There is, for example, no set-down contribution. In a sense, therefore, this method is complementary to that of Bowers. Neither approximatio. however, includes interactive effects of diffraction on the gradient terms.

A more accurate method, due to Pinkster (4) and described also in section 3 of this report, in which both wave diffraction and gradient effects are combined This method is the basis of a new computer program developed at NNI for

predicting low frequency wave forces. Terms I - V are obtained directly from the first-order diffraction solution (NMIWAVE) , and are calculated exactly. Bowers's approximation is used for the set-down term VI.

i The Bowers Method

Bowers (12) was particularly interested in the low-frequency surge motions of moored ships in short-wave head seas. He neglected all wave diffraction and radiation effect and consequently found that the mean force was zero. His low-frequency forces came entirely from the horizontal gradients of second-order quantities represented by termE

I, II, III and VI of section 3.1.

The surge force was described in terms of properties of the undisturbed free wave:

½pg1

2'

d + p (w2-u2)n1dS

-o

J50

where ç is the undisturbed surface elevation, u and w are undisturbed horizontal and vertical particle velocities in the wave, all according to first-order theory, and (2)

is the second-order velocity potential associated with set-down. n and n1 are components in the surge direction of unit normals to the equilibrium waterline L and hull surface S

o o

Bowers then assumed that the water was deep for first-order waves, but shallow for

set-down waves, in the sense that k d»l, but k - k ¡d « 1. In some offshore

m m n

applications, however, the water is also likely to be quite deep for the set-down waves, so that km - kd»l. Both deep and shallow-water conditions will therefore

be considered.

The first-order surface elevation is given by equation _{(2.1). In order to simplify the} presentation, waves will be assumed to travel parallel to the x-axis, so that i = O, as in Bowers's original paper.

The set-down pressure may now be approximated (c.f.full expression in equation 3.1.)

(2) N-1 N

(2) _{= __9}

amanfmn cos 1(kmkn)x - (Gm_On)t + Cmr1

p

=p

2

m=ln+1

where _mn = 1/d for lkm - kd .< 1

mn = km - kn for _{km - kn d > i}

The first (shallow-water) expression comes from Bowers's (12) original paper, and the second (deep water) formula comes from appendix5 . The approximation

2OnIOm - cIgk

- knj has also been used.

Bowers shows that the velocity-squared contributions to the force are small. The remaining two terms, for a ship of length L, beam B, draft h and rectangular section, may be approximated:

N N

pgB _{aman (lhfmn) sin (km_kn)L/2 sin}

(Gm_Gn)t + rn=l n=l

The set-down and wetted surface terms are now seen to have opposite signs. _{They are} of comparable magnitude, and therefore tend to cancel, _{if the vessel's draft is} similar to either the water depth d or of the wave group length.

m=1 n=1

Both terms become small as km approaches k, and the wave group length increases.

There is no force in purely regular waves.

4.2 Newman/Pinkster approximation

Newman (14) and Pinkster (15) approximated the slowly-varying force in terms of mean forces acting in regular waves. Their approach has much in common with an earlier numerical procedure described by Hsu and Blenkarn (16) , though is more

rigorously formulated. Hsu and Blenkarn treated each half cycle of an irregular wave history as part of a regular wave train with an appropriate height and period. They computed the slowly-varying force history from the mean force acting during the appropriate regular wave cycle. In effect they assumed that the wave form

varies slowly, and is almost regular for several successive cycles. This assumption is similar to Newman's:

G

- O

+ G

m n m n

where

0m' o are component frequencies of the wave spectrum.

The virtue of this approach lies in the fact that it is easier to compute or measure mean forces in regular waves rather than slowly-varying forces in either irregular or beating waves. There is now a considerable quantity of data on mean forces, but very little on slowly-varying components. What little there is tends to be of

doubtful accuracy.

In an irregular wave, represented by equation (2.l)the full expression for the slowly -varying force has the form (see section 3.3)

N N

-F(t) = E E a a {P(o , o ) cos (o - o )t - c

+ E

- m n m n - m n m

n-+Q(o

, o

) sin

(o - o

)t-c +E

m n - m n m

n-as shown in section 3.3. Newman arid Pinkster assumed that P and Q may be approximatE

0+0

0+0

m n

m n)

P (o , o ) P m n

2'

2 Q (o , o ) O m n

If P and Q are represented in terms of matrices

P

=P(o,o),Q

_=Q(o,o)

mn m n mn m n

then Newman's approximation is equivalent to replacing the off-diagonal matrix elements by those on the diagonal at the mean wave frequency.

This formula is widely used in offshore engineering, but in the spectral form devised by Pinkster (15)

w

S (o') = 81 S(c)S(o+c')P2(c+o',

0+0')

dc F

2

where SF(o ) , S(o) are power spectral densities of force and surface elevation respectively.

Pinkster's formula makes use of information about mean forces only, because a2P(o,o) is the mean force in a regular wave of amplitude a and frequency o. This method therefore takes no account of forces associated with the second-order horizontal pressure gradient, and is likely to be valid when wave diffraction effects dominate, and gradient effects are small.

4 . 3 Order-of-magnitude comparisons

Some insight may be gained into the relative importance of diffraction, gradient and set-down contributions by examining the forces acting on a fixed vertical cylinder. No attempt will be made to calculate the forces accurately. It is sufficient to identify orders of magnitude, and their dependence on the various parameters of the waves and structure.

a) Newman/Pinkster approximation

Havelock (17) obtained an analytical expression for the mean force on a circular cylinder in regular waves. His expression was later evaluated numerically by van Oortmerssen (18) . The most important features of this expression can be reproduced more simply by using the asymptotic long and short-wave limits (see reference 1).

The mean force in a regular deep-water wave of frequency o and amplitude a is then roughly

½pgDa2 K(kD)

where K = (kD)3 for kD< 1.26

K =

for kD1.26

and D is the diameter of the column. Figure 6 shows a comparison between this approximate expression and van Oortmerssen's exact numerical solution.

The simplest wave form which gives rise to a low-frequency force is the regular wave group, or beating wave. This consists of two superimposed regular sinusoidal wave trains with amplitudes a1, a2 and frequencies o 102 respectively (see figure 2)

Newman's formula

gives the amplitude of variations in the drift force:

pgDa1a2K(k1 21J) (4.1)

where k12 is the wave nunther associated with a regular wave of frequency 012

+ _o2)/2. According to deep-water theory

gk12 = (o

+

and g(k1-k2) _{= 2012(01_02)} (4.2)

b) The Bowers Method

A ship has two characteristic horizontal dimensions: its length L and beam B. General its length is much greater than its beam, and both dimensions will be

retained. It will now be assumed that both L and B are small compared with the group length, so that

sin (k1-k2)L/2 (k1-k2)L/2.

This approximation will be unsatisfactory if Ikr.k2IL 2: i.e. for a long ship in

ThenBowers's expression for the amplitude of variations in the low-frequency force consists of two terms:

pgBLa1a2 1k1-k2I (4.3)

representing changes in wetted surface area (term I) , and

½pgBLa1a2 1k1-k2 lb fi2 (4.4)

representing the set-down force (term VI), where hi is the ship's draft, andf12 is

thelargerof l/d and 1k1-k21.

o) Comparisons between expressions : regular wave groups

Removing factors common to all three expressions (4.1), (4.3) and (4.4), their relative sizes can be described in terms of ratios between three quantities:

2DK(k12D),

k1-k2BL and

k1-k2lBhLf 12

It seems reasonable to suppose that these three quantities will give an indication of the relative importance of wave diffraction, gradient terms depending on

first-order quantities, and second-first-order wave contributions to the second-first-order force respectively.

The diffraction term dominates when the structure is large in relation to wave length (k12D larger than about 1) , and the difference frequency (01_02) is small. It then seems likely that Newman's approximate formula will be valid.

At the opposite extreme, gradient terms dominate when D is small in relation to wavelength (le tubular members) orLis large (long ships in head seas), and the

difference frequency of interest is also large. Bowers's approximation is then likely to be valid.

Set-down contributes to the gradient term, but is only important if the vessel's draft is comparable with either the water depth d or 1/1k1-k21 (roughly of the wave group length).

In intermediate conditions both the gradient and diffraction terms may be significant The full solution is then likely to give a better result than either approximate method. The NMIWAVE computer program evaluates the full expressions for terms I-V, and includes both gradient and diffraction effects. Term VI is approximated in the same way as inBcwers's theory, and therefore neglects effects of diffraction on the second-order wave.

d) Irregular waves

The regular wave group is an artificial concept, useful to the researcher and for obtaining the matrix of drift force coefficients P , Q . As mentioned earlier,

mn mn

however, it can lead toundulypessirnistic estimates of system response in the sea. The above criteria are nonetheless of some value in giving guidance on the terms most likely to dominate the forces in an irregular sea.

Irregular waves will give rise to a fairly broad spectrum of wave forces. The only frequency which is likely to be of much interest, however, is the natural frequency of the system. In general therefore it will be sufficient to consider the case

= only.

It is not so easy to choose an appropriate typical wave frequency 0121 and this difficulty is compounded by the extreme sensitivity of the diffraction force to variations in when k12D is less than t. The best choice would seem to be the

frequency which makes the largest contribution to the mean drift force, and this will generally be near or above the modal frequency of the wave spectrum. Problems arise when the drift force peaks at a very much higher frequency: when k12D is small for frequencies

°12 around the spectral peak, and the wave spectrum is fairly broad-banded, with substantial energy at high frequencies. Not only are estimates of the diffraction force based on K(k12D) very sensitive to small changes in k12, but calculated and measured forces become very sensitive to numerical and experiment error. The practical relevance of these difficulties will become clearer in the context of some numerical examples, presented in the next section.

e) Sample calculations

Appendix 2 describes some sample calculations on a very large ship (200,000 DWT

VLCC), a conventional drill-ship (as described in section 7), a large semisubmersible and the 'Condrill' platform.

Forces and motions of both large and small ships in beam seas are dominated by the diffraction term. Newman's approximation is likely topredict sway forces and

motions fairly well in these conditions.

The criteria are also fairly well defined for ships in head seas. If the natural surge period is very long (greater than about 400 seconds for the VLCC, and 200

seconds for the drill-ship) , then the diffraction term dominates. If it is short (less than about of the above values) , then gradient terms dominate. In the latter case Bowers's method may work quite well. In many applications, however, the natural period will be between these limits, and both terms may be important. A method which includes both diffraction and gradient effects (eg the NMIWAVE program) should then be used.

The force regimes are less easy to define for semisubmersibles and similar structures. Their legs are generally of small diameter compared with dominant wavelengths. They cause little wave diffraction, but the gradient forces are also small. The

parameter k12D is typically less than 1, so that DK(k12D) is very sensitive to small changes in either the wave period or diameter. It varies, in fact, with the fourth power of D, and the sixth power of

Table i illustrates the problem. It shows conditions in which diffraction and

gradient terms dominate the second-order force on a fixed-vertical circular column of diameter 1Dm. The diffraction term is said to dominate when it is more than twice the gradient term, and vice versa. A range of typical mean periods between 5 and 10 seconds is covered. The results show the extreme sensitivity of the diffraction force to variations in this mean period.

The consequences are as follows:

likely to dominate (and equally whether Newman's orßowers'smethod is likely to be

valid) . The force regimes are very sensitive to small variations in either the structure or wave conditions.

ii) Reliable results can only be assured by using the full solution.

iii)The diffraction term becomes more important as the modal period of the wave spectrum decreases. The converse is also true.

iv) Broadening the bandwidth of the wave spectrum, so that, again, there is more energy at high frequencies, will have a similar effect.

Increasing the diameter of the main columns will also have a similar effect.

There are two other important consequences, which apply equally to the mean force components, and have been noted in reference 1.

vi) Because the diffraction force is very sensitive to high-frequency components of the wave spectrum, it is difficult to measure or calculate these forces accurately. Results are very sensitive to numerical and experimental high-frequency errors.

Vii)The diffraction force falls off rapidly as k12D decreases. Eventually drag force become dominant. This is likely to be true particularly of small tubular structures. References (19-21) discuss the effects of drag on semisubmersibles.

The above results generally bear out Pinkster's (22) conclusions on semisubmersibles, and explain why Rye, Rynning and Moshagen (23) found large discrepancies between Newman's theory and model tests on a 'Condrill' platform.

Focussing attention now on the two gradient terms, the above simple theory indicates that changes in the structure's wetted surface will usually have a larger effect than set-down. Numerical results, however, show that terms neglected above may sometimes cancel out the wetted-surface component. Set-down is then left as the single most important gradient term. The above simple criteria are not, therefore, always a reliable guide to the relative importance of set-down.

5.1 The Bowers Method

5. Theoretical Calculations on a Moored Drill-Ship

The purposes of the theoretical calculations on the moored drill-ship were twofold. The primary aim was to calculate the response spectra to compare with the experimental

results in chapter 7. In order to calculate these responses, the drift force spectra had to be calculated and this seemed an ideal opportunity to compare various methods

of obtaining these drift force spectra. The secondary purpose of the calculations, therefore, was to establish the usefulness and the limitations of the approximate methods

described in chapter 4. These were compared with the complete method, which was assumed to be the most accurate.

All calculations were performed at model scale (scale factor 1:20.09). The particulars of the drill-ship model were:

This method is described in the previous chapter for the surge drift force in head seas. Two corrections were made to the approximate formula. The first was due to the fact that

the formula assumes that the ship is a cuboid (or block) of beam, B, draft, h, and length, L. A correction to the force coefficient can be made in order to obtain the correct displaced mass. The force coefficient then becomes

F

=F

xc

mn mn B

where

c displaced mass pLEh

CB is in fact the block coefficient of the ship. Length: 4.703m

Beairi: 0.76m Draft: 0.288m Water depth: 7.62m Block coefficient: 0.655

angle to the surge axis. _{The correction for this was made by modifying the wave} numbers in the following way.

The force coefficient for a pair of waves of angular frequency o, a, propagating

along the surge axis can be expressed as

(k -k )L

F

=q(a

a ) sin { n m }

b

mn ₂

where L is the length of the ship. For waves propagating at an angle O to the surge axis, this becomes

F = q(o )

(k1 cos O - km cos O)L

mn m, n

2

The force coefficients for the drill ship can then be calculated using the method described in section 4.1. These force coefficients are shown in Table 6 (b). Using expression (3.2)the drift force spectrum can then be calculated. The expression for

the drift force spectrum is

SF(6f) = 8

IS(f)S(f+f)

q(f,f+ff)

Sfl

(k

L cos O)V df (5.1)

where

k' is the wave number of the wave component of frequency f+5f, k is the wave number of the wave component of frequency f and q(f,f+óf) is expressed in two ways as follows:

if (k-k')d i

then q = PB(- - 4rr2h(f+óf)óf) x CB

and if (k-k')d <i

where

I d = water depth

C5 = block coefficient.

The S(f) and S(f+óf) in the formula (5.1) are the wave power spectral densities at frequencies f and f+6f respectively.

5.2 The Complete Calculation of the Drift Force Spectrum

Section 3.3 describes how the slowly varying drift force coefficients are used to calculate the drift force spectrum. The slowly varying coefficients for the moored drill ship were obtained in the following way.

First order forces and responses for 10 wave periods were calculated using the NMIWAVE computer program suite as described in ref. 1. These quantities, together with the set-down contribution, were used to calculate the slowly-varying force coefficients

using the expressions described in section 3.1. These coefficients were calculated for all possible pair of the 10 wave periods, i.e. 55 pairs. This provided a matrix of force coefficients that were stored on file. These coefficients for the surge and sway forces are listed in tables 6 (a) and 8.

The slowly varying force power spectral density at the difference frequencies was then calculated using equation 3.2 in the form:

SF(Sf) = 8 J

OS (f) S (f-HSf) (F (f, f--6f))2 df

where F(f, f+5f) is the slowly varying force coefficient for the frequency pair f and f+6f. For any frequency pair, F was obtained by interpolation in the matrices of force coefficients (Tables 6 (a) and 8). This was done by using a standard 4 point bivariate interpolation formula (see ref. 13).

5.3) The Newman/Pinkster Approximation

This method for calculating the slowly varying drift force spectrum is described in section 4.2. It only uses the mean drift forces in regular waves. These mean drift forces were calculated as described in ref 1 for regular waves at 10 frequencies. The drift force spectrum was then calculated using:

SF(f) = 8

J

S(f)S(f+f) (F(f+½f, f+½6f))2df

where F(f+½Sf, f+½»5f) is the mean drift force coefficient in regular waves of

frequency f+6f.

F was calculated for any frequency by a linear interpolation between the 10 calculated mean wave drift force coefficients. At very low frequencies, the force coefficients were extrapolated to equal zero. Atveryhigh frequencies the coefficients were

extrapolated to a constant value, calculated by using Havelock's approximation for mean drift forces (see ref. 1)

:.4 Results

A)Surge Force:

All of the three methods above were used to calculate the surge drift force spectra in 3 irregular wave spectra. Two of these irregular wave spectra were obtained during runs 19 and 20(i) of the model test experiments described in section 7. The former was a Pierson-Moskowitz spectrum and the latter a JONSWAP spectrum. The third spectrum chosen was a theoretical JONSWAP spectrum with a significant wave height equal to the spectrum in run 20(i) but with a shorter mean zero crossing period. This

was chosen for reasons described in the discussion later in this chapter.

The surge drift force spectra are plotted in Figs 16(a), (b) and (c). The results are summarised below:

Frequencies less than 0.1HZ: Results obtained using the Bowers approximation are very much less than those from the complete formula. The Newman/Pinkster approximatiol produces results that are up to 17% less than the complete method. This frequency rangi is the most important one for present purposes because the natural period in surgefor thE moored drill ship is in this range.

The frequency ranqe 0.1 - 0.3 HZ: Peak values obtained n the Pierson-MoskowitZ spectrumof run 19, and using the Bowers approximation, are as much as 50% greater thai those obtained using the complete formula. In the JONSWAP spectrum of Run 20(i) the Bowers approximation gives peak values that are 250% greater than those obtained using the complete formula. This approximation does, however, agree quite well with the complete formula for the third spectrum. The Newman/Pinkster approximation,

on the other hand, does not reproduce any of the features of the complete formula. It produces results that re considerably lower than the

complete method in Figs 16 (a) and 17 (a), and it is considerably higher than the complete method for Fig 17(c). This frequency range could have been important if the drill

ship had been stiffly moored.

(c) Frequencies greater than 0.3 Hz This region is unimportant because the first order effects begin to dominate. The comparison of the three methods show that the Bowers _{method and the complete method agree reasonably above 0.35Hz, and give results} that are considerably lower than those given by the Newman/Pinkster approximation.

B) Sway Force

The sway drift force spectra were calculated for the two wave spectra of runs 19 and 20(i) using the complete formula and the Newman/Pinkster approximation. The Bowers method is not valid for the sway force. The results are shown in Figsl7(a) and

17(b) and are described below:

Frequencies less than 0.1Hz: The two methods agree well in this frequency range. Frequencies greater than 0.1Hz: The complete formula shows a peak that is

considerably higher than the Newman/Pinkster approximation. The complete method also decays faster than the Newman/Pinkster approximation.

5.5 Discussion of the Results

A) Surge Force:

The two wave spectra for runs 19 and 20(i) had mean zero crossing periods of l.68s (frequency = 0.60Hz) and l.82s (frequency = 0.549Hz). These periods lie in the range where combinations of wave periods of this order give rise to small surge force

coefficients (see table 6(a)). In this range the force coefficients obtained using the Bowers method are much larger (see table 6(b)) than those obtained with the complete method. The same is therefore also true of the force spectra, as seen in figure 16. However, it is significant that the results in the Pierson-Moskowitz spectrum

(Fig 16 (a)) agree much better than those in the JONSWAP spectrum (Fig 16 (b)).

and was a broad-banded spectrum, and therefore used coefficients from

the regions where the Bowers coefficients were of similar magnitude to those of the complete method.

The complete method gives such low coefficients because although the response terms (i.e. terms III and IV) are quite large, they cancel with other terms in the

expression for the force coefficient thus leaving only a small residue. The set-down contribution is a significant proportion of this residue as can be seen when comparing Tables 6(a) and 7.

The Bowers method gives rise to large coefficients because the responses are excluded and therefore the contribution from the free surface elevation term (i.e. term I) is overestimated. Term I should be calculated using the relative first order surface elevation. Bowers uses the absolute first order elevation. The former is considerably less than the latter at long wave periods relative to the ship. For wave periods around l.68s or l.80s, this was certainly the case for the moored drill ship. For larger ships this effect would appear at longer wave periods.

Several factors combine, therefore, to make the Bowers method overestimate forces in these particular conditions. Bowerss method is likely to prove more satisfactory for

larger ships in surge, or when the waves have a shorter mean period. A third set of calculations was carried out, therefore, in order to demonstrate this point, using a wave spectrum with a mean zero-crossing period of l.37s (frequency = 0.73Hz). In this example the Bowers and complete solutions agree fairly well for difference frequencies above 0.1Hz, as shown in figure 16(c). The Bowers method cannot, of course, predict rn forces or very low-frequency components because it makes no allowance for diffraction effects.

Comparing now the Newman/Pinkster approximate method with the complete solution, figur l6a and l7a show that the two force spectra agree very well at low difference frequenc The two curves diverge as the difference frequency increases: the approximate

spectrum tending to fall off almost continuously, while the complete solution reaches a second peak around 0.1-0.2Hz. This second peak is the combined result of a large set-down contribution (excluded from the mean force, and therefore from the Newman/ Pinkster theory) and of the growing importance of terms III and IV as the difference

frequency increases. All these terms make small contributions to the mean force. At very large difference frequencies terms III and IV become large enough to cancel

out other contributions to the force. As a result, spectra obtained using the complete solution die away more rapidly than the approximate spectra.

B) Sway Force

The results for the sway force are explained by the points made in the preceding two paragraphs. However the sway force has less contribution from the set down force and so the differences at medium frequencies between the Newman/Pinkster method and

the complete method are less marked. However, the response terms are still important and their contributions give rise to the differences seen between the two methods at large difference frequencies.

5.6 Conclusions

The conclusions of these calculations are:

The drift force spectra are very sensitive to variations in the individual terms I-VI, because these terms tend to cancel each other out.

The Newman/Pinkster approximation agrees quite well at low difference frequencies with the complete formula for the surge force. However, it is important to note that its results are lower than the complete formula.

For large ships on stiff moorings the Bowers approximation would give better results for the surge response spectrum than the Newman/Pinkster approximation.

For the sway drift force, the Newman/Pinkster approximation agrees very well at low difference frequencies with the complete formula.

Cv) For any ship on stiff moorings, the complete formula would need to be used to predict the response spectra correctly in either sway or surge.

6) Comparisons With Published Data and Analytic Results

There are few published theoretical calculations of slowly-varying

drift force coefficients. Only two sources of published data were discovered. These were as follows.

Theoretical calculations for a barge, a semisubmersible and a tanker published by Pinkster (22) . The barge was the simplest of these three bodies and

therefore the NMI computer program was used to obtain slowly varying drift force coefficients which could be compared with those published by Pinkster. These calculations are described in section 6.1 below.

Theoretical calculations published by Faltinsen and Liken (25) for various infinitely long, horizontal floating cylinders of circular or rectangular section. One of these cylinders was chosen for the NMI calculations, which are described in section 6.2 below.

To complete the theoretical validation of the computer program, calculations were performed on a rigid wall. These are referred to in section 6.3 below.

6.1 Calculations on a Barge Model

NMIWAVE was used to calculate first order forces and motions for the barge. These calculations are described in Part I of this report (ref 1). These quantities

together with the set-down term were then used to calculate the slowly varying surge drift force coefficients for head seas using thernethod described in section 3.3.

These coefficients are presented in table 2 in the form of a matrix. The entry in the mth row and the nth column represents the force coefficient, F , in a pair of waves of angular frequency _0m and

The angular frequency of this slowly varying coefficient would then be (a - 0n The figures on the diagonal of the matrix, therefore, represent the mean forces in regular waves, and as one proceeds away from the diagonal the frequencj of the

slowly-varying force increases.

Table 2 (a) shows the comparison between the NMI results and those published

by

Pinkster (22).

_{The mean forces (ie those on the diagonal) agree reasonably well}

and all the differences in these are accounted for (see ref 1)

.

However, there are major

discrepancies between the two sets of results in the off-diagonal terms.

Pinkster (22)

has used a different methcx1 of estimating the set down force but this difference

does

not account for all of the discrepancies.

This can be seen in Table 3,

which shows the comparison between Pinkster's results and the NMI results for the

set-down force alone.

In order to trace the source of these discrepancies the slowly varying coefficients

for terms I, II, III and IV in the formula for the drift force (see section 3.1)

were calculated.

These results were sent to Pinkster at the Maritime Research

Institute, Netherlands (MARIN)

.

He confirmed in a private communication that every

coefficient agreed reasonably except for term III where there was a sign difference

in the out of phase coefficients, ie the Q

(see section 3.3), between the NMI and

MARIN results.

A fault was subsequently discovered in the MARIN computer program

which explained this discrepancy. It now appears, therefore, that the NMI and MARIN

results for the barge agree except for the differences mentioned earlier in the set

down force. Table 2(b) shows the comparison with Pinkster's corrected results.

The calculations on the barge model compare two computer programs which use the saine

theoretical methods.

Both programs now appear to agree in their results, the only

differences now being due to different methods of estimating the set down force and

different numerical modelling of the barge (see ref 1)

(.1)

6.2 Horizontal cylinder of rectangular section

_-Faltinsen and Lken (25), both at Norwegian Institutes, have published slowly varying

drift force coefficients for various two-dimensional, infinitely-long floating

horizontal cylinders of different cross-sections.

They used an expression for the

drift force which,except for the set down

force,was a two dimensional version of the

Their first order velocity potential was obtained using the Lewis formtechnique, which assumes that the velocity potential can be written as a wave source or dipole plus a series of wave free potentials. The velocity potential is then made to fit the body boundary conditions using least squares techniques. The velocities were obtained by numerically differentiating the velocity potential. Faltinsen and Lken

noted that different differentiation procedures used in obtaining the velocities resulted in variations in term II of the drift force formula of up to 10%. They als<

stated that there were inaccuracies in their procedure at high frequencies because oi insufficient parameters used in the fitting procedure for the velocity potential.

In order to provide as good a comparison as possible between the NMI results and the Norwegian results, the NMI calculations were performed for a very long three dimensional cylinder in beam seas. The dimensions were governed by the calculation

limits of the NMIWAVE computer program. Faltinsen and L$ken provided structural dati for the cylinder in non-dimensional terms. The angular frequencies, o, were also expressed in a non-dimensional form,

o/Th1,

where h was the draft of the structure.

g

The cylinder used in the NMI calculations was made to fit all the non-dimensional data in order to provide a structure that was equivalent to that used in the Norwegian calculations. It was 200m long, had a beam of 20m and a draft of 10m. 84 facets were used to model a quarter of the structure and 44 points used to model a quarter of the waterlìne. This was quite close to the operating limits of the NMIW computer program which only allowed 150 facets plus waterline points at the time the calculations were done.

With these dimensions the wave periods for which the first order calculations were carried out ranged from 5.08s to 9.19s. The maximum wavelength was approximately

130m, which was of the same order as the length of the structure.

Diffraction effects around the ends of the cylinder probably became significant at

the higher wavelengths. This meare that at these wavelengths the NNI calculations were no longer equivalent to the Norwegian two-dimensional calculations.

The Norwegian calculations used a more exact method for estimating the contribution due to the second order velocity potential, je the set down effect. It was therefor felt that a better comparison between the two sets of results would be obtained if