The motions of a moored ship in waves

PUBLICATION No. 510

NETHERLANDS SHIP MODEL BASIN

WAGENINGEN. THE NETHERLANDS

Faculty WbMT

Dept. of Marine Technology

Mekelweg 2, 2628

CD Delft

The Netherlands

THE MOTIONS OF A MOORED

SHIP IN WAVES

THE MOTIONS OF A MOORED SHIP IN WAVES

THE MOTIONS OF A MOORED

SHIP IN WAVES

DR. IR. G. VAN OORTMERSSEN

PUBLICATION No. 510

NETHERLANDS SHIP MODEL BASIN

WAGENINGEN, THE NETHERLANDS

CONTENTS

INTRODUCTION 9

POTENTIAL THEORY DESCRIPTION IN THE FREQUENCY DOMAIN 16

2.1. A general hydrodynamic approach to harmonic ship

motions 16

2.2. The equations of motion in the frequency domain 21

2.3. The determination of the velocity potential 23

2.4. The potential for a ship along a quay 31

WAVE EXCITED FORCES AND HYDRODYNAMIC COEFFICIENTS 33

3.1. Numerical calculations with the 3-dimensional

source technique 33

3.2. Experimental verification 35

3.3. Discussion of the results 42

3.4. The influence of the water depth on added mass

and damping 56

3.5. The influence of a quay parallel to the ship

on added mass and damping 56

EQUATIONS OF MOTION IN THE TIME DOMAIN 67

4.1. Potential theory description for flow due to

arbitrary ship motions 67

4.2. Equations of motion in the time domain 71

4.3. Relation between equations in the time and

frequency domain 72

4.4. The behaviour of the damping for high frequency

motions 74

4.5. Numerical computations of retardation functions

and constant inertia coefficients 76

THE APPLICATION OF THE EQUATIONS OF MOTION IN THE

TIME DOMAIN 85

5.1. General 85

5.2. Numerical calculations 85

5.3. Examples of computed moored ship motions and

experimental verification 90

5.4. Analysis of the results 107

5.5. Extension to other systems 115

CONCLUSIONS 116 2. I. 4.

'5_

6,

APPENDIX I Spectral analysis of irregular signals 118 APPENDIX II Particular solutions of an equation of motion

with non-linear, asymmetric restoring force 121

REFERENCES 125 NOMENCLATURE 134 SUMMARY 137 139 141 -8-L a

CHAPTER 1

INTRODUCTION

Up till a few decades ago, the mooring of ships has been mainly a matter of practical experience. Ships were moored in harbours or sheltered areas only, where the external forces are in general limited to the rather steady current and wind forces.

In a few harbours, as for instance Long Beach and Cape Town, with an open connection to the sea, difficulties were encountered with moored ships. Sometimes moored ships showed erratic motions and even mooring line failures occurred in apparently smooth weather conditions. Such troubles may be caused by harbour reso-nance phenomena or seiches, exerting forces on the ship which, although they are small, result in large motion amplitudes because of their low frequency, which is close to the natural

frequency of the moored ship. These problems were the challenge

for investigators as Basil W. Wilson [1-1] to attempt to describe

mathematically the dynamic behaviour of a ship, moored in waves. With the development of the ocean industry and the advent of very large ships, which can only be accommodated in a few har-bours with sufficient water depth, the need arose to moor ships

in exposed areas. To this purpose special mooring facilities were designed to absorb the loads exerted by the environment on the moored ship. Nowadays a variety of mooring arrangements is in

operation. The appropriate system for a particular mooring de-. pends on water depth, weather conditons, ship size and the allow-able motions of the moored ship.

In general it can be stated, that the magnitude of the loads to be absorbed by the mooring system are the lower, the more

freedom the ship is left to move. How much the ship is allowed

to

move depends on the nature of the operation which has to take place during the mooring.

Transfer of fluid cargo, for example, can be done with floating flexible hoses, so that no stringent requirements have to be put on the restraint of the ship. Therefore, a frequently applied type of mooring arrangement for oil terminals is the single point mooring system: the ship is attached to a buoy or an articulated or fixed tower by means of a single bowhawser.

The ship has much freedom to move and will take an average

equi-librium position in which the external loads are relatively

small.

Sometimes a spread mooring system is used, in which the ship

is moored by means of a symmetric system of mooring lines, with

or without buoys. A disadvantage of this system is that the ship is not free to rotate to find a favourable heading with respect to waves, wind and current.

If the allowable motions of the ship are small, for instance

when cargo has to be loaded or discharged by means of land based

cranes, the ship is usually moored to a jetty by means of mooring

lines and fenders.

Other arrangements of interest are: ships moored to rotating

floating piers, or to other ships or storage vessels. In these

cases as well as for the single or multiple buoy systems the

loads in the mooring lines are determined by the motions of the

moored ship as well as those of the floating body or bodies to which she is moored.

Because of the short history and fast development of mooring

in exposed areas, the design of terminals can not be based on

empirism. On the other hand the problem is too complicated for

an analytical treatment. Therefore it is common practice to study

the behaviour of a moored ship by means of experiments with small

scale models. Although model testing provides an effective tool

to determine mooring forces and maximum motions of the moored

ship for design purposes, this method inheres a few drawbacks.

First, model tests are expensive and time consuming. The

test set-up is complicated, it is essential that elasticity

prop-erties of mooring lines and fenders are simulated very carefully, and sophisticated facilities are needed to simulate the relevant

environmental conditions. For this reason test programs are

usu-ally restricted to final design configurations and selected weather conditions which are assumed to be the most critical.

Further, the fundamental insight gained from model tests on

these complicated systems is limited. Only the resulting output

is measured without learning much of the mechanism which causes

this output. As an example the low frequency motion of a moored

ship observed in tests in irregular waves may be mentioned. Some

investigators believe that it is caused by second order wave

for-ces, others think that non-linearities in the elasticity of

moor-ing lines are the reason for it, while some assume that it is a free vibration caused by transient phenomena. A definite answer is hard to give, since for instance first and second order wave forces can not be separated in a model test. Therefore it would be helpful to have the disposal of a computer-based simulation

method, which gives more flexibility in this respect.

This thesis will be devoted to the formulation of a mathe-matical model for the prediction of the behaviour of a moored ship in irregular waves in a purely theoretical way, which can be used for practical calculation on a computer. The study will be restricted to systems with six degrees of freedom, which means in practical terms that the ship is moored by means of mooring lines and fenders to a rigid structure (jetty), while the elastic characteristics of the mooring system may be non-linear and

asymmetric.

It is not the intention to be complete, since each model, however complex it may be, can not be more than a poor reflection of nature, and a very complicated mathematical formulation does not necessarily learn us more than a more approximate one, of which the solution is feasible. Important is, however, that the model reflects the typical behaviour of moored ships and can be extended and adapted whenever this appears desirable from com-parisons with model experiments or prototype observations.

Although in first instance the excitation of the ship will be restricted to linear forces due to long crested waves, other external forces such as wind and current forces and loads in-duced by passing ships can be incorporated in the model as well.

The basis of the equations of motion is the law of dynamics

of Newton:

d(m*)

-dt r

or, since the inertia m of the ship may be regarded as constant:

The external force F is composed of

arbitrarily in time varying forces due to the waves;

hydrodynamic and hydrostatic restoring forces, which are a function of the motions of the ship;

restoring forces due to the mooring system, which are a func-tion of the instantaneous posifunc-tion of the ship.

In the classical ship motion theory, it is common practice to formulate the equations as follows:

(NI + a) x + bk + cx = F(t) (1.3)

a, b and c are coefficients which describe the hydrodynamic and hydrostatic restoring forces.

In fact, (1.3) is not a real equation of motion, in the sense that it relates the instantaneous motion variables to the

instantaneous value of the exciting forces. It can only be used as a description in the frequency domain of a steady oscillatory motion, since the hydrodynamic coefficients a and b depend on the

frequency of motion.

Analytical work on the moored ship problem published so far has been based on equation (1.3), where three categories can be discerned with regard to the simplifying assumptions made.

Some investigators, as for instance Kaplan and Putz [1-2],

Leendertse [1-3], Muga [1-4] and Seidl [1-5] linearized the

elas-ticity characteristics of the mooring system. The restoring forces of the mooring aids can then be incorporated in the hydro-static term cx and the equations (1.3) of motion in the frequency domain can be solved easily, with the restriction that only har-monic excitations can be used.

Others, as Abramson and Wilson [1-1, 1-6] , Yang [1-7] an

Kilner [1-6] add non-linear terms to equation (1.3) to account

for the restoring forces of the mooring system, and solve the

equations by means of the method of equivalent linearization, assuming that the excitation is pure sinusoidal and that, as in

the earlier mentioned method, the response of the ship is simple harmonic too, with a frequency equal to that of the excitation.

This is not realistic, since observations both in model and full

scale situations have revealed that also other modes of motion

-may occur.

The work of Wilson and Awadalla [1-9, 1-40], Lean 11-111,

Wilson [1-12] and Bomze [1-13] belongs to the third category,

which is characterized by the assumption that the hydrodynamic, coefficients a and b in equation (1.3) are independent of the

frequency, so that this equation is regarded as an actual differ-ential equation. The solution, which is found either by approxi-mate analytical methods (ref. [1-9] , [1-11] ) or by finite

differ-ence integration in the time domain (ref. [1-10] , [1-12] and

[1-13], ), may contain components with frequencies lower (subhar-monic) or higher (superhar(subhar-monic) than that of the forcing

func-tion.

It will be shown in chapter 3 that the assumption of con-stant hydrodynamic coefficients can not be justified: especially in shallow water these coefficients appear to be very sensitive to changes in frequency. Consequently, a time-domain description of the behaviour of the moored ship is needed which takes into account the frequency dependency of the fluid reaction forces. A possible approach would be to use the impulse response function technique. If for any linear system the response R(t) to a unit impulse is known, then the response of the system to an arbitrary force F(t) is:

:x!(t)? =j 124 F 1'01 dr (L.0

The moored ship as a whole may, of course, not be thought of as a linear system, but this difficulty can be overcome by isolating the free floating ship in still water, for which system the assumption of linearity holds true, as long as the motions re-main small. The non-linear mooring forces can be incorporated in the external forcing function. A drawback of this impulse re-sponse function is, that it relates the input and output signals of the system without reflecting the physical processes behind it, the system of ship-fluid interaction is regarded as a black box.

In this thesis it is therefore proposed to use the equations of Motion in the time domain as they have first been formulated

by Cummins [1-14] , and which can be considered as true

tial equations; they give the instantaneous relationship between the motion variables and the external forces. In these equations the various factors governing the response of the ship are sepa-rated into clearly identifiable units.

The only assumption involved is linearity of the hydrody-namic restoring forces. Non-linear and asymmetric mooring char-acteristics can be dealt with, and the exciting force may be arbitrary, which means that besides first order wave forces also slowly varying drift forces and wind- and current forces can be included in the forcing function, although in this thesis only the problem of first order wave forces will be discussed.

In the equations of motion in the time domain the hydrody-namic forces are expressed by constants and functions. It is not feasible to obtain these directly from the potential theory. However, there is a theoretical relationship between the equa-tions in the time domain and those in the frequency domain, and therefore the second chapter will start with the potential theory description of harmonic ship motions at zero speed in waves in shallow water. Since terminals are very often located in shallow water, it is essential to take into account the effect of the nearness of the sea bottom on the wave forces and the hydrodynam-ic coeffhydrodynam-icients. Also the effect of a quay parallel to the ship will be included. Since the two-dimensional strip theory, which

is widely used in naval hydrodynamics, is not applicable to the case of small underkeel clearance, the three-dimensional source technique will be discussed as a tool to compute wave excited forces as well as hydrodynamic restoring forces.

In chapter 3 numerical results will be presented of calcula-tions of the wave forces and hydrodynamic coefficients of a large tanker, together with experimentally determined values. From

these data the accuracy and limits of applicability of the cal-culation method will be discussed.

Chapter 4 deals with the equations of motion in the time domain and their relation with the description in the frequency

regime. Further it will be shown in that chapter how the unknown coefficients and functions in the equations can be derived nu-merically from added mass and damping data.

The numerical solution of the set of equations of motion in

14

-the time domain, applied to -the moored Ship problem, will be die-.

cussed in chapter 5. Example computations will be given for a large tanker moored in oblique, beam or head seas to a jetty, and the results will be analysed and compared with measurements, ob-tained from model experiments.

Finally, a review of the main conclusions is given in

CHAPTER 2

POTENTIAL THEORY DESCRIPTION IN THE FREQUENCY DOMAIN

2.1. A general hydrodynamic approach to harmonic ship motions

In this chapter a general formulation of the ship motion problem will be presented in the frequency domain. This general formulation, in terms of the linear potential theory, is valid for deep as well as for shallow water.

A complete theoretical derivation of the formulae will not be given. This has been done before by other authors, and for a thorough mathematical treatment reference is made to their work (see for instance Wehausen and Laitone [2-1] , John [2-2), Tuck

[2-3]). Here, only a summary of the most important equations will be given, which are needed for the derivation of the equation of motion and for the numerical computation of the wave exciting forces and the fluid reactive forces.

Since the aim is to formulate a mathematical model for the motions of a moored ship, the analysis can be restricted to the case of zero forward speed, which means a considerable simplifi-cation of the problem.

In this section will be dealt with pure harmonic motions and therefore the mooring system will be left out of considera-tion.

The ship is considered as a rigid body, oscillating sinus-oidally about a state of rest, in response to excitation by a long crested regular wave. The amplitudes of the motions of the ship as well as of the wave are supposed to be small while the fluid is assumed to be ideal and irrotational.

In the theory of hydrodynamics it is common practice to define a system of axes with the origin in the free surface. For the description of ship motions, however, it is more conven-ient to use the centre of gravity of the ship as a reference point. Therefore, to prevent transformations from one system into the other, the description will be given here in a right handed, space fixed coordinate system as shown in Figure 2.1.

.

The oscillating motion of the ship in the 3th mode is given by:

x.

c.e

3 -iwt = I, , 6 (

fl

-16-=

-_u

1

Figure The coordinate system:

in whichc.is the amplitude of motion in the modeland-Co the circular frequency.

The motion variables x _x2 and

_x3

stand for the

transla-tions, surge, sway and heave, while x4, x5 and x6 denote

rota-tions around the GX _GX2 and

GX3 axis respectively.

In naval hydrodynamics, it is usual to introduce a set

of

three independent angular displacements, the so-called Eulerian angles: yawing, being about the absolutely vertical axis GX2,

pitching, around the rotated position of the GX2 axis, which

remains in the horizontal plane, and rolling, about the positon

of the

GX1 axis after the previous two rotations. Since only

small motion amplitudes are considered, these Eulerian angles coincide with the angular displacements about the space fixed axes (see Vugts [2-4]).

The free surface at great distance from the ship is defined by: iKtOc cos = a + x2 sin eV-iwt

*

Co e 1 where

co

= amplitude of the wave

K = wave number = 2u/A, where A is the

wave length a e angle of incidence ('2 .2) 1 x3 _{WAVE DIRECTION} 2.1. jth

The flow field can be characterized by a velocity potential

4.(x x x t) = (p(x x x )e

1' 2' 3' l' 2' 3

The potential function tp can be separated into

contribu-tions from all modes of motion and from the incident and diffrac-ted wave fields.

Following Tuck [2-3] a convenient formulation is obtained

when writing

7

= -iw

E

c.

j=o 3 3

with the convention {2 = r . 'o

The case j = o corresponds to the incident wave potential:

cosh K(x3 + c) . , 1 e

iK(x

1 cos a + x2 sin a) LP -0 V COSh K d in which v = w2/g

c = the distance from the origin to the sea bed

d = water depth

The relation between the wave length and the wave frequency is given by the dispersion equation, which follows from the free surface condition:

V = K

tanh K d (2.7)

The cases j = 1, 2,..., 6 correspond to the potentials due

to the motion of the ship in the jth mode, while

47)7 is the

potential of the diffracted waves.

The individual potentials are all solutions of the Laplace equation V2t.P. =

+2

+a2)14). = o 2 . 2x3 3 d 1 Dx2

while the following conditions must be satisfied: - the linearized free surface condition

(2.3) (2.4) -18-(2.8) (2.5) (2.6) x3

1 aW.

__1 =

an

f

BP'

₌

_{on x3 - d - c}

ax3

the boundary condition on the sea floor

DO'

_{= o}

_On

_{= -c}

ax3

- and the boundary condition on

the ship's surface. Due to the

linearization, this boundary condition may be applied to the

surface S in its equilibrium position.

7

an)

-iwt

= E ri

,c-i

w, c e 40

on, S

42,1511

an

_j=o

in which nl through n6 are the generalized direction

cosines on

S, defined by:

= cos (n,

xial

n2 = cos (n, x2)

n3 = cos

x3)

m4 = x2n3

x3n2

ns = x3n1 - x1n3

96 = x1n2

x2n1

tor the incident and the diffracted waves

one finds:

DPo DP7

_{= 0,}

, an

an

When. defining.",

1 =

7

The potentials gl through 97 should satisfy moreover the

boundary condition at infinity, the radiation condition, which

states

(.2.9)

pLax4i

g Z.121

Ir2.43')" L21.1:54

= -n7 =

_3n

then (2.11)1 may simply be written as:

- -19-(2.14) 0, 1, ,

This matrix can be considered as a transfer function which trans-forms motion variables into force components (see Ogilvie and Tuck [2-5]).

Equations (2.18) can then be written as:

lim r (- - iv4) .) = o j = 1, 2, , 7 (2.16)

3

Supposing that the unknown potentials pl through 44)7 can be determined, the pressure on the surface S can be found from Bernoulli's theorem. The linearized hydrodynamic pressure is given by:

p(x., X2, x3, t) =

-p

4t

7

= P

w2 P.C. e-iwt

j=o 3 )

The oscillating hydrodynamic forces (k 1, 2, 3) and

moments (k = 4, 5, 6) in the kth direction are:

Fk

= - If p

nk dS 7 -iwt er = -0 w2 E e _{jj P} nk dS j=o

Now the following matrix is defined:

. -p w2 Tk3

fh

nk Lpj dS 7 Fk = E _{Tk3 .} 3

et

7=0

From (2.19) and (2.15) it follows that:

314), Tkj = -p

w2 if

tPj dS

s

'"

hence: DLPk 3%.1). Tkj - Tjk = -p

w2 fr (---

-

--2

IdS an 3 an k

20

(2.17) (2.18) (2.19) (2.20) (2.21) (2.22) E = =

6 d

t m 11-31j] X'

."

a

[ kj dt -k

1=1

Xk' the total external force in the kth mode, consists of hydro-static and hydrodynamic' restoring( forces and of wave exciting forces.

!I2 .26(1 The surface S can be closed by adding the free surface, the sea bottom and a vertical surface at infinity.

For the cases k, j = 1, 2, , 7 the integrand vanishes

on the extra surfaces, so for these cases we may apply Green's theorem, which results in:

=

jk K2.23),

For k, j = 1. 2---6 this relation means that a force

.

in the kth mode due to the motion in the 3th mode is equal to

the force in the jth mode due to motion in the kth mode.

Similar relations for the ship with forward speed have been

de-rived by Timman and Newman [2-6] and Newman [2-7]. For j 7,

equation (2.23) yields the Haskind relation:

kg -7k -p w2

ff

n

c

dS 7 k - 2 =

ff

no c'k

as

2='--Tok 12.24)

The total wave exciting force Amplitude per unit wave amplitude is given by:

1

,-Tko ko (2 .25)

2-.2. The equations of motion in the frequency domain

The differential equations which describe the motions of the free floating ship in response to simple harmonic waves can be found from Newton's law of dynamics:

k, j = 1, 2, , 7 =

+T

k7 =

-T

=

-21-Mkj is an inertia matrix. Since the origin of the system of axes coincides with the centre of gravity of the ship in its rest position, it is found that

Ik = moment of inertia in the kth mode

Ikj = product of inertia

6 2 . + C .)( . = (T

- T)/

Mk) k3 k3 '3 ko ok .o 3=1 T .k3 = w2 a .k3 + i w bkj.

-22-The rate at which the mass of the ship changes is very small, and its influence may be neglected when considering the motions during a period of time which is small compared to the time required to load or discharge the ship.

Therefore, one may write for the steady oscillating condition

6 d2 -iwt E (M 2 + ck3 .)c. e Fk k = 1, 2, , 6 (2.28) kj 3 j=1 dt

Ckj is a matrix of restoring force coefficients. Besides the

hydrostatic restoring forces, Ckj may also include restoring forces due to a mooring system, as long as this mooring system has linear load-excursion characteristics. For the hydrodynamic force Fk in equation (2.28) use can be made of expression (2.20). Hence

(2.29)

It is common practice to separate Tkj for k, j = 1, 2, , 6

into real and imaginary parts

(2.30)

The real parts akj are called added mass coefficients, the

imag-Mkj = where _ in o o o o o c m o o o o o o m o o o o o o 14 o -146 o o o o 15 o o o o -I46 o 16

in - mass of the ship

(2.27)

inary parts damping coefficients. When using these quantities, the following real representation of the equations of motion is obtained:

6

, 2

X

t-w

(Mkj + akj) sin (at

+ C.)

bkj W cos (Wt c,)

j=1

+ Ckj sin (wt +

e.)1r.

= Xk sin (at + di() (2.31)

3 '3

in which Xk = wave excited force in the kth mode

k = phase angles.

Thus a set of equations has been obtained which are not real equations of motion, but merely a set of algebraic equa-tions, fixing the amplitudes and phases of the six oscillations of the ship under the action of a train of regular sine waves

at one specific frequency.

2.3. The determination of the velocity potential

In the foregoing sections it has been shown that the flow around the ship is completely defined by the velocity potential. Once this potential function is known, the wave exciting forces, hydrodynamic forces and unrestrained ship motions in harmonic waves can be obtained easily.

For the computation of deep water ship motions, the so-called strip theory, which was first formulated by Korvin-Kroukovsky and Jacobs [2-8], has proven its usefulness. This theory takes advantage of the fact that for ships the longitunal dimension is large relative to the lateral and vertical di-mensions. For such a slender body the three-dimensional problem can be reduced successfully to a local two-dimensional problem. After its presentation, the method has been refined by many

au-thors and the results are in general reliable. A drawback is, however, that no information can be obtained about the surge mode.

Unlike the deep water problem, very few studies have been presented on the motions of a ship in shallow water.

Kim [2-9] has adapted the strip theory for a restricted wa-ter depth. For the vertical modes of motion this approach yields useful results, but it can not be used for lateral motions,

especially in the lower frequency range, since the strip theory is basically two-dimensional, requiring that the flow of water passes entirely underneath the keel of the ship. In shallow water, however, three-dimensional effects become important.

the water flows partly underneath the ship and partly around the ends. In the extreme case, the ship sitting on bottom, water can move only around the ends of the ship.

A three-dimensional approach has been presented by Newman [2-10] but he neglected the effect of the free water surface.

A very interesting contribution to the shallow water problem

has been given by Tuck (see Tuck [2-3], Tuck and Taylor [2-11]

and Beck and Tuck [2-12]. He has derived an approximate solution of the linear velocity potential for the case that the wave length is much greater than the depth of the water. Application of this theory is therefore restricted to long waves and low

frequency motions, which poses, especially for mooring problems, a rather stringent limitation.

A method of solution, of which the validity is unrestricted

as long as linearity is ascertained, is provided by the

three-dimensional source technique. This technique has been applied successfully during the last few years for the computation of

wave loads on large offshore structures (see for instance Daubert

[2-13], Garisson and Chow [2-14], Van Oortmerssen [2-15], Boreel [2-16], but it can be used just as well for ship shaped bodies.

According to Lamb [2-17] the potential function kpi

can be

represented by a continuous distribution of single sources on a boundary surface St

1 rr

Wi(xl, x2, x3) = jj 03(a1 , _{ay, a3 )y.(xl' x2 , x3, al' a2' a3)dS}

s 3

(3 = 1, 2, , 7) (2.32)

where y.(x1 , x2, x3' al' a2, a3) = the Green's function of a

source, singular in al, a2, a3

al' a2' a3 = the vector, describing S

ai(al, a2, a3) = the complex source strength

-24-1

For the Green Es function y we can choose either a complicatr ed function, being the solution of the Laplace equation which satisfies the boundary conditions at the sea bottom, in the free, surface and at infinity, or the simple fundamental source func-tion 1/r for an unbounded fluid. In the latter case the funcfunc-tion( a is such, that W.: satisfies all the boundary conditions. The boundary surface S consists of the ship's surface, the free wa-ter surface, the sea bottom and, to close the surface, a cylin-drical vertical surface at great distance of the ship. By apply-ing Green's theorem to this boundary surface, an integral

equa-tion is obtained for the unknown funcequa-tion ID, in terms of its

boundary values and its normal derivative on the boundary. By the method of discretization this integral equation can be re-duced to a set of linear algebraic equations with the unknowns being the values of the source strength function a at a discrete set of control points along the boundary. For more details of

this method reference is made to the work of Yeung [2-18] . A

drawback of this method is that the number of elements, required to schematize the entire boundary surface, is very large, which results in an evenly great number of equations to be evaluated numerically. A favourable feature is the possibility to take the, bottom topology into account, while the approach can be extended to the case of a fluid domain of finite extent, for example a canal or a

basin-In the present work the other approach will be used, in oth-er words a Green's function of more complicated form will be ap-plied on the surface of the ship only. The water is assumed to have a constant depth.

The Green's function of a source, singular in

(a1, a2' a3) which satisfies the boundary conditions in the free surface, on the sea bottom and at infinity, is given by (see wehausen and Laitone Z2-111)x

n. 4.

r r1

-Cd

in 2(C + v)e cosh Cifa3 + c) cosh E(x3 + c4)

* 8.17 f

_RR/dC

sinh Ed - v cosh Cd Jo 4

0

, 2 2

2x - V ) Cosh K(a3 + c) cosh KtX3 +

Jo4KR1 4,2.334

no

in which i. _{if(x1 - a1)2 +} (x2 - a232 + (x3 -

_a3)2

r1 = /(x/ - a1)2 + (x2 - a2)2+ (x3 + 2c + a3)

'

R =

/TX-

_{- a1)2 + (X2 - a2)2}

John [2-2] has derived the following series for ye which is the analogue of (2.33):

K - V

Y

_2z2

2 2

cosh r(a3.+ c) cosh r(x3 +

Kd - v2d + v

- (Y.o(KR)! - i Jo (rR)},

2

co' 4(pi + v2)

+ E cos p. (x3 + c) cos p. (a, + ci Ko (p.R)

i' i 3 i

1=1 do? + dv2 - v

i (27.34)

where pi are the positive solutions of:

pi tan' (yid) + v 12.351

Although these two representations are equivalent, one of the two may have preference for numerical computations, depending on the values of the variables. In general, equation (2.34) is the most convenient representation for calculations, but when R=0 the value of Ko becomes infinite, and therefore equation (2.33)' must be used when R is small or zero.

The unknown source strength function 0 must be determined

such, that the boundary condition on the ship's surface S is fulfilled:

1/2a.(x x x3)

3 l' 2' 3

a

4. 1

ff

a.(al,

a2, a3),T1-1-y.(x1, x2, x3, al, a2 a3)d3" 4x S

(2.36)

To solve equation (2.36) numerically, the surface S is subdi= vided into a number of finite, plane elements. The boundary

condition will be applied in one control point on each element. being the centre of the element. In each. of the control points,

-26-=

= c)

n =

the potential is given by

tPi = G

jm 4n

n=1

hthere':.Yjm is the value of p. in the

mth

control points 3

(finis the source strength on the

nth

element.

ynm is the contribution of the

nth

element to the Green's

function in the

mth

control point.

hSn

is

the area of the

nth

element.

Integral equation (2.36) now reduces to a set Of N algebraic

equations for the N unknown source strengths:

-11aim E y AS n 7

jn, an n = _im = 1

n=1 a

in which

nim

is the value of

n

in the centre of

theath

element',

Equation '('2. 38) can be written simply as:

a. .A = rt.

nm 3m

For each of the modes j = 1, 2, 7 this formula represents

a set of N complex linear equations in the N unknown source

strengths. The unknown source strengths can be found by inver-sion of the complex valued

N x N,

matrix A=

-1 aoi. 1=A .,n

)n

rim jm' 12.3711 *2.3811 1(2.39) (2.,(011'

The complex matrix A can be transferred into a real

2N x 2NI

ma-trix, which can be inverted with standard numerical routines. In general, the Green's function ynm may be computed with sufficient accuracy as if the source strength is concentrated in

the centre of the element. There are, however, two notable ex-ceptions to this rule. First, a difficulty is encountered when

evaluating y and

lel

when n = m. In that case, ynm has a

singularity of the type 1/r, as can be seen from (2.33). This singularity can be removed by spreading' the source uniformly'

ti, in 1 + . = 1 ,

over the facet. The total Green's function can be split up in a regular and a singular part:

1

Y = Yi

Where y' is the regular part, which can be computed without dif-ficulties when r = 0. The potential and its normal derivative due to the singular part can be found as follows.

Consider the potential LP' at P in Figure 2.2 due to a source of unit strength spread over the plane area AS. The normal from P

meets AS at Q, where the distance PQ = e. The distance from Q to the perimeter of AS is f (0), a function of the angular position G. Then:

= 1

ffAS dS 47 r =

ffido

ff(o) _ldi 4a 0 /;24-712-0 27 r

[2

f2,w)]k --eld0 (2.41) 47 J 0 Figure 2.2.

In the limit as den° it is found that:

1 27

f f(0)cle

47 0

For a circular facet, this results in:

where = _Viiff1 {1n(q+42-+1) + q in 14-beci

+1I

in which q is the aspect ratio of the rectangle.

28-(2.42)

(2.44)

tf)' = (2.43)

For the rectangular element:

-+

=

Figure 2.3 shows the value of Q1 on a base of the aspect ratio. The normal derivative of the potential in P becomes:

= 1

rr

a 1 Aq an 4w JJ an

T

--AS aw' ae 271 r 1 1 {eLe`+f'(0)J- -1}de-2 47 0

Here the limit for e-4-0 is

It is remarkable that, unlike the potential itself, the value of the normal derivative is independent of the shape of the surface element.

A second case which gives tog

some trouble is when a surface element is situated close to

the sea bottom, for instance the bottom elements of a ship with small underkeel

clearance. For this case, 095

P in Figure 2.2 represents the point where the potential is evaluated, while Q is the centre of the reflected

sur-face element. The contribu- 0.90

10

tion of the reflected element to the potential is given by

(2.41), in which e

is now

a constant, being the distance from the element to its re-flection. For a circular

element f(0) = a = radius of the element and consequently (2.41) yields As 4we atr an 2 a2 ' + 1 e2 20 20 (2.45) (2.46) (2.47) 30

Figure 2.3. Correction factor for rectangular facets as a function of the aspect ratio

This result means, that when the source strength is taken as con-centrated in the centre of the element, a correction has to be applied, which amounts to:

"

for a circular element.

pt In the same way a correction factor is found for

being: 2 -3

a

a2

v "

+ 1 + a

The factors gl2 and S23 are shown in Figure 2.4 on a basis of From this Figure it appears, that the influence becomes signifi-cant when ! = 0.25, or, when the keel clearance equals the ele-ment size. In case of non-circular eleele-ments the integration of equations (2.41) and (2.45) has to be done in a numerical way. When, after solving the set of equations (2.38) the source strengths are known, the wave exciting forces and added mass and damping coefficients are found using equations (2.20) and (2.30).

Finally, the motions of the free floating ship can be cal-culated from the equations of motion in the frequency domain

(2.31). 1

09

08

Figure 2.4. Correction factor for image facets.

07 -30-as 10 a (2.49) 15 n2 = (2.48) from

2.4. The potential for a ship along a quay

So far the hydrodynamics of a ship at zero forward speed in shallow, but otherwise unrestricted water have been considered. The method of solving the velocity potential can, however, easily

be extended to the case of a ship near a vertical wall. It is ob-vious that the influence of a quay is present in a lot of mooring situations.

Ih

WAGE _\

I - -

-I

Figure 2.5. Definition sketch.

Consider a vertical quay near the ship, defined as x2 = h, as shown in Figure 2.5. For sake of simplicity the wall is chosen parallel to the ship, which covers most practical cases, although a wall which makes an angle with the GX/-axis, can be tackled in the same way.

The velocity potential for this case can be found by the method of images. The ship is reflected with respect to the wall.

For the numerical calculation this means that the number of

sur-face elements is doubled, while the number of unknown source strengths remains the same, due to the symmetry of the flow field. To the Green's function of each facet the Green's function of its reflection has to be added:

1 v

oinlynm(xl, x2,

x3,

al, a2, 33)+ 4)jm = 411 n'l

ynm(xl, x2, x3, al, 2h-a2, a3)ii\Sn

kPj = 1 7

m _(2.50)

S.:11/././IWW4 QUAY

.1

The incoming wave will also be disturbed by the wall. The problem of Progressing waves in an infinite ocean bounded on one side by a vertical wall when the wave crests at infinity may make any angle with the shore line has been treated by Stoker [2-19] .

It should be emphasized that he was not able to decide whether the waves are reflected back to infinity from the shore, and if

so, to what extent. From observations, however, it is known that

a wave is reflected by a vertical wall, and by adding the image

in the plane x2 = h to the incoming progressive wave, a wave system is obtained which fulfils the boundary condition at the

quay (see Lamb [2-17]):

cosh

_k(x3

+c)

1

(iz(xlcos,a +

x2sin a)'

4)(7 v 0 cosh xel

Feiz(x1cos a

+ 2h, sin

a -

xsin

a)1

In most practical situations a quay forms part of a harbour

basin and therefore the undisturbed wave will in general be a

much more complicated system of progressive and standing waves.

Al

(2-S11

-32-CHAPTER 3

WAVE EXCITED FORCES AND HYDRODYNAMIC COEFFICIENTS

3.1. Numerical calculations with the three-dimensional source

technique

With the three-dimensional source technique described in 2.3, the wave excited forces, hydrodynamic coefficients and free floating ship motions were computed numerically for a large

tanker of the 200,000 tdw class. For these computations a FORTRAN

program was used on a Control Data 6600 computer.

Particulars of the ship are listed in Table 3.1 while a small scale body plan is given in Figure 3.1.

TABLE 3.1.

In the calculations, the ship had to be represented by means of a composition of flat surface elements. For a ship shaped body, it is rather obvious to use triangular and quadrangular Main dimensions 200,000 tdw tanker.

Length between perpendiculars 310.- m

Breadth 47.20 m

Draft 18.90 m

Volume of displacement 235,000 m3

Block coefficient 0.85

Midship section coefficient 0.995

Prismatic coefficient 0.855

Distance of centre of gravity to midship section 6.61

Height of centre of gravity 13.32

Metacentric height 5.78

Longitudinal radius of gyration 77.50

elements. For a proper choice of the number of surface elements the following considerations must be kept in mind.

Figure 3.1. Small scale body plan.

First, as is quite clear, the accuracy of the results will increase with increasing number of elements, since smaller elements can describe the curved geometry of the hull better,

and also because a finer distribution of sources will approximate the pressure gradient along the hull more accurately.

The accuracy of computation

will

depend also on the

frequen-cy considered: for short waves (or high frequenfrequen-cy motions) more elements are required than for long waves (or low frequency mo-tions). Although it is difficult to predict a minimum acceptable ratio of wave length to element size, it may be expected that appreciable errors will occur when the wave length becomes small-er than five times the length of a surface element. Due to the size of the ship, the range of frequencies which are of practical interest, is confined to Cl<w<0.8 rad sec-1. Therefore, the length of an element should not be larger than 17 metres.

Further, it is desirable to prevent large variations in the size of elements, and to keep the aspect ratio as close to 1

(which means a square shape) as possible.

Finally, the computing time increases progressively with

the number of elements. For this reason, the number of elements should not exceed 200.

Between these conflicting requirements a compromise has

been sought applying a number of 160 elements. The subdivision

in facets of the ship's hull is shown in Figure 3.2% As can be seen from a comparison with Figure 3.1 the shape of the sections has been simplified considerably. The bilge radius was neglected

at all, and the rudder was deleted.

Computations of hydrodynamic coefficients and wave forces, and motions were carried out for a water depth/draft ratio 6

amounting to 1.2, frequencies ranging from 0.07 to 8.0 rad.sec-j

and wave directions of 180, 225 and 270 degrees. Also a series of

calculations was performed with a wall parallel to the ship at

a distance of 16.50 m to the ship's side, representing a solid

,jetty.

The results are presented and discussed in Section, 3.3.

3.2. Experimental verification

In order to verify the numerical results obtained with the three-dimensional source technique, a series of model experiments has been carried out in the Shallow Water Laboratory of the Netherlands Ship Model Basin. This experimental basin measures 210 metres in length and 15.75 metres in width, while the water depth is variable with a maximum of 1.00 metres. At one end of the basin a paddle type wave maker is installed, capable to generate regular as well as irregular waves, while the beach at the other end can be adjusted in height to match the water depth,

The towing carriage, which accommodates the measuring and recording equipment, was positioned halfway the basin for these

tests..

The ship model was made of wood to a linear scale ratio of 1:82.5 according to the lines shown in Figure 3.1. Rudder, pro-peller' and bilge keels were omitted on the model.

The experiments comprised measurements of wave forces and motions, and forced oscillation tests to determine the hydrody-namic coefficients.

The tests were carried out im accordance with

Prouder&

law,

Of similitude.

-ss-2 3 4 6 6 19 16 16 17 18 19

Figure 3.2. Subdivision in surface elements of a 200,000 TDW

tanker hull.

4

The wave loads and motions were measured in regular waves. The undisturbed waves were recorded before the start of the tests at the position of the centre of gravity of the model by means of a wave probe of the resistance type.

During the tests the measured signals were recorded simul-taneously and continuously on ultra violet paper chart and on analogue magnetic tape. The evaluation of the results was carried out on a computer. To this end, the analogue records were con-verted into digital records by taking samples at time increments corresponding to 0.28 seconds on prototype scale.

During the wave force measurements the ship model was mounted to a 6-component dynamometer, which was mounted rigidly to the towing carriage as shown in Figure 3.3.

Figure 3.3. 6-component measuring rig.

The dynamometer essentially consisted of two large frames, con-nected by means of 6 force transducers of the strain-gauge type. Different angles of wave attack were established by rotating

the ship model around its centre of gravity.

The signals of the 6 measured forces were transformed on the computer into 3 forces and 3 moments in the space-fixed system of axes as defined in Figure 2.1. From those new signals, the

ampli-tudes of the first harmonic component were obtained. Also a har-monic analysis was performed on the signal of the incident wave, and the transfer functions were found by dividing the first har-monic components of wave forces and moments by that of the wave

elevation.

During the measurements of ship motions, the ship model was kept in its position by means of two long steel rods, which acted

as soft springs, soft enough not to influence the motions in the

range of wave frequencies applied.

Inertial and stability properties of the model were adjusted according to the values stated in Table 3.1.

The rotative motions were measured by means of gyroscopes, translations by means of a pantograph, which instrument trans-forms a translatory motion into a rotation of a potentiometer. A sketch of the test set-up is given in Figure 3.4.

TELESCOPE UNITS

FLEXIBLE RODS

-38-PANTOGRAPH

Figure 3.4. Test set-up for motion measurements.

Again, a harmonic analysis was performed on the measured

signals, and only first harmonics were used for further analysis.

To determine the hydrodynamic coefficients (added mass and

damping) the ship model was forced to oscillate with a prescribed

excitator, which main component is a Scotch-yoke mechanism. This excitator is equipped with two legs, spaced 1.00 metre apart, which can both perform a harmonically oscillating translatory motion either in phase, or with a certain phase difference. The

motions of both legs are measured by means of potentiometers. An

electronic control system is used to keep the number of revolu-tions of the driving motor constant, which is necessary to a-chieve a pure harmonic motion.

A six component force transducer was mounted between the model and the legs of the model excitator. A sketch of this transducer is shown in Figure 3.5.

STRAIN GAUGES

Figure 3.5. 6-component transducer for oscillation tests.

Six forces are measured by means of strain gauge transducers. The test set-ups for the oscillations in the various modes are depicted in Figures 3.6 through 3.9.

The model was always placed in the basin in such a position

that the generated waves traveled as much as possible in the

longitudinal direction, to minimize the influence of reflected waves from the basin's side walls.

The rotational motions yaw and pitch were achieved by

ad-justing a phase difference of 180 degrees between the motion of

the two excitator legs. The pivots were placed in such a way,

CARRIAGE

6 COMPONEN MIME TRANSDUCER

6 COMPONENT FORCE AAAAA DUCER

CARRIAGE

I

ill..

OSCILLATOR LEG

Ili OSCILLATOR LEG

-40-Figure 3.6. Test set-up for surge.

Figure 3.7. Test set-up for sway and yaw.

Figure 3.8. Test set-up for heave and pitch.

Figure 3.9. Test set-up for roll.

L

CARRIAGE

6 COMPONENT FORCE TRANSDUCER

CARRIAGE

HINGE

COMPONEPiT FORCE TRANSDUCER OSCILLATOR LEO

4i I"OSCILLATOR LEO

1104

Lea=

of gravity. During the tests in roll direction, the model was attached to the carriage by means of two hinges at the height of the centre of gravity.

The records of the six measured forces were, after

digi-tation, transferred into three forces and three moments in the

space fixed coordinate system GX1X2X3. Subsequently, a harmonic

analysis was performed on these signals as well as on the motion signal. Added mass and damping could then be obtained using the equations of motion in the frequency-domain (2.31).:.

(X cos r,.)/C.3 -M akj kj (3.1) bkj X sin 5_10 wCi 0.2)

where is the phase lag between the force Xk and the motion

Sl

4

The hydrostatic coefficients Cki were determined

by

means of

static measurements.

The moments of inertia in air of the model with measuring equip-ment (Mkj) were measured by means of oscillation tests in air.

3.3. Discussion of the results

The computed transfer curves of the wave excited Moments are presented in Figures 3.10 through 3.12, together with the experimental results.

In general, the agreement is good. The only notable excep-tions are the surge force and pitch and yaw moments in beam waves

(see Figure 3.12), which originate from the asymmetry in the hull shape. The discrepancy between theory and experiments can be

attributed in these cases to the simplified representation of the

bull

in the computations.

The experiments were carried out in waves with amplitudes corresponding to 1.5 metres. At frequencies w' = 2.25 and 3.38

three wave amplitudes were applied, ranging from 1 to 2.5

me-tres. In most cases the results obtained for these three wave

-5000

2 500

'AS

THEORETICAL EXPERIMENTAL

Figure 3.10. 'Transfer functions of wave excited forces and

moments; 6 = 1.2, a = 180 degrees.

=43

A 20.000 10.000 1000 000 500 000 25 50 25 50 25 50

5.000 2 500 0 F 0 54.A 25 100.000 50 000 1000.000 -44-1000,000 500.000

Figure 3.11. Transfer functions of wave excited forces and

moments; ei = 1.2, a = 225 degrees.

1

.

ilk

50 25 50 THEORETICAL EXPERIMENTAL 0 25 50 2.5 50 J. 25 50 10.000 5000

('0

X LA 500000 :81

a

120. ja

NO

X LA 1.000 EXPERIMENTAL THEORETICAL o a SO 200.000 100.000

to

d i o /"C"....-"E""4:e...__21...0...c.. 1 _ 0 0 i e m a o m ya 0 a Or_13 a Wirt -2-5 50

Figure 3.12. Transfer functions of wave excited forces and

moments; 6 = 1.2, a = 270 degrees. w1.1. 25: 50 25 50' zs 0 200000 100.000 .0 20000 10.000 17k-9 500 0 0 0 25 50 0 0 0 25

amplitudes are very close, from which it may be concluded that the wave loads are linear up to a wave amplitude of at least 2.5 metres.

The added mass and damping coefficients, as obtained from computations and measurements, are presented in the Figures 3.13

through 3.24 in a non-dimensional way, as defined in Table 3.2. In general, the agreement found for the main hydrodynamic coefficients is satisfactory. Relatively small differences as

they occur in the surge, heave and yaw modes may be explained by the simplification of the ship's geometry. More serious are the discrepancies found in the added mass moments of inertia in pitch and roll mode, and in the damping coefficient in roll. A reason for the higher measured damping coefficient in roll may be sought in viscous effects, but no reasonable explanation can be given for the differences between measured and computed values of a'44 and a'55' although it should be remarked that it is rather diffi-cult to obtain reliable experimental values for these coeffi-cients in the low frequency range. To find the total inertia coefficient, the hydrostatic restoring moment has to be subtract-ed from the total in-phase measursubtract-ed moment, according to equation

(3.1). The remainder is then divided by a very small value, w2. Thus, a relatively small error in the measured moment may result in a large proportional error in the inertia coefficient.

There are a few reasons why more discrepancy may be expect-ed in the coupling coefficients. First, the computexpect-ed values will be less accurate, since coupling effects are influenced largely by details in the geometry, and the simplification in the shape is therefore relatively more important. Second, the experimental error will be larger, since the experimental coefficients are found from small measured forces.

For the coupling coefficients, there are two sets of experi-mental data, as found from tests in different modes. Since from the linear theory it follows that

akj = ajk

and b = b.

kj jk

-46-the difference in -46-the two sets of data gives an indication of the experimental error.

Fortunately, the hydrodynamic forces due to the coupling effects are rather small and consequently even a large

inaccura-cy in a computed coupling coefficient will not have much impact

on the computed ship motions. This may be illustrated with the

following example.

The most important discrepancy between a measured and com-puted coupling coefficient is found in case of the coupling between sway and yaw. Supposing that the measured values are the most reliable, it must be concluded from Figure 3.22 that

there is a large error in the computed mass coupling coefficient

of yaw into sway at small frequencies. The resulting error,

however, in the total hydrodynamic restoring force in the sway

mode, will not exceed 7 percent in this extreme case.

The motion amplitudes as used in the tests are indicated

in the Figures. In the sway and heave modes three amplitudes

have been applied, to check the linearity. From the results it

appears that the hydrodynamic coefficients are linearly depending on the motion amplitude in the range of amplitudes tested. In the sway mode the maximum amplitude applied corresponded to 2.475 metres, in the heave mode to 1.24 metres which is quite large

in comparison with the available underkeel clearance.

The motion transfer curves as found from theory and model tests are presented in Figure 3.25. The agreement between experi-ments and theory is good, except for the roll motion at reso-nance, where viscous damping plays an important role. Since the

roll motion at resonance is overestimated by the potential theo-ry, small humps occur at this frequency in the computed sway and

yaw transfer curves in beam seas, due to the coupling terms.

The motion measurements were performed in waves with ampli-tudes ranging from 0.8 to 1.9 metres prototype scale. At three

frequencies, (w' = 1.7, 2.25 and 2.8) two wave amplitudes were

applied. Again, the results demonstrate linearity under the circumstances considered.

Resuming the results obtained, it may be concluded that the numerical results obtained with the three-dimensional source

technique for hydrodynamic phenomena in shallow water show a

0 2 _-02 2 2 cafr-g w THEORETICAL EXPERIMENTAL ci 0 825 m 4 CuCu Cu 4 2 EXPERIMENTAL-o C2 0 825 m C 2 1 650 m 0 2.475m THEORETICAL 6

Figure 3.13. Added mass and damping

Figure 3.14. Added mass and damping

coefficient in surge; 6 = 1.2. coefficient in sway; 5 = 1.2. 2 CO IT= 0 2 4 0 0 0 0 6 2

10 5 oo 10 THEORETICAL EXPERIMENTAL 2 corE 002 001 oo 0 0010 0 000 0 THEORETICAL 0 EXPERIMENTAL 004 rad. 0 2 c 3 0 413 m C3 0 825 m 1 238 m

Figure 3.15. Added mass and damping

Figure 3.16. Added mass moment of

coefficient in heave;

6 = 1.2.

inertia and damping coefficient in roll;

IS = 1.2. 0 o Q 0 0_11 4 6 2 0 0 4 a 2 4 4

0.2

A

dl 02 01 oo 04 041 2 4 w11-: THEORETICAL o EXPERIMENTAL C6 001 ,ad 0 0 fr--g

Figure 3.17. Added mass moment of

Figure 3.18. Added mass moment of

inertia and damping coefficient in

inertia and damping coefficient in

pitch; ô = 1.2. yaw; d = 1.2. 0 THEORETICAL EXPERIMENTAL c5 0.01 rad 2 0 C' 2 4 9

0

r

THEORETICAL

EXPERIMENTAL: FROM HEAVE TEST FROM SURGE TEST

02 0 _{-0 2}

2

0

THEORETICAL

EXPERIMENTAL: FROM SURGE TEST FROM PITCH TEST

0 o 0 I 0 o 1

Figure 3.19. Coupling coefficient

Figure 3.20. Coupling coefficients

of surge into heave;

6 = 1.2.

of surge into pitch; 6 = 1.2,

.11 0 1 0 0 2 6 6 02 0 4 6 2 0 0

N.H

0

02

EXPERIMENTAL:

FROM SWAY TEST FROM YAW TEST

THEORETICAL .Zr , ea 0 e

°

8 .

195.1

0 I 0 THEORETICAL

EXPERIMENTAL: _{FROM SWAY TEST FROM ROLL TEST}

Figure 3.21. Coupling coefficients

Figure 3.22. Coupling coefficients

of sway into roll;

= 1.2.

of sway into yaw;

E 1.2. -o loci 6 /T-a 0050 _{-0 025} 0 - 0 0500 6 2 9 4

02 0 1 0 o 0 0 0.004 oo 0002 w

Figure 3.23. Coupling coefficients

Figure 3.24. Coupling coefficients

of heave into pitch;

6 = 1.2.

of roll into yaw;

6 = 1.2. 0 0 0 4 a w 2 4 6 w THEORETICAL EXPERIMENTAL

FROM HEAVE TEST FROM PITCH TEST

0.004

THEORETICAL EXPERIMENTAL FROM YAW TEST

0002 0 0 0

-0 0 r

rather rough subdivision of the ship's hull into 160 surface elements is sufficient for a prediction of wave loads, hydrody-namic coefficients and motions in the frequency range which is of practical interest (up to a frequency w' = 4.5).

In section 2.3 it has been stated, that the strip theory will yield wrong results in case of shallow water, in particular

for the lateral modes of motion. How serious the deficiency of the strip theory is, may be deduced from a paper by Flagg and Newman [3-1]. In this paper the authors present data on the

two-dimensional sway added mass coefficients for rectangular profiles in shallow water, computed with a rigid free-surface condition, which means that their results are solely valid for the case w = 0.For the subject ship, with 6 = 1.2, one arrives at a sway coefficient a2 = 11 on a basis of their data, while Figure 3.14 indicates that at zero frequency the real added mass coefficient will lie in between 3 and 4.

TABLE 3.2.

Definition of non-dimensional added mass and damping coefficients..

added mass damping mode

, =

!N

J bk. _

-k= 3

k = j k = j

k =1,

= = = j 2 3 = 3 pVig/L b k = j = 4 kj j . lit k = j k = j = = 5 6 ._, pVI2 fo/L2/4/L k = 4, j = 6 5 4 6 -ak j ,, KJ bk. -k = 1, k = 2,

k = 2,

j j j = = = pVL

pVLAWE

k = 3, j = ₅ a"

01 0 10 5 2 a Theory E xperrment 180° 225° 270° 0 0 25 w hi

\

A Surge Pitch 0 a \ 50 20 5.0 25 1 1 A A' A -6-0 Wk/T-g Yaw

functions;

6 = 1.2.

1 1 1 1 1 E

Figure 3.25. Motion transfer

50 25 25 w frA 240 1.0 0 Sway

\ \

\ \

Heave

r

/a

_\

\

\

3.4. The influence of the water depth on added mass and damping

To investigate the qualitative influence of the water depth on hydrodynamic coefficients the oscillation tests in the sway and heave mode were carried out for in total four values of the water depth to draft ratio, 1.05, 1.1, 1.2 and 2.0. The added mass and damping coefficients for sway and heave are shown in

Figures 3.26 and 3.27.

From Figure 3.26 it appears that the added mass in the sway mode increases with decreasing keel clearance in the low frequency range, while the reverse is the case at high frequen-cies. The slope of the curve increases considerably with de-creasing water depth, while the hump shifts towards lower fre-quencies. The damping in the sway mode is also higher in shallow-er watshallow-er, but at high frequencies the curves approach each othshallow-er asymptotically.

Although not tested, it may be expected that the coeffi-cients in the other horizontal modes, surge and yaw, will show a similar picture.

The shape of the curves of added mass in the heave mode remain more or less the same, as can be seen from Figure 3.27. The curve shifts upwards with decreasing water depth. Also the heave damping increases with decreasing water depth. This result is in accordance with the data presented by Kim [3-2].

It is expected that the same trends will occur in the other vertical modes of motion, roll and pitch.

In general, it may be concluded that from the data present-ed in this section it appears that the influence of the water depth on the added mass and damping is extremely important. More-over the data show that the frequency dependency of the coeffi-cients is obvious especially in very shallow water.

3.5. The influence of a quay parallel to the ship on added mass and damping

A frequently occurring mooring situation is with the ship against a quay or solid jetty. Therefore the influence of a vertical wall parallel to the ship's centreline on the added mass and damping has been investigated both by means of calculations

4 ----2\\

2

--...

6 r.

2.0

6 :12

5 7 1 1 5 7

1 05

0

2 ₆

Figure 3.26. Added mass and damping coefficient in sway for different water depths.

w

4-/

/

/

/

10.0

5.0

II Ic 2

758-

----'1/4%

---4

W

-I6

2.0

6

1.2

6 1.1' :-.

1.05

(Id

11-Fijure

3.27. Added mass and damping Coefficient in

heave for different water depths.

"L.

I

=

and experiments. Only the sway and heave motions have been

con-sidered, under the assumption that they are representative for

the horizontal and vertical modes of motion respectively. The model tests and calculations were carried out for a

water depth to draft ratio of 1.2. The calculations were

per-formed in accordance with the method described in section 2.4,

while the same schematization of the tanker was used as

de-scribed in section 3.1.

Figures 3.28 and 3.29 show a comparison of calculated and measured values for the case that the distance between ship and

quay is 16.50 metres (0.356). The agreement is good. Figures 3.30

through 3.33 show the experimental results for various distances between ship and quay.

The presence of the quay has quite a remarkable influence

on the hydrodynamic coefficients of the ship. As can be seen from Figures 3.30 and 3.32 the effect of the quay on the added

mass disappears at very low and very high frequencies. But in

the range of frequencies which is of interest for ship motions in waves, the added mass is influenced significantly. Most interesting features are the occurrence of sharp peaks and negative added mass values. From observation during the tests it appeared that the peak values may be associated with the occur-rence of standing waves between quay and ship with nodal lines perpendicular to the quay.

A physical interpretation of negative "added mass" is diffi-cult. However, it should be kept in mind that the quantity under consideration is just the in-phase component of the fluid reac-tive force, and the denomination "added mass" originates only

from the practice to combine this component with the inertia term in the equations of motion. It could also be combined with the displacement term, and called then "hydrodynamic spring coeffi-cient". Anyway it is obvious that in the range of frequencies where the added mass value is negative, the water between quay

and ship acts like a spring. Consequently, a ship floating freely in waves near a quay, may experience a resonant motion at two frequencies in the sway mode. In the heave mode the number of resonant frequencies may even be greater, due to the presence of more peaks in the curves of added mass and due to the pres-ence of a hydrostatic spring constant.

10

20

10

-60-

wfr--9

THEORETICAL

0

EXPERIMENTAL

0

0

0

-6

0

2 4 ₆

Figure 3.28. Added mass and damping _{coefficient in sway;}

distance between ship and quay 16.50 m, 6 = 1.2.

4

20

0

2°

40

20

W

Figure 3.29. Added mass and damping coefficient in heave;

distance between ship and quay 16.50 m, 6 = 1.2.

0

THEORETICAL

0

0

0

0

o

EXPERIMENTAL

2 4

0

₀

₀

r

0

2 4.

6

5 0

2 5

-2.5

- 5 0

0

2

DISTANCE BETWEEN SHIP'S SIDE

AND QUAY:

-62-825 m

ie. 50 m

25.75 m

33 00 m

41 25 m

4 C/) CO

VT=

Figure 3.30. Measured added mass in sway as a function of

20

15

;I 0

5

2

w

Figure 3.31. Measured damping coefficient in sway as a function of distance between ship and quay;. 6 =

;

DISTANCE BETWEEN

AND QUAY:.

SHIP'S SIDE

.8.25 m

11,6.50

M

24.75m

33.00 m

41. 25 m

co

rm

1.---

--[

1 /

/ I I, II 1 I I [ 11 [ [ iiI I

i

1 \A

i

\

t % . ,...,... ... --... ... .. ---....:---.... ..._ ....

--

... ... 1.2.

2

10

10

20o

DISTANCE BETWEEN SHIP'S SIDE

AND QUAY

Figure 3.32. Measured added mass in heave as a function of distance between ship and quay; 6 = 1.2.

-64-8 25 m

16.50 m

24.75 m

33 00 m

41.25 m

corn

en

0

-in

40

30

2 .1 2 w

17-g 4

Figure 3.33. Measured 'damping coefficient in heave de function of distance between ship and quay; AS =

N _

DISTANCE BETWEEN

AND QUAY:

51-11P'S SIDE

825m,

il 6.50 m

25. 75. m

33.00 mi

41.25 nn',

CO tyl .---==-1

--,--

-ft

A 1

fil(11\

I

AI

iv :

1 I

/

I. IIV ! 1 I 1 \1 1 I

/

/

\\ \

\...2.>

... .. ... ... ... \ I a 1.2.

From Figures 3.31 and 3.33 it appears that the hydrodynamic damping increases considerably near a quay.

Finally it is remarked that the results presented here for heave show some resemblance with the results found by Kalkwijk

for a ship, oscillating in a navigation lock [3-3], and the

re-sults found by Lee et. al. [3-4] for heaving catamarans. It is

evident that the problem discussed here of a heaving ship near a quay is theoretically equivalent to the heaving of a twin hull ship.

CHAPTER 4

EQUATIONS OF MOTION IN THE TIME DOMAIN

4.1. Potential theory description for flow due to arbitrary ship motions

As has been outlined in the Introduction, it is desirable to formulate a set of equations of motions which relate instan-taneous values of forces and motions. The obvious problem is then to describe the reactive forces of the fluid due to arbitrarily in time varying ship motions. To solve this problem, the approach

of Cummins [4-1] is followed.

Cummins describes an arbitrary motion as a succession of small impulsive displacements. His basic assumption is that at any time the total fluid reactive force is the sum of the reac-tions to the individual impulsive displacements, each reaction being calculated with an appropriate time lag from the instant of the corresponding impulsive motion.

Consider a ship, floating at rest in still water. The space fixed system of axes is defined as in section 2.1. Suppose that the ship is given an impulsive displacement in the j-th mode, amounting to A x.. This displacement is achieved by moving the ship at a constant velocity Vj for a small period of time At as shown in Figure 4.1. At vi t =to t toAt Figure 4.1. so that Ax. rf V. Mt

During the impulse the flow can be characterized by a velociLy (4 .1 )