Motions of large structures in waves at zero froude numbers

DET NORSKE VERITAS

Cht&sffication and

Registry of Shipping

Publication No. 90

- Sept. 1975

MOTIONS OF LARGE STRUCTURES IN

_WAVES

AT ZERO FROUDE NUMBERS

OM. Faltinsen and F.C. Michelsen

Motions of Large Structures in

_{Waves at}

Zero Froude Number°

0. M. Faltinsen* F. C. Michelsent

1 INTRODUCTION

The offshore activities in the North Sea have resulted in many offshore structures that are rather large in volume and not amenable to two-dimensional analysis. As examples may be mentioned the 'Ekofisk oil_storage

tank and the 'Condeep' platform. For those large

volume three-dimetisional forms one cannot rely on the

Morison-type equation (Morison et al. (1)) or strip theory to calculate hydrodynamic wave forces and

mo-ments. One has to resort to numerical schemes based_on

three-dimensional source techniques or the use of Green's

theorem. This has been done by Lebreton and Cormault (2), Garrison and Seetharama Rao (3), Milgram and Halkyard 4) and van Oortmersen (5).

In this paper the source technique has been applied

to a floating object in regular waves and the added mass

and damping coefficients, the wave exciting force and moment and the motions in sx degrees of freedom and

the pressure distribution have been calculated. The hori-zontal drift force and moment have also been evaluated.

Notation

(Additional nomenclature used in the Appendices are defined only as they appear.)

Added-mass coefficients(j, k = 1, 6).

Area of water plane. Damping coefficients. Restoring coefficients. Element of area. See equation (37)

Exciting force and moment components

(j = 1, 6).

Green's function. See equations (23) and

(24)

Gravitational acceleration.

Water depth.

Moment of inertia injth mode.

Product of inertia.

Bessel function of first kind of zero order.

Modified Bessel function of second kind

of zero order. Wave number. Mass of the body.

Unit normal into the fluid.

See equations (19) and (20); i = 1,6

Displacements,(j = 1,2...6refertosurge,

sway, heave, roll, pitch and yaw, respect-ively; see Fig. I).

Source densities, / = 1,7. See equation (22)

The MS. of this paper was acceptedfor publication on 5th February 1974.

* Senior Research Engineer, Der norske Veritas, Postboks 6060, Etterstad, Oslo 6, Norway.

t Professor, The Technical Uhiversizy of Norway, Trondht'im, Norway. References are given in Appendix 4.

R

R'

r; 0, z, Cylindrical polar co-ordinates with

x = r cos 0 and y = r sin .0.

)Z

+(y

r Position vector.

S _{Average wetted surface of the body.}

S _{Vertical circular cylinder of large radius.}

t Time variable.

V _{Displaced volume of water.}

V,, V9, V.. _{Fluid velocity components in cylindrical}

polar co-ordinate system.

Co-ordinate system as defined in Fig. 1.

Bessel function of second kind of zero order.

z-co-ordinate of centre of buoyancy. z-co-ordinate of centre of gravity.

Direction of propagation of incidentwaves

= 0 means propagation along the

positive x-axis).

Wave amplitude of incident waves.

Wavelength..

= (w2/g) =ktanh kh.

Co-ordinates of a point on the surface of

the object.

Mass density of water. See equation (37) Velocity potential. See equation (11), / = 0,7 Circular frequency of wave.

= .,J[(x

_+(v

)2

_+Jz

_{_()21.}

= J[(x )

±(y )2+(

+2h +

2. THE EQUATIONS OF MOTION

A right-handed co-ordinate system (x, y, z) fixed with respect to the mean position of the body is used, with

positive z vertically upwards through the centre of gravity of the body and the origin in the plane of the undisturbed

free surface. The body is assumed to have the xzplane

as aplane of symmetry Let the translatory displacements

in the x, y and z directions with respect to the origin be

2and ?3 respectively, so that i is the surge, 12 is the

sway and i is the heave displacement. Furthermore, let

the angular displacement of the rotational motion about

the x, y and z axes be

, i and 6' respectively, so that

)74 is the roll, is the. pitch and ₆ is the yaw angle. The

co-ordinate system and the translatory and angular displacements conventions are shown for the case of a

ship in Fig. 1.

The linear frequency domain equations of motion in

regular incident waves of small amplitude can for a body

symmetric with respect to the xz plane and under the assumption that responses are linear and harmonic be written as folloWs:

p

(M + A11)1 + B111 + A133 + B13i73 + (MZ0 + A15)j5 + B15,z5

_{= F1 e"°'}

AJk F(0)

F e°t

G g h Ii 'jk Jo K0 k M Ii ni 1j x, y, z Y0 ZB /3 2 V p q5(0) QJj a)

/I3?7j + B311 + (M + A33)3 + B3i3 + C.33ij3 +A,35 + B35 + C3

= Fe"°'

(2) (MZG + A51)i1+ B511 + + B53i3 +C53?13 + (1 + A55)5 + B55i5 + C55?75 = F5e_wz .(3) (M + A22)2 + B22.2 + (A24 - MZG)4 +B24i4 ± A26?16 + B26ii6 = F2 e°'

(A42 - MZGY2 + B42j2 + (A44 + 14)774

+B44j4 + C44?74 + (A46 - '46) .

+B466 = F4e_i(0t A622 + B622 ± (A64 - '46) 774 + B644

+(A66 ± '6)'16 + B66i6 = F6e°t

r13

ne _y

-tional acceleration. A,, is the water plane area. V is the

displaced volume of water and Z8 is the z co-ordinate of the centre of buoyancy. The integration is over the water plane area.

The added-mass and damping coefficients and the

exciting force and moment are derived in the next Section.

3 THE HYDRODYNAMIC

BOUNDARY.VALUE PROBLEM

Viscous effects are neglected and we assumethe fluid to

be incompressible. The depth h is finite and constant

and the free surface is infinite in all directions. The motion

of the body and the fluid is assumed to be small so that we can linearize the body boundary condition and the

free surface condition.

The problem can be formulated in terms of potential

flow theory. We assume that steady-state conditions have

been obtained and write the total velocity potential as:

6

4)

= 4e'°t +

4) e_'°3' + 4)i

...(ll)

i=1

where 4) e°' is the velocity potential of the incident

waves, which can be written as 4)

gcosh k(z + h)j(,Cosp+kysjflp_wt)

0

_{- w}

_coshkh

Here is the wave amplitude and /3 the direction of

propagation of incident waves. k is the wave number, which is related to the frequency of the waves by the

dispersion relationship

=ktanhkh

...(13)

g

Further, 4

e1°" is the diffraction potential for the

restrained body and = 1,6,is the contribution to the

velocity potential from the j th mode of motion.

It can be shown that4),,j = 0,7, must satisfy

a2 _a2 a2

__++--'=0inthefluiddomain

...(14)

ax ay ôz

+ g;J =

_onz=0

...(15)

onz=h

Further, 4,,ft,,,j = 1,6,and 4). e°3' satisfr a radiation

condition and the following body boundary conditions

on the average position of the wetted surface of the body

an

a4)0

(18)

an an

Here a/an is the normal derivative in the direction of the outward normal n to the surface of the body.

Further n,,,j = 1,6, is defined by

n = (n1,n2,n3)

and

r x n

(n4,n5,n6) ...(20)

(12)

i, Surge q Heave Pitch

172 Sway Roll 176 Yaw

Fig. 1. Sign convention for translatory and angular

displacements

Here M is the mass of the body, I. the moment of inertia and 1 the product of inertia. 'the inertia terms are with respect to the co-ordinate system shown in Fig. 1. Further ZG is the z-co-ordinate of the centre of

gravity. and are the added-mass and damping

coefficients and F,, are the complex amplitudes of the wave exciting force and moment, with the force and

moment given by the real part of F e"°' (It is understood

that real part is to be taken in expressions involving

e0t.). F1, F2 and F3 refer to the amplitudes of thç surge, sway and heave exciting forces, while F4, F5. and F6 are

the amplitudes of the roll, pitch and yaw exciting moments w is the frequency of the waves and is the same

as the frequency of the response. The dots stand for time derivatives so that ?1k and , are velocity and

acceleration terms.

The body is free-floating so the restoring coefficients follow from hydrostatic and mass considerations. They

are given by /

C33 = pgA .. .(7)

C35 = C53 = _pgjjxds

C pgV(Z8 - ZG) + pg

J

y2 ds

C55 = pgV(Z - ZG) + pg jSx2 ds (10)

Here p is the mass density of the water, g is the

gravita-.(4)

.(6)

where r is the position vector

r=xi+yj+zk

...(21)

It is possible to show that the solution of cfr, (3 = 1,7) can be written as

= J$ Q, ,

') G(x, y,z.; , ,, ) ds . . .(22)

This has been shown by Lamb (6) for the infinite fluid case. The integration in equation (22) is over the average

wetted surface S of the body with (,

, ) being the

co-ordinates of a point on S. Q is the unknownsource

density function and G(x,.y, z; , , ) is the Green's

function for the problem, which can be written in two ways as follows (see Weeháusen and Laitone (7)).

2irIv2 - k21

G(x, y, z:

i 0

₌

_khvh+v

_{2 2}

'

cosh k(z + h)

x cosh k(ç + h) (Y0(kr1) - iJ0(kr1))

+ i p2

v2h

V (Pk(Z +h)) x cos_(Pk( + h)) KO(pkrl) .. .(23) or

G(x,y,z,,) =

+ + 2P V

(p + v) e' cosh p( + h) cosh p(z + h) J0(jir1)dp

x

p sinh ph - v cosh ph 2it(k2 - v2) cosh k( + h) cosh k(z + h)

J0(kr1)

+1

k2hv2h+v

(24)

In equation (23) Pk is the solutions of the equation

Pktan Pk'l + v = 0 . ..(25)

J0 is the Bessel function of the first kind of zero order;

Y0 is the Bessel function of the second kind of zero order; K0 is the modified Bessel function of the second kind of

zero order; and

v=_

...(26)

...(27)

R' _...(28)

MOTIONS OF LARGE STRUCTURES IN WAVES A-U ZERO FROUDE NUMBER

r = J[(x

)2

(y 11)2]' (29)

In equation (24) PV indicates a principal value integraL

For practical purposes equation (23) is used when kr1

_{0,1 and equation (24) when kr1 <0,1. It}

_was

found convenient to rewrite equation (24). This is shown in Appendix 1.

The source densities Q in equation (22) are found by satisfying the body boundary conditions (17) and (18).

This results in the following two-dimensional Fredhoim integral equations of the second kind over the surface S

2irQ(x,y,z) +

ni

an

when] = 1,6

when] = 7

.(30)

This is smi1ar to the infinite fluid problem formulated by Hess and Smith (8). In equation (30) one has to exclude the integration of the source part of the Green's function over _{the immediate neighbourhood of each point}

(,

, ) (x, y, z). on the surface S where the integral is

evaluated. The contribution to the normal derivative from the immediate neighbourhood of (x,y, z) is taken care of by the term - 2itQ .(x, y, z).

Equation (30) is solved' by approximating the body

surface by a large number of plane quadrilateral elements, over each of which the source density is assumed constant.

This transforms the integral equation into a set of linear

algebraic equations in the unknown values of the source

density on the elements. The approach is the same as used by Hess and Smith (8), the only difference being that we have selected the ceñtroid of each quadrilateral

as the point where the Green's function and its derivative are evaluated, while Hess and Smith used the null point,

i.e. the point where the velocity component in the plane of the surface element due to the source distribution of

that element is zero. This difference is not significant and it is not necessarily a more correct refinement to use the

null point instead of the centroid. The formulae for the integrated values of the derivates of the sources over a quadrilateral have been derived by Hess and Smith. We adopted the criterion given by them to determine

when the quadrilateral can be replaced by a source alone.

Otherwise we used the integrated values. The same procedure was followed for the images of the source over

the free surface and the sea bottom. For the other parts of the derivative of the Green's function we assumed constant values over each quadrilateraL The numerical

work can be considerably reduced by taking intoaccount

symmetry properties. If the body has one plane of

symmetry (which is the xz plane) the source density

has to be symmetric about this plane when] _{= 1,3 and 5.}

Further, when ]

2, 4 and 6 the source density is

asymmetric about the same plane. When_{j = 7 we can} split the source density into a symmetric and an asym-metric part. When the body has both the xz-plane and the yz-plane as planes of symmetry, then the source

density will be symmetric about the xz-plane and

asymmetric about the yz-plane for] = 1 and 5. Further, when] = 3 the source density is symmetric, about both

the xz-plane and yz-plane. When.] = 2 and 4 the

source density is symmetric about the yz-plane and asymmetric about the xz-plane. When] = 6 the source density is asymmetric about both the xz-plane and the

yz-plane. Finally, for j = 7 we can split the

source

density into four parts in the same way as mentioned above. Thus, when there is oniy one plane ofsymmetry

it is necessary to only satisl' the integral equation for positive x-values on the body surface S. When both the xz-plane and yz-plane are planes of symmetry it is only necessary to satis& the integral equations for

The integral equations (30) may not always yield a solution. For certain irregular frequencies (see John (9)) the method fails. This problem has been studied in the two-dimensional case by Faltinsen (10). However, as long as the body has no forward speed and there is no current, it is expected that the irregular frequencies will not present any problem. The irregular frequencies are

furthermore expected to lie above the frequency range of

interest. It should be jioted that there exists no irregular

frequencies for totally submerged bodies.

When the source densities Q have been found, the

normalized potentials cb may be obtained from equation (22). The integration procedure is similar as explained in connection with the solution of the integral.equation (30).

We then need a method to integrate sources over a

quadrilateral. This has been shown in Appendix 2.

We can now use Bernoulli's equation to Obtain the

pressure and, by definition, the added mass and damping coefficients AkJ and BkJ are as follows

AkJ = - p Re {Jj k k ds} . . .(31.)

Bkf _pwIm.{SJJnkds} ...(32)

Here Re and Tm mean the real and imaginary part,

respectively. The indices k and j go from 1 to 6.

The wave exciting forces and moments F, i = 1,6, are obtained from 4 and 4

e'0 by using the

linearized Bernoulli's equation to obtain the pressure and integrating this pressure properly over the body surface S.. We can now go back to the equations of

motion (l)(6) and solve for the motion. Having obtained the motion the velocity potential (11) is now determined.

This enables us to find, for instance, the pressure at any

point on the body. This may be used as the dynamic load

in a quasi-static structural response calculation. Further,

we can find the free-surface elevation at any point. The

fluid velocity and acceleration can also be readily

obtained. These may, for instance, be used in Morison equation-type calculations of forces and moments on small objects attached to the main body as appendages. The motions, velocities, acceleration and pressure in irregular seas may now also be described. If we know the wave spectrum for the sea state we may use a linear superposition technique to Obtain the response in an

irregular sea (St. Denis and Pierson (11)).

4

DRIFT FORCES AND MOMENTS

So far we have studied the linear response in regular

waves and neglected terms that are of higher than first ordt of magnitude in wave amplitude. However, in

some cases the higher order terms are

'important,

especially the second order horizontal drift force and

moment. These terms may be used to calculate the mean

drift force and moment on a body in an irregular sea (see for instance Gerritsma et al (12)) and can also be used to calculate slowly varying excitation forces and

moments on a body in an irregular sea (Hsu and Blenkarn (13)). Even ii these forces are small they may cause large

excursions of a free-floating body, since in such a case there are no restoring forces in the horizontal plane.

Further, the frequency of the slowly-vafying forces may

very well lie in the resonance frequency range of an

anchored body with the fatal consequence that the anchor

system fails. Thus, for dimensioning the anchor and dynamic positioning systems the study of drift forces and moments in regular waves is important.

Newman (14) has derived an exact expression for the horizontal drift force and moments in regular, waves.

He assumed infinite water depth, but his expressions can easily be generalized to finite water depth. Thus according

to Newman

ikcos0iksin0 ds

= f,j[p cos 0+ p Vr(V, cos0 - V0 sin 0)] r dO dz

. .(33)

= JJ[psinO +pV(l'.sin.O +

V9cosO)]rd0dz

. .(34)

M =

-

L1 I'.V0r2dOdz (35)

where the bars denote time average and the integration is over the surface S of a vertical circular cylinder of large radius r, that is extending from the free surfaèe down to z = - h. F and are the x- and y-components

of the horizontal drift force and M is the drift moment about the z-axis. We have used (r, 0, z) as cylindrical polar co-ordinates with x = r cos 0 and y = r sin 0. Vr and V0 are the radial and tangential velocity

com-ponents, respectively, and p is the dynamic pressure. We will now approximate equations (33), (34) and (35)

and only retain contributions that are of second order

in the incident wave amplitude. To do this we only need

to know the velocity potential to first order in wave amplitude. Using equations (11), (12),

(22) and an

asymptotic expansion of the Green's function expression (23), we may write

g4' cosh k(z + h)

eb0Ysi_t)

w cosh kh

+F(0) cosh (k(z + h))

ecoo)

.(36)

r Here F(0) is real and F(0) e° is given by

2ir(v2 - k2)

/9)_I3/4

F(0) e'° = k2h - v2h + v irk x $$ Q(, ii, 4')cosh[k(4'+ h)] e S (37) Further 6 Q(', ii, 4') =Q7 + (38) J4ere i is defined by =

e°t

...(39)

where i = 1,6 are the six modes of motion.

As shown in Appendix 3 we will get the following

expressions for drift-force and moment

p W4'a

/(21t\L

kh

1FYS =

2sikh

k)4s11Th2

1

x2F(f3) cos

(u)

Icos 6)

-

k sinh 2kh + ) { F2(0) dO 2 \

-

ri sinh 2kh

hi

I w /f'2x M

= [

± sinh kh

tk.

x F'(/J) sin +

- sinhkh

x''(I3)F(fl)cos((fl)

+

-x F2(6) '(0)dO} . (41)

Where 4'($) means d/d6 evaluated at 0. = /3.

5

COMPUTATIONS

A computer programme. NV459 based on the three-dimensional source technique has been developed at Det norske Veritas. For a fixed structure in regular waves the programme calculates the total linear

hydro-dynamic forces and moments, horizontal drift forces and moments, pressure distribution on the body and pressUre, fluid velocity and acceleration for any point in the fluid. No assumption about geometrical symmetry is necessary.

Floating objects do, however, usually have at least one plane of symmetry. For such an object the programme calculates the added-mass and damping coefficients and motions in six degrees of freedom. The object is geometrically described by using offset points on the wetted. surface of the body. Following the procedure of Hess and Smith (8) one then creates plane surface

elements approximating the wetted surface of the body.

For an object having two planes of symmetry and using 48 plane elements to describe the total wetted surface of the body, it takes in the order of one minute

CPU time on UNIVAC 1108 to solve the hydrodynamic

problem for one wavelength. To solve the problem for the same wavelength with a different direction of pro-pagation only a little additional CPU time is required.

For an object having only one plane of symmetry and

using a total of 48 plane elements, the computer time is 2 mm CPU on UNIVAC 1108 for one wavelength. Similarly, for no plane of symmetry and a total of 48 plane elements the computer time is 4 mix - CPU. The computer time increases approximately as the square of

the number of plane elements.

We have compared the computer programme with other analytical solutions, computer results and experi

ments and in general we have found very good agreement using a total of 48 plane elements. But in some cases, when

calculating moments, we found it necessary to increase

the number of plane elements.

The computer programme NV417 has also been used.

This programme is based on the method of Salvesen, et al. (15) and calculates ship motions and wave loads for regular waves of any direction of propagation. For zero speed it reduces to a conventional strip theory. The two-dimensional velocity potentials are calculated using either the Lewis' form technique or the Frank Closefit

Method.

In this report we show computations for two floating

(40) boxes. The bôxés have length and beam equal to 90 m

and the drafts were 20m and 40m full-scale. Further details are presented in Table 1. The water depth is infinite. The directions of wave propagations were /3 = 0° and 45°. The range of periods T chosen for the

incoming waves was from 8-20 s. full-scale.

The strip theory programme NV417 was used for

/3 = 0°. The strips were placed in the lengthwise direction. A similar procedure has been used by Kim et al. (16) for

the motions of a barge of length/beam ratio equal to 1.5

and with good results.

Table 1. Geometrical Data for Floating Box. L = 90 m,

8= 90 m

The added mass and damping coefficients are

pre-sented in Figs 3-16.

The agreement between calculated values by the strip

theory programme NV4 17 and the three-dimensional programme NV459 is generally not good and this was to be expected The difference between the yaw-added

masses of the two programmes is especially large. The exciting forces and moments are presented in the

form of amplitudes F and phase angles

;. They can

be written in a time-dependent form as,

I1Isin(cot+ c)

The incoming wave can be written as, sin (cot - kx cos/3 - ky sin/3)

A positive phase angle, therefore, indicates that the

force leads the wave height at the origin.

Exciting forces and moments for the box are presented in Figs 17-26.

The motions are presented in the form of amplitudes

,i1 and phase angles x. They can be written in a

time-dependent form as,

j sin (cot + )

and we note that a positive phase angle indicates that

the motion leach the wave height at the origin. Motions of the free-floating box are presented in Figs 30-37.

Drift forcesare presented in Figs 27-29. The asymptotic

values for small periods are also indicated Expressions

for these have been derived by Maruo (17). Assuming that

the waves are propagating along the positive x-axis and that the body has vertical sides at the free surface, Maruo derived the following asymptotic expression for the drift force in the x-direction when T-+ 0

F =

pg

1bsin2 dy

Here is the slope of the tangent of the waterplane curve with respect to the x-axis. Further, 2b is the beam of the body and the integration is along, the y-axis. In our case, for a heading angle/3 = 0°, we get,

F = O.5pg L -C. G (xG, YG, zG) 0,0, 10.62m 0,0, 8.82 m k 33.04m 37.32m 32.09m 33.30 m k. 32.92m 40.08 m

MOTIONS OF LARGE STRUCTURES IN WAVES AT ZERO FROUDE NUMBER

pv 0-5 20 1.5 A33 pV . 1.0 & 20 __._ NV 659 68 elements, total ---A---NV 417 8 offset points

* Expeñmentaf valueis,ampl=3m

---NV459 68e[èment0 -.---NV45g '108 elements

---NV6i7 8 offset points Experimental values ,amp1.*3m

II ii ampl,.z6m __4z__NV 659 68 elements total .._ NV417 8 offset points Experim?ntal values 0.5 16 16 16 20 10 12 Period, i S

Fig. 3 Surge added-mass coeffiôients for floating box

(L B d'= 90m x 90m x 20m). Infinite water depth

10 12 14 16 18 20 PriDd,T

Fig. 4 Surge addedrnass coefficients for floating box

(L B x d =90rn x 90ni x40rn). Infinite waterdépth

8 10 12 16

Period,r

Fig.. Heave added-mass coefficients for floating box

(L x B x d = 90m x 90m x 20m). Infinite wter depth

2.0

-o- NV 659 48 elements -u,- NV 459 108 elements

1-5 ----NV'417 8 offset points

A33 Experi'mertal values

0

Fig. 6. Heave added mass coefficients for floating box

(L x H x d = 90m x 90m x 40m). Infinite water depth

Oct -007 0.05 - -0.05_ r'L' _0.04 0.03 -0.02 0,.01 ,0 05' 006 0.03 002 ---t1V 159 ltetements 0.01 . NV 659 108 elements --b--NV 617 Boflset points Exerimenta['value 10 12 16. 16 - 16 20 Period,,r

Fig. 8. Pitch added-mass coefficients fOr floating (L , B x d =90 no 90 rn x 40m). Infinite water depth

i 12 16 16 18 20 Period -,--0- . 0 -o-- 58 elements --- B offset points Eoperftnental vdlues 10 12 14 15 20 Perid r

Fig. 7. Pitch added-mass coeffiients for floating box

(L x B d, 90 m x 90 m x 20 m). Infinite water depth

A 66 a VL 0.07 005 10 12 14 16 18 20 Period,T

Fig. 9 Yaw added-mass coefficients for floating box

(L x B x d = 9Dm x90m x _{20m). Infinite water depth}

O05

003

0. 01

.-o- NV 459 00 68 elements total ----NV 417 00 8 offset points Experimental values oNV459 48elements 0 NV 59 108 elements -- NV 417 8 offset points Experimental values

MOTIONS OF LARGE STRUCtURES IN WAVES AT ZERO FROUDE NUMBER

0 - a

--10 12 14 16 18 20 Per iod,r

Fig. 10. Yaw added-mass coefficients for floating box

(L x B x d=90m x 90m x 4Dm). Infiñitewaterdepth 20 NV 459 48 elements NV159 lO8elernents -... NV 417 8 offset points Experimental values 1.0 0 5 NV 459 68 elements total NV 417 8 offset points 833 10 12 14 16 18 20 Period,r

Fig. 12 Surge damping coefficients for floating box

(L x x d = 90m x 90m x 4Dm). Infinite waterdepth 20 1.5 1. 0-5 0 . -- L I I - I 8 10 12 14 16 18 20 Period,r

Fig. Ia Heave damping coefficients for floating box

(L x B x d = 90m x 90rn x 20m). Infinitewater depth heaveheave damping coefficients

e--NV459 48elements NV459 106 elements ---NV 417 8 offset points Experimental vaLues _...NV 459 68e(ement total ..JNV 417 8 dftset points Experimental values -- A---I 0 0 18 20 _{6 10 12 14} 16 18 20 Period, T

Fig. 11. Surge damping coefficients for floating box _Fig. 14 Heave damping coefficients for floating box

(L x B x d = 90m x_{9Oni x 20m). Infinite water depth (L x B d = 90rn x 90m x4Dm). Infinite water depth}

1.5 811

10 12 11. 16

0008

0007 _{NV 1.17 8cffset points}

0006

-Fig. 15.. Pitch damping coefficients for floating box

(L x B xd =90 m x 90 mx 20 m). Infinite water depth

® NV 1,59 58elementS

....NV 1.59 1.8 elements

NV1.59 108 elements

NV 1.17 8 ott set points

.10 12 11. 15 18 20 Period, T 2.1

iil

Vá/L _2.0 1.6 1.2 0.8 04 180 r 0.6

0.1 459 Heading angle 450 68 elements total

0 I- .

6 10 12 14 15 18 20

Period,r

-_- NV 1.59 Heading angle00 66 elements total

NV 117 u 0 8 offset points

10 12 11. 16 16 20

Period, I

I -I

Fig 17. Exciting force on floating box (L x B x d =

90m x 9Dm x 20m). Infinite water depth. amplitudesand phases

1.6

0

160

-Fig. 16 Pitch damping coefficients for floating box

(L x B xd 90m x 90 m x 40 m). Infinite water depth

90 .,l o

cIø,

0 I I.

Fig. iS Exciting force on floating box (L x B x d

90m x 90m x 20m). Infinite water depth. amplitudes and

phases

8 10 12 11. Per iod,1

Fig.

ia

Exciting force on floating box (L x B x d = 90m x 90m x 40rn). Infinite water depth amplitudes and

phases

-180 -- -.

180

0

e---NV 159 Heading angle 0° 48 elements total

B NV 459 ii 0° 108 n ---NV417 0° 8 offset points

10 12 1!. 16 18 20 Period, T

MOTIONS OF LARGE STRUCTURES IN WAVES AT ZERO FROUDE NUMBER

90m x 90m 40m). Infinite water depth, amplitudes and

phases

Fi 21. Exciting force ofl fIôating box (L x B x d =

90m x 90m x 20m). Infiflite water depth, amplitudes and

phases

20

...0_...NV 459 Heading angle 00 48 eLements totdl

NV459 ii 00108 1.6 - ---NV 417 ii 0°. 8 otfet points V3l 180 90 0 -90 08 04 0 180 90 -- -I I 10 12 14 16 16 20 Per.iod,r

_-.o--...NV 459 Heading angUe- 006.8 etements total

---&---NV 417 ii 0° 8 offset paints

I

-10 12 14 15 18 20

Period;

Fig. 22 Exciting force on floating box (L B x d =

90 m x 9Q m 40 m). Infinite water depth, amplitudes and

phases

2.0

1.5

Fl

v--- NV 459 Heading angle 450 48 étements total

I -,gVa/L 1.2 08 0.4 180 go 0 .0 Fig. 20 I -10 12 14 Period ,i 16 18 20

Exciting force on floating box (L B d 0 0 2.4 2.0 1.6 F 3 pgVCa/L 17

- 0.20 F 5 024 012 P0.06 0.06 0 160 go 0 a, 10 12 it. Period, T 16 18

-._NV 659 Heading angle 0° 18 elements totat

---.s---NV 417 ii 0° 8 offset points MARINE VEHICLES 20 0.24 0.20 006

...0...Nv.45g Heading angle 0° 18 elements total 001 rn NV 459 ii 0° 108

NV 417 ii 0° 8 offset points

10 . 12 11. - 16 16 20

Period, T

Fig. 23 Exciting moment on floatiflg box (L x x d = Fig. 25 Exciting moment on floating box (L x B x d =

90m x 90m x 20m) Infinite water depth amplitudes and 90m < 90m x 40m) Infinite water depth amplitudes and

phases phases

0-26...

020 pg

__,NV 459 Heading angle 45° 66 elements total V

0-16

0.12

0-08

v--- NV 459 Heading angIe 65° 48 elements total

0-04 10 6 10. 12 16 16 -. 18 20 Period, T 160 -90 0 90 180

Fig. 24 Exciting moment on floating box (L'x B x d = Fig. 2& Exciting moment on floating box (L B x d =

SOm x 90m x 20m) Infinite water depth amplitudes and 90m x 90m < 40m) Infinite water depth amplitudes and

phases phases 160 90 0 a, 016 Fj I

ga

0.12 -180 -160

pgL _a2

02

0.1

Asymptotic value

45°

MOTIONS OF LARGE STRUCTURES IN WAVES AT ZERO FROUDE NUMBER

0.5

NV 459 Heading angle 0° effect of motions included,48 elements total

90m x _{90m x 40m). Infinite water depth}

0.4 0.3 02 -* Asymptotic ° 1.., * 20 _{0.1 -} vluei:45° 0

lmptotic

value 8 -- Th -. 12 - 18 20 Per iod,r

Fig. 29. Drift forces on floating box -(L x _{B x d =}

90 m x 90 rn x_{40 m). Infinite water depth}

3.0 25 1.5 10 0.5 180 .90 0 -go

NV 1.59 Heading angle 0° 68 elements totat

2.0 ___-i-__NV 417' ii 0° 8 offet points

10 12 14 'i's 16 ' 20

Per iod,T

180 ._ I I

I-Fig. 3Q Motions of floating box (L x.B x d =

90m ' 90m x 20rn). Infinite water depth, amplitudes and

phases

+

m

--.--NV

NV 1.59 n 0° v 108

459 ii 0° effect of motions not included, 48 elements total

NV 459 ii 0° u 108 Experimental values 06_{- Asymptotic volud} .VxI 05 0l. 03 02 '0.-S. S 0.1 a I I 8 10 12 14 16 Period T 18 20 o S _= Fig. 2 Drift forces on floating box (L x B d =

05

0.1,

.-NV

o NV 459 Heading angle. 0°-effect of motion included, 68 elements total

NV 459 ii 45° .11

'1.59 _{0° effect of motion not included,68 elements total}

1, 59 m n__ 45° Asymptotic value 07 06 .__NV ---NV

1.59 Heading angle 1.5° effect of motions included, 48 elëménts fatal 45,9 Heading angle 45° effects of mat ions not included,

48 elements total

-* Experimental vatues 10 12 14 16 16 Period, $

Fig. 27. Drift foEce.(L x B x _{d = 90m} x _{90m x 20m). On} floating box infinite depth

Fig. 31. Motions of floating box (L x B x d 90m x 90m x 20m). Infinite water depth, amplitudes and

phases 3-0 2-5 Ill 2-0 1-5 0-5 180 0 H90

0-- NV 459 Heading angLe 0° 68 elements total NV459. n 00 108 ii

£ MV 617 ii 0° 8 offset points

Fig. 33 Motions of floating box (L x B x d

90m x 90m x 40m). Infinite water depth, amplijudes and

phases 3.0 2-5 11131 Ca 2-0 1-5 1-0 0-5 180 90 0 'a -90 10 .12 14 16 18 20 8 PeriodT a

MV 1.59 Heading angle 0° 68 elements total --=-- NV 417 0° 8 offset points I I -I 10 12 11. 16 18 20 Per iod,r -1 0 - - . -- 180 I I - I

Fig. 3Z Motions of floating box (L x B x d = Fig. M Motions of floating box (L X B X d =

90m x 90m x 40m) Infinite water depth amplitudes and 90m x 90m x 20m) Inf.n te water depth amplitudes and

phases phases -180_ - 160_ 90 9 w 0 C - V. -9 '. -180 I t i- I -160 _i I _-f-- I I -30... 2..5 -0

-'---

elements 3.0 2,5 I 1111

---Iteadiflg angle 45° 48 elements tOtal

I -I-2-0 1.5 1.0 0-5 2-0 1..5 1-0 - 0-5 - 0

NV 459 Headingangle 45° 68 total

I I I - I 6 -_ 10 12 16 16 18 Perio&T 20 8 10 -12 14 16 10 Perio&T . 20 S - s_

a 3.0 180 90 0 -90

Fig. '37. Motibfls of floating box (L x B x d =

180 I I I I

90m x 90m x 40m). Infinite water depth amplitudes and

Fig. 3S Motions of floating box (L x B x d phases

90m x 90m x 40m) Infinite water depth amplitudes and

phases'

MOTIONS OF LARGE STRUCTURES IN WAVES AT ZERO FROUDE NUMBER

10 12 14 15 18 20

Per odT

0.-...HV 1.59 Heading angle 0° 1.8 elements total

u NV 159 ii 0° 108 ii

--A---NV 117 n 0' 8 offset points

+ Experimentat' values

..____ NV 459 Heading angle 0° 68 elements total

459 H 45068 II,

10 12' 1!. 16 18 20

Peribdj

Fig. 36. Motions of floating box (L X B x d =

90m x 90m x 20m) Infinite water depth amplitudes and

phases

3.0

2.5

In5

alL

o--NV 1.59 Heading d'ngle 00 48 elements total

1.5 vNV 459 450 48 _ii s NV 417 II 0° 108 ii 1.0', 05 180 '90 0 9.0 -180 10 12 14 Period I

FOr both boxes this asymptotic value agrees. _{cruite well}

with our calculations.

Introducing an x'-axis along the direction of Wave propagation we may use Maruo s formula also for

heading anglefi 450 We then get,

F=!pgL

for the drift force along our permanent x-axis. Again the. asymptotic value agrees very well woth our computations

We may further note that the effect of the motion on the drift forces is small for the lower periods. For the floating

box of draft 40 m and close. to the heave, resonance

frequency, there is a strong dependence on the thotions. This may be expected since the maximum immersion of

the body is increased due to large relative vertical motion between wave and body For the floating box of draft 20m there is a strong dependence on the motions

for the larger periods. The body moves more. or less as a

water particle for these periods and the drift force is

therefore very. Small.

6 _CONCLUSIONS

A theoretical methOd has been presented that describes the motions, in all six degrees of freedom, of large floating structures m waves when the Froude number

is

_{zero The method also gives the hydrodynamic}

pressures and the horizontal drift forces.

The method of analysis Used is based on a three-di.. mensional source technique With the sources 'distributed

on the surface of the body. The drift forces are derived

from momentum equations.

Numerical calculations and experimental results for the case of a floating box of dimensions 90 m x 90 m and a draught of 20 m and 40 m show very good agree-ment in the measured quantities. On the basis of this comparison one may cOnclude that the method does indeed provide a solution to the dynamic analysis of large three-dimensional structures in waves, subject, of course, to the usual limitations of the linearized

theory.

It may seem attractive to try to apply a two-dimen-sional strip theory to the analysis of the structures of

concern here. This has been done for the box mentioned above and the results show that this method of approach cannot be used in the dynamic analysis of such structures. Only a three-dimensional method of analysis, as given in this paper, will be adequate.

Experiment

For the purpose of verifying computed values of added mass and damping coefficients of the 90 m x 90 m box at draughts of 20 m and 40 m a series of tests were

per-formed in the model basin of the University of Trondheim

with a model built to scale 1:100. The model was oscil-lated in heave, surge, pitch and yaw at amplitudes of 3 cm and 0.05 rad respectively. Some tests were also made at twice these amplitudes to check on linearity.

The experimental results agree generally very well With the theoretical predictions. Where deviations from

theo-retical values are most pronounced it was found that these were probably caused by the fact that it was not

always possible to provide a pure one degree of freedom

motion to the model. This was true in particular for the

cases of surge and yaw in the. range of low periods.

In regard to damping, deviations from theoretical

values may seem to be significant in this range of periods.

This may be due to viscous effects, but it should be

pointed oUt that damping is small and difficult to

determine experimentally with any great degree of

accuracy. Only a few degrees of error in the measuremei)t of the phase angle will, for instance, result in large

varia-tions in the damping force. For the same reason it was found that damping in yaw and pitch were much higher than theoretical predictions indicated, and, no

experi-mental values are therefore given.

A model to scale 1:60 was also built for the purpose of

determining excitation forces. These experiments were not successful due to instrumentation failures, and the tests will therefore have to be repeatcd. A free-floating test did, however, provide us with experimental data on geave motions and drift forces.. Again one can con-dude that correlation between theory and experiments

is good.

7 ACKNOWLEDGMENT

The. authors are particularly indebted to Mr A. Løken of Det Norske Veritas for his contribution in programming

and numerical computations. Also the assistance of the staff at the Ship Model Basin in Trondheim is fully acknowledged, as is the work of Mr T. Mikkelsen and

Mr J. Knudsen, students at the University in Trondheim, in analyzing experimental data.

APPENDIX 1

Numerical evaluation of the Green's Junction Equation (24) can be rewritten as

G(x, y, z: ,;i, )

1 1 1

2(ku + v)e'cosh [k(C +

+ x cosh(k(z + h)u)J0(kr1u.)du

u sinh (khu) - (v/k) cosh (khu)

(uj

(z + _{)kuJ (kur )k du} e

2v J0(kr1u) du

+

_uv/k

2ir(k2 - v2) cosh [k(C + h)] cosh [k(z + h)] J0(kr1)

+

_k2hv2h±v

.(42)

We have here disjoined the image source with respect to

the free surface by making use of the fact that

((x )2 + (y ,)2 + (z + )2)_4

= j

eJ(,pr1)dp

...(43)

(see Gradshteyn and Ryzhik (18)). The limit of integration u1 in equation (42) must satisfy the following conditions

(i)

u1 ?2,

due to numerical methods used in the evaluation of

the principal value integral,

(ii) u1 ? 4,5/kh

so that we may with sufficient accuracy set cosh khu

- for u

sinh khu J

The principal value integration in equation (42) has been

calculated using the 'midpoint rule' and the procedure proposed by Monacella (19). The infinite integral in

equation (42) can be converted into a finite integral

by using the integral representation

.J0(kr1u)

=

ei0sOdO

.. .(44) (see Abramowitz and Stegun (20). We may then write

2v J0(kr1u) du

.)UI

uv/k

=

IdO

d z++ivri cosfl

11

E1 [- (k(z + () + ikr1 cos 0)

(1

)]

.;.(45)

where E1 is the exponential integral as defined in

Abramowitz and Stegun (20). The integral in equation

(45) has been evaluated using the 'mid-point rul&.

APPENDIX 2

IntegratiOn of sources with constant density over a quadri-lateral

We consider., a plane quadrilateral source element lying

The x-y or co-ordinates ofthe four corner points defining the quadrilateral are (, i7), 12)' and

(, 114). It is. desired to determine the velocity potential induced by this source element at a general point P n space having co-ordinates x, y, z. The value of the

source density is set constant over the quadrilateral. The potential then becomes proportional to

lid _ll

dcd

J

JAJJ[(x)2+(y_

.(46)

= IA

Fig. 2. Plane quadrilateral source element

Following a procedure similar to the one used by Hess

and Smith (8) for the velocity components we obtain

= .- $d iog(y

1112

The first integral of equation (49) can easily be integrated analytically (see Gradshteyn and Ryzhik (18)). The second

intergrand of equation (49) has no singulanty in the

integrand and no difficulties in the numerical integration

are encountered. Difficulties with singularities in the integrand of the other integrals of equation (47) are handled in a similar manner.

APPENDIX 3

.

Derivation of second order drift force and moment

We shall show here how one can pass from equation (33), (34), (35) to equations (40) and (41), which are correct to second order in wave amplitude.

Using Bernoulli's equation we may write

p dz =

25_i, I dz ...(50)

Here is the free-surface elevation and Vthe fluid velocity

vector which has the components (I', F, V) in the cylin-driôal co-ordinate system. It is possible to show that

P2

_{± a1(0)05h(1th)}

2

22

g

x cos(kr(l . cosOcos/3 - sinOsin/3) + 4(0))

+yF2(0)cosh2khr_1}

(51)

2g

p[lsinh2kh

h Jl w2 cos (/3 '- 0)

+ F2(0) r3 + F2(0) k

X 12 sinh2 kh 1

cos(fl-0)

2a sinhkh

F(0)r x sin (kr (cos (/3 - 0) 1.) CO5

- 0F(0)kr

sinh kh

x cos [kr(os(fl 0) - 1) -

_(0)]} ...(52) 5

V dz =

+ r3(F'(0))2 + r 3(F(0)'(0))2 +

x sin(0 /3) rF'(0) sin [kr(cos(0 - /3) - 1) - 0(0)]

.(47)

silh

sin (0 - /3) r F(0)0'(0) ..(48) X cbs [kr(cos(0 - /3) - 1) - 0(0)]} ...(53)

p[1snh2kh

_hill

w

j

V dz =

2[k

4

2fl2sinh2kh

+ r

1F2(0)k2 + sin! hr4F(0)k (49) .

x cos[kr(cos( /3)

-I

dz=

p [1 sinh 2kh +

hi fl w22 j2 (0 - /3)

2[k

4

_2j12

sinh2kh

) -

0(0)]} ...(54) lsinh2kh h 1 w22 +

]{2flh2h

x cos(0 /3)sin(0 - /3) 2Sflhkhc0s(O /3)

x rF'(0)sin [kr(cos(9 /3) - 1). 4(0)]

MOTIONS OF LARGE STRUCTURES IN WAVES AT ZERO FROUDE NUMBER

+j[(y_11l2)2+(x_)2

z2)]

- $dlog(y

+ 1123)2 + (x )2 + z2)]

dlog(y-34

0 +

- 1) + (x

z2)] s: d log (y - 1141 + 1141)2±(X -Here = 11 + ;; = :;: ('

We note that the integrandofthe first integral is singular

when z = 0,

= x and y

- 1112

<0. In this case we

change the integral to read

J d log ((x )2 + z2)

- J d log [(y

- 1112)

+ J[u - 1112)2 + (x )2 + z2]]

ç

+ 2 sinh kh cos(0 - /3) rF(6)4f(6) x cos[kr(cos(0

_{- /3) -}

1)-a _r- _{F(0)sin(6 - /3)} 4 sinhkh x sin[(kr(6 /3) - 1)

(0)] -

r 3F(0)F'(0)

+ kr

2F2(o)(o)'

siniI

r - F(6)k sin (0 - /3) x cos [kr (cos (6 - /3) - 1) - 4(0)]} .. .(55)

Here F'(0) and 4'(0) mean dF/dO and dcb/dO, respectively.

Now by applying the method of stationary phase

(Erdélyi (21)) we may write for large r

fg(0)cos

[kr(cos(0

_{- /3) -}

1) -

(6)] dO

()

{g cos

($)

±

+ g(/3 + it) cos (- (fi + it)

+ -

2kr)}

where g(0) is some arbitrary function. By using equations

(50)(56) we may write the drift forces and moments in

the form Of,equations (40) and (41).

APPENDIX 4

REFERENCES

(I) MORISON, J. R., O'BRIEN, M. P., JOHNSON, J. W. and SHAAF, S. A.

'The forces exerted by surface waves on piles', Trans. Am. Petrol. InsI. 1950 189 (No. 2346).

LEBRETON, J. C. and CORMAULT, P. 'Wave action on slightly

immersed structures some theoretical and experimental con

siderations', Proc. Symp. Research on Wave Action 1969 4. GARRISoN, C. J. and SEI1tAMA RAO, V. Interactions of waves

with submerged objects J Waterways Harbours Coastal Engng Div., ,ASCE 1971 97 (No. WW2) 259-277.

.(56)

MILGRAM, J. H. and HALKYARD, J. E. 'Wave forces on large objects n the sea', J. Ship Res., 1961.

VAN OORTMERSSEN, G. 'Some aspects of very large offshore

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LAm, H. Hydrodynamics, 6th ed. 1932 (Cambridge University

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WEHAUSEN, J. V. and L.rroNE, E. V. 'Surface waves', Hàndbuch der Physik, 9. 1969 (Springer-Verlag, Berlin).

HEsS, J. L. and SHrill, A. M. 0. 'CalcOlation of non-lifting potential flow about arbitrary three-dimensional bodies',

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JOHN, F. 'On the 'motion of floating bodies II', Commun. pure

appl. MOth. 1950, 3.

FALTINSEN, 0. 'A study of the two-dimensional added-mass and

damping coefficients by the Frank Close-Fit method', Det

norske Veritas, Oslo, Norway, Rep. No. 69-l0-S,, 1969.

ST. Dsr'iis, M. and PIERSON, W. J. 'On the motion of ships in

confused seas', Troils. SNAME, 1953, 61.

Gaitrrsi,e.&,,J., BOSCH, V. I). J. J. and BEUKELMAN, N. 'Propulsion

in regular and irregular waves',, mt. Shipbuilding Progress 1961.

Hsu,F. H. andBI 1ARN, K. A. 'Analysis of peak mooring force caused by slow vessel drift oscillation in ratidom Seas', O.T.C. 1970, Paper No. 1159.

NEWMAN J N The drift force and moment on ships m iaves

J. Ship Res. 1967.

SALvESEN, N., Tuck, E. 0. and .FALT[NSEN, 0. M. 'The ship

motions and sea loads', Trans. SNAME 1970, 78.

Kia, CH. and CHOU, F. 'Hydrodynamic characteristics of barges', Oflshore TechnQlogy Conference, Houstón 1971.

MARuo, H. 'The drift of'a body floating in waves', i. Ship Rès. 1960.

GRADSHTEYN, I. S. and Ryzmi, .1. M. Tables of integrals, series and products 1965 (Academic Press, New York and London). MONACELLA, V..J. 'The disturbancedue to a slender ship

oscillat-ing in waves in a fluid of finite depth J Ship Res 1966 10

(No. 4), 242-252.

ABRAMbWITZ, M and STEGUN, I. A. Handbook of mathematical

functions 1964 (National Bureau of Standards Mathematics

Series, 55, Washington, D.C.).

ERDELYi, A. Asymptotic expansions 1956 (Dover PublicatiOns, Inc., New York).