Seakeeping applications using a time domain method

TECHNISCHE UNIVERSITEIT Laboratodum _voor

Sthom

Mekeiweg

2,

2028 CD Deift

TsL 015-786873 F 015.781838

Seakeeping

Applications

Using

_a

Time-Domain

Method

Allan

Magee,

BASSIN

D'ESSAIS

DES

CARENES

ABSTRACT

The seakeeping problem for _a ship in _a ideal in-

compressible fluid of infinite depth is solved with

the aid of the Green function for _an impulsive

source in arbitrary motion. The Green function

satisfies the linearized free surface condition, _ra-

diation condition, and the appropriate initial _con-

ditions. Thus, singularities need to be distributed

only over the hull surface to satisfy the _{no pene-}

tration condition. Green's second identity is _ap-

plied to the fluid domain to obtain an integral

equation for the unknown potential _on the body

surface _at each _instant of time. This equation is

solved by time stepping and the results _are trans-

formed into the frequency domain by Fast Fourier

Transform for comparisons with other calculation

methods and with experiments.

With the above described solution method,

several problems of importance _can be treated,

the most important being the linearized seakeep-

ing problem for _a ship with forward speed. By

resolving the equations of motion in the time

domain, the effects of unsteady geometry _can

be taken into _account in the so-called finite-

amplitude formulation. Nonlinear external forces,

such _as

that

due to _an autopilot _can also be in-

corporated in _a straight-forward _manner.

INTRODUCTION

The fundamental solution for the time-domain

Green function,

that

of

_a

_source moving arbi-

trarily beneath _a linearized free surface, _was first

elaborated by Brard, 1948. The Green function

employed in the present method is developed in

Wehausen and Laitone, 1960. Time-domain

_so-

lutions _were presented by Adachi and Omatsu,

1979 and Yeung, 1982 for two-dimensional bod-

ies, and Newman, 1985 for the axisymmetric _ver-

tical cylinder. Three-dimensional solutions began

to _appear with Liapis and Beck, 1985 and Liapis,

1986 followed by King, 1987, King et al., 1988,

and Korsmeyer, 1988.

Ferrant, 1989 presented _a solution to the _so-

called finite-amplitude problem, in which the free

surface condition remains linear, but the body-

bound _ary condition is applied on the moving sur-

face. Beck and Magee, 1990 and Magee, 1991 also

presented _a finite-amplitude formulation, and Lin

and Yue, 1990 solved the finite-amplitude _equa-

tions of motion in the time-domain for _a floating

body. The problem of

_a

submerged body, acceler-

ated from _{rest was} also treated by _Ferrant, 1990.

More recent work using the time-domain Green

function has been presented by Magee and King,

1992, Bingham et al., 1993, and still others _are

being presented here.

A similar development has occurred in

the frequency-domain beginning with the

_two-

dimensional strip theories. Three-dimensional

_so-

lutions _{at zero} forward speed have become widely

accepted and computer codes based _on these

methods have reached _a highly developed state,

suitable for industrial applications. These

_zero-

speed solutions _were first extended to the _case of

forward speed using the so-called frequency of _en-

counter approximations for _small forward speed.

Subsequent work has been devoted _to approxima-

tions which _{capture some} of the forward speed

effects, _{while retaining the advantages of the}

_zero-

speed Green function, _e.g., Coudray and LeGuen,

1992.

Despite the complexity of computing the _re-

quired Green function, their have been several at-

tempts to solve the three-dimensional frequency-

domain seakeeping problem for ships with mod-

erate forward speed. Chang, 1977, Inglis and

Price, 1981, Guevel and Bougis, 1982, Iwashita

and Ohkusu, 1992, and several others have pre-

sented solutions for this problem. For finite-

amplitude problems in the frequency-domain, Fer-

rant, 1989, presented _a numerical solution for sub-

merged bodies. Rankine-source based methods,

such _as presented by Nakos and Sclavounos, 1990,

in the frequency domain, are an alternative to the

Green-function-based methods.

PROBLEM

FORMULATION

In this section, the formulations of two problems,

namely the fully linearized problem and the

so-

called finite-amplitude _or body-exact problem _are

given. The time-domain Green function is used in

both _cases.

The

Linearized

Problem

- - - - - -

For the linearized hydrodynamics problem _a

_co-

-

cJ

C'

calm

wa1.surface

_z 0, with the z-axis verti-

cally upward. The origin is translating with the

steady speed of the ship U0. The _case of _zero for-

ward speed _can be recovered by setting Uo

₌

0.

The fluid domain _is bounded by the _mean free _sur-

face S1, the _mean three-dimensional body surface

S0, and _a control surface at infinity

S.

The unit

normal is defined _out of the fluid domain. An

ideal incompressible fluid is assumed, _so

that

the

Laplace equation governs, and we can formulate

the problem for the total velocity potential

tT.

If

VT is the total fluid velocity, then

VTV'T,

1T=°

(1)

The

total

potential is separated into several

components as follows

y, z, t) =

fJ0x

+ o(z, y, z)

+

h(x,

y, z, t)

±

'k(z,y,Z,t)

(2)

k=1

with,

(Icr

+ o

=

the steady flow potential

çj

=

the incident wave

çbk ₌ the six radiation potentials due _to,

respectively, _{surge, sway,} heave, roll, pitch,

yaw and the diffraction potential (k=7)

On the _mean free surface, we apply the lin-

earized free surface condition

(a

lLoJ

a2

+g-

Ot

3xj

3z

The elevation of the free surface is given by

i7(x,y,t) ₌

ia

--(--U0--)

(4)

=0

Because the disturbance originates _near the origin

V---0

_asrco

k=1,..7

(5)

The unsteady potentials _k, k

₌

1, ..7 also

satisfy the initial conditions

ôck

=

O

ast--cc

k=1,..7

(6)

Green's second identity is applied _to the fluid

domain _to obtain _an integral equation.

JJf(G_Gz)

=fJds(-_c)

₍₇₎

where S

₌

S1 U So _U

S.

The Green function for

the transient problem is given by

(i

i\

- G(P, Q, t,

r)

₌

(:--

7)

(tr)+H(tr)G

(8)

=0

(3) z=0 2 where, Ò(P, _{Q, t,}

r)=2fdJgk

sin((t

_-

r))e()Jo(R)

P

₌

(x(t), y(t), z(t)) Q

₌

r

r,

₌

R

₌

(t)

₌

delta function

fdt

_6(t)f(t)

=

H(t)

₌

unit step function

fo

ift<0

-

i

ifi>0

and Jo is the Bessel function of the first kind of

order 0.

Using the properties of the Green function,

it _can be shown

that

the integrals in (7) _over

vanish, and those _over Sj _can be reduced to line

integrals around the _curve F0 defined by the inter-

section of the _mean hull surface and _z

₌

O plane

(cf. Liapis, 1986). The resulting integral equation

for the potential _{at a} point in the fluid

P

is found

to be 4

(P,t)+f

dsQ

(Q,t

(

_-

so

-

L0

dsQ

(r

-

1'\5

(9)

-

f

_J-cc

drf

dsqí(Q,r

-

so L

-

[dr

¿dQ

LT

gJ

Jr

uo

(Qr

_-

The boundary conditions to be applied _on So

are where, (k (k k k

₌

1, ..6 (10)

=

applied displacement in mode k

=

applied velocity in mode k

=

ñ

k=1,..3

=

Fxi

k=4,..6

=

_(.V)

k=1,..3

=

-(ñV)(ixW)

k=4,..6

8çt'k

=

flkCk+mk(k

Vçbi

. so

an

87

ôn

and W is the velocity due to the steady flow Define the Fourier transform pair _as

W ₌

_Uoi+Vo

o is the solution to the classic Neumann-Kelvin

problem ôn U6 32x

+g--

2 _, .*.0 Fjk ₌

J

dSPn

(13)

_JdrKjk(t_r)k(r)

(17) so

co

Non-Impulsive

Inputs

and

Impulse-

Response

Functions

The choice of boundary conditions in (10) is _con-

sidered here. The most obvious choice is

that

of

a

6function

velocity in time. This input _con-

tains equal amplitudes

at

all frequencies. King,

1987, proposed the choice of a non-impulsive ve-

locity input, in order to limit the high frequency

content of the signal, since the _response at _very

high frequencies is of little interest for the _sea-

keeping problem, and since waves of very high

frequency cannot be resolved within the finite _spa-

tial and temporal discretization used to solve the

problem. In addition, the behavior of the solution

at

the so-called irregular frequencies (see Adachi

and Ohmatsu, 1979,) _was shown to be improved

using the non-impulsive method. The input _ve-

locity takes the form of _a Gaussian and the _cor-

responding displacement is _{an error} function

= O

= U0n1 (11)

on S0

O

on z = O

The hydrodynamic _pressure is found from

the linearized Bernoulli equation

-

=

p(---+WVç)

(12)

The forces _on the body in the mode

j

due to _{an ap-}

plied boundary condition in the mode k _are found

from

Here, to is _a constant chosen _so

that

the value

of

((0)

is close to zero. The spreading parame-

ter a governs the width of the Gaussian function.

Taking the limit of _a

-

yields _a

6function

velocity, and a step function displacement, hence

the origin of the term non-impulsive.

where,

Pjk is the frequency-independent added mass

bk

is the frequency-independent damping

Cjk is the hydrodynamic restoring force

Kjk is the impulse-response _{or memory} function

The frequency-domain added _mass, damping

and exciting force coefficients and the impulse _re-

sponse functions of radiation and exciting forces

are related by Fourier transform. For radiation

forces,

Ak(w)

₌

_Pjk

₊

{FKjk}

Bk(w)

₌

bJk+Fc{JKk}

(18)

I<jk(t)

₌

J'{iw(Ak

_-

Pjk)

+

(B1k

-

bk)}

For the exciting forces, defined _as the _sum of

Froude-Krylov

(j

₌ 0) and diffraction

(j

₌ 7)

forces,

=

Fo(w)

+

Fj7(w) ₌

F{Ko

₊

I\7}

Ko(t)

₊

K7(t)

₌

'{F10

₊

F7}

Bingham, 1993, showed

that

the coefficients

of damping

bk

satisfy the relations

b k = O

forj=k

b _{k =}

bk

forjk

f(w)=f(t)

₌

f(t)

₌

1'f(w)

₌

J_t

f(t)e

JJ(w)e1

₍₁₅₎

The Fourier transform of the input function,

which is also _a Gaussian, gives the frequency con-

tent

=

eeZì

(16)

which shows the exponential damping of the high-

frequencies. Non-impulsive inputs _are also used

in the diffraction problem. For the details of how

the impulse-response functions _are derived from

calculations using nonimpulsive inputs, _see King,

et. al, 1988.

Cummins, 1962, shows

that

the forces _on the

body due _to radiated _{waves may} be written _as

F(t)

₌

/1k(k(t)

_-

bjkk(t)

_-

Cjk(k(t)

(t) _{= e}

a(to)2

Hence the off-diagonal terms are not in general

zero for bodies with forward speed. The terms

jij and bk are found from the solutions to the

simplified boundary value problems

O

1k,2kIonz=0

₌

O

Vt'1k,2k

0

_as (19)

a1

_k

=

an on S0

32k

=

an

on S0 as, Pj k

=

PffdSlkfli

₍₂₀₎ k

=

ffdS

_(2k

-

The Green function for these problems is (

-

The hydrodynarnic restoring force coeffI-

cients _are found from

_[PJfds.Vk)nj]

(21)

Discussion

of

the

Interaction

Terms

In developing this linearized boundary-value

problem, the implicit assumption is made

that

the

body geometry and forward speed _are such

that

the steady disturbance potential _o is small. This

must be _so in order to justify the linearization of

the free surface condition about the free _stream

velocity U0.

The steady flow has often been approximated

as W =

U0

î, assuming the perturbatin _poten-

tial to be of higher order, thus, avoiding the dif-

ficulty of having to compute the required second

derivatives of the steady-flow potential _to obtain

the rn-terms. With this approximation, _we have

ifi ₌

(0,0,0,0,

U0n3, _U0T1O).

The calculation of the _pressure in (12) _re-

quires the gradient of the unsteady potential. The

theorem derived in Ogilvie and Tuck, 1969, _can be

employed to calculate the force _on the body while

avoiding the need to take the gradients

fJdS[mj+nj(W.V)]

_d1

_ri(Ïxii).'

r'

(22)

Thus, the force _may be written

Fjk

-

pIfds

8k

I,

(23)

4

The line-integral _{terms appear to} be of higher _or-

der. (See Magee, 1991.) One of the hypotheses

of this theorem requires

that

the steady flow does

not penetrate the hull. If the hypothesis is _not

satisfied, _an extra term of the _same magnitude _as

the _one retained, and which _contains the gradient

which we hoped to surpress, remains in the sur-

face integral. Thus, it _appears incorrect to apply

the theorem, _as such, using only the free _stream

for the steady flow. In addition, the theorem gives

only the total force _on the body, and not the Io-.

cal _pressures, which _can be useful for

structural

calculations.

Alternatively, a source formulation can be

used to solve the integral equation. The formu-

lation for the linearized problem using _{a source}

distribution is given in Liapis, 1986. In this _case

the gradients _are available directly and _we do not

need _to resort to the theorem. Another possibility

is to use an equivalent source distribution,

that

is,

given the solution to equation (10), to _pose the in-

tegral equation for the source strengths

that

will

give the _same potential distribution. From this

distribution, the velocities _can be obtained. Fi-

nally, the possiblity of using higher order panels,

in which the potential varies at least linearly _over

each panel, allows _one to obtain the required _sur-

face derivatives by finite difference. This is the

approach which has been retained here and will

be discussed below.

The

Finite-Amplitude

Problem

In addition to the assumptions about the steady

flow, the amplitude of the motion must be small

because the body-boundary condition has been

expanded about the _mean position of the body

surface. In certain _cases, it _may arise

that

the

body's motion is such that the form of the im-

mersed portion changes dramatically _over time,

but

that

the disturbance _on the free surface _con-

tains only _waves of mild slope. Such would be

the case, for example, of a submarine, operating

beneath, but close to the free surface, _a ship _ma-

neuvering in moderate _{waves, or a} structure with

large flare _near the waterline. Second-order theo-

ries _may not be applicable in these cases, whereas,

the fully nonlinear solution _may not be required.

This leads us to formulate the so-called

finite-amplitude problem, in which the body-

boundary condition is applied _on the instanta-

neous body surface, but the free surface condi-

tion remains linear. We note

that

the form of the

Green function is the _same, regardless of the

_po-

-

sition of the _source and observation points. Thus,

tion in the _case of finite-amplitude problems.

For this problem _a globally-fixed coordinate

system is chosen, and

_a

single, perturbation _po-

tential is defined. The motion

starts

from rest _so

the conditions to be satisfied _are

Here, ç5r represents the incident _wave and 7

is the local velocity of _a point _on the body.

The integral equation to be solved for the

potential in the fluid is formulated _over the in-

stantaneous body surface, Sb(t), is (see Magee,

1991) 4

(P,t)

+

f

ds

d(Q,t

(

_-

8m

_r

_r

r

fi

1\3

=

Jds

-

(Q,

t) Sb(t)

r

r

8m

-

fdrf

dsQ[d(Qr

00 Sb(T) n

-

fdrfd1Q

[d(Q

r)]

VN(Q,

r)

g

_-

r(r)

8r

ör

where F(t) is the _curve defined by the instanta-

neous intersection of the body and the z

=

O

plane, and VN is

the

two-dimensional normal _ve-

locity in the _z

₌

O plane of _a point _on F.

It should be noted

that

in the fully linearized

problem the line integrals do not contribute in the

case U0

₌

0. However, for the finite-amplitude

problem, the line integrals have _nonzero values

whenever the shape _or location of the waterplane

changes in the fixed coordinate system. Also note

that

equation (30) is equivalent to equation (10)

for _constant forward speed in the steadily trans-

lating coordinate system using the _mean body _po-

sition. For _a body executing _an unsteady _maneu-

ver in the horizontal plane, equation (30) reduces

to

_a

form given by Liapis, 1986, Appendix A. The

pressure at a point on the body is most easily cal-

culated by numerically by following _a point _on the

body. The nonlinear Bernoulli equation _may be

written

P

₌

_P[(+.V+Vd.Vd

-

.vd+gz]

(30)

Here, V is the velocity of _a point on the body, and

the term (

₊

V

V)

represents the substantial

derivative of the potential following _a point _on the

body.

HYDRODYNAMICS

CODE

The computer code SIMSEA is further described

in Magee and King, 1992. It is used to solve the

integral equations (10) _or (30). The surface of the

body is discretized into _a number of triangular

panels. The potential is assumed _{to vary} linearly

over each triangle

where _s and t _are the local cartesian coordinates

situated _on the panel,

_a

is the index of the triangle

corners, and the shape functions are given by

Na(s,t)aa+bas+cat

(32)

where _a, b, and _{c are constants} determined from

the triangle geometry. The tangential derivatives

of the potential _are constant over each panel

3d 8s (31) 3d

daba,

_-

a=1

Taking into _account the contributions from

the singular integrals, the given boundary condi-

tion is applied and the integral equation is solved

for the unknown values of the potential at points

on the body. The solution proceeds by time step-

ping from rest. Once calculated, the potential and

velocities _are used in the Bernoulli equation (12)

or (30) to determine the pressure which is in

turn

integrated _over the body _to determine the gener-

alized forces.

The numerical evaluation of the right hand

side of the integral equation is performed by col-

location not at the panel _corners, but

_at

the _cen-

troids. Since there _{are more} panels than nodes, _an

over-determined system of equations results. This

is solved by singular value decomposition, Press

et aI., 1986, _so the usual back-substitution step of

LUD decomposition is replaced by simple

matrix

multiplication. The corresponding distribution of

singularities consists of constant strength _sources

and linearly varying dipoles which gives _a consis-

tent order of truncation for the flat panels being

employed.

The method of Cantaloube and Rehbach,

1986, is used to integrate

1/r

and

1/r'

and their

normal derivatives _over panels. The method _uses

3 da _Ca (33) o2

tT=Vç,

a,

Lç=O

=0

z=U (24) (25) 8n -: -. ad1 on Sb(t) (26)

n--

3m

Vç-0

ascx

k=1,..7

(27) 0,

-

O _{as t}

-

-

(28)

-

Stokes theorem _{to convert} the surface integrals

over flat panels into line integrals around the

panel edges. The required integrals _are given by

FxdÎQ

r(.r(ix.d4

+1

(34) r

J

r(ri.

f

i /dsç,

_-

r

where,

e±7

(36)

The sign of

i

is in general arbitrary, but _one

should have

i

₌ when F is in the plane of the

panel and it is preferable to avoid the situation

where

_r

_-

i.

F becomes too small. Given _{an ap-}

propriate choice of

i

the integrands _are not sin-

gular unless the field point P is on the cont6ur,

which is _never the case here. The line integrals

are computed using a standard Gauss quadrature

rule with different numbers of integration points

for large and small separation of _source segment

and field points. The analytical formulas of New-

man, 1986 _can also be used, and _appear to be ef-

ficient for triangular panels which _are elongated.

The integrals of the _wave terms _over the panels

are approximated by a one-point Gauss rule be-

tween panel centroids.

MOTION

SIMULATIONS

The resolution of the equations of motion in the

time-domain _presents several advantages _over tra-

ditional frequency-domain analysis. For strictly

linear problems, the time and frequency-domain

solutions _are related by Fourier transform, and

hence _are complementary. By contrast, nonlin-

ear external forces such as the action of moving

stabiliser fins, viscous and eddy-damping, _or, for

example, changes in the hydrodynamic and hy-

drostatic coefficients due to finite-amplitude _ro-

tations and translations _can be included in the

transient solution in _a straight-forward _manner.

In order to address this problem, _a tempo-

ral simulation code (see King, 1990,) has been

developed. The code consists of driver routines

(which integrate the equations of motion using the

method of Bulirsch et Stoer, Press, et al., 1986,

and subroutines (which calculate the forces _ex-

erted _on the body). Modularity is thus ensured,

and the force modules _can be modified and _up-

dated _as needed. Such _a modification _was needed

for the calculation of the Froude-Krylov and hy-

drostatic forces, _since the previous calculation

fssQ

was based on a strip method which, while simple

and efficient, lacked precision for pitch motions in

head _seas. The strip method _was exchanged for _a

three-dimensional distribution of panels and the

subsequent predictions _were improved. The _new

code has been rebaptised RATANA. Other exter-

nal nonlinear forces _can also be included in the

simulation by the addition of subroutines calcu-

lating the required forces.

The original approach taken is similar to

that

of Oakley and Paulling, 1974, in which the effects

of finite-amplitude movements, including the Eu-

ler angles for rotations, can be taken into account

in calculating the Froude-Krylov and hydrostatic

forces, while

_at

the same time the forces due to ra-

diated and diffracted _{waves are} calculated based

on the linearized theory. Generally, the former

two forces taken together give the largest contri-

butions _to the total force acting _{on a} ship. If

they _an be correctly predicted, the overall simu-

lation should be improved with _{respect to} linear

theory, without augmenting the calculation _cost

unreasonably.

In order to resolve the finite-amplitude dy-

namics problem, _an axis system

O'

_e», with its

origin at the center of gravity, and which _moves

with the body is defined. If the body movements

are linearized, the fixed and moving systems cor-

respond. The double prime system is useful for

dynamic calculations because the inertia matrix

remains constant in this frame. If the position

of the body is given by the generalized displace-

ment vector (z1, ...x6), then the transformation of

a point in the body-fixed system to a point in the

global system is given by (see King, 1990):

{

[T]

₌

X y z

x1+Uot

I _X3

where the matrix T is obtained by _{a sequence} of

transformations about the x"-axis, the intermedi-

ate axis V-axis, and then z-axis respectively, and

is given by E _C5C6 C5 S6 S4S5C6

-

C.i.S6 548555

+

C4C6 c E cos x, S, E sin _x, C4SC

+

₄₆

C4S5S5

-

S4C6

The incident _wave potential is defined _as

1(,t)=

i z Wn n (38) S4C5 C4C 8

(1

[/i(ixr.dîQ

8nr)

Ic

r(ri.)

-

(i.

r)

(j

xV)

.jln(r_i.rdQ

z,, } (37)

r is the amplitude of the wave component

ß, is its direction of propogation with respect to

the x-axis (ß

₌

_ir indicates head seas)

=

w/g

is the _wave number

Irregular seas are considered as a sum of the n

individual components and it is possible to _con-

sider multi-directional _seas. The Froude-Krylov

pressure combined with the hydrostatic pressure

is given by

oc1'

PI

₌

_p----

_-

_pgz

and the elevation of the free surface is

7/i

=

g Ot

1 Odi

z=U

(40)

To obtain the Froude-Krylov and hydrostatic

forces, _we integrate the respective _{pressures over}

the immersed hull surface. This integration _can

be performed either _over the _mean (linear) _or

instantaneous (finite-amplitude) body surface in-

cluding the finite-amplitude rotations and trans-

lations. We note

that

the expression (39) _can be

integrated analytically _over flat panels.

Much _controversy exists about the _pressure

above the _mean free surface. Linearized theory in-

dicates

that

the integrations should be performed

up to the plane z

=

0. King proposed to inte-

grate _up to the surface P1

₌

0. In addition, it has

been found useful, for _waves which _are long with

respect to the ship, to measure the coordinate z

starting

at

the linearized free surface z

₌

ij,

and

to integrate _{up to} this surface. Since the precise

form of the pressure is not known above the _mean

free surface, the option is left _open to the _user.

Ignoring the effect of the changing body ori-

entation, the radiated _wave force _may be written

in the form of Cummins, equation (17). The _re-

quired terms _can be obtained using the results of

a time-domain code or by Fourier transform of

the results of _a frequency-domain hydrodynamics

code such _as DIODORE with the formulae:

2r°°

Kk(t)

dw(Bk(J)

_-

bk)

coswt (41) 1 1

r'

Ak()

(42)

f1jkAjk(k)+

I _d

4/7J

i is the highest wave number for which the

added _mass is known.

Equation (42) is the result of an asymptotic de-

velopment due _to Greenhow, 1986.

(39)

7

The diffraction forces _are calculated in _a sim-

ilar _manner (again, ignoring the effects of chang-

ing body position) as

7]nFDn(Wn,ßn)e (43)

n

where PDn is the vector of generalized exciting

forces, The ramifications of calculating the diffrac-

tion force _on the _mean body surface, while the

Froude-Krylov force is calculated using the in-

stantaneous position remains _a point for further

investigation.

The differential equation system to be solved

is given in King, 1990. Note

that

the Froude-

Krylov and hydrostatic forces in the right hand

side of the equation _system depend _on the position

of the body while the radiated wave force includes

convolution integrals which depend _on the body

velocity. Using a predictor-corrector method, the

position and velocity _are unknown, _{a przorz,} and

are determined only in the _course of the solution.

Since the major

part

of the CPU time is spent on

calculating these forces, it is important _to min-

imize the reqired number of function calls. In

order to reduce the calculation time, the future

values of the velocity and position _are predicted

one time step in advance. From these values, the

maximum entropy, _or all-poles method, Press, _et

al., 1986, is used to predict the future values of

the external forces, which _are then interpolated

linearly _over the intermediate time steps. This

method has _proven accurate and efficient.

The eventual goal is to integrate the large-

amplitude hydrodynamics code into RATANA _as

a subroutine in order to determine the finite-

amplitude effects of the radiation and diffraction

forces _on the motion predictions. While this work

is still in _course, much has already been _accom-

plished in the re-gridding needed to calculate the

incident _wave forces.

NUMERICAL

RESULTS

The emphasis here is _on the linearized hydrody-

namics code, and on the finite-amplitude effects

of Froude-Krylov and hydrostatics. The finite-

amplitude hydrodynamics problem has been

treated in Beck and Magee, 1990 for submerged

bodies, and in other references cited in the intro-

duction.

In what follows, the coefficients _are made

nondimensional by _p, the density, _g, the accel-

eration of gravity, L the nominal length of the

ship (L

₌

121.9nz), and the _wave amplitude A.

The phases of exciting forces are given in degrees.

The _wave frequency is noted _w, the wave number

k

_{_}

w2/g, and the _encounter frequency _We.

Figure 1 shows the nondimensionah heave

and pitch added _mass and damping and _{cross cou-}

pling terms as functions of the nondimensional

frequency for _a Series 60 (CB

₌

0.60) hull form

at

zero forward speed. The Fourier transform

of the time-domain calculations, (labeled SIM-

SEA in the figures) _are shown for two distribu-

tions of 792 and 490 triangles _on the half body.

Frequency domain calculations obtained with the

code DIODORE _are also shown for comparison.

The magnitude and phase of the exciting force _co-

efficients in _surge, heave and pitch for head _{seas as}

functions of the _wave number _are given in figure 2.

This ship _was chosen because experiments _are be-

ing performed _at the Bassin d'Essais de Carènes

by J.F. LeGuen and at St. John's University,

Newfoundland

_one objective being the valida-

tion of seakeeping computer codes. Only motions

were measured, so no

data

is available on the in-

dividual hydrodynamic coefficients.

The _agreement between time- and frequency-

domain results is good for the heave-heave and

pitch-pitch coefficients, but slightly _worse for the

B35 and B53 cross-coupling coefficient. The time-

and frequency-domain results should be identical,

but slight numerical differences _are still present

at this level of discretization. The coefficients

obtained with the time-domain method _converge

rapidly, and little difference _can be _seen between

the two discretisations shown. The frequency-

domain calculations _are shown only _up to a value

of non-dimensional frequency of about 5.5 because

the _presence of irregular frequencies disturbs the

results in the higher frequency _range. While also

present in the time-domain calculations, the ef-

fects of irregular frequencies _are less pronounced.

The exciting forces in head _{seas are} relatively

small above the first irregular frequency, for this

relatively slender ship

(L/B

₌ 7.5), _{so as} far _as

motion predictions _are concerned, the irregular

frequencies do not cause any problems since there

is _no significant _movement in such short _waves.

In figures 3 and 4, the radiation force _co-

efficients _are shown for the _same ship at

Fn

₌

0.275. Here, three _sets of calculations _are shown

from three different computer codes. TIMEDV is

the time-domain code described in Magee, 1991.

Constant strength singularities and fiat quadri-

lateral panels _are used. The Ogilvie-Tuck

theo-

rem is used to calculate the forces due to inter-

action terms. As decribed above, SIMSEA _uses

flat triangular panels with linearly varying dipoles

and constant strength _source singularities and the

8

-

derivative of the potential is calculated by differ-

entiating the shape functions for the potential.

DIODORE _{uses a} potential formulation with

constant strength singularities over flat quadrilat-

eral panels. The boundary conditions applied _on

the body surface _are modified with the _assump-

tion of slow forward speed to include _some of

the interaction terms due to the forward speed,

(notably in pitch) but the _zero speed frequency-

domain Green function is employed. For de-

tails, _see Coudray and LeGuen, 1992. The theory

lies somewhere between the classical frequency

of encounter approximations and the fully cou-

pled frequency-domain approach, which _uses the

Green function with forward speed. This lat-

ter formulation _was used for comparisons _pre-

sented in LeGuen, et al., 1992 for modified Wigley

hulls. Unfortunately, the method is not yet reli-

able enough to be employed in the _same fashion

as the slow-speed method, and the CPU cost is

relatively high. Thus, it _was decided to _use this

show-speed formulation here.

In comparing the results of TIMEDV and

SIMSEA, the general character of the _curves is

the _same. There is _a peak at the frequency which

corresponds to Brard number

_r

₌

Uow/g

₌

1/4.

In general, the results of SIMSEA show fewer os-

cillations, which _may be due to the

treatment

of

the interaction terms, and due to the fact

that

the

results _are better converged. The discretization

employed in the two calculations has the _same

number of nodes distributed _over the body, but _a

different number of triangles than quadrilaterals

results. It _appears

that

the _cross coupling coeffi-

cients _are anti-symmetric about _{a non-zero mean}

value.

The results of DIODORE _agree generally

with the time-domain results for the heave added

mass, but lack the distinctive peak, while the

pitch added _mass and damping coefficients show

quite a different character. In figure 4, the A35

and B35 coefficients _appear close to the time-

domain results for high and low frequencies, while

the A53 and B53 terms, (which _are equivalent to

the zero speed results of figure 1) _are different.

The exciting forces in head _{seas are} domi-

nated by the Froude-Krylov forces which _are iden-

tical in both time- and frequency-domain meth-

ods. It is thus of less interest to show _a compari-

son of the total exciting forces, which show good

overall agreement between the two methods. The

results of SIMSEA again _appear to _converge more

rapidly with increasing number of panels, but this

is not _a major _concern since the Froude-Fírylov

tian, and hence is not _a

major

part

of the _com-

putational cost. It _may be economical to use a

larger number of panels to calculate the Fraude-

Krylov force and hence to obtain

better

overall

convergence of the transfer function, without _aug-

menting the cost incurred in calculating the other

hydrodynamic coefficients _{on a} refined grid.

In contrast, the diffraction forces, while small

in magnitude, _can play _an important role in de-

termining the phase of the exciting forces. The

diffraction forces for the Series 60 in head _seas at

Fn

₌

0.275 using SIMSEA and DIODORE _are

shown in figure 5. Large differences, _as much _{as a}

factor two, can be _seen between the results. The

frequency domain results have been truncated just

before the first irregular frequency. For this ship

with forward speed, the irregular frequency is

found at a relatively low value of the _{wave num-}

ber because of the frequency of encounter shift.

It _occurs in the _range where the diffraction force

has not yet decayed to _zero and thus adversely af-

fects the solution. The time-domain results show

no apparent effects of irregular frequencies.

Given the coefficients

pj,

bjk, Cjk,

K,

and

the diffraction force coefficients _as input, the

equations of motion _are solved directly in the

time-domain using the code RATANA with the

linear _or finite-amplitude approximations. For

finite-amplitude calculations, only the Froude-

Krylov and hydrostatic forces _are determined _on

the instantaneous body surface, with the Euler

angles taken into account.

The results of _a typical calculation for the

Serie 60 at

Fn

₌

O in regular head _{seas are pre-}

sented in figure 6. The dimensional time histo-

ries of _surge, heave and pitch using the linear and

finite-amplitude options _are shown. The right-

hand set of figures is _an expanded view _near the

end of the record. The frequency considered _cor-

responds to the peak in the pitch _response, and

the _wave length to height ratio is

\/(2A)

is _ap-

proximately 24.

After _an initial transient, during which the

wave forces are augmented gradually to avoid

a sudden

start,

the motions settle down _to

_a

steady-state condition. The linear solution is _per-

fectly sinusoidal, but several features of the finite-

amplitude result show nonlinear behavior. There

is _{a nonzero mean} value, particularly remarkable

for the _surge, but also present for the heave and

pitch. An autopilot is used to control the low fre-

quency surge motions. It is described in King,

1990. and consists of _a spring and filter. With-

out the autopilot, the ship receives _{an initial} im-

pulse and continues _to drift in _surge due to the

effects of the initial transient. Higher harmonics

are also present in the finite-amplitude solution.

For this _case, the second order components are

approximately 7% and 12% for heave and pitch

respectively, and 1.5% and 1% respectively for the

third-order _components.

The RAO's _or linear transfer functions for

the zero-speed _case, obtained by Fourier trans-

form of the results of RATANA using the lin-

ear approximations, are shown in figure 7. For

comparison, the results of frequency-domain cal-

culations and three points of experimental

data

are shown. The effects of the autopilot _can-

not be not included directly in the frequency-

domain solution _so the _surge transfer function is

affected _at low frequencies. However, the heave

and pitch transfer functions obtained from the

time- and frequency-domain calculations do _agree

with nearly graphical _accuracy, validating the cal-

culations of exciting forces, radiation forces and

integration of the equations of motion. The _re-

sults _agree with the experimentally observed val-

ues to a reasonable precision, justifying the use of

frequency-domain codes for bodies without for-

ward speed in head _seas.

To _see the finite-amplitude effects, figure 8

shows the linear transfer function _{as a} function

of _wave amplitude for _a single frequency, _near

the peak in the pitch _response. Note

that

the

vertical scale is exagerated. As expected, the

finite-amplitude results tend to the linear _ones,

in the limit of small amplitude. This limit is _not

exactly respected, but the _{errors are} sufficiently

small to provide _a base from which _{to pursue}

finite-amplitude effects. The fact

that

the _exper-

imental results fall within the _range of predicted

values and

that

the agreement is improved with

the finite-amplitude model (except for surge) is

encouraging, although the single amplitude avail-

able in the experiments is _{not a} sufficient test.

The trend of decreasing transfer function with

increasing _wave amplitude also _seems physically

reasonable.

In figure 9 the comparisons between calcu-

lated and experimentally observed transfer func-

tions for the Series 60

at

Fn

₌

0.2 in head _{seas are}

given. Two sets of experiments _were performed:

with _a heave staff and under auto-propulsion. The

latter

corresponds _more closely to the calcula-

tions using the autopilot. No phase information

is available for the auto-propulsed _case. Over-

all, the agreement _appears to be good, except for

the over-prediction of the pitch transfer function,

which is typical of most seakeeping codes. The

duce the transfer function _{appears to} be _a step in

the right direction.

CONCLUSIONS

The time-domain method is viable for seakeep-

ing calculations for ships with zero and non-zero

forward speed. Although the present study was

limited to the Series 60 hull form, previous work

indicates

that

the method _can also be applied to

more modern forms. The code SIMSEA offers

the advantage of direct calculation of the _spa-

tial derivatives of the potential _so

that

the lin-

ear and nonlinear forces on the body can be ob-

tained without resort to the Ogilvie-Tuck theo-

rem. Three-dimensional frequency-domain codes

also work well for _zero forward speed, and _are

more highly developed for industrial applications,

but suffer _more from the effects of irregular fre-

quencies which _can be detrimental because of the

frequency of encounter shift at forward speed.

The time-domain equations of motion

present a wide range of possible applications, in-

cluding the effects of finite-amplitude motion _on

the hydrostatic and hydrodynamic forces, and

the influence of external nonlinear effects. The

coupling of the finite-amplitude hydrodynamics

code and the equation of motion solver will al-

low conclusions to be made concerning the effects

of changing geometry _on the motions. Future _ap-

plications include the effects of moving stabiliser

fins which _can be taken into account directly in

the transient solution, without first linearizing the

forces, _as is typically done in the frequency

do-

main.

Acknowledgements

This work is the result of research supported

partly by DGA/DRET, under contract number

93/2029J. This _support is gratefully acknowl-

edged. Thanks also _{go to} the original Ratana.

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2.5 2 1,5 0.5 o _{0.08 0.06 0.04} 0.02 DIODO RE SIMSEA NP=792 SIMSEA NP=490

-o 2 4 w IL/g 6 8 0.12 0.1 0.08 0.06 0.04 0.02 2 4 w/L/g

Fig. 1. Added IflSS and d amping versus frequency for a Series 60, C11=0. 6 huh form, Fn().

6

8

FUL pg V A IF5 I pg V A 1.5 0.5 20 15 F3 IL pgVA L5 0.5 5 10 15 kL DIODORE SIMSEA NP=792

-

SIMSEA NP=490 O 180 go -90 -180 180 90 -90 -180 180 90 o -90 -180 20 25 0 5 10 15

Fig.

2.

Exciting

force

coefficients

_versus

_wave

number

for

_a

Series

60,

CBO.6

hull

form

in

head

_seas,

Fn0.

pV DIODORE SIMSEA NP=792 _{TIMEDV NP=423 O} 02 0.15 0,1 0.05 0 02 O

Fig. 3. heave and pitch added

mass and damping, Fn=O.275.

pVL2

0000

0.2 _0.18 0.16 pVL2 0.14 0.12 0.1 O 0.08 0.06 'e

.

0.04

.

000000000

0 o 2 4

WeL/g

2.6 2.4 2.2 2 1.8 1.6 1.4 12 08 0.6

08 0.6 0.4 0,2 0 _{-0.2 -0 4 -0 6} -0.8 0.8 0.6 0.4 0.2 O -0.2 -0.4 -0.6 -0.8 o -'

.

-DIODORE A SIMSEA T'P=?92 - _{TIMEDV Nr=423} O _O 2 weJL/g 4 6 0.8 0.6 0.4 0.2 O _{-0.2 -0.4 -0.6 -0.8} 0.8 0.6 0.4 0.2 O -0.2 -0.4 -0.6 -0.8 o 2

Fig. 4. Cross-coupling a(l (led

mass a ud (la unping for heave and pitch, Fn=O.27.

w/L/g

DIODO1tE SIMSEA NP=792 -TIMEDV NP=423 O pV L B pc7L/g/L

I' IL pgVA IF. I pg V.4 1.4 1.2

i

0.8 0.6 0.4 0.2 O O SIMS EA DIODOR.E -180 180 90 -go -180 180 90 O -9° -180

Fig. 5.

Diffraction

force

coefficients

in

head

_seas,

Fn=O.275.

1.2 180 90 0.8 F1 IL 0.6 _o pgV4 0.4 -90 0.2 20 25 5 10 15 kL 0 5 10 15 20 25

2 Tli O -0.05 -0.1 -0.15

f

I

f!

f

f

f!

I _i I 0 50 100 150 200 250 TIME (SEC) 77i o -2 -3 -4 -5 1.5 0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08 -0.1 230

Fig. 6.

Motion

simulations

using

RATANA

in

regular

head

_seas,

Fn=O.

-6 2 1.5 L L LI L _i 0.5 773 0 -0 5 -1 -1.5 -2 0.15 0.1 0.05 775 0 773

I'I

II?IIIILI

775

f'

1ff!

f!

240 TIME (SEC) 250

'Il A '73 A 2,r A 2 1.8 1.6 1.4 1.2 0.8 0.6 0.4 0.2 0 0.9 0,8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 ç) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 O o O

a

0

4' 'aa& 4

AI

1. FREQ DOM RATANA LIN O EXP BA

+

3 L ¡g 4 Si .4 Si 2,rA 0.48 0.46 0.44 0.42 0.4 0.38 0.395 0.39 0.385 0.38 0.375 0.37 0.365 0.625 0.62 0.615 0.61 0.605 0.6 0.595 0.59 o

Fig. 7.

Linear

transfer

function

versus

Fig. 8.

Transfer

function

_versus

_wave

freuuency.

Fn=O.

amplitude

for

a

single

frequency,

Fn=0.

0.01 0.02 0.03 0.04

.4 1.4 1.2 0.S 0.6 0.4 0.2 O 1.4 1.2 _-

rL

0.8 2wA 0.6 _- 0.4 0.2 _- O

.

N' -50 -100 -150 -50 -100 -150 150 100 50 'lis O -50 -100 -150 -

/

Fig.

9.

Transfer

function

for

the

Series

60,

CB=O.6,

Fn=0.2.

RATANA EXP BA STAFF EXP BA AUTO O

+

'1'l 150 100 50 ° 0.8 .4 0.6 0.4 0.2 9 :3 Lfg 4 k3 150 100 50 O 1.4 1.2