The drift forces and moment on ships in waves

(1)

THE DRIFT FORCES AND. .NT ON SHIPS IN WAVES

by J.. N. Newman

David Taylor Model Basin Washington,. 'D, C.

June .965

ARCHIEF

y.

SchpLbouwbnde

Technische Hogeschool

(2)

ABSTRACT

The second-order steady: horizontal forces and vertical moment are

derived for a freely floating ship in regularwaves. The fluid is assumed to be ideal and the motion is linearized Momentum relations are used to

derive general results for an arbitrary ship or other body., in terms of the Kochin function, or far-field velocity potential of the body. Explicit

results are derived for slender ships.,. based .upon the ássumptions of.. slender body theory. Cbmputations. are made for a series 60 hull and are comparedwith experiments.. The analysis of the vertical moment permits

the prediction of stable heading angles in oblique waves, .and it .is shown

that unless the waves, are very short., the ship will be stable only in beam waves.

(3)

F' X F g H(e) n

r

t Cartesian coordinates

Heading angle of ship to waves (head sea =180°) Free surface elevation

Incident wave amplitude

:Shjp beam

Longitudinal force

Lateral force

Gravitational accéleration

Kochin function, defined by equation (20)

Waterline moments (n 0, 1, 2,)

Angular momentum vector

Wave number, K. = w2 1g

Ship, length

Linear .- momentum vector.

Vertical (yaw) moment

Unit normal out of. fluid

Flui4 , pressure

Polar radiús, :

'Position. vector'

Time ' '

Norma1. velocity, of' control surface "Fluid velOcity vector

(4)

Heave amplitude Pitch amplitude

e Polar angle, .e =.tan (y/x)

X Wavelength

p Fluid. density

+ Velocity potential

(5)

INTRODUCTION

When a ship is floating in the presence of an incident wave system,

pressure forces and thoments will be exerted on the ship and will influence its behavior. These forces and moments will include not only the conven-tional unsteady exciting forces, which give rise to the oscillations of the ship in waves, but also higher-order steady forces due to various non-linear effects., These latter forces are generally.too small to influence

the oscillatory thotions of the ship but they can nevertheless be important

in certain circumstances, particularly in considering those degrees of freedom of the ship which do not have a static restoring force (surge,

sway, and yaw). If the ship has a substantial forward speed then

effective restoring forces exist in these modes, through the actions of the propeller, leeway angle, and rudder, respectively, but at zero or very low

speeds large excursions will result from even a small wave-induced force

if it is steady over a long interval of time. We shall refer to this situation loosely as Hdriftingtl, and to the drifting forces and moment as

those second-order steady components of the wave-induced force in the horizontal plane and of the corresponding vertical (yaw) moment.

Clearly the drift velocity of the ship will be governed by the mag-nitude and direction of the drift force, while the drift moment will

govern the heading angle of the ship with respect to the waves. The prediction

of the transverse drift force and the drift moment are of importance, for example, in the design of bow-thrusters [1], if it is desired to maintain the mean position and heading of an unrestrained ship at zero speed in

(6)

predicting the stableheadtug angle of.the ship.

A theoretical analysis of drift forces has been .provided by Maruo [2], but the corresponding moment was not. included

a4

no.coputations were ma4e On the other hand, Suyehiro _[31 analysed the yaw moment.in order. to predict t!e stable heading angle of a drifting ship, .but in that analysis

hydrÓ-dynamic effects were. neglected entirely in favor of gyrostatic moments

From the experimental viewpOint an .thvestiga.tion. was undertaken by; Spe,ns and Lalangas [4],inwhich meaSurements.were made of.the transverse drift .force and the drift momént on aSeries 60 model atone heading angle

In,the present...paper.we shall.derive general analy4cal equa.tions for. the drift forces and moment on an atbitrary body, and specific formulae for a slender ship. Calcúlations will be presented to illustrate.the

application of these results to a Series60 hull form, and comparison will be made with.the experiments of Spens and lalangas. This cOmparison of

théGry -ad-perimeñt :is only fair, and additional .experimental . conf irma

tionis needed.

Our analysis makes use Of the usual techniques of. linearized water

wave, problems, assuming the f luid to beincoesstble aúd inviscid and the wave amplitudes to be smalicompared to the wavelength and

character-istic ship dimensions. MOmentum relations are utilizéd to express the drift forces and moment in terms of. the farfield disturbance of the ship, thus avotding the necessity to determine the secondorder.disturbance in

(7)

We write

wh e r e

THE DRIFT FORCES AND MOMENT FOR' ARBITRARY. BODIES

vZ

I)

J

ReL

' V(xJy 2)

is the (linearized) potential of. the incident.wave.system'and

+B is the disturbance due to the presence of. the body. Here g is the acceleration of gravity, K =

uP/g

is the wave number, A is the wave amplitude, and

the.angle ofwave incidence relative to the x-axis.

The potential +B is determined as the solution of.the boundary

condition

(1)

(2)

(3)

We consider, a floating body with no forward velocity, on the surface of an ideal incompressible fluid andin the, presence Of an incoming, incident

plane prorèssive.wave system. Let (x,.y, z)be Cartesian coordinates with

z = O the

plane.

of.the undisturbed free surface and the z-axis positive

upwards. The fluid motion is assumed to be harmonic in time with the

frequency of. encounter w/2rr _-J

The fluid vefocity canbe represented by. the gradient of a velocity

(8)

--dt(7»,

-s+sps.t.s,oQ

7Ccc(Ili) \)

P( co:(ny,,/

(6)

±I+u

(4)

on.the ships surface,.where is the .normal.velòcity of.thatsurface, .and of the boundary condition

--o

o,

O

(5)

Equation (5). is the. linearized free surface condition (cf.Wehausen and Laitone [5]).

In addition. +B.st.5a.sfy

equation throughout

the fluid domain,.a radiation condition of outgoing waves at infinity, and

yield vanishing velocity, as z -. -.

general the solution of this boundary value.problèrn is unknown and

we must restrict ourselves to. simplified bodiessuch as-thin or slender

ships in order to find

+6 explicitly. - Before doing so, however, we can obtain certain general results for arbitrary bodies, and óur.task is

simplified considerably by employing the. far-fiéld asymptotic form of. the

potential .+.valid.at.large distances from the body.

In order to..relate the mean force, and moment acting on the body to

the far-field potential w consider the rate of change of. linear and angular. momentum within the fluid domain bounded by the shipVs wetted surface, _{free.surface, SFS} , and a control-surface at.inf.inity,

As deriyed in Appendix 1.the rate of change of horizontal.linear momentum is given by .

(9)

and the rate of change of the vertical component of angular momentum is

dt 2

Here p is the total. flUid pressure, p is the density, eos(n, x) and cosn,, y) are the direction cosines with the direction of the unit.. vectoi'

taken as positive out .of the closed surface

_{5B +}

+ S , is

the normal velocity of the surface., Ir is the position vector, and the

subscripts x, y, z denote componènts of vectors, 0m the body surface

SB and' the free surface SF thé normal velocities of the body and fluid

(p(rxnp(irxv)(vuj]ìs

' (7)

are equal','or

V - U

=.0 We.shall take the control surface S to be

n

.n'

' ' '

fixed, so on tl± surface U= O The contribution fromthe pressure t.er ofl 'S is just-the desired force or moment:

cg

ot)

¿Ç

c.oi(y)

C/S

,i.=ffpo1rxc1

Finally, on 'th'e free surface 'the pressure v4nishës. Thus we obtain the föllowing results for the above forces:

-(8)

(9)

(10)

Next we také time averages of the above equations, to ôbtain the mean

forces and moment. There is no contribution from the. last terms since the -motion is periodic arid there can be no. net increase of momentu.in the

control. volume from one cycle to another Thus

(Ir)(n,)

p(iV)

Vj0Lc

-

IS

[f

-\JV,

j 0Ls

-

f1x)

VV]

Nèwman [6]).

-- We now take the control surface. S

-. - .

-R?

_-

_ff

(i, j

*f

V 7

Equations (l+. -.1.5) can also e derived from conservation, of energy. (cf.

to consist of a circular cylinder (.12)

(-13)

of large tad-ius about.the z-axi _{and extending from the free surface. down}

to z.=-

(The.contribution from.a horizontal..closure Of can

be neglected sinëe the fluid veloàityis required to vanih .there)

(14)

('15)

(11)

We take (R, 8,.z) as polar coordinates, with x=R cos e and

y=R,sin 9

. After.making the appropriate substitutions..in (14.-16) we

then obtain, the results

SiMs

\,v

e1Gh

The above -equations are exact since no. linearization has. been employed,

.but.to.proceed further.we must assume that the wave height is small., and

we shall only retain contributions to the forces and moment which are of

se.cond. order in the incident wave amplitude.

The pressüre is giien .by Bernoulli's equation

8 -V9

ç,

e)]RcIe

(17)

(19)

and the velocity components are

t,

(.21)

V

1

(22)

+

(12)

The far-field potential. 'is of the form

where

expfr[

I,? c.s,'(2-eJ]

and where

*

and. thus, it follows that

/7'('ir#9)

_{¿'p (c ('kfri7r/4) .}

(23)

(25)

Here H(e) is the Kochin function and the integral is over the submerged

surface of the ship. Equation (25).is derived iñ Appendix IL

Since in the far-field both and _+B' have exponential..dependance

on thecoordinate. z ,it.is convenient first.td càrry out the..z-integration

d2

11Loe1 j-o()

-

_Tvi

J2o

*

The infinite lower limit does not contribute to the hydrostatic pressure

-.afterintegrating with respect to e

H(e=

-

d

(26)

in.equat.ions (17-19).. For-this purpose we 'note that the upper limit of

-the z-íritegration is the free surface

-'

-

Z:

(13)

Thus,.rieglecting.terms. of order

?

8KJ0

LLR

ae

r !#

-j

Ld,e

e r2rr

±_ ì±±*.

¿ g1-J0

L L,

-k ++ Re

cJ L ¿,

e

RI&

RK°

R

6

where we have used the fact that

+

-s

an4 asterisks denote.the complex conjugate.

Finally it is necessary to substitute the far-field potential (23 25) in (27 -29). Since(27.- 29) are qudratic in the. potential

(.29)

and its derivatives, thé restílt. will ïnvole terms which te quadtatic in .the potentials -and ..sep4rately plus cross-terms. involving products

of and . The contribution from alone must vanish since there

can be no forces or. moment associated with. the.undisturbed .iñcident wave

(14)

and

performing

the necessary operatiOns. Thus, we need only consider the cross terms and the terms. which àre quadratic in

integrals are

G) j co.c

9 c/

f

IC

9ir

Lf)A

R

(cosfl#c.os9)

H(n9)

exp

_{fíkR 1'-}

C-ii&- e)j-tí7r/j 01e

_{Q (R}

'/aJ

(30)

J/Hfrt9))2

,

9 19

f4I4

(/<)¼.

RJ«(g

_{) H(-e)}

exp'fi<RtI-o(fl-o)]*L1rÌdO

O(R)

The moment integral must be.treated more carefully. 'First let us 'consider

the cóntribution from 4uadratic terms in

JO

-

_Rej

i--'2:o

j

rQir

í:_

.ij

ye) d

9ir

o

(15)

where a prime denotes the derivative with respect to e Next we consider

the cross terms involving

4B and RfL

aì

-df Re

r2' #:

jo L ç2ir

-

(ì.,+ coi

ee) H'(r*9)

xp{/RL;-

co

fl9)]+7ri/í S9

Combining these results it follows that

,y

#r 8

-

]1ede

;;

/1 (9)fr/ 19) S

-

2Tr 1A

)

.rL (/-)1

H'(ir*)

exp{keL/-ô',,4-eJJ

1r/

d9

o(A7-'2)

The integrals in (3Q - 32) involving R can be evaluated by the method of stationary phase (cf. Stoker, [7]). Thus

e(e) ex? f-d*-e cs(/' 9))

(2ff

\YZ

=

)

_[e

* ff/4

+ e

--

o((KRy)

n)J

(32)

(16)

Applying this asymptotic approximation to (30 32) we.obtain the forma.la.e

=

/H9)j c9d&

-

A c&

Ir»

H(rifl.)

(33)

'

f2ir

.2.

/H(6))

cì9

pA

S/

_{Ih H (7r*f2)}

I

F(9)W'(6) c1

where Hv(rt ₊ ₎ _{is to be interpreted as} -i

Li.«Je

= 1-r + ,

These are the desired results for the drift.forces and moment on an

arbitrary body.

The force eqt.lations (33 - .34) can be put.in a

tore

compact.form,

ollowingMaruo [2], if we assume that there is no net work being done on

the bödy by external forces. Then from conservation of energy, (cf..

Stoker [7]) .

(34)

Re /7"(7rfl)

(35)

(17)

and direct reduction of this integral, by the same methods employed above, leads to the relation

Im

i

_/i-,'('e))

_1&

4,rA)

'4

Substituting (36) in (33 - 34) it follows that

pk2

ir

=

;; [

/H) (c

'9

/

7tr

JW(&J)

(18)

Let uß now âpply the results derived, in the preceeding sect.on. to a

slendér.' ship. The Kòchin fûnction is

and the potential satisfies the boundary condition

-

ç,

'(40)

where 'u is thé. normal velocity, on the body. For asleñder.'body y. n

z .are o(e) and moreover

«

(cf. Néwman and Tuck [8]). Thus in

the limit -the Kochin function is

!

)s

= LI

Q()e1

(41)

where Q =

- dl

is the flux through the. transverse section C

f the six amplitudes of the ship's motion are C.e iwt (j = 1, .6

for surge,. sway, heave, roll, .pitch,.and yaw,.respectively) it.follows

[8] that

THE FORCESAND'MOMENT''ON4.SLENDER.BQDY

g

xp

fr(

ti

yrt')]

d

=

and thus that

H)

-1

(42)

(19)

'Here B(x) is the beam of the ships waterline. Since , B(x) =,O(e)

H(9) = O() and the error in (43) is O(2 log. s) From (43)

I HfrJ

- -

Re fß(r) (-x)

/Çwce.fl

ax

j

f5(x) (-x-) &&

,]

z'(r-)&

+o(

(44)

but. fróm (36) 'the left-hand-side of (44) must be Q(s) , so that the'

integralj. in

(44)

must vanish, to order e , unless work is done on the ship

by external forces. This conclusion can be checked directly, for

uñrestrained motions in waves, where the first order equations of motion are [Ö]

f 2x (xr)dx

(46)

The drift forces

T

and can be found conveniently, from ('37) ànd

(38') if, we write

(20)

.Sincethis-nan even functionof e it follows that

3r

_(C*s

f

g()[ik

rJ

añd, similarly

k

3fe()

[

-,r

+7

(47)

4 k'f fß()[I

, ,

fe

*

/

_J

_e'

CO

9 k3

fe

f (J [e

(i-j

(48) li

í3esei

oC

tL 4-

k;4,

We note the similarity of these integrals with the damping coefficients for

a slender ship. Jndeed if = 900 _{the integrals in (48) are identtcal} to. those occurring.in the damping coefficients for pitch and heave in

ca1mwater. Thus it follows that f or.beam wàves

f&'t [A -tC4S(2

'''j d'

(21)

i.

g3

-

_2/(i

from the fact that in beam waves a .slènder shi,' follows the orbital motion of the wave.

We turn finally to the drift.momentacting on a slender ship. For (35) we. need,, from (43)

-

weotfl]alx

(/oJ.

1)k) OE35.

Refx&x(-)e'

J

(49)

where = heave dmping coefficient

B35 = pitch-heave cross-coupling damping coefficient

= pitch damping coefficient

However for free motion n beam waves, = iA and

=0 from the

solution of (45 - 46), and thus (49) vanishes, Physically this results

(50)

In general. this intregral is non-zero, unless the. ship is restrained. _Thus

there is a çontribution. to the d.tift moment for a slender ship..of the same

order as the beam, or

0()

:

(22)

[ß

.r, - 1;

) e' '

where the constants I are the waterline moments

n

f()

Substituting (52 .53). in (51), we obtain, the expressiön

r -zJ

[kr

-

f) co

(i)r

O (c

a)

In the simplest case of a ship with.rectangular wateplane,.so that B(x) = B.= constant ,.the constants I are

BL

Il O

= L'/12

(53)

'(54) For free motions in waves we can substitute the first order

displacements and in (51) . From (45 46) it follows that

(23)

where L is the 1ength Thus, in this special case,

Here

/

fl /

SL

cflf

L

3 a

-

k2g

_,

(/-spherical Bessel function.

The.essential features of (55) are.readily determined by. considering the product

j1(x)

j2(x) For small x

-g

(x8't')

X3

Fòr large X

ii

)á2()()

j(z) =

(z) is the

and j1 j2 = O at each of .the zeros of j1 and j2 The f irst few. such

(24)

zéros are, approximately,

x=4.49,5.76, 7.72,9.09, 10.90,12.32,

Thus we can draw the following conclusions regarding the dtift moment

0) =0

for 0<<

For short-er :wave lengths will change sigti.in,quadrant 0-< < 900 A typical graph of the drift moment as a .fuction of

of the form shown in Figure. 1.

Thé requirement that -the ship. be stable is that

and

O <. < 4,49

or

IL>

0.70

is therefore

M = O

and. the first

z

non-vanishing dèrivative is positive. Thiscondition is met by = 90

at allwavelengths, só that the ship is ahays stable in.bearnseas. For

X/L> 0.70 this is the only stable condition, but for 0.55.< .< 0.70 it.is alsostable in head or followingseas. For shorter wavelengths points of. stability will occur at intermediate angles. This solution may

be.compared with the studyof Suyehiro [3},who.considered the non-linear

gyrostatic coupling between roll. and pitch but ignored all hydrodynamic phenomena. Oi this basis Suyehiro concluded that ships:would be stable only in beam seas provided the frequency of èncounterdid not. satisfy the

inequality

w

(25)

while it would be stable only in head waves if

w

<w<w

roll pitch

Here w and w denote the natural frequencies in the two modes.

roll pitch

If the frequecy of encounter is equal to one of these Suyehiro predicts that the ship will be neutrally stable at all headings Experimental

evidence is given by Suyehiro with a small model but the data is dimensional

and the model length is not given. Thus it is conceivable that Suyehiro9s

experiments are in agreement with the present analysis, as illustrated in

Figure 1, and the agreement ;with his own theory is fortuitous This seems

likely to be the case for practical ships which are relatively slender, since the inertial forces considered by Suyehiro will then be of much higher order (in terms of the slenderness.parameter e) than the moment

which we have analysed This suggestion is confirmed by the experiments of Spens and Lalangas [4], who computed the gyroscopic yawing moment considered by Suyehiro, and found it to be very small compared with their experimental observations.

Finally we note the following symmetry properties for an arbitrary slender ship, with respect to heading angle, which are readily derived

from (47 - 54): MZ( =

=

-=

(-)

-= y(iT

=(rr -) (56) (57) (58)

(26)

CALCULATIONS FOR THE SERIES 60 MODEL

In order to determine the drift forces and moment for a realistic ship hull, a computer program was written for the I.B.M. 7090 digital computer, based upon equations (47), (48), and (54). Values of the heave and pitch amplitudes were obtained from the first order equations of motion (52, 53). Calculations were then made for the series 60, .60 block coefficient hull which was tested by Spens and Lalangas [4]. The results for the two forces

and the moment are shown in Figures 2 - 4, plotted against the wavelength

ratio X/L , and for various heading angles. Headings are only given in one quadrant, ranging from head to beam waves, since the results for other quadrants can be inferred from the symmetry equations (56 58).

The two forces have been non-dimensionalized by dividing bythe force

pgB2L(KA)2 , while the moment has been non-dimensionalized with respect to pgBLA9 . Effectively, the forces are non-dimensionalized with respect to

the wave slope, while the moment is with respect to the wave height. The reason for this inconsistency lies in the fact that for very short waves

(i.e. large K) the lateral drift force tends to infihity, for all heading angles, and the longitudinal drift force is unbounded for head and

following seas.

For head seas the longitudinal drift force is always negative, as is

evident from equation (37), and it attains a maximum absolute value for wavelengths of one-half to three-quarters of the ship length. For oblique

(27)

the shorter wavelengths No experimental data. are available

for

coparison. with this component of the drift force, although in fact this corresponds to the mean increased, wave resistance dùe to the waes, at zero forward speèd

The lateral drift force vanishes for. head and beamwaves. . (Thé latter. is a conequence of the sleñder bódy approximation, together with the fact that foi- beamwaves the ship will follow the orbital motion of.the wave

it-self, and will, therefore not influence the inc.dent.waves.to.the.. same extent

. for pther ..eadingsj The lateral orce is only signflant fòr.very short wavelengths., dropping off. shatply, as the wavelength approaches t:he ship

length. It can b.e shown that the limiting value of.thi.s force coefficient,

as the wavelength tends to zero, is

Experimental values of the lateral drift force are available, from Reference

4,or the heäding angle f l2O°ànd for X/L equal to one-half and one.

Thse are shown by the circles in i.gure 3. Two differént::experimental procedures were èmployed

_and

for.the shorter wavelength these led

to

results

which differ by.a factorof two, but the higherpair of t.hese experimental points is in fairly good agreement with the theory. At.the..l.onger wavelength the theoretical force has fallen off much more rapidly than the corresponding

experimental points. .

Figure 4 shows the non''dimensionalized drift moment for two heading angles. Other heading angles áre easi.iyinfèrredfroth the fact.that,,f'ro

(28)

function only of K'cos Thus, .exceptfor.the

smB

factor, the drift

.momentdepends on the heading. angleonly insofar as it affects the apparent wavelength in the direction parallel to the shipUs axis This moment vanishes for -head or beam waves, as in the case of the lateral. force The drift

moment oscillatesabout zero for short wavelengths, before risingto a

maxii and then téndirig toero. Figure 4 also showsthe corresponding experimentalPoints froth ef-érence 4 At the shorterwavelength, .,X/L= the agreement is excellent,but forthe longer.wavelength, X/L= l.,.the theoretical prediction is again substantially lower than the experiment

On the whole the experimeital confirmation in Figures 3 and 4 is nOt very satisfactory Possible reasons for this may rest with either the theory

ok/the experiments. Regarding the heory, it may be noted that the drift forces and moment aré only significant for very. short wavelengths, and it -is

known, [8], that the slender body theory does not give satisfactory predic-tions of the pitch and heave mopredic-tions under these circumstances On the other

hand,.the agreement is best at the shorter wavelength, where the drift force

and moment are much larger, and it seems plausible that at the longer

wave-length,,where the force and moment are much smaller, the experimental data maybe less reliable. It.is hoped that furthér experiments can be carried

out in order- to clarify this situation.

The results of Figure 4 can be used to predict the stable heading angle of:the Series 60 block 60 hull in waves, when it is ùnrestrained,.as

discussed in the preceeding section for the rectangular ship Itwill be

(29)

wavelength rat-ioexceeded OJO. From the value .of. - (X/L).sec. at the last

zeroin Figure 4.,.it.is apparent that the corresponding value of X/L for the Series 60 hull is 0.53. Thus for practical purposes in the ocean it would seem that the only stable position of significance is that of barn

seas!

ACKNOWLEDGEMENT

The author is indebted to Miss Evelyn Woolley. for the preparation of

the computér.program and the calculation of the results shown inFigures 2.-4. These computations. were performed on the iB.M. 7090 computer of the

(30)

REFERENCES

L. "Some Aspects of Bow-Thruster Design," by George R. Stuntz,Jr., and

Robert. J. Taylor, Trans. SNANE.,-Vol. 72,1964.

2. "The Drift of a Body Floating on Waves,".by Hajime Maruo, Journal of Ship

Research, Volume 4, No. 3,Dec. 1960, pp. 1.- 10,

"Yawing of Ships Caused by Oscillàtion Arnongs. Waves;" by K. .Suyehiro

Trans. Institution of Naval Architects, Volume 62,1920, pp. 93-101.

4. "Measurements of the Mean Lateral Force an4 Yawing Môthent on a Series

60-Model in Oblique Regular Waves," by- Paul G. Spens and Petros A. Lalangas, Davidson Lab. Report 880, June 1962..

.5 "Suface Waves," by J. V. Wehausen and E.V. Laitone, Handbuch der Physik, Volume. 9,. 1960, Springer-Verlag, Germany.

6. "The Damping.and Wave Resistance of a Pitching and Heaving Ship;" Journal

-of Ship Résearch, Volume 3, No. 1, June 1959, pp. 1 - 19.

-.7. "Water Waves" by J. J. Stoker, Interscience, ew York, 1.957. "Current Progress in the: Slender:Body Theory for Ship Motions," by

J. N. Newman and E. 0,;Tück, Fifth Symposium on Naval Hydrodynamics,

Bergen Norway, Sept. 1964.

9. "Theory of Bessel Functions," by G. N. Watson, Cambridge. U. Press, Second

(31)

and the continuity equation

(A2)

Here 'p is the fluid.pressure, ,p the density, .g is the gravitatJonl

acceleration and z is the. vertical coor4inae, dire&ted ..Siteards.

The rate of change of. linear.momentum in the volume V is

fT v

APPENDIX I

The Rate f: Change of Linear and Angular Momentu

Here we wish to derive equations (6) and (7).. We consider the fluid

in a volume V , bounded by. a closed, surface S , with unit outward normal

We allow this volume to vary with.time, and denoté the normal

velocity, of the surface S by U

The flu-id is assumed to be inviscid and incompressible. Thus the

velocity vector

V(x, y,

z, t), satisfies Eu1ers equations of motion

Vd

pß'VS

j+(vv)

y]

(Al)

(32)

However, from (A2),

(v/.v)v

= 7.V)VV(wv)= v)V

(A4)

where the operatôt V operates on all factors to its right. Thus the volume integral, can be transformed into a .sutface-.integai, using GusS

theorem:

.Th [?'f' +(V/'in)v3

_vus

where V

=Vi

. Wenote from (A3) that the contribution from the gravitational potential gz is a vector in. the vertical direction, and 'thus the horizontal components of the rate 'of changé of lineàr'momentum

need notinclude this term.

Next ve perform a similar analysis of the tate of change of.angular

momentum. Thus

dt

L pfffúi

Io

_1ff

í

x

Jo

V/) U d S

4f

»[V(r/ri) $

y)

*

flffrV

4V

(-uJ

:(A5)

VLc

(A6)

(33)

Since vxir

= O . Similarly,

x(VV\Y (V)(xV

+Vx

.\V(irx\y)

Since %/

x (\Y'V

)r= YxV

= O and thela.stequality follows as in

(Ä4). Thus the volume integral can againbe reduced from GaussV theorem:

»JJ

((rx;n(#/p?)

(\Yin(ir\v)7a/S

pffrx \V)

Uds

Howeyer.

=

(vv)(irv)

Vx(V.)

-

=(7.\v/)

_(wxV)

(47)

-

ffr

_{+ (V}

-U

(ir

_\V)JdS

(A9)

The similarity, of (A9) with (A5) i.s readily. apparent. We note that the term involving the gravitattonal ,otentiál may be deleted if we consider the vertical component of (A9), since \Vz is vertical and therefore the term

(34)

G

where

APPENDIX II

The Far-Fie1d Asymptotic Expansion of the Velocity Potential

FrOmGreens theorem, the .potential.at any;.point..in-the f:luid.due tö..

the presence of the ship. is

(AlO) s3

where the integration is over, the wetted surface of the.ship, .i is the .unit.normal into the shp,..and denotes the GreenVs function or. source

potential satisf.ying the free surface. condition. From Wehausen and Laitone

[5],.equation (l3..17),.this can be written in.the form

4t-4

(.t.I<O k (1) JA

+

1TLI

,./(1).

L

r

_{(y-)+ (.!.Ç]1"}

)J

I/

R

L(x2

is the Bessel function of the third kind (Hankl function), and K

(35)

We desire a far-field approximation to thevelocity potential, which

will..be obtained by substituting (All) in (AlO) and assuming.that the

parameter KR is large. First.let us consider the asymptotic approximation

of the Green's function itself. Since K(kR) is exponentially, small. for

, the factor (I + K2Y1 can be expanded in.powers of E , and.

since

k,(..kJ) dA

s:

ô(

'

it.. follows that

R

) 1(JR)V

.The.integral over k has, the value (cf. Watson' [.9], 13.21(10))

,(1R)k

and since

r

9- O(r-)

=

it. follows from (Al2) that

KeI<

_{H' (Ks}

-

K RY

(36)

Finally substituting the well-known asymptotic expansion for the Hankel function (ibid f7.2 (1))

=

2(py'a

e

K() ,.

i(k/?,.r,4)

-(A14)

We are interested in the situation when the field point (x,

y,

_{z) is} in the far-field and the source point

_(,

, ) is on the ships surface.

Thus we set

x= R

cos.;e

Y = _R:

si4e

and note that

[(RG

e

¿

(R

)J

.

o(R'

Thus, after substituting (A14) in (AlO) it follows that

(Als)

(37)

where

is the Kochin function.

Equation Al5) is the well known far-field asymptotic approximation for

the velocity potential, in terms of. the Kochin fuñction. It is important to

note thàt..the error associated with thisapproximation is samli of.order

R

3/2

.0

(38)

0.6

0.4.

0.2 _-0.4

_-0.6

0O

LEGEND

X

---= 0.63

i'

/

I,

\

%.

i

I

g I I I F i. I I I . I I I

The drift forces and moment on ships in waves

ARCHIEF

y.

SchpLbouwbnde

Technische Hogeschool

r

Polar radiús, :

a4

vZ

I)

ReL

uP/g

plane.

--dt(7»,

-s+sps.t.s,oQ

7Ccc(Ili) \)

P( co:(ny,,/

(6)

±I+u

(4)

--o

O

(5)

In addition. +B.st.5a.sfy

5B +

(p(rxnp(irxv)(vuj]ìs

V - U

.n'

ot)

¿Ç

c.oi(y)

C/S

,i.=ffpo1rxc1

p(iV)

Vj0Lc

-

IS

[f

-\JV,

-

f1x)

VV]

-R?

-

ff

*f

V 7

to z.=-

y=R,sin 9

SiMs

\,v

e1Gh

8 -V9

ç,

e)]RcIe

t,

V

1

expfr[

I,? c.s,'(2-eJ]

*

/7'('ir#9)

¿'p (c *('kfri7r/4) .*

d2

11Loe1 j-o()

-

Tvi

J2o

*

H(e=

-

d

Z:

?

8KJ0

ae

r !#

-j

Ld,e

e r2rr

_{5B +}

_-

_ff

_{¿'p (c ('kfri7r/4) .}

_Tvi

_{fíkR 1'-}

_{Q (R}

_{) H(-e)}

_Rej

_[e