First order wave loads in beam waves

TUDeift

Faculty of Mechnnical Engineeringnd Marine Technology Ship Hydromechanics Laboratoiy

DeIft University of Technology

First Order Wave Loads in Beam

Waves.

Journée, J.M.J. and A.P. van't Veer

Report Nè. 1027-P

ISOPE-1995, Offshore and Polar Eng.

Conference, The Hague, The Netherlands

The Proceedings

_of

the Fifth (1995)

_{International}

OFFSHORE, AN].

_POLAR

ENGÌNEERING

_CONFERENCE

VOLUME III, 1995

NUMERICAL WAVES, WAVE MEASUREMENTS,

WAVE BREAKING & STATISTICS, WAVE-BODY INTERACTIONS, HYDRODYNiJvfJC FORCES,

DYNAMIC RESPONSES, HIGfIERORDER EFFECTS, VORTEX & VIBRATIONS, COASTAL

HYDRODYNAJCS,.LORATORY & OCEAN MEASUREMENTS

edited by:

Jin S. Chung, ColOrado School of Mines, Gölden, Côlorado, USA

Hisaaki Maeda, University of Tàkyo,

Tokyo, Japan

C.H. Kirn, Texas A & M University,

_{College. Station, Texas, USA}

presented at

The Fifth (1995) International Offshoreand Polar Engineering Conference

heldin The Hague, The Netherlands, June 11-16,

1995

organized by:

International Society of Offshore and POlar

Engineers

sponsored by:

International' Society of Offshore and Polar Engineers (ISOPE)

Offshore Mechanics and Polar Engineering CounciÏ (OMPEC)

with cooperating societiesand associations

Internatiönal Society of Offshore 'and Polar

_{Engineers ISOPE)}

'P.O.

_{Box 1107 Goldén, Colorado 80402-1107}

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Copyright © 1995 by International.SocietyofOffshoreand lpolaï,Engines,

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Abstract

In many cases, the simple strip-theory method with the

clas-sic relative motion approach delivers a poor prediction of the

first order wave loads for sway, heave and roll of a cross

sec-tion of a ship in beam waves

A better prediction is given by the diffraction theory, which

calculates the diffracted wave system byusing Green's second

identity and the known incidentwave potential. A

disadvant-age of this method is that the wave load calculations are

re-latively time consuming, because these calculations have to

be repented for each wave direction.

-

Therefore a siinplecalculation method to obtain thewave

loads, without solving the diffraction problem itself for each

wave direction, would be welcome. As a first attempt for

beam waves only, this paper presents a very quick, simple and

accurate method to obtain time wave loads from the potential

coefficients and the radiated wave energy. Results of this

method have been- compared with computational results of

the diffraction theory and a perfect fit in beam waves was

found.

Also the radiated damping waves, produced by the

oscil-lating ship, have been used to account for the amplification.of

the incoming wave system. Calculated vertical relative

mo-tions have beeim compared with experimental data and a fair

agreement was found.

Key Nords: strip theory; beatmm waves; first order wave loads;

wave energy; orbital itiotions; vertical relative motions.

i

Introduction

If alt incident wave system encounters a body, a diffracted wave system will be induced by the prcsettce of the body. Tite disturbed flów pattern causes a pressure distributioti ott tIte body and will therefore introduce a hydrodynamic force, the difFraction Force.

The difFracted wavesystetim cati be calculated by applyitig Greett's serolt(l titcoretmi autrl using tite known incident wave potential. A

ttietlmnd to calculate tite two-diinetmsionai p0tetttiais, describing tite dif

FIRST ORDER WAVE LOADS 'N BEAM WAVES

J.M.J. Jou,:we and A.P. van 't Veer

Ship ilydrornechanics Laboratory

Delft University of Technology

The Netherlands

i

fraction problem, was first presettted by Framik (1967), and is generally known as the "Frank Close-Fit" method. Attother mnetitod, based on Lewis conformal mapping coefficients, is giveum by Keil (1974). These two and some other nmetlmods areimmipletimented iíi tite strip-theory PC-programnSEAWAY (Jourtiée.(1992a)), used here to obtain the calcu-lated data. The advatitage of the direct diffractiòncakulations is that the atmiplitutdesand phasesof the loads are obtained without any dif-liculty, because the in and out phase parts of the loadsarecalcuiated. 'FIje disaxlvantageis thatthecalculation speed of thestrip-theory pro-grams is slowed down,since thecalculations are complex and:tlme dif-fraction problemlias to be solved for each wavedirection.

in the classic relative tnotion theory, sectiottal averaged orbitalino-tions ofthewaterparticles are-calculated fronm'thepressuredistribmmtion in the undisturbed waves on the section coittour. With these average orbital accelerations and velocities amid tite potential nmas.s and damp-ing'coefficients, the in and out phase-parts of-tIme diffraction loads-are calculated. The advantage of using time relative ittotion approach is that time amplitudes and phases of tIme loads are obtaiimcd fronm thepo-tential coefficients, which are independent of time wavedirectiomi. Titis method delivers-a fairly accurate prediction of the 2-D first order wave loads for heave of a simip sailing in bow waves. But in beam waves the method gives in tmiatty cases a poor prc(liction of time wave loads for sway, heave and roll, wheti commipared witim results when using tite diffractiotm theory.

With the Haskind-Newmtman tmmetlmod, Newman (1962), tIme amp-litudes of tite overall wave loads omm a cròss sectiotm of a simip can lie obtained from time potential damping coefficients, witlmout solving time diffraction problem itself. Thisrelatiommshipcamm he derived analytically

frotn the radiatedenergy of time-cross section. lIowevem, mio informmmatiomm

concerning the pitases-can be obtained.

In this paper, the phases of time wave loads with respect to time mncomnmng waves are obtamned frommi time resuit.s of the potentrd mass and damping caiculations

A perfect fit with results of diffraction

calculations wasfouimd

In the classic strip-theory with time relative motion approach, tite

anipimficatmon of time mmmcotmmmtmg waves, dime to waves iiroduced by time

oscillating ship, Imas not beeim accoummted For. Based omm the radiated wave energyapproaclt,a very simple mnethod is givemi itere to calculate this amplification, often referred toas dymmaummic swell-up. Vertical

rel-aLive iiiutioits in regular head waves, measured at the bow of a model tif a cargoship with forward ship spce(l, arc compared with calculated data. A fair agreement was found.

2

2-D coordinate system

A right handed orthogonal axes system has been used here with tite vertical x3b-axis at the centerline of thecross section, positiveupwards, and the lateral x2-axis in the waterline, positive to the right.

Roll motions and roll moinentsare positive when turning froiu X2b to X3b. The wave direction ,z is zero iii following waves. Beam waves with u = 90° are traveling in the positive x2b direction.

ut tite caldulations two typical different cross sections of a 200,000 tdw crude oil tanker are used. The shape'of these two cross sections is l)re.seuIted in figure 1. Cross section 01 has a heeled side wall 'and cross section 10 lias a vertical side wall at the intersection with the waterline.

Figure 1: Cross sections 01 and IO of a 200,000tdw crude oil tanker

3

Potential coefficients

Suppose ali itiflitite long cylinder in tite still water surfaceof afluid.

'Flic cylituder is forced tocarry out a simple itaruitoitic motion about its initial position with afrequcutcy of oscillation w and asinall amplitude of displacenteuttx2 for sway, s3,, for Iteave and 54,, for roll:

= xj,co.swt for

j = 2,3,4

(1)

The 2-1) iiy(lrodynauttic loads X,j iii tite sway, heaveor roll directions i exercised by tite fluid on a cross section of tite cylinder, cati be

obtaitted from tite 2-D velocity potentials and tite linearised equation of l3erutotulli. The velocity potentials have been obtained by using an N-paraiuieter conformal ma)ping utiethod of tite cross section to tite unit circle.

Tite iuydrodyuainic loads are defined by:

2py,°

cos(wt + q,j) - siui(wt (2)

iii where j is tite modeof oscillation and i is tite direction of the'load. Tile piia.selag is defined as tite phase lag betweentlte forced itiotion or Lite cros.s section iii still water and tite velocity potentiaiof tite fluid. Tue radiated waves have an amplitude (j,, autd Ywl half tIte breadth of Lite section at tite waterline. Tite potential coefficients i1j and B1 and Lite 1)115.50 lags q, expressed in conformal mappiitg coefficients, are given by Ta.sai (1960), Tasai (1961) and Jotirnée (1992b).

These loads can 'be expressed in terms of in and out phase coni-ponents with the harmonic oscillations:

Peu

_9(1a2(

(AuQo,-i- B1Po)coswt +(AujPoj -

BjQoj)sinwt)

with a22 = 2, a24 = 4/y,, a33 = 2, a44 = 8, ii42 = 4!Jwl and for the tertus P0 and Qo:

xj,, w2

Poi =

WIwt 5!flihj

z,, w2

Qo = +1N?Jwl .cosc,,

g

Tite pitase lag between the velocity potentials of thefluid aiid the forced motions of the cross section in still water is now incorporated in the coefficients Poj and Qo and can be obtained by using:

1Poi

hj = arctan+Q

Equation (5) will be used further on to obtain tite wave load pitases. Generally, these hydrodynamic loads are expressed in potential mass and damping terms:

X,,, = Muï1

-= Mjw2x,, coswt + Nuwx,, sut wt (6) with:

M - b

_iiP,,

AjQo + BPo

_P1+Q

. -

. -

AP0

BQ0

N,, - pb,2 _{p2 ç)2} w

O

'0j

in wiuere b22 = b24=y1, b33 = b44 = and b42 =

Note that in the general notation of equiation (6) the phase lag information q,j ishost. Titis, however, is not the case when equation (3) is used.

Tasai (1965) has used tite following potential dam pingcoeflicients in his formulations of'the ityd rodyitamic loads:

N42 N24 = N22

in which I,,, is tite lever'of tite rolling moment. Because tite values of these two coefficients are equal, tite potential roll dantping coefficient N44 can be obtained from tite sway dauitping coefficients N22 and the cotipling coefficients N24 or N42 by:

(N24)2 (N42)2

N44

-

(9)

1V22 22

Computational validation of these relations for tite roll daunpiutg coefficients has been' carried out with the sLrip-theory prograuti SEA-WAY (Journée (1992e)) for alIcross sections of tite 200,000 tdw tanker.

Figure 2 shows an exauttple of these validations for two cross sec-tions'of titis ship. Tite figure shows a perfect agreeittent, as has been found for all sections.

From equation (9) it follows that tite potential daitiping coupling

coefficients N24

(=

N42) can be obtained from the potentialsway autd roll damping coefficients N22 and N44 by:'

11V241 = 1N421

=

IN22 N4

. (10) Tite sign of these'coupling coefficients follows froutt the course of N44 with frequoitcy. This re!atioit ('10) will he uused further oit, to obtain the wave loads for roll.

0 0.5 15 2 25 Oscillation Frequency (radis)

Figure 2: Example of roll damping coeílicient.s for two

cross sections of a 200,000 tdw crude oil tanker

4

Wave loads

Consider a fixed infinite long cylinder in regular beani waves with

a wave frequency w and a small wave amplitude (,,. The 2-D wave elevation at time cross section iii beato waves is defined by:

(=(,,cos(wt-kx2t)

(Il)

The wave loads for sway, heave amid roll respectively, olla cross

section consist of a Froude-KriJov part lj and a diffraction part Rmj: = F,,,1 + R1 for i = 2,3,4 (12) -J o Roll damping N44 Section 01 A N44 Section lO - N24 N24 I N22

Figure 3: Vector (liagraill of wave force-componeiits for sway 'l'ho figures 3 amld 4 show the in amId out pitase parts of the wave load comimpomielits for heave and sway. The roll moment vectors are similar to the sway force vectors.

Note that Lime sign conventions for the loads are:

X =

{X,,e«"1+')) = ((X10 + iXoug)eio)

= X,, tos (wt+)

= X,, cos cos wt - X,,sin: sin wt

= X,, coswt - Xo,a simm wi

-3-Figure 4: Vector diagram of waveforce components for heave

The Froude-Krilov load F,,, can be calculated by a simple imlteg-ration of the water pressureon tite cross section, where it is assumed that tile cross-section is fixed and not disturbing tile incoming waves. Methods to determimme the diffraction loads R1 are discussed in the following paragraphs.

4.1

Relative motion approach

in the relative motion approach in tile strip-theory (see for immstance

Vugts (1970)), sectional averaged orbital motions of time waterparticles

in the sway and heave direction are defined to obtain tile diffraction

loads in these directiomis.

The sectional averaged orbital motions in the sway and heave

(hir-ection in the origin are respectively:

( =(,,sinwt

( =(,,coswt

Tile (reduced) wave amplitudes for sway (i,, and heave (, are

de-termined from time pressure distributloim along tite comitoimr of tile croas

section in tite undisturbed wave (Journée (1992b)) by:

ro

suI(k2b sii ,) kx3bd

J-r

kx2siiig

(

=

fT X2bdX3b

{sin(kyw,siIl)

Ywl _{-ky,,,jsimi:p} (14)

k I

sin(-kx2bsin/t) kx3bd}

.

J-r

-kx2b sin /L

Thedepth of the cross section is T amid y,,,j is Imaif the breadtim of tite

waterline.

The diffraction parts of tile wave loads are determoimmed froni tite potential mass and damping coefficient.s and the sectional averaged orbitai velocity and acceleration of the waterparticles, using:

Rw2 = M22 + N22

R,,,3 = M33( + N33( (15) R,,,4 = M24 + N24

Note that in the sway force and the roil niommient tenus with ¿ and

will not appear since time Ihmid is assummied to be free of rotation.

Equation (15) can be written as:

= R,,,1,, cos(wt + a) j = 2,3,4

(16)

where tile 'amnplitudesof time diffraction wave loads are:

11w2a =w,/(wM22)2 ± N22)2 w3a =w/(Li)M33)2+ (N33)2 (17) = w,,/(wM24)2 +(-N24)2 (13) 400 E 300

z

w 200 to C o-E (u 100 o a:

and whore the pitases of tite diffraction ioads are calculated with: = arctan (fw M22 N22 = arctan ( N33

)

w M33 ¡w M24 = arctan N24

Originally, titis relativo motion approach was introduced to-obtain Lite heave and pitch itiotions in bow waves and this- showed a fair a.greciiteiit wiLli results obtained when using the-diffraction theory.

w2 w3 w4 (18) -100 O

Wave Frequency (radis) Figure 5: Calculated vertical wave loads on cross section 01 in head waves

05

15 2 25

Wave Frequency (adis)

Figuro 6: Calculated vertical wave loads on cross section 10 in Itcad waves

Figures 5 and 6 respectively show tite Frottde-Krilov force

(Fr-Kr) aitci a coin parison of the results of tite relative motion approach (RelMot) with those of tite diffraction titeory -(Diffrac) for wave load

-colti polleuts for heave in Itead waves of the two cross sections., 'l'he figures show a fair agreement between the two -theories. A perfect lit will be obtained for cross sections with vertical sidè walls, sttclt as soctioit IO. lii case of Iteeled side walls, sucht as section 01,

(levialions will be fonitd in tite diffraction part of tite wave loads at

Itigher froquteiicies

-4-However, very poor agreements between tite two theories can be found iñ beam waves, especially in the higher frequency region of tite wave loads-as-will be shown later on in the paper.

4.2

Radiated wave approach

Thedisadvantageof using direct diffraction theory calcttlatións

(Dif-frac) is that the strip-theory program is slowed down since tite

dif-fraction problem has to be solved for each wave direction, witile itt the relative motion theory the diffraction force amplitude and phas-ing are calculated directly with the previottsly obtained- hydrodynaunic coeflicietits.

Therefore a simple calculation ntethod to obtain the wave loads,

based on the diffraction theory but without solving the diffraction

problem itself for each wave direction, is obviously welcome. -As a first attempt a-method will be presented here for beatn waves only, to obtain the wave loads using only results of tite liydrodynainic potential coefficient calculations.

The overall wave loads for sway, heave and roil- are defined by: xw2 = Xu,2a cos(wt + (,u2)

xw3 = cos(wt + (,03) (19)

xw4= Xiu-i cos(wt ±-u,4)

4.2.1

Wave load

amplitudes

Comisider a cross section of an oscillating body, moving witit zero forwardspeed in the direction j with a frequency w atid an amplitude

in still water. The energy required for titis oscillation (left hand

side of the equation of motion), should be equal to tite eñergy radiated froitt this cross section by the dantping waves.

-So: 1 ,-T0,

-J Ni -idt

i = 2,3,4

(20) T030 o 2 in which: C = wave velocity = g/w O = acceleration of gravity

T030 = period of oscillation = 2ir/w

k = wave-number = 2ir/À

À = waveiength

N = 2-D potential damping coefficient

xi

= x, coswt, harmonic displacement in directiomi j = amplitude of the radiated waves

p = water density

= time

Froto substitution of equation (1) into equation (20) it follows that the ratio of tite motion atti plitude of tite body xi0 amid the radiated wave amplititde ( can be written in termos of the 2-D damping coefficient

as follows:

(,

w\fNj

I /pgc

j.=-2,3,4

- (21)

The motion amplitude x0 also represemtts a sectional averaged amp-litude of tite liarmnonicorbital mnotions of tite water particles, relative to the oscillating-cross section.

Consider now a cross-section of a fixed body with zero forward speed, subject to regular incomning beam waves (right hand side- of tite equation of motion), having an energy equal to equation -(20). Titen the incoming wave has an am plitude (,, and thesway and heave motion amplitudes x-0 fromo equation (21) now represents sectional averaged harmnonic orbital motion amplitudes of the water particles relative to tite fixed cross section. Because of tltesiiiiiiar energies (left

and right hand side ofthe equation of motion) these mnotiomt amplitudes

are equal. Note that tite orbital mnotiolt x4 of the water particles does not exists here, since the fluid is assumed to be free of rotation.

Heave, hJ=180 Fw3-In (Ft-Kr) Y Rw3-ln (Diltrac) A Rw3-out (Dilirac) - Rw3-ln (RetMot) - Rw3-out (RelMol) -AA --

..!

Heave, hJ=lBO' - Fw3-In (Fr-Kr) V Rw3-In (Oittrac) A Rw3-out (Diifrac) IRw3-in (RelMot) - Rw3-oul (RelMot)

05

15 2 25 250 E 200 U)

c 150

w C 100 50 u o u-o w > (O -50 E 400

z

300 U) C w 200 o-E o

o ioo

w o o LL w -100 -200

X0,4,, _/

=

VPYCN44 (a

So the tIm ree wave load amplitudes are given by:

V2 V2

' w2a w3a

N22 - N33

iv44 = pgc(

(23c)

(24)

Relation (24) between the exciting wave loadsand the amplitude of the radiated waves at infinity wasalready shown by Newman (1962), who (lerived this by using Green's secon(l identity. This method to obtain

the exciting wave amplitude frommi time potential damping coefficient is

known as the hla.skind-Ncwmmman immethmod.

4.2.2

Wave load phases

'lime pItase lags h2 and q,3are the l)hmascs between the velocity

poten-tials of the huid and time translations of time oscillaling body relative to

the water particles in still water. 'l'o obtaimi time l)ha.se lags w2 amid (3

between the velocity potentials of the fluid and the translations of time

fixed body relative to tIme water particles in beam waves, the phase lags

!,2 and q,3 have to be diminished by Ir. Because of reference of sway

to the vertical wave motions, ir/2 has to be added to w2

= q,2 -

ir +

= !,2

(25a)

= (,3 -

ir (25h)

In waves time body is fixed and time fluid is a.s.smmmned to be free of

rotation, so no relative roll motions of time water particles are present.

'l'bis immeans that for time determination of time phase lag 0,4 ilOuse can

be mimade of time phase q,4 between time velocity potential of the fluid

and the rotation of tIme oscillating body relative to the water particles

in still water.

Iteferimmg to figure 3 and using time equations (8) amid (24) it can be

slmòwmi that:

F'0,4

+

11w4out 11w4in N24

F,02

+

'1w2oui - 11w2i,, - N22

Frommiammalysing sway and roll data of cross sections with various shapes, calculated with the diffraction theory by the strip-theory program SEAWAY (,lourne (1992e)), it was found that the sign of these ratios

follows frommi the.sigmm ofN24:

¡,,4

+ 11w4.oui ¡?0,4_j,, (26) Iw2+ R0,2,,0t N22

-5--100 o 05 15 2

25

Wave Frequency (rad/s) Figure 7: Wave load coiiiponemmtsof sway for cross section 01 in beam waves

These figures shOw that the use of this radiated wave approach (RadWav) in beam waves with iimfornmatioim on the potential coeffi-ciente only, leads to a perfect agreement witim the results of the (11f-fraction theory Diffrac). Time results of tite relative motion approach

p=90 Fw2-oui (Fr-Kr) Y Rw2oui (Diffrac) A flw2in (Diffrac)

I EEE

::

'.

The amplitude of the exciting wave force for sway and heave due to the radiated datii iing waves is given by:

X,,,, =

= N22wx21, (22a)

X,03,,

=

iY33x3,,

= Nwx3,,

(22b)

This leads to the following expression for the wave force amplitudes for sway and heave in beam waves:

=

pycN22 (23a)

X0,2,, ,

X0,3,, _/

= vpgcN33

(23h)

(a

For an oscillating body iii still water x,,/(,, exists, but for the fixed

body iii waves it (loes not exist, because the huid is freeof rotation. TIle wave imidimient amflplitu(le for roll in this fluid is defined by:

Xu,4,, = = IN2ikiix2,, (22c)

With equations (10), (22a) and (23a) a similar relation asfor thesway and heave forces will be found for tIme roll moments:

where

R0,_0

denotes the in phase part and R,,,,_,,,,1 the out pima.se

part of the diffraction loads.

From calculated diffraction theory data in bow amìd quartering

waves, it appeared that time expression for the out phase part ol Lime wave loadsis valid for low freqûencies onlyand that the expression for

the mn phase part of the wave load is valid in the whole wave frequency

range for all wave directions:

- N22

N24

(t =

The wave moment for roll must be obtaimmed from the wave force for sway using equatiòn (26) and (23a):

N24

A0,4 =

-= N:4pgc/N22

X,,,2 (27) So the phase lag becOmes:

(4 = 0,2 = 1,2 -

(25c) When N24 becomes negative X0,4 will change sign and

4

is to be increased with ir.

5

Results of calculations

From the foregoing it follows that in time radiated wave approach the wave loads are defined by:

X0,2

= /pgcN22 .

(,,cos(wt + I,2

-X,,,3

=

pgcNiri.(,,cos(wt +

(I3 -

ir) (28)

X0,4 N241jpgc/N22. (,,cos(wt +(1,2

-Thesewave loadsin beam waves can be:obtaimmed using only the results of the potential mass and damping calcimlatioims where time pitase lags

q,, are obtained fromim equation (5). Time ligures 7, 8 and 9slmow all calculated wave load components in beam waves for croas section 01.

E loo

z

u, c 50 mu C o D. E o

o

O w C.) o u--50

250 E

z

U) i: o Q-E 100 D

o

50 -50 o 500 E

z

U) C 250 o o-E o

o

E o ai -250

Wäve Frequency (radis) Figure 8: Wave load components of heave for cross section 01 in beam waves

0

05

15 2 25

Wave Frequency (radis) Figure 9: Wave load cotuponeiit.s for roll on cross section UI in beam waves

(l{elMot) lit tue results of the diffraction theory (Diffrac) at very low

frequencies only. This was found for all cross sections of the 200,000 tdw tankeL

Frouii figures 3 and 4 it caui be seen that the in and out phase parts of the (liifraction parts of thin wave forces for sway and heave can be obtained Irotii the Fronde-K rilov force F,j and the overall wave force aiuiplitude X,,1, when the plisse lag c,,,j or the phase lag (,,j is known. From analysing plia.se lag c data, obtained by (hilfraction

the-ory calculations, it appeared that for tow frequencies tends to the deli nition given in equation (18), as used in the relative motion

approach. This is shown in figure 10 for sway and iii figure 11 for

heave.

Froni analysing plisse lag data, oh,tained from potential coefli-cient or diffraction theory calculations, it appeared that in the higher frequency region these plisse hagscan be-approximated by:

w2 w3 k. y,,, (for w - oo) (29) 'luis is shmowii in the figures 10 and Il too. For cross sections with a vertical side wall, such as section 10, this apprOXi!flatiOII is almost

perlecL.

-6-D) a) 3 540 C o o (D U) 360 ai (I) (a -C 900 720 -180 O

05

Figure 10: Phase lags of sway wave forces for cross sections 01 and 10 in beam waves

360

15 2 2.5

Wave Frequency (ad/s),

05

i

15 2 25 -360

Wave -Frequency (rad/s)

Figure 11: Phase lags of heave wave forces for cross

sections 01 and 10 in beam waves

6

Dynamic swell-up

Let an oscihiatiiig body produce damping waves, L(, with ammmphitude

/-(G When the vertical relative motions of a point fixed to the vessel with respect to the wave elevation are-calculated, the influence of the radiated damping waves must be added to the undisturbed incoming

wave. Then the vertical relative motions 83 in a point (Sib, x) can

be calculated using:

s3(xlb,-x2b) = ((xib, 526) + ( - 53(516, 526) (30)

where ( is thie elevation of the incoming waves and 53(x1b, 52b) is thie verticalmotion of the-vessel in (sib, 52b) and wherethie elevation of-the radiated waves due to the heaving cross-section (dynamic swell-up) is given by:

¿g-= ¿(cos(wt -

kIx2bI - ir) (31)

For zero forward speed, the ratio of the amplitude of the local

motion S3 at cross section 1b ofthme body and time amplitude of thie produced transverse-radiated waves i(,, followsfrouu equmatiomm (21):

530(Xlb) I wVN33 540 360 180 O 180 540 360 180 a) O)

o

C o o w U) a) (I) (U o-D) a)

o

C o o w o (I) w U) (U -180 Have, p=90 °-Fw3-In (Fr-Ku) V Rw3-in (Dlffraó) A-Rw3ouitt(DIIfrac) -RadWav - - Rw3-in (RéiMoI) Rw30U1 (IRelMol) --Sway, p=90 (Dw210) )Di)I) E-W2/Ot(Diulrac) Heave, p=90' -K- (DiHac) A E-w3!OI )DI)fta) (Dw31)O (OiJhc) V E-w3!lO (DlIf,ac) - Approxlmam)on

-lIIrR

- s'

Roll, p'90 Fw4-ouI (Fr-Kr) V Rw4-out (Diifrac) A Rw4-In (Diifrac) RadWav - - 5w4oul (helMet) - -- Rw4-In (RetMot)

,'

--'s - 's / F

-AII 900 720 D) w -o 3 540 C -o (-i a) (1) 360 a) U) (U 180 O

05

1.5 2 25

Using this equation, the ratio of the amplitudes of the radiated waves due to the heaving cross section and the incoming wave amplitude becomites:

L(a

Z3a(Xi) IN33 (a C,, Vpgc

An experiiiiental determination of tile vertical relative motions Of a itiotlel in a towing tank at zero forward speed is hardly impossible, because of tank wall iiiterference. A forward speed is required to

obtain reliable experimental data. Therefore equation (32), valid for zero forward speed, is extended with forward speed effects iìl a simple

mañ mier.

lu ca.se of a forward ship speed V, the section is oscillating with the emicounter frequency w, and therefore the wave number and the wave velocity of the radiated waves haveto be ba.sed on theeiicounter

frcqimeiicy a.s well. Thus:

k, =

and

c=---t..),

Due to the forward speed the radiated waves are swept back in the wake.. TIme wave elevation at a certain point (x1, x2) in the vessel fixed reference system is now a result of the radiated waves from a more forwardly located cross section. The zit-position of this cross section can be simply calculated using:

Sib = 51p + V (33)

I n.ca.se of a forward ship speed, the amplitude of the radiated waves can now be calculated using tIme damping coefficient based onthe encounter

frequency and equation (33) for xlb:

Tliis:calciml'ation method lias been verified with some results of

ex-periments iii regular head vaves carried out in the past by .Journe (1976), wiLli a seifprojelled 1:50 _{model of a fast cargoship.} In the ballast condition of thin imiodel, the vertical relative motions have been measured at four Froude mmumbers (0.15, 0.20, 0.25 and 0.30) at a section placed at 10%oI the vessel's length aft of the forward perpen-dicular.

Figure 12 showsan example of the results for I = 0;20 in head waves. 'l'Iiô vertical ship motions at forward speed have been calcu-lated with Lime well-known ordinary strip-theory method. The meas-tired relative motions are compared wiLli calcúlated motions with and

without the effect of the dynamic swell-up; Similar agreements as presented hiere, have been found for the other three Fronde numbers. Whemi accounting for the dynaimmic swell-up, the calculations show a

3troimgly immiproved agreement between the immeasured and the l)redicted relative imiotions, especially at resonance frequencies.

7

Conclusiòns

l'ue roil-roll dlaiuping coelliciemit can be obtained fromim time sway-sway amidi Lime sway-roll potemmtiai damping coefliciemits, in a very simple

man-mier.

When comn)aring the computational results of the relative motion approach (h{c!Mot) with those of the diffraction theory (Diifrac), fair

agreemimeimts have beeii found in head waves. In beam waves however, large discrepancies are foumid.

TIme commiputatiommal results of time radiated wave approach

(Rad-Way) show perfect agreement with results of time diffraction theory (Diifrac).

The addition of a simple method to determinethe dynamimic swell-Ill) of Lime waves iii time calculation of time vertical relátive mnótionsforward resmilLa imito a fair prediction of these mimotio,ms in regular head waves.

¿(,,

53,,(XIb)

/Nsj(xI)

(32) (34) C,,

(

V PUCe 6 o

05

15 2

Wave Length.! Ship Length.

Figure 12: Vertical relative motions forward of a fastcargo ship at F = 0.20

References

Frank, W. (1967), Oscillation of cylinders in or below time free surface of deep fluids, Techmmical Report 2375, Naval Ship Research and

Development Centre, Washington D.C.

Journée, J. M. J. (1976), Motions, resistance and propulsion of a

ship in longitudinal regular waves, Technical Report 428, Delft Uni-versity of Technology,Ship i iydromechanics Laboratory.

Journée, J. M. J. (1992e), SEAWAY-DELFT, User Mammual of Reinase

4.00, Dehft University of Technology, Ship Hydromiieclianica Labor-atory. Report No.910.

Jotmrne, J. M. J. (1992b), Strip Theory Algorithiiis, Revised Report 1992, Technical Report 912, Delft University of Technology, Ship

hi yd romechian ics Laboratory.

Keih, H. (1974), Die hydrodynamuischme kräfte bei der periot.hischien bewegung zweidimensionaler körper an der oberu]äclie flacher ge-wasser, Technical Report 305, Ummiversity oîflamburg.

Newman, J. N. (1962), "11m exciting forces omm fixed bodies in waves',

Journalof Ship Research 6(4), 10-17.

Ta.sai, F. (1960), Forniula for calculating hydrodynamnic force on a

cyl-inder heaving in the free surface(N-paranieter famnily), l'echuical re-port, Research Institutefor Applied Mechanics, Kytmshmu University,

Japan. Vol. Viii, No 31.

Tasai, F. (1961), Hydrodynamic force and immomitent lrodhmmced by

sway-ing and rollsway-ing oscillation of cylinders on Lite free surface, Tech-nicai report, Research institute for Applied Mechanics, Kyushiu Uni-versity, Japan. Vol. IX, No 35.

Tasai, F. (1965), Ship miiotiòns in beam waves, Technical report, Re-

-search institute for Applied Mechanics, kymishu University, Japan.

VoL XIII, No 45.

Vugts, J. II. (1970), Thin Hydrodynantic Forces and Ship Motions in Waves, PhD thesis, Deift University of Technology.

H,,ad Wave., F,, -020

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