The added resistance of ships in regular oblique waves

-I

1. Introduction

In evaluating the added resistance_{of ships} in regular head waves, it is general

_{that the}

calculations are carried out on the basis of Maruo's theory. 1)-6) However, the established method for calculating the added resistance of ships in regular oblique _{waves has never} been reported, because the theoretical ap-proaches are very difficult and troublesome,

and the added resistance of ships in

head waves is more severe and important for ship Operation.

Furthermore, experimental approaches in

obtaining the added resistance in

regular oblique waves have never been carried out, because the testing techniques

_{are so}

dif-ficult and the instrumentations_{are so compli.} cated.

Under these circumstances, the power pre-diction of ships in

_{waves is carried out by}

making use of the experimental results of ship models, self-propelling in regular and/or Irregular oblique waves.

* College

of Engineering, Univ. of Osaka Prefecture i

r.aboratarlwn voor

mmeths

Archief

Me1g 2, 2028 cD Deift

iiL oiaa - Fa O1818

1.

_{The Added Resistance of Ships in Regular Oblique Waves}

Ryusuke HOSODA*, Member (From J.S.N.A. Japan Vol. 133, June 1973)

Summary

A method for calculating the added resistance of ships in regular oblique waves was

developed, based upon Maruo's theory. _{It is assumed that the velocity potential of fluid} motions around the ship's hull is obtained _{from the conception of strip method. In order} to determine the strength, position and phase _{of the hydrodynamic singularities, which}

re-present the motions of a two-dimensional ship's section, Isolated Singularity Method is

used.

Numerical calculations of the added resistance in regular oblique waves were performed using digital computer. _{It is shown that the calculated results}

of the added resistance

coincide fairly well with the experimental results in regular oblique waves, and that the

ef-fect of lateral motions on the added _{resistance in oblique waves is almost negligible.}

Nevertheless, either in evaluating the short term distributions of power increase of ships,

or in predicting the nominal speed loss of

ships in confused seas, it is

_{neccessary to}

obtain the added resistance

of ships in

regular oblique waves theoretically.

In this paper, the formula for calculating

the added resistance is given by Maruo's

theory."

_{The H-function, which is indis.}

pensable to calculate the added resistance, is determined by the-distribution of hydro-dynamic singularities which are located on

the center plane of ship's hull. On the basis of strip theory, the strength of hydrodynamic

singularities is determined by the relative

velocity of fluid to the ship motion, which is

parallel to the ship's section.

The depth

and the phase of the hydrodynamic singular-ities are determined by assuming that _the wave height of progressive waves from the oscillating ship's section is equal to the two-dimensional exact value, at the infinite dis. tance.7)

The numerical calculations of the added resistance in regular oblique waves are

car-ried out using the data of ship motions

¡ I

t

calculated by Ordinary Strip Method (O.S.M.) for a containership of SR-108 ship form.

The calculated results of the added

re-sistance are compared with the experimental results of thrust increase obtained by self-propulsion tests in regular oblique Waves.°'° At the same time, the relation between the resistance increase and the ship motions are also investigated ih the study for the com-pOnents of the added resistance in regular ob-lique waves.

2. Fundamental assumptions and co-ordinate systems

Some fundamental assumptions are

necces-sary to develop the theory for calculating

the added resistance in regular oblique waves. The formula to calculate the added re-sistance in regular oblique waves given by Maruo's theory is determined by the velocity potential òf the fluid motion which satisfy

the linear boundary condition at the free

surface under the assumption of the slender body.

The velocity potential is determined by

the velocity of the two-dimensional

fluid

motion relative to

the motion of ship's

section.

We consider the ship motions with five

degree of freedom, and suppose that the

surging motion can not make any contribu-tion to the added resistance in waves.

We define the right handed rectangular

co-ordinate system as shown in Fig. 1. In the space-fixed co-ordinate system 00-XYZ, we put the origin O0 on the undisturbed free surface with Z-axis-vertically upward. Then, we consider a relative cordinate system o-syz which has the origin o on the X Y-plane of the 00-XYZ system and z-axis is parallel to Z-axis. This co-ordinate system moves

with its speed V towards its own x-axis,

which deviate with an angle of

from

X-axis. Next, we consider a co-ordinate system

G-xbybzb

fixed on the ship's body.

The

origin G is presented as G(xG, G,za+ha) in the o-xyz system, and the components xG, G and zG correspond to the ship motions of

Fig. i Co-ordinate system

surge, sway and heave, respectively. hG is the distance of G from undisturbed free sur-face as shown in Fig. 1. We remove the

G--XbbZb system along the zbaxis, and coincide

the origin G

with o', then we have

co-ordinate system o'-x'y'z'.

We consider a co-ordinäte systemG-xoyozo, which is parallel

to o-xyz system. G-xbybzb

is obtained by

rotating G-xoyozo about its own axes Xo, yo

and Zo. We define these angles of rotation

about .Xo,Yo and zo axes as roll co, pitch O and

yaw çb, respectively.

The incident waves

propagate towards positive direction of X-axis fixed in space co-ordinate system.

The relations

between the co-ordinate systems are obtained. as follows.

The

re-lations between the 00-XYZ and the o-xyz system are

x=Xcos+ Ysinx Vt

y=Xsin+ Ycos

(1)

z = Z,

and the relations betweén the o-xyz and

the o'-x'y'z' system are

x'=(xxo)cosçb cos O+(yyG)Sifl çl' sinO +(zzGhG)sin O

y'=(xxG)(cos çbsin O sin wsin çl' cos ço)

+(y ya)(sin çb sin O sin ç+cos çb cosço)

+(zza--hG)cos O sin ç

Z'(XXG)(COS çl' sinO cos çosin çb sin o)

+(yya)(sin çb sin Ocos ocos çbsin çø)

+(zzo---hG)cos çb cos ço+hG.

We consider that the ship has oscillatory motions with small amplitudes» then if we put

cos(ç&, O, w)=1 sin (çb, Û, ço)=(çb, O, ço),

the relations between the o-xyz and the o'-x'y'z' system become

x' (±XG)+.( y - yG)Çb+(zzGhG)O

y'= (xxe)çb±(yy)

+(zzGhG)ço

z'=(xxG)O+(y - yG)ÇD+(zzG).

3. Boundary condition on ship's surface and basic expressions of singularities

We should firstly determine the velocity of the two-dimensional ship's section, relative to the surrounding fluid, which is necessary

to obtain the hydrodynamic

singularities distributed on the ship's section.

If the ship motion and the motion of fluid around the ship are supposed to be periodic oscillations,

the velocity potential of the

fluid motion is

0(X, Y, Z; T)çb(x', y', z'; t).

If we define the boundary of the ship's

hull as

F(X, Y, Z; T)=f(x', y', z'),

which also includes the free

_{surface of the}

fluid, we obtain the

_{following equation on} the boundary

8F 80 8F 00 0F

80 8F0

4)

Now, consideting the relations,

'

+

"

+ dz' 0T

dx' dT

dy' dT

dz'

ai'

and

808F 808F 808F

OXOX+OYOY+ dz dZ d OqS 0f

L

-

Ox' dx'

dy' dy'

dz' dz'' we obtàin the following equation,

(O 6y' Of

dx' dT ) Ox' dy'

aT) dy'

+(0+ 8z'df

\8z' OT)dz'

from eq. (4), where 0/dT indicates the partial

differentiation operated in relation to the

space co-ordinate system. Ox'/OT, Oy'/OT and Oz'/OT in eq. (5) are expressedas

Ox' OT

Vxa+çb(yy)çby

ti(zzGhG)+OG

=ç(x--xa)+ V+bthGG

+ç(zzGhG)çôG

=O(xx) VO+OG--çb(yyG)

+GZG,

by using eqs. (I) añd (3).

Now, we decompose the velocity potential O into four parts ç'ii, qS2, ç4 and çSw, and put

c'=#1+cb2+l53+çS. (7)

b1 is the velocity potential due to the change of displacement by the symmetric motions of the ship. For this vélocity potential, we should consider that the wave surface is de-formed sinusoidally. ç5i is the velocity

po-tential due to the symmetric motions of an arbitrary ship's section. ç5s

is the velocity

potential due to the antisymmetrical motions

of an arbitrary ship's section.

çb»»,

is the

velocity potential of the incident waves. We may define the ship's surface as

f(x', y', z')=y'ys(x', z'),

and we will decompose eq.

(5) into three

parts in the same way as eq.

(7).

Con-sequently, we can obtain the following equa-tions for the boundary condiequa-tions that should be satisfied by the velocity potentials on the ship's surface.

In these expressions the term of the velocity by antisymmetrical motion is shifted from eq. (9) to eq. (10).

Now, if we suppose that the ship is a

slender body, and assume that the partial

differentiation d/dx gives the smaller value of the order of ship's slenderness ratio than d/dy and df dz, we can simplify eqs. (8), (9)

and (10) as follows, aç5i dçb1 dy

dx' ay

dy' dz' dz' dT dx' dy' dz' dz' \. dz' dT) dz'' dØ3 /dç

a7+W) 7

\ dy _(dç5 +dy'

\dy'

dT

Here, we should consider the deformation of the wave surface. If we make the origin of z'-axis onto the wave surface and put it z" according to Hanaoka's expression,'° we òbtain the relation

z' = z'

where z=zw represents the wave surface.

Consequently, we can obtain the equation of ship's surface approximately as follows,

y5(x, z)=ys(x', z')+-(zz').

(14)

Therefore, taking the relation (6) into ac-count, equations of linearized boundary con-dition are expressed using the periodic term of necessary order as

--- V(xOz+z),

dy dxdz dy (dçb dy dz dz = dz

+XOVOG,

J dz (16)

_(--+

v__)

+çb(Y_z).

(17)

We have made clear the boundary

con-ditions of two-dimensional boundary value problem. In the linearizing procedure

men-tioned above, we should take notice that

the operation d/dz is identical with the

operation dzfdT in considering the

defor-mation of the wave surface.

The above mentioned boundary conditions involve the displacement effect shown in eq. (15), which

is not involved in the Ordinary Strip Method.

We can solve the boundary conditions (15), (16) and (17) by the distribution of sources on the ship's surface. But, it is difficult to analyze the added resistance by using these source distributions. Therefore, we try to concentrate the strength of these sources on the center plane of the ship's body.

The procedure for calculating the longi-tudinal distribution of the concentrated

singularities will be presented in the next

chapter. In this chapter, we derive the basic expression of these singularities which should be modified by the modification factor. Now,

in order to simplify the derivation, we

as-sume the ship body as a thin ship.

(15)

(d

dy, dy'

(8)

dx dx'

aT) dx'

(dçi dy3 dz') dz'

(_'\ dys

₊

dx') dx'

dy'

( ds

dÇIm dz' \ dy5 9 + + 0 dz' dz'

dT) dz'

-(dç.3 \ dy

_+(-+--+

\dx')dx'

\dy'

dy' dT

j+w(YYa)coYa

=0 (10) 4 Ryusuke HOSODA

I

According to the assumption of a thin ship, we can derive the following equations from eqs. (15), (16) and (17),

[-1

= R: V!(xoza+z),

L6YJv=o dxdz

p95w

-.

"ayo

LaYLO(

±

a7

+xO VO Za)-,

F1

L

ai=0=

\ dz dz

(20)

where the double sign in eqs. (18) and (19) represents the distribution of sources on each side of ship's surface. Therefore, we may understand that eqs. (18), (19) and (20) are satisfied by the distribution of sources, and

that the sum of the sources is obtained by

integrating eqs. (18) and (19) with respect toz

as follows,

i=f[b (+xO_ VO_iS)

-The position of the concentrated source is represented as

d=1T, r=A/2bT, (22)

according to Maruo's method, where T, b, A

and -

are the draft, the half beam, the

sectional area and the sectional

area

coef-ficient of an arbitrary ship's

section,

re-spectively.

On the other hand, eq. (20) is satisfied by the distribution of

doublets which have

lateral

_{axis, and the sum of doublet}

dis-tributions is expressed as

M=-21---C3T (xc+ VÙa)

(23) and

(lai)8 +3as

(24)-where, coefficients

_{ai and as are given by}

the Lewis form transformation, and in deter-mining C3, we have to assume that the ship has slow oscillations, namely w-0, and that

the added mass obtained by M coincides

with

the exact value of two-dimensional

oscillation of the ship's section. _The posi-tion of concentrated doublet, in case of the antisymmetrical motion of two-dimensional ship's section is expressed as

d=7T, r=A/2bT

in the same way as eq. (22).

4. Isolated

_{Singularity Method and}

_the modification of the basic expression In case of the symmetric motion of ship, the strength of concentrated singularity and the modification of the basic expression has been reported.fl

In this chapter, only the

results of the analysis are represented. _The strength of source and doublet is

i=-b(° +xO+V',

for

(21) 2z _{\ dz}

_dxj

the strength of source, and

flZxì_bV(xO_zG+zw), for the strength of doublet,

respectively.

The modification factors are as follows, A=) i)r/z0x,b,

H0 23Te jiC

eß1log

ö0= Tan-' (og/o,),

where, Ac,,

r and ö

are the modification factor for the amplitude which modifies the amplitude of singularity strength, the modifi-cation factor for the depth which modifies the depth of the position of singularities and the modification factor for the phase which modifies the phase of periodic singularities, respectively. In eq. (26), H, indicates the half beam-draft ratio of a two-dimensional

ships section, eB=002b/g, A* is the exact

i

J,

(25)

value of AH obtained

by UrsellTasai's

method, and c and o are the real part and

the imaginary part of the source c,

re-spectively.

In case of the antisymmetrical ship motions, we define the two-dimensional ship's sectiòn

as shown in Fig. 2, and define that x-axis

lies on undisturbed free surface and y-axis extends downward.

y

Fig. 2 Two dimensional ship's section Now, we put a point doublet, which ha a lateral axis, on a point (0, d5) on the center line of ship's section. Then, we obtain the strength of the doublet as follows,

nz8 exp (iöy)=m8+ im8.

The stream function due toa point doublet of unit strength is expressed as

S(x, y; 0, ds)=S(x, y.; 0, d5) +iSs(x,y; 0,d3) (27) and S(x, y; 0, d5)- + +2K18, (28)

=X2±(yds)2,

TB - N t(y+d5) cos ux Io=lim\ du, p-o o

u-K+zp

where, p is the Räyleigh's resistance coef-ficient and K=we2Jg.

A method for

the analysis of. eq. (30) is shown by Maeda..") According to Maeda's method, we have

Io=sgn(x)[{E8cosKx

+(Es-2r) sinKIxI}e_K(v+as

_i2re(Y+ds) cosKjxl], ₍₃₁₎

where, E0 and E8 are to be obtained by the exponential integral.

In case of the swaying motion of two.

dimensional ship's section, if we employ the conception of thin ship hypothesis, the boundary condition on ship's surface is ç1

- y and equations of stream line which deter-mine the section form are

Øc=-y=m(0, ds)S8(x, y; 0, d8) -m8(0, ds)S8(x, y; 0,.d5)

çb8=0=rn8(0, ds)Ss(X, y; 0, d5)

-rn8(0, ds)S0(x, y; 0, d5).

We may determine the strength of doublet Inc and in8 by fitting the stream function in

eq. (32). But, it

is almost impossible to

satisfy eq. (32) on the whole position of the two-dimensional ship's surface, by using the Isolated Singularity Method Therefore, we determine mc and m8, so that the boundary conditions are satisfied on a typical pOint on

ship's surface, to simplify the method for

calculation.

If we take the bottom of the.

ship's center

line, (x, y)=(0, T), as the typical point, flB8 and m8 .are obtained as fol-lows, -TS0(0, T; 0,d5) iflc=s2(O _{T; 0, ds)+S;2(0, T; 0, d)°°17°} TS(0, T; 0,4e) "2°S0'(O, Ts; 0, ds)+S82(0, T; 0, d5)°°° Ya mos= _,/mc2 + ni82, (33)

where, Ya indicates the amplitude of sway-ing motion.

Therefore, the amplitude ratio of progres-sive waves, generated by this periodic doublet of strength m1s, is expressed as

As=2irjmnzsI Kede'ds. (34)

Here, we introduce As* as the exact value

of A, given by Tasai's method," and

posi-tion of doublet d8 is determined by coincid-ing.A with As* as follows,

i 2'r 'fl8sI

ds=lo

_As* (35)

We can obtain mc, m3 and d from the

simultaneous eq. (33) and eq. (35). A technique of successive approximation shall be used in solving the simultaneous equations numeri-cally.

Next, in case of the rolling,.if we employ the conception of thin ship hypothesis, the same assumption as the case of swaying, we can apply the same analysis as mentioned above. Accordingly, substituting the bounda-ry condition on the ship's surface

çb=(X2+y2) (36)

into the eq. (32), we can obtain the equations of stream line for rolling as follows,

_Fx2+y2)=,,lc(o d8)S(x, y; O, d)

m3(O,ds)S(x,y; O,d) ₍₃₇₎

O=fl2c(O, ds)Ss(x, y; 0, d)

m(O, d8)S(x, y; O d8) Now, we take the point with co-ordinate (x, y)=(O, T), for the typical Point at which the boundary condition should be satisfied.

Then, the strength of doublet is

obtained frbm eq. (37),

(1/2)T2S(O, T; O, d)

mc_S02(O _{T; O, ¿s)+Ss2(O, T; O, dg)}WeÇDa

(1/2)T2S(O, T; O,d8)

T; O, ds)+Ss2(0, T; O, dg)WcÇ/,

(38)

where, Oadenotes rolling amplitude.

after eq. (33), we may put

mR= ./mc2 ± m2

añd may obtjñ the amplitude' ratio of

gressive waves

AR =2r

We can apply the.

_{same technique as the} swaying motion of the two-dimensional sec-tion, and then we obtain the depth of doublet as Taking (33') pro-(39)

i

2rIrnK

dRlog

_AR*

From eqs. (38) and. (40), _{we can determine}

the strength and the position of the singu.

larity. .

Now, we should make some modification of the basic expression by using the value of m0, ms, d and dR, in order to obtain the modified expression for the calculation of the added resistance.

The modification in these cases _{ate made} after the same manner as the heaving motion of the two-dimensional ship's section. _That is:

1) _{Amplitude modification for the strength}

of doublet: A, A

From eqs. ¶:33), (36) and (25),

Av=Jrns/C3'T2, _{for swaying, (41)} and

A,= mLl?/ C8 T2, for rolling. (42)

2) _{Depth modification for the position of}

doublet: p, p From eqs. (35), (40) and (22), 1 _2jrImcIKea

K7T log A8* for swaying,

and 1

1 2rJmtR!KEa

.

K1T og AR* , for rolling.

3) _{Modification of phase between the}

singu-larity and the motion of ship's section: 3,, ô,

From eqs. ('33) and (33')

ö,= Tan'(m/ni)

(45)

ô= Tan'1(nz/iìz0). (46) 5. _{Calculation of added resistance}

5.1 Calculation of the H-function The H-function is given by.

a, t)= (x, y, z t)

x exp {icz i,c(x cos a+ y sin a)}ds(x, y, z)

where, (x, y, z; t) shows the distribution of singularities on ship's surface.

Making the distributed singularities on the ship's surface into the longitudinal line dis.

tribution of sources and doublets, we can

define the standard plane as y=O, and can define that ds(x, y, z)dx, because each

singu-larity is located at the point of mean draft

of the section.

Therefore, we obtàin the H-function as

H(ic, a, t)= SL c(x, O, da; t)

xexp(idHivcosa.x)dx

+i&cosa5 m(x, O, d; t)

x exp (,cd,,iic cos a.x)dx

+isina5rnv(xo d,,; t)

x exp (,cdmv---iec cos a-x)dx,

where, c shows a source, shows a doublet

with a longitudinal axis and m, shows a

doublet -with a latèral axis. dH, d and dm

show the dpth of source and doublet. We have obtained the singularities which are needed in calculating eq. (47). In case of the symmetric ship motion, putting y=0

and z=dH in the first equation of (25), we

have the source density as

u(x,O, dif;

and putting y=O and z=d

in the second

equation of (25), we have the doublet dis-tribution as

nz(x, 0, d,,,; t)=f V(oxzG+zw).

(49)

In case of the antisymmetrical ship motion,

putting y=ü and zdmy in eq.

(23), we have the distribution, of doublet as

m(x, 0, dmy; t)

'

C3T2(_xç&+Vq,_lo)

8 Ryusuke HOSODA

+CaTsb(4b2_hGT+T2).

(5Q)

Now, on the subsurface y=O and z--d,

we have the velocity potential of the incident waves as

Ç5w= i

x exp'[ kd+i{kx cos(k Vcosw)t}].

(51)

we consider thé ship motions as sinusoidal motions and define as follows,

-zoZaeiat1z,

for heave, for pitch,

for sway, (52)

for yaw, for roll,

where suffix a denotes the amplitude of the ship motions and c is the phase difference

'between the incident waves and the ship

motions.

Substituting eqs. (51) and (52) into eqs. (48), and (50), we obtàin the required singu-larity distributions,

b

Q = -_2r

X [icucxOae"o jw,,,ß'z + iw(ae k(d11- ix cos X) + i Vk cos XCaC -k(d11- ix cas X)}eiOct (53)

b

flix, V

2ir

x [XOaCo_ zeiz± ,-k(d11-ix coo

T2

22r

X [iweXçbaC'+ Vç1'ae'iweyae"Y +wC sin xe_k s_dI05 X)]eict

+tC3Tae(b2_hGT+

T2)elet, where, dH, d

and dR show the depth of

singularities for heave, sway and roll,

re-spectively.

Then, we obtain the H-function of time

independent form,

H(K, a)

=

5L b{xOoe'8zaez

+Cae_kdH_ ces Z)}(d11ixCOZ dx

-

jwcçøalrSin a T > T2) x eiFe-(dRX cos Z)dx. Now, if we put K COS a=m, (;n +oQ)2

-,

ICO

L sin

a=Lo/(Q+_!Ç

_=mi, L0/ Q

,cog/V2, 12rVw/g,

the H-function is obtained as follows,

J

x Ç b{xOae'o zei 'z+Cae (0ix CooZ))

X e(it-°')dx

i

I-\ CST2{wexbaei#+Vçbaeùs& 2ir JL

+Weyae'Y+ iwC sin Cos Z))

X

e_(s-m)dx

i

fliWeÇa C3T

>

(--i_-r+4 T2)ee_(R_imx)dx.

(58)

Here, we decompose the H-function into a

real part and a imaginary part in order to

calculate H(m)12. For this purpose, we de-fine the following integrations,

i I I Ksc}

i

f Abe_(k+')pzrT sin (kx c)s x) K00 2ir L ícos(lnx+ôo)Ìd X (sin (mx+ô5)) (56) L0'

15

_{AvC3T2e-"/'yTr} L0) 2x L fcos (nx+ö) dx X(sin(mx+ôv) J (57)

Mc}i

M0 22rL {COS (sn.i-i--. sin (mx+ô) N00} _{sin x} 5 AvC3TBe_(k+')pyrT

N - 2

L cos(mx+ö)}d

)< cos (kx cosx){i(Ö)

N00) sin 5 AvG3 T2e_'»yrT

N5521r

L fcos (mx+ö) }dx X sin (kx cos x) (sin(nzx+ö) Pc

if

PZ1JL

AC3T(--b2_hQT+--T2) (cos(mx+ö f dx. (sin (mx+o)) (59)

Coefficients A0, A, A, ,

p, ô, ò

and

ö are the modification factors for the basic expression mentioned above.

_{As m is an}

+ a-ir cosa VÇ b{xeae'ozae'

JL

+CaeH

COZ Z))_(dy_tz cs dx

Ic}1

Ç Is 27rz lsin(mx+Ö5) dx

JC}1

_{f Azbxe_rPzrT05(mx+}dx}

+K Sin a 5L C3T2{iwcxbae - Vb0ep + iw0y0e1'v Jo K

KCJJL

2tjz

tsin(mx+ö0)

i

f A0be_(k+n')pzrT cos (kx cos x)

COZ Z))

WCa sin xe

xe(dszcos Z)dx

(cos (nix+ö)

odd function of a, we obtain

H(m)2, by using eq. (55).

¡ H(m) l =(w + m V)2{(J2 +Js2)0a2 +(I + Is2 )2a2

+(K,,2+K2+K,2 +K2+2KK

2KccKss)Ca2 2(IcJo+ IsJs)O,Za COS £e

+2(I5JsIgJc)aZa Sifl ee5+2(JcKc

JcKss+JsKc5+JsKsc)Oaa COS o

+JsKss)OaCa Slfl Ce

- 2(13K,3 + 13L, + I5K - IcKs) ZaCaCOS E

+2(w2 V(L8M0+ L8M8)çba2 +we2(LcMc+L8M8)çbayaCOS eji

+we2(L5M0LcM8)çL'ayaSIfl £çbj +w52(McPc+M3P8)çl'açoaCOS , +we2(McPsMsPc)çbaa sin +weV(Lc2tLs2)çbayaCOS y +oieV(LcP0+L8P3)çbaçoaCOS Eq +wcV(LoPsLsPc)bçia sin +wc1(LcPc+LsPs)ya COS +w62(LcPsL8Pc)yaçøa sin We(.O(McNcs + M8N3 M8N88 + MsN53)çbaCa cos s +wew(M8NMcNu

+ M3N55 + M5N80)ØaCa Sin eç

- V(LCNCS +LcN35L8N + L5N) X ÇbaCa COS C +w V(LcN-JL5N88+L8Na+L8N85) XÇaCaSjfl C, wew(L5Nc5 +L8N58 L5N5 +L3N52) X yaCa COSLy +wew(L2Nc8 + L5N30+LN8,,L5N) XYaC,zSIfly w8w(PN+ PcNsc P5N55 + P3N83) XÇOaCa COS E, +ûew(PcNocP0N55+ P8N55+ P8M,) XçOaCaSflç,], (60)

where, &=e,

and ,

= Il

-5.2

Calculation formula of the added

re-sistance

According to Maruo's theory,8) the formula for calculating the added resistance in regular oblique waves is given by

2(1K3+ IK hK

(rr-0

r/2 3/2 + IsKss)ZaCa sin R4w=2rp1t I \ + _r/2 I L sf2 + fl212[w52(MO2 + M82)çba2

+ V2(Lc2+L22)çba2 +we2(L2 +L32)ya2

cos ak

X ¡H1(j,a)I _{/1-4Q cos a} da

+wei(Pc2+PsS)Wal

₊

Ç 2s- _{H1(2, a)}

+ (N2 + N2 + NO2 + N852 +2N85N88

2NccNss)Ca2 XK2(2 cos aIc cos x)da. (61)

1-4Qcos a

lo Ryusuke HOSODA

Now, making some transformations as shown in eq. (57), we have

e cos a=m, (i=1, 2)

(m +oQ)2

(i=1, 2)

m=-cosa±1_40s a

2 cosa

H(xi, a)H(m),

(i1, 2).

Then, eq. (61) reduces to

RAw=27rp +

+

I [ 5m3 _Ç52 754 mjJ (m+iroQ)2(m kcos x) X _{/( )T:02,82 -IH(m)I2dm} (63) where, mi') K

=(1-2Q±/1-4Q)

mi) 2

m3Ko(lWl4Q)

m4j 2

_J

(64) } (62)

We should take into account the limits of integration;

For Q>1/4, from the third relation of eq. (62), the integration reduces to

Co Çr/2 Ç3/2 _C'o

I

da+' daI da=I dm

J-,!2 J/2 +500

_{dm_5' dm+5dm}

- m3 Cm4 r0Q f

da--

_-

dm+\

.)0Q .jm4 (65)

therefore, we have the integral limits as,

oo<m<mo, rni<m<oo. (66)

For Q1/4, the integration reduces to

Çr/2 _Cm1

da

da = dm + dm J-0/2 Jr/2 Jo, Jm1

_dm+fdm

_{J m3} fm4 (m2

da=

dm+\ dm

O m2 Jm4

and, then we have the integral limits as

oo<m<ms, m4<rn<m2, mi<m<c.

5.3 _{Components and D-coefficients of the added}

resistance

The non-dimensional form of the formula for calculating the added resistance is

KAw R4w/pg002(B2/L)

1 1Ç j(2F0 rn' ±co)(jn'

(sr/p)cos_x}

2ra2 [

_j

../(2FThm'+wL)4_4,n'2

xIH(m')2dm' (69)

where, aB/L,

wL=/

_{p(2r/p)Fflcos}

_P 2/L and m'=L/2rn. _{If we put}

s(rn (2Fnrn'+wL)2{m'_(/p)cosx} ₇₀

v'(2Fflm'+WL)4_4mF2 ( )

the added resistance coefficient _{KAW reduces} to

25

KAW»

I

27ra2j1 L

According to this expression, we can

de-compose the added resistance into many

components relating to the contribution of ship motions and the incident waves as fol-lows,

KAs, = DHHZO2 + DO02 + D3 + DHCzO COSe

+ DRIZO sin + DO0 cos r

+D3Oo sin o+DHpczoOo cos eez +Dni'5zoOo sin oo+Dyyçbo2+Dswyo2

+DRRç'o2+DA+DyO cos e,

+Dv0çbo sin +Ds1y0 cos ey+Dssyo sin y + Dcçoo cos + Dapo sin ,

+Dvsç'oy0 Cos +Ds1b0y0 sin -pij

+DYR4'oçoo COS ,+Dnìsboçoo sinr10

+DsRcyoçoo cos £y,+DsRSyoçoo sin ey,

where, suffix o denotes the non-dimensional

amplitude of ship motions. From

_{eq. (72),} we can investigate the contribution of ship

motions and incident waves to the added

resistance. The coefficients DER,

...,

and D3.1 have been named D-coefficients of the added resistance, and they are

DHE=-5(2F, m'+wz)25(m')(Q+Q)dm'

D=-

5 (2F,,;n'+WL)25(m')(J,°-l-J1°)dm'

Ds=j

5(2F m'+WL)°S(m') X(Kcc°+Kcs°+Ksc°+Kss2_2JcKm +2Kc3Ks)thn'

DHC '

5 (2F,,m'+WL)°S(m') X (IcKcc IK00+ ¡sKcs+131Ç20)dm' Dfls=

-15

(2F,,m'+WL)°S(?fl') X (IcKm + I0K2 I,K,,, + 15K33)dm'

12 Ryusuke HOSODA X ( JcKccJcKss +JsKcs+JsKsc)dm'

D=1

5 (2F,, n'+wi)2S(in') X (faKes +JK55J5K5 +jsKss)dm' DHFC= (2F,, m'+WL)2S(m') X (1cJ5+13Js)dm'

Dnps='

_{5 (2F,,}m'-FWL)2S(m')

X (IJIsJc)dm'

5 m! 2S(m')(MO2 + 5 1ra2 8p x 5 m'2S(m')(L51t/I5+L,IV1,)dn' Dsw=__&L2 5 m2S(in')(L2+L2)d;n'

DRR_wL2

5 rn!2S(rn')(P2+P32)dm' DA=--s1n2x 5 m2S(rn') x (N2 + N2 + N52 + N2 + 2NCSNSC 2NccNss)dm'

D0=

_-

x .,J?_wL 5 n2S(;n'). x(McNs+McNscM8Nc+M5N55)dm'

a1fl x \/2rF

5 m'S(m') x (LCNC,+LCNSCLSNCC+ L,N55)dm' a 211.1 x 5 rn,2S(inh) X (McNccMcNss+ M5N5+ MsN3)dm' < 5 X (L5NL5N55± L5N5+ L5N5)dni'

D55

a2SinX

\/(IJL

_{5 m'2S(,n')} X (LcNcs+LcNscL3N50+L3N55)diii' D55 a2sifl x 5 m2S(m') X (LgNcs+ L5N55+L5N55L5Nss)d;n'

D5=4' /?Ez

5 m1'2S(rn') X (P0N3+ PaNsa - PsNcg+ PsNsa)dm'

D-2" X

5 rn,'2S(m') X (P8N08+ PaNsa + P5Nsa P5N55)dm' DySc=Fwa2 5 m1'S(m')(LM0+L5M5)dm' +pcvz2Fn 5 rnS(m')(L52±L52)din' a_2 5 ,n1'2S(rn')(L3M5L5M3)dm' Dy55 = =7ra2 2Ç OIL _{\ rn,'2S(rn')(M5P5+M5P5)drn'} 4p + _2p cúsF,, 5 m,'2S(rn') x (L5P5+L5P5)dm' D_{2'Rs=O)L \ m1'2S(rn')(M5P5M3P5)dm'}7ra2 2 4p +ira2OIL F,, 5 rn'2S(nz') X (L,PL5P,)d,n' Dsjac=a 5mi'2S(m')(L5P5+L5P5)dm'

D5=

a2

WL2 5 m1'2S(rn')(L5P5LsP5)dm'. 4p (73)

6. Numerièal applications and discussions

6.1 Numerical results of modificàtiòn factors

The calculated

results of modification

factors are shown in Fig. 3 and Fig: 4, in

the case of swaying and rolling, respectively. These coefficients are evaluated by the Iso-lated Singularity Method.

In case of the swaying, the value of A

has a tendency to increase slightly, with

the frequency of oscillation. For rolling, the

value of A varies much with the change

of section form.

In case of the antisymmetrical ship motion,

the modification factors are smaller than

those of the symmetrical ship motion,

be-caúse the strength of singularity is

deter-mined by àssuming slowly oscillating ship

0-5 0 1.0 0 e 0.5 05 0 1.0 H. 0.6 1.4 0.7 9 L 09 07 H.

0.6

1.0 . -14 00 _{10 20 3.0 40} 50 60

Fig. 3 _{Modification factors for swaying motion}

motions in the basic expression. Conse-quently, the Contribution of antisymmetrical ship motions to the added _{resistance is} ex-pected to be small.

6.2 _{Numerical results of the added}

resistance. Numerical calculations of

the added

re-sistance in regular oblique waves were car-ried out for SR-108 _{container ship of CB=} 0.559, L/ß=6.89 and _{L/d=20.6. The cases of} calculation are as follows:

_{F=O.2, x=l80°,}

150°, 120°, 90°, 60°, 30° and 0° and À/L=O.5, 0.7, 0.9, 1.0, 1.2, 1.4, 1.6 and 2.0.

The ship motions _{as the imput data of the} numerical calculation _{were calculated by} O.S.M., and _{are shown in Fig. 5.}

The calculation using the basic expression was also carried

_{out for the same}

_cases

0.10 a 't 005 05 o 1.0 o 1.0 ï °0.z f-."

\

H. 0.6 1.0 ¡.4

/

/

-I

\

/

//

.-"

\

--- ol.-

2.---20 4.0 50

Fig. 4 _{Modification factors for rolling motion}

mentioned above.

In Fig. 6,

_{numerically calculated results} of the added resistance _{are shown.}

_{In the}

same figure, experimental results of propeller thrust increase"" are shown. _Experiments at S.R.I. _{were carried out with the model}

ship of the same ship form, but the

ex-perimental results by Vossers et al.° are corresponding to the Series 60 ship form of Ca_ 0.56, L/B= 6.89 and L/d=20.6, which are the results of systematic self-propulsion tests in regular oblique waves. _Experimental re-sults of the added resistance in_{regular head} waves with the model ship of SR-108 con-tainer ship, which have been carried out at the experimental tank of Osaka University,

14 _{Ryusuke HOSODA} , 1.0 b o 1.5 05 o 0.5 Heave Vn r 02 Pitch Fn = 0.2

180120 . -90 -. 600 -1.0 05 Fig. 6 (b) Fnr02 ,' 9Q 1.5 1.5 b 1.0 05 o ¡.5 1.0 o 2.0 10 Yaw Vn r 02 X 1500

¡200 . -600 Sway Fnr 02 0

tI,

Roll / Fn=0.2i

1

- I_r-i

--15 20 2 0₀₅ 1.0 15 _{2.0 .kfL2S} (a) _(b)

Fig. 5 Calculated ship motions in regular oblique waves.

2.0

1.0-are also shown in the same figure.

From Fig. 6, we may appreciate that _the calculated results of the added resistance are

in good agreement with the experimental

results of added resistance or thrust increase in the condition of head or bow waves (900

Fn=0.2 Z=60 Q-Z 180e

/

10 ₂₀ Fig. 6 (e) Fn r 02 alcu1otjon with Lateral motion without Lateral motion without

\

\

Fig. 6 Comparison of Calculated and measured added resistance in regular oblique waves.

<xl80°).

On the other hand, in beam,

quartering or following waves (OoX900),

the qualitative tendency of the calculated

added resistance gives fairly good agreement

with the experimental results but

quanti-tatively, there is not always good agreement

between them. Particularly, in the condition of =3O° and 00, the calculated results show considerable large value compared with the

experimental results in the range of short

wave length, 2/L=0.5-4.Ø. For this results, we explain some reasons undermentioned.

From a theoretical standpoint, we

may consider that,

D3, the component of the

added resistance due to the diffraction of

the incident waves on the ship's surface, in-creases in following waves. From exper-imental standpoint, we should consider that the self-propulsion tests in regular oblique waves are carried out in different conditions with which we have made

use in the

theo-retical assumptions. _{Namely, in the} theo-retical analysis, we have assumed that the

drift of the ship is

negligible small, and

both the mean course of the ship and the

trajectory of the center of gravity of the

ship coincide with the

intended heading angle, butin experiments, the ship is steered

that only the trajectory of the

center of

10 3 Fi, o 0.2 X 1800 1500 1200 900 60° 300 0° o 05 10 15 20 Fig. 7 (a)

gravity of the ship coincides with the intended heading angle.

Besides, the phase differences between the ship motions and incident waves, calculated by O.S.M. do not always coincide with those measured by experiments.

Unfortunately, we find it difficult to grasp the influence of this discrepancies

_{on the}

calculation of the added resistance inwaves.

However, if there is any influence, it is

considered to be important in beam or fol-lowing waves, because the lateral ship motions increase in these heading conditions. After all, the strict comparison of the calcu-lated and experimental added resistance is

considered to be difficult, but from the practi-cal point of view, the theoretipracti-cally practi-calculated added resistance in regular oblique waves

ob-tained here gives satisfactory results.

6.3 The cornonents of the added resistance In Fig. 7, the calculated results of

D-coef-20 1.0 'J 3 n o -1.0 Fn02

N.

_{?/L 20} -

-FnrO2 ₁₈₀₀X 150° 120° 90° 600 300 00 O.. X 2800

150°

120° .

90°

-600

30°

-0°

2.0 1.0 t, o -io -20 Fn r FnrO2 1800 1500 --\

120°--\

90°-60°

30°--

-- -!.

05 Fnr02 Fig. 7 (c) X 180° 150° 120° 90° 60° 30° 00 Fnr02 1.0 2.0 1.0 o C

1800

-1500 1200 . 900 . -600 300 00 ... - --..

05.

;.5 15 AIL 20 r-Fig. 7 (d) 1.0 1.5 AIL 20 FnrO.2 X 1800 1500 1200 900 60° 3Q0 00 Fig. 7 D-coefficients of

the added re-sistance t t t

i

X ¡800 1500 1200 . -¡.5 _AIL 20

90°---too---

_0°

-Fig. 7 (f) 00 1.0 ¡.5 X 180° 1500 7200 900 600 A, 20 1.0 t o Fnr02 00

.

. .

-- -- --

7-,--/7

/

16 Ryusuke HOSODA 2.0 1.0

t

o 0.1 o 0.5 0 ¡.0 15 _AIL 2.0 -1.0 Fig. 7 (e)

ficients are shown, but

some of them are

negligible small, so that they

are omitted in the figure.

The components of the added resistance are shown in Fig. 8.

From these figures, we

can understand

that the longitudinal ship motions, heave

and pitch, mainly contribute to the added

resistance in oblique waves, but contributions

due to the lateral ship motions are rather

small. Among the D-coefficients due to the lateral ship motions, the values of Dsw,

DRA, _{and DSRC are large in the range of}

short wave length, however, in this range the lateral ship motions are small, then the

con-tributions to the added resistance become

negligible small.

In order to investigate the contribution of

the lateral ship motions, the calculations of

the added resistance due to only the longi-tudinal ship motions were carried

_{out and}

are shown in Fig. 6. _{From this figure, we} can find that the contribution of the lateral

¡-0 ¡.5 Fig. 8 (a)

5 A,,1 20

X

00

ship motions is small.

_{This result gives a}

theoretical support to the empirical fact that the contribution of the lateral ship motions

to the added resistance in

_{waves has been} said to be small.

The component D3, due to the diffraction of the incident waves on the ship's surface, increases in following waves, and takes most part of the added resistance in the condition of =O°.

_{It appears unreasonable that D3}

takes such a large value.

A theoretical

discussion for this is given below.

Calculating formula for the added

re-sistance is given,

RAW 27rp{

rr-0

ro/2 3o/2

= Mo/2 Jo0\

Li

6i(vicos ak cos X)d cosa Fn r 0-2 io Ñ -io L, o 1800 - 600 1500 -- 30°

¡20°--

00 900 . . 05 1-0 1-5 AIL 20 05 ¡o 1-5 _{AIL 2.0} Fig. 8 (b) ¡0 FnrØ.2 X 180° _60° 1500 300

120°.

00 900 .. -¡H1(1,a)2 Fn02 X 1800 ₆₀₀ 1500

--1200 . _{0° ...} 2Or Fn02 ¡800 1500

--1200 - . .900 .. 600 AIL 20 X 1800 1500 ¡20° 900 600 300 00 20 Fn 02 o 1-0

18 _{Ryusuke HOSODA} 1.0 X 1.0 1.5 Fig. 8 (c) Fig. 8 Components of + H1(2, a) 2

¿2('r2 cosakcos)

X _-da

from eq. (61). In this formula, the term re-lated to k cos y corresponds to drifting force, and the term related to cos a corresponds to the non-uniform wave resistance.

Now, if we put cosy=1.Ø in the above

equation, we obtain the negative added

re-sistance which is given by the integral of

the terms related to k cosy and i cos a. As

13/2

the contribution of the integration da is generally small, the term corresponding to the drifting force gives the negative _added resistance when the incident _{waves follows}

10 -1.0 j,0 g FnrO2 1.Q FnQ2 Fn0-2 X

90°--180° 60° 150° -- 30°

120°--

0° X 180° - 60° 150° -- 30°

720°--

0° 7.0- Fn02 s 0

7.05

a1 L 20 60'

30'

-0° X

the ship. _{This supports our physical} intu-ition.

_{On the other hand, when the ship's}

advance speed is equal to zero, only the

second term of the above equation

con-tributes to the added resistance.

In this

case, if we suppose the ship symmetric fore

and aft, the H-function reduces to

sym-metric, and we have a reasonable result that the term of 2 cos a has a different sign

ac-cording to the direction of the

incident

waves.

Next, we should investigate the component of the added resistance due to the diffraction of the incident waves on the ship's surface. Here, as the ship motions are assumed re-strained, the term corresponding to the drif

t-ing force is small. According to Havelock's theory, this term would be diminished.

780° 60° 150° -- 30°

120°--

0°

90° - -

180' - 60°

150' -- 30°

120°--

0° ... o 05 1.0 20 05 15 Fig. 8 (d)

the added resistance

On the other hand, the _{component due to} non-uniform wave resistance becomes to be mainly effected by the _{term ic cos a. This}

term is

_{always positive, and contributes}

surely

_{to the added resistance.}

In this' theory, we consider the _{displacement effect} due to the deformation of wave surface for the determination of _hydrodynamic singular-ities, therefore, the

_{component due to the}

non-uniform wave resistance brings _large value of D8 compared

_{with that given by}

Maruo's basic expression. J Fig. 9, the component D5 is compared with the value

20

F02o

with ¡solOled Singularity Method

- - with Moruos Approximation Have(ocks Ori! Ping Force 70

o ₀₅

10 7:5

\

\

Fig. 9 Added resistance in:following _waves

given by Maruo's theory.

There is a

con-siderable difference _{between' them, but we} must prepare for the _{accurate experimental} results to decide which method is reasonable.

6.4 _{The added resistance calculated by Maruo's}

basic e±pressio,zs

In order to _{investigate the applicability}

of

the _{Isolated Singularity}

Method for the modification of Maruo's _{basic expressions',} the added resistance was calculated by using Maruo's basic _{expressions. The calculated} results are shown in Fig. 6, compared with the calculated

_{results using the}

_modified expressions by I.S.M. or with the

ex-perimental results. From Figé 6, we can

¿p-preciate that in the range of short wave

length, /L=O.5.O.7, there_{is remarkable}

im-provement for the value of the

_added re-sistance in regular head _{waves. In oblique}

waves, the change of the

_{added resistance} v.s. wave length-ship length

_{ratio is very}

strange. _{It is consideréd to}

_{be caused by}

making use of Maruo's basic expressions for the hydrodynamic_{singularities of the lateral'} ship motions. _{Therefore, we may support}

the applicability of

I.S.M.

_{for the}

modi-fication of the basic _expressions.

7. _,Conclusions

The author has

_{developed a theoretical} method for evaluating the added resistance

in regular oblique

_{waves, and show the}

numerically calculated results, _compared with the experimental results. Consequently, we may find some conclusive matters as

fol-lows.

The hydrodynamic singularities used in the basic _{expressions are obtained by}

as-suming slowly varying ship motions. Ac-cordingly, some modifications, are necessary when the frequency of

_{the ship motions is}

high. _{Isolated Singularity Method is a} suita-ble method for _{modifying the basic}

expres-' sions and after this modification is performed, the calculated added resistance is remarkably improved.

In regular obliue

_{waves, heaving and} pitching motions mainly contribute to the

added resistance, but the

_{contrjbutiòns of} the lateral ship motions are rather small.

Therefore, for the practical use of this

method, we can calculate

_{the added} re-sistance in regular oblique waves by con-sidering only the _{longitudinal ship motions.}

In this study, the

_{experimental research} has not been carried _{out because the model} experiments in oblique _{waves are difficult.} But, the model experiments in oblique waves should be neccessary to discuss the validity. of this theory and the _{results of calculatiOns.}

Acknowledgement

The author sincerely wishes to thank Prof. Shoichi Nakamura

_{and Dr. Matao Takagi}

for their kind guidance and encouragement. The author also wishes to thank Mr. Masa-aki Ganno, Dr. Yoji Himeno and Mr. Kimio Saito for théir valuable discussions and help. The numerical calculations were perfórmed with the electronic computers in the Compu-tation Centers öf the Osaka University, the Kyoto University and the University of Osaka

Prefecture.

-References

H. MARU0: On the Increase of the Resistance 9)

of a Ship in Rough Seas, (1), J.S.N.A. Japan, Vol. 101, pp. 33-39, Aug. (1957) (in Japanese) H. MARUO: On the Increase of the Resistance 10) of a Ship in Rough Seas, (2), J.S.N.A. Japan, Vol. 108, pp. 5-13, Aug. (1960) (in Japanese)

H. MARUO Resistance in Waves, Researches

on Seakeeping Qualities of Ships in Japan, 11) Chap. 5, The Society of Navàl Architects of Japan, 60th Anniversary Series, Vol. 8, pp.

67-102, (1963)

S.' NAKAMURA and A. SH1N'rA.NI: On Ship 12)

Motions and Resistance Increase of

Mathe-matical Ship Form in Regular Waves, J.S.N.A

Japan, Vol. 118, pp., 24-35, Dec. (1965) (in

Japanese)

H. FUJ1I and T. TAKARASIU: On the Increases 13) in the Resistance of .a Ship in Regular Head

Waves, Mitsubishi Heavy Industries Technical

Review, Vol. 4, No. 6, pp. 644-650, Dec (1967)

(in Japanese)

R.F. BEcx: The Added Resistance of Ships in Waves, MIT Repoft, No. 67-9, June (1967)

M. TAKAGI, R.' HOSODA and H. SHIMASAKI:

An Improvement for the Calculàtion of Added'

Resistance in Waves, Journal of the Kansai

Society Of Naval Architects, Japan, No. 141, pp. 33-44, June (1971) (in Japanese)

The 108th Committee of JSRA: Investigations

into the Seakeeping Qualities of High Speed

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Waves and Theory of Wave-Resistance,

J.S.N.A. Japan, Vol'. 98, pp. 1-5, Feb. (1956)

(in Japa. nese)

MAEDA: Wave Excitation Forces on Two

Dimensional Ships of Arbitraty Sections, J.S.N.A. Japan, Vol. 126, pp. 55-83 Dec. (1969)

(in Japanese)

F. TASAI and M. TAKAGI: Theory añd

Calcu-lation Method of Ship Responses in Regular

Waves, Symposium on Seakeeping Qualities

of Ships, The Society of Naval Architects of

Japan, pp. 1-52, (1969) (in Japanese)

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