Minimization of resistance of slowly moving full hull forms in short waves

Lab. v. Scheepsbouwkunde

Tech nische Hogeschool

Delft

_{Minimization of Resistance of}

Slowly Moving Full Hull Forms in Short Waves

T. SAKAMOTO and E. BABA

Nagasaki Technical Institute, Mitsubishi

Heavy Industries, Ltd., Japan

ABSTRACT

A theoretical investigation was made on the minimization of resistance of a slowly moving full hull form in short waves.

First, the exact nonlinear free-surface condi-tion was split into steady and unsteady linear free-surface conditions under the assumption of small perturbation over the double body flow around a ship as done in the low speed theory for steady wave-making problem. The unsteady linear free-surface condition newly derived in the present study was found to coincide with that introduced in Faltinsen's asymptotic theory for prediction of added resistance in short waves. Then the sum of steady wave-making resistance calculated by use of the low speed theory and added resistance based on the Faltinsen's theory was minimized by the non-linear programming technique. It was shown that principal geometric parameters of a full ship have dominant effect on the added resist-ance in short waves as well as on the steady wave-making resistance. Under the specified values of principal geometrical parameters the added resistance does not vary sensitively with the local minor change of hull forms when it is compared with steady wave-making resistance. NOMENCLATURE Aw waterplane area ship breadth C b block coefficient ship draught gravitational acceleration

by incident wave height (hw ° 2Ca) k0 incident wave number

ship length resistance

Rw steady wave-making resistance

Raw added resistance due to wave reflection ship speed (uniform current)

598 X we wo

ARCHIEF

uro value of ur on z 0

double body flow velocity along the waterline

v =(u,v,w) velocity vector

vr-(ur,vr,wr) velocity vector of double body flow

vo=(u0,vo,w0) velocity vector of steady wave motion

v1=(u1,v1,w1) velocity vector of unsteady wave motion

value of vr on z 0

wave direction (a 0 deg. is head wave) wave height

amplitude of incident wave

fictitious wave height of rigid-wall flow wave height of steady wave

wave height of unsteady wave motion density of water

incident wave length total velocity potential diffraction wave potential incident wave potential

velocity potential for double body flow velocity potential for steady wave-making velocity potential for unsteady wave motion

wave direction (X =180*-a)

circular frequency of encounter waves circular frequency of incident waves

1. INTRODUCTION

A ship hull form which is designed to have the best propulsive performance in calm sea is

not necessarily be the best one in waves. In the case of large ships, the length of waves which they encounter with high probability is small when it is compared with ship's dimen-sions. In the short waves, ship motions are negligibly small and resistance component due to wave reflection at the bow part is added and increases as the fullness of the entrance part of a ship becomes large.

Therefore a more rational design objective for the large full ship form should be the minimization of resistance not only in calm sea condition but also in short waves. As a first step to this problem, a theoretical investigation was made in the present study on the optimum entrance form of full ships which has the minimum resistance in short waves within the framework of potential theory and neglecting the ship motion.

First, the assumptions used in the low speed theory for the steady wave-making

problems /1,2/ were extended into the unsteady problem, and consequently the exact non-linear free surface condition was separated into two independent linear free surface conditions, i.e. one for the steady wave-making flow and the other for the time dependent wave motion. It was found that the former is identical with the one in the low speed theory and the latter differs from the conventional free surface condition used in the linear ship motion theory hitherto. The characteristic of the time dependent free surface condition newly intro-duced in the present study is that the equation is formulated on double body flow around a ship as done in the low speed theory for the steady wave-making problem.

If turned out that the free-surface condi-tion introduced by Faltinsen for added resist-ance calculation in short waves /3/ can be derived from the newly derived free-surface condition when it is applied to the vicinity of hull surface.

Next, discussions were made on the calcu-lation methods of resistance when applying to conventional full hull forms. Finally, the optimum hull forms of least resistance in short waves were determined, where the resistance is

the sum of the steady wave-making resistance calculated by the low speed theory and the added resistance due to wave reflection calcu-lated based on Faltinsen's theory.

2. BASIC EQUATIONS

2.1 Coordinate System and Free Surface Condition

Let us consider a ship running in short waves so that the ship motion is neglected. In that case, it is convenient to define a coor-dinate system fixed with respect to the ship. The coordinate system is defined as shown in Fig. 1. x-axis coincides with the direction of steady uniform stream whose velocity U is identical with the ship speed. The origin of the axis is at the point of intersection of still water surface, midship section and center plane of the ship. y-axis is horizontal and

599

_

Fig. 1 Coordinate system

normal to x-axis, and z-axis directing vertically upwards.

Supposing a ship is in an inviscid, irrotational, incompressible fluid, the velocity potential 0, which represents flow

around the ship and satisfies Laplace's

equation V20 = 0, is introduced. The velocity vector v is expressed as:

= 170. (1)

The free surface is defined by its elevation z = C(x,y,t).

The kinematic and dynamic boundary conditions on this free surface are written respectively as:

1T1

(C-z) = 0 on z = C, (2)

where

D/Dt =- a/at VV, the substantial derivative and

0 (v0-v0) + gz = - U2 on z = g,

t 2 ₂

1

(3)

where g is the gravitational acceleration. Since the equation (3) is satisfied on the free surface for all time, its substantial

derivative can be set equal to zero. Combining this with eq. (2), the exact non-linear free surface boundary condition is expressed as

.tt + 2VO-V2. +

IVO.WV6.V.) +gz = 0

2

on z = ;, (4)

where ; is reduced from (3) as:

(x,y,t) ' lrf.i2-vo-VO) - t] on z = C- -(5) c _{i 12%"}

2.2 Basic Assumptions'

In the low speed theory for steady wave,. : making problems, the total velocity potential was assumed to be the sum of the rigid-wall potential and the steady wave potential Which represent the double body flow and the steady wave motion near the free-surface around a ship respectively. In the present study, the loiq

speed theory is extended to an unsteady .problem. Here, we assUme that the total Velocity potential can be separated into the

following three components:

0(x,y,z,t)

= k()c,y,z)

+

So(x,y,z)

+ 1(x,y,z,t) (6) where

Sk velocity potential representing double body flew in calm water. This Is regarded as a base flow on

which

the following perturbation potentials are superimposed.

so : velocity potential representing steady wave-making.

1 :

velocity potential representing unsteady wave motion.

And the velocity cOmpenentS corresponding Co

those potentials are

= (uk,vk,wk) = Vipr,

(u0,V.0?w0) = /7.0.

(n1,1710;i1) (7)

(8)

In the low speed--theory for steady wave-making problem, the following assumptions are made about orders of magnitude for the double body flow and the steady wave motion in the 1:6W speed limit Of ship speed U.

Sr(s.,Y,z) ° 0(U),

j)--

JL

'

By' = 0(1) when operating on $t, (9) ax az

then

ur. Vk, Wr = 0(U).

So(X,y,z) (go?,

a a.- a

'

z ° 0,(1'2) when operating on

$o,

(10)

ax'

Zy a

then

uo, vo, wo

= 0(U3)

In the present study the order of magni-tude

of

_{the Unsteady wave potnetial}

is denoted by c which should be-fixed later:

61(x,y,z,t)

= 0(c) (11)

600

For the differential operator, following assumptions are newly introduced in the present study:,,,,

-3 1 3

i; 0(U '), when operating on fl, (12)

then

u1, vl,

wl -

0(0172), (14)

where (12) implies that wave length Of the unsteady wave motion is as short as that

_of

the steady wave..

The. wave height

c is

_{also expressed as the} sum of two parts in the low speed theory for the steady wave-making problem,

to,09 a

Cr(x,Y)

+ co(*.y)

where Cr it the fictitious wave height due to the rigid-wall potential Sk,

Cr(xi7) ° -4[112 -

(111-2"r2'wr2)]_{on z =} (15)

and co is the steady wave height. In the present study, the wave height is also assumed to be the Sum of the above two components and the unsteady component

CI,

C(x.Y.c) = Cr(x.Y) + Co(x,Y) + Ci(x.Y.r). .(18) Considering (5), co and CI may be defined by the following two equations:

(uji,vr24.wr2) _{(.624.,e+w02)}

2(ukqevrvo+wiwo)] -

Cr,' on z _cr+CO. (17) _

(ur2+vr2+wr2)

_ (.,02+v024.w02) + 2(ukuo+vkvo+wkW0)

-

(u12+v12+w12) - .2(urui+vrvi+wkw1) - 2(uoui+v0vi+w0wl.))

01;1

Cf CO, 1 °n Z Cr+CO"Cl. (18)

From (15) the order of magnitude of

Cr is

found to be 0(U2)-

_{BY applying}

_{Taylor expansion} to eqs. (15), (17) and (18), the wave heights can be expressed as

. 1

ft2_-°

'2gt 0 yr ] 0(U6), ₍₁₉₎

C8(x.Y) °.-Ikr0.0+vf0valy.cr +.0(U6),

(20) ci(x.y.t) - ---Lurout+vavr _1+4,

+ 0(01).

(21)

ck(x,Y)

aur r( + 0(0). Zero wr0 a:;: + ,rsat ,z=0 + 0(U -Zero 0(U3)

where um and vro are velocity components of the double body flow at z = 0, and the order of Magnitude of c0 and CI are 0(114) and.0(cU-1) respectively. It should be noted here that the order of magnitude of the unsteady wave poten-tial 01 must be much smaller than

6(0)

in order to derive (21). Further, the order of magnitude of ci must be lower than 0(U6) which is the order of residual terms of Cr. and ;0. So, we assume the following relation hereafter,

e = Un, 3 < n < 7. (22).

2.3 Derivation of Linearized Free Surface Boundary Conditions

The exact free surface boundary condition (4) is rewritten as 91tt + 2(vr vlevrvle-v1. V1t) + Ivr-VQr+vroNevr.VQ1+v0-V0r+v0-VQ0+v0.9Q/

+111.9Qr+vi.VQ0+vi'VQ11 +

g{wr+w0+wl}

= 0

on z " Cr4-00+Cl" (23) In the case of steady wave-making problem, It can be written as

fvr.7Qr+vrVQ0+vo'VQr+v0-VQ01 + giwe 01 - 0 on z Cr+CO3 (24)

where Qr. Q0 and -Q1 are defined as follows,

12

2 2 Qr = i(uk +yr ), 1, Qo °

ikue,"ve

1-w02) + (uruevrvo+wrwo), Ql .(e12+v12+w1 ) + (uruk.+vrvi+rywo + (u0u1+v0v1+w0w1). (25)

Taking the lowest order terms, which are of order 0(U3), of (24) by use of the assumptions (9) and (10), equations for the wave heights

601 - awr Cr(iT)z=0' . Bur

Bvr-'""-Q(ax

f.By .

-we have he following

lineal...free

surface

-condition for the steady wave potential 90:

1:fur04 vrO*1290 4- 490 ' P(x,Y)

on z = Cr, where

a

D(x.Y) = j(uro4r) _--(vr0;r)°

ax 'ay

These equations are found to be identical with those derived by one of the authors in calcu-lating steady-wave-making resistance for low speed ships /2/.

Subtracting (24) from (23), and taking the lowest order terms, which are of 0(cU-2), by use of the assumptions (9) through- (14) and (22), equations for the wave heights (19) through (21) and the Taylor expansion, we have also the following linear free surface condi-tion for the -unsteady wave potential (pi:

Ir a

vroayi ci

a,c, u

on z = Cr. (28) It sholild be noted that in eqp. (26) and (28), terms including differentiation of double body flow components

'Ito

and vro, such as

avro a0

ava

uzo

j

in (26) or

uro-iiT

ay in (28) should he neglected as higher order terms.

In general, we have to solve the et's. (26) and (28) on z cr(X,Y) Simultaneously with LaOlace's equation, the body boundark condi-tion, and the radiation condition. Here how-ever for the simplicity of treatment, we I

introduce a following non-conformal transforma-tion of coordinates:

x' = x,. y' = y, z' = z - cr(x,Y)

The Laplace's equations for 90 and are written as a24,0 a24,0 324,0 ° 1;71

Wr/

i;77

0(U) ac, a24,0 - 2--t ax 3z

0(0)

a;r a290 + 0(U5) 3y" ay'az' (29)

In this derivation the following relations are (19) and (20), Taylor expansion, and the relation:

0 = a244

_ax'2

_ay'2

24)1

+.a2.1

ae2

o(Eu-")

aCr a20

-

-Dx 3x

az.

Thus, to leading order, 00 and 01 satisfy the usual Laplace's equation in terms of the new variables x'. Y'. z'. Taking the*loWest order terms and dropping the Primes on the new variables, the linearized free surface

condi-tions for the steady wave potential 00 and the unsteady Wave potential 01 are written as:

if

a 2L,2

luroal-vroa100

TIrf =

and

a

32

a

ur0Tc v

r1

y (P1 °

respectively. In these equations, the free surface condition for

-00

does not have the effect of 01, and vice versa. So these equations can be solved independently on the common base flow around a ship expressed by (uro, vro).

The equation (31) Corresponds to the basic equation for the steady low speed wave-resist-ance theory, and the eq. (32) has been derived newly in the present study, Which makes a pair with the eqUatihn (31) for steady motion and can be applledto furl ship faits. For,

refer-ence, when we put ur0+11, and Vi.0+0 the equa-tions (31) and (32) become the conventional linearized steady and unsteady free surface boundary conditions, respectively.

For reference, it is noted as follows that the free surface condition (28) is derived also by applying the present assumptions on Order

of

magnitude to the free Surface condition presented by Newman for smell unsteady motion which is superimposed on :a general steady flow around ship /4/. In terms of NeWmarOs original notations, the unsteady velocity potential is governed by the following first-order free, surface condition:

-(Tt

wo7(9)11 (Wpvw2) + 0(CU--1)

Ttt

2W-Vrt I. _4107(W-Vf) 0(cU-2) 0(c11-2) 0(c11-2) . 2 41207(w2) + /51'5 = 0 on z ° C. 0(e) 0(e11-2) , g'at + 0(e) (30) D(x,y) on z=0, (31) on z=0,

602:

where the following relations exist between NeMMan's notations and those In the present

study:

3, RESISTANCE FORMULAE.

As shown above the lowest order terms, which are of-0(cU-2), are written

as:-Ttt

+

2W.Vft

+ W.V(W.V?) + gc& °

on z = Cr, which Coincides With the equation (28) using the relation

3

W°7 urbj; vrgi-Y

3.1 Steady Wave-making Resistance

In the present study, steady and unsteady velocity potentials Superimposed on the double body base flow are governed by independent linearized free-surface conditions eqs. (31) and (32). In this linearized problem, the wave-making resistance due co steady flow and the added resistance, which is determined as a time average value

in

the periodic wave motion, can be calculated separately /-5/. Based on the solution of the equation (31) a formula for the steady wave-making resistance of a slow ship was given by

Rw = 1TPU2 f1C(e)+1S(6)12 cos30 de (34)

where

C(e)+1S(B) = seCaefigdn D(E,n)

. x exp[iA sec2e(Ecose+hilie)1.,

(15) and i denotes imaginary unit and p is the density of water.

Though the integral of (35) With respect to and n extends over the infinite domain outside of the h011 surface, we have an asymp-totic expression of the amplitude function in the low speed limit by partial integration of (35) with respect to E :

sec2e

XU fi u(to,no)expfiAsec2e

x (C0cose+n0sine)i dn0.. (36)

:

where _(E0,110) is a point

on

the intersection of the hull surface and the still waterplane.

C(8)+iS(8) = aCr a24,

As shown above, the steady-wave-making resistance may be evaluated only from the velo-city of double body flow on the intersection of the

hull

surface and the still water sur-face. It is considered, further,, that the steady wave motion of a low speed ship appears only

in

the region close to the still Water surface. Also for the unsteady wave motion, only the part of a ship hull close to the water surface will have an effect when the wave

length Is small and the wave motion decayes depthWise rapidly in the fluid. So we may replace the ship hull by a vertical, infinite-depth cylinder supposing the ship has suffi-ciently deep draught and nearly vertical sides.

Then the double body flow around the hull may be approximated As A two-dimensional flow and be calcillated by the conformal mapping

technique. Suppose that the intersection of the hull surface and the waterplane may be transformed into the Unit circle by a mapping function as C8 L/2 = nElaa cos(2n)B, nO N = nElaa sin(2-n)3, L/2

Where L is the Waterplahe length and 13 denotes

argument,. And then, the tangential velocity V of the double body flow on the hull surface it given by

V(8) N

2a1 sinB/1{ E an(Z=Ocos(1-n)612

n=1 , N

E aa(2,n)sin(1-n)81 211/2

n=1 ' - (38)

uro

and

vro

in (19) and (27) can be calculated from V.

Further, for the sake of simplicity of calculation, we assume that the Waterplane form is convex and symmetrical with respect to the y-axis. This simplification does not seem to effect the result so much, since our aim is to determine an entrance form Which has the least value of the sum of the steady wave-making resistance and the added resistance

in

short waves.

By applying. the stationary phase method in the integrals (36) and (34), we have a simple

formula for wave-making resistance /6/ 1

Rw .10u2L2 x

U6

'71717.

ne(13)d8

x 0i2F(43)2 {E '(3)}4{ne(13)}z ' (39)

where

'no'(B) = dno/dB

_Co'(3)

= (g0/d8

NB)

Yr8(E0iTIO) V dV U Ud8

(37)

3.2 Added Resistance in Short Waves

There is a trend that measured values of added resistance for full'ihip-mOdels in short Waves are greater than those predicted theo-retically under the assumptions of slender

ship. To improve the shortcoming of. the theory Fujil and, Takahashi introducedan.additional term, which takes account of wave' reflection at

ship's

bow /7/. In their study HaveloCk's formula for the drifting force bated on the wave reflection /8/ was used together with empirical...corrections.

. . . . .

A formula propoped by Fujii.and.Takahashi. for resistance increase Raw due to wave

reflection at bow part is given by

1 ,

Raw . al(14-a2)ipg4a4B sin28

1 B/2 , sin28 = I sinz0 dy,

0"B/2

where ol and a2 are empirical correction factors considering the effect of finite draught d and that of the advance velocity:

al _{w2112(R0d)/{52112(R0d)} R.12(Rod.l.) (42)

U 1/2 02 .

₅₍₇₇₎

8

where L and 1:1 are the modified Bessel functions, ko Is wave number of incident wave, B the breadth of the waterplane, Ca the ampli-tude of incident wave, and 8. denotes the inCli-nation of the waterline to the centerline as shown in Fig. 2.

V

Incident Wave

Fig. 2 Coordinate system along waterline (43)

The empirical constants in (42) And (43) were modified later based on additional experimental data obtained in the Nagasaki Experimental Tank as

. x2112(1. Skod)/

1.02112(1.54d) +1(12-(1.51E0)1. (44)

U 1/2

Though the above formulas are for added

resist-ance in head waves, they may be. applied also in

oblique waves by a modification of ;17.1T2T

as-1

sin2(6-x)

Isin2(8-x) dY.

where x is the wave direction as shown in

Fig. 2. and the integration is to be performed

in the non-shadow part.

Years later, an- asymptotic theory co

calculate added resistance of a ship in slow

speed forward motion in short waves was

prsented by Faltinsen 13/.

Looking into

wave motions near the ship's hull, he

intro-duced, a linearized free-surface condition for

diffraction velocity potential OD along the

streamline close, to the ship's hull side:

. a 2

(--V--) OD +

_.at

_{as Bz D} 0

on z

0,

where s is the coordinate along the waterline

and V the velocity along s as shown in Fig. 2.

It turned out- that this free-surface

condition (47) coincides with the equation (32)

which was newly derived in the present study.

At present, lhowever, the equation (32) together

with body boundary condition has not been

solved yet by the present authors.

But

Faltinsen. solved approximately this unsteady

free-surface problem.

He considered the effect

of flow distorti6n around a body on the

diffraction wave alone in solving this boundary

value problem.

Then an asymptotic formula was

derived to estimate added resistance in short

waves at low speed limit.

_{This theory is}

regarded as a theoretical extension of

Havelock's theory for a case of moving bodies.

In the present study, added resistance in

short waves is calculated based on this

faltinsen's theory.

In this theory added

resistance in short waves is given by

R = I Fnsine di,

F2![!

_n

2sin(8+a)].

_{gCa 2}

_Ito

- cos2(8+a) +

k2

,

(49)

kl

{we - Vk0cos(8+a)}2/z,

_

(50)

412 - k02 cos2(e+a).

(51)

where Fn is wave force acting on an infinitely

small vertical element of the hull. we is the

circular frequency of encounter, ko and ca

are

wave number and amplitude of the incident wave,

a.

_{m- x is the wave direction and a is the}

inclination of the waterline as shown in

Fig. 2.

kl and k2 are wave number components

of diffraction wave, which Is assumed to be

expressed in a form:

F

lpg

2

eiepNiz + ilkos cos(e+a) - nk2/]

where n and-s are coordinates normal and-along

the waterline.

The integration of (48).1-d

along the -waterline in the non-shadow Part as

shown in Fig. 2.

An asymptotic formula was

also derived, in the low speed limit for the

wave force on an Infinite small element Fn of

(49) as

2woU

2[sIn2(8+a)+-=-H1 cos6 cos(8+a)}],

(52)

where the following approximation is used:

V = U cose.

From (49) the authors have derived a little

different expression from (52) as:

Fn = logca2kin2(8+a)

(53)

2woU

Icosa - cos0 cos(e+a)1].

(54)

Now let us consider a case of a surface

piercing vertical circular cylinder.

In this

case, the added resistance can be expressed

analytically by use of the asymptotic

expres-sion in the low speed limit.

Exact- velocity of

the double body flow along the circular

cylinder is given by

V

2U cos8.

₍₅₅₎

Then from eqs. (48) through (51), we have in

the low speed limit

1 2 T4 wo

Raw -

rr.-dcosa *

-

U

_{(l+sin2a)/].(56)}

'

4

'

where r is the radius of the circular cylinder.

When using Faltinsen's approximation (53), we

(48)

have

1wOD

-pgc 2 r[L'fcosa +

_{2 a 3 . g}

(l+cos2a)11,(57)

Depending on the approximation of the flow

around a body, the effect of advance speed in

eqs. (56) and (57) appears differently, e.g.

the effect of advance speed in head wave (a m,

) is estimated twice as large as When use is

made Of Faltifiten's approximation (53),

In Fig. 3 the slim of steady wave-making

resistance and added resistance due to wave

reflection it shown for different incident

angles, where the steady Wave-Making resistance

is Calculated by the equation. (39) at:

Rw

_-pU2(2r)2

8192.

(----)

_U 6

'2

315

for a surface piercing vertical circular

50

L = 2r = 100m

A= 30m, hw = 0.5m

0

0

1-

2 3 I- I 0 0.05 0.10 I I I 3

4

5 U(m/s)

0.10

0.15 U/

5 1.1(m/s)

0.15 W at:

From the Fig. 3 it is found that the added resistance due to the wave reflection is dominant in the low speed region while the .

steady wave-making resistance increases rapidly With increase of Fronde number. As

con-sidered previously, the calculated values of added resistance appear diffeCently with respect to the Incident Wave angles depending on the approximation of the values of bate

flow

around the body. This difference is large for extremely blunt forms Such as the present Case

(circular cylinder) but it may be Stall for conventional ships as Shown later.

It is also a characteristic of FaltinSen's theory that the added resistance does not vanish even when an obstacle symmetrical With respect to midship section, such as a circular cylinder, is running in beam waves. This is attributed to the effect of advance sPeed.

Raw calculated by eq. ( 56 )

Raw calculated by eq. (57)

100

= 90°

Fig. 3 Resistance

of

surface piercing Vertical circular cylinder Rw +Raw Raw Rw I J 2 3 4

5 U(m/s)

0 0.05

0.10

0.15 W ar_

The effect of advance speed aPpears to make the wave field asymmetrical with respect to front and rear parts of the body due to the change of wave number by the Curved flow around a body. The change of wave number ki/ko and K214 are shown

in

Fig. 4 An the case of the surface piercing vertical circular cylinder in beam waves. In the figure, 4tI/It0 and k2/k0 are not drawn in the regions where circumferential angle

on

the cylinder is larger than around 120 deg. They are the regions where the value of k2 calculated by (51) is :imaginary and they ire .excluded from the, range of integration of

-(48). It should be mentioned here: that this sort of considerations about the value of k2 and the region of integration of (48) were not made

in

deriving the asymptotic expressions

(56) and (57).

OS

)7

To

=

2 5"

/4

S?

50

cbi = g(003 e-kmt exp ( koz+ikoy)

= Ae-i"texp kiz+if kos cos 0+ f

3.0

U/Vii

,= 0.2

- 2.0

1.0

1k0

k2/ko

30 60

_{90 120 150 180}

7 (deg)

Fig. 4 Variation of wave number around a surface piercing vertical circular cylinder in beam waves

3.4 Application to Ordinary Ship Hull Forms In Fig. 5 values of steady wave-making resistance calculated by eq. (39) are compared with the experimental values. In, the calcula-tion; ship

hull

forms were approximated by infinite vertical cylinders symmetrical with respect to midship section, which have the same waperplane curves as those of the fore part of original ship

'hull

forms. Assuming that contribution to the steady wave-making resist-ance from the run part is small, a half Of the

Experiment Fig. 5 _{Comparison of wave-making resistance}

of full hull forms

606

1.0

calculated resistance by eq. (39), that is the contribution from the entrance part only, is compared with the experimental value.

Considering the approximation adopted in the theorediCal.calculation, wave-making resistance coefficients of full ship models which have comparatively 0-shaped framfline and nearly vertical sides at the waterplane are shown in the figure.

In spite of the rough approximations mentioned above, it is observed that the cal cdlated Values of wave-making resistance agree well with the experimental values. Therefore, it is considered that the present

calculation method of steady wave-making resistance is applicable to the Optimization problem of a hull form which has U-shaped frameline form.

In Figs. 6 and 7, experimental values of added resistance of conventional full hull forms are compared with calculated'velues in _a non-dimensional form:

'

0.3,4 = kwa_4(pgc:2132/L) ₍₅₈₎

Solid lines represent aaw values _{calculated by} eqs. _{(48) through (51), where the basic flow} velocity V in eq.. (50) is calculated by the conformal mapping technique replacing the hull form by an infinite depth vertical cylin-der Which has the same taterplane curve as the original

hull

form. This method based on Faltinsen's theory is called the present _method hereafter for the sake of Convenience. _{On the} other hand, chain lines represent aaw values calculated by

eqs.

(48) and Faltinsen's asymp-totic formula (54), where V is assumed as Ucos8. _{Added resistance in short waves was} .calculated also by Fujii 7 Takahashi.'s formulas

(40),(44),(45) and (46). _{The results are Shown} by dotted lines in the figures.

In the region where

Offir

_{is less than} o 15 in Fig _{6, FujiiTakahaahi'S formula} predidis somewhat larger added _{resistance value} than that of the present method. _Faltinsen's asymptotic formula (54) gives the Values almost same as those of other formulas. _The

differ-ence of the calculated results is small compared with the scatter of the experimental data.

As for the results in oblique waves, the difference of added resistance for various wave directions is relatively small in the calcula-tion by the present method and the _tendency coincides with that of the experimental values. In the experiments shown in Fig. 6 _{and Fig. 7,} periodic ship motion was negligibly small _but measured Values of added resistance

in

_oblique waves may include some effect of steady drift and check helm taken to cancel _{yaw moment,} which are not considered in the theoretical calculation.

Though Faltinsen's theory Is an approxi-mate theory, it is found that it can predIdt the added resistance in the sate level of agreement with experiments. as FujiiTakahashi's empirical formula-. _{The present method of} calculation of added resistance based on Faltinsen's theory was used in the optimiza-tion problems of hull forms in the _followings.

2

2

Fig. 6-a Added resistance in head wave Fig. 6-b Added resistance in head wave

6-

6

-Ship E, Cb = 0.79

Ship E, Cb = 0.79

1/L=0.3

4 -

1/L=0.25, U/N51:=0.15

0'

0.1

6

-Ship F, Cb = 0.84

-x 4

A/L =0.3

4 %.3

0

0.3 607

2-90 120 180 180

6-0

'

X(deg)

Ship ,

= 0.84

F Cb

1/L=0.2, U/N/F=0.12

2

Ship C, Cb = 0.82

A/L=0.3

0 _{9' 0} 0 0.1 0.2 0.3

Ship 0, Cb = 0.80

A/1=0.3

0

0.1 0.2 0.3

U /

1 t

120

150 .180

X(deg)

Fig. 6-c Added resistance in head wave Fig. 7 Added resistance in oblique waves

ique waves

16.1 6 4 0 6 4 2 0

Ship A, Cb

=

0.84

A/L=0.3

I

0

Experiment

1

-1 6 -4 2 0 6 4 2 0 Present Method Faltinsen

---- Fuji-Takahashi

I 1 0.1

Ship B, Cb

=

0.83

1/1=0.3

o

G

0.2 0.3 U

/

Iii.

6 0:1 0.2 0.3 U

/

fgr

: 0

4. MINIMIZATION OF RESISTANCE

4.1 Optimization Method

.7 Optimum hull forms are determined in such a way that the sum of steady wave-making resis-tance and added resisresis-tance becomes minimum in a certain Wave condition. for thii calculation a non-linear programming technique, i.e. Hooke and Jeeves' direct search method with external penalty technique /9/ was used. The Objective function,

which

should be minimized in the optimization problem is the sum of the steady wave-making resistance calculated by the eq.(39) And the added 'resistance due to wave reflection'in short

waves

In the sane Manner as examined in the pre-vious section, a ship hull fort was replaced by an infinite-depth vertical cylinder and the double 'body flow V was calculated by the

con--fbrm-al mapping technique. For the sake of simplicity, the cylinder is Assumed to have symmetrical waterplane curve with respect to

-midship section, which is identical with that of the entrance nett of the subject ship, since our interest is in the entrance fort whose resistance, namely the sum of steady wave-making resistance and added resistance, is minimum. The constraint of constant waterplane area was satisfied generally

in

the following ploblem.

4.2 Optimization of Length-breadth Ratio of Elliptic Cylinders

Within the framework of the low speed wave resistance theory there is a tendency that the steady wave-making resistance does not necessarily decrease with increase of length-breadth ratio. 'Considering this tendency, length-breadth-ratio of vertical elliptic cylinders was oimized as a simple case in 'waves. The waterplahe curve of an elliptic

cylinder is expressed as - = a1cos8 + a3cosE, L/2 no = alsina - e3sinS, L/2

where al and a3 are the coefficients of confor-mal mapping function (37) And should satisfy

-al + a3 = I, (61)

z1

was chosen as a design variable of the opti-mization problem. In order to satisfy the constraint of constant water-pi-0e area, length of the cylinder L was determined by the follow-ing relation:.

L4

Aw = _711(2a1 - 1). (62)

The waterplane form of minimum Steady wave-making resistance was chosen as the initial form for the optimization procedure. The con-ditions

of

the calculation are

608

ROM

-1200 Rw

JO

Raw Raw - --- - -

---hw = 2.0m

Fig. 8 Optimum form for various wave heights (wave direction x = 180°, head wave)

1

L

Length (initial value)

L '185.6m

Waterplane area

Breadth (initial value):

441-7854 M

B 53.9 m

Ship. speed m/sec (14 Kn)

Wave length 60 m (A/L-0.323)

Wave. height hw -261 = 0.5 - 2.0 m X = 180° (head wave) Wave direction

IC,' (beam wave) Optimum Watetplane dUrVes are shown in Fig. 8 for various wave heights and'in":Fig-.; 9 for various wave directions.

From Fig. 8, it may be round that lengthr

breadthretio

of

the optimum forms increases as the wave height becomes large Though the steady wave-making resistance component of the

optimum forms is a little larger than that of the original form, the total resistance is smaller. This is due to the decrease of added resistance due to wave reflection:

. In Fig. 9, length-breadth.ratios of

opti-mum forms in oblique waves do not change much from that in head wave. So it seems that an optimum waterplane form designed. in head wave will be A nearby optimum' form also in oblique waves.

Optimum Form

X = 180° (Head Wave)

X = 150°

4.3 Optimization

of

WaterMlane Form -AN

1L1

Raw R(KN)

1200

Raw

= 90° (Beam Wave)

Fig.. 9 Optimum form for various wave directions (hw=1.0 m)

As a next step, waterplane form was opti-mized under the constraints of constant length, breadth and waterplane area. In the calucula

tion,

waterPlane fort was expressed by 5th polynomials /10/.

n = A + Bt + Cw, ₍₆₃₎

A = 1 - 442. + 744 - 345. (64)

11..7 -1.3(2'3Z)(14), (65)

C = 601'2(1)2. (66)

where and 7-1- are the. coordinates normalized by the length of entrance L. and the half breadth of waterplane B/2 as _shown inTig. 10:

Ix-x&!/t,

(i)7)

n = y/(B/2), (68)

w

is

the area coefficient of entrance and run parts, and t is the inclination of the water-plane curve at the fore and aft ends res-pectively: _

w = f

nci& (69)

0

t = d; at = 1

609

Fig. 10 Expression of waterplane curve

Waterplane curves expressed by (63) satisfy the following cenditions: -n = 1 at = 0,

4

42

= 0 at = 1

In the Present calculation w and t Were chosen as design variables and Z was determined by the following equation in order to satisfy the constraint of constant waterplane area:

L Aw

-(1 - --) = 1(1 - w)

2 LB

Further an inequality constraint was imposed so that the waterplane curve is convex everywhere between = 0 and E = 1 in order to calculate, the steady wave-making resistance by use of the stationary phase method:

42

d2;

< 0 for

0 <

<1

(72) Within this inequality constraint the optitut values of w and t are determined.

Calculations were made for the following conditions. Length and advance speed of the ship, wave length and incident wave height are fixed:

.

1

= 300 m, U = 7.2 m/sec (14 Kn) A 60 in (A/L = 0.2),

hw - 1.2 in (hw/) = 1/50),

while. length-breadth ratios and fullness of waterplane area are varied:

L/B= 6, 5,

A.,á/LB = 0.7, 0.8, 0,9.

Optimum waterplane forms were determined for the following three cases: .

Minimization of steady wave,taking-resist, anCe alone

Minimization of added resistance alone MitimiZation of total resistance,.i.e. the

sum of steady wave-making resistance and added resistance..

(71) Optimum Form

INN

In Fig. 11 optimum waterplane forms of the above three cases are compared. The values of resistance components for the optimum forms are also compared in the figure.

-It is observed that optimum waterplane forms of minimum total resistance are close to those of minimum steady wave-making resistance. On the other hand, the optimum waterplane forms

of minimum added resistance are somewhat different from the others' as shown figures.

It is found further that steady wave-making resistance Rw and added resistance Raw change largely with the variation of principal geometrical parameters such as L/B and Au/LB. However, no appreciable difference is observed for Raw with respect to the local minor change

L/B=

6

Rw Minimum Form

Raw Minimum Form

Rw+Raw Minimum Form

Aw

08

LB

L =300m

U =

14kn(U/fgt=

0.133)

A =

60m(A/L=

0.2)

hw = 1.2m

(hw/L=

1/50)

Aw 09 LB 610

of waterplane forms. In other words, Raw seems insensitive to the change of local form. On the other hand, Rw changes according to the variation of-waterplane form. Therefore, in the minimization procedure, the steady wave-making component Rw plays an important role on the determination of waterplane form of minimum total resistance (Rw+ Raw). As a result, the waterplane forms of minimum total resistance come close to those of minimum steady wave-making resistance.

From this study it may be said that under the specified principal geometric parameters the waterplane form of minimum steady wave-making resistance is considered as an optimum

form even in short waves.

miria

i

Raw

Fig. 11-a Comparison of 'waterplane forms of minimum resistance (L/B 6)

Raw

R(KN)

]100

5. CONCLUDING REMARKS

In the first part of the present paper, a linear free-surface condition for unsteady wave motion was newly introduced. This linear equation is formulated on the double body flow

around a ship and makes a pair with that of the low speed theory for steady wave-making

problem. In the region near the hull this equation coincides with that introduced by Faltinsen for a diffraction problem in short waves. In the present study, therefore,

Faltinsen's asymptotic solution was used for the calculation of added resistance. Recently, with relation to Faltinsen's theory, Ohkusu and Naito et al, discussed the effect of curved bow flow and breaking waves around a ship

/11,12/. Further study remains for deeper understanding of flow phenomena around a ship's bow. It is hoped that the free surface

condition introduced in the present study would be a basis for the study.

In the latter part of the present paper,

Aw LB

L = 300m

U = 14kn(U/IF:= 0.133)

A = 60m (A/L= 0.2)

hw = 1.2 m (hw/ 2= 1/50)

Aw

=0.8

=

LB

0'9

Fig. 11-b Comparison of waterplane forms of minimum resistance (L/B = 5)

611

R(KN)

100

50

0

attempts were made to minimize ship resistance in short waves by use of the nonlinear pro-gramming technique. The results obtained through the calculation are summarized as follows. Both steady wave-making resistance and added resistance due to wave reflection change widely with the variation of principal geometric parameters such as length-breadth ratio and fullness of waterplane form. So the principal geometric parameters of a full ship should be determined considering the added resistance due to wave reflection as well as the steady wave-making resistance. While for the local change of waterplane forms, the added resistance is rather insensitive when it is compared with the steady wave-making resist-ance. Therefore, under the specified principal geometric parameters, the waterplane form of minimum steady wave-making resistance is regarded approximately as an optimum form even in waves. It should be reminded that these results are applied only to the ships whose motions in waves are negligibly small.

Rw Minimum Form

L/B= 5

-- Raw

Minimum Form

ACKNOWLEDGEMENTS Kansai Society of Naval Architects, Japan, No.I97, 1985, pp.39-45.

The authors Would like to express their sincere gratitude to Mr. T. Takahashi, Project Manager of Ship Hydrodynamics Laboratory of Nagasaki Technical Instittite, Mitsubishi Heavy Industries for his instructive discussions in preparing this paper.

REFERENCES

I. Ogilvie, T.F., "Wave Resistance : The Low Speed Limit ," Univ., of MiChigan, NAVA1 Archi-tecture and Marine Engineering, NO:002, 1968.

Baba, E. and Takekuma, K., "A Study on Free Surface Flow around the Bow of Slowly Moving Full Forms," Journal of the Society of Naval Architects of Japan, Vol.137, 1975, pp.1-10.

Faltinsen, Minsaas, Liapis, N. and Skjordal, S.O., "Prediction of Resist-ance and Propulsion of 'a Ship in a Seaway.'" Proc. 13th Symposium on Naval Hydrodynamics, 1980, pp505-529.

Newman, J.N., "The Theory of Ship Motions," Advances in Applied Mechanic's, Vol.18, Academic Press, 1978, pp.221281.

Maruo,

H:,

"Resistance in Waves," 60th Anniversary Series, The Society of Naval Architects of Japan, Vol.8, Chapter 5, 1963,

pp.t7-102.

Baba, E., "Application of Ship Resist-ance Theories to the, Design of Full Hull Forms'," Centenary Conference on Marine Propulsion, NEC 100, North East Coast Insti-tution of Engineers and Shipbuilders, 1984.

Fujii, H. and Takahashi, T., "Experi-mental Study on the Resistance Increase of a Ilarge'Full Ship in. Regular Oblique Waves," Journal, of the Society of: Naval Architects of Japan, Vol.137, 1975, pp-132-137.

'Havelock, T.H., "The Pressure of Water Wave upon Fixed Obstacle," Proc. of the Royal Society of London. Series A, No.963, Vol.175,-1940, pp.409-421.

Parsons, M.G, "Optimization Methods for Use

in

CamputerAided Ship Design,"Proceedimgs of First Ship Technology and Research (STAR) Symposium, SNAME, 1975.

Gertler, M., "A Reanalysis Of the Original Test Data for the Taylor Standard Series,"

pris

RepMkt 806, 1954.

Ohkusu, M., "Added Resistance-

in

Waves

of Hull

Forms With Blunt Bow," Proc. 15th Symposium on Naval Hydrodynamics, 1984,

pp. 135-148.

Naito, S., Nakamura, S. and NishigUchi, A., "Added Resistance in Short Length

Waves on

Ship Forts With Blunt Bow," Journal of the