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Wave Resistance of Ships in Low Speed

Eiichi Baba*

From the flow measUrement arouAd the bow of a slOwly moving ship it was found that, except in the thin boundary layer near the free surface, velocity components agree well with the calculated values from the double body potential. In tt.e thin free surface layer the velocity components change depth wise very rapidly. Based on this experimental result an asymptotic formula to calculate wave resistance of s/ow ships is derived by assuming that the flow around the floating body can be expressed as a sum of two parts. One is the double body flow and the other is an additional thin free surface layer superposed on the double body flow The formula includes a local nonlinearity around the body and gives better agreement with experiment than the conventional linearized wave

resistance theories. Appliëations of the prèseAt theory to the hull form design are cOnsidered.

1. Introduction

In this paper the contents of the previous two papers by the present author (1975 a, b) are consOlidated with detailed derivations of the wave resistance formulas and

with a recent result of theoretical and experimental studies.

It is the objective of the present paper to introduce a wave resistance theory for slow ships. The theory was primarily developed in order to get better understanding

of the mechanisth Of wave breaking phenomenon observed

around the bow of full forms. It has turned out, however

that the theory is also used to predict the wave resistance of slow ships.

In Sec. 2 a brief review of the studies of slow ships is given. In Sec. 3 a result of flow measurement around the bow of a blunt form is presented. From the flow measure-ment it was found that except in the thin boundary layer

near the free surface, velocity components agree well with the calculated values by the use of the double body velocity

potential. In the thin free surface layer, on the other hand, the velocity components differ from the double body solutions and change depthwise very rapidly until they

reach the value of the double body solutions. Based on this experimental result .a wave resistance theory for slow ships

is developed in Sec. 4. In the theory the double body

solution is used as the zero-order solution and an additional

velocity potential is then derived in such a way that the sum of two velocity potentials satisfies the free surface

conditions.

In Sec. 5 the wave resistance of simple forms such as a vertical circular cylinder is calculated for a qualitative examination of the theory. In Sec. 6 the calculated wave resistance of a Wigley's paraboliô form is compared with experimental values which are determined by means of wave pattern analysis. In the low speed range (F < 0.25)

a good quantitative agreement is observed. In Sec. 7

appli-catiOns of the present theory to the hull form design are

considered.

* Dr Eng, Resistance and Propulsion Research Laboratory,

2. A brief review of the studies of slow ships

In 1968 Ogilvie reported the study on a two-dimensional

problem Of the thin free surface layer on the non-uniform flow which is determined by the double body soFution. Ogilvie assumed that the basic non-unifOrm flow varies slowly with space variables while in the thin free surface

layer physical variables such as velocity and wave height are

assumed to vary very rapidly. In a paper by Hermans (1974) a wave resistance problem in low speed is treated in the similar manner as Ogilvie. A theory to analyze the

free surface flow around a semisübmerged horizontal

cylin-der, perpendicular to the incoming flow is developed. Timman (1974) extended Hermans' theory further to study the free sUrface flow around a semisubmerged three-axial

ellipsoid with its middle axis horizontally on the still water

plane so that the free surface meets the bow over a relative-ly broad part of the front. The problem is then transformed

into tWo-dimensional one in the plane perpendicular to

the middle axis. Keller (1974) studied wave patterns of full

forms in low speed by a different manner from those men-tioned above, but based on the similar assumption on the free surface flow. Keller assumed that for small Froude number, the flow consists of the double body flow plus an oscillatory flow which represents the wave motion.

Dagan (1975) discussed the non-uniformity of the solution of thin ship expansions in low speed free surface problems.

In a recent paper by Newman (1976) a boundary value problem fOr slow ships has been developed. The free surface

condition obtained by Newman coincides with the free surface condition which was derived independently by the present author. The present study is a direct extension of Ogilvie's t'heory to the three-dimensional case. Originally Ogilvie did not intend to apply his theory to a surface piercing body. Here, however, a theory for conventional

ship forms in lOw speed limit is developed, since the results

of the flow measurements around a blunt form motivated

(2)

MTB 109 August 1976

piercing bodies.

3. Flow rneaJrement around the bow ota slow speed ship For the flow measurement by use of a 5-hole pitot tube, a simple hull form M.2201 as shown in Fig. 1 was used. The principal particulars of the ship model are shown in Table 1.

The flow measurements were conducted in two different load conditions. Fig. 2 and Fig. 3 show the flow patterns around the bow of shallow draft (200 mm) and deep draft (400 mm) respectively at speed U=1.089 m/sec (U/../L 0.1420). With an increase of ship speed short waves observed in Figs. 2 and 3 are transformed into rather

con-fused 'disturbed flow (breaking waves) as shown in Figs. 4

and 5 (U1.284 rn/sec. U/ ,/E=0.1674).

Since there are rather confused short waves on the free

surface, only the mean elevation of' the free surface from

the still water level was measured first (Fig. 6). Then the velocity components (u, v, w) were measured up to the free surface by use of a 5-hole spherical pitot tube of

7 mm diameter. The measured results are shOwn in Figs. 7 through 10 comparing with the velocity components

calcu-lated numerically from the double body velocity potential which satisfies the rigid-wall free surface condition. In those figures z 0 corresponds to the free surface. The rigid-wall solutions are extrapo!ated up to the free surface from the values on the still water level, since they change

depthwise very little.

From the comparison it is found that except in the thin

boundary layer near the free surface the measured velocity components agree well with the calculated values based on

'the rigid-wall solution. In the thin free surface layer mea

Fig. 1 Lines of M.2201

Table '1 Principal particulars of M.2201 FP

sured velocity components show rapid changes with an increase of depth while the calculated values show very

little change. 100 mm

EIH

Fig. 3 U 1 .089 rn/sec in deep draft Fig. 5 U =1.284 rn/sec in deep draft

- -ShaIIo draft Deep,draft

10

-500 300 0

(F.p.)

Fig. 6 Measured wave heights at U =1.089 rn/sec

uIu, v,u,w/u

-Free surface

- o-a-'-

100

0 - I 0 $ I

Oi

I

-

I

D.

t

l00.i.

rneoid cuIcuiated

j

0.5 4 LÔ2 Still -W?evlevel 100mm

Fig. 7 Velocity components at x 350rnm, y 0

Load ' Deep Shallow

Lpp('m) 6.000 6.000 B' (ml 1.000 1.000 d (mm) 400.00 200.00 n(kg) 2004.7 958.68

L/B

6OO 6.00 Bid 2.50 5.00 Cb 0.8353 0.7989 C, 0.8359 0.7999 Cm , 0.9993 - 0.9987

load U/U V/u W/U deep o Shallow

I

£

Fig. 2 Flow pattern around

bow of a full fOrm at U =1.089 rn/sec

(U//7J

=01420)

iii shallow draft

-LL

Fig.

4 U

1.284 rn/sec

(U//T=0.1 674)

(3)

a

Fig. 8 Velocity components at x -175mm, y 0

U/a, V/U,W/u Taking the rectangular coordinate system fixed on the

0.5 L00 body with the origin on thestill water plane weset x-axis

directing to the uniform flow U and z-axis directing

up-wards as shown in Fig. 11. Supposing a ship floating on an inviscid irrotational incompressible fluid we consider that the velocity potential fOr the free surface problem is - expressed as a sum of two parts:

-100

(x,y,z) = ø,(X,Y,Z) +(x,y,z),

(1)

N

where 0,. (x,y, z) is the potential fOr the rigi&wall problem, and 0 (x y z) is an additional potential to ,. (x y z) so that the sum satisfies the free surface conditions. Fig. ii -200 shows the modeled scheme of the present problem Though

05 10 we are considering short waves appeared aroühd a slow

ship, the surface tension effects on the wave formation is

neglected for the simplicity of treatment.

The boundary valuC problem for the present study is

written as follows:

0= 'I+ .I, +

(2)

U2=gHx,y)+-[+cI+1

onz'H(x,y),(3)

u/U, V/U,W/U

9---

as

as

u/U,v/U,w/U

as

100 1.0200 E E

Fig. 9 Velocity components at x O y 40Omrn

.00 where ,.(x, y) is the wave height due to the double body

potential, i.e.

100

N

1.0200

Fig. 10 Velocity components at x1 75mm, y=400mm

4! Theoretmal study

4.1 Bouncb.rv value problem

From the results of flow measurements we may assume

that the rigid-wall solution is quite accurate everywhere

except in the thin layer near the free surface.

o (//x2+y2)

for x <O,

where g is the acceleration of gravity. The last ohe is the condition insuring that waves only follow the ship.

According to Ogilvie, we assume that the wave height is expressed as the sum of two parts:

r(X,y)[U2_0x(x,y,0)_Oy(X,y,0)],

(7) since

,(x,y,0) = 0.

(x,y) is a superposed wave on

.-.-i-e. U

4(x,y,z)

rigid-wall olutiOn

Fig. ii Scheme Of tilemodeled Phenomenon 1(X,y,z)SoIurion of thin surface layer

MTB 109 August 1976

flr

ttt

21

I

2-o

I

II

O.I

15

'U

i-t

I I I I I

A______

0 I I I I I 0 0 ,n is a normal

-

Ux =

+Hj-

on z

Hx,z),

vector on the body surface,

O(h,41x2+y2)

forx>0,

as

x2+y2-°°

(4)

(5)

(4)

MTB1O9 August 1976

In the low speed limit we introduce the following

as-sumptions about orders of magnitude. The detailed reason-ing of the assumptions is found in Ogilvie's paper

Ø(x,y,z) = 0(U),

,(X,y) =0(U2),

a a a

-'

, -= 0 (1) when operating on

4)(x,y,z) or

ax ay az

Cd)

4)(x,y,z)=0(U5),

ix,y)0(U4),

p,

__.., --

0(U-2) when operating on 4)(x,y,z,)

ax ay az

orix,y).

These assumptions are interpreted physically as follows.

To an observer at the free surface around a slow ship, it appears that there is a uniform stream below him which has a speed equal to the local value (cbrx, øry). This

Ob-server can not see the free surface disturbance represented by ,(x,y), for it varies on a length scale which is too large. Only the small, superposed waves, represented by

(x,y), will be visible to him.

Under the above assumptions about orders of magni-tude, the lowest order terms of the free surface boundary conditiOns (3) and (4) are written respectively (see

Ap-pendixA):

-gix,y)+ 4)rx(X,9,0) Ø(x,y,z)+ 4)(x,y,0)

x cb(X,Y,Z) =0 on Z = r(X,Y), (8)

4) (x,y,z)

-jx, Y) 4),-

(x,y, 0)-

x,y) 4)(x,y,0)

= -

[r(x,y)ct)rx,y,0)] +[r(x,y)4)ry(X,y,O)],

ax ay

onz,(x,y).

(9) Eliminating (x,y) from (8) and (9), and taking the lowest Order terms, which are of 0(U3), we have the following

free surface condition (see Appendix A):

[4),-., (x,y,0)--+ 4),. (x,y, 0)-_-] 2 4)(x,y,z)

+4)(x,y,z)

D(x,y),

on z = r(X,Y) (10)

where

ar

-

D(x,y) = [,(x,y)4),(x,y,0)

ax

+ftr(X,Y)4)ry(X,Y,O)].

(11)

In general, we have to solve Eq. (10) on z = ,(xy).

Here, however, for the simplicity of treatment we introduce a following non-conformal transformation of coordinãtès.

x' = x,y' =y, z' = z -,(xy).

The Laplace equation fOr4) (x',y',z') is written as

0a2Øa24)

a2

ax'2 ay'2 az'2 order

[UI

-2--

a2

+0(U5).

ay ayaz

[U3]

Thus, to leading order, 4) satisfies the usual Laplace

equa-tion in terms of the new variables x', y', z'. Taking the lowest order terms and dropping the primes on the new

variables,.our boundary value problem is written as

(12)

[ø,x(x,y,0)+4)ry(x,y,0)f]20(x,y,z)

+Ø(x,y,z)=D(x,y) onz0,

(13) 0(1/s./x2+y2)

forx>0,

çb(x,y,z)=

asx2+y2-o

(14) o(h/Jx2+y2)

forx<0

+øry(x,y,0)4)y(x,y,0)}

(15)

In addition to those conditions the body boundary

condi-tion must be added. Here, however, this conditiOn is omit-ted, since Ø(x, y, z) is considered as the solution of the first

step of an iteration cycle.

In a recent paper by Newman (1976) a free surface condition which is identical to Eq. (13) has been derived independently., In hiS derivation the double body

poten-tial is also used as the zero-order solution for the slow ships.

When droppWig the terms including cbry, we have a free surface condition which coincides with the Ogilvie's two-dimensional expression. Ogilvie obtained an explicit solu-tion by the use of complex funcsolu-tions. In our three dimen-sional problem, Fourier's method is used with a slight modification.

4.2 Solution of the present boundary value problem

First we introduce a double layer potential 'I' (xy,z) which equals to D(x,y) at z 0 as

'4'(x,y,z)

_4LfkdkJ'dO

X1fJddflD(,).

(16) We then assumethe form of velocity potential 4)(x,y,z) as

Ø(x,y,z) .=

_f kdkf

do ekz+ixc0s05mn0)

xF(x,y,k,O),

(17)

where

F(x,y,k,O) is

determined in such a way that Ø(x,y, z) satisfies Laplace equation and the free surface condition

(13). From (17) we may consider that

F(x,y,k,O)

is

relating to the wave amplitude which depends on the in-tensity of disturbance acting on the free surface. In our problem D(x,y), the right hand side of (13), is considered as a disturbance acting on the free sürfacè. Then we may

a ,. a2

ax' ax'az'

(5)

assume intuitively thatF(x,y,k, 0) is expressed in terms of

the double body potential. This assumption means that the operation a/ax or

a/ay

on

F

does not change the order of magnitude due to the assumption (C). (x,y,z) is already assumed to be of

0(U5)

in the thin boundary layer near the free surface where

x,y,z=0(U2).

Therefore, from (17), we may deduce the following orders fork and

F(x,y,

k, 0):

k0(U2),F(x,y,k.0) 0(U9).

Then,

4 (x,y, z)=

kdkf

do e kz+ilc(xcosO+iisin8)

[_k2cos2OF+2ikcosO.f+.L

ax ax2

[U5]

[U7]

Eu9]

The second and third terms in the above bracket are of

higher order than the first term. The same rule is applied to and When taking the lowest order terms, we see

first that 0 satisfies Laplace equation. Then, substituting

them into the free surface condition (13),

F(x,y,k,0)

is determined as follows.

k0(x,y,0)

F(x,y,k,O)

lko(x,y,0)k k

xffdd7iD(n)e

-1kcosO+sinO) where

ko(x,y,0)g/fr,b(x,y,0)cos0 +Ø,(x,y,0)sinO

2 (20) We see that

F(x,y,k,0)

is expressed in terms of rigid-wall

solution. Thus

F

satisfies the previously assumed

condi-tions,

e. aF/ax, aF/ay=o(F).

Substituting (19) into (17), and taking the radiation condition (14) into account, we have the following expres-sion of our velocity potential:

dO k0(x,y,0)

x

pvf

dk

k0(x,y,0) k

+ ,.1T/2

XJ

ko(x,y,0)ekoOsin k0(x,y,0)&i}dO.

(21) where

W= (x)cos0 + (y) sin0.

The wave height due to the potential cb(x,y,z) is

deter-mined from (15) when taking the lowest order terms of derivatives with respect to x and y. Here, it should be noted that a/ax,

a/ay=o(1)

when operating onk0(x,y,O):

17/2 x

f

(x,y,0)cosO+0ry(X,y,0)sinO x

pvJ'

k(

0)k

ffddnD(,17)

(/2

cos1k0(x,y,0)}dO

f0 (x,y,b)cos0 +Ory (x,y,0)sin0 In the far downstream where cb,., -+

height is expressed as

(x,y) dO [S(0) sin -- sec20 (xcos0 +ysinO) 7/2

+ C(0)cos _sec20(xcosO +ysinO)} 1, (23)

where

C(0) +iS(0)

= ---sec0

irU3

i__sec2O(cosO+tsriO)

j2

Then the wave making resistance R is expressed as

RwlrpU2J

IC(0)+zS(0) cos30dO, (25)

0

where p is the density of water.

In principle, we can calculate wave making resistance of

any body shape, since D(xy) is determined from the rigid-wall solutions which are obtainable numerically or

analytically.

5. Waveresistance of simple forms at lowspeed

5.1 Wave resistance of a vertical circular cylinder and a semisubmerged sphere

Calculation of wave pattern around a ship by the use of

the present theory makes it possible to compare directly with the observation. However, this calculation needs lengthy numerical work even for a geometrically simple form. For the preliminary stage to examine the validity of the present theory, it may be enough to calculate wave

making resistance of simple forms.

For the convenience of analytical work, a vertical

infinite circular cylinder and a semisubmerged sphere are

considered. The double body velocity potentials are written as follows.

0(x,y,z)=0(x,y).=U(x+

x2+y2/22X

for a vertical circular cylinder of radius a, (26)

a3x

3/)

2(x2+y2+z2)

for a semisubmerged sphere. (27)

From (11) we have the following non-dimensional expres

sion forD(x,y):

dO MTB 109 August 1976 (22) U, cb 0, the wave (24)

(6)

MTB 109 August 1976

L_i_P(s)cosl3+J_Q(s)cos313 s 1 (28)

where the origin is fixed at the center of the cross plane of the cylinder or the sphere and the still water plane, and.

F = U//, x/a= scos3,y/a

=ssinl3, s= /2+y2/a,

P(s)=2s4s6,Q(s)=s2

for a vertical circular cylinder,

Q(s)=--s

fora semisubmerged sphere. Substituting D(s,13) into (24), we have

C(0)+iS(0)=i4asec3U

{cosOfP(s)Ji()ds

_cos3OfQ(s)f3(S20)ds

where the cross area of a body and the still water plane iS

excluded from the. integral range, and J1, J3 are Bessel

functions.

The asymptotic expression of the amplitude function in the low speed is derived based on the stationary phase

method:

C(0) + iS(0) = i-1--F[P(1)cosO +Q(1)cos3O]

x cos(J_sec2O___) + O(F5),

2F2 4

where P(1)1, Q(1)=-1 for a vertical circular cylinder, and P(1)=75/128 Q(1)=-27/128 for a semisubmerged sphere.

The wave making resistance coefficient C in low speed limit is expressed as follows.

C

Rw/PU2(2a)2FO(Fn)

for a vertical circular cylinder,

for a semisubmerged sphere.

C values are shown in Fig. 12 comparing With experimen-tàl values of a simple ship model M.2201 used for the flow

measurement in Sec. 3 and of a conventional full form of Cb=O.84. It is found that the order of magnitude. of

theo-retical values agrees with that of experimental values which are determined by Hughes method (CwCtotai

- (1+k)c1

Hughes' k is the form factor). It is also shown that the trend of theoretical C curves with respect to

Froude number resembles that of experimental curves. This result, encourages us to apply the present theory to slow ships. -4 4u10 - - - Experiment - Theory ypu'if

L Lead wateillae far M.2201 and Conventional form

L 2a for Cylinder and Seft,

,-sjbmerdhme

-_0

-0.10 0.11 0.12

Simpl, form M.220i Brarde theory deep draft> //ciroular

Sünpform Shallow draft Present theory

(5erd

a- Corwaimonal _-.- full form CBO.84 0.13 ..0.140.15 0.16 u4gr

Fig. 12 Comparison of wave resistance

5.2 Comparison of the present theory with conventional theories

For the comparison with other theories we calculate wave making resistance of a vertical circular cylinder by two other methods The first method is to calculate wave making resistance due to the surface source distribution

on the cylinder which

is determined by the.. so-called Zero-Froude-number approximation by Kotik and Morgan (1969). This' is the conventional method Of calculating

wave making resistance. The second method is the Brard's

one which includes the contribution of the line singularity around the intersectiOn of the, body and the still water plane In addition to the above surface source distribution

(Brard 1972).

In the conventional method the amplitude function due to the surface source distribution for a vertical, circular cylinder (b/a= e - 1 'in (39) of Brards paper'together'with c=L1) is written as

B (0) + iA(0) = -

i4aJ1 (sec20).

(32)

The asymptotic form of Cat low speed limit is'expressed

as

zi

It should be noted that the first term cancels out the amplitude function due 'to the surface source distribution. The asymptotic fOrm of the sum of both amplitude

func-tions (32) and' (34) in low speed is written as

flmory

(Circular Cfilnder)

-

8Fsin(-1-+--)+ O(F2). (33)

The order of magnitude of C is different from (30) by F Therefore it is evident that this formula gives practi

cally unacceptable high values in low speed.

In the second method, the amplitude fUnction due to ('31) the line singularity is derived as follows for a vertical

circUlarcylinder [ c - 1 in (52) Of Brãrd's paper

B3(0) + i&4(0) i4aJ1 (_ijsec20)

+ i8a.F,seo9cOs30J0(__t2sec2O)

(7)

B(0) + iA.(0)

+6B3(0) + iöA (0)

cos30cos (_!_sec20__IL)

+ O(F,).

(35)

2F, 4

This is quite similar to the amplitude function (29) derived by the present theory. The order of magnitude of both

expressions agrees with each other i e O(F) The asymp totic form of C at low speed limit is expressed as

C =

+ 32V7F,sin(+---) +O(F,),

(36)

which is identical to the wave resistance of a vertical cir-cular cylinder derived independently by Guevel et al.

(1974) in the frame work of the linearized free surface condition. C values expressed by (36). are also shown in

Fig. 12. C curve by Brard's theory is rather oscillatory. compared with that by the present theory where the oscillatory term is disappeared as a higher order quantity for the case of a vertical circular cylinder. The reason for the strong oscillatory property is due to the linearization

of the free surface conditions in Brard's theory as explained in the followings.

D(x, y)

of a vertical circular cylinder is rewritten from

(11):

iU

2

D(x,y)=

(U+rx)pxx(px+(Pry)I

+ c0ry cPrxy -

(i'

+ } , (37) where p is the perturbation velocity potential of the double body approximation:

p(X,y)=

Ø,(x,y) - Ux=

Uax

for a vertical circular cylinder.

Neglecting formally any products of derivatives of the

perturbation potential, we have

U2 .

2U3a2 x3+3xy2

g (x2+y2)3

The, non-dimensibnál expression of D(x,y) in the polar

coordinate system is written as D(s,13)

. S3cos33 (38)

4UF,

This expressiOn coincides with the second term of the right

hand side of (28). Therefore the asymptotic form of the

amplitude function in the low speed limit is expressed from (29) as

C(0)+S(0)

(39)

which IS exactly the same as Brard's amplitude function (35) which includes the contribution from the line singular ity. This theoretical result indicates that the reason for the

MTB1O9 August 1976

strong oscillatory property in Brard's theory is due to the

linearization of the free surface conditions. Recently Nakatake and Yamazaki (1976) proved that the expression

of the velocity potential, which ihcludès the line integral

term in addition to the Havelock wave source term, can be

derived from the velocity potential (21) of the present theory after the formal linearization.

In the lOw speed, however, the linearized free surface

condition becomes less and less accurate as Brard suspected.

Then we may say that the present higher order theory is one of the ways to overcome the lack of accuracy arising in the low speed problem with the linearized free surface

conditions.

In Appendix B wave resistance formulas of a

sèmisub-merged three-axial ellipsoid and a vertical infinite strut of

arbitrary across section shape in low speed limit have been derived.

6. Examination of the present theory by experiment For a direct comparison of the wave resistance obtained

by the present slow speed theory with experiment, a mathematical hull form defined by the following formula

is considered.

y = 2Bf(1---)11

(L)21

L/B= 10.0,d/L

= 0.0625. (40)

This form has been known as Wigley's parabolic fOrm. For this form it is expected that the viscous effect on the wave generation is very small compared with that of blUnt forms such as a vertical circUlar cylinder and a

semisub-merged sphere.

By the use of the expressions (B-6), (B7) and (B-8) in Appendix B, the amplitude function of the Wigloy's

para-bolic form and the wave resistance is calculated

numerical-ly. In this computation the velocity components due to the double body potential around the load water line are obtained first by Hess and Sniith method with 26 x 10

source panels lengthwise and dèpthwise respectively. Then the funätion F(L3) expressed by (B-8) is calculated, where the numerical differentiations are Used fOr a17/a13,

and The integration with respect to j3 in (B6) is carried out carefully by searching the zero points of the

integrand. The wave resistance values are then obtained by

the use of the expression (25), where the upper limit .of the integration with respect to 0 is about 750 The validity of the procedure of the numerical computation has been confirmed by applying the computer program to the case

of a sernisubmerged sphere fOr which the asymptotic value of the wave resistance is known analytically by the. expres-sion (31).

Fig. 13 shows the computed results comparing with :Michell's theory and the experimental results of Wigley's form of 5 meters long. In the range of Froude number less than 0.25 a remarkable attenuation of wave resistance

(8)

MTB 109 August 1976 .4 *10 5 Vt{A(e)}cos30/L2

FnU/(jt0.2O

N

. Wove pOttern analysis

Michell \ / (longitudinal Cut)

\

'I

iuJA A'ikA

20° 30° 40° 50°

e

Fig. 14 Comparison of wave spectra Of Wigley's parabolic model

curve with respect to Froude number is attained by the-present author's theory. A quantitative agreement is also

observed between the present theory and the experimental values determined by means of wave pattern analysis and by the conventiOnal Hughes method.

A comparison of wave spectra at Frouie number 0.20

is shown in Fig. 14. When comparing with Michell's

linear-ized theory, it is a characteristic of the préséht theory that

the wave spectrum corresponding to the transverse waves (small 0-values) is considerably reduced and approximately

agrees with the value détérmined -by the wave pattern

analysis.

This numerical study ii,dicates the validity of the

asymp-totic solution (21) of the lowspeed free surface condition (13) which takes into acôoUnt the nonlinearity of the flow

around a shiP

7. Applications to the hull form design

7.1 Characteristic of wave resistance coefficient in low

speed limit

In Sec. 5.1 it was found that the wave resistance

coeffi-cient of full forms with deep draft has the following form

in the low speed limit:

C = inF,,

fU2L2

Wigley's parabolic model Y 2B(X,&Xl_X4..)(u_(Z/d)5) %lO. d/LO.0625 Micheli

/

/L\

/1

/\__/

70°

Fig. 13 Comparison of wave resistance coefficients ofWigley's parabolic model

0

Wave resistance

-determined from towing test

(Hugher method)

-Wave Pattern Onólysie

(ion.itudinol cut) 0.25 0.30

F, U/

where m is determined by the body geometry alone. The exponent of the Froude number is different from that derived from Michell s wave resistance theory in the low speed limit, where C is proportional to F.

Hughes (1966) and Prohaska (1966) used the expression = nzF, in determining the form effect on the viscous

resistance from the towing test data in low Speed range. For

instance, Prohaska proposed the following formula for the determination of the form factor k,

C/C', (1+k)+mF,/C'f,

-where C is the total resistance coefficient, and is the

frictional resistance coefficient. At Hydro-Og Aerodynam-isk Laboratorium results of about 200 model experiments - were plotted based on this formula. For Froude numbers between 0.1 and 02 the Cf/C'1values for a great majority of the models did plot on straight lines against Ft/Cf

values. However, for full -forms, say withCb > 0.75, it was

found that the points may plot on concave curves. ihis indicates that either (1+k) or m or both are speed

depen-dent.

-- lnui (1968) studied the exponent of the Froude number for full forms in low speed assuming a form

C/C1= (1+k) + CW/cf. (42)

where c'=mF'1.

From analyses of a number of model experiments of full forms. lnui found that for deep draft condition n6 gives better fitting to C than n=4. In the shallow draft condi-tion, on the other hand, n=4 or n=5 gives better fittings. Fig 15 shows some examples of lnui's plottings Of the experimental data of a full form with Cb=O.8O based on

the fOrmula (42).

Fig. 16 shows another example of fittings. C-values were determined by Taniguchi (1966) from the towing

test data of geometrically similar models of 4.2, 7 and 10 m

long with CbO.BO. lnui obtained the following fittings for C-curves:

C (Full load)

Rw/i_pU2V

= 83i9F,,

(43)

Rv

(41) C(Ballast load) 28O7F,

xiÔ5 O4 0.3 0.2 0.1 0.15 0.10 0.20

(9)

I.e 1.6 1.4 Ct,cq Full 0

.

.

.

0 0 p 9, 0.0,0.12 0.l4 Ct/Cf (IK)m.F,!/Cf 0

.

0 S 016 0.18 0.20 F S S

.

S flc 6 S Fn 0.19 0.10 0.14 0.15 0.16 0 0.5 1.0

F/CI

l.5z102

Fig. 15 lnui's plotting based on Prohaska's proposal

Fig. 16 lnui's fitting of C-curves of a tanker model

-8.88Fcos[S(1+)--+--],

(44)

where c7 is the displacement volume, C Le/L, Le is the

length of entrance,

c=p/C, and

F

is the forward shift of bow waves as shown in Fig. 16. For this ship form C =0.26, 0.173. lnui's expression (43) forCs,,, in full

load condition well corresponds to the expression (41)

obtained by the present theory.

The water line form at the ballast condition of this ship form has a point where the curvature equals to zero.

For such a form the present theory suggests that C, in low speed limit is proportional to as explained in

Appendix E (E-5). Therefore lnui's fitting (44)

approxi-mately corresponds to the theoretical prediction.

Thus, the present wave-resistance theory can be used for a reasonable determination of the level of viscous

resistance curve in low speed limit.

7.2 Decomposition of wave resistance into bow wave

resistance and stetn wave resistance

For the sake of convenience, an infinite vertical elliptic cylinder is considered. C is expressed in a form (see

Appendix B)

I' -D

TT2,2_ r'6

I.

L -

mLrfl,

where the oscillatory term of O(F,) disappears, and

67T/

3

16(1+e)

r

2sin4Ocos8O(1+e2tan2O) 'adO, Jo

e =B/L.

Since mJ, is determined by a geometrical parameter e alone, the calculation of C is easy when mL is known for a givene. Table 2 showsmL values for variousL/B.ratios.

From the table it is found that for a fixed Froude number, the wave resistance coefficient C, is not always getting smaller with increase of L/B ratio. The reason for this tendency is explained as follows. Beyond a certain L/B ratio, say L/B'5, the change of velocity around the elliptic cylinder becomes larger with increase of L/B ratiO when

the ship speed U and the ship length L are fixed, since the velocity which was null at the stagnation pOint has to reach near the ship speed within a very short distance. ThiS means

that the Froude number defined by is very large

so that the disturbance on the free surface around the

bow becomes large and then contributes to an increase of wave resistance.

When introducing ship breadth B as the characteristic length instead of ship length L, the wave resistance

coef-ficient is rewritten as

C

,Rw/PU2B2 = mBFflB,

where mB = emL, and FnB=

U//.

mB values are also listed in Table 2. Fig. 17 showsmB

and mL values versus L/B ratios together with those of a simple full form having semi-circular bow and stern with parallel middle bodies of different lengths. It is observed thatmB values tend to reach a constant value with increase

of L/B ratio while mL values increase. This calculation

indicates that the ship breadth B is an appropriate

charac-Table 2 mLandmB mL

2

MTB 109 August 1976 1.00 0.60 0.40 0.20 0.10 0.05 L/B 1.00 1.667 2.50 5.00 10.00 20.00 mj, 26.01 11.43 9.543 13.38 30.76 92.34

mB=mL

26.01 6.858 3.817 2.676 3.076 3.117 Bo.L Lost S CCf 1.8 1.6 .4

(10)

MTB1O9 August 1976

/

/

30

Simi circuLar bow

m6 a.nd ;tsrn

6 7 8 9 10 Ii 12 13

Fig 17 Comparison of rn-values

teristic length rather than the ship length L for a compara-tive study of wave resistance of full forms which usually

have long parallel bodies.

Figs. 18 and 19 show C values versus F,,. The figures show that C, values tend to reach constant values for a fixed FflB with increase of L/B ratio This implies that the interference effect of potential flow between bow and

stern form in generating waves is negligibly small. In other

words, wave resistance may be separable into two parts, e. one due to the entrance part and the other due to the

run part, whenL/Bratio is greater than, say, 6.

In order to verify this theoretical prediction, towing

test data of several models having the same bow form and

stern form with different length of parallel body are exam med as shown in Fig. 20. There. is no appreciable effects of the difference of L/B ratios on wave resistance C'

except in the higher speed range (F, > O.5) where inter-ference effects between bow and stern may exist. Thus those experimental results support the prediction of the

theory in the low speed range.

Further, Fig. 21 shows other experimental results of models having different stern forms with same bow form and parallel middle body. The effect Of difference ofstern forms is very small. Usually the afterbody of full forms is designed to have a relatively fine form compared with bow forms in order to prevent flow separation which has

unfavorable effects on propulsive performance. In addition

to this

the viscOus boundary layer and wake behind a

ship may reduce the waves generated by the stern. There

fore wave generation from the run is relatively small com-pared with that from the bow Then we may say that the wave résistánôè characteristics of full forms are mainly

depending on the bow geometry.

In Appendix F a design method of full forms based on

these experimental and theoretical evidences is described.

In summarizing the present section we may conclude

0

a

0

Elliptic cyLinder

L, 1.0

Fig. 18 C-valuesfor elliptic cylinders of variOus

L/B ratios

Fig. 19 C-vaIues for full forms of semi circular bow

and stern

Fig. 20 Comprison of among the 3 ship forms having the seme entrance & run with different parallel body

that the bow form can be treated separately from other parts of the body when L/B> 6 Therefore, we may con-centrate our effort upon the improvement Of bow forms

in order to reduce wave resistance of full forms of

conven-tional type.

For instance, Fig. 17 is considered as a comparison of mB values of two different water line forms, i.e. one is a semi-circular water line form and the other is an elliptic

020 015 - -010_____________________________________

EI

C52

/

infinite cylinder with ómi circular bow

and Stern/ Fe': Fn!U/J -;---.-FnB=WJ

,

,

,

mL , WptiC __' cytlfld.y 0.50 ass 040 04B oso 0.25

(11)

Fig. 21 Comparisons of C among the models, having the same entrance & different run

water line form. The latter gives smaller mB values. This means the elliptic water line form is better than the semi-cirëular water line form in reducing wave resistance. In

practice we have to determine the optimum water line forms under the given design constraints. When we apply the present theory to general full forms of finite draft, we may obtain information about the effect of not only of water line forms but also frame line forms on wave

resistance.

7.3 Effect of protruding bow in reducing wave

resist-ance

7.3.1 Problems arising in the case of shallow draft

Discussions on the formula of wave resistance coeff i-cient C have been concerned with cylindrical full forms of deep draft. In this section we consider full forms of

shallow draft.

Calculations of C-values for various Bid ratiOs are car-ried out fOr semi-submerged ellipsoids of

L/B1.Q. As

shown in Fig. 22 ellipsoids of large Bid ratios give high

Cs,-values with strong oscillations with respect to Froude

number.

To find the reason for the increase of C-values for the

case of a large B/d ratio, calculations of the function D(x, y), the right hand side of Eq. (13), which is the dis-turbance induced on the free surface by the double body velocity potential, are carried out. Fig. 23 shows com-parisons of D(x, y)-values on the x-axis for various Bid

ratios.

It is observed that peak values of D(x, y) increase and come c!ose to the body with the increase of B/d ratio. In Fig. 24 calculated wave heights represented by 0)

for various Bid ratios are shown When comparing Figs 23 and 24, it is found that steeper waves give higher peak values of D(x, 0). Those calculations indicate that the reason for the enormous increase of C values in the shal-low draft condition is due to steepening of waves. Further, it is understood that large values of D(x, 0) at x,'a =1.0 contribute to enlarging the oscillatory term of C, , the

third term of formula (B-14), since there are relations:

F(0)=D(a, 0)/UF,, and F(ir)

D(a, O)/UF,.

When waves induced by- the double rnodeI'-potentiaI

become steeper

it

is anticipated that the assumption of the present theory, i.e. the condition of slowly varying basic flows, is violated and the free surface becomes less

L1

B/d=l0C UFU B/a'4.0 'O.O(d'oo)

-

-ao U

Fig. 23 D(x,0)-values for semi-submerged ellipsoids of variousBidratios

- U

12 4 0 X/a. 0 B/d'00 (doo) -1.2 Lon9itudülaL center pLOne

(L/B'l.0)

-20

'jdoo)

Longftudinat center pLane

(LIe' 1.0) MTB1O9 August 1976 '0.0 'd'2.0 B/d'40 BId' 10.0 ,d'0.0 0 20

U__rrnrn

Ce#0.82

C_________

-Fig. 22 C-values for semi-submerged ellipsoids of various Fig. 24 rx,0)-vaIues for semi-submerged ellipsoids of

Bidratios variouSBidratiOs

0.25 &50

0.35- 040

ass 050 FnB'Wfj 0.13 0.07 0.00 0.09 0.10 0.11 012 0.14 0.15 0l 0I 050 0

(12)

MTB 109 August 1976

and less stable, so that even a small perturbation induced by the additional potential (x, y, z) results in breaking

waves.

From (22) the wave number of superposed waves on (x,y) is known as

kO(x,y,O)g/4,(x,y,O)cosO +Ø,.(x,y,0)sinO}2.

(45)

This expression indicates that wave length becomes very short near the bow, since the flow becomes stagnant. Furthermore steep local waves enlarge the amplitude of

the superposed short waves, since the amplitude depends

on the intensity of D(x,y).

According to the theoretical studies by Longuet-Higgins and Stewart (1960, 1969), short waves riding on long waves tend to be both shorter and steeper at the crest of the long

waves than they are in the long-wave troughs; If the lông

waves become steeper, the steepening and shortening of the superposed short waves are drastic. This is considered as one of the triggers of appearance of breaking waves (white-caps) on the crest of waves in deep water.

Therefore we may consider that in the vicinity of blunt bows there exist similar free surface phenomena to those

of ocean waves.

-When wave breaking occurs, the present theory is not applicable any more. However, we may consider that the function D(x, y) is used as a measure of wave breaking inception. Although a correlation between D(x,y) and the

incipient breaking of bow waves has not been derived yet, it, is considered that wave breaking phenomena will be suppressed by reducingD(x,y)-valües in front of the bow.

- 7.3.2 Reduction ofwave resistance byprotruding bow From both model experiments and service peformances

of full forms, the effectiveness of the protruding bows in

reducing wave resistance has been confirmed.

To explain this effect a protruding bow is considered separately from the main body and is replaced by a

sub-merged sphere for the sake of simplicity. Then calculations

of D(x, y)-values of the submerged sphere for various

im-mersions are carried out.

Neglecting the effect of induced velocities on the sphere

by its image system above the free surface, the velocity potential for the submerged sphere in the uniform flow is

written approximately as

Ua3 x

ør(x,y,z) = Ux +

x22+(z_f)2

+ x 1, (46)

x2+y2+(z+f)2

where f is the immersion of the sphere.

After substitution of velocity components derived from 0,. into (7) and (11), an expression for D(x, y) on x-axis

at the free surface is obtained as

D(x,O)_6(a3-4af2)

(1

j2)

UF, (a2+72)%

i2(3af2-3o3)i

where F=U//, and a=x/a,f =f/a, and

2

1+ 1

(1 3a

(a2+f

234

2+j2

Fig. 25 shows D (x, 0) values for different immersions.

It is shown that a negative peak value appears in the front part of the sphere and it becomes larger with a decrease

of immersion.

This simple calculation suggests that a submerged sphere

or a protruding bow works in cancelling D(x, y)-values

induced by a main body in front of the bow, i.e. the

protruding bow is effective in reducing steepness of local bow waves. Further it is suggested that the shallower

im-mersion gives a greater effect of cancellation. Then we may

consider that in the ballast load condition the protruding bow contributes to the reduction of wave breaking resist-ance which is attributed to the steepening of bow waves. In the full load condition, on the other hand, it is

con-sidered that the effect of the protruding bow is relatively small compared with the effect in the ballast load con-dition. This theoretical prediction coincides with our ex-periences about the protruding bulb which are mounted

near the bottom of bow.

For the improvement of resistance characteristics in the full load condition Couch and Moss (1966) developed bow forms with protruding bulbs which are raised up to

the middle part of the stem (R-series). Experiments showed

a marked improvement in the full load condition. This is a good experimental evidence to verify the above prediction

about the bulbs of shallow immersion.

Fig. 25 D(x,0)-values for a submerged Sphere (41)

(13)

8. Concluding remarks

From the studies in the previous sections we may expect that the present asymptotic solution of the non-linear free surface problem is applicable to the prediction of wave resistance characteristics of conventional ship forms which have not always been tractable by Michell's

linearized wave-resistance theory.

For instance, the asymptotic expression CmF, may

be used in determining a reasonable level of viscous

resist-ance curves of full forms in low speed limit, and the wave

The author wishes to express his deep appreciation to Mr. K. Tamura, manager of Resistance and Propulsion

Laboratory, Nagasaki Technical Institute and Mr. K. takekuma, senior research engineer of the laboratory, for

their stimulating and encouraging discussions. Thanks are

also due to Mrs. M. Hara and Dr. Y. Kayo for their efforts

Baba, E. and Takeküma, K. (1975a), A Study on Free-Surface Flow around Bow of Slowly Moving Full Forms, J. Soc. Nay. Arch. of Japan, Vol.137, 1-10.

Baba, E. (1975b), Blunt Bow Forms and Wave Breaking, The First STAR Symposium on Ship Technology and Research, The

Society of Naval Architects and Marine Engineers.

Brard, A. (1972), The Representation of a Given Ship Form by Singularity Distributions When the Boundary Condition on the Free Surface is Linearized, Journal of Ship Research, Vol. 16, No.1, 79-92.

Couch, R. B and Moss, J. L. (1966), Application of Large Protrud-ing Bulbs to Ships of High Block Coefficient, Trans. SNAME, Vol. 74,392-441.

Dagan, G. (1975), Waves and wave resistance of thin bodies moving

at low speed: the free-surface nonlinear effect, J. Fluid Mech., Vol.69, part 2,405-416.

Guevel, P., Vaussy, P. and Kobus, J. M. (1974), The Distribution of Singularities Kinematically Equivalent to a Moving Hull in the Presence of a Free Surface, International Shipbuilding Prog.,

Vol.21, 31 1-324.

Hermans, A. J. (1974), A matching principle in nonlinearship wave theory at low Froude-number, Delft Progress Report, Series F., Vol.1.

Hughes, G. (1966), An Analysis of Ship Model Resistance into Viscous and Wave Components, Parts I and II, Trans. of RINA,

289-302.

lnui, T. (1968), Separation of Ship Resistance Components, Symp. on Ship Resistance and Propulsion, Soc. Nay. Architects of

Japan, 39-53.

Keller, J. B. (1974), Wave Patterns of Non-Thin or Full-Bodied

Ships, 10th Symp. on Naval Hydrodynamics.

Kotik, J. and Morgan, R., (1969), The Uniqueness Problem for Wave

Resistance Calculated from Singularity Distributions Which are Exact at Zero Froude Number, Journal of Ship Research, Vol.13, No.1, 61-68.

Acknowledgments

References

MTB 109 August 1976

resistance formula developed in the text is used for the development of full bow forms of least wave resistance by

taking into account the effect of protruding bows.

However, when the waves are broken, the present theory is not applicable to a study of the free surface flows around ships. Then a special analytical treatment to analyze

turbu-lent vortical flows around the bow has to be developed.

Before an accomplishment of such analytical means,

detailed investigations into the. free surface flow around

the slow ships are required in the future works.

in developing the computer program for the calculation of wave resistance of arbitrary body shape. The author also wishes to express his appreciation to all members of. Nagasaki Experimental Tank for their cooperation in

car-rying out this investigation.

Longuet-Higgins, M. S.. and Stewart, R. W. (1960), Changes in the form of short gravity waves on long waves and tidal currents, J. Fluid Mech.. Vol.8, 565-583.

Longuet-Higgins, M. S. (1969), A nonlinear mechanism for the

generation of sea waves, Proc. Roy. Soc. A.311, 371 -389.

Milne-Thomson, L. M. (1960), Theoretical Hydrodynamics, 4th Edition, 506-511.

Moriya, T. (1941), A Theory of an Arbitrary Wing Section, J. of Soc. Aeronautical Science of Nippon, Vol.8, No.78,1054-1060. Nakatake, K. and Yamazaki, R. (1976), DiscussiOn on the line integral of wave resistance theory, The Proceedings of

Interna-tional Seminar on Wave Resistance, Tokyo, 445-446.

Newman, J. N. (1976), Linearized Wave Resistance Theory, The Proceedings of International Seminar on Wave Resistance, Tokyo, 31-43.

Ogilvie, T. F. (1968), Wave Resistance: The Low Speed Limit, Univ. of Michigan, Naval Architecture and Marine Engineering, No.002.

Prohaska, C. W. (1966), A Simple Method for the Evaluation of the

FOrm Factor and the Low Speed Wàvë Resistance, Proc. of 11th ITTC, 65-66.

Taniguchi, K. (1966a), Study on Scale Effect of Propulsive

Per-formance by Use of Geosims of a Tanker, J. Soc. Nay. Architects of Japan, Vol.120, 19-3.5.

Taniguchi, K., Watanabe, K. and Tamura, K. (1966b), On a New Method of Designing Hull Form of Large Full Ship, based on the Separability Principle of Ship Form, J. Soc. Nay. Architects of Japan, Vol.120,36-45.

Taniguchi, K., Tamura, K. and Babe, E. (1971, 1972), Reduction of Wave-Breaking Resistance by "MHI.Bow", Mitsubishi Juko Giho (Japanese), Vol.8, No. 1, or Mitsubishi Technical Review (English), Vol.9, No.1.

Timman, R. (1974), Small Parameter Expansives in Ship Hydro-dynamics, 10th Symp. on Naval Hydrodynamics.

(14)

MTB 109 August 1976

The dynamic and kinematic free surface conditions are

written respectively:

_!_1j2=

gH(x,y)

+ on

z =H(x,y),

(A-i) o

42(x,y,z) onzH(x,y),

(A-2)

where zH(x,y) represents the free surface.

In our problem we assume that the total potential

'(x,y,z)

and the wave height

H(x,y)

are expressed as follows:

c1(x,y,z)= ø(x,y,z)+ø(x,y,z),

(A-3)

H(x,y)

=

r(X,Y} + ix,y),

(A4)

where

x,y) =__[U2_Ø, (x,y,

0)

Ø,,(x,y, 0)].

By the substitution of (A-3) and (A-4) into (A-i) and (A-2),

the free surface conditions are written as:

U2

(x, y,

0)

Ø(x,

y, 0)1 +gix,y)

24rx(X,y,Z)øx(X,Y,Z)

+q(x,y, z)+ ci(x,y, z)-i- z}cb(x,y.z)

+

ø,z(X,Y,Z)+(x,Y,Z) +

2ct2(x,y,z)Ø(x,y, z)],

(A-5)

0

[rx(X,YHx(X,Y)]

[Ø,(x,y,z)+Ø(x,y,z)]

+ (X,y) + y(X,y)] [øry (X,y, z) + ø(X, y,Z)]

,z(X,Y,Z) ø(X,Y,Z). (A-6)

Based on the Ogilvie's assumptions on the order of

magni-tude (a) through (f) in the text, the following Tayler expan-sions at z=O are derived for ØPX(X,Y,Z), øry(X,Y,Z) and

rz(X,Y, z):

rx(X,Y,Z)=ørx(X,Y,0)+(r+irxz(X,Y,0)

+_(r+)2rxzz(x,y, 0)

= 'P,x,Y,

0) 0)+

[U]

[U5]

P(x,y,z}=

øpy(x,y,

0) +t(x,y)t1ryzz(X,y,

0)4.

[U] [U5]

z)= r(x,y)ctrzz(x,y, 0)

[U3]

+ (x,y)cb,,(x,y,

0) +...

[U5]

where the following relations are used

ørz(X,Y, 0) = 0, 0)= 0, ryz(X,Y,0)= 0.

On the other hand, the Taylor expansions at

z=,(x,y)

are derived for cb (x,y, z), Ø,, (x,y, z) and ØZ(X,Y, z):

øx(X,Y,2)x(X,Y,r)(X,Y)øxz(X,Y,r)" [U3] [U5]

Appendix A. Derivation of the free surface condition (10)

øy(X,Y,Z)y(X,y,r)+(X,y)cbyz(X,y.,)+."

[U3] [U5]

Ø2(x,y, z)= z(X,Y,

+ (x,y)gzz(x,y, r)

[U3] [U5]

By substituting those expansions into (A-5) and (A-6),

and taking the lowest order terms, we have

0=

gix,y)

+ ØJ.X(X,Y, 0)Ø(x,y, ,)

[U4] [U4]

+ ø,y(x,Y, 0)ø(x,y, ,), (A7)

[U4]

0=

ø(x,Y, ,) +

(x,y)t1(x,y,

0)

[U3] [U3]

+y(X,Y)øry(X,y, 0)

+(x,y)ø,(X,y,

0)

EU3]

+ry(X,Y)çbry(X,Y,0) r(X,y)ørzz(X,Y, 0).(A-8)

[U3] [U3]

By the use of the relation

ørzz = ørxx The equation (A-B) can be rewritten:

ø(-,Y, ,) -

x(X,y)ørx(X,y, 0)

(x,y)çb,(x,y,

0)

[r(X,Y)rx(X,Y, 0)] +---[,(x,y)cb(x,y, 0)] .(A-9)

In order to eliminate (x, y) from the equations (A-7) and

(A-9), we take first the derivatives of (A-7):

0= g(x,y) +crx(x,y, O)4xx(X,Y,r)+øry(X,Y, 0)çbyx(X,y,r) [U2) [U2] [U2]

+ ø,xx(X,y,

0)çt(x,y,

+ 0)cb(x,y, ,.)

[U4]

+Ø,(X,Y, O)(x,y,

r)+cbry(X,Y,0)øyz(X,Y.r)f

[U4] [U4]

[U2] [U2]

[U2]

+ rxy(X,Y, 0)øx(X,y,r) +Øryy(X,Y, 0)(x,y,

[U4] [U4]

+ ørx(X,Y,0) z(x,y, ,)

--+ çb,(c,y,

0) yz(x,y,

r)

[U4] ' [Un]

By neglecting the terms of

0(U4)

and substituting x(X,Y) and y(X,Y) into (A-9), we finally have

_!_[,(x,y,O) ---+Ø(x,y,0)

] 4(x,y,)

+ctz(x,y,r)

D(x,y),

(A-b)

where

D(x,y)

=

---[Ø(x,y, o)(x,y)]

ax

+_[ct(x,y, 0),(x,y)].

(15)

The amplitude function of the wave represented by 1X, y) in the far downstream is expressed by (24) in the text:

C(0)+iS(0) sec3OffdxdyD(x,y)

x exp[i_2jsec20(xcos0+ysin0)] (B-i)

By the partial integration with respect to x, an asymptotic expression of the amplitude function in the low speed limit is expressed as a line integral around the intersection of

the body and the still water surface:

C(0)+iS(0)=__sec30,fdy

[

(-1)

e0lx0

irU

n0

pfll

x sec20

,fdyD(xo,yo)eY0+Px0+O(U6)

[U4] (B-2) where D(n) ax p=i-2---sec0 q=i__.sec20sin9 U2 U2

and (x0, y0) is a point on the intersection of the body and

the still water surface.

When introducing the stream line coordinate system along the intersection, the value of D(x0, Yo) is rewritten as: D(x0 ,Yo) = [ Ø,(x, y, 0) øry(X,Y, o)

j

r(X, Y) ax ay r(x,Y)ct)rzz(X,Y, O)] y=yo

= - sgn (yoh/Ø+ Ø,'- ,(x0, Yo)øzz(Xo, Yo 0),

(B3)

where 2 is taken along the intersection. Since

sgn(y)Øry(x,y, 0)

dy = - dQ

O)+(x,y,0)

the amplitude function (B-2) is written:

C(0)+iS(0) =

irU J

aQ

Ju+v

x exp[i-

sec20xo(Q)cos0+yo(Q)sin0 } 1. (B-4)

where u = px(Xo,Yo, O),v øry(Xo,Yo, 0), = ørz.XO,YO, 0).

Appendix B. Wave resistance formula of a semisubmerged

three-axial ellipsoid and a vertical infinite cylinder of

arbitrary crose section shape

x

a0 = ah

r

(sin2j3+ 2cos2f3)2

MTB 109 August 1976

-J0 (á2+X)s,/(a2+A)(b2+X)(c2+X)

and a, b, care radii, = b/a, 8 = c/a.

ii) Vertical infinite cylinder of arbitrary crosssection shape (see AppendiA D) F(13)

sinj3 {nBcosf3} i +2nB,

+ nBsinn13}G(f3)1PU3)}_3, (B-la) where G(j3)

--cosj3+

n2Bcosn13P(13) +

nBsinnflQ(I3),

P(f3) =+sih,213 +{nBcosnj3}2,

- EnB, cosnf3}{En2Bsinnf3.

When the equation of the intersection is expressed by Fourier expansion through the relations:

--=---cos13, -'--=BsinnI3,

(B5)

L 2

L'

where L is the length of a body, and BA are the Fourier

coefficients, the amplitilde function is rewritten as

C(0)1S(0)_

1 sec OF2f dj3F(13)

x

exp{i0 ç(i(13,0)}

(B-6)

where F =U/.../j Froude number, and

1i (13,0) cos13cos0 + (2EBsinni3)sin0, (B-7)

F(13) =

[--(1 U2-52)

(1 j22)

J__sin2i3+( nBcosn13)2

+ I ,(B-8)

.Jj2+.2 sgn(sin13)

F(j3) is determined by the double body potential and is

depending on the body geometry alone:

I) Semisubmerged three-axial ellipsoid (see Appendix C)

F(j3) - 2

2 2 2

[(2

)2

2a0 sin 13+e cos13

2a0

2 cos2j3 si n2j3

+jcos2fl li(

2ao' sfr2

05213

(B-9)

where

(16)

MTB 109 August 1.976

+

2-s/F,'F( 0) F(ir)

1+21EnB2+2

(-1Y'nB2

sin(--+j-)

The velocity potential for a translatory motion of an ellipsoid with velocity U in an infinite fluid is given as

Ux

a0

Appendix C. DeriVation of F(a) for a semisobmerged ellip-soid

integration with respect to a in the expression (B-6). In this case the lowest order term of C, is proportional to F7 instead of F as shown in Appendix E.

The following are the expressions of C for simple

forms in the low speed limit:

Semisubmerged three-axial ellipsoid

c,=4F6fJFL

d0+(

2

Jo /1+etanO

,'TT 2-a0

x (B-16) since = 0(n 2),a2 =a1 Hp"(a1,o)I= where

F(0) = 2 )3 2 cos4O sin2O (1 + e2tan2O)

2 a0 e

2

+---(

)cos 0 ii - (

) sin 0

5

2-an

2-a0

It should be noted that the second term of (B-16), the

interference term, decreases with an increase of draftS. Semisubmerged sphere

CWrF6

++\/F,;7sin(-j-+---)+O(F,),

(B-17)

which is obtained by substituting e = 1, 5 = 1 and a0 = 2/3

in (B-16). Th.is expression coincides with (31) which is

derived in a different manner as shown in the text.

Vertical elliptical cylinder

C=16F,

(1-I-c)6

xJ'sin40cos80(1 +e2tan2O) hdo

+O(F,),

(B-18)

which is obtained by substituting 5 = 00, a =2e/(1+e) in

(B-i 6).

Vertical circular cylinder

lily2

= i6F,26 j sin4o cos5Od0 + O(F8)

0

8192 L'6 1)IE'8 315

'

which is obtained by substituting = 1 in (B-18), and co-incides with (30) derived in a different manner as shown in

the text

where

dX

x abci

, (C-i)

(a2+ Xh/(a2+ A)(b2+ X)(c2+ X)

- .

(C-2) + X)(b2+X)(c2+ X)

0 (a2+XhJ( Semisubmerged sphere

Substituting e = 1, = 1 and a = 2/3 in F(j3) of the

semisubmerged ellipsoid, we have

F(j3) __cos2f3(_ sin2j3 + 1).

(B-il)

Vertical elliptic cylinder

Substituting = 00 and ao

2e/(1+e) in F() of the

semisubmerged ellipsoid, we have

F(13)- 2 2 2 3 2e3(1+e)3cos2l3sin2fl

(sin 13+e cosj3) (B-12)

which is also obtained from (B-b) by the substitution of

B1=--e,B=O(n2).

Vertical circular cylinder

F(j3) = i6cos2j3in23 (B-13)

The wave resistance of those forms are obtained by means of stationary phase method when carrying out the

integration with respect to a and 0 as shown in Appendix E.

A general form of wave reSistance coefficient C, is ex-pressed as cw = pu2L2

6 r

FW1) dO+ 2F6 {F(132)}2cos0dO

'J0

lii"(j3, 0)1

J0

hp"(3,O)

(B-14) where j3, a2 are the solutions of the equation

-sinj3cosO + (2nB cosnj3) sinO = 0,

for

0j3

: aft body,

a2-ir :

forebody,

and

"ia,o) =-cosj3cosO-(2n2Bsinnj3)sinO. (B-15)

The expression (B-14) can be applied to full forms having

convex water lines, such as semisubmerged ellipsoids. When the load water line has a concave part, there exists a point where 0) =0. At this point the third derivative of

(17)

a,b, care radii of the ellipsoid.

There are relations between A and (x, y, z):

a

y

az_

z

ax

2(a2+A)'

ax

2(b2+x)'.ax 2(c2+X)

Further, the following relations can be derived

(Milne-Thomson):

oX 1 ax

ax_

1 Oy

ax_

i

a ax h OX'

ayh ax' azh

OX' where

[1

4

(a2+x)(b2+X)(c2+x)

and (A, , v) are the ellipsoidal coordinates.

In order to calculate F(j3) defined by (B-B) for merged ellipsoid, the values [OØr/OX] z = 0, z= 0, and O2Ø/Oz2Jz= 0 are necessary. Those values on the ellipsoid (X0) are obtained:

00, aø,

a,.

ax

[1z=o

0x

ax

Ox Uabc

r

dX

2-a0j0

(a2+x)/(a2+x)(b2+x)(c2+x) Ux abc 2-a0 (a2+x)v'(a2+X)(b2+x)(c2+A) Ox X0

2U

2U

b2c2 x2

2-a0

2-a0

a2 pv

2U

xy a2c2 Oyz=o.

2-a0

a2

pv

_abcUx

2b 1

0z2 Z0

2-a0

ac iiv where iv i6 given: pv

a2b2+---(

2)x2

+(c2 -b2)y2.

a

Introducing a new variable 13 as

x=acosj3, y=bsinj3

together With the definitions =b/a, =c/a,we have

.2

2cosI3

2-a0

2-a0

sin2I3 + e2cos2f3

When the section shape is expressed by

x 1

y.=

---cosj3,

=-j-=

EB1sinnj3,

the values of and (1 -U2 j2) are obtained by the use of

(C-3)

Appendix 0. Derivation of F(13) for a vertical cylinder of

arbitrary section shape

(D-1)

2 eosj3sii13,

2

(C-li)

2.-a0 sin 13+e cos 13

=çb,(x,y, 0)

U

2-a0

cc!s13

sin2l3 + e2cos213

From those expressions we have

itz2-2 = 1

( 2 )2 sin j3 (C-13)

2-an

sin213+2cos213 2._ )2 2esinf3cosl3

2-a0

(sin2J3+e2cos23)2 sgn(sinj3)sinj3 . 2

Jsin2j3 + e2cos213

2-a0

By substituting those expressions into (B-8) and using

B1 --e, B=

0 (n 2), we haveF(j3) for a semisubmerged ellipsoid as

F

2 2 2 2cos2l3sin2I3

sin2fl +e2cos2j3

2-a0

(sin2f3+e2cos2j3)2

2

2

+Cs13

1 (_.

)2 sin

2-a0

sinj3+eco

By the use of the relations:

.3r= dx

2

l.=i.aaj

(a2+x)5/2 3

F(j3)for a semisubmerged sphere is obtained as F(f3)= (i.)35213sifl2j3+3COS213

75

27.

= 4cos13[-jjcos3--j--cos3I3].

By the use of the relations: = 00,

a0 =lirn

abf

(b2+

x)'

(1+)'2

2

2r(--)r(i)

2

1_e2

(1_2)I(f)

1+e

We haveF(f3)for an infinite elliptic cylinder as

F

(13 -)_.3(1+e)c052135in1213 (sIn2+e2cos213)3 a 0.13 x

dx

I5 =-sinl3(-}-+

nB)(nBcosn13)

MTB109 August 1976 (C-12) (C-i 4)

(Ci5)

1]

(C-16)

conformal mapping method developed by MOriya (1941)

for 2-dimensional wing sections:

1

(X-i4(A-v)

a semisub-[OOr/Oy]

(C-b)

(C-17) (C-i8)

(18)

MTB1O9 August 1976

P Po

pU2

The amplitude function expressed by (B-6) can be eval-uated approximately by the use of stationary phase method at small Froude number.

C(0) S(0) sec2O L L

-

2i

x expi

S i]i(13,8)}df3 secO i F(13m) 'II2ti"(13m,O)I 0) + sgn(iV'(37, (E-1) where (13ni3O)'c0si3mcos0+ (2BflsinnI3th)sin0, P"(13m, 0) a2(13, - cos0cOs13 (2En2Bnsinnl3m)sin0, (E-2) áhd f3, 132 are the stationary points which satisfy the

equa-tion

aF(13,e)

--a.-

-

sin13cos0

+ (2nBcosn13)sin0 = 0. (E3)

By the definitions = Bsinn13, -j-cos13. (E-2) ôañ be

rewritten:

\/"(13m, 0) sin0 d , (E-4)

f2lT

Appendix E. The derivation of wave resistance in low speed

limit

sin /3 + 2nBsinn(3 } Q(/3)

P(f3) =

(i)2 +

r_sin2/3 +

nBcosn$32,

+

¶sin/3cos/3

nBcosn/3

n2Bsinn/3L

When using B1=--e,B1, 0 (n 2) in (0-5), We haveF(/3)

for an elliptic cylinder:

F(j3) 2e3(1+E)3cos2l3sin2/3 2/3 +2cos2/3}3

which ôoincides with the expression (C-18) derived as a

limiting case of a semisubmerged ellipsoid.

(D-6)

the amplitude function ôan be applied to a ship form whose

water line curvature dOes not change its sign, i.e. d2/d2

When there is a point where d2/d2 =0 in the water line, the asymptotic expression of the amplitude function

is evaluated approximately:

C(0)S(0)

cos0

r(4T)./9

L L 2ir

x

F(j3)

e1Fi21vO),

(E-5)

where I() = 26789385347, 13 m is the stationary point

which satisfies the equations aiJi(j3, 0)

0 82/i(j3, 0)

0

aj3'aj32

In this case L"(/3m, 0)is expressed as

1 . 0

1I (13m.0)=5m 13m5n0

2cos2j3m

sin0-(E-6)

sin/3m

For the sake of simplicity we consider a ship form whose water line has a convex curve such as a semisub

merged ellipsoid For this case the wave resistance coeffi cient iS written as follows by the substitution of the

expres-sion (E-1) into (25) in the text.

2r

2

pu2L2

J0 L L

which indicates that the asymptotic expression (E-1) for

Cw1 +C2+C3,

(E7)

/1()2 + (_)2},

(D-2)

= 1 I72_i2= 1 sinj3 + nB sinnl3)2

/f(i)2

(dfl)2}

(D-3) where p-pô is the static pressure on thecylinder.

Since = 0 for a vertical cylinder we have from (B-8):

F($) = (D-4)

Substituting (D-2) and (D-3) into (D-4), we have

F(13) sinf3 1 + 2 nB cosnf3

x-F sinf3 + EnB sinnj3 G(13)P(13)13 (0-5) where

(19)

where

C1

= d ,

li"(j3,O)l

=

2Ff

)I70

dO 6 (/2 F(131)F(132)cOsO

C3-4F,

Jo s/Ls(131,.0)'(132,O)I

x cosE,{I1i(j31,O)-,(f32,O)}

contribution from

the after body, (E-8)

contribution from

the fore body, (E-9)

+-f(sgn/"(3i3O)-sgnt4i"(j32,O) )]dO,

(E-10)

where

C3

is an interaction term between the fore body and the after body For the evaluation of C we can again use the stationary phase method when carrying out the

integration with respect to 0. Let us define first

f(0)=

SeC20{'(13 0)

i(3O)}

.. (E-1i) The statiOnary point is the solution of the equation

df(0)_sec3OsinO

P1,O-2,O)

2

sec

cosf3sinO. + (2EBsinn131)cosO

+ cosl32sinO

(2Bsinnf32)cos0 = 0,

(E-12) since there. is the relation between 0 and (j3, 132):

a,D(13, O)_ sin13cos0 + (2nBsinnj3)sin0 0

at13 =l3 and 132.

When we put 0 = 0 in the equation (E-13), We have (E-12)

A design method for full fOrms was developed based on

the following three experimental and practical evidences

which were. derived from the analyses of towing test,data of

more than 200 full forms (Taniguchi et al., 1966). It has

been served as a routine method since 1963 in Mitsubishi

Heavy Industries, Ltd.

.i) For a full ship (U//E< 0.20,

Cb

>0.80) with a

well designed run part the wave resistance characteristics depend mainly on the geometry of entrance part, and the contri.bution from the parallel part and run part is

negligi-bly small in generating waves.

ii) The propulsion factors depend mainly on the geom-etry of run part Practically there exists. a limitation of fullness for the run part to prevent wOrse propulsive

per-Appendix F. Outhne of a design method of full forms

developed in Nagasaki Experimental Tank

MTB 109 August 1976

j3= 0,

132= r. Thus in the case of 00,

isestab-lished. Therefore, we may say that 0 = 0 is the. stationary

.point. Then the approxiinate evaluation of

C3

is given:

F(0)F(7r) -cw3 =

Fc

x cos[2,2(O,O)-i(,O)}

+sgn i/i"(0, 0) ---.sgn

)"(ir,.0)}

+?Jsgn (g)]

4 dO (E14)

Further we can derive the fol'owing relations

d2f1

_i.

1

dO2' OO,j3O, 2ir 2F

[2nB, 2

We have then

F(0)F(i}

C3

= '.2ii-F, -. = =

'JT+2

F24

Thus we obtain a geneaI expression of C for a ship form

which has a convex load water line.curve:

C= 2F,

r{F1312s0

dO+2F,

(T/I1322c050dO

J0

lii"(j.3,0)I

J0

h/i"(f32,O)I

2-s/F

sun(L+2t_).

\/1+2nBnJ2+2{(_1)nmnB,j2

F, 4

formance and undesirable flow phenomena around the

stern.

iii) A reliable formula to estimate the form factor by

use of given geometric parameters has been developed.

In this method the bow forms are designed so as to reduce wave résistáhce, and stern forms are designed to obtain better propulsive efficiency and the parallel parts are designed to satisfy the required displacement So the

experimental data. such as wave resistance and self propul sion factors are stored-together with geometrical parameters

of each part For instance the parameters for wave resist

ance data are

Froude number:

FflBU/s,/

Breãdth.draft ratio: Bid

(20)

MTB1O9 August 1976

Fineness factor: Cm He/B

where Cm is the midship area coefficient and He is an effective length representing fineness of entrance defined

by

He=(lC'pe)/Le,

where Cpe is the prismatic coefficient of èntränce patt, Le is the length of entrance. When a new ship form is

designed a proper bow form and a stern form are selected

independently from the stored data so as to achieve a

desired propulsive performance.

The design method mentioned above was originally

developed based on the experimental evidences. Its validity

has been confirmed by a number of model experiments.

By the present theoretical study in the text the assumption

of separability of a body into parts has been verified theoretically In addition to this a way of reasonable deter

minatiOn of the viscous résitancè in low speed limit has been provided This may be an important contribution in

Obtaining reliable wave resistance data from model experi-ments.

As far as the bow form design is concerned, the

f011ow-ing method has been Used

A ship like a Oil tanker or bulk carrier usually serves both in full load and ballast lOad conditions with almost same frequency Therefore one must design the optimum entrance not only for full load condition but also for bal

last condition. In the full load condition it. is more econom-ical to Choose shorter entrance under the design constraints.

In the ballast condition one of the measures to quantify the propulsive performance is the difference of service speeds between fUll and ballast lOad conditions with

con-stant horsepower To satisfy the required speed difference

say 1 knot, the entrance length in ballast load condition must be longer than that in full load condition so as to prevent enormous- increase of wave resistance. Thus the Optimum entrance length in ballast !oad condition differs from that in full load condition.

To meet this requirement the MHI Bow was invented The key idea of MHI-Bow is to combine two entrance forms each Of which is optimum in full and ballast load

Cytaty

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