Numerical evaluation of a wave-resistance theory for slow ships

1. Introduction

A wave resistance theory which takes account of the nonlinear effect on the free surface condition in low speeds has been developed by Baba and Takekuma1121. In this paper a procedure is presented for the numerical calculation of conventional ship forms based on this theory.

The theory is an extensiOn of Ogilvie's two-dimensional low-speed wave resistance theory131 to the three-dimension-al bodies piercing the free surface. It is a characteristic of the theory that the double-body velocity potential is used as the zero-order solution. As the next order solution a

surface-layer velocity potential which represents a wave mo-tion is determined in such a way that the sum of both velocity potentials satisfies a free surface condition. The free surface conditiOn used in the present paper is a linear equation for the surface-layer potential. However, products of derivatives of zero-order potential are included. There-fore the coefficients for the equation are dependent on space variables.

In section 2 an outline of the present theory is given. It is then shown that wave resistance derived from the present theory consists of three parts. One is due to the singularity distributions over the body surface, the second is due to the singularity distributions around the intersection be-tween the body and the still water surface (the so-called line integral term), and the third is due to the free surface disturbance expressed in terms of products of derivatives of double-body potential. The last one is considered as the

nonlinear effect on the free surface condition.

In section 3 a quantitative discussion on the character-istics of the present theory is given by applying the theory to a vertical circular cylinder piercing the free surface. It is then shown that the contribu.tion from the singularity dis-tributions on the body surface is cancelled out by the lower order terms of the contribution from the line integral term. The remaining higher order terms are the same order of magnitude as that of the third part of the free surface

dis-Dr. Eng., Resistance and Propulsion Research Laboratory, Naga-saki Technical Institute, Technical Headquarters

"Resistance and Propulsion Research Laboratory, Nagasaki

Tech-z y

Eiichi Baba* Midori Hara**

A procedure is presented for the numerical calculation of wave resistance of conventional ship forms. Wave resistance theory used

in the present calculation takes account of the nonlinear effect on the free surface condition. Because of this a remarkable attenua-tion of the humps and hollows of wave resistance curve is attained in the practical speed range of convenattenua-tional commerical ships Taking a semisubmerged sphere as an example, each stage of numerical calculations is examined by comparing with the analytical values. Finally wave resistance of conventional ship forms is calculated and compared with experimental values. Within a practically

acceptable order of magnitude wave resistance can be estimated by the present theory.

turbance. It is further shown that this third part contributes especially to the reduction of transverse wave component. As the result, humps and hollows of the wave resistance coefficient curve are attenuated remarkably.

In section 4 a procedure to calculate wave resistance of conventional ship forms is explained. In the computatiOn an asymptotic expression of the amplitude function in low speed limit is used. Necessary quantities for the computa-tion are double-body velocity components around the load waterline. They can be obtained by finite element method such as the one developed by Hess and Smith for nonlifting bodies141. Details of numerical calculation ofwave resist-ance are described. In the procedure validity of the

numer-ical calculation is examined by applying the computer pro. gram to a semisubmerged sphere whose wave resistance is obtained analytically by the present theory.

Jn section 5 several examples of wave resistance calcula-tion for convencalcula-tional shipfbrms are shown comparing with wave resistance determined by the towing tests.

2. Outline of the theory for slow ships

Taking the rectangular coordinate system fixed on the body with the origin on the still water surface, we set x-axis directing to the uniform flow U and z-x-axis directing upwards as shown in Fig. 1. The total velocity potential is

- The contents of this paper was presented to the Second Inter-natiOnal Conference on Numerical Ship Hydrodynamics,

Septem-(x,y,z): Surface-layer + pOtentiol 4.(x,y,z) Double-body potential Fig. 1 Coordinate system

defined:

ør(X, y, z) +(x, y, z),

where _{r is the double-body potential obtained from the} rigid-wall problem. is the surface-layer velocity potential which represents a wave motion.

The boundary value problem presented by Baba and Takekuma11 for the surface-layer velocity potentialcz'(x, y, z) is written as

o =cO(x, y, z) +q(x, y, z) +q(x, y, z)

z<0,

(1)

Erx(cV,0)

+øry(X,y,O)]2co(X,y,Z)

+P(x,y,z)=D(x,y)

onz=0,

(2)

where g is the acceleration of gravity, rx(X, jf, 0), 4ry(X, )I, 0) are the velocity components at the still water surface.

[øry(x. 1.', 0) y)J, (3)

y)

1[U2

2(x y,O) _0rc y, 0)].

(4)

In equations (1) and (2), z = 0 corresponds to the free sur-face elevation expressed by the term r(X, y). Besides the equations (1) and (2), the radiation condition should be satisfied byco(x, y, z). It should be noted further that in the present theory the surface-layer potential ç(x, y, z) does not satisfy the body boundary condition. The zero-order potential y, z) alone satisfies the condition. Recently Newman5 derived independently the same boundary value problem for slow ships as the equations (1) and (2). The details of the derivation of the above boundary value prob-lem and the following results are found in the reference6.

An asymptotic solution cD(x, y, z) in low speed limit is obtained as çO(x, y,z) "72

xf

d0ko(xv0)fdk

ekZcos(kZ.)

k1(x,y,0) k

+

+JJdx'dY'D(x'Y')

xf dO k0(x, y, 0)

ezko()cY.o)sin[ko(x,

_{j.', 0)],}

(5) where k0(x, y, 0) = g/[4,.(x, y, 0) cos 9 + Ø,(x. y, 0)sinO](b)

=(xx1 cosO +(y_y') sinO

Wave resistance derived from the velocity potential Ø(x, y, z)

is expressed as

"2

R=1TPU2f

_{IA(OH2cos3OdO,} (8)

where p is the density of water, A(0) is the amplitude function:

'

3OjJdxdvD(x,y)

A(0) =----sec

irU xexp[ivsec20 (xcosO+ysinO)], (9) where V =g/U2.

It is a characteristic of the present wave-resistance formula that the amplitude function is expressed as an integral of disturbance D(x, y) over the free surface. Recently Maruo has derived independently the same wave-resistance formula

as (9) and

The disturbance D(x, y) is rewritten as the sum of two parts:

where q5 is the perturbation velocity potential of the double body, i.e. Ø(x, y, z) = ør(" y, z) Ux,

u

a _1A24,h2

2g ax "'x

't'y

x2y]

+

It should be noted that N(x, y, 0) is expressed in terms of products of derivatives of the perturbation potential. Within the framework of the linearized theory, this term is usually neglected as the higher order quantities as done by Guevel et al. In the present theory, however, it is shown that the contribution from N(x, p. 0) plays an important role

on the wave resistance in low speeds. The present authors consider that this term represents a nonlinear effect on the free surface condition.

Substituting (10) into (9), we have

A(0) = sec30Jfdxdv y, 0)

x exp [ivsec20 (xcos0 +ysinO)]

_! sec30ffdxdy N(x,

y, 0)

x exp [/vsec20 +(xcosO ysin0)]. (12)

Excluding the cross section between the body and the still

(7) water surface from the integral range, and integrating by

parts, the first term of (12) is written:

-

-J-- sec3efdvø(x, y, 0) irU

x exp [/vsec2O (xcosO + ysinO)]

+ -h sec3O (1

vseco)fdy

cb(x, y, 0)

xexp[ivsec2O (xcosO +ysinO)]

xexp[ivsec2O (xcosO +ysinO)], (13) where c is a curve of the intersection between the body and the still water surface.

When a body is expressed by surface source distribution a(x, y, z) and normal doublet distribution i.z(x, y, z) so as to give zero-value of perturbation potential inside the body, we have the following relations.

4(x, y, 0) = 4ir [a(x, y, 0) cos (ox, n)

+ sin(ox, n)} on c,

Ø(x,y;0)=-4iru(x,y,o) on c,

J'fdxdy [-i- +

x exp[i vsec2 0 (xcosO + ysinO)]

-

4ir

vsec2O exp [vz'sec2O +1 vsec2O(x'cosO + y'sinO)].

Thus we have

A(0) _{As(0) +AL(0) 4F(0)'} where

As(0) = -.f.sec3O,fj;dSa(x,Y,z)

x exp [vsec2O z + i(xcosO + ysinO)I I + - sec3

off

_{dSji(x, y,}

x exp[vsec28 lz+i(xcosO+ysinO)(],

-+-.

AL (0) = - -j-sec3o1fdv[a(x, y, 0) cos(ox, n) + ii'secO i(x, y, 0) sin(ox, n)] x exp [i vsec20 (xcos0 + ysin0)].

AF(0) = ---s sec3offdxdv

N(x, y, 0)

(15)

_J2(2;2

sec2O)J,

AL (0) =2/a_{_/ (2;2 sec2}ü)

ía

2F2 sec2O [Jo(-;;. sec2O)

_J2(22 sec2O)]

+ 8/a F2 sec 0 cos 30 Jo(-j sec20)

It is understood that the amplitude function consists of three parts. One is due to the surface singularity distribu-tions over the body surface. This term gives the conven-tional wave-resistance formula which was derived by Have-lock for submerged bodies.9> The second is due to the singularity distributions around the intersection between the body and the still water surface. This term is the so-called line integral term. The third is due to the free surface disturbance expressed in terms of products of derivatives of double-body potential.

3. Wave resistance of simple forms

In order to evaluate numerically the characteristics of

the present theory, the above mentioned three parts of the amplitude function are calculated for a vertical circular cylinder piercing the free surface.

The density of source and doublet distributions for a

vertical circular cylinder of radius a are given as follows.

a(x, y, z) =

4lTa' p(x, y, z) =

Ux Ux

4

The double-body potential Ø(x, y, z) for a circular cylinder is written as

(x, y, z) -

_{x +y}

'

2' X + y

> a

Ua2x 2 2 2

Substituting (17) into As(0) and AL(0) of (16), we have

As(0);; 2/aJ1

2sec2O)

sec 0 [J0(- sec2O)

n

(19)

+ 4jsec3O (ivsecO) dydx çb(x, v, 0) _{x exp[ivsec2O (xcosO +ysin9)] (16)}

_ff(xv'z')[3+ 4]dS,

(14) where n is the outward normal to the surface at the inter-section c, and Q is the tangent. S is the surface of the

sub-merged part of the body. Here, the body is considered

which has a vertical hull surface at the intersection c.

r = [(xx')2 + (V)")2 + (z_z12]h/2,

+ (yy')2 +

.2 ilO 5.0 4.0 5.0 1.0 A.(9) + (0) F= 0.20 60 -1.0 -2.0 -3.0 -4.0 -5.0

Fig. 2 Comparison of amplitude functions of vertical circular cylinder in low speeds

On the other hand, the wave resistance due to the sum of three parts As(0), AL(0) and AF(0) is obtained as

8192

315 F,,6#O(F,,e).

Fig. 3 shows a comparison of the wave resistance curves for both cases. From this figure it is found that AF(0), which represents the nonlinear effect on the free surface condition, plays an important role on the attenuation of

humps and hollows of wave resistance curve in low speed

range. 2.0 1.5 1.0 0.5 0005 u07

\LFQ

Vertical circular cylinder

Linearized theory

(24)

Fn U//

0.09 0.11 0J3 0J5 0.17 0.19 021

Fig. 3 Comparison of C-vaIues of vertical circular cylinder

- 32/a F,,4 cos 0 cos 30 J1 ( 2F,,2 sec20). (20) where F,, = ii/...J , and J0, Jj, J2 are the Bessel

func-tions of the first kind. The first term of (19) is the con-tribution from the surface source discon-tribution, and the second is from doublet distribution.

It is observed that the amplitude function due to the surface singularities As(0) is cancelled out by the first

two terms of the amplitude function due to the line

integral term AL(0). This fact was first pointed out by Brard within the framework of the linearized theory(1O).

The asymptotic expression of the sum of As(0) and AL(0) is written in the low speed limit:

As(0) #AL(0) cos3O

cos

2F,,2 sec20 + O(F,,5). (21)

Next, substituting (18) into (11), we have N(x, y, 0) as follows.

2a5x a7x N(x, y, 0) = 4U F,,2 (-76-

-for r=/x2+y2 >a.

From AF(0) of (16) we then have

0O 2 1' 2a a6

AF(0) = 4/ sec 0 / (7

-x J1 (yr sec2 0) dr .16a sec20 ir

cos0 cos(-- --j)

4-O(F,,5). (22)

It is thus shown that in the low speed limit As(0) +AL(0) is of same order of magnitude as that of AF(0). Fig. 2 shows a comparison of the asymptotic values of A5(0) + AL(0) and AF(0) at F,., 0.20,

where

s(0) As(0)/2ai, AL(0) =AL(0)/2ai, AF(0) =AF(0)/2ai.

From the figure

it is found that AF(0) contributes

especially to the reduction of transverse wave component (small 0-values). As the result, humps and hollows of the wave-resistance curve are attenuated remarkably.

In the low speed limit wave resistance due to the sum of As(0) and AL(0) is obtained by means of stationary phase in (8): R1,, W _LpU2(2a)2 2 1 1T)

-

6656Ffl6+32%ñFn7sin(j-j

4

- 315

+Q(F,,8). (23) Rw (20) Preseni theory

An additional example of calculation is shown in Fig. 4, where the wave resistance curves of a semisubmerged sphere are compared. The wave resistance coefficient

ob-tained from the sum of two parts As(0) and AL(0) is

given as follows. "0 1.5 1.0 0.5 -0 cw = 224 6

the semisubmerged sphere is given by

R +PU'(20)' Linearized theory

#72yF

sin( + O(F8). O(F8). Semisubmerged Sphere c,= F+72 jirF:sin(- +.-) + -f-) 4

On the other hand, the wave resistance coefficient ob-for tamed from the sum of three parts As(0), AL(0)

= -F6

+!jF7

_'F,2

-=

Present theory

F= u/

on the contribution of each term of amplitude functions

to the wave resistance. In the present section a procedure is explained to calculate numerically the wave resistance of arbitrary forms. For convenience of computation, the expression of amplitude function (9) is used directly in-stead of the expression (15).

By the partial integration with respect to x in (9),

an asymptotic expression of the amplitude function in low speed limit is obtained as a line integral around the

intersection of the body and the

still water surface: .sec2O I'

A(0) -

_'

ir

ydv D(x, ')

x exp [/sec2O (xcos9 + ysinO)]. (27) In this calculation the cross section between the body and the still water surface is excluded from the integral range in (9).

When the equation of the intersection is expressed by the following relations

x 1

ye0

= --cos(3,

L 84,7 sin(n(j),

where 4 is the length of a body and B, are the Fourier coefficients, the amplitude function is rewritten:

A(0) sec2O 21T

I--- F

L

-

2ir 2fdi3F(13)₀

where F = u/v'iiJ,

= cosflcosO # (2B-, sin n(3)sinO, sec2 o

x exp[, ,.,

F((3)

V_pWZr5,

p(j3)=

_2

u=1 #cb(x,y,o)/U

_{v=Ø(x,y,O)/U,}

w = çi(x, y, 0) L/U

The amplitude function can be calculated by use of the velocity components around the load waterline. The wave resistance is then calculated by (8).

Taking a semisubmerged sphere as an example, the validity of each stage of numerical calculations is examined as follows. _{First, the integration with respect to (3 is} carried out in (28) by use of the exact expression of F(13) for a semisubmerged sphere26:

F(13) cos2(3 (. -sin2fl+ 1). (29) For this numerical calculation Simpson's rule is applied with the interval (3 = irF2/47. At 0 750, this interval

provides more than 15 ordinates in a range from one peak to the next peak of the highly oscillating integrand. Next, the integration with respect to 0 is carried out (28)

0.08 0.09 0.10 0.11 0.12 0.13

Fig. 4 Comparison of C.values of semisubmerged sphere

In this case, not only an attenuation of humps and hollows

but also a large reduction of the basic term which is of order F6 are attained by the addition of the amplitude function AF(0). The details of the derivation of (25) and

(26) are explained in Appendix.

4. NumeriCal method to calculate wave resistance In the previous sections analytical studies were given

in the wave resistance integral (8). In this computation zero-points of A(0) with respect to 0 are searched first and then each interval from one zero-point to the next

zero-point is numerically integrated by Simpson's rule

in such a way that the number of ordinates in one interval is not less than 10. It is confirmed that the value of

wave resistance usually converges to a certain value when

the upper limit of the integral range is close to 0 750 as shown in Fig. 5; an example of calculation on the semisubmerged sphere.

\C.G

a'

Semisubmerged sphere

at Fn0.15

-, e

Upper limit of integral range

10° 20° 30° 40° 50° 60° 70° 80°

Analytical

Fig. 5 Variation of C-vatues with respect to the upper limit of the integral range

Table 2 Comparison of numerical and analytical values of v(13),p(13),dp Id13, w(j3), dy/Ld(3, and F($3)

In Table 1 the computed wave resistance values are shown comparing with the analytical values which are obtained by the method of stationary phase with respect to $3 in (28) and 0 in (8) for a semisubmerged sphere as expressed by (26) in the previous section.

Table 1 Evaluation of numerical integration with respect to $3 and 0

This result shows the validity of the numerical integration for a practical use.

In the above mentioned examination of the numerical integration, the exact expression of F(fi) is used. In

practice, however, F(j3) for an arbitrary body has to be obtained numerically.

Representing the body by finite number of surface elements on the body, velocity components at the null point of each surface element can be obtained. Taking

again the semisubmerged sphere as an example, computed values of v($3), p(j3), dpldj3, w(13), dy/d13 and F($3) are compared with the exact values at the load waterline:

v($3)= --cos$3sinj3, w(j3) -3cos$3,

Froude C1

C2

C,1/C2

number Numerical Analytical

0.15 0.116463-3 0.115927-3 1.005 0.20 0.635963-3 0.640767-3 0993 0.25 0.189489-2 0.177887-2 1.065 Numerical

v()

Analytical p(13) Numerical Analytical dp/d3 Numerical Analytical 0.14245 -0.19096 -0.21080 0.96319 0.95465 -0.58765 -0.63239 0.27564 -0.40290 -0.39283 0.82567 0.83333 -1.2215 -1.1785 0.44442 -0.58736 -0.58226 0.57906 0.58410 -1.6852 -1.7468 0.61628 -0.70839 -0.70751 0.25025 0.24830 -2.0385 -2.1225 0.78920 -0.74859 -0.74998 -0.12401 -0.13355 -2.1624 -2.2499 0.96259 -0.70103 -0.70340 -0.49861 -0.51 539 -2.0299 -2.1102 1.1362 -0.57045 -0.57286 -0.82839 -0.85112 -1.6548 -1.7186 1.3100 -0.37198 -0.37370 -1.0735 -1.1004 -1.0807 -1.1211 1.4839 -0.12913 -0.12969 -1.2040 -1.2331 -0.37549 -0.38907 Wz(j3) dy/Ld13 F(13)

Numerical Analytical Numerical Analytical Numerical Analytical

0.14245 -2.8776 -2.9696 0.49269 0.49494 1.4778 1.5364 0.27564 -2.8769 -2.8868 0.47999 0.48113 1.6323 1.6204 0.44442 -2.7145 -2.7086 0.45056 0.45143 1.6980 1.7313 0.61628 -2.4527 -2.4481 0.40732 0.40802 1.6941 1.7497 0.78920 -2.1115 -2.1132 0.35165 0.35221 1.5267 1.5879 0.96259 -1.7031 -1.7142 0.28527 0.28570 1.1808 1.2319 1.1362 -1.2500 -1.2631 0.21021 0.21052 0.72631 0.75816 1.3100 -0.76229 -0.77355 0.12874 0.12893 0.29665 0.30920 1.4839 -0.25609 -0.26036 0.43357-1 0.43394-1 0.35119-1 0.36527-1 10 1.5 1.0 0.5 0

p(/3)= 1_sin2,3, -=-cosj3,

dp(3) 9

-

2sunj3cosI3.

In this computation, half of the surface of submerged part of the sphere is approximated by 18x18 surface elements. The source density is assumed constant over each of the elements. Table 2 shows the comparison of analytical and numerical values.

For the calculation of dp/d3, p(j3) is approximated first by a set of parabolic curves determined by p(3) at the

null points of surface elements at the load waterline.

Then dp/df3 is obtained by a curve fitting through the derivatives of p(j3) which is approximated by the parabolic curves For the calculation of w(13,) in the present paper,

w(j3) is obtained first at the point which is a little apart from the surface element closest tothe load waterline. The distance from the load waterline which is suitable for calculation was investigated

in the trial and error

process. As the result, the most reliable values for w were found in this case for the point which is 3% of the half breadth of the body in y-direction from the upper

edge E and 3% of the draft of the body in

negative z-direction (=) from the z=0 level as shown in Fig. 6. w(I3) is then determined by dividing the value of w(j3) by 5.

Fig. 6 Calculation ofw1-va!ues

In practice, rather small number, say 350, are used from the economical view point. Therefore, the present authors consider that such a method mentioned above is necessary in determining the values of w(13) efficiently. It is needless to say that the suitable distance of the point where w(3) is calculated should be searched in accordance with the numbers, the size and the shape

of the surface elements.

Finally, Table 3 shows a comparison of wave resistance obtained by analytical method with the wave resistance obtained numerically in all the steps of calculations.

Table 3 Comparison of wave resistance numerical and analytical I0 2.0 Semisubmerged Sphere Rv 4-ptf(2a? naIyticol Numerical I.0 0 0.13 0.15 u/J 0.17 0.19 0.21 0.23 0.25

Fig. 7 Comparison of numerical and analytical C-vaIues of semisubmerged sphere

Fig. 7 shows a wave resistance curve obtained analytical-ly compared with the values obtained numericalanalytical-ly. From this study it is said that the present procedure for numerical calculation of wave resistance in low speeds has sufficient accuracy for practical use.

5. Calculation of wave resistance of conventional ship forms

In this section results of numerical calculation of wave resistance of five different ship forms are shown. Table 4 shows the particulars of the ship forms.

Table 4 Paiticulars of ship forms and numbers of surface elements

The first example of the calculations is for Wigley's parabolib forms M.1719, and M.1720 which are

geo-Froude number C,,.,3 Numerical C,,.,2 Analytical C,,.,3/C2 0.15 0.111583-3 0.115927-3 0.963 0.16 0.164280-3 0.169636-3 0.968 0.17 0.188316-3 0.194107-3 0.970 0.18 0.300873-3 0.318895-3 0.940 0.19 0.406422-3 0.410392-3 0.990 0.20 0.605154-3 0.640767-3 0.944 0.21 0.613495-3 0.632632-3 0.970 0.22 0.114399-2 0.112825-2 1.014 023 0.146388-2 0.154081-2 0.950 0.24 0.135996-2 0.149167-2 0.912 0.25 0.178262-2 0.177887-2 1.002

M. No. Cb LIB Bid Surface

elements 1719,20 0.4444 10.000 1600 30x10 1955 0. 5576 6.720 2.581 27 x 18 2330C 0.7391 6.770 2.381 27x19 1360 03754 6.966 2.510 23x 12 19146 08624 6.358 2;581 23x14

1.0 P(p) 0.5 Surface elements 34 x 10 Surface elements 34 = 2.5' I0' Surface elements 30 x 10

Fig. 8 Arrangements of surface elements near the fore and aft ends of Wigley's form

z=0 Surface elements 32 'VL= 0.5. I0 Semisubmerged Sphere Surface elements lB. 18

Wigleys parabolic form

V4 94 Analytical

/

_Computed o Surface elements 30. 0 C ₃₂₅₁₀ 345 AP FP

aft ends is 0.5% L as shown in Fig. 8. In addition to this case, two other cases, i.e. the breadth of the elements

being 0.25% L and 0.05% L are also studied so as to know a change of wave resistance with respect to the breadth of the surface elements.

In Fig. 9 the calculated values of p(fl) are shown com-paring with the values for semisubmerged sphere. It is

Understood that near the fore and aft ends p(j3) of Wigley's

form varies rapidly. It is also shown that p(j3) of Wigley's

form is almost unchanged for a large difference of the breadth of the surface elements.

Fig. 10 shows a comparison of wave spectra at F,, = 0.20. With a decrease of the breadth of the surface element, wave spectrum decreases. However, its change is small. Fig. 11 shows the result of integration of wave spectra with respect to 0. From this figure it is found that wave resistance is not so sensitive to the difference of the

breadth of the surface element used in the present study.

-4

Ito

2.0 .0 " 100 _{20 30}

60

70'

- e

Fig. 10 Comparison of wave spectra with different breadth of surface element at fore and aft ends

-4

I0

0°

Wigley's parabolic form

at Fn0.20

21t I A(e)Iecos3e,L2

Breadth of element

at fore and aft ends

X/L

5.0 X

2.5 X io

0.5 x l0

Fig. 11 Change of C, with respect to the breadth of surface element

1.0 Wigley's form Cv at FnCO.20

0.5 Cv _LI7UZL'

Breadthof element atforeand of endst

-

X/L 0 0 2 3 4 5.I_oa SS V2 SS 8'2 9 9 02 0.3 04 (rad.) 0.5 0.49Th 0.4900 0.4777 04605 '/L --COSP AP (FP)

Fig. 9 p(a)-values near the fore and aft ends metrically similar and defined by

y=[(I)2 _()2][1_()2],

LIB = 10, d/L = 0.0625.

In this computation 30x10 surface elements are used. Near the fore and aft ends smaller elements are used, because velocity components vary rapidly with position in those regions. The breadth of the elements at fore and

-4 110 4.0 R cw -3.0 4PU2L2

Wigleys parabolic form J_)2_ ( X )2) [I- (Z/d)2)

2.0 Y=2B((2

T

L/6100, d/L 0.0625

l..o Michell

Fig. 12 shows the computed wave resistance for a wide range of Froude number comparing with the wave resist-ance obtained by Michell's linearized theory. In this figure wave resistance determined from the towing tests of geometrically similar models of 8 meters and 5 meters are also shown. A remarkable attenuation of the humps and hollows in the range F < 0.20 is attained by the present theory. A quantitative agreement is also observed between the present theory and the experiment. It should be noted, however, that in higher speed range the present

theory gives a poorer estimate than Miôhell's theory. Fig. 13 shows a comparison of wave spectra at F 0.20. When comparing with Michell's theory, it is a

charac-teristic of the present theory that the wave spectrum corresponding to the transverse waves (small 0-values) is considerably reduced as observed for vertical circular

0

0

A

Present theory (Surface elements 30 10)

0.05 0.10 0.15 0.20 0.25 0.30 0.35

Fig. 12 Comparison of calcuJated and measured wave resistance of Wigley's form

-4 110 30 2.0 1.0 0.06 M. NO Ce L/B B/d O 1955 0.5576 6.720 2.581 o 2330C 0.7391 6.770 a381 360 0.7764 6.966 2.5 10 © 1914B 0.8624 6.358 2.581 form factor 0.294 0.296 0.336 0.51 7 R, W V displacement volume Present theory 0.08 0.10 0.12 0.14 0 Experiment 0.16 0.18 a20 0.22 0 0.24 0 i0 20 0.26 a 0 Experiment 8m Model ₁₇₁₉ 5m Model ₁₇₂₀

Wigley's parabolic form

at Fn0.20

I

0.28 0 a 30 40

Fig. 13 Comparison ofcalculated wave spectra of Wigles form

Ce 0.56

0

° Experiment

0.30

Fig. 14 Comparison of calculated and measured wave resistance of conventional ship forms

cylinder. This is the reason for the attenuation of humps and hollows in wave-resistance curve.

Fig. 14 shows the computed wave resistance of four conventional ship forms (LIB 6.4-7.0, Gb 056-0.86). For those ship forms, the number of surface elements in the longitudinal direction corresponds to the number of uare stations which are used for the drawing of lines in routine. It is observed that in a wide range of block coefficient the calculated values of wave resistance are in the same order of magnitude as those of experimental values which are determined by Hughes method, where the form factor is determined by a5suming that the total resistance is the viscous resistance in very low speed

range (F0.06).

When comparing carefully the computed wave resist-ance with experiment, it is noted that theoretical curves are shifted a little toward lower speed range. This tendency is also observed in the case of Wigley's form in the range

F > 0.20.

6. Concluding remarks

In the present paper a procedure is explained for the calculation of wave resistance of ships in low speeds. The surface-layer velocity potential used in the present study is an asymptotic solution in low speed limit. n accordance with this fact, an asymptotic wave-resistance formula in

The authors wish to express their deep appreciation to Mr. Kinya Tamura, manager of Resistance and Propul-sion Laboratory, Nagasaki Technical Institute and Dr.

Baba, E. and Takekuma. K. (1975), 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, 1-10.

Baba, S. (1975). Blunt Bow Forms and Wave Breaking, The First STAR Symposium, Washington, D.C.

Ogilvie, T.F. (1968), Wave Resistance: The Low Speed Limit, University of Michigan, Naval Architecture and Marine Engi-neering, No. 002.

Hess, J.L. and Smith, A.M.O. (1964), Calculation of Non-lifting Potential Flow About Arbitrary Three-Dimensional

Bodies, Journal of Ship Research, Vol. 8. No. 2. 22-44. Newman. J.N. (1976), Linearized Wave Resistance Theory, Proceedings of International Seminar on Wave Resistance, Tokyo, 31-43.

Baba, E. (1976). Wave Resistance of Ships in Low Speed,

Acknowledgements

References

low speed limit is

used for the calculation of wave

resistance.

From a number of computation based on the present theory it is found that in practical speed range the wave resistance of conventional ship forms can be estimated within a practically acceptable order of magnitude. The

wave resistance of those ship forms has not been tractable

by the thin ship theory.

It is the breakthrough which has been achieved by taking into account the nonlinear effect of the free surface condition in the present theory. From the practical view point, it is expected that the present theory is used to find a ship form of small wave resistance in an early stage of development of ship forms. It is also expected that the present theory can be used for the determination of the level of viscous resistance in low speed range. Then a reliable value of form factor is determined. This contributes to the increase of accuracy of power prediction of ships from the model tests.

There is, however, room for an improvement of the present asymptotic theory. In the future a correction should be added to the surface-layer potential so as to satisfy the body boundary condition. By doing this it is expected that a better estimate of wave resistance may be possible. Further, trim and sinkage of a ship should

be considered.

Yoshio Kayo for their stimulating and encouraging dis-cussions.

Mitsubishi Technical Bulletin. No. 109, Mitsubishi Heavy

Industries, Ltd.

Maruo, H. (1977). Wave Resistance of a Ship with Finite Beam at Low Froude Numbers, Bulletin of The Faculty of

Engineering. Yokohama National University, Vol. 26. Guevel, P., Vaussy, P. and Kobus, J.M. (1974), The

Distribu-tion of Singularities Kinematically Equivalent to a Moving

Hull in the Presence of a Free Surface, International Ship-building Progress, Vol 21, 311-324.

Havelock. TH. (1932), The Theory of Wave Resistance,

Proceedings of the Royal Society, A, Vol. 138, 339-348. Brard, R. (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.

The perturbation velocity potential of a semisubmerged sphere in the rigid wall problem is given as

Ua 3x Ø(x, y, z) = 2(x2+y2+z2)3'2

where a is the radius of the sphere. The first term of D(x,y) defined by (10) in the text is obtained as

1 a3U3 9x 15x3

--Ø(x,y,0)=

_{2g r7}

4UF2 [(-3-- :j-7) cos3-j-57 cos3!31

for

1,

(A-2)

where

F=U//a,r=Jx2#y2

,s=r/a,

cosl3=x/a, sinj3=y/a.

On the other hand, N(x, y, 0), the second term of D(x, y) is obtained as

9 9 177 201

15 63 39

xcos3#( lss4*?_i28s1o)c?53]

for s>

1. (A-3)

The amplitude function due to the surface singularity distributions and the line singularity distributions around the load waterline is expressed by the first term of (12):

AS(0)#AL(0)=__Jsec3/Tdxdvl_iøXX(xv,O)I

x exp [ ivsec2O (xcosO + ysinO) _1. (A-4) The amplitude function due to the free surface disturbance N(x, y,0) is expressed by the second term of (12):

AF(0) _-

seceff dxdy N(x,

y, 0)

x exp [I vsec2O (xcosO + ysin8) ] - (A-5) Substituting (A-2) into (A-4), we have in the low speed limit:

(A-i)

Appendix Wave resistance of semisubmerged sphere

.16a 72

As(0)#AL(0)= , F13[ cosO

1

120

cos3O_Icos sec2O -j-)

Substituting (A-3) into (A-5), we have in the low speed limit:

.16a 147 93

AF(0)

Fn3[jcosO

+-jjjcos38_I

1 2 5

X cos sec 0 --i-) + Q(F 1.

The sum of three parts As(0), AL(0) and AF(0) is thus written:

As(0) +AL(0) +AF(0)

.16a 75 27

=

_{-r- F3 [-j-}

_{cos0 --j-jcos30]}

xcos(22sec20

IT) _(A-8)

Substituting (A-6) into (8)

in the text and using the

stationary phase method, we have a wave-resistance formula of a semisubmerged sphere when the free surface condi-tion is linearized:

W

# 72'/F,,' sin (-

+.-)

(A-9)

When the contribution from the free surface disturbance N(x, y, 0) is included, the wave resistance is obtained

by the use of the expression (A-8):

cwöFn6

+fF

sin(

+Q(F8).

(A-b)

1 iT