Study on separation of ship resistance components

Study on Separation of Ship Resistance Components

Eiichi Baba*

This paper describes theoretical and experimental studies on t/e separation of ship resistance components,

namely the wave resistance and the viscous resistance. The theoretical analysis gives the asymptotic formulae of both components with linear approximation. _{The experimental results are analysed by the use of the} asymptotic formulae.

The sum of both components, each measured separately, _{is in good agreement with the total resistance} measured by dynamometer. _{The interaction term of these components can be neglected as a higher order} quantity.

A new resistance component was found from the viscous resistance derived by the wake _{survey method.} The component is caused by the breakdown of bow waves and is much _{greater than the wave resistance} derived by the wave analysis method. A theoretical approach has been made _{to interpret the nature of the} new component.

Introduction

In 1951 TulinCi) suggested the possibility of separation of ship total resistance into wave resistance and viscous resi-stance by means of the wake survey. In 1962 Eggers(2) presented a theoretical study on methods of calculation of wave resistance from measured wave patterns. Since then, several investigators have tried to obtain more refined tech-niques of wake survey and wave analysis. In the recent years Sharma developed a method of wave analysis based on

the measurement of waves along a longitudinal cut. Land-weber and others") also developed a technique of wake

survey and discussed the variation of viscous resistance as it relates with Froude number.

Unfortunately, however, only a few examples have been given in which both measurements of viscous resistance and wave resistance were used to determine the total resistance Recently, in Mitsubishi Experimental Tank, extensive tests have been made on a variety of ship forms including Wigley's mathematical form and geosims of tankers to

measure each resistance component separately.

A consistent method of theoretical analysis, based on the linearization of velocity field, have also been developed. The linear theory gives asymptotic formulae of both com-ponents.

The sum of the viscous resistance and the wave resistance analysed by this theory was found to be in good agreement

with the total resistance measured by dynamometer. And a new resistance component was found from the viscous

resistance derived by the wake survey method.

The component caused by the wave breaking at the bow

of fuller ships. Especially in ballast condition of full ships, the component occupies the most part of the so-called wave resistance calculated by Hughes' method. In the following

is presented the outline of those recent investigations.

Theoretical Study on Separation of

Resistance Components

2. 1 General expressions of ship resistance

Applying the conservation principle of mass and

momen-* Experimental Tank, Nagasaki Technical Institute. Technical Headquarters

turn to viscous incompressible fluid in steady motion, follo-wing basic equations can be written.

pq=O ₍₁₎

p(qv)q=vcb+pF

(2)

where q is the velocity vector, p the density of fluid, F the external force per unit mass sUch as gravity, and 0

the stress tensor.

The stress on the surface of fluid is given by

='pn+p(2(n)q+nxw)

(3)

where p is the pressure, p the coefficient of viscosity, n

the unit outward normal, and oj=pxq the vorticityt7).

In addition to the basic equations (1) and (2) the

boun-dary conditions should be considered. On the surface of

the fluid nq=O is satisfied. In viscous fluid the dynamical boundary condition q=O on the hull surface and

nO=O (4)

on the free surface should be considered, where the surface tension is neglected. If p=O, the condition (4) is identical with the condition of constant pressure on the free surface of the inviscid fluid.

Now consider a ship floating in a uniform stream U= Ui.

Fig. 1 shows the coordinate system and other symbols to

be used in the following discussion. x-axis coincides with

Fig. 1 Coordinate system and control surfaces

The main part of this paper was presented to the Spring Meeting of I the Society of Naval Architects in Japan held in May 1969

the direction of uniform flow with its origin at F.P. of the

ship, y-axis being horizontal and normal to x-axis, z-axis directing vertically upwards. is the surface of the paral-- -lelpiped surrounding the ship, of which Sf is the free surface, S0 and S1 are perpendicular to the uniform stream. A is

the surface of the ship and w the sectional area of the

viscous wake.

By Gauss's theorem, the integration of equation (2) over the volume V surrounded by E atid A gives the following equation, since nq=0 on A,

.A

pFr

+J (n - p(nq) q)ds

(5) where d is the volume element.

The integral on A is the resultant force acting on the ship, because neb is the stress on the hull surface.

The relation (5) indicates that there are two ways of

ex-pressing ship resistance, i.e. the oe by the integration of stress on the hull surface and the other by the integration of stress and rate of momentum on the control surfaces at

a distance from the ship, since there is- no contribution

from the free surface because of the boundary conditions nlb=0 and nq=0.

By the substitution of the relation (3)

in the

left hand side of the relation- (5), total- resistance becomes

R1=-pcos (n, i) ds_S p (nXw) ids

(6)

where i is the unit vector of x-direction Derivation of (6) is described in AppendixJ. The first integral is the pressure resistance R and the- second integral is the frictional

resis-tance Rj, since nXw is tangential to the hull surface. Now consider the resistance which is expressed on the

control surfaces at a distance from a -ship.

To handle the expression on the control surfaces in more usual form, the- total head of the flow is defined by

p 1

H

+z+

q2 (7)

- 2g

Then the momentum equation (2) becomes

-qxw==-gH-op'Xw

- (8)

where v=p/p is the kinematic coefficient of viscosity, and

g the acceleration of gravity. For an irrotational flow,

since w=O,

VH=0 - - (9)

which iniplies that the total head has the same value throu-ghout the fluid. For the rotational flow, on the other hand, it is assumed that the flow can be divided into the regions inside and outside the viscous wake, and outside the viscous wake the flow is assumed to be irrôtational, i. e. H is

con-stant and equals the total head of the uniform streamdefined

by

1

U2 (10)

pg 2g

Inside the viscous wake, however, the total head is regarded as a function of position because ofthe presence of vorticity. Fiom the right hand integral of (5) the total resistance

expressed on the control surfaces becomes-

-(Ho-H) ds+-e2j2wdy

-+

5(vz-+wz_(U_u)zdE

(11)

where w is the wave height at the control surface Sj. and

(u, v, w) are the velocity components in the direction of

x-, y- and z-axes respectively. The derivation of (11) is

described in detail iii Appendix.

In general the- total resistance expressed by (11) is not divided further, since the pressure and the velocity

com-ponents must satisfy the non-linear partial differential equa.

tibn (8).

In order to divide the total resistance into its

components, it is necessary to introduce some approxima-tion.

2. 2 Linear approximation

To divide total resistance expressed by (11) into resistance components, Oseen's approximation at a distance is used. The control surface is taken at a distance from a ship so

that the velocity is expressed by

q=Ui±q'

(12)

where q' is the small deviation from the uniform velocity Ui.

Then the momentum equation (8) becomes

u

P

_{+gz_vxw}

-- - (13)

dx

_{\ p}

/

The solution of this equation (13) can be expressed as the

linear combination of two kinds of velocity

q'=qr+j7çr -- - (14)

where ço is the velocity potential of irrotational motion which satisfies Laplace's equation and qr is the velocity vector of rotational motion and satisfies

(15)

Vqr=O

-- (16)

Outside the viscous wake, since w=O, the solution is given

by

q'=j7'

- (17)

Under the linear approximation the pressure is expressed by the- velocity component of irrotational motion both in

the viscous wake and outside the wake.

p= -pgz-pU

- - (18)

Now split the wave height formally into wave height of potential motion p and wave height of rotational motion

-, i.e.

wp+-r. By the substitution of the pressure

and the veloity components expressed by (12), (14), (17)

and (18) in (11), the total resistance becomes

\2( a \2( ip

2JsjL\\dy)

\dz)

\\dx))

+

''

₇

C2rdy+pgrCpdY

urdSp

U2rdS S

(ir)°ds

_pUJ urds

(19)

The velocity potential çô satisfies the following linearized

free surface -condition, which is derived from the conditions

nq=0 and nø=0,

+ q dço =0, for z=0 - (20)

ax- LIZ dz

At a distance X far behind a ship, ço

is of O(X). By the

similarity hypothesis of velocity profile in the turbulent wake<8 it is assumed that Ur is of O(X-) and the- sec-tional area of the viscous wake e, is of Q(X).

In this case it is assumed that the kinematic coefficient of viscosity includes a fictitious kinemtic eddy viscosity. As X-'oo, all the terms of (19) vanish except the first, the

second and the last terms.

Then the total resistance is expressed asymptotically = 0 (,' dço \2 / dço \2 / dço \2'

SL

dy)

t

dz)

a±)

)-+ 7

52pdy pU5urds

(21)

No term showing the interaction between rotational and it-rotational velocity components appears in (21). In the present

(vz

investigation, wave resistance Rw is and the second integral of (21), i. e.

Rw = _-SdYS

_[&-)

()2

+?9_S2pdy (22)

On the other hand viscous resistance Rv is defined by. the third integral, i. e.

Rv=-pUu,.ds

(23)

And the interaction term of wave resistance and viscous resistance can be neglected as a higher order quantity.

The wave resistance is the component of resistance at-tributed to the irrotational fluid motion associated with

gravity and the viscous resistance is the component of

resistance attributed to the rotational motion in the viscous

wake.

For the ships with small

breadth / length ratio, e. g. Michell's thin ship, formal application of linear approxima-tion gives from (6)

L

033

R5= -2pU) dx)

---dx

(24) 0 -CO dxdx L 0 _dii

Ri=2)dx

dz ₍₂₅₎ 0 -d 3)!

where i denoted the breadth of the ship, L the length and d the draught. Since R is expressed by the velocity po-tential and Rf by the rotational velocity component, it is

written that

R Rw ₍₂₆₎

R1=R

(27)

For ordinary ships, however, the relations (26) and (27) do

not exist because of the non-linear fluid motion around a ship.

But the present theory indicates that

is the sum of pressure resistance and and at the same time the sum of wave

cous resistance

R1 = R + R1= Rw+ Rv

Method of Wave Analysis and Wake

Survey

The methods of calculation of wave resistance and viscous resistance from the measured data are described.

3. 1 Method of wave analysis

In the section 2. 2 the expression of wave resistance has been obtained in terms of velocity potential (ci. eq. (22)) which satisfies the liñearizd free surface conditions. To determine the velocity potential from measured wave patterns, equivalent doublets are distributed on the keel line as a

representative of a ship.

Present method of wave analysis determines first the

density of the doublets corresponding to the wave patterns

measured along a longitudinal cut. Then the amplitude

function and velocity potential are determined according to the elementary wave concept developed by Havelock(9).

Sharma's longitudinal cut method of wave analysis deter-mines the amplitude functions directly from measured wave

patterns by means of Fourier transform. In the process of Fourier transform Sharma needs correction for the

trun-cation of wave patterns in order to improve the accuracy

of calculation of wave resistance

In the present method of wave analysis, however, such correction for the truncation is not necessary, since the amplitude functions are not determined directly from measur-ed wave patterns.

Let us assume a point doublet M located at the point (x4, i, Zj) in the free stream U. By the linearized wave theory

defined by the first

the total resistance frictional resistance resistance and

vis-(28)

the wave height

, at a point (x, y) due to the doublet

is given

211'I /2

Cp(x,y)= ₎ de

7rU Cr/2

oo n2 (n cos flZj+ Co sec2 sin nxJ e -nI! dn -

n2+CsecO

4°M rr/2

$ sec4

e°

522 0sin(Co sec2 0) dO where

o=tan-1(- (x-x4)/(y-yo)J,

= (x-x) cos 0 + (y-y) sin 0

(30)

and 1to=g/2.

The first integral of (29) is called the local wave pattern and the second integral is called the free wave pattern.

If the wave pattern of a ship can be expressed by the

linear combination of a limited number of wave pattetns due

to the doublets which are distributed on the keel line, the

wave height is given by

N

Mi (xi, 0, -d) F(x, y;xi, 0, -d)

_"(31)

0=0 where

F(x,y;x, yt, Zj) =

2 r'12 .00 j2( cos flZI+ Kosec2 0 sinnxjjen Io

I d0j dn

irU .1,r/2 .O fl2+KO Sec4 0

0 sec 0sin(Co sec2 8)dO (32)

Since the function F can be calculated previously, the

den-sity of doublets M1 (i=1, 2, 3,..., N) can be determined from the measured wave patterns by the method of least squares. After the density of the doublets being determined, the wave patterns at a great distance from a ship is con-sidered.

_{As (x-xj)/(y-yj)-s.00, the term of local wave}

pattern vanishes. Then the wave height is given

asympto-tically

N

M(x1, 0, -dFoo(x, y;xi, 0, -d)

(3)

where

Foo(x, y-;xi, yj, zj)

sec 0e00eco8 ain(oo3sec° 0) dO (34)

U 'r/2

By the change of order of and fin (33), the wave pattern far behind a ship is expressed by

r rr/2

p(X,y) =_.' [S(0)sln(Ko sec2 0 x cos O+y sin 0 ) - C(8) cos (0o sec2 0 x cos O+y sin 0)) dO - (35)

where S(0) and C(0) are the amplitude functions

} =

4

(xi, o, - d) sec4 eCna'2 (I

coo

(o sec2 0 x O+yi sin 0) sin

Then the velocity potential çô is expressed by r/2 L 'ozsec S

çe(x,y,z)=Uj

e ,r/2 (29) (36)

cos 0(S(0) cos (° sec2 0 xcos O+y sin 0) + C(0) sin (o sec2 0 x cos O+ si O)JdO (37) Substituting the gradient of r in (22), the well known

expression of wave resistance is obtained

o.rr/2

Rw=lrpU2t [{S(0)}2+{C(0)}2) cos°OdO (38) In the routine practice, wave patterns are measured by

resistance type probes which are fixed transversely across the tank.

The probes are usually located at 1. 5 meters distance from the center line of 4. 2 meter length model and at 3. 2

100mm Water head - 50 - 2cm 0

1

v V

V'

V

V V V V V

5=1.112m 1 m

Theoretical wave pattern of S.2ô1

Fitted wave pattérh by prCsent method

L=2.777rn

-d=0.1388

y 2B(X/L)(1x/L), BIL 0.2, d/L 0.05

Fig. 2 Wave analysis of theoretical wave pattern

meters distance for 7 meter length model. About 300 points

of fiieasured wave height are used for the computation.

Number of doublets are 20 for models of block coefficient less than 0. 60. On the other hand for the fuller ships, i. e. Cb>0. 60, 30 points of doublet are distributed not only on keel line but On the other two lines located at a half breadth

from the keel line on the bottom. A good fitting of

meas-ured wave patterns is obtained by the three lines distribution Before the application of

the present method to the

measured wave patterns, a hufnerical check was carried out by use of the theoretical wave patterO. Fortunately Sharma presented a few theoretical wave patterns for the numerical

check°. A theoretical wave pattern of Inuid S-201, which is generated by a continuous distribution of sources and

sinks, was used for the present numerical check.

The result of wave aialysis by the present equivalent sin-gularity method is shown in Fig. 2. The fitted wave pattern gives a good approximation of the theoretical wave pattern.

The amplitude function derived by the present method is

compared with the result calculated directly from sources and sinks distribution of S201.

Fair cdihcidencC can be seen except in the range of higher value of 0. The reason for the discrepancy

is due to the.

assumption of submerged doublets for a floating ship, since

(H0H) F,=O.289, =-50mm

(H0H) F,=0.222, =-50rdm

Measured static pressure X 0.5 L1. behind

Calculated static rissure by using of the result of wave analysis

F,= 0.289

_---F,=0.222 N

Fig. 3 Comparison of static pressure in the viscous wake

A.P. I .5X.10 m 2n s(0) ± C( 9 ' coslO/L

A

- Exact value Present method

________F:

= 0.224

VII

30 60 90 0

the amplitude functions of submerged singularities decrease exponentially at the higher value of 0. This is one of the

weak points of the present method. The accuracy of the

calculation of wave resistance is about 92 percent. This degree of accuracy is probObly the same order as that of

Sharrpn's longitudinal cut method with truncation correction. 3. 2 Method of wake survey

In the section 2. 2 the following asymptotic expression of viscous resistance has been obtained.

Rv==_PUS u,.ds (39)

To make this expression more practical, the following

relation at a great distance behind a ship is used

pgj(Ho_H)ds.._pUjurds

. (40)

so as to derive viscous resistance from measurable quantities. Then an equivalent expression for (39) is given

Rv=p$ (HeH)ds

(41)

In practice, however, .a control surface is chosen at a finite

distance behind a ship for measuring the total head. The

position of the control surface is determined in the follow

iiig manner. Measure! data qf total head and static head

oh the several control surfaces were compared.

At the nearest position, 0. 15 Lpp behind A. P.,

the measured static pressure had rather strong

effect due to local flo*. Therefore the. control

surface at 0. 5 Lpp behind A. P. was selected

as a standard where the influexce of local flow

on the static

pressure was hardly observed. Tinder the assumption of linear treatment of

flow, the static pressure p is expressed in terms

of the velocity potential. When the static

pres-sure is meapres-sured by manometers, the hydrostatic

pressure is eliminated and eq (18) can be re

written as

(42) Since the velocity potential çø can be deterined

from the measured wave patterns, numerical

check is possible to see whether the static

pres-1.0

0

0 -2cm

- - Keel -line Three-lines

Fig. 5 Examples of wave analysis

/

Fig. 4 Comb-type wake traverse equipment

sure measured by pitot tubes agrees well with the static

pressure calculated from the measured wave patterns. Fig. 3 shows the comparison of the Static pressure. The calculated static pressure gives rather good approximation of the measured one.

Therefore it can be said that the

assumption of linear treatment has been verified on the control surface.

In the routine practice of wake survey, a comb-type wake

traverse equipment is used with 20 static head tubes and 20 total head tubes as shown in Fig. 4. The domain of traverse is 3 meters wide and 1 meter deep. To determine

upper boundary of the region of viscous wake, wave heights are measured by means of a wave height probe.

4.

Experiment

4. 1 Models used

Both measurements of viscous resistance and wave

resis-tance were performed on 6 models with 12 conditions in total. The particulars of models are tabulated in Table 1.

M. 1719 and M. 1720 are geosims of Wigle"s parabolic

M. 1720 measured at y=2Pm 3cm

2

I 5m

A

iOm1

_4

15ni 20m 25n,

Measured wave pattern Fitted wave pattern (Keel-line distribution) 5cm it 4 3 I M. 1870 measured at y=1.5m 2 Sm, -1Pm

A A

ismA A

1V

V

M. 1862 measured at y=3.Om,

£

wvwvy

.IornA A

50 x io- 2rS(0)±C(0)cos8/L 5.0 2.5

Ship form Geosims of tanker

model with 8 and 5 meters in length respectively and the

form is expressed by

y=2B(x/L)(1-x/L)[1-(z/d)°J, L/Bi0, d/L0.0625

They are similar to the model used by Lackenby (11) at N. P. L. irs the study on ship resistance components.

M. 1870 is a cargo liner model of C6=0. 73 with 4.2

Table 1 Particulars of models used

12.5X 10°

2Pm 25m

M. 1719 M. 1720 M. 1870 M. 1870 M. 1870 M. 1870 Load cond. designed designed full 1/2 DW full 1/2 DW Lpp m 8.000 5.000 4.200 4.200 LWL m 7.984 4.990 4.305 4.152 B mm 800.00 500.00 586.90 586.90 d mm 500.00 312.50 257.54 178.52 V m3 1422.22 347.22 465.25 306.76 S m2 9.408 3.675 3.655 2.964 Roughened model Co 0.4444 0.4444 0.7329 0.6971 C 0.6667 0.6667 0.7432 0.7113 Cm 0.6667 0. 6667 0.9861 0.9779 Trim 0 0 0 1%aft Watertemrat 13.0°C 15.3°C 22.6°C 27.2°C Ship form Wig1eys model Cargo linermodel

M. 1862 M. 1862 M.1715A M.1715A M. 1483 M. 1483 Load coed, full ballast full ballast full ballast

Lpp m 4.200 4.200 7.000 7.000 10.000 10.000 LWL m 4.265 4. 196 7. 147 6.891 10.203 9.845 B mm 802.70 602.70 1004.52 1004.S2 1435.02 1 435.02 a mm 233.82 115.51 373.03 192.51 532.91 275.02 V m3 452.52 224.39 2 095.0 1 038.9 6 107.91 3028.72 S m2 3.739 2.787 10.381 7.743 21. 186 15.802 Co 0.7984 0.7670 Cp 0.8050 0.7793 Cm 0.9918 0.9842 Trim 0 2%aft Watertp.at 27.8°C 27. 9°C 15. 0°C 19. 2°C 24. 6°C 24. 3°C 20 40 60 80 19

meters in length. This model was tested also in a

roughen-ed surface condition after the test in a usual smooth

surface condition, in order to obtain quantitative information on wave-wake interaction.

M. 1862, M. 1715-A and M. 1483 are geosims of a tanker of Ct = 0. 80 with 4. 2, 7 and 10 meters in length respectively. Wave measurement of 10 meter model was not earned out.

4. 2 Test results

To present the results of experiments, the following dimen-sionless coefficients are introduced

C =R/ pU°V

total resistance measured by dynamo-meter

Cw==Rw/pU2V : wave resistance derived by wave

analysis

Cv=Rv/ pU°V1 :. viscous resistance derived by wake

survey

F = U/i/Lj

: Froude number

where is the volumetric displacement, LWL the length of a water line.

Temperature correction is done on the measured total

resistance by use of 1957 ITTC friction line so as to compare it with the sum of viscous resistance - and wave resistance at the same temperature.

Since the wake surveys of models smaller than 5 meter length are conducted at the smaller tank with 120 meters in length and 6. 1 meter in width and models longer than 7 meter length were tested at the larger tank with 165 meters in length and 12. 5 meters in width, the blockage correctiOn (10) was made on both C and Cv values. However, Cw was not corrected for blockage effect, since all wave.

measurements were conducted in the larger tank.

In Fig. 5 are show'n three examples of comparison of

measured wave patterns and those which are approximated

by the series of (31). The measured wave patterns are shown by solid lines and the fitted patterns by chain line. Fitted wave pattern of Wigley's parabolic model M 1720 gives good approximation Of measured one.

The first 4 meters

of the fitted wave pattern of a cargo liner model M. 1870, do not give a good approximation of the measured one. A

(H,-H) 20mm Water head 75 100 150 200 250 400 450 500 550

broken line shows the wave patterns fitted by distributing doublets not only on the keel line but on 'the other two lines on the bottom. Improvement of fitting and slight

change of the amplitude function are observed. Wave

pat-tern of a tanker model M. 1862 was measured at 3 meters distance from the model center line and analysed by the

three-lines distribution of doublets as the cargo liner model M. 1810.

Figs. 6 through 10 show the wake survey results. Figs. 6

and 7 show the distribution of total head loss (H0I1) of

8 meter Wigley's model. The distributions show a usual

bell-shaped pattern at both lower and' higher Froude num-ers. 0 Depth mm _.o___.-o 50 75 100 1 50

....

. ,200 250 300 350 400 -

0'0 0

Q 0 0450 500 550

o----0---0-0-O-0oo0°oOtr00

600 f.-- Model .breadth -1.0 -0.5 0 0.5 1.0 p m

Fig. 7 Head loss distributions of Wigley's

model M. 1719, FñO.27

1 20mm Water head (H,-H) 20mm Water head (H,-H) Depth mm 25 50 1 00 1 50 200 250 300 350 400 450 500 550 600 700

Fig. 6 Head loss distributions of Wigley's parabolic Fig. 8 Head loss distributions of a tanker model in

model M. 1719, Fn=O.17 ballast condition, Lpp=7m, Fn=O.13

1 .0 0,5 m 600 Model breadth -1.0 -0.5 0 p 0 0 0 0

0000e

QQOOO 0

0'0

0 -H 0.5 F- Model breadth 01 m 1.0 1.5 -'1.5

-'1.0

-05

(F10-H) 20mm Water head 0 0 0 0 0 0 0 O-C----0---0---O--0-0 0 0 0 0 0

00000

00 0 a 0 0 50 100 -0-0----0---0----0----0---0 1 50 0 -0-0 0 a 0 -0400 a Q 0450 500 550 0600 0 700 0 0 00-0-0 0 0

Table 4 M. 1870 wake survey and wave

analysis results ; Full load

Table 6 M. 1862 wake survey and wave

analysis results ; Full load

Depth mm D-0 25 200 250 -0 300 0 350 -1.5 (11,-H) 20mm ] Water head 0 .á-0----0---G 0 (1 0 0 e 0

00000000

H -1.0 -0.5 Model breadth _H

e0 Q-0 a

150 -°---0-0---°----0---0---°----0- 200

-00 0

0 0 0 0 250 300 350 400 450 500 550 600 700 0 0 0 0 0 000000-0_0_0--0--_0--__0 0 0 0 -0--0.- -0----0- 0 0

Fig. 10 Head loss distributions of a tanker model in ballast condition, Fn = 0.24

Table 3 M. 1720 wake survey and wave analysis results

Table 5 M. 1870 wake survey and wave analysis results; 1/2 DW load

bers.

Figs. 8, 9 and 10 show the head loss distributiOns of 7 meter tanker model in ballast condition. The head loss distribution at Fm=0. 13 (Fig. 8) shows the bell-shaped

pattern as Wigley's model. As Froude number increases (Figs. 9 and 10), however, unfamiliar head loss zones appear near the free surface and outside the usual wake belt. The side peaks are related with finding of a new resistance com-ponent which is described in the following section 5.

The viscous resistance is calculated by the integration of

7 Fm Cv Fn Cw Cw 0.077 0.02940 rou6hened 0. 108 0.02961 surface 0.160 0.02751 0.200 0.02660 0.200 0.00056 0.00056 0.223 0.02682 0.223 0.00170 0.00111 0.249 0.03001 0.249 0.00388 0.00330 0.267 0.03232 0.266 0.00735 0.00577 0.290 0.03597 0.274 0.00925 0.00836 0.289 0.01944 0.01673 Fn Cv Fm Cw Cw 0.079 0. 110 0.03357 0.03164 roughened surface 0. 162 0.02618 0. 170 0.304 0.02424 0.204 0.00106 0.00075 0.227 0.02519 0.228 0.00192 0.00166 0.253 0.02653 0.255 0.00401 0.00338 0.271 0.02816 0.272 0.00751 0.00634 0.295 0.03154 0.280 0.01037 0.00924 0.295 0.01569 0.01626 Fn Cv Fm Cw 0. 130 0.02748 0. 150 0.00017 0. 179 0.02675 0.173 0.00036 0.200 0.02832 0. 180 0.00038 0.220 .0.02955 0.200 0.00066 0.210 0.00174 0.220 0.00357 -0-C--Q 0 0 0 0 -0---0---0--0 0 -oo 0 0 0 0 0 0 e 0 0 0 Fn Cv Fm Cw 0.071 0.03165 0. 110 0.02838 0. 170 0.02770 0. 170 0.00102 0.222 0. 02008 0.223 0.00309 0.249 0.02622 0.250 0. 00500 0.266 0.02535 0. 267 0.00580 0.289 0. 02501 0.289 0. 00813 0.321 0.02464 0.320 0.01043 0.354 0. 02338 0.354 0.00989 0.400 0.02540 0.397 0.01538 - Model breadth -1.5 -1.0 -0.5 0 0,5 1.0 1.5 if 01

Fig. 9 Head loss distributions of a tanker model

in ballast condition, Fn=0.20

Table 2 M. 1719 wake survey and wave analysis results

Fm Cv Fm Cw 0.070 0.03307 0. 108 0.02448 0. 169 0.02447 0. 170 0. 00139 0.222 0.02400 0.223 0.00311 0.249 0. 02422 0.250 0. 00583 0.267 0.02391 0.267 0. 00507 0.288 0. 02459 0.289 0.00902, 0.319 0.02354 0.320 0.01240 0 0 0 0 0 0 00 0 0 0 0 0 00O-25 50 mm 100 0 0.5 1.0 1.5 if 01

C

2 0.06

0.06

Table 7 M. 1862 wake survey and wave analysis

results Ballast load

Table 9 M. 1715-A wake survey and wave

analysis results ; Ballast load

ca-C, Hughes - C-

---0=-u--I-

-O--- ' 0.10 0.14 0.18 0.22 0.26 0.30 F,

Fig. 11 Resistance components of Wigley's parabolic

model, Lpp8m, G5=0.44

0.34 - C of a roughe?

-C Cv±Cu --8

C/

.4-Cw of a roughened model 0.10 0.14 0.18 0.2 0.26 0. 0 0.34 F,

Fig. 14 Resistdnce components of a cargo liner model in

1/2 DWloadcondition, Lpp4.2m, C0=O.79

Fig. 12 Resistance components of Wigley's parabolic model, Lpp=5rn, C5=0.44

7

Table 8 M. 1715-A wake survey and wave

analysis results ; Full load

Table 10 M. 1483 wake survey results

C 0aroughefle---odel ---Cw±C1 -0--c'. TIC. / Cu. of a ioughened model C-7-5-I I

the head losses with planimeter. The non-dimensional values

of v and Cw are tabulated in Tables 2 through 10. Figs. 11 through 20 show the results of direct measure-ment of viscous resistance and wave resistance. Total

resistance coefficient C1 measured by resistance dynarnonieter is shown by solid lines. The sum of viscous resistance and wave resistance is shown by black circles.

It is evident from these results that the sum of viscous resistance and wave resistance agrees well with the total resistane measured by dynamorneter not only for the fine models (e. g. Fig. 11) but also for the fuller models (e; g.

4 0 :< 2 Fu Cv Fn Cw 0. 100 0.02671 0. 131 0.02658 0. 157 0.02665 0. 155 0.00037 0.184 0.02557 0. 180 0.00032 0.202 0.02738 0.200 0.00077 0.212 0.02825 0.210 0. 001.59 0.222 0.02904 0.220 0.00351 Fn Cv Fn Cw 0. 131 ' 0.03302 0. 183 0.00061 0. 182 0.03568 0.203 0.00052 0.223 0.03843 0.214 0.00061 0.224 0.00093 Fn Cv Fu Cw 0.099 0.03024 0102 0.00003 0. 129 0.02969 0. 157 0.03030 0. 153 0.00009 0. 179 0.03137 0.198 0.03229 0.203 0.00061 0.220 0.03489 0.230 0.00186 0.238 0.03631

Full load Ballast load

Fu Cv Cv 0. 130 0.02152 -0. 132 0.02825 0. 180 0.02235 0. 182 0.03049 0.201 0.02321 0.224 0.03280 Cu ±C 2

00-'P-I I' -0 F'''

_-u__

I r I 0.06 0.10 0.14 0.18 0.22' 0.26 0:30 0.34 0.38 0.06 0:10 0.14 0.18' 0.22 .26 0.30 0.34 F,

Fig. 13 Resistance components of a cargo liner mold in

full load condition, Lpp=4.2 m, C6 =0.73

0.10 0.14 0.18 0.22

0.06 0.42

Resistance components of a tanker model in full load condition,

Lpp='4.2m, C5=0.80 Fig. 15 6 5 4 0 x 2 6 5 4 C

Fig. 16 Resistance components of a tanker model in ballast condition, Lpp-4.2 rn, C6-O.77

4 0

U2

4

I

3 0 X2 I.T.T.C. C,.

Fig. 18 Resistance components of a tanker modelin ballast condition, Lpp 7 in, C1 = 0.77

r.

I.T.T.C.

-0.06 0.10 0.14 0.18 0.22 0.26

F,

Fig. 20 Resistance components of a tanker model in ballast condition, Lpp=10 m, c5=O.77

Fig. 17). These. experimental results show the validity of the theoretical analysis in the section 2. 2. The total

resis-tance can be divided into the resisresis-tance components by wave analysis and by wake survey.

Viscous resistance curves of Wigley's model have the same tendency as the 1957 ITTC friction line with respect to

Froude number (Figs. 11 and 12). This experimental result also supports the validity of the theoretical result ; Cv=C1 for the ship with small breadth/length ratio (cf. eq. (27)).

On the other hand the viscous resistance curves of the fuller models (e. g. Fig. 13 or 17) tends to increase in accord-ance with increase of wave resistaccord-ance. Especially in ballast condition of tanker models, Cv begins to increase at low

Froude number (e. g. Fig. 18). More detailed discussion on the increase of viscous resistance curves is described in the section 5.

4. 3 Wave analysis of a roughened model

In order to obtain quantitative information on the wave-wake interaction, wave analysis and towing tests of a

rough-E E C + C Cr 4 3 0 -

----'JL

-010 0.14 0.18 0.22

Fig. 19 Resistance components of a tanker model in full load condition, Lpp==10 rn, Cb=0.80

Fig. 21 Scraper for roughening a model

I"

Fig. 22 A roughened model of a cargo lines

ened model were carried out by use of 4.2 meter cargo liner model M. 1870.

The whole .surface of the model was roughened by scrap.

ing with a saw-shaped steel plate along the frame lines. The shape of the scraper is shown in Fig. 21. Fig. 22 shows the roughened model.

Towing test of the roughened model shows that the total

resistance measured by dynamometer increases about 50 percent as shown in Figs. 13 and 14. Fig. 23 shows fit

9 0.06 0.10 0.14 0.18 0.22 0.26 Cw F, TTc

-Hughes c-I.T,T,C, 2 5 4

0.06 0.10 0.14 0.18 0.22 0.26 _{Fig. 17 Resistance components of a tanker model in}

F, _{full load condition, Lpp=7 nz C1= 0.80}

006 0.10 0.14 0.18 0.22 0.26

Smooth surface

example of wave analysis results of the roughened model. The wave resistance of the roughened model is slightly smal-ler than that of the smooth model. The difference of wave resistance, however, is negligibly small compared with the increase of total resistance (Fig. 13). This implies that the effect of viscous wake on the potential flow is rather small. To summarize the above experimental results, it is con-cluded that the total resistance can be divided into two resistance componets, i. e. viscous resistance and wave resis-tance. The interaction between the two components can be

neglected as a higher order quantity. And the present

method of theoretical analysis is successfully supported by experirlierts.

5.

Finding Of a New Resistance Component

In the section 4 it has been observed that the viscous

resistance curves of fine ships (e. g. Wigley's model) agree approximately with the ITTC friction line (Fig. 11 or 12).

On the other hand the viscous resistance curves of fuller ships have different tendency with respect to Froude number; the viscous resistance derived by wake survey increases in accordance with increase of wave resistance (Fig. 13 or 17). Especially in ballast condition of a tanker model it begins to increase at comparatively low Froude number (Fig. 18). This is against the conventional idea that .the viscous ance is approximately proportional to the frictional resist-ance of flat plates and independent of Froude number.,

5. 1 Comparison of head loss distribution

In order to find a reason for the increase of viscous

re-sistance with respect to Froude number, the distributions of head loss (Ho - H) in the viscous wake are compared.

The distributions of Wigley's 8 meter model at Fm = 0. 17

and Fm 0. 27 are shown in Figs. 6 and 7 respectvely. It has been already pointed out that the distributions show

bell-shaped pattern.

On the other hand the distributions of 7 meter tanker model in ballast condition have different tendency. As Froude number increases unfamiliar head loss zones appear near the

free surface and outside the usual frictional wake belt as

shown in Figs. 9 and 10.

The side peaks of head loss curves are located at about

the same distance as the breadth of thodel from the center

line.

They seem symmetrical to the center line of the model and immediately disappear as the :depth increases. The

usual frictional wake zone (the central peak of head loss

curves), however, does not show such an immediate change with the depth.

From these comparison of head loss, the author convinced

that there is a close relationship between the increase of

viscous resistance and the side peaks of head loss curves.

F,= 0.249

Fig. 23 Wave analysis of a roughened model

5. 2 Definition of the new component

In order to calculate the component of resistance attributed

to the side peaks of head loss curve, the author splits the zone of viscous wake into two parts as shown in Fig; 24. The one is the zone denoted by we, which is regarded as

the usual frictictional wake zone.

The other the zone of

the side peaks denoted by wj.

By the separation of the wake zone, the asymptotic expres-sion of viscous resistance (cf. eq. (41))

RvpgJ (HoH) ds

(43)

can be expressed by

R=Rv0+Rv1

()

where Rvo is the resistance component due to the central

peak of head loss curve;

Rve'p

(HoH) ds

(45)

and Rvo is the component due to the side peaks;

RvipgJ (HoH) ds

(46)

The new component of resistance is defined by the integral (46).

5. 3 Separation of viscous resistance

Denoting the dimensionless quantities Rve/ pT]° V by

Wi

x

15m 20m

Usual frictional wake zone

Fig. 24 Division of wake zone

5

4

Wake zone relating to the new component

Fig. 25 Separation ofviscous resistance of a tanker model M. 1715-A in ballast condition

M. 1870 measured aty=1.5m

A

5mj

A

w

_w

(0

0

AmptltUde functions derived by wave analysis

-5.0X104 I I I +25_ Roughened surface . Co 20 40 60 80 0 0.26 0.10 0.14 0.18 0.22 F, 0.06

0 L)

I

- Cw+Cv

C

Fig. 26 Separation of viscous resistance oj a tanker model M. 1715-A in full load condition

F,= 0.184 1 20mm Water head (H0 -H) Depth Model breadth _-H -1.0 -0.5 0 0.5 1'O S m

Fig. 27 Head loss distributions of a tanker model in full load condition

Cvo and Rvi/ pU2 by Cvi respectively, Ct'o, Cvi and Cw of 7 meter tanker model in ballast condition are plotted in Fig. 25.

The sum of Cw, Cvo and Cv is in good agreement with C, the total resistance coefficient measured by dynamometer. Tendency of Cvo curve agrees approximately with that of ITTC friction line or the curve derived by Hughes' method

and shows the same tendency as the viscous resistance curves of Wigley's model (Fig. 11).

On the other hand the new component of viscous

resist-ance Cvi begins to increase at low Froude number (F= 0. 15) and increases with the increase of Froude number.

The magnitude of Cvi

is much greater than that of cw

25mm 50 100 150 25mm 50 100 150 25mm 50 100 150 200

Table 11 Resistance components of tanker models in ballast load condition and form factors

Cv°Rv/IpU2VI, CwRw/pU2Vf. Fn=UI -/9LWL.

Rn° ULWL/e 15°C temp.

Table 12 Resistance components of a tanker model M. 1715-A in full load condition

derived from the measured wave patterns.

Fig. 26 shows another example of the separation of viscous resistance in full load condition of the same model M.

1715-A. In this case the new component begins to increase at F,,=O. 18. Fig. 27 shows the head loss distributions in full load condition. _{The distributions near the free surface are} compared in various Froude numbers. The side peaks ap.

pear and increase with Froude number.

The viscous resistance Cvo also agrees approximately with the viscous resistance by Hughes' method. The values of

Cvo, Cvj and Cw are tabulated in Tables 11 and 12. From these examples it may be concluded that one of the reasons for the increase of viscous resistance with respect

to Froude number is attributed to the new component

defined by the integral (46). In ballast condition of fuller

ships the new component begins to increase even at a low

Froude number and occupies most of the so-called wave

resistance calculated by Hughes' method.

Flow Traverse by 5-Hole Pitot Tube

To make clear the source of the new component the flow traverse tests are conducted by a 5-hole pitot tube along the side of the 7 meter tanker model in ballast condition.

Three control surfaces are set perpendicular to the uni form stream at the entrance part (square station 8. 5), the fore shoulder part (s. s. 7) and the aft shoulder part (s. s. 3) respectively.

6. 1 Traverse results

By use of the equipment shown in Fig. 28, the flow field

was measured at 7 point breadthwise and 3 points depth. wise;

The domain of traverse is about 0. 6 meter wide and 0..1 meter deep. Measured head loss near the free surface at

each control surface is shown in Fig. 29. It is observed that the head loss generated at the entrance part leads to

11

Fn Rn Cv Cvo Cvi K°Cvof Cw

0.131 3034106 0.03463 0.03463 0. 0.377 0. 182 4. 185 0.08714 0.03427 0.00287 0.450 0.00052 0.223 5. 133 0.03981 0.03326 0.00655 0.462 0.00093 0. 129 6.388 0.03016 0.03016 0.0 0.381 0.000090 0.1.57 7.822 0.03074 0.03015 0.00059 0.432 153) 0.179 8.841 0.03180 0.02988 0.00192 0.451 0.000610 0. 198 9.825 0.03271 0.02995 0.00276 0.430 °0. 203) 0.220 10.932 0.03530 0.02975 0.00555 0.498 0.00186 0.238 11.851 0.03670 0.02992 0.00678 0.528 _=0.230) 0. 132 11. 149 0.02910 0.02910 0. 0.470 'eas 0. 182 15.435 0.03127 0.02898 0.00229 0.548 ured 0.224 18.918 0.03354 0.02814 0.00540 0.555 F,, Cv Cvo Cvi Fn Cw 0. 100 0.02671 0.02671 0 0. 131 0.02658 0.02658 0. 0. 157 - -0. 02665 0.02665 0. 0. 155 0.00037 0. 184 0.02557 0.02557 0. 0. 180 0 00032 0.202 0.02738 0.02561 0.00177 0.200 0.00077 0.212 0.02825 0.02551 0.00274 0.210 0.00159 0.222 0.02904 0.02553 0.00351 0.220 0.00351 0.06 0.10 0.14 0.18 0.22 0.26

F.P.

Head loss due to turbuIeit motion at bow

(HH) ata=-5Omm o.o 0.05 0.10 * Estimated. 0 effective breadth o:i 0 - 50

75

S.S. 8.50.15 - 50mm 0.05 0 p. to 0.05 mm 0.254 0.05 z--125mm 0 tOO 300 500mm

Fig. 28 An equipment of 5-hole pitot tube traverse

the one which appears outside the usual wake belt at the

control surface of 0. 5 Lpp behind A.. P. Measured head loss disappears immediately as the depth increases.

Fig. 30 shows a turbulent motion which has been caused

by the breakdown of waves at the bow of the modeL It is

M.1 715.A Water Lime in 2% aft trim condition

(H0H(at z=. 25mm

- 100mm

Distance from the side wall of model (500mm from t)

0+ 20 40 60 80 mm mm

-

i2/pg//m(Ho_H)dyd: 1.049kg rn/S - 44.4mm S.S.7 0=-2Ornm z= 50mm .Jfu(HoH)ds 0+ 20 40 pgffu(H0 H) dydz= 1.222kg.rn/s =-7.1mm mm 150 (H0H)at=-25mm tomm 0.10 (1

50

S.S. 3 z=-25m - 50mm

50

pgffu(H0H) dydz= 1.2306gm/s 100 (OOo64mrn

Fig. 30 Breaking wave at the bow of a tanker model in ballast condition, Fn=O.24

supposed that the head loss is generated by such strong

turbulent motion.

6. 2 Calculation of kinetic energy loss

Calculation of dissipation of energy due to the turbulence was tried on the basis of calculating the kinetic energy loss across the control surface. The rate of kinetic energy loss per unit time ia obtained by

pJu(Ho_H) ds

(47)

where u is the velocity component normal to the control surfaces, wi the sectional area of the wake related to the

new component of viscous resistance. The process of

in-mm

Fig 29 Flow traverse by 5-hole pitot tube 200

A.P. 0.10 0 0.05 0.05 0 0 - 0.5L31 Head loss due to turbulemt motion at bow

(HorH(ata=-25nm

pg uff(H0H)dydz= 1.2306gm/s

,= 7.0mm

Head loss due to friction on the hull surface

0.05 0.5L,,. from A.P. z= 25mm 50mm - 100mm - 150mm iflJu(Ho_H(dy 20 40mm +5_. 2 40.50mm 0.10 _100mm 200 400 600mm 200 400 600 S

U: Velocity cdmponkmt perpendicular to the comtrol surface H0 Total head of uniform flow (in mm water)

H; Total head in wake. Model speed U= 1.973/o, F,= 0.237

z0: Undisturbed frek surface, w; Height of water surface

0

0 200 400 600 800 1000mm

- Distance frem the side wall of model

tegration is shown in Fig: 29. Heights of the free surface are estimated from the measured Static pressure near the surface.

At the control surface of 0. 5 Lpp behind A. P.,

neglect-ing the higher order terms of velocity components, the

kinetic energy loss is calculated by the asymptotic formula

PPUS (HoH)ds

(48)

Energy 'loss measured at the control surfaces s. s. 7, s.s.

3 and 0. 5 Lpp behind A. P. shows good coincidence, i. e. 1. 222 kgm/s, 1. 230 kgm/s and 1. 230 kgm/s respectively. Energy loss measured at s. s. 8.5, however, is 1.049 kg rn/s and about 15 percent less than the values of other control surfaces. This may be due to the shoulder wave breaking which is observed between the control surfaces S. s. 8. 5 and S-s. 7.

To summarize the above experimental investigation it may be concluded that the loss of total head relating to the new component of resistance has its origin -at the entrance part

of a ship and the most of it is attributed to the turbulent

motion generated by the breakdown of waves at the bow. 7.

Theoretical Interpretation on the New

Resistance Component

7. 1 Basic expression of dissipation of energy

In order to obtain a theoretical interpretation of the new

component, the basic expression of dissipation of energy

due to internal friction is derived first.

Applying the conservation principle of mass, momentum and enetgy to incompressible, viscous and heat conducting fluid, following Oquations are written

--pdv=O

- (49)

p q= ± pF

_pcTwI+KpzT

where P the stress tensor, c the specific heat, s the coef-ficient of heat conduction, T the absolute temperature, and WI is the dissipation function (

w=2 p e2 , e,-,=

(

a

+ aq,_{) =esr} (52)

f.S, 2. 3 ax3 dXr

After scalar multiplication by q, the equation of motion (50) gives the relation between the rate of dissipation of

energy and the kinetic and potential energy

wcfr = [_J(pn)qa's_

1(nXw) qds]

_4q2dv__pgzdv

₍₅₃₎

where S is the surface of fluid of volume V. The formula

(-53) indicates that the rate of dissipation of energy is the difference between the rate of work done by stress on the

fluid surface and the rate of change of kinetic and potential energy.

In the fluid V within a fixed closed surface S, there holds a relation

S WI d =$w2 dv

where w2 i the vorticity squared.

7. 2 One-dimensional mathematical model

At the bow of fuller ships churning-up and breaking-down of the fluid surface is often observed as shown in Fig. 30.

A rough sketch of the breakdown of waves at the bow

-is shown in Fig. 31. Complete mathematical presentation of the phenomenon will be very difficult. Although in this

case the water itself is not shallow, this phenomenon is (54)

P0

Fig. 31 Sketch of breaking waves at the bow of ship

Fig. 32 One-dimensional model of surface discontinuity

very similar to the hydraulic jump in the case of shallow water. On the other hand the results of flow traverse by a 5-hole pitot tube showed that only the water near the

free surface is relating to the turbulent motion due to wave

breaking (Fig. 29). Therefore the author assumes that there exists an apparent critical depth of water on which

the breakdown of waves ocëürs.

Then he trys to apply the shallow water theory to the wave breaking phenomenon and introduce a very simple

mathematical model of the phenomenon.

Taking the (xz) plane of a right-handed rectangular

coordinate system with x-axis along the direction of uniform flow and z-axis with its positive sense vertically upwards, consider a region made up of the water lying between two

vertical planes x=a0 (t), and x=a, (t) with al>ao. These

planes always contain the same water particles. Let us sup-pose there appears a finite discontinuity in the surface

eleva-tion at a point x=e (t) within the column of water bet-ween x=ao (t) and x=a, (t),

as shown in Fig. 32.

The apparent critical depths h0 and h1 are not predetermined but determined by the condition in which the surface disconti-nuity is created.

Following the assumptions of the shallow water theory

it is assumed that

( i ) -the horizontal velocity component u is Constant

throughout any vertical plane.

(ii)

z-component of the acceleration of the water has a negligible effect on the pressure p.

On this assumption the pressurep is given as in hydrostatics

by p=pg(wz) and p=0 on the free surface.

Applying the basic equations (49), (50) and (53) to the simple one-dimensional model (Fig. 32), the following equa-tions are obtained, since dvCw dx,

d ai(t)

-3-

pwax=0

(55)

(At _ao(t)

-d a,(t) h3 h1

__5dt ao(t)

pCwudx=5 Podz_5 ptdzpgh

0 0 2

-(56)

J= ----

dSal(t)(

(pCu,-+

U° pg

, dx+

) hopouodz

dt ao(t) 2 2

-r u ci,..

(57) where J denotes the rate of dissipation of energy.

Following Stoker(12) the limit case is considered in which

the length of the column tends to zero in such a way that

the discontinuity inside the column. Then the relations (55) and (56) become

phi (ui-e) = pho (uo-)

(58)

phiui (ui_E)_phouo(4o_e)+pg(h_hi)

(9)

Using the relations (58) and (59) the expression of energy becomes

(h1-ho)3

J=pgho(uo-)

(60)

4,1 0

where e is the velocity at a point x=E (t) and pho (uo-E) denotes the mass flowing across the sin-face of discontinuity. Since the dissipation of energy due to internal friction has the nature of irreversible change, J must be always positive. Therefore, if there is the surface of discontinuity, h1 is greater than h0. If h1=ho, no dissipation of energy occurs. The relation (60) also indicates, in conjunction with (54), that vorticity is generated at the surface of dis-continuity.

The generation of vorticity yields loss of total head and contributes to the increase of water temperature by the

equation of conservation of energy (51).

In steady motion the following relation is obtained from and (59)

- /

fh1\(hi+ho)

2

-If the height of surface discontinuity is given, since is

identical with the ship speed, the critical depths h0 and h1 can be determined -by (61) and it becomes possible to calculate the dissipation of energy due to the surface of discontinuity by (60).

7. 3 Comparison of theory and experiment

In section 6. 2 the kinetic energy loss was measured on

the control surfaces along a ship mOdel by use of a 5-hole pitot tube.

It is possible to make a rough comparison of the

experi-mental results with estimated energy loss from the height

of surface discontinuity by the elations (61) and (60).

First the height of surface discontinuity was taken from

the photograph of breaking waves at the bow of the 7

meter tanker model. Froth Fig. 30 the height of surface

discontinuity (hi-ho) is taken, i.e. 0.131 metets at the ship speed E= -1.973 rn/s. From equatiOn (61) two solutions of

h0, i. e. h0=O.137 meters and 0.063 meters are obtained

respectively.. Remembering the assumptions in section 7. 2,

since the draught of the model at the bow is 0.136 meters in depth ho=0 137 meters is chosen as the solution Esti mated rate of dissipation of energy per unit breadth is J=

4i33 kg/s. In the calculation following values are used

U0=O, e =-1.973m/s, q=9.8 rn/s° and p=101.97kgs2/rn4.

On the other hand the measured rate of dissipation of

energy at the control surface of s. s. 8. 5 is 1.049 kg rn/s as indicated in Fig. 29.

Dividing the measured rate of dissipation of energy by the estimated rate of dissipation of energy per unit breadth, the effective breadth of surface of discontinuity can be obtained;

The effective breadth of surface of discontinuity

= -0.254 (meter)

-(61)

The derived effective breadth of surface of discontinuity should be compared with the breadth of the actual wake at

the control surface of s. s. 8.5 for examining whether the estimated breadth gives approximate breadth of the measured wake zone.

From Fig. 29 it is observed that the estimated breadth

(0.254 meters) is comparable with the breadth of measured

wake zone at s s 85 On the other hand smaller value of h0, i.e. h0=0.063 meters gives 0.080 meters of the effective breadth. Evidently this is too small.

To summarize the above comparison, the one-dimensional

model of the breaking waves gives approximate value of dissipation of energy and provides -a basis of analytical treatment of the phenomenon.

7 4

Scale effect on the new component

From the theoretical approach to the bow wave breaking,

an important suggestion on the scale effect is obtained; Although the new component of viscous resistance is caught

by means of the wake survey, the theory implies by the

relation (60) that the component is associated with gravity

and follows Froude law of similitude. To examine the

validity of this consideration, the values of the new comp-onent obtained by the wake survey are compared in respect of geosims of tanker.

The geosims are 4.2 meters, 7 meters and 10 meters in

length respectively, whose particulars are given in Table 1. Comparison of Cvo (the viscous resistance coefficient due to the usual frictional wake) and Cvi (the new component) of three geosims in ballast condition are shown in Fig. 33. The values of Co and Cvi are tabulated in Table 11. The scale effects on Cvo are clearly observed among three geosims. On the other hand Cvi, the coefficient of the new component,

shows rather good coincidence. This experimental result

supports the theoretical prediction. The wave resistance of 7 meter model is in good agreement with that of 4.2 meter

model.

Next the form factor K is calculated by means of

divid-ing Cro by Hughes' frictional resistance coefficient. Fig. 34 shows the form factors of three geosims. Form factors of

o iOrn model a a 7m model - .04.2m model L,,.4.2m .C-, 7m 7m _{ITTC fnction} line 1 On, Cr

Fig. 33 Scale effect of the new component

0.7 0.6 a 0.5 _a a 0 0.4 0.3 0.2 _{o 4.2m model} C's A 7m model 0.1 O lOm model 0.12 0.i4 0.16 0.18 0.20 0.22 0.245 F,

Fig. 34 Form factor of gebsim models

4 3 0

x2

j

0.10 0.14 0.18 0.22 0.26 F,

4.2 meter model and 7 meter model are in good coincidence. The form factor of 10 meter model, however, gives slightly

larger values than those of the other two models. The

reason for this difference is not yet clear.

However, the form factors of the three models have the

same tendency with respect to Froude number. They show gradual increase in accordance with the increase of Froude

number. This implies that in the usual wake zone there still exist some effects of Froude number which are presuma-bly due to the effect of changing trim and sinkage. 8.

Conclusions

From theoretical and experimental studies on the separation Of ship resistance components, the following results are obtained.

The sum of the wave resistance and viscous resistance

derived by the present method of analysis is in good agreement with the total resistance measured by a

clyna-mometer.

The wave resistance component is practically independ-ent of viscous resistance, i. e. the wave resistance of the

smooth model and the roughened model is practically equal.

The viscous resistance of the finer ship model is in

fairly good agreement with the calculated viscous resist-ance by Hughes' method including form effect.

However, the viscous resistance of the fullet ship

models, especially in ballast condition, is greater than the

calculated viscous resistance at higher speeds in which wave resistance becomes appreciable.

M. P. Tulin: The Separation of Viscous Drag and Wave Drag By Means of The Wake Survey, DTMB Report 772, (1951)

K. Eggers: tlber die Ermittlung des Welienwiderstandes eines Shiffs. modells druch Analyse semes Wellensystems, Shiffstechnik Bd. 9-Heft 46. (1962)

S. D. Sharma: An attempted Application of Wave Analysis Technique to Achieve Bow Wave ReductiOn, 6th Naval Hydrodynamics Sympo-sium .(1966)

K. T. S. Tzou and L. Landweber: Determination of the Viscous Drag of a Ship Model, IIHR Report No. 101, (1967)

S. D. Sharma: Zur Problematik der Aufteilung das Shiffswiderstandes in Zähigkeits und Wellenbedingte Anteile, J. S. G. 59 Band. (1965) K. Taniguchi, T. Fujita aOd E. Baba: Study on the Separation of the

Derivation of the formulae (6) and (11)

The formula (5) in the text shows that the resultant force R acting on a body can be expressed by two ways, viz, by

the integral of the stress over the surface of the body and

by the integral over a surface enclosing the body, thus

d.c

(A-i)

=Jp+$ Cn p(nq)q)as

(A-2)

Substituting nø= +pn+2p(ng)q+pnXw

in (A-i) and

(A-2), we obtain

R=-pn+2p(n)q+1inxw)ds

(A-3)

(-pn±2i(n)q+nxw- p(nq)qJds

(A-4)

Since q=0 in the incompressible duid, the following

rela-References

Appendix

tion is obtained.

(nv)q+nxw=(nv)qn(vq)+nxw=(nxv)xq

(A_5)

By Stokes' theorem we obtain

$xv)><=

dLxq

(A-6)

(nxv)xqds=fdLxq

(A-7)

where c is the intersection between the free surface and the hull surface. Since q=0 on the hull surface, the line inte-g±1s (A-6) and (A-7) vanish. Then from 3) and

(A-4) we obtain

R=S(_pn_,inxwjds

(A-8)

pntnxw p(nq)qJ d.c

(A-9) By taking the x.component of (A-8),

R1=

_{i)ds_IIS(nxw)ias}

(A-b)

The reason for the increase of viscous resistance is due to a new resitance component.

The component is generated by the expenditure of wave energy due to breakdown at the entrance (especially at the bow) of ships.

The component can be caught by the wake survey

method, i. e. by integratiOn of head loss which appears near the free surface and outside the usual frictional wake belt.

By the application of shallow water hydraulic jump

theory, the dissipation of energy due to the breakdown of waves can be approximately estimated. The theory indicates

that the new component follows Froudes' law of similitude as wave resistance;

In ballast condition of the fuller ship models, the new component corresponds approximately to the difference between the. viscous resistance by wake survey and the one calculated by Hughes' method. And the component is much greater than the wave resistance derived by the wave analysis method.

Acknowledgement

The author wishes to express his gratitude to Dr. K.

Taniguchi, Manager of Nagasaki Technical Institute,

Mitsu-bishi Heavy Industries, Ltd. and Dr. K. Watanabe, Chief

of Mitsubishi Experimental Tank of the Nagasaki Technical Institute for their continuing guidance and encouragement.

The author also wishes to express his appreciation to all members of Mitsubishi Experimental Tank who cooperated in carrying out this investigation.

Resistance Components, 11thITTC,(1966)

L. M. Milne.Thomson : Theoretical Hydrodynamics, 4 th Edition. (1960)

Birkhoff and E. H. Zarantonello: Jets, Wakes, and Cavities. Aca. dernic Press INC., (1957)

T. H. 1-tavelock: The Calculation of Wave Resistance, Pro. Roy. Soc. A vol. 144, (1934)

K. Taniguchi and K. Tamura: On the Blockaje Effect, Experimental Tank (Nagasaki), Technical Report No. 307. (1958)

Lackenby An Investigation into the Nature and Interdependence of the Components of Ship Resistance, RINA, (1965)

which is identical with the formula (6) in the text. Now let us consider the integrals (A-9). It is assumed that outside the viscous wake the fluid is irrotational, i. e.

w=O and inside the wake the fluid is rotational The fol lowing definitions of total head are introduced

H=

+z+

U2= UI; total head of the unifoiin

pg 2g 2g _flow

H= - +z+ q2

pg 2g

p 1

total head in the viscous wake The total head H differs from that of uniform flow. Denot

big the difference of the total head (Ho.-H) by HL, we

obtain the following expressiOn of pressure p

1 1

P= --pu2-

.pq2_pgz_HL

(A-li)

Substituting p in (A-4) and using the following relations

jpFdr= SI+A

(A-12)

and

J80UdS =Suds ; the continuity condition ..(A-13)

we obtain the expression of the total resistance as

x-cona-ponent of (A-9)

Ri=pgj' HLdS+e_-$ (vZ+w2_(U_u)2)ds 2 si

pgSzcos(n, i)ds_pg

zcos(n, i)ds

p$ (nxw)ids

(A-14)

Sf()

where Sf() indicates the free surface of the viscous wake.

The third and the fourth integrals can be rewritten further

i)ds_pg5zcos(A, i)ds

= Zwdy (A-15)

- 2 si

By using the dynamical boundary condition n(I) = 0 on the free surface and the

relation (A-5), the last term of (A-14)

becomes

p

J(nxw)ids=2pJ

Sf() si

--ds

(A-16)

It is assumed that this term can be neglected compared with

the other terms in (A-i4) at large distance behind a ship

Then we Obtain the required formula (Ii)

R1 = pgj (HoH)ds+ (U_u)IJ js

+--

(,dy

(A-17)