A study of wave resistance characteristics through the analysis of wave height and slope along a longitudinal track

ARdge70

K30-6

Onderafdelin *bliotheek van de . - , ,e.sbouwkunde Isc e Hogeschoo , 1 DOCUMENTAT1E : DATUM:

A Study of Wave Resistance

Characteristics

Through the Analysis of

Wave Height and Slope

Along

a

Longitudinal Track

BY

F. C. MICHELSEN

and

S. B. S. UBEROI

1_117.;;j @Ci.),[nbnligaJCf.)n.':18

DANISH TECHNICAL PRESS

iH

Hydrodynamics

Section

Report No. Hy -15

.

August 1971

Lab.

_{v. Scheepsbouwkunde}

Technische Hog eschool

IDOCUMENTATIEI

Delft

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Hy-1 PROHASKA, C. W. Analysis of Ship Model Experiments

and Prediction of Ship Performance 5,00 (Second printing)

Hy-2 PROHASKA, C. W. Trial Trip Analysis for Six Sister Ships 6,00 Hy-3 .SILOVIC, V. A Five Hole Spherical Pitot Tube for 6,00

Three Dimensional Wake Measurements

Hy-4 STROM-TEJSEN, J. The HyA ALGOL-Programme for Analysis of Open Water Propeller Test

6,00

Hy-5 ABKOWITZ, M. A. Lectures on Ship Hydrodynamics 20,00 Steering and Manoeuvrability

Hy-6 CHISLETT, M. S., and STROM-TEJSEN, J.

Planar Motion Mechanism Tests and Full-Scale Steering and

12,00

Manoeuvring Predictions for a MARINER Class Vessel

Hy-7 STROM-TEJSEN, J., and A Model Testing Technique and 12,00 CHISLETT, M. S. Method of Analysis for the Prediction

of Steering and Manoeuvring Qualities of Surface Vessels

Hy-8 CHISLETT, M. S., and BJORH EDEN, 0.

Influence of Ship Speed on the Effectiveness of a

12,00

Lateral-Thrust Unit

Hy-9 BARDARSON, H. R., WAGNER SMITT, L., and

The Effect of Rudder Configuration on Turning Ability of Trawler Forms.

20,00 CHISLETT, M. S. Model and Full-Scale tests with

special Reference to a Conversion to Purse-Seiners

Hy-10 WAGNER SMITT, L. The Reversed Spiral Test. 10,00

A Note on Beth's Spiral Test and some Unexpected Results of its Applications to Coasters

Hy-11 WM. B. MORGAN, VLADIMIR S'ILOVId, and

Propeller

Lifting-Surface Corrections

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PROHASKA, C. W. Interlocking and Overlapping Propellers

HYDRO- OG AERODYNAMISK

LABORATORIUM

Lyngby - Denmark

A Study of Wave Resistance Characteristics _Through the Analysis of Wave Height and Slope _{Along a}

Longitudinal Track

by

F. C. Michelsen1) and S. B. S. Uberoi2)

University of Michigan

Hydro- og Aerodynamisk Laboratorium

Hydrodynamics Department

August 1971

1

Introduction

The system of waves generated by a ship has a direct

relation-ship to its hull form. The purpose in analysing the wave system is therefore to determine this relationship and thereby its wave resistance

characteristics. In particular such an analysis could produce ships

of minimum resistance. We have witnessed how the bulbous bow has found widespread application in ship design in relatively few years. The

merits of such a bow have been well known since the days of D. W. Taylor's pioneering work on hull design, but it should be recognized that the rational approach to bulbous bow design was introduced by

Wigley and Weinblum and recently again by Inui. The method of approach used by Inui and his colleagues to achieve their remarkable results was to a great extent dependent upon a careful study of the wave systems

being generated by the models. The study was made of the entire wave system through a painstaking and time-consuming process of obtaining

contour maps by photogrammetrical techniques [1].

It is possible in principle to compute from theory the wave pattern generated by a ship, but at present the only practical theory available for this purpose is that of the thin ship. This linear theory is not sufficiently accurate except possibly for submerged simple forms

such as bulbs. One may therefore estimate the effect of a bow bulb by adding its theoretical wave system to the system of the parent form which has been obtained experimentally. For other more complicated

hull changes it is most convenient to investigate the effect of these by studying the differences between wave systems as measured

experi-mentally. Considering the potentials of the analysis of wave patterns it is therefore fortuitous that it has been shown to be sufficient to know the wave height and/or slope along certain paths.

It appears that the first presentation of a method of obtaining the wave-resistance of a ship from its wave profiles was made by

Eggers [2] in 1962. This publication was the result of a suggestion by Korvin-Kroukowski that all information on the wave system generated by a ship was contained in the wave profile taken along a single path

-

2-one perpendicular to the direction of motion, the other parallel to it.

These are usually referred to as transverse and longitudinal cuts

respectively.

-Eggers' longitudinal cut method was not well understood and

suffered, furthermore, from the need of a rather long record to achieve

sufficient numerical accuracy. The emphasis was therefore placed on

the transverse cut method, which also has the advantage of being more

rigorous in approach since it includes the reflections of waves from

the basin walls.

In practice it has turned out that the transverse cut method

has certain drawbacks. Firstly it is necessary to take several parallel cuts and sanial,yse these in pairs. It is then customary to take a mean

of the calculated values derived from these pairs to be the true value

of the wave-resistance. Secondly, the cuts cross the wake where

viscous effects are significant. A problem of a more practical nature

from the experimenter's point of view is the difficulties encountered in obtaining the profiles in a moving coordinate system fixed with

respect to the model.

A remedy to these complications can be found in the application

of longitudinal cuts. No less than five papers proposing or using

ana-lysis of such cuts appeared at the International Symposium on Theoretical

Wave Resistance in 1963. _{[ 3]P L41 [5]1}

171.

The methods presented by Sharma and Newman are mathematically equivalent, but only that due

to Sharma has been used to any great extent [8]. It is shown in the following section, however, that a straight forward change of variables

leads to significant simplifications in the numerical work of the Newman

method. Furthermore one finds that the wave profile or wave slope

enters the resistance integral In the form of an auto-correlation

function. This type of function is much used in other fields of

mathe-matical analysis so that its properties are well known. A simple method

of studying wave making characteristics of hulls in greater detail

is therefore now available. Separate studies of correlation functions

of wave cuts are expected to reveal significant features of the wave

WAVE PROBE

Ar'

WAVE FRONT REGION OF BOW-WAVE REFLECTION

7/7/77/ ///7////////7/////7//7////

//7//////1//7//////,

I.RECORDERS

Figure 1. EXPERINENTAL SET-UP.

It is believed that the modified Newman's method presented here offers the most simple and straight forward method of wave system analysis and it has therefore been adopted as a method of analysis of resistance characteristics of ships at the Hydro- and Aerodynamics

Laboratory.

The derivation of governing equations given in this report is

somewhat different from that used by Newman. For this reason, and for the sake of completeness we shall start the analysis with the known Green's function for the wave system generated by a Havelock source. The pnelysis will be restricted to the steady state case, i.e. the

velo-city is constant, and it will be assumed that the towing basin is sufficiently long so that transients have died out where the wave

pro-file is being measured.

The basic equations used are those valid for the ship moving In

waters of Infinite extent and depth. Theoretically it is therefore

necessary to have a wide towing tank. This is also needed to provide

a

sufficient length of cut before the reflected wave from the tank wall

is reached. The wave profile is taken from the time record of a

sta-tionary probe, measured as the model moves past. The record is termi-nated before the reflected bow wave reaches this probe (see Fig. 1).

.Assuming that this point is sufficiently far aft of the model a simple one component transverse wave is taken as an analytical continuation

of the measured wave cut. The analysis leads to a numerical inte-gration of the wave cut. It becomes necessary to truncate this inte-gration and to use only a finite length of the cut. .Estimates of truncation errors have been made, however, and corrections can there-fore be made to account for the finite length of the measured profile

plus its analytic continuation.

Longitudinal Wave Cut Analysis

The model has moved with a constant velocity for a sufficient length of time to allow the fluid motion to be considered stationary with respect to a coordinate system fixed in the model. This coordinate

system is taken with the z-axis positive upwards and the x-axis positive

in the direction of motion.

Assuming potential flow and linearized free surface conditions the Green's Function for a Havelock source located at (xl, yl, -z) is

given by (cf. Wehausen and Laitone

Clop

G(z,pe,xt,

=

-where 71/2 cd

÷

jrateidik.ek(E+zj

CosA()ces83.Gm;144-104,444

0 0

cosle

81c2-7/2

+

f

,de

28.

el/c(°4-Ej "29.

sw,' Ea/c2Cx- xi) Sec eg.

G2

0

X

c05ive-(1-1,)

_e-

_seGzei

,2

r.

(

ift

-4)

,g

r2.2

(z--A)2

4-

(1-

YI)2

4. 1)

and where # denotes a principal value integral.

For a distribution of sources,

K(xl,

yl, z1), on the surface S

the total potential then becomes

4,0c,

y,

z)

=fizr(x.i,yi,zi)G(x,y,z;x1,yi,z3.) de

The wave elevation follows from the linearized boundary

that

The wave resistance can be written as

*2

Ria = 16 f

see6l1P2(e)

+

Cr(

03 de

(4)

0

where

p(e)

(6)

=

fYoci,

p,

x

expki;

se&94-i(x4seci;41,

see6 s, 03 de

(5)

Substituting (1) in

(3)

the wave height for a Havelock source of

unit strength is obtained as

=

a

469 41k

keke

56

ic (-Xs)

cos

e3costk4-VOss:00

77-c Jo

- 842 .

sex,2e

X12

342

El. 4-

141 f do.

sec36 e

c

IC COS 2/c2

- X)

63. costgica-(1-1.1).5444.5,2ej

(6)

For a surface distribution of singularities the wave height then

becomes

=115.(x,y,

Is

(7)

Taking the Fourier transform of '4: with respect to x it follows

DO

,

₍₍

t. (Alt)

z-

f

(74,1).e

_{otx =}

00

c

ti,4) els.

a1/2 e*0

ke

_{k cos el. cosa(tito}

""'"

xf

dB

e

k ws20- 2.)

0

144

tsz0-26c,,so cos (kx, ws 8 )- cos(k4ocos)

_{sin (4}

xf. cede ).1.111 4 4. oe

-A4

xf719,21)-Y (xi,

)zi)

T/2

3)e,. seoze

+tt ffr(zi,

ds

pa.

_{stoe e}

_{ces[0(1?-11).}

54,19.sec26].

cs

(a)

g

f iXx

X a*/

_e

+1.

[Sin (A-0 sea) M

N-2ssecil

CX,1)

=

L1

-tol-* 71G

The first integration on x becomes

tin I

[sin( kx

cos

cos( I, .o.s

cos(kz case) ui (kg, cos 9)j4z,

m-0 Do

=

ibn ( acos Xx

sin

(14 x CDS

e).

cos (k

xi ad

a)

- cosAx cos (kx. cos e) 4.00 _Ad

Sim( k ac, cos 0..7

i [sin ADc sin

(km. cos

(bet

COS

)X.

COS(kPCCA2S9) Sir I

(1

ICI

cos9)31 1x

The first and last terms of the integrand make no contribution

because they are odd with respect to x.

Hence

r.

{_Isla(

-

k cAs (9) m

(

k (Asa) pi j. sis

0,0 co5 9 N4- k CeSD

4. 4...Viol (A

-

k cos E) M

X -k cos 0

sin (X + k cos 0) It43.

_{cos (kc,}

_case)}

_(9') CoS0

In a similar manner one obtains Af

12=

.107

eg

cosf0(

sezej cix

/44-ocio

=

L'Art {.1"

(A-

sec

M

s

M-304

L

1....2)scc e

SeGO &I].

_{ces(Dzi sea}

X4.2)sxce

Vim (14.a. sec e)

_szvt(a)aci

_sec05}

X+a)sace

The Fourier transform of 4 can now be written

742

pc)

ifrot,

%b.

_2441e9

0

o

k co9 ces

y- vs) s

im

t

k cosa 0 -1.)

L

X-kc.ose

sA4k c°s0 t'4

_{s;,"(k. 74, c.ose)}

A-1.kcos0

_J

iisima-kcpsOM

(X4'1"°s0"4] cDs(kx, vase)} as

L

_{-Ir. case}

74kco50

1)E., seN

C3

/*colt ,e)

fae

sec.30.e

0

eke,

lemma

Or

-7

x

cos

[33(1.-3,1)

56,0

S

t sias

))

sec

e)m

slm

0.

xl

sec

e)

1,41

-

sece _7.4.)secti

x cos (1)xt sea.) +

[

Sim(A

_{SeCe) M}

x

69.4(91

Some of these integrals can be evaluated if we use the following

to tr

F(k)

tik)

_clk

= 7r

F7( 14..)1

-t+

1r(k)

I ca (kcal

6

_cos

_cts(k)

F

t

0

to

0.)

where k is the root of

0

g(k) = 0

Equation (11) is defined in the region -034

4

₊₀₀ Taking

to be negative we note that the terms in the first integrand of equation

(11) which will make a contribution to t _{according to this lemma}

have

g(k) = +kcos

8

:

ko

= -

cos

e

g (k)= cose

,

I

(k0)1

.

cos e

It

is

easily shown that when )1 is positive then the contribution

comes from the terms where

g(k) -kcose : ko =

cos

The results will be identical so we shall restrict ourselves to

positive.

In the second integral of (11) we consider

g( e ) = 1)sec 8 : sec e =

2)

Go

I

(e0)1

-1 2.)

= cos

( X. 1)=

EL) C2 = 1) sec eotan Bo

31)7x2.

(N4s) sece)141

4--3) sec()

Also

(A,1)

tan

e

= t and it becomes

'Again similar results are obtained for

X

negative.

Applying the lemma to equation

Cu)

Nxiyyr,2,)etsfn[eAret.

cos(AN-41)tawle.)

,(- s4,xx.

4L

cosXx

j j

(Geoci

12

')

{..21.reig1(t).

I

cos°

case ij

e3

--

0),413.3

+

cosE0(1-1JATii9

/

_os

024177775r

_(i5)

In the

first integral of this equation let

xcose-).)

0

A irsec ar =

l+tT

i

AO =

dt

(140e)

A-17

_

if

_{y(xi,11,e) e}

_e

_{cos(A(1-totias}

The integrand is singular at t = to Where to is the root of

-= 0

jf

2

t0=

+2> 3Ar---142

234,17;77--

k qy) -1-t)( 2-1+t)

sib.(1)7c1304

ds

71V2 c-rits c,,s

(741-1,yto.a)

fibe(xt,

eg:Xx.,(do eE,

cote ()- a) sec e)

it

I _)oci

42T

_{(24-)7.---1 )10c,}

et)

e &:

os[7,(1-1)F4:1 1e

'As

c

(2i_fr.t

12)

We shall now write

XA

ITer

c.c's

(IX(

14)t)

op GLt

AA

F477-)`-').F7t2.

For

t12*1

it is readily Shown that the

major

contribution to

this integral comes from the integration around the singular point t = to.

The path of integration must lie in the upper

half

of the complex t-plane.

. Thus

f_or

A(21)

r )!.

t

2 )--3-7-0---._.132 ext) 17-) ZI.7k(r (-;)Nr7F-1]

= 7r

_{exp( 2z)}

_{. sky.}

_A(-1

13

Substituting

13

into.equation

(12)

gives

1-Vt

13

46.4)

7--1±2___ ((Nei

PP2

2jc

N

4.t.(X

-X(1-14011 )]

With

2-=.= sece

We note from (5) that

P(0) +

(1(0) -

case. e

(,g,$)e

ax.

(15)

Lk 7r

jai 344 v

%A mfte Oa

iojesite

-

-06

After squaring the real

and imaginary

parts and adding one obtains

2

p2(e)

ce(e)

oleo E[

_{1;(-3c,1) Ges(2)2c 5.4}

cu]

0.3

[ f

(;y11).

seal)]

(16)

Do

cl

_36,29

eia)x sec eaz

2

16 7e.

(14)

Where

x(2

Kz,Li 'CO = (-

,

coseI

cbs te)

_e

_ _

0

Pot,tug2 .i)0c4),ff2'

-so that

girt =

and substituting in (18)

the

_{non-dimensional variables} A

x =

it follows that

.r

tob

Substituting

.(16)-

in (4)

7v2 DO_

e

7T -o _-04

Equation: (16) can also be

written

02(ej 1:12.(oN

c

- -2,k

stAN ty COS_;:0

-1

tete

The .expression '

for:,

the :wave..

resistance,

''then :becOnied.

. . _ .

This last integral is

by

definition the auto-correlation function of the wave height

4;04).

Both K_1(T) and

lit

(r)

are symmetric with respect to

D.

Thus

00

IIw

=

K

cif'

(22)

Integrating (22) by parts we find that

FZ,A

= -

114 ("a

j

(v)

(r)

d-r

where

0(.04

(,(v)) =---j

i)

and

-44

7172

KIn(a)

_(e)

=

ces8. sLA(ti sea)) 40

(23)

Since both of these functions are asymmetric with respect to 1:1

it follows that

FLIZt.4rc;

(r) Ko(r)

IL 2-41 where 59,1

=1(

i)

o+v,

This function is

by

definition the cross-correlation function of

and .

Integrating by parts once more the wave resistance can also be written

22.214

where 00

0

2

= -

go(1722.

CO Klee) de

0

.0

9L(°

f

51c+r, 3) 0121 (27) ..+00

i.e. the auto-correlation function of the longitudinal slope of the non*

dimensional wave height.

Of the three kernel functions defined above only K_1(T) has a logarithmic singularity at 'V= O. It can be shown that the functions

TICO

vr) and

₉₂₂₀₉

are also singular at 'V = O. This

diffi-culty is removed if we consider the slope of the wave surface in the

transverse direction. This slope goes to zero rapidly with respect

to distance behind the Model. It is given by the expression

1/2

k2eke,

5;41(x-24). 0,5e]. sim[k(N-s)

7 oo

43. Aldo

Sec

_{k- I sec2e}

7r/2

1-1.at

_

lq f

410 siefe.s

"18.

cos 4.6

(x-x0

544196-sinEli4-Yi

.5.1.19-s4c26 0

Taking the, Fourier transform. of.gi and proceeding as in the of the Wave height it is easily Shown: that

co'

-=

: -

(r)

0t-r

241 f

w

3 33

where is allavelock P=function definesi-br_ case

vIR/2

2

P2vt-I

(Z) = (-1)

C"

"7103

P23, (e)

=(-1TIEcosit

t'le

and where

12

-(z sfte) de

i.e. the auto-dokrelation funbtion'of-the.wave.slOpe

-.So far four different expressions have, been given-for the wave

resistance-based on Information obtained from the wave surface along 4

longitudinal. path. The quantity. most easily Measured is probablythe wave height, although there ie,evidendelhat the slope can be.leasUred

With equal accuracy. Only experience will tell which formula will be

the easiest to use.

(30)

Corrections for Truncation Errors

It has already been mentioned that the wave cut must/be ter-minated before the reflected wave enters the record: If

22>4

and

A A .4 A

also M y, where is is the end point of the record, it is assumed that

A

4

cos (

A _A

A- A

te, 7c >Inn (32)

This follows from the asymptoticexpression for the wave height behind a ship for large values of [10].. The value

ofiie

and

is found. from the wave trace at the end of the available record.

Substituting. (32)

in (24) We find that

4s,(0)

diverges for

all values of 77 The logarithmic infinity of 56,-(1) at

rto

is a

consequence of the assumption that the analytic continuation. has a con-stant wave length. It therefore becomes necessary to consider a finite length of cut and then make an estimate of the truncation errors. New-man

[61

has already done this and we Shall make use of some of his results.

The procedure used here is as follows:

The total wave cut, including the analytic continuation, is

di-vided into two parts, i.e.

4 .^

e(2)0

n(119)

.4 A (33)

ti2( g 14

Substituting in (24) we find that the auto-correlation function reduces to A _A 5;1(0) =

L 4,(1',a)

(2

,i) dx

13 -ao,,t

(atril

i)

ea

2

2

We therefore write (1) (2)

'new =

"At

-4"

411 -14 where

Pc,.

0)

218

fy")01) K (e)

7r2)

0

II

R(2)

"I

_7r2)3

8 f°4912(.0

14 .

(r) at?

()co

(34)

(35)

Newman [6] has shown that

AR =

s

qopp.

dill 12sy

7r2.0

,

If (35) is written in the form of (36) then

w( 2 = ff 2fS CZ

J2

4

K_1(2 -A)

4(2

Due to symmetry of K...1(1') it follows therefore that

CO43-

7r

= AC

S. _{-70 .27r2i3}

74-The numerical integration of 0Achi terminates at a value of 17

such that the truncatian.error is less than or equal to 6/ 7/46,21, which is the order of the terms neglected in (37). This is accomplished

by extending the value of

V

to where the contribution from integration between two successive zeroes of the integrand to the value of ilgj° is less than

e.

The analytic continuation of the wave height given by (32)

re-sults in similar behaviour of

VII

and 5f22(r)as that of

_{9, (-c)}

For the evaluation of 12eN and

12g4

the wave cut is therefore also divided

into two parts as indicated by (33).

Because the kernel functions are different, new estimates will have

to be made of truncation errors. The derivation leading to

(36)

can, readily be modified to account for these differences. Thus

and

42

214F

3 A

II

I,

°27T2t

iCi3

-

14

-44

71r2 (36)

(37)

It is noted that these truncation errors are of smaller order of magnitude than that of

ii<4.

One should therefore expect that it would

be preferable to calculate

_22eik,

provided, of course, that

Ute.4)

A 4

n00

is

as accurate as

('(Z4).

The truncation error 33,,14 will be an'order'of magnitude less than (39) and therefore negligibly small for any reasonable

A value of M.

where On

(E)

.For small values of z

3(E)

2-#

I- ti

Ef+

4.39(f)_[,%(

4+0(0.27a account

of_(o)

of:, 44

bsz.

_(n()

ons0

-.15-and with the aid

of (30), it follows

that

(-21 )2n"

{iv%

3

I

1

-[ftropl-f)

4V-zR (43i;-,+,111-11324-gt

(2r1

I)

pso

For large values of 2 the

following asymptotic

valid [11].

(e) tze

-ALSO

-

4.,

0. If 43 ) _c(0-125

_z

0-0732

o

_{a 5}

)]

Po (4

_ALC(J

*

9;-4r)

S

-

tffic inj

2.695+ 4..245)

0V.

2.

0

-331

..

220- 9 )

Et

a q

E5

r

2216I3.5 )1

E:5

a

42-96

19801 1

ES

4-

J

23.

4.

iititt)

sy:±21

el

4

-5

C

cos (2

7714) expressions are

(40)

(44)

8-31z8

26971

2-

S

7

(45)

The Kernel Functions

From the integral definitions of the K-functions and P-functions given by (19), (23) and (30) it immediately follows that

(E)

air)-4 (a)

4-

F3ni-I (a)

K2,3 (E) = P2r, (E)

+

1.2. (Z) Also one notes that

den'

do'

The Havelock f-functions have been treated extensively by Lunde

Kr

(0)

=

(Z)

5R. = Pr-na (g)

(41)

[11]. He gives the following relationship

01. 2)2n+I

ea .14-1

tri( if)

_Lr(n-to

_24-ii}

(42)

(e) =2.

ft.0

(o, )!

where

A

a-15(2'632-5

-

16

-Newman[6]has given the series expression for K_1(z) as

°°

cirotier

K_1(2) =

{gen

a)

LY(141)

1-

j1

6.02014

2v1

Substituting (45) in (40) the following asymptotic expansions

of the K.-functions for large z are obtained.

41-1B21

_e

_{(I 9.256 I,. 1.122}

ex

e j

j

(46)

Ko (e)

-

A

-1[c(4.122

s131_44:05.. NI]

(47)

--i

K (

A e.

re)

LS (

5.78 227-aS)

0

C

175"

_r

)3

Es

Lunde [11] gives a recurrence relation for the P-functions as

n

P,

(2) = z

{P_, (e)

(e)]

-

(e) ;

%. (48)

By using (40) one notes that this relationship is also valid for

the K.-functions.

We note from (45) and (47) that the K.-functions are approaching

zero an order of magnitude faster than the P-functions as

The GIHR-ALGOL program for the

K.../(x),

K0(x) and Ki(x) is based

on the relationship of the Havelock P_1(z) to the Bessel function and on

numerical methods given by Luke [12] . By definition

P__l

(4

-

Nf. (a)

(49)

Then

po (a)

g

(e)

z

The recurrence relationship

(10)

then gives

P()

=

[ Po (a) +

Y1 ()]

P2. (E) =

Li [at P, (e)

4.

P_, (z)} -

P0 (e)]

(e)

Le t

P2 CE) - Po

(e)1

-

(A)]

Then for calculating the three kernel functions

K_1(z), K0(z)

and L1(z) the following functions have to be evaluated.

(z)

-(49a)

-9 10 .

17

-Yo (E) ; (E) 5

cola

j

Neo(e) az

This is done through the following expressions:

Jrt-i (2) 1- 3 v1+% (a) =

.3,"(2)

Jo (a) + 2 32(e) + 2.

.1-4(Z-) 4- 2 Ji

(a) 4. -

-Nfo(2)

z

V(4- 11" (I))

Jo

(a)

-

4Z

(-')" o% nal

(a) =

EL 3 (a) +

_{z 0}

(V) 4:71-0J,(s) 4-1 -I)

where

I

=4 0.5772156649.... 9 Euler's constant.

In (52) the series is terminated at jr(?)where

3r(g)>6).71.4"

E being the allowable error of j"(). jr(z)is then arbitrarily set equal to unity and from the recurrence relation (51) all lower orders of the

Bessel function are calculated. The correct values are then determined

by a normalization through (52).

The relative error in the values of

NW)and Is

(4) is of the order

a

The values of the integral 1,(A6,4 are Obtained .from the

fol-lowing polynomial approximations.

(LA

)1 30 CZ)Gtk

0 zk-vi "

(t)

+-

(z)

2k+1

21'(It

-

- 6k! (

) 112)

0

(54) un (-e)] (53) L

where

k

(E)11.3.0-9;

6(z)

1-10-9;

and 04 zZ. 4 and

ak

bk

o

4.00000 0000-

1.07661 1469 ]. 5.33333 3161 2.56725 0468 2 3.19999 7842 2.28731 7974 3 1.01586 0606 0.90475 5062 4 0.19749 2634 0.20338 0298 5 0.02579 1036 0.02960 0855 6 0.00236 2211 0.00303 4322 7 0.00013 3718 0.00023

5002

8

0.00001

3351

- where - E (4)

I

.6-10

_

- 18

,-= 0.124611038

.

0.0312.8 08.48

0.02364 4978

,0.02200 7499

-,0.01623. _6617

0.00739 0830

,0.00149 6119

()

(4] az

=-- e . 6

7

L .

.0

and 8

gz 06 .

(56)

b

0.79784:8790

0.04963.5§33

'

0n02366 4841

0.01825 5209

0.01242 2640

0.00543 4851

0.00107 _6103

0.79788' .45600:

-0.01256_ 42405

0.00178 70944

_

0:00067.40148

0:00041 00676

.0:00025 43955

0.00011 07299

0.00002 26238,

ak

70.96233 47504

1

''.0.00404,03539

2

0-.00100 89872

_ . 3

0.0005'5 66169

4

0.00039 92825

5

0.06027:, 55057

6

0.00012 70039

7

0.00002 68482,

19

-Numerical Evaluation of Wave Resistance

The numerical procedure outlined in this section applies

spe-cifically to They are similar for

ne.

andtlew except

for the special treatment of the logarithmic singularity of = 0.

The 17 -functions are slowly oscillating functions of

r

and so are the K- and P-functions. Numerical integration by trapezoidal rule is therefore used.

The correlation functions

iv),

Z(r)

and

1 (r)

₂₂ are evaluated in the following manner.

10(0-44I)

=

ex+V-04Z) difle

AV

(k)

.

a .(ki

where a(k) = * for k = 1 or k = Ng a(k) = 1 elsewhere.

N = total number of discrete points in the record.

AD

= uniform spacing of the discrete points.

= 11

2, 3,

f(k) = finite wave record (including analytical extension).

g(k) = wave record extended infinitely.

The final integration for the wave resistance is performed by

means of the Simpson's first rule.

The region 0

4, r_

10-/ is treated separately in the case of

att..;

as follows:

The function

561(0)

has a maximum value at = 0 and is

sym-metric with respect to

C.

Assuming a parabolic form

61

(0

-

e),0-9

(

61

;

04. r4.

₍₅₈₎

10-2

In regard to

_K-1(2?)

it is sufficient to retain only the first

term of (46). Thus

KIM rz-

izt

I) 5

o 41'4 to"-!

(59)

How close an approximation this is to a more accurate estimate

of

K..1(7!) is

shown in Table

I

below where

(59) is

compared to ic_ice)

to six significant figures.

R(1)_....

(D)

59)

3.7.4270 5.721102

10 1.429561 1.418519

Table 1. Comparison of K

The expression.tor the .wave resistance now becomes

-071 &V

211r.(I)eo

(0)

-

00

P)

f

tra)

0-1 2 If= Artis and Eq. (59). .0) -18345766

t.(9),,:e

95095

T1 (.0.1 (11) sr r (60)

-The truncation: of the ..remaining integral has been treated earlier. It is :suggested that IE is estimated from (39) such that

'LO

10-1 kg.

Application of the Wave Cut Analysis

-A most difficult physical ,concept to define is that of the wave-resistance of a' ship. The residuary resistance obtained through the appli-cation of various friction lines, such as the.ITTC line, is not entirely

due to wave-resistance. The introduction o; a form factor, as proposed by Hughes, may produce a residuary resistance more closely related to

the wain-resistance, but it certainly is not a pure wave-resistance corn,.. ponent either.

- There exists today no theoretical method by which the

wave-resistance can be described with sufficient accuracy even if a definition

of wave-resistance had been possible. In particular, the linear wave-resistance theory is known to deviate significantly from the residuary

21

-resistance obtained in the usual manner from model experiments. An

exception is the thin plank, as has been shown by Weinblum. For normal hull proportions the viscous effects on the wave-resistance is undoubtedly

of some importance. Unfortunately these effects are not well

under-stood, which complicates further the problem of defining a wave-resistance

component.

It is reasonable to assume, however, that the ship generated waves, at some distance from the ship, can be described by the linear

theory of propagating surface waves. The method of analysis described

in this report should, therefore, account for the component of resistance

caused by the transfer of energy taking place through the wave system.

The resistance so determined can quite properly, and without ambiguity,

be named a wave-resistance. Subtracted from the total resistance it leads to the definition of a residuary resistance which is primarily caused by the tangential shear stresses acting on the hull, and the pressure re-sistance due to the growth of the boundary layer, but which also con-tains eddy resistance and the influence of form on the frictional

re-sistance. Furthermore, the formation of breaking waves will produce a normal wave-making force component which does not contribute to the free

waves of the ship. In view of this it becomes clear that the application of the longitudinal wave-cut method can lead to a new procedure of

extra-polation of model test results. How accurate such a procedure may be-come can only be determined through the evaluation of a large number of

experiments, both model and full scale. In principle it should not be

too difficult to measure the wave profiles full scale. This fact needs to be exploited vigorously.

Application of the longitudinal cut method of analysis to the de-termination of optimized hull forms is immediately possible. In recent

years it has become commonplace to

try

to optimize a hull form by fitting a bow bulb to the hull. Usually the procedure followed to arrive at the best location and size of bulb has been one of trial and error. How-ever, the longitudinal cut method of wave-resistance analysis permits

one to approach this problem in a more rational manner as follows:

The model is tested with a bulb in a position estimated to be

close to optimum. The hull without the bulb is also tested and the wave profile along a longitudinal cut is subtracted from the wave profile of

the bulbous bow model. The difference, is then the wave generated by the bulb. Assuming a longitudinal shift of the bulb in steps, the bulb wave

'May"be.sgeOMeiricaliyadded-tothat.-af.the-patent hull. The autodorre-'

lation-fUnction and the wave-resistance are calculated from each of the

.

_resulting wave profiles. -;,Since the -blab wave is proportional to its -Size it

is

a simple tatter'ta.repeat,the process for several, bulb sizes.,

Rence71t_is possible tb calcillate the WaVe-residtance for a series of configurations from information obtained fromtwo model tests only..,

This method of analysis was first proposed by Sharma [8]. Figure 2 is a graphical-output-bf-a computer program written for thispurpose

at the University of Michigan.

.0_04,

°

. _Jo"

LjJ

CC'

PREDICTED CONTOURS OF BULB INFLUENCE FACTOR

ETA IN THE -,p7c1 PLANE , (S01_ ID LINES). .

0.

-08 06

:

2--.04

L.

.13002

FIELRTlyEBULEI LOCRT fON

0-.06

Figure .2.

.Graphical Output for Optimising Bulb' Location

and Size.

Experimental Results

-The present method of analysis has been applied to a number of

cases where model towing test results were available. Results for three

such cases are presented here. Particulars of the models are given in

Table 2.

Table 2. Particulars of Models Tested.

The calculated wave resistance coefficients are shown in figures 3-5 together with curves for residuary resistance coefficients evaluated from towing tests by the use of Hughes basic line multiplied by a form

factor. (l+k) determined by the method of Prohaska [13].

Figure 3. Cw and Cr. Mariner Large and Mariner Small. Model Identification Mariner

Large Mariner Small 710311 698735 Length b.p.

m

6.437 3.218 4.196 6.904 Breadth 'maximum

M

0.926 _{0.463 0.773 0.953} Draft mean

m

0.298 0.149. 0.311 0.284 Displacement m3 1.065 0.133 0.463 1.099

Wetted Surface Area m2 6.658 1.664 3.917 7.087

Block Coefficient Form Factor (1 + k) 0.6 ' 1.29 0.6, 1429 0.46 1432 0.59 1.38

Location of probe -model C.L.

m

3.00 3.00 3.00 3.00

Breadth of

Towing

Tank 12.0 m

11

-

108 6 4 2. -0-' ).104 r Cw C C Mariner Large Mariner Small Mariner (1+k)

+

1.29

:Depth of Towing Tank 5.5 m

0.22. 0.24 0.26 0.23

1+1c)": del .69 C;735 ...____6

r

LiOdel 698735

'(1+k)

1 . 38 '

Cw and Cr

Modal

:7103.11:-..:

0.24

_

0.26

25

-From these figures it can be seen that the longitudinal cut wave-resistance is lower than the residuary wave-resistance obtained by the use of

a form factor (l+k). In a recent paper Brard [143 considers the "wave

making viscous resistance", i.e. the resistance due to the waves generated

by the boundary layer and wake. The discrepancy between experimental results may in large part be due to the neglect of this "wave making

viscous resistance" and also the neglect of wave breaking.

11:1

Bow

Figure 6. Typical longitudinal Wave Cut. Mariner Small. Fn = 0.279.

or-Bow 1 sec. 1_1 ! _L 1 sec

IHI

III

11

1 1 1 1 1 1

1 111H.411111

Figure 7. Typical Longitudinal Wave Cut. Mariner Large. Fn = 0.277. --7xpected Reflecti

-

26-Figure 6 shows a longitudinal wave cut record for the small Ma-riner model and Figure 7 the corresponding record for the large MaMa-riner

model.

Figures 6 and 7 are typical for the model sizes shown. It can be seen that it is easy to terminate the record for the smaller models be-fore the wave reflected from the tank wall is reached, while for the

nor-ma]. sized models the reflected wave is reached before a sufficient length of record is attained. _{This gives difficulties in terminating the} re-cord, and results in some arbitrariness. _{The same has been observed in} other tests with normal sized models. It is of interest to note,

how-ever, the agreement that exists between the results obtained with the two

Mariner models.

It is too early to draw any conclusions on the basis of the limited

number of tests described in this report. The results do indicate,

how-ever, that the linear wave theory may, in the case of relatively fine hull forms, satisfactorily predict the major part of the wave-resistance

characteristics of a hull from Longitudinal Cuts. For full hull forma it is possible that wave-breaking becomes a major component of resistance which does not show up in the form of a free wave pattern.

What has been presented is mainly a detailed description of the method of analysis and the experimental techniques together with a few

numerical results. It is hoped that the Longitudinal Cut method will

prove successful in the future.

From the experimenter's point of view it is important that the analysis is very simple to apply, does not require extra time or extra personnel in the tank and that with suitable recording equipment it is

27

-APPEEDIX I

Test Equipment

The wave height and slope is measured by three conductive wave probes, one centre probe (x1), the other two displaced by 20 mm in the direction of the tank (x2) and transverse to it (y).

From three separate 3 kHz oscillators with floating outputs the current is passed through the gilded wires in each probe. As the current is a linear function of the water level, it represents the wave. After

transformation, rectification and filtering the signals are fed to operational amplifiers which contain gain and balance controls.

The wave signals (y, xi, x2) and the differences (y-xl) and (x2-x1)

are recorded on an analogue tape recorder with P.M.-modulation. Tape

speed is 60 inch/sec.

From pre-set sensors, remote-signals for the tape recorder are

given by the passing carriage.

After the run the wave-signal is data-logged with a reduced tape

speed of 7 inch/sec.

By the data-logger the wave level is read at equal time intervals

and recorded on an 8-hole punch-tape. (BCD-code with 5 digits).

The test information is now ready for processing on a digital

3kHz

Osd..

!Balance'

29

-REFERENCES

1 KAJITANI, H.: "Wave-resistance obtained from photo-grammatical analysis of the wave pattern".

Proc. International Seminar on Theoretical

Wave Resistance, Ann Arbor, Mich., Aug. 1963.

2

EGGERS, K.:

mUber die Ermittlung des Wellenwiderstandes eines Schiffmodells durch Analyse seines

Wellensystems".

Schiffstechnik, Bd. 9, 1962.

3 PIEN, P.C., MOORE, W.L.: "Theoretical and experimental study of wave-making resistance of ships. Proc. International Seminar on Theoretical Wave Resistance, Ann Arbor, Mich., Aug. 1963.

4 SHARMA, S.D.: "A comparison of the calculated and measured

free-wave spectrum of an Inuid in steady

motion".

Proo. International Seminar on Theoretical Wave Resistance, Ann Arbor, Mich., Aug. 1963.

5 GADD, G.B., HOGBEN, N.: "An appraisal of the resistance problem in the light of measurements of the

wave pattern".

Proc. International Seminar on Theoretical Wave Resistance, Ann Arbor, Mich., Aug. 1963.

6 NEWMAN, J.N.: "The determination of wave resistance from

the wave pattern".

Proc. International Seminar on Theoretical Wave Resistance, Ann Arbor, Mich., Aug. 1963.

7 SHOR, S.W.W.: "A Fourier transform method for calculating

wave-making resistance from wave height on

a line parallel to a ship's track".

Proc. International Seminar on Theoretical

Wave Resistance, Ann Arbor, Mich., Aug. 1963.

8 SHARMA, S.D.: "An attempted application of wave analysis

techniques to achieve bow-wave reduction".

6th Naval Hydrodynamics Symposium,

Washing-ton, D.C., 1966.

9 KIM, H.C., MICFRLSEN, P.C.: "Experimental wave component

analysis as applied to ship wave systems,

Part 1, Analysis of available methods and

evaluation of some experimental data".

University of Michigan, Office of Research

WEHAUSEN,. J.V.-,LAITOBE

9 E.V.:

115urfaCe

Vol.

9, tandlitich

der Pb,yeik, Springer-Yerlag.

_

,

LUNDE J.K.:.,"A. note on the linearized 'deep water

theory_ ofr-rwaVe;

Profile

and wave

resistance:

.

-University of Cal., Series No.

82,

Issue

,

-2

,

,

PunCtionic".

:f-

McGraw4gillf:,:1962,i._.(pp. 60-69).

,E3OHASKA,.C.W. :,' "A .iiimple:_methodl 'for the evaluation'Of

-

the form

.faCto*,

and the ,low speed wave

, ,

..-..: --,:resistance.:- :,:...._ - ,:-*,,: -. ' '.:.- '--. - ,

,.._.. .. _{.. .}

PiaOc . 11th International Towing ' Tank

Conference,

Tokyo,-1966

_{_.,} (Pp.:

65-66).'.

..

-a.. "Viedatiity-,17akeg and

ShiP-, Waves".

Journal of ship Inieearch-, r' 6