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Report lOO August 1969

DEVELOPMENT AND EVALUATION OF A SECOND-ORDER WAVE-RES I STANCE THEORY

by King Eng

Prepared for the Office of Naval Research

Department of the Navy Contract Non r 263(65) Mod. 5

This document has been approved for public release and sale; its distribution is

un-limited. Application for copies may be made to the Defense Documentation Center,

5010 Duke Street, Alexandria, Virginia 2231+.

I

\/

DAVIDSON

LABORATORY

tab.

y. Scheepsböuwkunde

R-lLOO

Technische HoeschooI

De If t

(2)

DAVIDSON LABORATORY

Stevens Institute of Techno'ogy Castle Point Station

Hoboken, New Jersey 07030

Report 1400

August 1969

DEVELOPMENT AND EVALUATION OF A SECOND-ORDER

WAVE-RESISTANCE THEORY

by

King Eng

Prepared for the Office of Naval Research

Department of the Navy under

Contract Nonr 263(65) Mod. 5

(DL Project

3507/102)

This document has been approved for public release and sale; Its distribution is unlImited. Application for copies may be made to the Defense Documentation Center1 Cameron Stat in, 5010 Duke Street, Alexandria, Virginia 223l4. Re-production of the document in whole or in part is permitted for any purpose of the United Stàtes Government.

Approved

lx + 70

pages

3 tables, 8 figures John P. Breslin, Director

(3)

This report is taken in part from a dissertation submitted in partial fulfillment of the require-ments for the degree of Doctor of Science in the Department of Mechanical Engineering at Stevens

(4)

R_ILIOO

ABS T RA CT

A new second-order wave-resistance theory for floating bodies is developed, and then assessed by application to a parabolic strut. The problem is treated as a potential-flow problem with a centerplane

distribution of sources to represent the body. However, the kinematical boundary condition is satisfied on the surface of the body. lt is

supposed that the beam-length ratio , t , is small and that the square of

the disturbance velocities which appeared in Bernoulli's equation on the free surface is given in terms of the components of the first-order potential. lt is also assumed that the solution of the source density,

a , Is in the form of an asymptotic series in t , and the solution for

a is obtained up to the order of t2. The wave resistance is then

computed on the basis of the improvedsource density and the correction arising from the Improved representation of the free surface.

It is found that at low Froude number the expansion scheme used by Sisov, by Maruo, by Vim, and by Eggers, cah give negative resistance if the beam-length ratio is not small enough. Therefore, a new definition is adopted; that is, the sécond-order wave resistance i.s based on the

improved disturbance potential without expansion of the amp!ltude functjon of the wave-resistance integral.. The result is a marked improvement over the result obtained from the Michefl theory, and correlates rather well with experiment. lt is also found that the second-order correction to

the kinematical boundary on the body is as important to wave resistance as the non-linear free-surface correction. This is not the case when the conclusion is based on results for submerged bodies.

KEYWORDS

(5)

1.2 Potential for a Source R-1400 CONTENTS Abstract List of Figures Nomenclature INTRODUCTION

SECTION 1 THE FREE-SURFACE BOUNDARY VALUE PROBLEM . 3

1.. 1 Statement of the Problem 3

6

VI

SECTION 2 DISTURBANOE POTENTIAL OF A PARABOLIC STRUT 14

2.1 KInematic Boundary Condition on the Body Surface 14.

2.2 Solution of the Integral Equation 16

SECTION 3 WAVE RESISTANCE ... ... . . 21

3.1

DefinitIon of Second-Order Theory 21

3.2

Wave Resistance of a Parabolic Strut

...25

SECTION 4 EXPERIMENTS 30

4.1

Description of Experiments . . . .

...30

4.2

Model Characteristics and Test Conditions 31

4.3

Results . . . 32

SECTION 5 DISCUSSION OF RESULTS AND CONCLUSIONS

...33

APPENDIX A. Source Density Due to the Effects of Froude Number. . . 34 APPENDIX B. Amplitude Function Derived From the Correction

on the Free Srface . . .. 39

APPENDIX C. Amplitude Functions Due to the Froude-Number-Dependent

Sources 47

APPENDIX D. Second-Order Amplitude Function

(Zero-Froude-Number Correction)

...55

REFERENCES 57

TABLES (l-3) 59-61

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R-lL+OO

LIST OF FIGURES

1. Ratio of Second- to First-Order Wave Resistancé ofa

Parabolic Strùt According to the Expansion Procedure 62

2. Wave Resistance of a Parabolic Strut Versus Froude Number 63

3. Parabolic-Strut Model Used in Experiment 613

14A. Mounting of Drag Balance. in Midsection òf Strut (Photo) 65 LIB. Experimental Setup (Photo) . .

. 65

5A. F Ó.259;. Draft, D = 7.5 Inches. (Photo) . . 66

5B. F.=O.259;Draft, D =12.0 Inches (Photo) . 66'

6.

Data Obtained From Drag Balances .67

7. Resistance. Coefficient of a. Parabolic Strut 68

8. Total Drag Coefficient of an 800-FootParabolic Strut 69

9. Effective Horsepower Requirement Per Ton of Displacenent

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'b' ' (L,w) (L,W) J (x) k o R-1400 NOMEN CLATURE a k sin e C e1

D draft of top section of strut (Fig. 3)

F urn

. 2gL

g gravitational acceleration

H harmonic part of the Green's function

(Eq. [1.26))

h draft of the body

si-1

Second-order amplitude function of wave

resistance integrai due to zero Froude number, to free surface, and to finite Froude-number correct ions.

Bessel function of first kind, n' order

k radius in cylindrical coordinate

(dumy variable)

g , wave number

i 2

L finite-Froude-nuinber source density (local part,

Eq. [2.151)

L semi-length of strut

m density of a single source

P,Q amplitude function of the wave-resistance integral

(8)

u, y, w

Ui, Vi, Wi

R-1400

S/u v2+ w2 , magnitude of disturbance velocity

wave resistance coefficient normalized by

2,2

2pU

x'

wave résistance

distànces (Eq

[1.251)

flrst- and second-order wave-resistance coefficient

beam-length ratio of strut

(x, y, z) components of disturbance velocity (subscript 1 denotes first approximation)

x,y,z Cartesian coordinates (subscript i denotes

x1,

, ziÇ the Fourier transformed space)

Um free-stream velocity

w finite-Froude-number source density (radiation

part, Eq. [2.16])

equation, of strut offset y0

equation of the frée surface

tan' ()

tant

(1+X)

-

tan [t(1-x)J

t

sec2 sin 9

t

sec291 sin 8i 0.577215.7 (Euler's constant) tan1 [t('i+x)] R0, R1, R2 R1, R2

(9)

U

2F

X

Ksec29

X1 t sec2 8

sec 8 tsèc 6

specific gravity of fluid sóurce density (distributed) total velocity potential

w

R-11+OO

Dirac delta function small parameter

Cartesian coordinates (durmy variables) coordinates of a point source

angular variables of cylindrical coordinates (wave angles)

disturbance potential (1 , 2 denote first and second approximation)

prescribed function on the free surface (the right-hand side of Eq. (1.11])

(x-') cos 8 + (y-11) sin e

(x-')

cos 8 - (y-11) sin e

(l+x)

e+

e (l+x) cos 9 - y sin e

(l-x)

cos

e + y0

sin 8 (l-x) 8 - y0 sin 8 cp, cp1, 4r(x,y)

(10)

R-lLOO

INTRODUCTI ON

The resistance experienced by a floating body is composed of a viscous and a wavé-making component. According to the commonly accepted Froude's hypothesis, the viscous drag is considered to be a function of Reynolds number only, and the wave resistance Is considered to be solely dependent on Froude number. This hypothesis assumes that the viscous effects and the wave-iñduced effects do not interact and therefore permits us to study. each component independently. Thewave-making component Is the subject of this study.

lt was In 1898 that Mlchell1 gave the first approximate solution of the problem of wave resistance for a thin ship-shaped body moving on the surface of a perfect fluid of Infinite extent. This formula successfully depicts the nature of the variation of wave resistance with Froude

nUmber, but the result appears to be in error when compared with

experimental evidence. Many attempts have been made to Improve Michell's formula by some rational: approximate corrections.21 However, the

success of these undertakings has been very limited. In an effort to correct the situátion, theorists in the field have developed a higher-order wave theory in terms of a systematic expansion in small parameters such as the beam-length ratio of the body.'2'13 The first-order term of the expansion is identical with MichelPs formula, and the higher-order terms enable us to improve the boundary condition on the body and on the free surface. This technique, which was introduced by Peters and

Stoker,12 can give a successive approximation up to any order. However, the second-order term becomes so complicated that numerical computations seem scarcely possible.

The treatment of wave-making theory for submerged bodies such as two-dimensional symmetrical airfoils, cylinders, spheres, and spheroids

Is relatively simpler. By using the linearized free-surface condition and an image method, Havelock, in 1928,1 gave a second approximation for a submerged cylinder and, in his later 1936 paper,15 was able to construct

(11)

RlLeOO

a complete formal solution to the problem up to any order. Again by using the linearized free-surface condition, Bessho16 presented a higher-order solution for a submerged cylinder and spheroid. Tuck, in 1965, showed the work of Havelock and Bessho to be inconsistent and proved that the most important contribution to the second order is the free-surface non-linearity which they had neglected In their analyses. Tuck's

conclusions were strengthened by Salvesen's results in the case of the submerged two-dimensional foil. 18

Sisov, in 1961,

formulated a second-order wave-resistance theory for a thin ship,19 as did Maruo, in 1966, In a more general way.2° Vim, in

1966, treated the problem of higher-order wave theory in slender ships,21 and Eggers, in 1966, made another approach to the same problem.22 At

present no definite conclusion can be reached as to the merit of the works cited above, because of the difficu'ty involved in numerical evaluation.

The present study Is concerned with the assessment of second-order theory in predicting the wave resistance of a body. Specifically, the analytical work is concerned with finding a disturbance potential of a body up to the second order In beam-length ratio, and determining corresponding wave resistance. The analytical result is compared with the experimentally determlnèd wave resistance instead of with the usual

residuary. The procedure is applied to a strut-like model with a syrwnetrical parabolic cross-section. The wall-image approximation Is used to calculate the free-surface correction to wave resistance, so that the numerical work can be made more tractable. For most surface ships, the operating Froude number is small, and in these cases such an

(12)

R-1400

Section 1

THE FREE-SURFACE BOUNDARY VALUE PROBLEM

1.1 STATEMENT OF THE PROBLEM

Consider a body which is at rest in a semi-infinite fluid through which a stream of constant velocity U in the negative x-direction is flowing (see sketch below). Throughout the discussion which follows, a right-hand Cartesián coordinate system with the z-axis positive upward and the origin at the undisturbed level of the free surface at the center section of the body is used. The equation of the body surface and that of the free surface are denoted, respectively, by

and

The fluid Is assumed inviscid and incompressible and the flow is considered Irrotational.

(13)

R- 11OO

Under the assumed conditions there exists a velocity potential and a disturbance potential p related by

=

cf(,,'r)

+.:1L

(1.3)

u

which satisfies continuity in the form of Laplace equations

= o

i

and expresses theve!ocity. ata point of the fluid In the forjn

'

:

where: u,v,wareothponents o.fthe:disturbance velocity..

The boundary condítiór Ì t,hÎchthere is no'f 1owacross the surfaces

y = y(xz)

and z = Z(x,y) yields

and

o

The Bernoulli equation in the. present cale ¡s

ifD +-[('r+

Z)+

2:

(1.8)

I

(1.9)

and the condition p = Oat the free surfce yields

= tWoi4.

(14)

we then obtain

or

Differentiating Eq. (1.9) with respect to x and noting that

(j

t (U\

+

--

t

\)

Z j) - C3

R-1400

Similarly differentiating Eq. (1.9) with respect to y , we get the

result

Eliminating Z from Eqs. (1.7), (1.9a), and (1.9b), we can write

the exact boundary condition at the free surface as

uo +)

.$

z

+

J)

Thus, we are required to find a solution of Laplace's equation which satisfies both the linear equation (1.6) on the body surface and the hon-linear equation (1.10) on the unknown free surface.

The free-surface condition (1.10) is inconvenient, not only because

it is to be applied on a surface of unknown location, but also because

It is non-linear, so that the superposition technique is not applicable. The condition may be transferred to the plane z O by means of a Taylor". expansion, such as

('x1)

l;iL

(,o) + i:;

+

( .9a) (l.9b)

(,o) +

T

(','',o)

Z!

o)+

.

etc.

(15)

R

ikoo

which, substituted into (1.10) with (1.9) applied, gives terms up to the second order In disturbance velocity:

¿1.11)

Equation (1.11) is still non-linear. However, If we employ the method of successive approximation, it is possible. to obtain a solution for the system. Let cp1 denote a solution which satisfies (1.11), with the

right-hand side equal to zero; that Is,

Then, letting the, right-hand side of (1.11) be a fuñction of cp1 , one

obtains, to the second order,

(4!L

À1Li4

\b)

- iJ ò . b'

ØaI5 (4.

This procedure, which was suggested by Professor. L.. landweber, can be continued to obtain linear approximations to (1.10) of higher order.

1.2 POTENTIAL FORA SOURCE

Let (x,yz) be a right-handed coordinate system as shown in the sketch on page .3. iet a source of strength m be located at the point

P(11,C)

in the fluid

(<o)

, and let the right-hand side of Eq. (1.11) be a prescribed function i(x,y) ,,on the surface z = O Suppose the coordinate system, the source point, ad ,(x,y) are moving

(1.13)

(16)

as

=

V'b

-r ('ì.306

where cp Is a harmonic function everywhere in the fluid region except

at P , and behaves like mIR In the neighborhood of P (R Is the

distance from P). By virtue of the properties of the Dirac delta function, the disturbance potential p satisfies the equation

V24

-

4)

Th

-')

(i.c

This is to say that cp Is harmonic everywhere but Indeterminate at the

point P(0,110,C0) in the fluid. The condition at the free surface is of the form

R-l+00

(&\

-)

'.)

The other conditions for p are

o

- o'

rrt_.4) ,w

(;-)

, -

» o

(1.17)

Although the solution of this problem is known,13 It seems desirable to exhibit the derivation here to clarify the specific form of the complete solution that is desired In this analysis.

The Fourier transform method Is chosen to solve the boundary value problem as posed by Eqs. (1.14) to (1.17). Reference 23 states that If F* (x1) Is the Fourier transform of F(x), then the following relation Is true:

F

< > F()=

(1.15)

(17)

If the Fourier transforms of p and i are denoted by

t4)

(2.. .

**

t-')c

=

R-1400

=

(

,

)

e.'

&

i

ç

'P

(' ')

) e..

c fr

L1+,)

=__

then the transforms of Eqs. (1.14) and (1.15) with respect to x and y can be written as

-zm (-L)

e00)

(1.18)

=

4

(tc,,1)

(1.19)

cr4=o

(1.20)

(18)

R-11+0O

becomes

4

j;)

The inverse transform of with respect to z1 , cp , represents

the particular solution of Eq. (1.18):.

($(,. ;L

))

=

¿(+ Q;;)

d

yru

(1vL).... t_+'?)

(1.21)

The homogeneous solution of Eq. (1.18) is of the form

=

Equation (1.20) requires that B(x1,y1) = O , because z ¡s always

negative. Hence, the general solution of Eq. (1.18) is

7

e'

I.ç0\j:)

1.22)

where A(x1,y1) ¡sto be determined fromthe boundary condition (1.19). Since Ç<0, then in the neighborhood of z = O ,

-= -o>o

(19)

R_1LOO

q4

+

L**i .\

.4

S where

=

The Inverse Fourier transform of xi,yi,z) with respect to x1 and yj. will yie1d cp(x,y,z) of the system of Eqs. (i.l) tó (1.16).

1

-T'

111) d

'4L

Changing the variables

x1,y

by letting

1L

,(-L))}

dj

and noting that thé integral representation of

hR

Is of the form

/

--if

#(Xcose4Yne))

(-'i

dd

(20)

we can write R-1400

=

(-&

+

+

--

;\

o

+

+

f

=

R

1 (-- ('e-1

+ (

whe re

(1.25)

Ç, k:)

=

-i-J

Íek{)_

(%-(oSO #(1ne)

&ce

.tÓ

When integrating with respect to k from k = O to k = in the function H , we must go around the singular point of k = k0 sec2 8

Since H

is a harmonic function, it is necessary to interpret the

24

integral properly. Havelock suggested a method of artificial viscosity which interprets H as follows:

I.

Ltj7Ll

i

.

dkde

()O

To fix the singular point, we can fix the sign of the 8 integration by

TI 2

reducing the range of integration from

8=0

to 8=

jd

-\

L

(e..4-a

.'

+

o

(k

ei)

G (1.26)

k-

- (sce))j

(21)

W= (-)(Qs- ('-)'ne

= (- (oG (-) 6:flQ

>'

0ec

It can be shown that H decays exponentially in x far upstream

(x

> > o)

of the source point, and behaves ltke

/

-

_j

-\)cbsO)

os-i4.e.

Q

far downstream (x < < O) . Thus, the asymptotic behavior of q far

upstream is

and far downstream it is

-im1)2..

>(hSQ'

b(-cose)

c4-t0' n

)e.

î-t

1'

0(i-) C.os)

COS )

¿0

(1.27)

(1.28)

With H defined by Eq. (1.26), the disturbance potential Cp given by Eq. (1.25) satIsfies (1.17) automatically, and therefore it is the solution of the boundary value problem which is posed by Eqs. (1.111) to

(1.17). An interesting feature of this solution is that it shows hi the non-linear free-surface correction (x,y) enters the problem. It

Is as though a distribution of sources with an associated strength were distributed on the surface z = O . Note that t(x,y)

is an approximation of the right-hand side of Eq. (1.11), and that no additional constraint is placed on this function in the derivation of cp

(22)

R-lkOO

The term mIR1 In Eq. (1.25) is a potential solution ofa point source In infinite fluid. The quantity (mIR1 + mIR2) Is the solution of the problem If the free surface Is replaced by a wall (I.e., cp2 = at z = O); thus the term m/R2 is comonly known as the wall Image. Equation (1.27) indicates that no gravity wave Is present far ahead of the disturbance, whereas Eq. (1.28) shows that gravity waves are present behind the disturbance. Equation (1.28) is usually identified as the free-wave part of the disturbance potential.

(23)

R_1LOO

SectIon 2

DISTURBANCE POTENTIAL OF A PARABOLIC STRUT

The main concern of this section is to determiné a centerplane distribution of sources as a disturbance potential for a parabolic strut. However, the kinematical boundary condition is satisfied on the surface of the body, rather than at the centerplane as is true In usual thin-ship theory. It is supposed that the thickness-to-length ratio, t , is

small, and.that the right-hand side of Eq. (1.11) is given by the first-order disturbance potential. It is also assumed that the solution of

the source density , , ¡s In the form of an asymptotic series in t,2°

and the aim of the present analysis is to obtain up to the order t2.

Throughout the discussion, all the length dimensions are normalized by the semi-length ,

. ; the source density and all the velocity dimensions

are normalized by the free-stream velocity, U

2.1 KINEMATIC BOUNDARY CONDITION ON ThE BODY SURFACE

From Section 1.2, the disturbance potential for a distribution of sources on the plané y = 0, -1 < x < , - < z < O , moving at a

constant speed below the surface z = O on which the right-hand side of Eq. (1.11) ,

4i(x,y)

, is prescribed, is given by the expression

=

-k

;

+

(24)

where where

R0

J

j ('-+

+(-=

j (-+

(#?

k (;

7c

--,°

It has been shown in Section 1.2 that cp satisfies the conditions

c+Iktó

=*Ç1)

o

For the solution of the complete boundary value problem as posed in Section 1.1, one condition remains to be satisfied. This ¡s the "no-flow

through" boundary condition, which is given by Eq. (1.6). The specialized form of (1.6) for a parabolic strut is

[_

=

R-lkOO

Ke

(2.3) ..4J o

k +

se.e)

L.dO(4') k

'c

zl:

(2 2a) .(2.2b) (2.2c) (2. k) (2.5) (2.6) (2.7)

(25)

where

= t-t (,ç

O M.)

EquatIon (2.8) Is a Fredholm Integral equation of the first kind in which the source density , a , is the unknown. The homogeneous

solution is usually taken to be Identically zero for thin bodies and this solution Is adopted In the present analysis.

2.2 SOLUTION OF THE INTEGRAL EQUATION

The function

4(,11)

Is of the order t2 , because it is composed

of the square of the disturbance velocities, and for bodies with a y-plane of symetry, it Is an even function of 11 ; that ¡s,

R-11+00.

Introduction of the disturbance potential p , given by Eq. (2.1), to the kinematical boundary condition, given by Eq. (2.6), yields the expression

lì.

=

4

-

iç%st

ad

The terms of order t2 in the last integral of Eq. (2.8) are

(2.9)

+

?J6

t

4(+(i1

(26)

R-l'400

4*3+

(3))

(2. 10)

But the. quantity

l.#V-'3

)=Q

is an odd function of 11 ; hence, Eq. (2.10) ls identically zero. The

remaining terms of the last integral in Eq. (2.8) are of the order t3,

and will be omitted since we are only Interested in obtaining the solution to the order t2 . Equation (2.8) reduces to

a- = (2. 12)

Equation (2.12) is the well-k.nown.thin-body approximation.

O(t3)

(2.11)

It should be noted that the kinematical boundary condition on the body surface is independent of the free-surface correction up to the second order.

Equation (2.7) shows that both y0(x) and

y(x)

are of order t

Using an iteration technique, we find that the solution of a up to the order t (i.e., a= ta1) is

(27)

R-l#OO

Replace a in Eq. (2.li) by [_2y(x) + t2a2(x,z;1t)] it in the following manner:

-J4

t'-od It

--

:

'(t)

dçd)

Ic

4ijjI%

I)

Ici

tt can be shown that the last integral of (2.13) is of the order t3

Keeping terms up to the order t2 we can write Eq. (2.13) as

=

and write /I

{O1b(

()-()\

-

I

ShÇ0

-o

1u

Thus, the solution of Eq. (2.8) for the source density a of a thin strut up to the order t Is

'3_,= tG7+ taj

(2.14)

where

=

(2.15)

Çç;t»

0(t3)

(28)

R-lkOO

___

+

)

:)

..Jto

/

V

Çi:c.

For the special case of a parabolic strut , ci and 2 are of the form

(2.17)

c.oste 51(&)+

Z

co s (ta)

+ -

(.k)

6-Q

+

k)

w

=

:p

(tf)) O(COS -)

-

Z%

+

Cos where Q..

te

¡J

A

cete

>

.k.ec3G..

2.16)

(2.18)

(2. 19) (2.20)

(2.21)

where

J= -+

(i.3(11

))

+L

+

w

L+W=

(29)

R-1400

For the purpose of numerical analysis, the function .' sin Bt(l-x2)

Is approximated as follows:

+ (2.22)

where J0 and J1 are Bessel functions of zero and first order, respectively. The right-hand side of Eq. (2.22) behaves like (l-x2) where t Is small, and it becomes a bounded oscillation when t

becomes large. This substitution particularly suits the need of Eq. (2.21), because the main contribution to W is at the range of 8 where (2.22) gives the best approximation (i.e. , t is small); and It has

the advantage of uncoupling the 8 and x variables. Inserting (2.22) into (2.21),.we obtain W in the form

:i: a[;ft

(4) '' (.)

4 z os ('-)

(4)

(_z4s

-Z%

J -

-.CO5

*))}

(

(30)

R-lkOO

Sectioñ 3 WAVE RESISTANCE

3.1

DEFINITION OF SECOND-ORDER THEORY

The wave resistance derived from an analysis of momenta at control surfaces taken at large distances from the body is given by the

equa t on13

where Z is defined by the pressure condition on the free surface

(Eq. 1.9), and Cp is the disturbance potentiaL The asymptotic

behavior of cp Is given by Eq. (1.28), which can be written as

(O

LjJQ

(*-) c.us(4(ii)

6tG ¿:;)

do

--

*1

JL

-))

CbS

e

¿a

(3.2)

At large distance downstream (i.e. , x

« o)

, only the square terms of

the disturbance velocities will give a non-zero contribution to the wave resistance.2° Hence, Eq. (3.1) is reduced to

RJf(

(-

()Z]

(31)

ralo -.

Í((5#

7..

-

t'

-R-1LOO

11-.

L

11cr-

n-)

+')

4-A.IÌ

));

0jJi

rt(-s

dch

For symmetrical bodies with centerplane distribution, 2 and P1

(3.3)

Introducing Eq. (3.2) to the above integral and making use of the Fourier double-integral theorem, we have the following final wave-resistance. express Ion :15

(++#i)

sec

de

(3.4) where cÓ

(-ç

(t

ds

c4 -

4(.1l) coS(..)

e

dclrk

(32)

where

The definition of second-order wave resistance used In References 19-22 is obtained by making use of the expansion

o

"P4. t.(=

1i(cr,';k)

#))

e

j1

R_1L#OO

a by Um , and all lengths by the semi-length L , R in. Its

non-dimensional form can be written as

R =

(?+

)

G

0

(3.5)

JJ)

os

(ß1)

od

. . . (3.6) a = ta1 + t3a2 + (3.7a) ta,2 + (3. 7b) p

tp1 + tp +

(3.7c) Q = tQ1 + t2Q +. (3.7d) R

tR1+ t3%

+ (3.7e) where (3.8

(33)

R-11400

The first-order wave resistance , R1 , Is positive definite, but the

second order , R2 , can be negative. Unless t can be made arbitrarily

small, then at some Froude numbers the.second-order correction (I.e., R = t2R1 + t3R2) will make R negative (see Fig. I and Table l).

If we examine the procedure more closely, we find there ¡s no assurance that the third, or higher, order can eliminate the possibility of R's becoming negative. Although the procedure is mathematically consistent,

It Is unrealistic. Let us state the definition of second-order wave resistance adopted ¡n the present analysis.

j+-t+ (Q-tQ) cose1çj

(3.10)

o

where

=

k)

¿

(3.11) Lo s

(4L.)1«

14 L

=

G(Ç

;

k) e.

¿. ¿7

+

JE

e.

dd

04

This, however, retains terms up to t4, but it is. correct, since wave drag Is a second-order force.

(34)

R-1400

3.2

WAVE RESISTANCE OF A PARABOLIC STRUT

On this section, the aim Is to.developthe necessary formulas for computing the second-order wave resistance of a parabolic strut according to Eq. (3.10).

The source densities a. and a2 are given by Eqs.

(2.17)

and (2.18), respectively. The quantity In needof further discussion Is the free-surface correction , i(x,y). From Section 1 ; $(x,y) Is defined as

__

-((+

(4+

(t)

(3.13)

where p. Is the first-order disturbance potential and is given by the

expression . .

-I-ø

t ,ù

=

d1

kk

ö;ç (3.1's).

To compute the wave resistance, we have to take the Fourier transform of the function x,y) . Because of the large amount of computer time

required, for such a calculation, the wall-image approximation Is used to estimate this contribution. For most surface ships,. the operatIng Froude number is small, and in these cases such an approximation is reasonably valid. For the present case, the wall-image velocity potential is defined by the expression

,,

' (3.15)

(35)

=

R11+OO

Substituting Eqs. (3.16) through (3.18) into (3.13), we can write $(x,y) as

I

1f_

7t

4

Applying the expansion given in Eq. (3.7b) , 4' (x,y) Is defined as

tç') =

j

-ii.

r'

It should be noted that $2 (x,y) is defined only when (x,y) is

outside the body.

The first-order amplitude function P1 is zero because of symmetry, and Qi is given by

=

(3.21) (3.16) (3.17) (3.18) (3.19) (3.20)

(36)

Jw

Ji--flÇ

fl(b)-b))

ir )U

(P-û R- I '400

Let (P2+ ¡Q2) of Eq. (3.12) be of the form

ç.QL;)= (ib+1w*IL+Ir)#443)

(3.22)

From Appendices B to D, the quantities

1b 1L and are identically

zero, and the remaining quantities are given as follows:

b=

3:. +

c1O7L)ICOS())

(z5

35

44q4.

golz\

,

(\

___

-

L

+

V

=

11j

{ t

+

A .4

(A+

b)--:ì

F1(-)

(+)

J+

.ces

(bJ

(37)

= Cos() {

where

(__(

\

(L_Cz) s

-

C

4 (i+

1g-î)

-

({-+

L +

#b3

(!-e-)

/

(z-T)c

4

4

C + 1J

-

+

i Z t*cN

+

/

ZC C

\

_z(:

(-eJ

4y)

et

-F C (i- c) 1))

+

_-

(i-2.e).

-R-lkOO

-C;

(3z)

- .ft

+ sn (

s:r () +')

+

¡it

(38)

b -

+

4ì()

+

- o.ozzz

b3=

= o.c4tzt-j

The term

(Ib+IJb)

can be identified as the contribution to the amplitude function from the zero-Froude-number correction to the kinematical boundary on the body. Correspondingly,

[(IL+Iw)

+

is from the finite Froude number and

(I+iJ)

is from the free-surface correction. The wave resistance versus Froude numbers for three

different sets of theoretical results is shown in Fig. 2 and tabulated in Table 2 (the sets are the first order, the second order, and the second order without free-surface correction Eiie. , I +iJ is omitted in

p p R_lLOO

A=

-

ze.os(4)

ti -l-'--

\zii\) t

F(') =n.() - (o()

---Cos() . the calculation]). z 1).

T

-

J_J4 '-4;::c

tft43

u,=

V-

(39)

-ossde

o

which can be written as

°°

-where K and K1 are the zero and first-order modified Bessel functions

of the second kind, and D is the draft of the top section. Hence at

sufficient depth of submergence and low enough Froude number, the drag of the center portion is predominantly due to the effect of viscosity (i e

R_lLOO

Section L EX PERI MENTS

DESCRIPTION OF EXPERIMENTS

The model used in the experiment was a parabolic strut whose principal dimensIons are shown in Fig. 3. The strut consists of three separate sections assembled so that the drag of the surface-piercing portion and thé center portion can be measured independently. This setup is necessary becausé the méjor contribution to the wave resistance comes from the portion of the strut that is closest to the free surface. For the purpose òf model design, a rough estimate of. the wave resistance of the center ection, in percentage of the total, is

-ft

D

I

-;t O

(40)

R-11400

of the upper part, and establishing the viscous drag coefficient from the center section, we can determine the wave resistance of the upper section under the assumption that the viscous effects and the wave-induced effects do not interact.

Li.2 MODEL CHARACTERISTIcS AND TEST CONDITIONS

The model consists of three fiberglass parabolic cells with

aluminum ribs Inside each cell, at about one-foot intervals, to prevent the cell from warping and the water from sloshing (see Fig.

M).

A drag balance was installed on the top section and in the center section of the strut (Fig. kA shows the balance mounted in the center section; Fig. 3 shows the 0.12-inch gap between sections). The cells were flooded with water to provide the weight needed to keep the strut submerged in water. A Hama-type boundary-layer stimulator, which measures O.O'42-inch In height, was placed k Inches from the leadin.g edge to stimulate turbulence. Figure 4B shows the experimental setup, with the model

restrained in trim and heave but free to roll.. The free-to-roll provisi:on Is designed to prevent the model from developing any side forces and roll

moments due to model misalignment, which could cause damage to the top drag balanceañd the model itself. The roll axis is inclined 15 degrees forward. This makes the model stable in roll wheñ it is towed in the

forward direction (see Figs. 5A and

5B).

For all runs, the roll angle measured by the accelerometer wasless than

0.5

degrees, which was considered an acceptable model misalignment for the experiment. The

strut was tested at two different drafts (of the top sectIon), 7.5 and 12 inches. Figures 5A and 5B show the strut béing towed In Tank 3 at the Davidson Laboratory.

The span of the center section is 1,5 inches and the measured resist-ance of the section includes the resistance of two gaps. For the case of

the 7.5-inch draft, the top section has a span of 7.5 inches and the measured resistance includes only one gap. Therefore, if we normalize.

the measured resistances of the two sect ions by, their respective wetted areas, and assume that all the gap résIstances are equal, we fiAd that.,

(41)

R-1400

when we take the difference ¡n drag coefficient of the two sections, the gap contributlonis automatically eliminated.

4.3 RESULTS

The resistance measurements from the two balances for two different drafts of the. strut are shown in Fig. 6 and tabulated h, Table 3. The corresponding drag coefficients normalized by their respective wetted areas, (the area between the gaps Is excluded) are shown in FIg. 7. Each balance Is accurate to within ±0.005 lb, and all measurements can be repeated to within the same range. Figure 7 shows that the free-surface effect begins to reach the midsection at Froude number above 0.26 for

7.5

inches of midsection submergence. At 12 Inches of submergence, this effect decreases; considerably, but it persists. However, with the aid of the ITTC and Schoenherr friction curves, data from the center balance are sufficient to establis.h the viscous-drag coefficient for the strut..

When this newly established viscous drag is eliminated from the top balance, the remainder is the uncoupled wave-making component, under Froude's hypothesis. The wave résistance which appears above Froude number 0.26 , in the center balance at 7.5 inches of submergence, can be

obtained in a similar way. The sum of the wave-making components of the two sections is taken to be the wave resistance of the infinite strut; and the correspon:ding coefficients based on the wetted area of the top section are. shown at the bottom of Fig. 7.

FIgure 8 shows the total drag coefficient for an 800-foot parabolic ship at standard condition, and Fig. 9 gives the corresponding effective horsepower, requirement. The form drag appearing In Fig. 8 is obtainèd by eliminating the contributions from the gap, the turbulent stimulator, and the Schoenherr friction at test condition from the viscous drag in Fig. 7. The resistance of the Hama stimulator Is given in Reference 26, and the contri bution from the. gap is assumed equal to the difference between the Schoenherr and the ITTC friction.

(42)

R- 11+00

Section 5

DISCUSSION OF RESULTS AND coNcLusioNs

According to the expansion procedure that was used In References 19-22 (see p.23), the second-order theory can give negative wave

resistance. This is revealed by the result of the parabolic strut shown In FIg. 1. Taking a strut with a beam-length ratio of 0.1 as an example.,

the wave resistance , R =t2R1 + t3R2 , becomes negative at Froude

numbers 0.205 < F < 0.215 . This is physically unrealistic. Therefore

a new definition Is adopted in the present analysis; that is, the second-order wave resistance. Is based on the improved disturbance potential without expansion of the amplitude function ¡n the wave-resistance integrai (see p. 21+). The results are shown in Fig. 2

together with the results from application of the Michell theory and the data from tank testing. A remarkable différence between first- and

second-order theoretical resistance is revealed by the figure. Depending on the Froude number at which thè comparison is taken, the ratio of the

first to the second order can be as high as 3.63 at Froude number 0.21. The intermediate result of the second order (without free-surface

correction) indicates that the correction to the kinematical boundary on the body is as important as the non-linear free-surface correction. This conclusion is contrary to the one which was based on the results for submerged. bodies)'18

Although the second-order resistance in genéral is still higher,it corresponds rather well tothé data obtained from tank,testiñg. Such correlation is not surprising, because the effect of the boundary layer near the body surface dampéns the waves generated by the body. This Is particularly true for the stern wave system. In the Froude number range 0.27 F 0.33 , the free-surface correction becomes

insignificant and the agreement with experiment lessens. This may be due to the wall-image approximation used in computing the disturbance velocity on the free surface. . . .

(43)

R-1400

Appendix A

SOURCE DENSITY DUE. TO.THE.EFFEcTS OF FROUDE NUMBER

The portion of the source density which depends. on Froude nùmber according to Eq. (2.19) Is

where H is defined ¡n Section 2 ár,d can be written as

1h1r4+

î:

¿+

k ()-tsecQ)

The limiting process is to be taken after the reva1 .of the singularities. The singularities which may lie on the first or the fourth quadrant

depending on the sign of w. and i will consequently require a

different contour fór evá.luation of their corresponding residues. Since the procedure Is straightforward, the detall is omitted in this appendix.

Ça I

. (x-i) > w > O

and a> O

N=

do

¿ de +

e

e)

¿i.

(A .2).

LW

=

j:

(44)

and

R-1OO

Case 2:

(x-i) < O .-'-

< O and < O

.L, ) e

d

'dG

A..UQe) e

¿

e)

c K

de

where ,

J s4

#ç'}

os(kCs+j

(-)

C.oS +

'ii, se

('c.oe

-Note that A(k,e) = A(k,-e) , and that iii and 2 can be written in

a common form Ì which is the same expression as Eq. (A.2) In another form. That is,

=

tJÍJA()

de

+JA(k1e)

kdG

-j

de

H

(A .3)

Denoting

and recognizing that

Jcos()ds

=

tk(k)

i

(45)

R-lkOO

;;L

3:Ç7

d

p

d

TL) (

J kcoso

J

kose

_____

&

+

{

c.cs(

k(#>)

dk

-

e.

)--I

(A.k)

We have a closed solution for the last integral of (A.k), and we now integrate the remaining integral by parts, obtaining the final expression for in the form

f

+

(&+kcos)

)

o

I (t4koso)

2k

d+[(icos)_Fe

)

-it

-

4Cos()

+

¿G -

4.jos

(x

6

áe

where

F

- --

kc.os (k) +

(q))

;

o = J. S?SZ0(

4

-_( «-

(A .5)

whe re

:i5=

_t)ose_Lse)

,.

(46)

:::À.

((ose)-áde

t

. JL z

_Q&#ìcos

V?&O;,

)OEec1G

-Iii

dg

-+.jjj4kos) v:A:

R-1OO .

j

(.f4f)

+ (°s

+

,1-\

4+f ic)

£&10

¿ULd)dK

ed

(A. 6) (A. 7)

j(4Qe

de)

k

jci_k:e

ZóSO

dojdk

ed 4

¿de

(47)

R_lLOO

For small t , Eq. (A.8) can be written

1T

-JÍ(i-)

os6 s

(&) z%se

(&&°

(-ke'

dkcSe

4OCtl)

{srre

c..os

#-

rn.

)O6G

+ ii'-)

w

=

4{te

(tos

-))--Z)

-

- QOS

L+V=

(*)

tt

\-

(A.8)

u yo

(48)

=

j(z

-

(-) Cos

-o

Appendix B

AMPLITUDE FUNCTION DERIVED FROM THE CORRECTIONON THE FREE SURFACE

Definition of Terms In Appendix B

=

-

j

(i-fl

;4 (z4))

R-JOO

X3

= J+2

d

X4

( Cos1(t

39

(49)

(._\)

Qos(»1(_)')

K3

4Z

DeinItion of Terms in Appendix B (Cont'd)

(z

4

(

(z

s (

i:

R-1400

t)

s() ¿d%

(50)

R-l00

14

tr)

B. )

where $2(x,y) is defined by Eq. (3.20), whIch, when introduced in the above equation, yields the expression

_T'.1.j

''

\ßs(.it)

¿)(

d?(

kT

j J1_1

Ç)14)_i.4)

+

4.>'

L

c.os(

e'c1«

JO& Ò)1

j

'SU-t

(8.2)

the last Integral of Eq. (B.2) is omitted because it is of the order t

integrating the reñaining integral by parts with respect to x , and.

then interchanging the order of integration to evaluate the y Integral,

gives the result . .

_____

____

_ß1bC-t

¿1;h1)

The real part of Eq. (B.3)

,

I , Is identically zero, and the

imaginary part , , is given by the expression

(51)

whe re o b? R_lLÖo

tI

k-'c.os

Có51(-

(B 6)

Replace x by Ex +

(l-I-i)]

, then by (l-2) , in Equation (B.5);

and replace x by

Ex -

(l-)]

, then by (2g-l) , in (B.6). Then

q(1)

and q(2) assume the following form:

pI

J(+z

n

cos(K+1

o

Consider the integral

jj(+) i& (

os (s+o)

dd

û It can be written as A1 jI (8.5) (B. 7 )

)

(zO (zj- )

n j

csj1(-))

(B.8)

(52)

'I=

(K1-çT(

È'1

=

Ê

¡'t

k

\t

(t4Zb)\ ('h)

4)e"

chc

o R-lkOO

rA2

= >

() co-)

(#-)

&d

K5<5')- A1

(t

A=

5n

(

5(ß4&

¿

+ ,

(

os(1(-

dc

u1(Ì1- 4)-A(Z4

'Q +

(ç+ tç-L +

-143

g;)

*

42)

)i

(:.) C.G5(J(#) e

+

(Ç+ KçZ4)

A

= Li1 (1+

-K) +

(53)

..1 R- 1k 00 (' (*4

(4

(;\-iP;) + (A-z,Ç)

=

where

- -'i-

(K4# K

'B= ti1(K-(3') +A(K-K + .t)1p1(K3+\Ç)

):'.1% ¡

B=

tJ1(LK

+

(2

- u

(i+

RI

-

- ,

(-))i

155= Z

( K-zi)

--.

(tJ) K,

=

4s

cos())+

-

J% cos(1')

o

i

(?-p) si.(') .k.)

ßÇ

d'

- z cos

- z(31 s

= -.

5i{{-z(i\O)+()}

+

+(i-.)(«1)+ï)}

where

y = 0.5772157

(Euler's constant)

'J

(54)

Jeos(.i)

R-11400

co s (i)) -

sì n.(ì)

(i-_))

¿os (i.ij (z- i)

()

(i-44+

os(P)

-

(-i))

cos()

-

+

o(4('

.)

Sur\.(t.)

D3

-

(i4)1s()

-134=

u(i+

'. sn(Lç

-

--Qos(j. t-2n ed

(z.-t) S

(-I) QQS

(55)

For integration, the function xn(x) may be fitted by another curve with equal area ànd the same end conditions; that Is,

-xLn(x)

2x-3x2+x

Integrating by parts to get B4 and 85 in terms of -xLn(x) ,wè then replace the latter by (2x-3x2+x) , to get the final expressions

of B4 and 85

B4,=

os(IiI{ii.z

+

{

Jr

i

Jfl4.&\

Ñ1

)Ç R-140Ò

The sum of the Bn's yields the quantity [q:t1)+q2)J whIch appeared in Eq. (B.4). Thus the amplitude function can be written

os({z(i_fr)(()+Y)_

(z_j))

.4

(tJ

(iI«%) i.

ï)

+

4

4

(t

X

* I

>t))

j

{1i4 .t1+ t))(4 (X) +)

-3

.

(a+

.

+.) +

,$

ß4

(56)

R-1400

Appendix C

AMPLITUDE FUNCTIONS DUE TO THE FROUDE-NUMBER-DEPENDENT SOURCES

The contribution to the amplitude functions from the Froude-number-dependent sources is defined as

C)

Iw+.I.:Jw=

1k4

:i;.

);..

t'4

(c.2)

where W

and L are given by Eqs. (2.23) and (2.20), respectively.

Ìntroducing W and L to the above expressions, and then Interchanging hè order of integration, we get this result:

4

p" 4 qj:

ÇI4)

j).(3-...

(U)

(44)_)

u ()-ti)

(#A)

)

(c.4)

+

(P.t4#

-çe

(C.3).

(57)

where.

+

=

A

(iV

=

0s ()

=

-

Sn(

R-1400

4.= *+.+y

r[-tc.

+

')

(4C)

kcos

s13 ¿'d

e

(C.9a)

(c.9b)

=

i.

L2' 4_3ZCF"

(58)

-R-1OO

yi=

;no 3ck

(c.9c), t.u

4c

=

( f(i+z+

G$$

13de

L

5Lc

\ so

24.

2()

e

s&the

(&!*)j

(a3+

')

(4

(+

sO)

When the integrand is rearranged takes the form

ko

ì

()6

(Ci4))

(cA.ø;L)

'

(i#&e_

Z6'

¿ce

(C.9h)

+)

j . (c.9d) (C. 9e) (C.9f) (C.9g)

2se'ckce

()

(59)

R- lkOO (i')

c3.

kY1L3G (s;r

-

c') ke

Js

-

j

(t# t) (ie +

+

6

(

.-ÇQ ( HQ¼s 5 o

-6

;flo

ò-.

cQc:E

)

(tt) z)(+ (j')

I

e+

i

l#'

'j

Ik6no4-c

Q+C3 430

All the inner integrals of Eq. (C.lO) can be evaluated by the residue theory, and r5C1) reduces to

-

j

-i-

+

G

¿e

43Clí

___c.

CJ+4d

c.zs. J

M+&

C.

(#i) jkcik

J +e>

2C.J i+c.)

(4

t)2

(C. 10) (C. 11)

We then change the variable on the first two terms of the last integral of (C.11) by letting

(60)

(d

15

\

1dk

+

de

j

+

4%)

4Ñ)J (4

j

t?j

(j o

+

3Z$C

Ç

(zf

-\).Z.

)

d'-- àL:.7- (,A_

zth-ç I t QL'Y7L I C. t

''° 1

)

kjcr

I)_ iç_(_

c .1

- .yM)L

J'ItJtP)

R- 11400 c fi

-

j)%_+

4-'

We can obtain, r (2) , r (i) , and r.

s c C

fol lowing results:

- ZCn (z

4

t()

öl

¿'3.

+ì4-(.)4

-..riE)

4&[.4+

:c;

in a similar way, with the

(4#zt.')

j

(.4;'Ç)

s -

.

T'-.)

{-c

(C.12)

(C.13) The final result of s of the form

+

3z e

(61)

R_lLOO C.t

ik

A(ZC)

ja

(c4#

1

jj:'

ItJ4L

+

n (zc.) - ___

e1)

+

n

(c+ji')--f 4-L

Q -

.4. "

+

+

(i 4c+zc4

(1

114- iiE"\

4\3

3ce1

(c.k) (C. 1 5)

In Eq. (,C.6) the major contributions of f5 and are

(r5(1) + r5(2j

) and (rc(i) + rc(2) -

J)

respectively; the contributions

from ÇF (»

+ ) and (r + F(2) ) are comparatively small. To

avoid the lengthy computation of small quantities, we can closely approximate these integrals by using the technique described below.

Consider an Integral A , which Is a typical form of the integrals

that appeared in Eqs. (C.9e) to (C.9h).

t

I '

o#(cz._6;o')sw.o e

=

>

I

j S

(fl

y

+

)

Since O c 1 and C2X1 =

cv1 =

> i , the main contribution of (C.16)

(62)

o_e R-1400

=

F I 6)c-

£flO

TtO (4.);) ( 4.

t)

( b4)

A (Q

= --

>;

(s;nOf

cc-

-zke

Jô'()

tM*

de

k

(C.17)

(C..18)

'

:'

ddk

Equation (C.20) should béa good approximation to (C.l6), because the

-2k sin e -2k8

error introduced by replacing e by e us compensated for by replacing [k2 sin2 e + y12) (k2

e+2)]

by (v1t)2 Using the above-described technique to evaluate Eqs. (C.9e) to (C.9h), we have the following:

(c.2o) where 81 and 62 are constants between zero and one. A parametric study of Eqs. (C.16), (C.17),and (C.18) reveals the relationship

AL(0)

Au()

A J9)

Equation (C.19) is true for a range of values of t and c , espécially

(63)

._.(t) .._.(z

Ç

Ti+ J

=

3(ZC-i)4 4b1(i-c3)-

4b)

+

({-ze)# O(i)

(c.21)

-

iz&

zC

-(C.22) where

=

o.riszi4-ir

t,

= -o.o3zzz3.

3

=

The final expression of (C.7) and (C.8) is

13+_

1j.. _Çj

5Cv'

-f= z-cT-cV)Wzc)4(j-)

[-

-')

c + i4civ-

r)

+

¿'eI '

4(C(zQ)+13

ji:J

+

*c._Iz(i+cW+.

Cz(ZT

(t4c)T)

where zC

(t2-+)+

cL(zc.)

+_ZC..t

i

i'\

U,

Q3ê1

+

c(1-cU)

+

-

(-.z)

4

(64)

-Appendix D

SECOND-ORDER AMPLITUDE FUNCTION (ZERO-FROUDE-NUMBER CORRECTION)

The contribution to the amplitude function from the zero-Froude-number correction to the kinematical boundary is given by

4(

(i-3

(1#

In Eq. (D.l), the real part

1b

is identically zero, and the Imaginary part

b can, when integrated by parts twice,

be written as

cs1)

D .2)

The non-oscillatory part of the Integrand In Eq. (D.2) is fitted by a polynomial. That is,

(

t5+5+3'')

which, when substituted in (D.2), yields the final expression for J1,

-

-

3Z

f (

3

qoz' os(Ò

(65)

R-1400

REFERENCES

MICHELL, J. Hi,, ,"The.Wave Resistance of a Ship," Phil. Maq., Vol. 45, 1898, p. 106.

HAVELOCK, T. H., "Calculation Illustrating the Effect of Boundary Layêr on Wave Resistance,":Trans. Inst. of Naval Arch., March 948. WU, T. Y. T., "Interaction Between Ship Waves and Boundary Layer,"

International Seminar.ón Theoretical Wave Resistance, Vol. III, August 1963.

L,. WIGLEY, C., "Effects of Viscosity on Wave Resistance," International. Seminar on Theorètical Wave-Resistance, Vol. III, August 1963. WIGLEY, C., "The Effect of Fluid Viscosity on Wave Resistance Using

a Modification of Laurentieff's Method - A More Accurate Calculation," C. Wigley, Flat 103, 609 Charterhouse Square, London EC1, England,, May 1966.

INUI, T., "Study on Wave Making Resistance of Ships," Soc. Naval Arch., Japan, 60th Anniversary Serles, Vol. 2, 1957.

BRESLIN, J. P. and ENG, K., "Calculation of the Wave Resistance of a Ship Represented by Sources Distributed Over the Hull." Paper presented at the International Seminar on Theoretical Wave

Resistance, University of Michigan, Ann Arbor, Mich.., August 1963 (DL Report 972, July 1963).

EGGERS, K., "On the Determination of the Wave Resistance of a Ship Model by an Analysis of its Wave System," Proc. International Seminar on Theoretical Wave Resistance, Ann Arbor, Mich.., August 1963, pp. 1313-1352.

KOBUS, H, E., "Examination óf Eggers' Relationship Between Transverse Wave Profiles and Wave Resistance," J. Ship Research, Vol. 11,

No. L1, December 1967.

SHARMA, S. D., "A Comparison of the Calculated and Measured Free-Wave Spectrum of an muid in Steady Motion," Proc. International Seminar on Theoretical Wave Resistance, University of Michigan, Ann Arbor, Mich., August 1963, pp. 201-257.

IKEHATA, M., "The Second Order Theory of Wave-Making Resistance," J. Soc. of Naval Arch., Japan, Vol. 117, 1965.

(66)

R-1400

PETERS, A.S. and STOKER, J. J., "The Motion of a Ship as a Floating Rigid Body ina Seaway." Communication in Pure Appl. Math, 10, 399-490 (1957).

WEHAUSEN, J. V. and LAITONE, E. V., "Surface Wave," Encyclopedia of Physics, Vol. IX, Fluid Dynamics III, Springer-Verlag,

Berfln, 1960.

1k. HAVELOCK, T., "The Vertical Force on a Cylinder," Proc. Royal Soc., A, Vol. 122, p. 387 (1928).

HAVELOCK, Ï. "The Forces on a Circular Cylinder Submerged in a Uniform Stream," Proc. Royal Soc., A,.Vol. 157, p. 526 (1936). BESSHO, M., "On the Wave Resistance Theory of a Submerged Body,"

Soc. Naval Arch., Japan, 60th Anniversary Series, Vol. 2, 1957. TUCK, E. O., "The Effect of Non-Linearity at the Free Surface on

Flow Past a Submerged Cylinder," J. Fluid Mech., Vol. 22, Part 2, June 1965.

SALVESEN, N., "Ön Second-Order Wave Theory for Submerged Two DimensionalBodies," Proc. 6th Nava.1 Hydrodynamics Symposium, Vol. II, Washington, D. C., 1965.

SISOV, V. G., "Contribution to the Theoryof Wave Resistance of Ships in Calm Water," Izv. Akad., Nank, SSSR, Dept. of Tech. Sci., Mechanics. and Machine Constructions, No. 1, pp. 75-85, 1961

(Bureau of Ships Translation No. 887).

MARUO, H., "A Note on the Higher Order theory of Thin Ships," Bulletin of the Faculty of Engineering, Yokohama National Univ., Vol. 15., Yokohama, Japan, 1966.

YIM, B., "HigherOrder WaveTheory for Slender Ships," Hydronautics, Inc., TR503-1, Feb., 1966.

EGGERS, K., "On Second Order Contributions to Ship Waves and Wave Resistance," Proc. 6th Naval Hydrodynamics Symposium, Vol. II, Washington,. D. C., 1965.

SNEDDON,. I. N., Furier Transforms. McGraw-Hill, 1951.

2k. HAVELOCK, T., "Some Cases of Three-Dimensional Fluid Motion," Proc. Royal Soc., A, Vol. 95, p. 355 (1918)..

LUNDE, J. D., "On the Theory of Wave Resistañce and Wave Profile," Skipsmodelltanken Meddelelse NR. 10, April 1952.

TAGORI, T., "A Study of the Turbulence Stimulation Device in the Model Experimenton Ship Form," Ship Model Basin Lab., Univ. of Tokyo, Japan 1963.

(67)

R-11400

TABLE 1. RATIO OF SECOND- TO FIRST-ORDER WAVE RESISTANCE OF A PARABOLIC STRUT ACCORDING TO THE EXPANSION PROCEDURE

R = t2R1+t3R2+... R2 = R2(0)+R2(p)+R2(F)

R2 (o) , ZERO-FROUDE-NUMBER CORRECTION

R2 (F) , FINITE-FROUDE-NUMBER CORRECTION R2(p) , FREE-SURFACE cORRECTION F R2(0)/R1 R2(p)/R1 R2(F)/R1 R2' .20 -0.792 -4.054 -4.808 -9.654 .21 -1.986 -4.562 -5.214 -11.762 .22 -0.362 -3.931 -4.70,9 -9.002 .23 -0.030 -2.611 -4.224 -6.865 .24 -0.997 -2.750 -4.281 -8.028 .25 -1.848 -3.323 -4.667 -9.838 .26 -0.477 -2.787

-4576

-7.840 .27 1.451 -0.756 -3.716 -3.021 .28 1.239 -0.087 -3.488 -2.336 .29 0.655 0.102 -3.370 -2.613 .30 .047 0.183 -3.290 -3.060 .31 -0.578 0.192 -3.329 -2.715 .32 -1.175 0.055 -3.525 -4.645 .33 -1.405 -0.410 -3.816 -5.631 .34 -0.371 -1.307 -3.901 -5.579 .35 1.947 -1.976 -3.490 -3.519 .36 3.542 -1.835. -2.991 -1.284 .37 3.977 -1.375 -2.713 -0.111 .38 3.914 -0.951 -2.569 0.391+ .39 3.720 -0.620 -2.461 0.639 .40 3.515 -Ò.364 -2.353 0.798

(68)

R-1400

TABLE 2. WAVE-RESISTANCE COEFFICIENT x 1O OF A PARABOLIC STRUT

Beam/Length = 0.1

F First Order Second Order

Second Order Without Free-Surface Correct ¡on .20 0.2794 0.1028 0.1734 .21 0.2639 0.0726 0.1400 .22 0.2022 0.11714 0.1616 .23 0.14805 0.21496 0.33714 .214 0.6239 0.2547 0.3729 .25 0.141458 0.1621 0.2516 .26 0.3261 0.2169 0.2578 .27 0.5567 0.5047 0.5129 .28 1.0413 0.8904 0.8818 .29 1.14878 1.1828 1.1627 .30 1.6751 1.2512 1.2246 .31 1.5635 1.0882 1.0681 .32 1.2615 0.8050 0.8115 .33 0.9401 0.5631 0.6166 .34 0.7585 0.5026 0.6159 .35 0.8246 0.7062 0.8809 .36 1.1857 1.2013 1.4270 .37 1.8371 1.9757 2.2324 .38 2.7384 2.9951 3.2560 .39 3.8298 4.2138 4.41483 .140 5.01456 5.5810 5.7577

(69)

R-1400

TABLE 3. DATA OBTAINED FROM DRAG BALANCES

Draft = 7.5 in. Draft = 12.0 ¡n.

Velocity ¡n fps Drag in lb Top Mid.

3.822

.975 .989

3.717

.928

3.620

.876 .911.

3.526

.810

.870

3.417

.714

.820

3.320

.643

.770

3.217

.601 .731

3.140

.566

.704

3.052

.533 .669

Velocity

in fps

Drag in ¡b Top Mid

3.852

.702 1.043

3.720

.665 .994

3.617

.607 .924

3.523

.573 .890

3.419

.507 .836

3.313

.442 .761

3.217

.405 .730

3.134

.374 .704

3.040

.361 .659

r

2.908

.347 .616

2.808

.308

.576

2.673

.288

.535 -2.593 .268 .495

2.502

.243 .459

(70)

w

z

4

I-(n (n w o: w >

4

o: w

o

o:

o

(I) o: IL

o

I;.

o

z

o

u

w U)

u-o

o

-8

o:

R:t2 R1+13R2

R2: R2 (O )+ R2 (F) R2 (pl R2(0),ZERO-FROUDE-NUMBER CORRECTION R2 (F)1 FINITE-FROUDE--NUMBER CORRECTION R2 (p), FREE-SURFACE CORRECTION 4-0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 FROUDE NUMBER

(71)

2.4 2.0

o

X i-I.6 z

w

C.) IL.

u-w

o

o

1.2 "J a' .j:-I (I, U)

J

"J > 4 0.4 e

s e

EXPErIMENT4L RESULTS SECOND-ORDER THEORY

SECOND ORDER (WITHOUT FREE-SURFACF CÖlPFCTIAr

I

- - - --

FIRST- ORDER THEORY

-..',

j BEAM/ LENGTH: O. IO

/1

/

li

/

Ryj /

r-

2 2 a

2pUP

I'

t

/

t

/

I

/

/

0.20 0.22 0.24 026 028 0.30

FROUDE NUMERF

¡

1,

/ 1/ /

£

/

,

0.32 0.34 0.36 0.38

(72)

R-11+O0 60.00 W.. L. 0.12 GAP-1

I

t

0.12 GAPI

HALF BODY 0F REVOLUTION

I

-ALL DIMENSIONS ARE INCHES D:DRAFT OF TOP SECTION

WETTED AREAs 0.8388 FT2/IN OF SUBMERGENCE

DISPLACEMENT: 8.68 LB/IN

FIGURE 3. PARABOLICSTRUT MODEL USED IN EXPERIMENT

t

6.00

i

t

15.00 48.24 f 15.00

(73)

FIGURE. Li.A., MOUNTING OF DRAG BAINCE IN MIDSECTION OF STRUT

.':

---/

I, L - _,; ) R-1h00

(74)
(75)

.2 ¡.0 0.8 0.6 0.4 Q2 O R-1400 DRAFT 0F TOP SECTION 7.5 ¡N. 12.0 IN. BALANCE 'ro P

o

MID

s

A 2.4 2.8

32

3.6 4.0

MODEL SPEED ¡N FEET PER SECOND

(76)

w

08

tu t-w

o

C 1.0

< 0.2

o

D DRAFT OF TOP SECTION

ST: TOPI

1-SECTION WETTED AREA

SM:MIDJ (EXCLUDED GAP AREA)

BEAM! LENGTH: 0.10

SM:12.58 FT2

0.__-.----_._

0.22

\ITTC FRICTION

SCHOENHERR FRICTION

WAVE RESISTANCE DERIVED FROM

EXPERIMENTAL DATA (BASED ON S1:6.29 FT2)

VISCOUS DRAG 0F STRUT

INFINITE STRUT

-7.5 IN. SECTION

FROUDE NUMBER

0.26 028 0.30 0.32 034

D(IN.)

ST(FT2) TOP

MID

7.50

6.29

0

S

12.00 10.07 A I .2 1.4 1.6 1.8 2.0 REYNOLDS NUMBER X IO

z

o

0.6 w (n N

(77)

I.0

4

w

4

0.8

I-u

z

o

o

0.6

4

w w

o

X

I-z

0.4

u

u. u. w

o

u

o

4

Q -J

4

I-o

I-C O EXPERIMENTAL RESULTS SECOND-ORDER THEORY

FIRST- ORDER THEORY BEAM / LENGTH: 0.10

WETTED AREA:I.61x105 FT2(BOTTOM AREA EXCLUDED)

0.20 0.22 0.24 0.26 FROUDE NUMBER REYNOLDS NUMBER X

I

/

/

/

/

/

/

/

/

SCHOENHERR FRICTION PLUS FORM DRAG

SCHOENI-IERR FRICTION (59° F SEA WATER)

(78)

25 Commanding Officer

OFFICE OF NAVAL RESEARCH Branch Office

Box 39

FPO New York, New York 09510 6 J. S. NAVAL RESEARCH LABORATORY

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U. S. NAVAL RESEARCH LABORATORY Washington, D. C. 20390

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CHIEF OF NAVAL RESEARCH Department, of the Navy 'Washington, D. C. 20360

Attn 'Code LI38 Code 473 Code 468 Code 421 Code 461 Code 466 Code 463

CHIEF OF NAVAL RESEARCH Department of the Navy Washington, D. C 20390

Attn Code 481 Director

OFFICE OF NAVAL RESEARCH Brañch Office

495 Sumer Street

Boston, Massachusètts 02210 Director

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219 Dearborn Street Chicago, Illinois 6060k

R-1400

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Cop i es Copies

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New York, New York 10011

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OFFICE OF NAVAL RESEARCH Branch Office

1030 E. Green Street

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