A new interpretation of the wave resistance of point sources moving at constant speed below the surface of an infinitely deep fluid

v

Scheepouwkunde

Techriische Hogeschod

.&Ift

N

4

I

DAVIDSON

LABORATORY

TECHNICAL MEMORANDUM 1LJ4

A NEW INTERPRETATION OF THE WAVE RESISTANCE OF

POINT SOURCES MOVING AT CONSTANT SPEED BELOW

THE SURFACE OF AN INFINITELY DEEP FLUID

by

John P. Breslin

TECHNICAL MEMORANDUM 1i1i

A NEW INTERPRETATION OF THE WAVE RESISTANCE OF

POINT SOURCES MOVING AT CONSTANT SPEED BELOW THE SURFACE OF AN INFINITELY DEEP FLUID

by

John P. Breslin

TM_1L14

TABLE OF CONTENTS

Pa qe

ABSTRACT

I NTRODUCTI ON

VELOCITY POTENTIAL AND GREEN'S FUNCTION FOR THE LINEARIZED FREE-SURFACE PROBLEM

EVALUATION OF THE GREEN'S FUNCTION IN UPSTREAM AND 7

DOWNSTREAM REGIONS

THE FORCES ACTING.ON ANY PAIR OF ELEMENTARY SINGULARITIES 1L

ARIISING FROM THE GENERATION OF WAVES

AN INCORRECT PROCEDURE WHICH LEADS TO THE INTERPRETATION 18 OF MUTUAL UPSTREAM AND DOWNSTREAM INTERFERENCES

5. SUGGESTED RESEARCH ON WAVE-RESISTANCE

21

CONCLUSIONS 22

REFERENCES 23

ACKNOWLEDGEMENTS 23

ABSTRACT

The wave resistance of any pair of sources M at (,Th-C) and M' at

(t,.nt,_ct) with II being upstream of M' when moving at constant. speed U beneath the surface of an incompressible, inviscid heavy.fluid is known to be given by R =

sec3ee_2c2de

M' 2sec3ee_2C 'sec2e J2

2p2

iM'

j

cos(_')sec8}05\Iv 11_y'

sinesec2e]e')5ec8sec38d8

0

where = g/u2, g being .the acceleration due to gravity, p the mass density of the fluid. The first two terms are the resistances of each source were

they alone, and the last term is an interference term which depends upon

their relative separation. Since this interference term is proportional to twice the product of the soürcè strengths and is independent of the

trans-position of thecoordinatés, there has been a tendency to think of this term

as a mutual interaction. It is the purpose, of this analysis to. show that

It

arises solely from the action of the waves produced by the forward or upstream source on the downstream source and consequently the downstream source con-tributes nOthing to the interference term, as would be expected from physical

Intuition. ThIS simple fact can be exploited to study.the separate

contribu-tions of the seccontribu-tions of a ship to its wave resistance and to isolate. the origin of favorable and unfavorable interferences, provided that a floating

ship can be represented, by distributions of sources. A new experimental and theoretical effort is proposed to study the applicability of surface

distributions which involves theisolation of the resistance of the forward

half-length of models. Such a study will reveal the usefulness of the

con-cept of the local wave-resistance intensity and, as a by-product, would pro-vide insight to therole Of stern form resistance.

TM-l4

INTRODUCTION

There have been many analyses published in the literature pertaining

to ship hydrodynamics which deal with the waves and wave resistance produced (theoretically) by moving singularities. An excellent summary of this work is to be found in the treatise of Laitone and Wehausen1. The wave resis-tance of an ensemble of singularities, such as point sources, is of interest

when the source strengths are adjusted according to one approximation or another to represent the hull of a ship or a near_surface-running submerged body. This resistance can be ascertained by considering the flux of energy.

into the far wake and it can also be secured by calculating the pressure distribution on the body or by calculatin.g via the Lagally theorem the force

induced on the singularities in the interior or on the surface of the hull by the flows of the "image" singularities In the upper half-space. Each. method must, if correctly carried out, produce the same result and they do.

The character of the answer is that the wave resistance of an ensemble of

sources m1 is a sum of terms proportional to the squares of each m1 , so

that for two sources there is a term proportional to m1 , another proportional to rn and a third term proportional to twice their product

2nm.

This cross product term reflects the interference ef.fects of one

element on another and is responsible for the significant "humps and hollows."

In the theoretical wave resistance curves when plotted against Froude number. Similar, but greatly muted or attenuated, variations in the residuary resis-tance curves derived from.model tests are also noted.

There has been some tendency to consider these interference terms as arising from a mutual interference of the waves from longitudinally displaced

singularities, or that half of the term in the case of a pair of equal sources

(or a source and an equally. strong sink) is contributed by the action of the

after element on the forward element. This is physically unrealistic since It is known that the wave pattern generated by a travelling point singularity

lies behInd, or downstream of, the point. Since it would be instructive to

know both the local contribution of any section of a ship and the additional

contribution arising from the interference of all o.ther ship elements at that station to the total wave resistance, it appears worthwhile to isolate

the.wave-associated flOws from elementary singularities so one can definitely

-1-state the origin and character of these Interference effects. The following development shows In detail the structure of the wave-associated flows due to sources and reveals,through the use of the Lagally theorem, that the In-terference term comes èntlrély from the action of forward elements on after

elements and, consequently, that after elements cannot lnduce wave resis-tahce forces on those forward. This clarification may be of significance

In current studies dealing with the problem of minimizing the wave resistance

of ships.

This work was made possible by Internal research funds awarded to Davidson Laboratory by Stevens Institute of Technology.

-2-*

CP(x,y,z) =

See Appendix A

TM-l4

1. VELOCITY POTENTIAL AND GREEN'S FUNCTION FOR THE LINEARIZED FREE-SURFACE PROBLEM

It lsknown that the velocity potential function for the motions generated by a floating vessel moving at constant speed, which satisfies

restrictions as to small waterline slopes and to which are Joined

restric-tions of the small perturbarestric-tions of the free surface, can be expressed by

- _{_)G(x,y,z;,ThC)dS}

where G is the Green's function given by

(1.2)

+G'(x,y,z;,.T,C)

[(x

)2 +(y)2

+ (z+ç)?]u/2

v = g/U2, g the acceleration caused by gravty, U the free stream speed; : here G' Is regular in the lower half space.

In (I) the region S is the hull surface below the.waterflne and L Is

the undisturbed waterline intersection. The double integral in (1) may be Interpreted as the. sumof a"free-surface" or. Kelvin source distribution of strength density

p/n

over S and a Kelvin doublet. distribution with.

axes normal to S and of strength-density rp . The single integral may be

thought of as a distribution of longitudinally directed doublets of strength cp and sourcesof strength cp/ distributed along the waterline.

In addition to the kinematical conditions on . S, the potential cp

must satisfy .. . .

.2

2

(1.3)

x2 y2 z2

In the lower half space, excluding the interior of the boundary S and the

combIned linearized pressure and kinematical conditions on the undisturbed

free sUrface z 0: . . .

(---+)cp'O

To find G, or more properly G' , the additive function to the simple source in (1.2) located at x = , y = 1], z = -, we may proceed in several ways. One procedure Is to define a function

2

H(x,y,z)

=

C-z .r (1.7)

and to note that H must be harmonic and vanish in the plane z 0. It

must have the same singularity at the _{sam point in the lower half space as} so we may write

(1.8)

z r

and, since H _{vanishes at z=0,this singularitymust have a negative mirror}

Image of this singularity at _{z =} . Hence H' may be taken as

so that where and H' = 2 2 +

oI

-';; Vi/T

v-i. ,

where _{v = g/u2, U being velocity of free stream parallel to the positive} x-axis. _{The form of (1) then implies that the Green's function must} satis-fy the same relations,.(I.3) and (1.14.), i.e.

V2G = 0 _(1.5)

z0

(1.6)

Inserting (1.10) into (1.7) yields the following differential _equation:

+ v )G' 2 = H' -c )21/2 r'

_[(x)2

+ (y 11)2+ (

If we now apply the Fourier transform, defined by the following pair of relations, (y,z;w) =

.L

(1.14) f(x,y,z) = (1 15) (whence f(x,y,z) =

j

ff(x,y,z)e+1W_X)dxtcko) (1 16) to (1.13) to achieve where where TM- 144 -w w2z = SZRt(y,x;w1

The complete solution of this equation is

2

wX

'

d +

s a solution of the homogeneous equation, viz.,

2 h

h

Applying the inverse transform (1.15) then, one obtains

=

$e_Xt(yx;w)dxcJ,

+ (1.20) (1.17) (1.18) (1.19)

G(x,y,z) _IGh(W,y,z)e iWx&, (1.21)

which _Gh itself has to satisfy

V2G =0

and

+v=O on

Z0.

h ₂ Z

Use of the expression for H' in (1.20) given by (1.14) provides

=

-Se

(x)

_{e_;(22}

)r,(;y,x)tdX+ Gh

To evaluate the x' Integral, it is convenient to use the Identty

r'

_{[(x_)2+}

_{(y-T)2+ (z_C)2]1'2}

r

=IIk[(

e

z-c )+ I (x- )cos8+(y-i

)s I ne

dkd8(z<O)

-IT 0

Let us compute the x'-integral

J_ r-a)'(_2

_-

Jfe{(X_C)+I(Xt)cos9+(Y1)sIne] dkdedx

2,1

-

_-rio

SSfIT

k{(X-C)+i[(x'_)cos8+(y_11)sIne]

dx'dkd8

e

O(k2cos2e

_vk)e

-1.ro-Now

fe'c0cixt = 2,18(k cose-w)

where 5

is the Dirac-delta function.

So we are left with

zw2

= +

Lff(k2cos2e+vk)$eC1e\

_{6(k cose-w)e'}

k{-C+i[_

cos8+(y-TI)sinO] }

ddkdO + Gh

The w-iritegral

is now trivial, yielding

Ti z

r202

k2cos28

G'_Gh =

..L [f(k2cos2e+vk)

'

.ek{[CO5_Shh1]dxdkde

The x-Integral may now be effected to give

=

SScos

e+vk)ek[(z_C) +IW+]dkd9

k2cos2O - v k

-ITO

where = (x-)cose + (y-)sinO' and, by adding and subtracting -2vk in the numerator, this last expression can be seen to go over to

0

Because of the singularity in the k-integral, the principal value Is

taken as giving the value of this, integral. Elementary manipulation of the

0-integrals can be employed to reduce the expression for the total G to

where

= k + Im

= - 3'

-

$.v..{

_.1

The first three terms agree with those presented in the literature (see page li81i, equation 13.36 of. Ref. 1). However, in order to exhibit the

dif-ference in the structure of G forward arTd aft of the point (,ii,c), it is necessary to evaluate the double integral in (1.211.) carefully in these two

regions and this will reveal the necessary selection for Gh.

2. EVALUATION OF THE GREEN'S FUNCTION IN UPSTREAM AND DOWNSTREAM REGIONS

The double integral in, (1.211) can be expressed as the sum of two

integrals: -1

=-

tsec2 IT _J

I

0

=-

rsec2e

17J

0

I

'. k.cos20-(1.211.)

where these are to be taken as the 'real parts of

C-)

ii11ai k[('z_C)4 iw k cos20-v e 1w.,. + -c dK dO K- sec28 i +z_c)dK dO K- sec28 G 2.1)

= (x-)cose + (y-r)sIne

(2.5)

= (x-)cose - (y-)sIne

(2.6)

and the contours C of Integration in the complex K-plane are quadrants of Infinite circles (center at K = 0) including the real k-axis, the selection of the upper or lower quadrant being dictated by the necessity of achieving

convergent Integrals along the Im. axis. This selection Is dependent upon the

sign of the products and 1Kw since they appear In the factors

and e-. The contributions from the Integrals In (2.2) and (2.3)

then depend upon the signs of w4 and w_ or ultimately on the relative magnitudes of (x-) and (y-T).

It is, therefore, necessary to look at two regions, one In which x -

<

0, y _- > 0, i.e., forward of the source and to the starboard or

right side, and the other In which x - > 0, y - '11

>

0, I.e. aft or

down-stream and to the starboard of the source. The values in the areas to the left or port side must be the same as their starboard counterparts because of the inherent symetry of (1.24) in the quantity (y-fl) as should be

ex-pected physically.

Upstream

[(x_)

< 0 ; y - >

In this region

= (x-)cos8 + (y-1)sIne Is less than zero for

0

<

tan(-

=

vanishes for e =

and Is greater than zero for

< e

Also In this region

= (x-)cos9 - (y-i)sIne

<

0 for , -8-(2.7) (2.8) (2.9)

oe/2

(2.10)

where and The contours

v see

= tan f(m,e,z-C,v) m cosm(Z-C TM-l1

for I(+) in this region are: i) the lower quadrant

for 0 e < since w.v. < 0 and

1(0 + lm)w+ = < 0 when - < m < 0

and ii) the upper quadrant for <O < ,.T/2 since In this .e-range

Wf> 0

and i(o + im)w < 0 fOr

0m<.

The contour for

_{IF_)}

In this region

is the lower quadrant since w_ < 0 for,

o<e</2.

k.-Taking the principal value of the integral and the contribution of the pole on the real axis in each case being careful to go around the

con-tours in a consistent manner, yields the following lengthy expression for the

double Interal in(l.?k):

=

_2V{sin(wsec29) sec2ø

ez_5ede

+ _{Ssin(vw+sec2e)sec2} e\)(z_C)sec28dB

-fsinvw+sec2e

sec2e

ez_9ede}

- dmd e

J2

+ _{fsec2e .f f(m,e,z-C,v)e"} dmde - Jsec2e

note that

0<</2)

+v sec29.slnm(z-C) +,2sec48

-flb _{drnde (2.11)}

cw = g

1{2

x 2G G h . = 0 jsin(vwsec2 0' TM-14 1.T%2

in[v(x_ )sece1cos[v(y_T )s i n8sec28

sec28 e

z_)sec edO}.h

sec2B

eC

)sec28d8

(2.16)

In order the 0 for all x',y In the forward region, it would

seem necessary that

U2

= +

I\) cos[(x_)sec9]cos[v(Y_1)sinê sec28lsec38

eC sec8d8

rr/2

- 1.\) cos[vw+sec28]sec39 secede on Z = 0 (2.17)

0'

(where account has been taken of the fact that w(c) =

(x_)cos(y)5ito'O,

and hence the term involving the derivative of the limit vanishes).

However, it is also necessary that VGh = 0 and

on

z=O.

The first

integral,

in (2.17) will meet these conditions, the second will not. The second term is evidently not a complete wave function, but rather is a

truncated or. incomplete wave function being made up of waves with crests perpendicular to lines w.v. = a constant only over direction angles between

-1

tan and ,.y/2 . Vt becomes complete at x = and y

, but

vanishes rapidly as x Is made negatively large, and also vanishes along

the line y = i where , = ,.y/2 . Vt would seem that this term should be

considered with the local effect terms with which in consort it must stisfy the free surface condition and the Laplace equation since the double integral

I does. Therefore, to disallow all true waves

emanating forward, it is sufficient that

Gh = 1i

Ssi{(x

sec8ljcos[v(y_i

inesec2e]sec2e

eC29de

0 . (2.18)

-11-and then the Green's fUnction G for points upstream (u) or forward reads: G 1 1 r 0 sin(vwsec2e)sec2e

ez_sec2ede

sec28

e- dmd9 -

fsec2e

I

dmd9 m2 +v2sec48

This result does not differ with what has previously been found, but

it is not easy to find this explicitly stated

in

the literature.

Downstream (d) x -

> 0

;

y -

i

> 0

In this area, the quantity w Is greater than zero for all e In the interval of integration over e . Thus the contour for I() is the upper

quadrant.

Thequantity w_

has the following values:

> 0

e

< tan

= B

(2.22)

w_.

= 0.

0, B < è

Consequently the Green's function in this region has the following expression:

rr/2 + 2 {

J' sjn(,wsec29)sec2e

v(z-)sune

+

fsIn(vsec2e.)sec2

-12-ez_c

de 0 + sec2e ₁ e_rfll)+ _{dmd8 (2.19)} 0' 0 where =

tan1

{- :--}

x -

0 y - 'fl 0

(2.20)

and rn cosm(z-() + sec2B slnrn (z-) (2.21)

region I5 Gd = r1 -, + :ln[v(x_C)sece]cos[v(Y)sIne

sec2e]sec2e eZ_S28de

0

-

14\, S5mn(5ec2e)sec29

ez_sec2ede

B

{i5ec28

S

dmd8 - ,sec2e

ff

e°dmd8

+ sec2B

I

e-

dmde} B 0 where B = tan = y-T TM-l14t

fs1n_

sec28) sec2e

ez_5ec29d8

-{

$.e''4

dmdO - Isec2o

-rflj)_

+ fsec2e

_jf

e dmde (2.23)

B

where f Is given by (2.21).and Gh by (2.18). Substituting from (2.18)

and manipulating as before, the Green's function in the downstream or after

-13-(2.211.)

(2.25)

Inspection of the separate functions G and Gd reveals their

dif-ferent character. In checking the.11mltlng values as the field point x,y Is brought directly over the point , it is necessary to realize that o'

and _B are zero for this point since and w_ will have one sign over

the entire range 0< e < rr/2 . In can be shown that

G(,1) = Gd(,11) = 0

and that the surface elevation in the case that the Green's function Is In-terpreted as a source of strength M Is given by

0

Here K and iç _{are modified Bessel functions of the second kind of}

0

orders zero and unity. Tables of this integral have also been especially

2

tabulated.

The foregoing expression may nowbe. used to calculate the wave

re-sistanceof any one element (or source) In theprèsence of any other element. In this connection, there will be a need for the x-derlvative.of the Green's function induced by the images in the free surface at the singular point it-self. This Is given b,y

e2

secede _(2.27)

0

3.

THE FORCES ACTING ON ANY PAIR OF ELEMENTARY SINGULARITIES ARISING FROM THE GENERATION OF WAVES

Let there be two sources of.strengths M and M' located

respec-tively at and _',fl',' with the proviso that M is upstream or

forward of M' or simply that < '. We seek to find the forces necessary to hold these singularIties in the presence of a uniform stream U (from

left to right) with a free surface at any wave number v The potentials of these source, designated by are given by the respective Green's function multiplied by M/141r and

M'/lli-r.

The longitudinal force

acting on M is by Lagàlly's theorem

F =+pM

x and that on M' by -u

u_-U

Gd - g x - g

211%

2

= + U M sec38

eVC

secede

TTg =,y=1,z=-C ',y=T' x

-U M e

2 (K(f.)

Lg

-14-())

₂ (2.26)

(3.t)

(3.2)

Since interest Is centered on the fOrces generated solely by velocities induced by the wave-associated "image" terms only, such terms will be employed

from the Green's functions. Furthermore, It Is known that the local-effect

parts do not contribute to the wave resistance and hence they will be

omit-ted. in calculating the forces.

Then the only possible term from the image terms In cp In (2.19)

(when. multiplied by M'/lsrt) which might contribute i,n (3.1) Is

are:

i-r/2

u 2M' r

séc3O e2' secBdB

0

At this JunctUre, the force on the :leadlng source is then

tan

cos((-')sec9)

vli_il,lsjnesec2e)t)sesec3ede

The second term in (3.1) which represents the force Induced on the

soUrce M by Its own wave-associated image will involve, from (2.27),

Ti-TM-l14-4

= _MM'

+ 2M2

fsec3e -2 seced9

The corresponding terms occurring In the force on the downstream source M'

(2[?._)sece1cos[v(ti)s1ne

_sec2e]e'

= +

2p2

MM' 114 p\)2MM' cos[v('_)sece-.v

'2

_]-i-'i + pv2M'

f

sec38

e_2tsec28de

-15-i1-1i'l sinB sec2e

I

''i I

(3.3)

c3.I)

-'(c-'-c ' )sec28sec38dø

(3.5)

sine sec2e]e

t+C)secez3ede

(3.6)

1-r _'J

y=tan-where absolute va1ues have been Inserted on _T'-fl1 and

h-ii'

because the induced velocities must be even functions of the lateral distance from the

singularities because of the basic symmetry of the source flow.

From an inspection of (3.5) and (3.6), it is clear that. the terms arising from the "truè,catedt' or incomplete wave functions vanish, if M and

M' are on the same longitudinal line, i.e. if T1..= Ti'. Then

F = Pv2M2

f:cse

e2

secede

0

and there is absolutely no interference term arising from the aft singularity;

the force on the after source is, in contrast,

.22

F' - " sec3e e x 0. rr

2pv2MM' fos[v(t_)ece]

Ct+C)sec2esec2ede ;

C.',C >

0 C'

sece

-16-*

(3.8)

where the last term represents the tota.l interference of .the forward element on the after element. Thus It is now clear, at least for singularities in.

the same longitudinal line, that the interference term arises from the action. of the forward element (or, more precisely, from its wave-associated image)

on the after element and that the ,after element produces no wave-associated force on the forward element. This is certainly consistent with physically

motivated intuition. .. .

In

the

more general case where the sources are not in line, I.e.,

T1

fl',

it

is clear, upon summing the forces

F and F'.

that the terms

X

arising

from the truncated integrals (involving tan

-and

tan1

in their limits) sum to zero This is consistent with the

result

which would be obtained by considering the waves far aft since the

contributions of such terms to the surface deformat ion rapidly attenuate

with distance from the singularity and thus do not participate in the trans-port of energy into theultimate wake. Such forces as given by the first term of (3.5) and the second term of (3.6) mUst be regarded as internal forces arising from local flows and are similar to those forces which would arise

from M/Ili-rr acting on M?/ll1.Trt (here r' is not the negative image óf r

*

The first integral in. (3.8) can be expressed in terms of the modified Bessel function K0(vC') and l (uC') and the second integral. in terms of a series of

TM-I114

but the source at (E',i1',-') as well as the forces which are mutually

in-duced by the local effect terms (those involving the rn-integrals and the

function f in (2.19), for example).

It Is therefore possible to state that the longitudinal force density

on any point of a ship which contributes to the wave resistance Is made up of the wave resistance of that element itself plus the interference-resistance of all elements forward of that element (all elements aft of that station do

not contribute to the wave resistance density there). Thus it is possible to calculate a wave-resktance intensity curve which would reveal the separate

importances of the resistance of any slice of a ship and the constructive or destructive interference of the portion of the ship forward of a transverse slice.

Specifically the wave-resistance force-density at any station x' for' a ship represented by a surface distribution of sources is given by

/2 'dF' 2 x Pv = -;;-fl'M'(T',C';xt) sec39 e2\'529dO ds' fM'('c';x')ds'f

$

MI(,C)cos[v(x'-)SeCe]

5'

s=LO

cos[ih'_lij sinesec2e]e_\)

')s9

sec38d9 dds

where H' is now considered as a distributed source density over the

wetted hull and where the s' and s integrals are taken around the wetted girth; the -integral runs from the bow at x = to the general station at x

= Xt.

(Here the ship is considered to be running from left to right or held stationary

in a stream flowing from right to left.) A calculation of this local wave resistance-intensity curve would reveal much about the contribution of verti-cal section shape and the relative importance of the interference from all

forward elements. Such calculations are best limited to the forward half

length since the effect of viscosity becomes crucial in the stern regions.

-17-=+pMW

_dx

xI

x=

xP

y1

y1

z-C

Unfortunately, there is doubt that even at low Froude number the use

of source distrlbutión,s found from the hull geometry (ignoring the influence

of the free surface on the boundary condition) is sufficient or adequate since for a floating ship the velocity potential is made up of not only sources of

strength proportional to but also of a surface distribution of sources and x-di.rected doubletsa1ong the undisturbed water line as exhibited by (1.1).

Thl.s seeming deficiency of source distributions and the lack of realism in the

application of potential thebry over the entire ship leads to the following

proposal for. additional research.

)4 _{AN INCORRECT PROCEDURE WHICH LEADS TO THE}

INTERPRETATION OF MUTUAL UPSTREAM. AND DOWNSTREAM INTERFERENCES.

It might be thoUght possible to avoid the lengthy evaluation of the

double, principal-value integral In the Green's function as cited in (1.211.)

and insert the homogeneous solution from (2.18) to give the potential func-tion for a source strength M

iii

the form:

M l{(.x )2 _{+ (Y-1}

)2+

(z _+ç )]h/2 )2 _{+ (y-11}

)2

_(z-C

)2]1(2

cos[k(x_)cos]cos[k()sine]

_{e_dk d}

-P

j

. k cos8- ', 0

rr/2

+ 2.

f

si ri[v(x_)sece]cos[v

(y-)sin8 sec e]e''

(zC)sec

sec

0

An exactly similar expression can, of course, be written for the

pOtential cpt äf a downstream source M' at '

with. the

under-standing that . . ,.

The Lagally force acting on the upstream source is then presumably

given by

-18-(1k. 1)

TM-144

and the corresponding force on. the downstream source Is

F'= pM' + pM' (4.3)

x X

x'

_x=

y=i1'

z=-C'

z'

In evaluating the various x-derivatives, the contributions of only the last

two Integrals ln(4.l) will be taken since the forces of interest here are only those dependent upon the wave number . Thus, of one blindly

differ-entiates the last two terms in (4.1) and evaluates as directed by (1.1.2)

(remembering to change to and

c'

and M. to M' to give

the contributions from M'), the following expression for F Is obtained:

F

p242

j'sece e2

secede

pvjlM'

f

k

sin[k_')cose]co4k(mi1')sine]

e_k')dkde

IT

cose

rr/2 + py2MM'

rcos(-' )secO\cOs1v_1P)sinesec

e]eC')seede

I.) '-

-0 (1.1.1+)

Here the

first

term is the self-wave resistance of the forward source which

may be considered to arise from;the flow induced by Its own Kelvin image in

the free surface. The other two terms are presumably interference terms

arising as they do from the action of the wave part of cp' On cp (or M' on

M). Thl.s result does not agree with (3.7), and there is certainly no physical basis for the presence of the last term of (1.1.4).

The corresponding force on M' Is

F'

py2Mt2 $ sec3 e

e2

sec 8d9

rr/2 _{_1.1..t__.\l_SSk(.flu_)sine]} + p',MM'

r

K 2 _J 2 it IT!2

2MM' fcos[v('_)sece]cos[v(1'11)sit8 sec2e]e529sec3ede

0 . (45)

-19-Upon adding F and F' , it Is most curiOus to note that this

er-roneous procedure gives the correct total frce or rr/2 F + F'

secO

e2tS2

x x r _j 0 0 IT!Z + 2pv2MM'

jcos[v )sece]cos[v (li-il' )sin9 sec2e]e

This is obtained because the second terms in each separate force annul each other upon addition since

sir[k('-)cos8]

= -sIn[k(-')cose]and. inCail other terms, the transposition of primed and unprimed coordinates makes no

change.

The foregoing methcd clearly Implies that halfof the surviving

in-terférence. term comes from the action of the downstream sOUrce on the up-stream one and. the other half from the action. of the one upup-stream on the downstream source. Since this interpretation is Inescapable and, at the same time, Is physically untenable, it Is necessary to find the mathematical

flow in the calculation which gives F and F( through the use of (1.l).

ThemathernatIcalerror In applying (l.1) to obtain cp/x and

cp'/x lies in the interchange, of the differentiation and integration

processes in the.double interal term'of (4.i). As has been shown in the detal)edeVáluationof that double, principal value Integral, the 0-limits and the character of the contribution depend upon the relative position of

the point of interest x,y and the singularity location ,T1 Thus It

would appear that, the function represented by the double integral in Q-. 1) cannot be differentiated without exhibiting its dependence on x first

-20-d

i-ri2

+ py2M'2

_{'sec8 e2528d8}

J

2

secO

3

TM-l4J

5. SUGGESTED RESEARCH 014 WAVE-RESISTANCE

All the previously made compari.sohs between computed wave resistance and residuary resistance deduced from longitudinal force measurements on ship models have been beclouded by the lack of precise determination of the viscous

form drag of the model. Recent work now makes it plausible that the wave

pattern at different Froude numbers changes the viscous form drag so that the

residuary resistance as deduced by subtracting out the skin friction (with

an allowance for form) very probably has a different coUrse with Froude number than does the true wave resistance.

It is therefore highly desi rable to measure and compute only the re-sistance of the forebody. The measurements can be made by using a split model

carefully Joined by a flexible membrane at the jOint. TheforebOdY force can be sensed by an electrical balance which requiresonlY a few milli-inches of

displacement against a knowr spring.. The entire apparatus and after-half length

can be made free to heave and trim as is dOne normally. If the models selected for this work possess a reasonable amount of parallel middle-body then It' is certain that the potential flow pressUre distribution which would exist in the presence of a rigid free surface will not provide a sensible force as the

forebody will act very much like a half-body which of course has ,zero form drag. (This would not be true of blunt-bpwed tankers, for Instance, because. the

bluffness of their form in th.e region of the.forefoot can cause separation

and hence formresistance.) The residuary resistance of the forebody can then be estimated by approximating the skin friction. The theoretical wave resis-tance of the forebody can .be computed from a source distribution calculated

by the Douglas3 program. A comparison of this with the residuary resistance

of the forebody should be more meaningful than comparisons of the

correspond-ing values for th complete model. The Isolation of the wave resistance from

the total measured forebody resistance can be Improved bymeasuring the corresponding force on the model in the presence of its reflected image In a

wind tunnel. In addit.ion,a check of the reasonableness of the assumption

that the undisturbed pressure distribution gives virtually zero form drag can be made by integrating the theoretical distribution as provided from the

Douglas program.3

-21-Should this comarison be more favorable than previous work it may be-taken as a substantiation of the usefulness of source distributions. If

serious discrepancies still per.ist then efforts must be made to evaluate.

the, relative importance of each of the; terms in (1.1), and finally Of the

importance of solving the;boundary value problem at each Froude number. The importance of eachof the terms in (1.1) could b.e ascertained by using

a simplified form such as a. strut for ihich the uhdisturbed cp on the boundary is known (the undisturbed ;iS of course always known once the geometry and

forward speed are specified). In summary then, there are many incisive studies which shOuld be made with the. linear Green's function by improving t.he hull

representation even though these may be somewhat. inconsistent with there-tention of the linearized free surface condition.

A very important by-product of the above program, if applied to

several bow and stern combinations, would be the isolation of the contribution of the stern to form and wave resistance, and their-mutual interaction in the

stern regions. This is particularly re.levent to ships of high afterbody block coefficients. ;

CONCLUS IONS

It Is concluded that the wave resistance of an ensemble of sources (or doublets or higher order singularities) is made up of the sum of the separate wave resistance provided by each singularity plus the interference stemming only from each forward singularity, (no interference contribution

from downstream elements). This means that the relative importance of each can be assessed and the origin of favorable and unfavorable interferences

can be traced; this type of analysis has never been attempted.. A combined theoretical and new experimental program has been outlined which if pursued with care should reveal more definitively than heretofore the applicability

of existing techniques. for the prediction of the wave resistance of ship bows An important by-product of the' proposed isolation of bow resistance would be

the revelation of the role of the stern in form resistance and possibly the

action of the waves from, the bow on the stern form resistance.

-22-J

REFERENCES

Wehausen, John V. and Laitone, Edmund V.: "Surface Waves",

Encyclo-pedia of Physics, edited by S FlUgge and C. Truesdell,

Springer-Verlag, Berlin 1960.

"Tables of

e_0'e sec'1e.1

(8

sece)de" Admiralty Research Laboratory, Teddlngton, Middlesex, England, A.R.L./T.9/MathS 2.7, November 1956.

Hess, J.L.and Smith, A.M.O. "Calculation of Non-Lifting Potential

Flow About Arbitrary Three-Dimensional Bodies," Douglas Aircraft Report

No. E.S. 1i0622, March, 1962.

ACKNOWL EDG EMENTS

The assistance of Mr. King Eng, Research Scientist, in checking mathematical details and Professor T. V. Davies, Visiting Senior Scientist

In careful review of the final draft and.provislon of the Appendix is

gratefully acknowledged. Miss Jacquelyn Jones is thanked for her pains-taking work In typing this difficUlt manuscript.

-23-APPENDIX A

The application of Green's Theorem in the problem of the steady motion of a ship

The submerged surface of the ship i.s denoted by S and C is the closed

curve around the water line of the ship which bounds the surface. S. The

free surface of the water outside C is denoted by S. Let r denote the

semi-infinite volume, in

z< 0

and below S . Let P be a point

of

.

The function cp. satisfies the following conditions

(I) 72cp= 0

in

and isregular

at

all points of ;

(Ii)

._S2_o.oflS*.

(:1 1 1)

- -U on

S , where (L,m,n) are the direction cosines of the normal to S drawn into the water;

(iv) cp -. 0 at infinite, like hR.

and we require to find at the point P

Consider also a function G which satisfies

v2G

= 0

in

r- X, where

is a small spherical volume of radius

with center at P, a being the surface Of the sphere;

+G' ,'near P where R2 =

(x-)2

+

(y-)2+ (zij)2

and GT is regular near 'P

0 on S

G -.0 at Infnity like l/R

The condition to be satIsfied by G on S or on the projection of S upon

the free surface will be discussed later.

We then obtain from Green's Theorem the result

$

$Jp

Gvp)d.r =

(q . - G

-4- hemisphere at

where /n denotes differentiatiOn along the outward normal..

The triple integral Is zero since p and G are harmonic in The integral over the hemisphere at infinity will be zero and if P Is a

point In the water, it can be shown In the usual way that the integral over will tend to as the radius shrinks to zero. Hence we obtain

1ITTCPp

jj- G)dS+

it should be observed that if P is a point lying on the surface S , the

first term must be altered to

If we replace

/z and

G/z from the above stated condition,

we have

$J(-

G.)dS*=.1

S9J{-

cp .dxdy

We canapply Stokes' TheOrem in (2) to this double integral to convert it

into a line integral around C,. there being no contribution from the circle at. infinity, and we thus obtain

1Pp

-

G)dS-TM-1

A-2

in order that G be defined uniquely, it is necessary to state its behavior either. on S or upon the project ion of S upo.n z = 0 Peters and Stoker

and (iv) stated earlier and, in addition., satisfies 2G/x2 + v

G/Z

0 on z = 0 . This is one convenient way of defining G and It represents a Green's function which Is quite Independent of the ship's geometry. An alternative procedure would be to take G/n = 0 on S . This has the

advantage that the term G/n will disappear from the surface Integral

over S leaving the formula . .

In the plane

.S2

so that

The disadvantage of this procedure Is that the determinatIon of G then

becomes much more difficult and can be resolved only when the ship's hull S

takes simple shapes such as a hemisphere, etc.

It we accept the first of the above definitions of G , namely that

in which 2G/x2 + v G/z = 0 at z = 0, equation (l),can be regarded as

an integral equation for p when the point P is brought, back on to the

surface of the hull 'and Is changed to as mentioned earlier.

if It is assumed that the normal to the hull at the waterline C lies In the free surface z = 0 , this integral equation can be expressed In the, following form. Let (L,M,0) be the direction cosines of the tangent to .0

= L - M z = 0; then

L+M

.2+ i2

and similarly for G . We thus obtain ' .

hull +

£UG)ds_!(GL.

- GM

_{L.+cFl.i)Mds=0}

and. since on the waterline also in this case

-LU

we have

A-3

TM-1

!

_GLM .2 -

!

ui2u ds

\) _U V

-Qcp(LM

M2)ds=O

The first line integral can be integrated by parts and, if we use the

re-sult that p and G are single valued functIons, this result becomes

hull

+ jUG)dS

+1

_{.I Gui)-LM - M2

hill 1

=

2

_{dGLM ds}

(6)

it will be seen that _WIhull satisfies an integral equation provided the

normal to the hull at the waterline lies in the free surface., if this nor-mal dOes not lie, in the free surface, the reduction involved I (3), (Ii.)

and

₍₅₎

Is no longer possible.

+

IsIhull

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