Some applications of the slender body theory in ship hydrodynamics

SOME APPLICATIONS OF THE SLENDER BODY THEORY

IN SHIP HYDRODYNAMICS

SOME APPLICATIONS OF THE SLENDER BODY THEORY

TN SHIP HYDRODYNAMICS

P RO E FSC H RIFT

TER VERKRIJGINO VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP AAN DE TECHNISCHE NOGESCHOOL TE DELFT OP GEZAG VAN DE RECTOR MAGNIFICUS DR. R. KRONIG, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE,

VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN

OP WOENSOAG 4 APRIL 1962 DES NAMIDDAGS TE 2 UUR

DOOR

GERRIT VOSSERS

SCHEEPSBOUWKUNDIG INGENIEUR

GEBOREN TE SEROOSKERKE (w.)

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR:

PROF. DR. R. TIM MAN

STELLINGEN.

De golfweerstand van een slank schip wordt bepaald door de vorm van de kromme

van spantopperviakken en bet getal van Proude.

2

Peters en Stoker hebben de mogelljkheden van de theorie van de slanke lichamen

voor het gedrag van een schip niet voldoende onderzocht.

A.S. Peters en J.J. Stoker - Comm. on pure and applied math,

10, 1957, p 399-490.

3

Het berekenen van de hydrodynamische krachten op een slank schip in golven

door gebruik te maken van bet concept van de relatieve beweging tussen schip

en wateropperviak, is een correcte methode voor een schip dat in recht

voorinkomende golven vaart met een golflengte van gelljke orde van grootte

als de scheepsiengte.

4

Het in rekening brengen van de snelheideafhankelijkheid van de dempingekoppeling

tussen de stamp- en dompbeweging is niet in overeenstemming met een gebruik van

een striptheorie voor bet berekenen van de hydrodynamische krachten.

B.V. Korvin-Kroukovsky en W.R. Jacobs - Tr SNAME,

,

1957,

p 590-632.

5

Door toepassing van het theoreme van Green en een geschikte Greense functie,

kan men een logische afleiding geven van de theorie van slanke lichamen in

een onbegrensd medium.

6

Het bewijs dat in golven recht op de kop geen vermogenevermindering ten opzichte

van bet stil water vermogen kan optreden, is nog niet geleverd.

Discusele van Weinbium op Swaan en Voesers, TRINA, 103, 1961,

p 297-328.

7

De optimale plaats van bet drukkingepunt uit bet oogpunt van weerstand is

afhankeljk van de modelachaal.

8

Ret ontbreken van representatieve golfepectra voor verechillende oceaangebieden,

is een ernetige belemmering voor bet beoordelen van bet gedrag in seegang van

een scheepeontwerp.

9

De aanwezigheid van moderne rekenmachinee kan er toe leiden dat grote

bereke-ningen worden uitgevoerd op grond van twjfelachtige hypotheses.

Prootte G. Vossere,

Delft, 1962.

lo

De negatieve waardering van de invloed van de techniek op nene en maatschapplj door sommige cultuur filoeofen bernet in vele gevallen op een onvoldoende kennic van de techniek.

il

De bezorgdheid van enkele leden van de Berete Kaner omtrent de afwezigheid van goeete.wetenechappelljke faculteiten aan een technische hogeechool, dient zich

evenzeer uit te etrekken tot sen bezorgdheid omtrent de afwezigheid van een technische faculteit ann een univerciteit.

"Versing van de beeprekingen in de Eerete Kamer ter gelegenheld van de etichting van een derde Technische Fogeschool,

Aan n1jn Oudere.

CONTENTS. Summary.

Chapter 1. Introduction and general outline.

_pl

Chapter 2. Formulation of the problem. Linearization. p 17 Chapter 3. Velocity potential of a slender ship in steady motion

for high Froude numbers. p 33

Chapter 4. Wave resistance of a slender ship In steady motion. p 57 Chapter 5. Velocity potential of an oscillating slender ship at

zero speed. p 60

Chapter 6. Velocity potential of an oscillating slender ship at

forward speed. p 70

Chapter 7. Equations of motion of a slender ship. _{p 76} Chapter 8. Excitation forces on a slender ship in waves. p 87

References. p 94

SUMMARY.

The theory of a slender-body is applied to the behaviour of a elender ship on a free surface. A slender-body la defined as having a shape with both beam and draft small compared to the length and the problem is attacked by formai developments with respect to this email beam-length parameter.

In the tiret place the case of the steady motion at constant forward speed la studied and a formula is derived for the wave resietanc. of a elender ship in calm water.

In the second piace the unsteady behaviour is dealt with and it is ehown that for a certain range of frequencies of motion and forward speeds the hydrodynamic oscillatory forces on the body can be calculated by means of a strip theory, integrating the two-dimensional forces per unit length on transverse sections of the ship. Por the behaviour at forward speed in bead seas with a length of the same order of magnitude as the ship length, the excitation forces consist of the buoyancy effect of the displaced volume by the undisturbed wave and a strip-wise correction factor depending on the sectional added mass and damping values.

CHAPTER 1. INTRODUCTION AND GENERAL OUTLINE.

1.1. Preliminary remarks.

A complete treatment of the behaiiour of a freely floating rigid body

on the free surface of an infinite ocean is in general beyond the

capabilities of modern mathematics and in ari early stage severe restrictions on the type of equations and mitai and boundary conditions have to be imposed.

One of the first restrictions normally being made is that the treatment remains within the realm of the potential flow theory of an incompressible medium. This excludes consideration of viscous effects. A second restriction is that the amplitude of the surface waves will be email, which requires certain conditions for the shape of the body, the forward speed and the amplitude of motion of the body. Under these assumptions a suitable linearization of the problem can be achieved.

Various restrictions on the shape of the body have been used. Possibilities within the class of ship-shaped hulls are illustrated in Fig. 1.1.:

thin ships , having a beam small compared to the length of the ship

flat ships , having a draft small compared to the length of

the ship

yacht-type ships, being a combination of a thin and a flat ship slender ships , having as well a beam as a draft small compared

z to the length of the ship.

X z y y y

(b)

z (C)

_(d)

FIg. 1.1.

The study of the thin ship originated with the famous paper by Kichell (1898), who investigated the wave resistance of a ship oving in etui water at a con.tant forward speed. A sequence of notable papere followed this original paper. Por a review see Wehausen (1957).

The study of the unsteady behaviour of a thin ship in waves, started with a paper by Ra.kind (1946) and was followed by a eerie. of paper. by himself (1947), (1954) and by Hanmoka (1957).

However, Peter. and 3tok.r (1957) showed that for a thin ship the force. due to hydrostatic pressure and body inertia ars of lower order with respect to a beam-length parameter than the hydrodynamic forcie

x.prse.nting the co-called damping and added mase terms. This difference in order of magnitude has not been taken care off In Haskind's and H&naoka's a priori lineariaation.

Newman (1960) tried to circumvent the difficulties by introducing the assumption that for resonance condition the amplitudes of motion of the

ship are of lower order with respect to the beam-length parameter than the amplitude of the incoming waves. Although he obtains in this manner a consistent theory, the assumption of the amplitude of motion of the ship being of lower order than the amplitude of the incoming waves is questionable in view of available experimental evidence of actual ship-shaped bodies.

On this basis it must be concluded that the thin ship approximation is not a satisfying model for the investigation of the unsteady behaviour of a ship, although for the study of the wave resistance in still water considerable success has been achieved.

A next step to formulate a tractable attack on the unsteady behaviour of a ship-shaped body would be an approximation in the direction of a flat

ship. This approximation corresponds to a given pressure distribution on the watsr surface. Hogner (1924) studIed in connection with the wave resistance of a ship the effect of constant pressure distribution moving at a given forward speed. Peters and Stoker (1957) and MaoCamy (1956),

(1958) formulated the problem of an oscillating flat ship on the surface of a fluid. Por such a flat ship there is a connection with the diffraction of water waves against a flat plate - the finite dock problem. Unfortunately, the mathematical analysis of flat plate problems Is rather complicated, smc. the solution of a singular integral equation is involved. No numerical results for ship problems are available. Although the difficulties with the thin ship concerning the relative order of magnitude of the hydrostatic and hydrodynamic forces are avoided, it is not sure whether this type of

approximation corresponds sufficiently with the behaviour of an actual ship to warrant a detailed numerical treatment.

The yacht-type model has been proposed by Peters and Stoker (1957) as a more ship-shaped approximation than the thin or the flat ship. The difficulties in arriving at suitable numerical results will be even larger for this type of approximation than for a flat ship and it is not to be expected that this approach will lead to usable resulte in the near future.

The final suggestion of a ship-shape approximation, as given in Pig. 1.1., is a elender ship, having a. well a beam as a draft small compared to the length. One may wonder why this type of approximation bac received euch a small attention in the hydrodynamics of the behaviour of a ship, since it appears to lend itself perfectly to the presentation of the geometry of a ship. Practically all existing ships have length-beam ratios between 5-12 and length-draft ratios between 15-25.

In the aeronautical literature the concept of a slender-body has been very powerful, since it originated with the paper of Munit in 1924 on

The aerodynamic forces on airship huile".

-2-A multitude of papere has appeared in the mean time, as well for subsonic flow as tranesonic and supersonic flow. For a review of the slender-body

theory in aerodynamics see Ward (1955), Miles (1959) and Lightbill (1960). The principal idea of a slender-body theory is that for a body with transverse dimensions small compared to the length dimension, the flow in the neighbourhood of the body can be approximated by neglecting the first term in the potential equation

+ 4,yv+ 4,zz

where the x-direction corresponds with the direction of elongation of the body. The flow near the body is adequately described in that case by

4,yy+4,zz = O

(1.2)

With a suitable boundary condition on th. body itself, and eventually a free-surface condition, the equation (1.2) poses a two-dimensional probi.. in the transverse plane. The solution will vary from section to section, i.e. with x and can be written as 4' (x;y,z). Por the forces and moments on the total body the two-dimensional results are integrated in the x-direction, the so-called "atrip-theory". In this way a considerable simplification of the problem bas been achieved, since for the two-dimensional plane the well-known methods of conformal transformation of the complex function theory are available; also a direct numerical attack appears to be more promising for the two-dimensional problem than for the complete three-dimensional case.

The development of a slender-body theory in aerodynamics is nearly completed. In the field of ship hydrodynamics, however, only a few attempt. for developing a slender-body theory are known. The presence of a free surface condition complicates the solution.

Intuitively some of the concepts of a slender-body theory, namely that the flow in each cross-section normal to the longitudinal axis, is independt of the flow at other sections, has been applied already in the calculation of the oscillatory forces on an unsteady ship: see Grim (1953), (1959), (1960), Haskind (1954), Korvin-Kroukovsky and Jacobs (1957), Kaplan (1957) and Pay (1958). Some justification of the use of this strip-theory has been given by Grim (1959), (1960), but no treatment has been offered with the same rigour as Peters and Stoker (1957) carried out exemplary for the thin ship case.

Such rigour is necessary too for the slender-body theory, especially for the forces in the longitudinal direction. It is known from _the slender-body theory in aeronautica, that one should be cautious in considering the solutions _{(x; y, z) of the two-dimensional problem as the comp]ete} eolution of the original three-dimensional problem. Por the velocity potential of a slender-body in an unbounded medium it is obvious that a term g (x), depending only on the x-coordinate, can be added to 4(x; y, z) without imparting the two-dimensional formulation.

The only satisfying way to derive the function g (x) is to start from the original three-dimensional formulation and to derive the required first order term by an asymptotic expansion from the solution of this three-dimensional problem.

Such a program has been carried out for supersonic flow for bodies of revolution in an unbounded medium by Von Karinan (1935). (See aleo Ward (1955)). In this study we will carry out a similar program for a slender-body on a free-surface for a certain choice of the order of magnitude of the forward speed parameter anó frequency parameter.

1.2. Outline of th. analysie.

We consider in this paper several coordinate systems; the principal ones are indicated in Pigure 1.2.

z X z

-4-V Pig. 1.2

The system X, Y, Z is a system fixed in space with the X, Y-plane in the undisturbed free surface and the Z-axis vertically upwards. The z, y, z-oyetem is a moving system with the z, y-plane coinciding with the I, Y-plane and moving with the mean speed V of the sh4p in the positive X-direction. The system of coordinates x', y', z le fixed in the body. The origin of thie oscillating system will be represented by a vector with the components

x0, Yo' z0,

denoting respectively surging, swaying and heaving of the ship. The angular displacements around the x, y, z-axes are indicated with the modified Etlerian angles p , , X

denot4ng respectively rolling, pitching and yawing of the ship. The system z', y , z is assumed to coincide with z, y, z-system when the ship and the water are at rest in their equilibrium position. It is assumed tlat the center of gravity of the ship coincides with the origin of the z', y , z

-system; a lese restricted approach may be formulated with the result of introducing some additional terms which are irrevelant for the purpose of our study.

We introduce the equation of the ship's hull with respect to the primed system of coordinates in the form

where f1 (z', z') is a two-valued (corresponding to the port and starboard side of the ship) function of the coordinates with continuous derivative, except, possibly at the bow.

Purthermore we introduce dimensionless coordinates, as well in the primed as the unprimed system:

X Y z _(1.4)

L/2 O

The equation of the ship's bull will be represented in this dimensionless system of coordinates by

1C,z').

o4 fi(.')

(1.5)

As a characteristic length the half-length of the ship has been

2

chosen. The parameter o is a measure of the transverse extent of the

boundaries of the ship and corresponde to the beam-length ratio of the ship:

a B/L. Por a elender ship the vertical and the transverse extent of the boundaries of the ship are of the same order of magnitude and both of order o compared to the longitudinal extent of the boundaries.

The motion of the water is assumed to be given by a velocity potential $ (I, y, Z, t), which is a solution of Laplace's equation

$yy+'t' = O (1.6)

and appropriate boundary conditions at the free surface, on the hull of the ship and at infinity.

The boundary conditions on the hull of the ship depend on the motion of the ship, which in its turn can be found by determining the forces acting on the ship through the pressure of the water and by the solution of the differential equations for the motion of a body with six degrees of freedon.

We want to linearize the problem and think of the velocity potential $ and the quantities which determine the motion of the ship as functions of the slenderness parameter o

In our application the velocity potential $ (I, Y , Z; t, o ₎

(z, y, z; t; o ) is assumed to possess the development

Z;t; 0)

024»m(xy,z;t) 4o$(x,y.z;t ) (1.7)

The free surface condition in this moving system of coordinates will have the following form for the first term iW

+v2_2v+go

z=O

With the introduction of a dimensionless velocity potential

) 4'(,iÇ;)

2

and a dimensionless time

= (1.10)

L/2

(1.8)

which reduces for harmonie motion to

= ut

where

X - - (1.12)

2V

represents the reduced frequency, we can write the free surface condition as:

e Ç=O (1.13)

where ₀ represents the forward speed parameter

- 2V2

oir

The parameter

L is a dimensionless frequency parameter important

for the generation of surface waves. Sometimes lt is more convenient to use

= (1.17)

Th. parameter 8 bac been introduced by Haskind (1946) and le of iapo$ance for the description of the wave pattern generated by a moving oscillating body. Por a value of 1/ the generated waves are confined to a sector behind the moving body; for a value 1/ waves are found before the body too.

With these parameters the free surface condition can aleo be written as

-6-(1 .14)

(1.16) We can identify in the free surface condition some well-known

dimensionless parameters:

If we choose the order of magnitude of - O (i) (or L = O C Cl1 ))

and p0 0 Co ), the free surface condition becomes, deleting terme of

O C ci ) and higher:

4;= O (1.19)

With Laplace's equation in dimensionless form, also by deletion of terms of O

(o2)

4lit

ÇÇ=O O (1.20)

and a suitable boundary condition on the body the equations (1.19)-(1.20) pose a two-dimensional problem where the i-wise change of the flow pattern

is not present.

If the order of magnitude of _Po 0 (1), the equation (1.19) still holds; however, terms of O ( ch/2 ) are deleted, which may cause too much error. A choice o'

B = o

( a ) degenerates the free surface condition

to

which applies for the low-frequency approximation. In the same way a choice = O ( o ) degenerates the free surface condition to the

high-frequency approximation

(1.22)

Therefore it is of so interest to restrict the choice of to values being of 0 (1), since in this range the most important contributions arising from hydrodynamic damping due to the emission of surface waves ars to be expected.

The choice O (i) corresponds for the case of forward speed

Po -

O

(0 )

to .O(cf1) ; =0(1) and we pose

x=OM)

Pi0(1) (1.21)

Since

2

, with 1 the length of the generated waves, the choice = 0 (1) corresponds to

= 0(1)

or the length of the generated waves is of the same order of magnitude as the beam of the ship.

In terms of the well-known dimensionless Proude number

Fr = V

(1.23)

the choice O ( o ) corresponds for a beam-length ratio o between 0.08-0.2 to a value

Fr

O2-O3

which covers practically the whole speed range of existing ships.

Por the case of steady motion the free surface condition degenerates with p0= o ₍ o ) or even O (i) to

(1.24)

We discuss In some detail the boundary condition on the ship's hull. e assume the linear displacements of the C of the ship and the angular rotations of the ship to be small of order 02 and we put for harmonic motion:

Xo=

o24 0e'

yo=o2_L90e

zo_o24

0e_I

= o2e

_o2e

x a2

(1.25)

Using the appropr4ts transformation formulae for the change of the primed system z', y', z into the unprimed system of coordinates z, y, z, the kinematical condition on the hull of the ship that the particle derivative should vanish, can be formulated as follows, deleting terms of

O ( o ) and higher:

n1+fi+f

fiç}e

(1.26)

on

-8-where n is the normal on the hull.

This kinematical condition consiste of a time-1nependent part, given by f1 , the longitudinal elope of the hull, and a time-dependent part,

comprising the swaying ( ), yawing (X0), heaving ( ) and pitching (q motion. The rolling and surging motion are not present; they will appear for a slender-body in a higher order approximation.

The equation (1.26) completes the set of equations (1.19)-.(1.20) (or (1.20) and (1.24) for the time-independent case) required for the two-dimensional solutionB in the transverse planes perpendicular to the

direction of elongation of the body. They do not serve us for finding the function g (z), which may be added on the two-dimensional solution,

although they are of some help to indicate whether two-dimensional solutions are to be expected. Por a complete solution cf the problem we have to return to the original three-dimensional Laplace equation and the free surface condition (1.8).

The normal procedure for formulating an integral equation for the unknown velocity potential, is to apply Green's theorem

G*_G%)dxdydz=_OG1_Gn)dS

(1 27)

to the velocity potential and the adjoint of a Green's function G, appropriate to our problem. The boundary surface S around the volume R consists of the wetted surface of the ship S, the free surface outside the ship and suitable surfaces at large distance from the ship, which are allowed togoto infinity. Because the singularity of G in xi, n, z1 is of type 1/r, with r2 = (x_x,)2

+ (y-yi)2 + (z-z1)2, we find, if the Green's function satisfies the free surface condition, with the classical argument the following integral equation for the unknown velocity potential:

(x1,y1 z1)=-

/f(G'n_on

_{) dS (xy,z )}

_(1.28)

4Tt

s

The total velocity potential consists in principle of the following terms:

(1.29)

where

the velocity potential of the ship in steady notion in still water

i

: the velocity potential of the plane progressing

incoming waves

2 : the velocity potential of the disturbance created by the

Por the time-independent motion we can write the equation (1.28) by introduction of a dimensionless Green's function

G"= -- Í- (1.30)

L

and use of the dimensionless notation (1.4), (1.5), (1.9) and the boundary condition (1.26):

-

//{.o(rfir

)- dd+ O(o) (1.31)

where the integration should extent on both sidee of the bull and appropriate values of the integrand should be chosen on each side.

Por the tine-dependent case we have in a similar way, by deletion of the time factor e

jfl

(1.32) d (1.33) where is given by =-\/1.e- f1 + _iç (1.34)

The value of 2n suggests to split '2 in the following componente:

- e i

= _{+ P2}

and for each component we can write an integral equation

1=,1,X, lo

-(1.35)

(1.36)

G0- - + 2 2 (x_Xi)2+(y_y1)2+ (z-21)2 (x-x1) + (y-y ) (ztzl)2 +j Re

lt

f

p[(z+z1 )+i(x-1) CO5J

d/dP e

_{cos[p(y-y1)sinI} J

22

L _{i-p!_ cose} g (xx)2 _(yy )2 2 (-xi)2

_(-i

)2 (z-Z1 2 + 2/dP

eP21

Jo[P(x-x1)2(y-Yi)2 j (1.40) (1 .39) (1.41) with

pX_g

; (i .38)

ii=

_//

+f1 r)

+f1 } ddç. O(e)

The velocity potentials represent the velocity potentials due to the oscillatory motion of the ship in the a direction. The velocity potential 4 represente the influence of the ship form on the incoming waves or the diffraction potential.

When suitable Green's functions are known, the integral equations (1.31), (1.37) and (1.39) can be solved.

For steady motion the following Green's function bas been derived (see for instance Havelock (1923), Timman and Vossers (1955), Peters and Stoker (1957)):

The path L encircles the pole p=--2sec2 with a small circle via the positive halfplane.

Por unsteady motion at zero speed we have (see Peters and Stoker (1957)):

where the path L encircles the pole via the negative half plane. Finally we have a rather complicated Green's function G2 for the case of unsteady motion with forward speed, for which we refer to (6.6)-(6.7) of Chapter 6. (Peters and Stoker (1957)).

It will be the purpose of our analysis in the subsequent chapters to derive an asymptotic expansion of the integral equations (1.31), (1.37) and

(1.39) with the appropriate Green's functions G0, G1 and G2 for small values of the parameter o . Retaining only the first order terms in the developmet

we can simplify these integral equations considerably and we shall be able to identify a function g (z), if there is any.

When the integral equRtions are solved in this way, we can calculate the pressure on the hull by means of Bernoulli's law

P(x.y.z.t )= _pgz_ o

{()2+(.))+

)2 (1.42)

which can be written by means of the dimensionless notation

P=_.pgL{c+o2x

(1.43)

Por the case of steady motion we can calculate the wave resistance of the ehip by means of this pressure with the formula

R=

_{_/fPFxdS}

(1.44)

Herewith we are able to give a new formula for the wave resistance of a elender ship. (See Chapter 4).

For the case of unsteady motion we have to use the pressure on the hull to formulate the dynamical equations of motion of the ship, since the

change of momentum of the rigid ship ecjuals the force acting on the ship and the change of angular momentum equals the moment acting on the ship.

These principles allow us to write four equations of motion in the oscillatory components

y0,

_x z0 and t4

M+myy21.Nyyi2+

_myx+

ye_)t

dt2 dt2 dt dt2 dt

+m NXY

dt2 dt2 dt dt2 dt

12

M--m

d2z0 dz0 d24 dt2

zzr

+Nzz_+mz,_+Nz4+BzzZo +Bzi4ze

dt dt2

'j

(1.46)

rn2

B,4i+ Bzzo=4e

wt dt2 dt2 dt2 dt

or their dimensionless aequivalents.

M represents the

maas

of the ship, I the longitudinal mass moment of inertia, while the coefficiente map and

Nap

follow fron the integration of the

dynamic

pressure over the hull surface. Por instance we have

.1 0

mzz+iZ = o2p4

/d1/di

)

.1 ¿-d

(1 .47)

where

_rnzz

is called the added masa for heaving and Nzz the damping coefficient for heaving.

The coefficiente Bap follow from the hydrostatic contribution of the pressure and represent the restoring forces and momenta, which are only present in heave and pitch. Finally the coefficients

Ga

denote the excitation forces and moments and follow from the contributions

and

_O

in the velocity potential.

The equations (1.45)

_and

(1.46) represent two systems of coupled equations of notion; the first system is a coupled system for sway and yaw and the second system a coupled system for heave

and

pitch. From these

equations the amplitude and phase of y0, _x and 4m , or

and in (1.36), can be calculated, which completely determines the solution of the unsteady problem.

Finally we like to discuss in this intioductory chapter somewhat 'ore in detail the excitation forces. The first part of this force is derived from the velocity potential of the incoming waves, which can be written In the x, y, zsyatem of coordinates

=_2 hWe

Z(05

_

_(i

where

= o21

representa the wave amplitude,

i

is the wave

frequency, a the direction of travel of the waves with respect of the x-axis and w the frequency of encounter

with the dimensionless wave amplitude

V

R(1)

(1.51)

L/2

and the wave frequency

L

(1.52)

The expression (1.49) may be written in the following dimensionless form

In the dimensionless system of coordinates, we have

i __Ve

t°

i(co5

a+o11SIfl

.LV

(1 . 50)

(1.53)

Since we specified in our applications the frequency of encounter to be of order o , it is clear from (1.53) that in general should be of order o-1 too or in other words

X/L=0 o (1.54)

In that case we find the exciting forces in beam seas (

a

= 900) to be

of the same order of magnitude as the other hydrodynamic forces. However, in head seas ( a 1800) the exciting forces are of order o compared to

the other hydrodynamic forces and in the equations of motion no excitation will occur. This i clearly understandable since for a wave length small compared to the ship length, the contributions in th excitation forces along the length of the ship cancel each other.

If we suppose 01<o<0.2 , we may assume in bead seas with a ship at forward speed that =0 (1) , although L=0(o1) which le illustrated in Fig. 1.3. Under this assumption the velocity potential of the incoming waves is written as

e°'i.0(o)

(1.55)

-and we can expect to have a contribution in the excitation forces of the same order of magnitude as the other hydrodynanic forces, i.e. added mase and damping.

The sane argument applies for excitation coming from the diffraction potential , for which we refer to Chapter 8.

Restriction of the value of the parameter o on the lower side, limits the accuracy which can be obtained in the calculation of the excitation forces in head seas. However, with the limitation of o 0.1 still a reasonable approximation can be achieved.

15

10

In the subsequent chapters the outline given in this introduction will be discussed in more detail.

In Chapter 2 we deal with the formulation of the problem and a suitable choice of parameters in order to be able to indicate when a series of two-dimensional problems can be expected.

In Chapter 3 the velocity potential is derived for a slender ship in steady motion with a forward speed parameter o= O (i). Herein a

derivation of the function g (z) i given, which allows us to formulate en integral for the wave resistance of a slender ship. (Chapter 4).

As a second example we apply our analysis to the velocity potential of an oscillating slender-body at zero speed with a frequency of motion

- 0 (1). In that case we find a strip theory to be applicable and the results published for the damping and added izase of two-dimensional

sections can be used for the prediction of the hydrodyriamic forces on a slender ship. (Chapter 5).

Even at forward speed with a speed parameter 3,= O ( o ) it appears

in Chapter 6 that the oscillatory hydrodynamic forces are given by the same equations as for zero speed, although the nature of the generated surface waves is completely different for this case ( = O (i)) from the situation at zero speed ( ¡ - O). In this case no first order contribution in the function g (z) le found.

In Chapter 7 the equations of motion are discussed in more detail. Pinally In Chapter 8 the excitation forces are calculated. It appears from this Chapter that in head seas we can calculate the influence of the ship on the wave (the diffraction potential) by means of a strip theory. This effect le a.quivalent to the concept of the relative motion between ship and water, which has been proposed intuitively by some authors.

Our analysis has not been exhaustive, since a large number of other choices of the order of magnitude of the frequency and forward speed parameters is possible. Por instance the choice í3- O (i) and _L0t1) may be of some interest for high speed ships in long waves.

-CHAPTER 2. FORMULATION OP THE PROBLEM. LINEARIZATION. Coordinate systems.

For the formulation of the problem we need several coordinate $ystem5 which we choose as follows:

Coordinate system OX, Y, Z (Fig. 1.2), fixed in space, with the X,)' -plane in the undisturbed free surface, the X-axis in the direction of

travel of the ship and the Z-axis vertical upwards (all coordinate systems are supposed to be right-handed).

Coordinate system O-x, y, z (Pig. 1.2), wIth the x, y-plane coinciding with the X, >'-plane and moving in the positive x-direction with the

constant speed V of the ship.

The transformation of the O-X, Y , Z-system Into the O-x, y, z-system le 'therefore given by

= X.-vt

y=Y

(2.1)

z= Z

e) Coordinate system O-x", y", z", with the axes parallel to the x, y, z-axes, but with the origin fixed to the center of gravity of the ship. The center

of gravity is supposed to make the following motions: surging :

swaying : y0 heaving : z0

O-x",

y", z"-system

It is assumed that the center of gravity of the ship coincides in the rest position with the x, y-plane in order to be able to omit some extra coupling terms in the equations of motions. The analysis might be carried without this assumption, but the extra terms are irrevelant for the purpose of this study.

d) Coordinate system O-x', y', z' rigloly attached to the ship. The orientation of this system relative to the O-x", y", z"-system is defined by the angular motions of the ship, for which the modified Eulerian angles X , d and p are used:

yawing : X

pitching :

rolling :

These modified Eulerian angles can be visualized by transforming the system O-x", y", z" into the system 0-x', y , z' by means of three successive rotations.

The transformation is given by

of the O-x, y, z-system into the

X" =

x-x0 y" z" =

y-y0

= z-z0

(2.2)

Firstly, a rotation through an angle x (positive in accordance with the righthanded coordinate system) around the z"axie bringe the coordinate system

01N,

zN into the system 0x" , y'" , z"

A rotation

4 around the y axis bringe the coordinate system 0x" , y' ,

Z"

into

the system Ox , . Lastly, a rotation through an angle w

around the x axis brings the coordinate system into Its final position, the Ox', y', z'syetem. The total transformation of the Ox", y", z"-system Into the Ox, y', z'z"-system is therefore written in matrix

notation:

X' X"

[L]

y"

(2.3)

z"

with the matrix [L]:

[L] =

cos w cos X

siflpsin

coSx--coscp sir

coswsin,cosX+

+sir' 'p in X

In the subsequent analysis, it will be assumed that the motions of the

ship will

be

small, say

X0=tx

y0=ty

(2.5)

(I)

20=tzo

X't

where

t

represents a small parameter, to be identified later on. Inserting these developments in the abovementioned transformation

formulae, we find the

following traneformation of the Ox, y, zeystem into the Ox', y', z'system:

(1) (i) (1)

x'=x4t(-x0 +X y_4k z)0(t2)

Y' Y+t(- y0 +V

(1) (1)

z-X

(1)X )+ 0(t2)

(2.6)

(1) (1) (1)

z'=z(-z0.qJx_wy)*0(t2)

Fundamental equations. Boundary conditions.

We assume the fluid to be nonviscous and irrotational. The motion can be described in that case by a velocity potential (X, Y, Z, t), giving for the speed in the

I,

Y and Zdirection respectively:

UX

Vy

WZ

It is known

that

satisfies in the halfspace Z O the Laplace equation:

18

-cos

45inX

-sin

sintpSirSinX+

s w cos 4

cos w cos X

coswSini5iflX_

coswcoSi

-sin

wCOS)

$XX

YYZZ

(2.7)

Prom the equations of motion for an irrotational flow, lt le readily verified that the following form of Bernoulli's law le valid, which has to be used as a dynamical boundary condition:

P +gZ= C (t) (2.8)

In this equation p represente the preesure in the point X, Y, Z;

p is the density of the fluid and g the acceleration due to the gravity. The constant C (t) may depend on t but not on the space variables X, Y, Z; we will take C (t) = O.

A second boundary condition is formulated by using the general law in continuum mechanics that once a particle i situated on the surface of the fluid (either an interface or a rigid boundary), it remains on this surface. This kinematical condition is written, if F (X, Y, Z, t) = O represents the formula of the surface, as:

FIt+XFIX+ yFiy+ZFiZ

(2.9)

Both boundary conditions (2.8) and (2.9) have to be applied as well on the free surface of the fluid as on the boundary of the ship. These

boundaries are, however, not fixed, but movable. The free surface is determined by the incoming waves and the motion of the ship, while In turn the motion of the ship follows by integrating the pressure on the hull by means of Bernoulli's equption (2.8) and solving the resulting ix equations of motion for the ship.

It is necessary for arriving at an unique solution, to stipulate some additional conditions for the behaviour of the velocity potential at

infinity. In the subsequent analysis it will be assumed that the water depth will be infinite, which imposes the condition z= O for On the free surface for Y and

X _e

more trouble some conditions are required for guaranteeing an unique solution. In the case of harmonic

oscillations it is customary to Impose a "radiation conditions, which allows only waves to be radiated from the body to infinity and suppresses any solution giving travel of energy from infinity to the body. In the time-independent case, one requires that the motion far ahead of the ship vanishes. No problems arise wIth the formulation of these conditions, when the problem is considered as an initial value problem and the motion prior to t = O is supposed to be zero. The right solution for the harmonic oscillation or the time-independent motion follows by the asymptotic behaviour if the time t -.

In our analysis no direct use of either of these two methods will be made, since the Green's functions belonging to the various problems ars supposed to be known.

-A derivation of these elementary green's functions has been given by Peters and Stoker (1957) and in their analysis the appropriate conditions at infinity have been used. Therefore there la no need in our analysis to consider this matter once more.

Linearization of the free surface condition.

In order to linearize the boundary conditions, lt le assumed that the velocity potential 4' can be expanded as a power series in terme of a small

perturbation parameter ô

=

(i)

624'(2) (2.10)

Herein we follow the suggestions of Stoker (1957) and Wehausen and Laitone (1960). We are not identifying at this stage the parameter 6

We assume that the free surface can be expanded in a similar power serles:

F(XyZ,t )= Z_zw(XYt )=Z-6z-

o2zV_...= o (2.11)

Insertion of the equations (2.10) and (2,11) in the Laplace equstlori (2.7) and the boundary conditions (2.8) and (2.9), results in the following equations, collecting the coefficients of the first powers of ô

(1) (1) (1) 4IXX + (1) (1) 4't-gzw 0 1) (i) ZWt+4'z O z< o

It le good to remember that for 4'(1)

, which has to be evaluated in

(2.13) on the free eurface (2.11), the following expansion is assumed to be valid

(1)

+'"(X.Y,Zt)

= 4'(X,Y,o. t _{) + Z4'Z}

o,t ) +

Ci) (1)

$(X,Yo,t )+§Z (XYo.t ) +

Thie indicates that the condition (2.13) has to be satiefied on the fixed boundary Z - O inetead of the moving boundary Z - z.

2

-(2.12)

Zzw(X,YZt)

(2.13)

Combining the equatIons (2.13) we arrte at the well-known free surface condition:

(1) (1)

Vtt+gcIZO

z=o

(2.15)

It is convenient to transform the above-mentioned Laplace equation and boundary condition into equations for the moving coordinate system O-x, y, z, using the transformation formula (2.1).

We introduce

X ,Y,Z,t )

x+Vty,z,t)

)

(2.16)

Using the relationships for change of coordinates, we rewrite the Laplace equation and the free surface condition as follows:

+ (2.17)

+vV1)

xx-2V+gz=O

(1) Cl)

zO

Boundary condition on the ship's hull.

We suppose the equation of the ship's hull in the coordinate system rigidly attached to the shIp, O-x', y', z', to be presented by

(2.19)

We assume (x', z') to be a two-valued (corresponding to right and left sides of the ship) function of the coordinates x' and z' with

continuous derivatives.

With (2.6) we transform this equation trito the one for the coordinate system O-x, y, z, which in virtue of the smallness of the motion parameter

t

becomes:

F(x+tp,y tq Z+tr)=F(x.y,z )+tPFx+qFy+tr F

Q(2)

(I) (I) (J)

(I) i (II (I) (JI (i)

z-x x)+(_X0.xy_

_{Z)fix+t(-Zo+_y)f,zr}

=F(x.yZt) =0

(2.20)

In this moving system of coordinates the kinematical boundary condition on the ship's hull is given as follows, by transforming (2.9) and using (2.10):

(i) (1)

F2t_vFx+ôx F2x+64F2y + o$z

F ₌₀

(2.21)

on F2(x,yz,t)=0

Thin- and slender-body approximations.

by

Furthermore we assume the equation of the ship's hull to be presented We introduce several types of linearization, using a method of

analysis of Miles (1959).

We introduce the following dimensionless coordinates:

X (2 2 2a) (2.22b) E L/2 (2.22c) L/ )tVt (2.22d) L/2 '(x,z) = o-f1(Ç

(2.23)

22

-Pinally we introduce the dimensionless velocity potential

Por an oscillating ship at zero speed it is inappropriate to nondimensionalize the velocity potential with the forward speed, and therefore we introduce

,wg *(.Ç)

2

The following connection between both potentials exists

=

e

As a characteristic length the half-length of the ship _L/2 has been chosen. The parameter o is a measure of the horizontal extent of the boundaries of the ehip and corresponds, therefore, to the beam-length ratio of the ship o = B/L.

The parameter i _{denotes the vertical extent of the boundary auch} that

Z/

_{L = O (i) in the neighbourhood of the 8hip. Two choices of the} parameter i are of interest:

j« o , which indicates a thin ship (Michell-ehip)

'=O(o), which gives a slender ship, having breadth and depth dimensions of the same order of magnitude. (See Fig. 1.la and d).

The parameter x _{le a parameter defining the time rate of change and} corresponds

to the

reduced frequency

)t =4

2V

(2.24)

(2.25)

if the motion is emole harmonic.

In the foregoing section we introduced a small parameter

_t

_defining the order of magnitude of the motions of the ship and a small parameter for linearization of the velocity potential. The dimensionless parameters

x ,

t

and § are to be determined by the boundary conditions and in relation to the, by definition, small parameter _o , denoting the transverse

Subetitution of the dieneionlee notation (2.23) and (2.24) in the Laplace equation (2.17), free aurface condition (2.18) and kinematical boundary condition on the ahip hull (2.21), reeult8 in the following uat ion.:

1toxr-3 pF1)-

C,f1.)+

EOf

X+9M+

riçi

Cfi-T 'Cfi-T

-

c$(fi-tx+tc

Af1g49

CF1

-c

c° r+t_toFg . 12

AFiç

+ OIl ₁₁₊ 2V2 gL

-

24 -' o (2.26) (2.27) (2.28) (2.29)

since

The parameter _ß0 i connected with the Proude number

Fr =

vt

(2.31)

130=2Fr2 (2.32)

We require both eides of the equation (2.28) for the time-independent case to be of the same order of magnitude, for which we choose the following relation for

o (2,33)

Substitution in the Laplace eqiation

indicates that two possibilities of the choice of â and o are of interest: 2 2 6_ 6 b)

<1,

â2 62

o2i2

-

25 -(2.30) (2.34) (2.35) (2.36)

The flst possibility gives

5-o

and i= o (i), which, in view of the two choices of the parameter , discussed on p 23 , indicates that

thi, possibility le applicable to thin ships.

The second possibility gives

5 =oi

and c1 , which indicates that this possibility is applicable to elender ships.

3fore we substitute these two possibilities in the boundary condition on the hull, we should assume on order of magnitude of the motion parameter

t

L reasonable assumption about

t

is that it is of the sane order of magnitude as S and we choose

t

5 This is aequlvalent to the assumption

that the ship motions are of the same order of magnitude as the wave height. Elem.ntary considerations on the motions of a ship on the free surface make this assumption plausible. Por a thin ship other choices have been proposed, which were discussed in the introductory chapter.

Substitution of the above-mentioned estimates of the order of magnitude of the parameters i , 5 and

t

in the equations (2.26) and (2.28), gives the following equations, omitting ternis of O ( o ) and higher:

a) Thin ship

-B+ I-x

OW

b) Slender ship _Q

_{f1= xLL(B+ Cí)+ f}

26 -(2.37) (2.38)

Por both types of ship the free surface condition (2.27) remaIns applicable.

We should make an assumption concerning the order of _{magnitude of x} and _{. Por the thin-ship approximation the only interesting possibility} Is )4-- O (i) and

_ß0

_{= O (i). Other estimates lead to a degenerate free} surface condition 4) = O or 4)ç O, which corresponde to the high-, respectively low frequency approximation. One exception muet be made:

t= O C o

);

Po _{0 (1), which, however, corresponda to the steady}

motion of the ship.

With x= O (i);

_P°=

_{O (i) we have, posing}

.L = 1:

Thin ship. (Michell-ehip).

O

(2.39)

For the slender ship another estimate of the order of magnitude of x and o is of interest: x= O ( o ); = O ( o ). Putting

x = o , = o 3p and = o we have

p

Slender ship.

(I) (I)

z 4),i1 4)çuIç XYo

L/ f1(Ç)=O(2.4O)

L/2

j

or using the dimensionless velocity potential

4) (see p 23 ), we bave

(I) _')

B

B(-: -I

on

ç=o

which shows that this approximation is aleo applicable for V = O. We identify the following wellknown dimensionless parameters:

2p_w2L

pw2BBOL

=

2.41)

(2.42)

Yor the thin ship L and are of O (i); for the slender ship we

investigate the cases L= O ( o 0 (1); O and = O (i).

There is a nonessential difference in the formulation of the boundary condition as given in (2.39), compared with the formulation of Peters and Stoker (1957). In our formulation the center of

gravity of the ship is oscillating in the swaying direction with respect of the coordinate system

Ox, y,

z; in Peters and Stoker's

formulation the coordinate system is following the swaying motion; therefore in their analysis the term with

Yo le not present.

It is clear from the boundary condition on the ship's hull that only the swaying motion, the yawing motion and the rolling motion are

contributing to the first order approximation. This is clear from an eleaentary analysis, by visualizing the thin shIp as a vertical plate with length and draft of the same order of magnitude and the tbicknes email compared to the other dimensione. The other motions are not present in the first order approximation and therefore no contribution In the velocity potential, i.e. added mase or damping terms of the pitching, heaving and surging motions are to be expected. These remarks of Peters and Stoker are very important and affect the foundations of Haskind's (1946) and

Hanaoka's (1957) calculations.

It

is

known that

the equatIons (2.39) for

the steady behaviour of a thin ship with zero yaw angle were already used by Mlchell (1898):

-= 0

i1=fig

i =0

(2.43)

p0+o

=0

The equations of the slender ship (2.41) show that a part of the determination of the unknown velocity potential is reduced to the solution of a series of two-dimensional problems in the transverse planes

g = constant, since g -wise change of the velocity potential is not shown in these equations.

The boundary condition on the hull discloses that hydrodynamic contributions to the velocity potential may be expected not only due to swaying arid yawing oscillations of the ship (which is similar to the

boundary condition for the thin ship) but also due to heaving and pitching. This boundary condition should not be applied on the plane y = 0, as is the case for the thin-ship approximation, but on the ship hull in the rest position, since the motions are of O ( o2 ) and the transverse extent of

the bull of O ( o ).

The slender-body theory for time-independent motion of the ship

=0

ri+rr=fg

+r(g,)= O _(2.44)

cIç=o

has been used by Cumins (1956) to investigate the wave resistance of a

P

slender ship. Although he was the first to realise the importance of the aeronautical studies of slender-bodies for the wave resitarìce of slender ships, his analyses are inconclusive and are not used irr our investigation.

It appears that the orders of magnitude of the various parameters used in our analysis are reasonable in view of practical application to technical problems.

-I

I

We have assumed O(o) or O 0(1) (2.45) 0(1) or O

The parameter o indicates the beam-length ratio and varies for moat

ships between 0.07-0.20.

Then = O C o ) corresponde with

Pr '-s 0.20 - 0.30

in which range moat ships are operating. Wave heights and motion amplitudes should be of order o2. Pull scale waves of practical importance have in general dimensions aequivalent with

ti 0.01 - 0.02

X

where represents the wave amplitude and X the wave length and these values are in agreement with our aesumptione.

Planar-body approximation.

Although the planar-body concept is not used in our investigation, it is of some interest to pureue the 8ame systematical analysis of the

boundary conditions as for the thin and slender-body.

A planar or flat 8hip (cf Pig. 2.1) can be represented by the following equation

z=(x,y)=o 5f(i1( (2.46)

The parameter o denotes the vertical extent of the boundary

a) (b)

Pig. 2.1

-of the ship and corresponds, therefore, to the draftlength ratio -of the ship o =2- . The parameter , which has been introduced in (2.22b), denotes the horizontal extent of the boundary of the ship, such that

Y _{_O(1)}

in the neighbourhood of the ship. Three choices of c are of interest:

.O(o); _« i , which is equivalent to the slenderbody approximation

given above

c » o £O(1) , which indicates a planar ship (Fig. 2.la)

;« i , which we calla slender planar ship (Pig. 2.lb).

The details of the analysis are omitted, but on similar lines as has been done In the foregoing section, it can be deduced that the assumptions b) and c) lead to the following equations

b) Planar ship

)4.0(1)

3O(1)

c) Slender planar ship

4r L/ x2p on A oF Ç=O =0

oußideAof0

on -a<y<a oF =0

on y>a andy'-a

of=0}

(2.47)

The equations (2.47) were already derived by Peters and Stoker

₍₁₉₅₇₎

and used in their formulation of the motion of a planar ship. It can be seen that in the first order terme of the perturbation series of the velocity potential contributions of the heaving, pitching and rolling

motions are found. A similar formulation of the problem has been put forward by MacCamy

(1956, 1958);

however, essential differences with the approach

of Peters and Stoker are present, which require further investigation. A combination of the planar and thin ship approximation has been proposed by Peters and Stoker

₍₁₉₅₇₎

as a suitable scheme for receiving hydrodynamical damping and inertial contributions in all motions, except

surging. They call this a yacht-type approximation. (See Fig. 1.1).

The equations (2.48) were used, in the case of tine-independent motion of a slender planing craft by Tulin

(1957).

The form of equations (2.48) suggest that an extension of his analysis may be possible for an oscillating slender planing ship, where again a two-dimensional approach may be

suitable.

-CHAPTER 3. VELOCITY POTENTIAL OP A SLENDER SHIP IN STEADY MOTION POR HIGH FROUDE NUMBERS.

As a first application of the concepts introduced in the second chapter, we will calculate the velocity potential of a slender ship in steady motion.

We start with the application ofOreen's theorem for two functions, one of the functioe being the adjoint of the Green's function G0 belonging to the free surface boundary condition for steady motion, the other function0 being the unknown velocity potential:

3.l)

R s

R denotes a three-dimensional region and S its boundary surface. Let

ue

first discuss the Green's function appropriate to this problem.

This function, G0, is a solution of the equations:

G0+

+G022 = -4

TtS(x-x1) 5(y-)

5(2

-Z)

z O + G =O z-O (im G O z -limG0 O X

Herein G0(x, y, z; z1, y1, z1) represents the required Green's function, which depends as well on the location of the observer (z, y, z) as on the

location of the singularity (z1, y1, z1); 5 is the Dirac delta function. The free surface boundary condition follows from (2.18) by equating to zero terms depending or the time derivatives. The third condition of (3.2) ensures the boundednese of the solution at infinite depth.

A solution of the equations (3.2) has been given by different methods. (See Havelock (1923), Tirsoan and Voseers (1955), Peters and Stoker (1957)). We give the result only:

G0(xy,z;x1,y1,21)_

Vx_x12(y_yl)2+cz_212

1 + Vc X-x1)2 + (Y Yi)2+ (2+ 2 (3.2)

Since lt/2

+JIdfdP e+Z1)+t_05cos[p(y_yl)sjnJ

o L

i__.2cos2

g

(3.3)

The path of integration L is given in Pig.

3.1

I

p -plane

(-\

.-p--

sec y2

Pig. 3.1

-

34

-G0(xy,z

x,.y1,21) # G0(x,y,z,; x.y,z)

(3.4)

we should be careful in applying in Green's theorem the adjoint of G0. The first term of (3.3) represents the Greens function corresponding to a point source in x,, y1, z1 for an unbounded medium. The second term of (3.3) follows from an image source with opposite sign in the upper half space, reflected with respect of the free surface. The integral term in (3.3) eommatee the contributions giving rise to the surface wave pattern.

Application of Greens theorem.

We apply Greens theorem (3.1) to the volume R bounded by the surfaces (see Pig. 3.2):

S- _i-- S24- S3 (3.5)

where

S0 : part or the surface of the ship below the water plane z = O part of the free surface between the waterline of the ship L0 and the arbitrary circle L1

the surface of a vertical circular cylinder between the circle L1 in the free surface and a horizontal plane at the bottom

53 : closing plane under water of the vertical circular cylinder.

Pig. 3.2

Prom Greens theorem and the classical argument involving the singularity in x1, y1, z1, we have

-G0)dS

(3.6)

On the free surface S1 we find, since the free surface condition of (3.2) applies as well to as 4itg

11.,

j](oGoz

G40z)dXdy

SI , 1

(+oGox Go0)dy

4Ttg L0

The integral along L1 vanishes when the radius of L1 goes to infinity. Also the integrations over ₂ and S3 vanish for these surfaces going to infinity, since and n tend to zero

of sufficient order for

infinity.

Therefore the only contributions left from (3.6) are the integration over the surface of the ship S and along the intersection

o± the ship with the free

surface L0.

On the surface of the ship the following relations are valid:

$

GFx+G'yFy+Gz

VFX2+ Fy2+Fz2 dS0

VF2FY2FZ2

dxdz

4F

+ F,, +

_(3.10)

V Fx2+ Fy2+ F2

with

F(x,yz) = y+'1(xy)

(3.11)

being the equatijn of the ship's hull.

-

36

-L

-

G dy (3.7) (3.8)

3.9)

Therefore the integral over the surface in (3.6) can be written as:

_IIOCG;FX+G;FY+G;FZ_GOXFX+OYFY+OZFZ1dXdZ

(3.12)

where x-wise integration should extend from

_-4

till

₊₄

, L being the length of the ship and the z-wise integration over the longitudinal plane of the ship from z = O till z = T with T being the draft of the ship.

Following the argument of Chapter 2 we introduce the following dimensionless notation:

x=4

; 11(x,Z)

oL f(

xy,z) = 2VL

) 2 2 B G*= 2V2

-gL

Since f1 ( . ) is a two-valued function, we assume for a symmetric

ship

) -f( )

for

=0

= +f(, )

for ,

O (3.14)

Furthermore we assume the ship to have pointed ends, which gives

f1(1) = f1(-1)

=0 (3.15)

It is not necessary to assume this condition in the slender-body theory; especially square-ended sterns may be tolerated and are of some practical interest.

s

The parameter denotes the beamlength ratio of the ship:

+1 -d

+1 0 = - 1d Td

K+K)-

1d

K+K)

4itJ

_J

4rtj

_j

-1 0

-i

-d ± ±

s*1

K1

40(,±f,Ç)

[r ;

f(c)rj

K = ±

rtfÇ)

(3.19)

The Green's function (3.3) transforms into

r

r

i.

= 12,3

(3.20)

r01

V

2 2 2 2 2 +

(i-i) +

(Ç-Ç1)

V_1)2 2(

)2+ u2(Ç+ Ç1)2 Tt/2

cos(qe-111)sn

]

roz3=ípefdjdq

j

° _L

1-0qco52

In (3.17) the integral (3.7) along L0 in not present, since it is of order a

38

-(3.16)

L

With this notation we may write for the equation (3.6), using for the boundary condition on the ship's hull the expression (1.26) and deleting terms of order and

higher:

(3.17)

(3.18)

with

with

with

We introduce the following notation:

= ± Finally we call +1 0

L1 = ;

4h]

j 1 -cl which reduces (3.17) to

4>0 = ZZL1 +LL1

The tern

L1

is presented by:

= 1,2,3 2 2 2 P3

(f-i11)

4. +1 o 4. L21

1

Id1d

4h]

_I

V 2 2 2 -1

_-d

°' ₃ k = 1,2 t = 1,2,3 (3.22) (3.24)

It will be the purpose of the subseQuent analysis to develop systematically the integrals of (3.24) in an asymptotic series of in order to find the first order terme which have to be retained in the final formulation.

+

Asymptotic expansion o the contributions

Lkl

; k-1,2; 11,2

(3.25)

(3.26) and

= = 1,2,3

-d

.0

= -

'a1f(-1,)1og2

i+l) + f(1) og 2(l-i)-

_2f1;Çflog

_{op3_}

41t1 L -1

o o i

-

I

Íd

Id

log 2()I

_{IdÇ Td}

log 2()+0(o2)

4J

J

4J

I

-d -1

-

40

-for

{iog-

+V-

)2 2 2 }

+P

V + 2

22

+ p3 for

we may write for L;1

o 2 2 2 1 P3 L 21 = -

-j

d

-d

+ff

() d

Log ( +

_-

)2

2p2

1 + O(2) =

o 2

22

=_1

_)/

₊ 2 2

2/

f(Ç)

+ p3 )/ +

I1

2

22

+fd

f

+0 p3

)

--1 f

() log(

_{i V}

_1)2+ o2p

)}+

0(02)

=

(3.27)

Using

and

wit b

+

With the same argument we can reduce L22 and we bave o

(3.28)

-d

with (3.29)

P42-

(f1)2+

(Ç+Ç1)2

The function f and f should be evaluated in and not in , and, therefore, the expression (3.28), being independent of , repreeente a part of the two-dimensional solution of the problem.

The contribution L has the following form:

+1 0 + i

T1

L11

{_i2

2 2

3/2 4. (3.33)

which transforms by means of integration by parts into n (U +

il

L11 =]d

-d

(3.34) p32

The contributions

L21

+ L2

can be transformed in a similar way into o - i

L1 +L22 =

f(1

(3.30) -d with 2 2 2 P1

= (f+1)

(3.31) and

2_(f)2(

Ç+1)2

(3.32)

Therefore it is clear that

with

iL+ IELkI=

)c=12 I-1,2 k-1,2 I 1,2 o

!Íd1G1

G4

it]

2 2 2 2

-d

4Pi 4P2 P3 4P4

+f(1)(1/2Logp2+ 1/2 log

p4

-G2 = 4i _{_} f _{-111 +}(Ç +

Asymptotic expansion of the Contributions

L3

+

The term L23 le given by:

+1 0 /2

/d/d fdjd

e

q(

Ç1) 1jc05

cos[q1-nsi

j

-1 -d O L

1-

0qcos2 -

42

--

1/2

_{log o'p3)}}

(3.35)

(3.36)

which we write e

+ + +

L23

= L231 + L232

+1 Q /2

L21 = - --

ffi idIdq

e

q(11 )cos-.(1--f)sir)

-1 -d O L 1- f30qcos +1 _{0 /2} + Re

L232 = - -

_fdjdfd.fdq

1-0qcos2

2it -1

-d

O L

We assume the point , to be situated on the hull surface,

since we are interested in the development of the potential on the surface of the ship.

We divide the integration area , over the surface of the hull in

four different areas (cf PIg. 3.3)

I II z

=O

III - - o

-

o IV - o

-

f -o

Fig. 3.3

11

(3.37)

1'

where

In the area IV the expression

w =

(1-)cos +oii -f)sir0

or

for

tg1 -

ki-Ia(11i-f)

Por w. coRplet. th. path L in th. negative half of the q-plane b asan. of an arc with radius R and the negative inaginary axis. (See Pig. 3.4

The contribution along the arc vanishes for R - , sinçe

..0

along this

arc. e encounter during this process th. pole q = 1/p eec'

Si.ilarly we coaplete the path L in the positive

half

of the q-plane, where In the area IV the expression

=

for 0<-Tt/2

(3.40)

and we coaplete the path L again in the negative half of

the

q-plane.

Pig. 3.4

00

L2

Reifdfdcf(c)1__L_ 1d1dp

ipiP(+1)

[2n2aJ

_1-ip

Qcos2

_i_. /d

'dp

_e

p+ip(

1

11dp

e

2n2aJ

J

, o

1_ip

cos2

2iG 'o

'o

l+ip

co52

_j__

/dsece

I2

1 2

r

sec

[i+c(Çi.1))

1

idec2

_{e Po}

sec2 [i'+o( i)]l

-.

44

-(3.41)

th±cíw

I

(3.38)

We reduce the four-fold integrals of (3.41) by introduction of a change of coordinates. In the first integral of (3.41) we choose:

y = -pucos

_J

(3.42) and we have

-ipt - V_p2t2+ s2(t2+y2)

-K(t)

52

cos2

((t-'

Kp)sin1 -

ycos1J2

= M(t)

u2K2

22 22

2

(t.y)

V p

t s (t +y

) wL(t)

ò (p)

with

s2=1_2+ci2j_f)2+c2(+Ç1.)2

u2₌

_-

f 2

2()2

In

the second integral of (3.41) we introduce

t = p

_{- ip(ÇI-1)=}

pusIn(-1)-ipp

y = pucos(-1)

_I

and we have

p = - K(t)

cos2

= M(t)

afty) - L(t)

ò(p)

(3.43)

with

Pinally we introduce in the third integral of (3.41)

t =

pw + ipo( Ç+) = pusn(+1)+pp

= pucos(+)

which gives

p=K(t) ccs2

=M(t)

(ty)

_{_L(t)}

ò(p)

Insertion of these transformations in the first three integrals of (3.41) enables us to write for the sum the following expression

(3.45) 23

- dte0Ht)

Ht) = 1d 1d (1+ì

Kp0M

)L

2it2_{J J} o

By integration by parts we find that the first term of the asymptotic expansion of (3.44) for

.-O

is given by

(3.46) L23 = f(0)+O(a)

and, therefore, with

z =

-

¿-M(0) (3.47)

46

i

_fdJdc

f(ç)

2ît

V1

2 2

-) -i-r

Similar transformations can be applied to the other areas of integration

I-III

and we finally receive from all the contributions along the imaginary axes the expression:

+1 0 +00

-

_/dfd f() [dz

-

sJ

-00

f(,Ç )

+3 2

f f

L23

= -- d

d

+0.r)

2 2 -1 -d

V1_2+021_

+a

Adding the similar expression for the negative side of the hull 11.O we have with (3.27): o +a -a L23

L23 =

-d o

01

+1

_it)

1dçd

fog2(-1)

j

-d

j

(3.48)

(3.49) (3.50)

where

L23 = - Re

tg2=

_oi1

= _L L23

= -Re

L

/d/d/dsec2e

O EßoJiv

We change the sequence of integration of and

'2

i 1d

ld'/d

-

sec2.ti+a(ÇÇ)J

sec2

ePo

T3oJ

0

-

48 -(3.52) (3.53) The development in (3.50) is non-uniform in and we cannot apply lt for Starting from the original integral, we find for the end-points:

o

-L23C1

-1) +L21 =-1)

_/d [_f

-1,

1og+1)+f(l,)tog4

--d

+1

_fd

) log 2 (1 +

+ O ()

(3.51) O +1 't

+fdf()log2(1_)+O()

Now we return to the three-fold integrals of (3.41), the contributions from the residue terms

Firstly we investigate the -integration. Since

2

-

wec

30sec L. _ec2

dsec2e Po

= de

1+aftg

we bave by integration by parts:

L23 =

RJ

dsec

-

_dÇf(1.Ç)J

1+of1)tg

o

__i_ sec2[ifl+Ç+Ç1)]

e PO r

Idça

I

1af(1)tg

dsec

e o

sec2[i()a(Ç+Çi)1

- JdÇfd)fd

f sec o g

(1+f ()tg)2

Depending on the value of the denumerator of (3.54). we have to interprete the integrals of (3.54) in the sense of a Cauchy principal value.

The first term In the asymptotic development of (3.54) gives

Tt/2

i!.i_i.sec

+ffb1 p=

I

(1 Ç) dsece

Po

L23

dÇf

L

(3 .54) i _{sec 2[i)+aÇ+ Ç1))} e

which can be written, using Watson's (1922) notation, as a Hankel function (2) 1-Io I2 (2)

Ho(x)=i/dsece_9e

o

+b1

L23 (3.55) (3.56)

where Y0 (x) represents the Bessel function of the second kind. The second integral of (3.54) will be cancelled by a similar integral In the area of integration I and we will not discuss this integral furthermore.

Por the third integral of (3.54) we derive in a similar way as for the first Integral that the first tern in the asymptotic development will have the following shape

+b3

1

L23

fdfd

f

(

)yji)

The second residue term of (3.41) can be handled in a similar way, as well as the contribution from the area of integration III.

The contributions of the residue terme in the areas of integration I and II are somewhat different. Only contributions for or

are found, which cancel against similar terms from (3.54). The terms with the Bessel functions Y0 do not show up in the contributions from these areas.

Therefore, adding all terms together, including the terms from the negative side of the hull -=O , which are dealt with in an exactly similar way, we arrive at the following expression for the first term in the asymptotic

development of L + L

o

L3

+

L3

= -

jd{r-1)tog

2i1 )+ f

(1,Ç)log2 (1ei)

50

o o

-d -1 -d

0

01

-

2fdc

f(1)0

) +

2fdjd

f(Ç) (LJ\

° _3o

-d

By returning to the original equations, we have, together with (3.51) the following developm.nt for

+1 o

L2(1=-1)+L23(1.1) =

n 0 +1

()

° Po -d -d

-,

-1)+ L

1) =

4+jf(.)LO92(1-Asymptotic expansion of the contributions L

These terms have the following shape

+1 0 It12 + Re L13 =

fd/dfdJdqa

e

q1 )+

-)cos.

- f)sin

-'

d o L G+

1- p0qcOs2

+1 0

It,,

2it2 J

_-i

_{1- p0qcos 2} -d o L G (3.58) (3.60) (3.59)