Propulsion theory of a ship in regular waves

ARCHIEF

Lab.

y.

Technisclt.

,'.

c

With the Compliments of the Author

Propulsion Theory of a Ship in Regular Waves

by

Ryusuke YAMA ZAKI

Reprinted from the Memoirs of the Faculty of Engineering

Kshu University, Vol. 46, No. 4

FUKUOKA JAPAN 1986

Propulsion Theory of a Ship in Regular Waves

by

Ryusuke YAMAZAKI*

(Received September 2. 1986)

Abstract

An ordinary displacement type ship, which is composed of three parts, i.e., a hull, a propeller and a rudder in order from the front, is usually advanced with an almost constant speed in a seaway disturbed by waves and winds. Then the ship continues surge, sway, heave, roll, pitch and yaw motions, and is requested at first to be safe from the standpoints of motion and strength and next to have a main engine on board, whose output is sufficient to maintain a given speed in waves and winds. In this paper, we develop the hydrodynamical theory to estimate the required main engine power of a ship with a given speed in given oblique regular waves.

Table of Contents

Introduction

The Motion of a Ship Velocity Potentials

The Simplified Propeller Theory The Thin Rudder Theory

The Linear Flow Theory around a Hull

Procedure to Estimate the Propulsive Performance

Concluding Remarks References

Appendix A The force Acting on a Body Appendix B Calculation of F'° and

1. Introduction

Usually a displacement type ship with a propeller and a rudder sails by rotating the

propeller with an almost constant angular velocity on the sea surface accompanied by

waves and winds. The ship oscillates and runs with an almost constant speed almost

straight in the designated direction by controlling the rudder. When the output power of

the main engine to rotate the propeller is kept constant, the ship tinder the condition of waves and winds loses in speed compared with that in calm sea. This speed reduction

is called nominal speed loss. While, the captain of the ship sometimes decides to drop

the engine power for reducing the ship speed from the viewpoints of safety in ship motions and structual strength. This artificial speed reduction is named delivered speed loss'>. However, from the hydrodynamical standpoint of propulsive performance of the ship, the problem in regard to delivered speed loss agrees basically with that concerned with natural

speed loss, because the former corresponds to the latter for the given output power of

the main engine.

In this paper, we consider the case where the ship composed of the three parts, i.e., a hull, a propeller and a rudder2>3> is advanced in a oblique regular wave, and neglect the effect of winds on the ship motions. Then we try to estimate the forces and moments

acting on these three parts, where the mutual interactions among them are taken into

account by superposing the flow field caused by each part2>.3). Here, the following assurnp-tions are introduced in relation to the flow fields:

The oscillatory motions of the ship are very small.

The propeller and rudder are fully submerged under the free water surface. The cavitation and the air-drawing are not generated on the hull, propeller and rudder.

The surface waves made by the ship do not break.

Thus, under the condition that the ship and the incident wave are given, we develop the procedure to calculate the oscillatory motions of the ship and further the relations among the incident wave, the output power of the main engine, the propeller revolution per unit

time and the ship speed by substituting the above estimated forces and moments acting on the three parts into the equations of ship motions.

2. Motions of a Ship4>

We treat in this paper a ship with a single propeller and a single rudder, which is advanced with a constant mean speed floating on the water surface accompanied by a

regular wave as shown in Fig. 1. At first, we define the moving rectangular coordinate

system G_x*y*z* fixed to the ship. Here, the origin G is a point fixed on the center

plane of the hull, and the x. and z*axes are taken to be in the forward and upward

directions, respectively. Then the still water surface can be represented as = - Z0. Further, the velocity components of G in the y* and z*.directions are denoted as (V1, V2, O). respectively. Here, V1 and j2 are independent of time t. Next, we consider the ship oscillates with small amplitudes around the Gx*y*z* system. In this case we

denote the new moving rectangular coordinate system fixed to the oscillating ship as

G0-xyz. Then the relations between the coordinate systems G.x*y*z* and G0-xyz are expressed as

X+E*yÇb*+ 20*. y*=y+;7*_z*+x*, zIz+ '*_x0*+yçb*.

(1)

where 7/* * ç2* 0* and ç denote respectively surge, sway, heave, roll, pitch and yaw motions with respect to the G*x*y*z* system, and are functions of time t. Thus the absolute velocity components (Vex, Vey, vez) of a point (x, y, z) are expressed approxi-mately as

VyV2_Vjçb*+7*_Z*+Xçi*,

vez=V,0*_V2ço*+*_x,*+y*,

(2)

where the quantity symbol with a dot on top indicates the time derivative d/dt

Now, in order to treat the flow field around the ship hydrodynamically, let us define

'Y

g

A ¿

Fig. i Coordinate Systems of the Ship

surface SH is symmetry with respect to the xz-plane, and is expressed by using the parameters x and as

SN: x=X, YYH, z= z,1, (3)

where

YHYH(X, 7)), ZHZH(X, 7)),

yh'(x,7))=yff(x.-7)), 2,(X,7))=2H(X,-7)).

(4)

Here, denoting the hull surface S, in the

still water by Sr,, we can consider S, to

coincide with S, at the averaged position with respect to time. Denoting the half breadth

and the draft at the x-section of the hull in the still water byb(x) and d(x). respectively, we have

X

where ±o indicates the o-value for the still water surface. Thus we define as

XaXXf,

o7o'

d(x)Z0zZ0, d(x)d,

where d denotes the draft, and Xj

and Xa are the forward and aft ends of the hull,

respectively. Then, from Eq. (5) the bottom line near the aft end is rewritten as

x=xr(z),y=0, dZ0zZo

(6)

Denoting the directional cosines of the outward normal nR to SR by (n11, n,, ny),

we have i i a11 i 9YH HR

a(x,)

_{HR ò}

11Z=

HR 9

-

ö(x,)

)2( aZH

R )2}1/2

HH{(

ö(ya,zH)

We assume the propeller center line to be parallel to the x-axis, and then define the cylindrical coordinate system (x, r, O) as

x=x, y= rsinO,

Z= Zp+ rcosO, (9)

where , represents the position of the propeller center line [vide Fig. 21. The propeller

takes number of blades N, radius

r0 and boss radius r,.

The chord length and the effective pitch of the blade section are denoted by c(r) and 2ra(r), respectively. For simplicity of computation from the standpoint of propulsive performance, we assume the

skew, the rake angle and the blade thickness to be zero, and then replace this original propeller with a new propeller which takes an infinite number of blades and maintains the radial distributions of Nc(r) and 2,ra(r) of the original. Thus the shape of the new propeller is represented by the circular disc S, i.e.,

Sp: x=xp, r8rr0,

O8<2jr

. (10)

z

X?

Fig. 2 Coordinate Systems of the Propeller

Hx=

The rudder is assumed, for simplicity of computation, to be a rectangular thin wing.

Using parameters u and z, and denoting the height, the chord length, the thickness and

the helm angle by hR, CR. tR(u, z) and a, respectively, the mean camber plane SRO is

expressed by

SRO:

X=XRO, YYR0, ZZ,

where

XRO=XR ucosa+yo(u, z)sina, YR0= USifla+yø(U,Z)cOsa,

UL<U<UT,

ZlZZU,

CJUT+UL, hR=ZUzl

. (12)

Here, the line (x=XR,y=O) indicates the rudder post center line, and - UL, UT,¿j and

z represent the leading, trailing, lower and upper edges, respectively [vide Fig. 3]. Further

a is assumed to be independent of time. Then the both side surfaces SRL, where K = i

and 2, are represented as

SRK: X=XRK, Y=YR. z=z (13)

where

(-. i)

( 1)

tR(U, z)sina, YRR = YRO + tR(u,z)cosa

2 2

Fig. 3 Rudder Section of the z-section

Next, denoting respectively the mass, the coordinates of the mass center, the moments

of inertia and the products of inertia of the ship by m, (xc, O,

Za), (I,,

Ii,,, I,,) and

(I = = O, =

= O. I = I,), the components of the hydrodynamical force and

moment acting on the ship by (Fr, F, F) and (Mi, M, Me), and the components of

the force and moment caused on the ship by the gravity by (Ft, F, Ffl and (Mi, MI,

Mf), the equations of motions of the ship are obtained as

m('*+zG*)=Fx+F, 1xx*+1*_mzc)*=Mx+

m(i*+xcç1i'*_zcç*)= F+F, Iyy

m(xG *_2GE*) M+ MJ

m(*_xG*)=Fz+F

Izzç'*+Jzx*+mxc;,;'*=Mz+M!

(15)

where the quantity symbol with two dot on the top indicates the time derivative d2/dt2. Further, denoting the moment of inertia of the propeller shaft system, the propeller angular velocity, the propeller torque, the output power of the main engine and the transmission efficiency to the propeller by I, (t), Q, PE and respectively, the equation of rotary motion of the propeller system is expressed by

where aç Ux= ax WHX Wpx, ¡Az = - 1'ez + az +WHZ+Wpz Uy = Vy + - .,( ïÛ)t

ww-wwe

Further, hereafter, for simplicity of discriptions we rewrite as (.* 71* * * 8", Ø*) _{(Ei» E2 E3' E4» Es» E6)et,}

(Fi, F,,, F, M, M,,, M)(F1, F2, F3, F4, F5, F6),

(Fi, F., F, M, M5,

Mf)_=

(Fe, F, Ff, Ff, F, Fif),

+Wjy+Wpy

(16)

(18)

Here, the non-potential flows are assumed to have no effect directly on the stream in

front of the hull or the propeller. Further, denoting the dencity of water, the acceleration of gravity and the atmospheric pressure p, g and Patm, respectively, from Bernoulli's theorem the pressure PH on SH is obtained as

P

-O)

H (19)

where the quantity symbol with bar on the top indicates the conjugate value of the

complex function, and the symbol Re[ I means the real part of the complex function. Hereafter, for simplicity, let us omit the symbol Re[ I and write, for example, the above

equation as

(22)

dÖ (t)

dt QE71T--QP Qg=PE/14P(t)

where (t)

is a function of time t.

Usually, when (t) varies, QE is constant for

Diesel engine and PE is constant for steam turbine.

Denoting the velocity potential of the incident regular wave by w, and the velocity

potentials due to the hull, the propeller and the rudder by , ç',, and R. respectively, the total disturbed velocity potential around the ship is represented by

(17)

While, denoting respectively the velocity components of the non-potential flows behind the

hull and the propeller in the x-, y- and z-directions by (wHX, WHY, WHz) and

wp,

Wp), the flow velocity components (

u, i,,, u) around the ship are expressed as

*_XO*+VHcp*) ₍₂₀₎ PH

_p[

a ( a

+Vey+V

a

+'s' a

2(

a \I2+ a

)2]

=

-at ax ez àz J 2 _{ax J} \ ay az (21)

and the symbol [ ]ín indicates the quantity in the brackets at a point on SH, i.e., x=xJf, YYu» z= zJf.

where

*

=

fff dxdydz, X =

fff xdxdydz,

2B =

fff zdxdydz,

2

'

= 2 _f b(x)dx, XF= _f b(x)xdx, BMT

3*

f b(x)3d,

Jx "WdX,

i

xl

L

BML= b(x)x2dx,

mT=BMT+ZBzG, mL= BML +ZBZG

(29)

and fff (,)dxavdz indicates the volume integral of («) with respect to Here, y" is

the displacement volume in the still water, and m7- and m, are the transverse and longi-tudinal metacentric heights, respectively. The frictional terms EH/I of Eq. (25) is expressed approximately as

PHI I = - RHf, FHf 4 = - 1(1)e E4R4 e w FHJ2 = PFif 3 = F1115 = EH6 = 0 (30) (nHx, flJfj, MHz, YflHz - Zflffy, Zfl Fix - Xfl, tHY - yn1)

(flH1, 112 H3 H4, fl115 flH6), (23)

and represent the terms relating to the hull, the propeller and the rudder as the subscripts

H P and R, respectively, and the terms concerning with the pressure and the frictional

stress as the subscripts p and f respectively. Denoting the components of P1 relating to the hull, the propeller and the rudder by F,, F1 and FR1, respectively, and the i-component

of the frictional stress on the hull surface SH by rHJ, we have

F1 = PHI + F1 + FR1 for

j=1,2,. 6,

(24)

where

Fif = F111 + FHfI, FHPJ =

_-

_{ff P}

Hi dSp, FH! J = ff rj dSH, (25)

and ff («)dSH indicates the integral of (") over SFi. Subsitituting Eq. (19) into Eq. (25), F111 is expressed as

FHPJ = F$1 + PHPJ for (26) I

where

-

fL P2nHJdSH.

FHPJ=ffPffnHJdSH

(27)

Neglecting approximately the effect of the wave height on the wetted surface of S, F$1 of Eq. (27) can be calculated. While, we can calculate Ef referring to the relation between the direction of gravity and the G0-xyz system. Thus, denoting the mean volume closed

by S

and the plane z= - Z as

we can obtain

Fj+ F= - pgAW(3xF5)et,

Fj4+ Pf = - mgmT4e't,

WHx_ WPx,

Here, RH., is the frictional resistance of the hull, and indicates the damping coefficient

of roll motion of the hull and relates to .

Then, dividing the velocity potentials and the flow velocities into the steady and the fluctuating terms individually, we have

Hc,+ç5Het,

+pei0t,

R+5Re"t,

S+eiwt

q5S=

_qS+ç,

_W+H+F+QR

from Eq. (17), and, similarly,

¿L'Hx Wix+ wHXe1. Wjjy Wfy+ wHYe", WHz= WSHZ+ WHZet,

w

w + w, e0

w + w,

w + w

u+ ue'°, u

u +

u

u+ uet

3q5 = V1

_{- ax}

W - w,

= - V2 + + + Wy,

u+wz+wz,

zx=iwe(eiyE6+zE5)---.V E61+

_a+Y+WPY

WeJ J V

_\5+yeJ+

+WHZ+WHY ZWeI J

from Eqs. (2) and (18).

Similarly, Hpj of Eq. (26), RHf of Eq. (30), and FRI of Eq. (24) and Q, of Eq. (16) are represented as FHPJ Fj,9 + ENPi +

for j= 1, 2,

6, RHI

R1

+RHf

e1,

F1

T + Tpe" , = Q + Qp e'°', + FP

et

for

j=2,3,-- ' .6

for j=1,2,',6.

(34)

Now, we consider the case where the incident regular wave . the angle çb0 between

the direction of this wave and the x*.direction. and the mean angular velocity Q of the

propeller are given. Then we assume approximately the mean velocity components

(V1, V2) and the helm angle a. Substituting Eqs. (23) through (30) and Eq. (34) into Eq. (15), and then dividing these equations individually into the steady and the fluctuating equations

with respect to time, we can derive the six equations of equilibrium with respect to the

steady forces and the six equations of oscillatory motions. The equations of equilibrium

for j=3, 4 and 5, which correspond respectively to floating or sinking, heel and trim,

hold automatically by small variations from the averaged positions. The remaining

equa-tions of equilibrium represent foward motion (,j= 1), drifting (j= 2) and turning (1=6). Therefore, letting the unbalanced forces for jrzrl, 2 and 6 temporally to be -

F,

F2° and F60, we have

T(R1 Fyp1 - F1) Fr,

F2+F2+ Fj2= F',

+ F,6 ± F6 = F

(35)

The equations of oscillatory motions for j=1, 2, -, 6, i.e., surge, sway, heave, roll, pitch and yaw motions are expressed as

- mw(Ei +zcEs)=

7p(Ry1

HP1

-mw( XGE5)+ pgA(3 XFE5)= FH+ J03+ FR3,

w(I6±IZXe4±mxGE2)=HP6±P6±R6.

(36)

Expressing O(t) and QE 71T as

QETiT=_QTiT+QETiTeiWt, (37)

Eq. (16) is rewritten as

Q= QSp/7., QE=(Qp+ weIpQ)/Tir .

(38)

Now, in

case where çb, Q, V1, V2 and a are known, we can

estimate T%, Tp,

for j=2 and 6,

.P,J for

j=2,3, ',6 and Q

by applying the simplified

propeller theory, Fj

for 1=1,2,6 and P. for j=1,2,, 6 by using the thin rudder

theory, and for 1=1,2,6,

for j= 1,2,..., 6, Rf, R

and R4 by the linear flow theory around a hull. Then, we can obtain ¿ for 1=1, 2,.. . .6 by solving Eq. (36),

and further the left-hand side of Eq. (35), i.e., F. F and F60. Further, if the ship is in the complete self-propulsion state in the given direction, the following relations should

hold:

F=F20=F6'=0

(39)

Accordingly, V1, V2 and a are determined uniquely from Eq.(35). That is, when ç5

and Q are given, we can obtain V1, V2, a, and further Q. Further, assuming S=0,

we can calculate the mean output P of the main engine by

PQQ=QQ/TiT

(40)

3.

Velocity Potentials

At first, we represent the velocity potentials q5w, , çb, and çbR with respect to the G0-xyz system as

ç5,

and 4 with respect to the Gx*y*2*, respectively. Now, expressing , ç5 and ç5 generally as , this velocity potential satisfies Laplace's

equation. the free surface condition on the plane f = - Z0 and the radiation condition.

Denoting the wave height due to ç by and assuming to be a positive infinitesimal

quantity, the dynamic and kinematic free surface conditions are expressed as hm [ ÒÇA

s+oL 3t

ax* 2

±°±g21

=0,

(42)

Further, , and can be expressed by the singularities distributed on S,,', Sp, SR

and the symbol [

indicates the quantity in the brackets at a point on S.

By substituting Eq.(1) into ç5"', ç5v. ç5, ç and q5, we can obtain the velocity poten-tials ç5,ç5 w. ç5,,'.ÇS,. and ç5p with respect to the G0-xyz system. Accordingly, Eq. (46) is

expressed as Eq. (21). However, hereafter, we adopt ç5"'. ç5'1,, ç5,,,qS7 and çS at (x*, y* z"', t) instead of ç5, ç5 w, ç5n,çbp and 5R at (x, y, z, t), and then rewrite newly these velocity

potentials at (x*, y* z"', t) with respect to the G.x*y*z* system as qS w,_{ç5n, ç5p} and

ç5 at (x, y, z, t). That is, Eqs. (43) through (47) are rewritten as

ç5=çet,

çw=

(48) wo 'w

-PHPLT

Çw=ihwexp{iko(xcosçbo+ysinçbo)}, [ aç5

_y

a1s

y

aò

1f aØ \2

( aç5 \2 f aç5 \2fl

fj,,'>

(49)

(50)

2

y

ax)

_ay)

and their trailing surfaces. Here, these surfaces fluctuate with time in the frame of the

Go*x*y* z" system. Therefore, for simplicity, we distribute the singularities on the surfaces

at the averaged positions with respect to time, i.e., S, S, S0 and their trailing surfaces.

Thus, the boundary conditions on S,,', S and SEO are replaced approximately with those

on S., S

and S0, respectively. Then, the pressure of Eq. (21) is expressed approxi-mately as

[aØ*

a*

V

a*

i Ifaø*\2 ía*\2 fa*\211

+a)

_+dz*)

Jj(H')'

(46) at a 2 where ç5*=

,+

(47) (43) (44) (45) exp{ko(z*+Zo)+iko(x*cosØo+y*sinço)}, ihwexp{iko(x*COsçbo+y*sinçbo)}, WoWeko(V1COSÇbØ+ V2sinçb0)

a-*

A 0SA

_y

a ] =

[

at ' ax* 2 ay* az*

Thus, we have a \2 um

r,

a y1 y2 av* ) A

=0,

(41) 1 2

¿

)+g

_az* z=-Zo

i

'

a

-

V1 a

-

¿ )]z.Zo

Eq.(41) is the free surface condition for i.e., , ça,, çf

and q.

Denoting the length, the amplitude, the velocity potential and the height of the incident

wave by ,,

q5j and , respectively, and giving the angle Øo, from Eqs. (41) and

(42), we have

=çÇVe1t, _v=

1'=

*w=

where

where

FHp= fL.HnHidsH

where ff .(«)dSH indicates the integral of («)

over S.

by , from Eq. (42) we have

1[a

- vi--

àç5 V2

g L dt

dx d x=-0

(51)

Accordingly, the total velocity potential 0 is expressed as

0= Vx V2y+çb= V1x V2y+q5w+q5H+q5p+R.

(52)

Further, PHPJ of Eq. (27) is rewritten approximately as

=±í

v1_-+

v2_)++)l

gL'

dx j z=-Zo

=_i[(ie_Vi

_{V2a)(W+H+P+R)l}

_j

_z-Z0 (56)

Finally, let us obtain Green's function, which is necessary to express the velocity potentials due to the singularities on S, S, S and their trailing surfaces. Denoting the velocity potential at (x,y, z) due to the pulsating point source at (x',y', z'), whose strength

is e', by (1/4r)G(we)e"t, we have4

i

I(x_x)2 +

_(yy')2

+ (z z')2

i tir

+,,

.r

_p---.o./-7rhm I d9

ç' {k( V1cosr.9 + V2sinY) we}2+ kg+ ii° {k( V1cosc9 + V2sini9) e}

Xj0

{k(Vicosi9+ V2Sjfl9)we}

xexp[k(z+z'+2Zo)-t-ik{(xx')cos9+(yy')sin9}]dk -

(57)

Here, G(w5) satisfies Laplace's equation, the free surface condition, i.e., Eq. (41) and the radiation condition.

4.

The Simplified Propeller Theory

Denoting the velocity components inflowing to the propeller in the x-, y- and z-directions

by ( u, up, u), respectively, and referring to Eq.(18), as the input data we have

Up =Vex - w +

i+ R)

WHX,

Denoting the total wave height

Here, . 15H çbp and çbR in Eq. (51) are represented as shown in Eqs. (31) and (48). Therefore, Eq. (54) is transformed into

= ₍₅₅₎

Here, the symbol [

](P) indicates the quantity in the brackets at a point on SF i.e.,

X'Xp

h(r')ç, y'= r'sin(+ 0'), z'

Zp+ r'cos(+ 0'), where O<q<oo. Further Q° is

the angular velocity of S' relative to the blade, and is assumed to be nearly equal to

Q. Then, since the vorticities are distributed in the flow field on and behind S, i.e.,

S*PF, there exists a non-potential flow in this domain. Assuming Q Q, the non'potential velocity components (wp, Wpy, Wp) in Eq. (18) are expressed as

r

_{{ps(r, 0)+ P(r, 0)e1t_(_x')>}} w+ zJpxeut,}

h (r)

= - {r(r, 8) + P(r, 8)e1w_(_x'm}cos0 wy + l;pyet(ût,

vp6=p(t)r+[upY](p.)cos0{upZ](p)sin0

v6+z6e,

(60)

where the symbol [ indicates the quantity in the brackets at a point on S*, i.e.,

XXp,Y rsin0,z=z+ rcosû.

Now, denoting the strength of the bound vortex by

r(r,01,t), i.e.,

¡"(r, 8, t)rs(r, 0)+P(r, 0)e'°,

and using Eq.(57), the velocity potential qp due to the propeller is expressed as = q + where 5

_{9')Gp(r', 0',}

0)d,

(61) (62) (63) ; =

_fr

r'dr'f2d0'f(r'

0')e-1'Gp(r', 0',

; we)d,

(64)

G(r', 0',

; w)=

(r'

+--

, )[G(we)1 Up = - Vez+

w + H + R)+

WHZ (58)

The propeller is represented hydrodynamically by the vortex system. i.e., the bound vortex arranged in the radial direction all over the propeller disc S* and the free vortex shedding from the bound vortex. And the free vortex is approximately distributed on the trailing

helical surface SF with the pitch 2,ii-h(r) without contraction. Referring to Eq. (9) and (10), we define Sp and SPF as

X= X, y= rsinû,

Z Zp + rcos8,

S: X=Xp

h(r)p, y= rsin(ço +9), z= Zp+ rcos(p+ 8),

(59)

where

09<2ir, O<q'<oo.

Then, denoting the components of the relative inflow velocity onto the propeller blade in the x-, r- and 8-directions by (- vp,, vp,-, Vp6), respectively, and dividing them similarly to Eq. (33), we have

r

i s -i_ iWt vPXuPXJ(P.=vPX-1-vPXe

where

rs (r, 8) =

r, 8) = O for

X> X, or

for X< Xp, T> T0, 0 < r< TB.

Since the propeller is considered practically to have the finite number of blades and

(65)

Then, from Eq. (66) and Eqs. (60) through (62) we have

a(r) v;?5 v r

(2/r2+a(r)2

r2+h(r)2

_{)f(r 8)}

]

h(r) [ d

Nk1c(r) 2rh(r)KN(r) ' L dx .> r L r(O8

Eqs. (68) and (69) with respect to h(r), rS(r, O) and f(r, O) simultaneously, we can obtain

V, P, Vp, Vp, V

and V from Eqs. (70) and (71).

Finally, denoting the section drag coefficient at the r-section by CPD, and then the

x-, _{r- and 8-components of the force due to the pressure on the elemental area rdrdû}

and those due to the local frictional drag on the elemental ring dr by ( d2F5, d2Fppr,

Further, since the pitch 2 r h( r) of the free one, we have

h(r)=k2rV/Vo

where a(r)

-

-= (68) hydraulic (69) r VPx

vortex is approximately equal to the

rr(r, e,

t)

_[

dP i

V+ Vpxet

2h(r)(r)

L dx

r(r, e,

t) + [ a

_i

V0+ Vpôe° (70) VP0=VPO 2KN(r) 22r

(

V dû,

r(Oj(P.)

i 2ir V

f

V dû

(71) = 2 jr =

---and k2 indicates the correction factor depending on the chord length-radius ratio. Solving

i

h(r) r

_i

r2+ h(r)2

(2/r2+a(r)2

Nk1c(r) 2rh(r)XN(r)

)rsr

'

8)

_[ (Ox r _L

rae

the finite

where

chord length, the kinematic boundary condition

dçp

on S = h(r) is expressed as (66) (67)

r2+h(r)2

(2/r+a(r) +

)r(r O

t) Nk1c(r) 2rh(r)KN(r) ' ' [ dx

r

a( r)

_VpVp,

L rdO j(p*)

r

/r+h(r)2

r0 / 2h(r) }

KN(r)=cos1exp{_N(1

ki=[1.07-1.05c(r)/ro+0.375c(r)2/rflr,r,

reo0.7ro

d2F0) and (dFpf

, dFpf r, dF10), respectively, we have d2F d2Fppr

₌₀

d2F0

_{= pI(r, G, t) V} = pr(r, O. t) V,0, rd rd O rd rd O ' rdrdû dFpfX

_{--pCpDNC(T)1+(h(r)/r)2 V; V0,}

dFpf

r =0,

dr 2 dr dFpfO

i

dr

Here, CPD is considered to be independent of time t approximately and to b estimated empirically by some method or other. Then, of Eq. (24) and Qp of Eq. (16) are expressed as

F1= fro

I

f

2,r d2FPPx rdO+ dFPfX}dr Jro l.'o rdrdO dr

QP= f

f ro I

_{'27r d2F,}

rdO +

dF0 }rdr

Ir8 l'o

rdrdü dr fro f2Jt d2F90 Pro

2,r d2F0

F2=j

drj

cosOd8,

F3=

I

dr]

sinO rdt91 ro o rdrdt9 .ìr8 o rdrdû F4 = - Qp - Fp9Zp P I

d2F0

F5=

(ro1

(rcosO+z)+

_rdrdû

sinOx}d8

Ira O i rdrdû

Pro P22rf d2FPPX

d2F0

rsinû + cosûxp}dO

F6= /

rdrJ

_{)_}

dra o rdrdû rdrdt9

Accordingly, substituting Eqs. (72), (61), and (70) into Eq. (73), from Eq. (34) we have

F1=T+ Dpet

Qp=Q'+Qpe'°

=

e'°»°

for j=2, 3,, 6,

where

=

fro{12

(r O) Vp9rdO - CpDNc(r)1+ (h(r)/c(r))2 Vp°x

Vo}dr,

Q=p fro,

I '27r

J

O

fs(T, O) VrdO ±Cp0Nc(r)1+(h(r)/c(r))2

V}rdr,

pr0 P2or

Fj2=pJ

rdrJ

_{r5(r,o)Vcosodo}

ra O

Ffto=_pTOdrfOPs(r.0)VosinOdO+F2xP

pro t2X

Tp=pJ

dr]

ro, O Pro p2,r

Q°'°J

_r

rdrJ

{Vs(r,O)Vpx+F(r,O)Vx}rd8

O pro r2or

F2=pJ

rdrJ

ro O pro P2,r

F3= -J rdrJ

(F°(r, O) V+F(r, t9)V}sinOdO

ra O PCPDNc(r)./1+ (h(r)/r)2 V (72)

Fp4=QpFp2Zp

Pro t2,r

-Fp5pJr2drJ

_TB O

{ps(r,8)Vpo+p(r,O)V,o}cosOdO+yp2P_FP3xp

F6 = -

_{P fr:}

r2drf2t{rs(r,

8) Vp9+ f(r, 8) V8}sinOdO + (75)

The procedure of calculations based on the simplified propeller theory developed in this section is called Propeller Calculation U, whose flow diagram is shown in Fig. 4.

When the geometrical form of the propeller is given, and V1, V2, Q.We,

j for 1=1,2,

6, çb,

H' R and (WfiX, WHY, wHZ) are known, we can obtain Q and

ups, u)

of Eq. (58) as the input data of Propeller Calculation U. _{Thus, according to this procedure} we can calculate ,(u'p, Wpy, Wp),

for j=1,2,»., 6, Qp and P as the output data.

Propeller Calculation U Input: Propeller Form

Q; u,

Upy, Up vPx, vpy.Vpz 't. Assumed h(r)

I

['(rOt)

q5; w,, a'p,, w; v8, v;, V,8; h( r)

I

Ih(r)Assumed h(r)I<i YES Estimated C0 Output: çz5; wpx, wry, w; for 1=1, 2 , 6; Op. Pj NO

Fig. 4 Flow Diagram of Propeller Calculation

5.

The Thin Rudder Theory

Denoting the velocity components inflowing to the rudder in the x-, y- and z-directions by ( URX, URY, URZ), _{respectively, and referring to Eq. (18), as the input data we have}

S0 and S'i'RF

where

x, = XR - ucosa + y0(u, z)sina, y = usina + y0(u, z)cosa

zlzzU,

u,.<u<c, yo(u,z)=Yo(uT,z)

for U>UT (78)

Here, the domains - UL < u< UT and UT < u< correspond to S and S*RF. respectively. Denoting the strength of the bound vortex by TR(U, z, t), and the strength of the source by aP(u, 2, t) similarly to Eq. (61) we have

TR(U, z, t)

y(u, z)+ y(u, z)et

z, t)

oj(u, z)+

z)ezt

(79)

Thus, by using Eq. (57) the velocity potential q5R due to the rudder is expressed as

c5R=+çbRe"'

, (80) where

fZfUTfon[

,

G(0)]Hodu'

+ 4

f'dz'f(u'z')[G(0)](R.(HOdu'

i fZ

_{dz'f du"fR(u". z')e_f»}

u')1V [ G(we)]

Hdu'

4Jr Z -Ui. ¿L aflRO (R)

i'Z,i

PUT.... J

dz']

CR(U',Z')[G(0)e)])R)HOdU' , (81) -Ui. a

,a

, a , a -

.

= nox ,

+ nv

a'

+

a'

y(Ur, z)=0,

YR(UT, z)=0,

XXpj, y)j, 22,

i

ay ROx

ii

11R0 i a ROY _{H au}

ay )2( ax )2( a(XRO,YRO)

)2}112 d(u, z)

CH

tz, tUT = I

d j

{uXJ(R.(dz R Jz d-ui. i

a(,y)

ROz _{H 3(u,z)} (77)

URX Vex(w+ g+ p)

WHX

u+

URyVey+(w+H+p) wH,wpYuY+uRYe'

URZ - Vez

p) w wp

u+

e1wet (76)

Then, the rudder is considered to be a thin rectangular wing, and the flow field around it can be represented hydrodynamically by the vortex system and the source distribution.

This vortex system is composed of the bound vortex on S and the free vortex on its

trailing surface SF. Referring to Eqs. (11) and (12), we can define the surfaces S and SRF such as

(flox,noy,floz)[flRox,

ROy, ROzI(u,z)-(U,z

al

o.(u,z)=_ä__t[uxcosa+uys1na](R.tR(u,z)

R(U,Z)

dU{c0+Ysm*)tu,z)}

Here, the symbols [ ](R) and [ ](R) indicate respectively the quantities in the brackets

at a point on S0, i.e., X=XRO, Y=YRO. z=z and a point on S0 and SF' i.e., X'=x0,

y'=y,z'=z', where

Next, the kinematic boundary condition on S0 is expressed as

nRox[

URX+

dR 1

_{dx j(R)}

+flRØY[URY+ dR] +flROZ[URZ+

dR 1

=0

dz j(R)

ày i(R)

that is, from Eqs. (78) and (82) we have

[ÒR sina+

_dx

dR cosa1

= [uRX](R.)

(sina+

dY J(R)

dyo(u, _cosa)

[URy](R )(cosa

dYO(uZ)i)+[

_du dyo(u,z)_du

Accordingly, similarly to Eq. (68), from Eqs. (84), (76) and (80) we have

c0s

.x](R) sina+

du cosa)

[

d . d l (

X ay j(R')

- [uy](R )(cosa

dyo(u, z) sina)+{ dy0(u, z)

du URZ](R*) dz

[

d .

dR

i

(

=[ÏiRX](R.) Sifla+ _du

cosa)

x dy

-

[Y1R.(cosa

dy0(u, z)_du

sina)

+ [URZ](R.) dy0(u, z)

dz

Substituting Eqs. (80), (81) and (83) into Eq. (85), and then solving the obtained equations

with respect to y(u,:) and

R(u, z), we can calculate d/dx, dçb/dy, dR/dx and

diR/dY from Eqs. (80) and (81).

Finally, denoting the section drag coefficient by CRD, the components of the forces and moments acting on the rudder i.e., FR in Eq. (34) are obtained as

FRJ=FRPJ+FR/

for j=1,2,..,6

, (86) pz, pur [ FRP4=PJ zdzJ ?'R(u,z,t)I 1RX Zj L dcbR] dx (R)HROdu,

FRps0

(z,, PUr EuRx

àR

_{(xRucosa)HRodu}

FRP6=PJ dzJ

7R(U,Z,t)

dx j(R)

Z, -Ut where

FRpj=_pfZdzfp(u,z,t)[

3R

HROdU + URY

dy J(R)

UT [

FRp2=_pfZdzf

_YR(u,z,t)

i

FRJ1=---p I

_{2 2,} CRD4XCRdZ,

Here, CRO is considered to be independent of time t approximately and to be estimated empirically or theoretically by some method or other. Accordingly, substituting Eqs. (76), (79) and (80) into Eq. (87), FR of Eq. (86) is rewritten as

FRJ=F!J+ Pjez

for j=1,2,..,6

, (88)

where

El]

Zu

Fj1=pf'dz r7(,)

_{dz, J-ut ay}

Hodu2Pf CRDUXCRdZ

Fj=_pfZdz

7(U,z)[U

a HROdu Z, d-ut

ax J)R')

Pur

H

ai

F6=pfZdzj

y(u,z) xX]ROi2R

¿ZI d-ut

FRfJO for 1=2,3,4,5,6

(87) PZu ¿1r1 E aÇsRl

FR6=p/

d-7f

y,z) URx

ax dZ, d-ULI

+(u

z)[ux

a]

_{ax i)R*)J}

ïHd+

FR3 FR5 = O (89)

The procedure of calculations based on the thin rudder theory developed in this section is named Rudder Calculation U, whose flow diagram is shown in Fig. 5. When

Rudder Calculation U Input: Rudder Form

URX. >RY UR cx

OR(U, Z, i). YR(L z, t)

Output: 'R; FRJ

for j=1, 2,6

Fig. 5 Flow Diagram of Rudder Calculation

,-z,, U r

i= pl dz

_Jz,

r

YU.Z)[RY+

d-uLl

aR

_+7R(UZ)[4Y+

:

i _THR0du J (R'> j(R')J PZ Pur

FR2=P/ dz!

_{dz, .1-ut}

{7(uZ)[iRx

aR]

_ax +7'R(U,Z)

H

ai 'HRodu,

_{ax j(R'(J}

FR4=P f zdz f

_{dz, J-Ut} I aÇ6R +7R(U,Z) (R')

H

a THROdU (R.)j ax ] ax ]

aR

Estimated (-RD

the geometrical form of the rudder is given, and V1, V2, a, We, ¿ for

j=

1, 2,

, 6, w, qp,(WHX, WHY, wHZ) and (wp, Wp, wp) are known, we can obtain (uRX, URY,

URZ) and a as the input data of Rudder Calculation U. Then, according to this procedure

we can calculate çbR, FR for

j=

1,2.', 6 as the output data.

6.

The Linear Flow Theory around a Hull

As stated in Section 3, we adopt the Gx*y*z* system and the velocity potential

qY'. which are rewritten newly as the G0-xyz system and . Then, the hull surface S

is represented as Eqs. (3) and (4), i.e.,

S: x=x, YYH 22H

(90)

where

YHYH(X,77),

2HZH(X,71), XaXXf,

dz+Z00

Denoting the intersect between S, and the plane z= - Z0 by C''. and letting the length along this closed curve C'' to be s, we have

C*: x=x, Y=Yc, z=Z0

, (91)

where

x esx11(x, ± jo)Xc(S), Yc esYH(X, ± o)=yc(S), ds st(dxc/ds)2+(dyc/ds)2

(92)

Further, the trailing surface S1F is assumed to extend straight infinitely rearward in the

xz-plane from the aft end of S, which is expressed as Eq. (6). Therefore, we have

SF: <XXT(Z), yO, dZoz_Z0

. (93)

The velocity potential çb is represented by the singularities distributed on S and

S,F, and çbj is expressed by those on S, and S,F. Similarly, denoting the velocity potentials due to the singularities on S and SF by ÇbHB and ÇbHF, respectively, we have

HÇ'HB+ÇbHF (94)

Thus, from Eq. (52) we have

Ø== VxV2y+çb.

(95)

Dividing kHB and Ç'HF individually into the steady and the fluctuating terms, we have - - ,I.S .Z iwt .j.

-

.&s

i

ja

YHOV'HB yqe

, PHFÇHFi-1HFe (96)

Then from Eqs. (95), (48), (62) and (80) we have

ø=V1xV2y+95

Ç5++p+

ß+(QW+p+ÇR+HF+Ç5HB)et

. (97)

Further, dividing HB and JÍF individually into the diffraction and the radiation terms,

we have5

6 - -. - 6

çbyBçbBD+ E amBm. Ç6HFÇ5FD+ E ûfflÇ/Fm (98)

1 L an11 j(H) a aflH a t7 an11

aBm

3nH

= nHX Vex + fljjy Ve, + filz Vez,

where the symbol [ Jjj .

indicates the quantity in the brackets at a point on S, i.e.,

XXH.Y=YH, z=z. Accordingly, by using Eqs. (23), (2), (100) and (99), Eq. (101) is rewritten as5 - V1n111 + VZnH2 an11 F) on S a

-

an11

on S

aFm

_{on S} =lWeflHm an11 '-H

While, denoting the total velocity potential for the internal domain of the hull by 0,

and referring to Eq. (100), we suppose

= - V1x V2y+ 5()

6

+

E am(Fm+

m= i

(103)

Here, q, çj,j and

are the internal potentials due to the hull. Further, since the fluid flow

is at rest

in

the inside of the hull as a matter of course,

(2) '> must be

zero6, that is,

=V1x+V2y(+

_+fF),

Therefore, from Eqs. (102) and (104) we have

a2

aq5sm

aj

an11 afl11 ' an11 - an11 ' an11 an11 ZU)efHm

(101)

(102)

Ç1Fm

(104)

on S

. (105) Now, expressing (ç5fB, (BD.

j) and (ßm,

T)

by (R »j>) generally, and

applying Green's theorem to the external and internal domaines composed of S and the

plane z = - Z0, we have7

where

a1=e1+

a2=52

a3=E3+

Vi

IWe iOje IWe ZOJe

a4=E4, a5=5, a6=E6

(99)

Here, IF' /1p and Fk are settled according to 1B' cIBo and respectively. Therefore,

0 of Eq. (95) is obtained as

0= V1x V2y+,

=++ F+B+{W+

(100)

where where 7v, d. +

dy )f a'

an>'

2je('

-

d'

2 ds'

a'

as'

/

+ (

-

5W))( , a + 2 a )G(we)l ds' , (106) ax'

ay'

_j(C)

dS= H,,dx'd', Hf=[Hx](X,.)X.,),

_{'15»](x,y,z.$)-(x,Y',z'.'>}

(yb, z) = [YB, ZH](x, 7)-.(x', ), (x'C, y'c)"[Xc, YcIss

Here, the symbols [ ]>'> and E

]c

indicate respectively the quantities in the brackets

at a point on SZ,

i.e., x'=x',y' =y, z' = z'11 and a point on C, i.e.,.' =x,y'

=Y'c

z' =

Z0, and ff()dS

and J(')ds indicate respectively the integral over S and the line integral along C*. Further, Eq. (106) is rewritten as

a

G(e)i

dS1

an'

Jill> a

+ 2 a 1

ax'

ay' )

G(We)] ds' , (107)

(C)

-

ÇbBI(f1.), (ci', ei') [c, /1]>x.)-.>x,)

dyc

_dc\fIvdYc

y

dxc\

ds

-

V2__).tI\

-

2 ds

+ ( y,

c + 2 dyc \ [ a

{1c.}

uS

ds )[

85j)C)

-

'(y dyc

_-

y2

dc)[J

,

(ci'L,p'L)=[L,PLIss

PL _{g ds}

ds

and the symbol [

]c.> indicates the quantity in the brackets at a point on C, i.e.,

x = xc, y = y, z = - Z0. Further, we consider the hull to be a thick wing, which sheds the free vortex at the aft end x= x(z) along S,F. Then, denoting the velocity potential of the free vortex or free normal doublet corresponding to by F and neglecting the

effect of the line integral in Eq. (107) on F' we have

i

Zo XT(Z) _a 4

f_dZof_

pF(z)e)x

_xr)z>)/v[ ay'

G(We)]d'

(109) where (108) PF(2) =

[I+ -

[ ]>±> hm { ]X»X7(Z)+1 (110) r1-.+0 Y=±2 E2 + O

and V2 is the x-component of the averaged velocity behind the hull and is nearly equal

toy. Substituting( pS

e) and (m,m'

(_

dy y2 dx'c dy

-

y2 ax's

)f

a

a>'

+ 4

.'c\ gA

ds'

-

_{as' JL\}

ds' an'11 an's /

[(a

a'

>'

G)

i dS

JSL' an11 ank.

)G(we)_('

3fl'lI _J(H)

E

a B >>B

¿1Lm,/JLm,I1Fm(Z) We) into (o',/i,CL,pL,J1F(z); We) of (Ç5B,F) expressed by Eqs.(107)

and (109), we obtain

(sD, ç) and (am, ÇbFm), respectively

(1) The steady terms (jB, '141F)

In this case We=O holds. From Eqs. (108), (104), (105), (107) and (109), we have

where

=[pS]1 )-.(X', ') (of.', =[ox,

(2) The diffraction terms (BD» IFD)

From Eqs. (108), (104), (105), (107) and (109), we have

where

/i'D =[IJD]x, )-(x', ') , (6D,f4íD)= [OLD, PLD]s-.s'

(3) The radiation terms(Bm, cIFm)

From Eqs. (108), (104), (105), (107) and (109), we have

_[a8rn

aTQ1

Cm L a a

j

Pm=

[am -

Bm](H') : unknown, H (H)

0D1

-íaBD

--I(vycv(v+v[

=0, PD=[ AsD]H.)

ds J\

' ds : unknown, I - 21We[14I(C.) L dfly aflH j(H) ds _{dS J)c')} -

l'1(

tzcfrc\

- -.

ILD =

-

2 »

PFD =[iDJ -

LL7DI(_ (114)

ds

)J)c

and a G(We)1 dS,» BD=

4',rffS.[

aflH Çxr(z') + 4

f[{n

a dy')}G(we)](C) G(We)l ds' , dx' (115) +PD(1l aax, (x'-x)z))/V' [ = FD _/

dz'/

PFD(z)e , (116) r J-d-Z, a Jy'=O and

ckcc)(Vdxc+V dy\[a51

' (111) (112) (113) V2 g ds

-

cls 2

ds )[ 6s i(0)

y, dyc dxc =[RS] [pS])1 = ds

rn=4'ff

_.'S,L 1 -Zo

-

2 ds

)[s]

p(z)

a G(0)1 dS an11 _J(H) d, + 4

f[{o{+(vi

, +v2 ay )}G(o)](C') XT(Z) _a dz

G(0)]dx'

=

f

f

(z')[

a'

a5i»]

=0

_{p5=[ç)_q5,ß](H.)}

: unknown,

[a

6 = 911 aflH

and

Bm_4JrIf[(c1m

_'sL'

+pma,)G(we)l

, a dS J (H) + 4

f

[{m+Ii'Lm(Vl a, +V2 a,

y h

G(e)1

_J(C)ds' , (118) Fm 41 tXr(z) = I

dz'J

Fm(Z')e_xT»v[

¿,

G(we)] dx' , (119) 7 _./-d-Zo where (Cn, j4n)[crm.

/.írn](x. )-(x,)

(Cm, /iím)=[Lm, /1Lm]s_s

Further, denoting the strength of the total normal doublet on SF by i HF(x, z, t), we

amFm(z)}et+_xT

!V}

Then, Kutta's condition must be satisfied at the aft end of the hull. That is, referring to Eq. (101), on S7 we have

i

11ml Veyl

e-.+0[ ÇIY _{J x=xr(z)-r}

y-0

By using Eqs. (2), (100) and (99). Eq. (121) is rewritten as

=

r-+0 L JX=Xr(z)-e_y=0

um1 a

₌₀

c-.+0 L a) JX=XTZ)-E y-0

Eqs. (111) through (113) into the first equation of (102), and then solving the above obtained equation with respect to _j so as to satisfy the first equation of (122). i.e., Kutta's condition. we can obtain pS. _{Similarly, we can obtain PD ¿ and 7i from Eqs. (114) through (119)}

and the second and the third equations of (102) and (122). Thus, we can obtain (RS,

ci.ji,p.(z)) from Eq.(111), (PD,LD,LD,PFD(Z)) from Eq.(114) and(o.m,

Lm,/iLm.

ílFm(2)) from Eq. (117). Then we can obtain ç and 0 from Eq. (100).

Now, introducing the integral operators A and AH, i.e.,

um1 a

(Fm+Bm)1

¿(Ve lWZ for for

m=2

n=4

-.+0L ay JX=Xr(z)-e_{y=0 jWeXr(Z)}

for m=6

O

for m=1,3,5

₍₁₂₂₎

Now, we consider the case where q, , , ç and ç are known. Substituting /Lm _g ds

i

( dyc

_{- y2}

dxc

){( y,

- y2

dx

)[-]

+(v

ds +

2)[ml

as

j(C,)em1}

( vi dyc V2 dx

)[mJ(c)

_{PFm(Z) = [m](+)}

_{- {mJ(-)}

(117)

--have m= i

A=ffdSH/.

+js{+(vi-_+ v2-)}

AH= ff

dSH{±amm+(JD+

amiÌm) _àflH }

+ f

ds{(w+

E amLm)+(PLD+ EamLm)( Vi

(Ai', A1) [Ah, AHJ(x, .$)-(x. .s') where

dSff=Hfldrd,i

from Eqs. (112), (113), (115), (116), (118), (119) and (98) we have

9IB=

41-AG(0),

HB and

4 -f

dz'f

(z')[ ,

G(0)]'

i P-zo txr(z) 1 6

HF7

_{'i7r .1-d-Zo}/

dz'J

/1FD(Z)+E am/JFm

- I m=1

(z')}e1'_xT))/ V

Further, neglecting the higher order terms of çb, we

a1

PHP[

_{at -}

_{' ax -}

2 dyj(H.)

At first, let us consider

for j=2, 3,..., 6.

of ship oscillatory motions, we adopt Eq. (129) as

have approximately

(129)

According to the elementary theory the expression of Pii. Substituting Eq. (129) into Eq. (127), and using the integral operators E and i.e.,

xlr a G

(we)]dx'

(125) L ay'

Thus, 0 of (97) is rewritten as

0=V1xV2y+

++ F +

41AG(0)+

{

+ P±

HF +AG(we)}e1 . (126) Finally, let us treat the forces and moments acting on the hull. p and H are

obtained from Eqs. (53) and (50), i.e.,

!1pj =

_-

ff Pii nm dS11 where

ao

_{11/a0\2+(ao2+(a0}

at

ax)

a /

\ az

)2}1V2 Vn]

a\2

f a\2(a

21

- V V2

_{ax) + ay)}

_{az) i(H)}

at

dx

(H )

E=Pff,dSHnHl(Vl

_{.1.1 Su ax}a

_ay!

a \\,

i=

ff ,dSHflHf(iw_Vl öXV2

)

(130) FHPJ is obtained as

FA1 + F1 e1°t for

j = 2, 3, , 6

, (131)

where

FA,.'1 = F1f +

+ FfS) + FS)

6

-F

HPI - JiPi _{m= i}

F=Eç5, Fj=Eç, FiIEÇffF, FEçbjß

F=E1g, FPJER,

FI1ÊfHF

P' - E .

_{HP.' -} j V' ED HP.' -(m) - E -i W Bm

Here, involves the added mass and the damping force, and PJJ71) and coincide with the results of Salvesen et al.51

Next, let us consider Fi,,. for j=1, i.e., P,,1. The magnitude of P is very small

compared with the other components. However, this component plays the direct role for the propulsive performance of the ship. Therefore, it is necessary to calculate ac-curately by substituting Eq. (128) into Eq. (127). In this paper, instead of Bernoulli's theorem we adopt Lagally's theorem, which is equivalent to Bernoulli's theorem [vide Appendix

A]. That is, is expressed as

1pA

_P{_iwe*el+iweAHx+AH}e1t

(133)

Substituting Eq. (126) into Eq. (133), and neglecting the terms containing e2'°

and e2"°t,

we have

= F,91 + e" (134)

where

F41=FA1+F,1,

Fj°1

=PAV1_F)_PA(a

+ a +

aHF)

ax ax ax

Fl=_F)_F2)_{AH(aP + aÇbR + öHF\1(ô/JP aR +aHF)T

ax ax

ax /

ax ax j

pl=piwe(V*l_AHx)+pAHVl_F_F_pA(P+

ax

dR

ax+

aHF\1

dx!

/ a

a aÇb-fF \

PAH

_dx +

dx + dx )

(Ai' aG(we)+IA,

dG(we))

ax

16'

H ¡f

ax H _ax

F)=(A11

df

+AH

w).

i'3

4

(AA'H

dG(we

_+AHA'

dG(o))

F41 = (136)

(132)

where

Gx(we)=B(we)exP[k(Z+ z'+2Z0)+ ik{(xx')cos9+(yy')sin8}]

. (139)

Then, from Eq. (138) we have

Gx(We)=Gx(We), G(0)'G(0)

(140)

Defining Kotchin's functions H0(k, 9) and H1(k, 9) as

H0(k, )=

_{Aexp[k(z+Z0)ik(xcos+vsin)]}

H1(k, )₌ AHexp[k(z + Z0) ik(xcos +

ysin)J

Eq. (136) is rewritten as F,°=42rpB(0){Ho(k,L9)Ho(k,9)} F11= jrp{B(we)+B(cve)}{Hi(k, 9)H1(k, 9)}

F-Fjj3=4'rp[B(we){Ho(k, 8+jr)H1(k, 9)}+B(0){H1(k, 9+,r)Ho(k, 9)}] i4rkogh

F»p

W H0(k0,çbo)cosçb0 Wo

Now, assuming We>O and defining

V1=Vcosß, V2= Vsinß,

K0=g/V2, AoVw/g(>0)

and then referring to Appendix B, and FjI'> of Eq. (140) are rewritten as P jrl Z

F°=8rpKJ

_{- ,r/2}

_{H0(K0sec29+ß)2sec49cos(9+ß)d9}

( P-0

f,r/2

_{f37r!2\ kcos(9+ß)IHi(ki, 9+ß+,r)I2}

"1-4A0cos9 d9 7r12 o Jr/2 f2,r_30 kcos(9+ /3)1 H1(k2. 9 + /3 + 2 d9 , ₍₁₄₅₎

+2rp

1-4A0cosa

xx'

(xx')2+(yy')2+(zz')2

xx'

(X_x)2+(y_y)2+(Z+z+2Zo)23

+ Gx(we) (138)

Further, referring to Eq. (57), and introducing the integral

_{operator B(e), i.e.,}

B( We) =

im

_{th9 f} ° dk ik2gcos 9

X{k(v+

_{V2sjn)w)2kg+i{k(Vicos+V2sin9)we)}

(137) we have

Input: Hull Form

Incident Regular Wave ç w, ; Fp

for j1,2,

6

cR;FRJ for j=1,2,-- .6 Assumed V1. V,,a pS, _{(); PD.LD.I1 LB} I FD(2). k.Ijk,0Lk,i1Lfr,/iFk(z) Assumed for j=1, 2,-, 6 ÇbH,FjqpJ for j=1,2,---.6 Estimated R1,

R;

WHX. W11y, ¿( for j=1,2, ---,6 Pp1 for 1=1,2---6; F, F, I'T V1, V,. a in Self-Propulsion State

Output: V,, Vz,a,E for j=1,2,-.-,6 We. ÇLFj. 'Hx, Way, WH

Fig. 6 Flow Diagram of Hull Calculation

Hull Calculation U , (147) NO V1Assumed V1HZE, f YES i. Fcos-' for o for

2cos

{1_2flocos±

1-4fl0cos }

(146)

Here, FJ° of Eq. (144) represents the steady wave making resistance, and the sum of

F,,» of Eq. (145) and FJ2 of Eq. (142) corresponds to the added resistance of the hull in

a regular wave, which is obtained by Maruo8'9 [vide Note 1]. Further, using Eq. (137),

F3 of Eq. (142) is rewritten as'°

where

where

Fj=i4pKo ('Td9 r

k2cos(9+ß)Ho(k, 19+/3)H1(k, 9+ß+7r)

j-or io

_{(kCOS+e/V)2Kok}

dk

/ P_do

fin2

_{f3lr/2\ kcos(19+3)H0(k1, 9+3)H1(k1, 8+ß+r)}

+4jrp( j

+

-)

11-4zi0cosâ \./,r12 a OnZ + 4rp Lo

2,ro kcos(9+ß)H0(k2, 9+ß)H1(k2, 9+ß+r)

d9 1-4flocosr.9

Fjd =- i4pKoj'd9(

ksec2l9cos(19+/3)H1(k, 9 +j3)H0(k, 8+ß+,r) on o kK0sec219 d19 +

4pKfmnZ

_{[H1(Kosec2, 9+ß)H0(Kosec2 19, 19+j9+ r)} - 7n12

+H0(Kosec2l9, 19+ß)H1(K0sec219, 19+ß+jr)]sec419cos(19+ß)d8 , (148)

where f')d19 indicates the Cauchy's principal value.

Further, it is necessary to estimate the viscous quantities, i.e.,

w w) and

(Hx, ü, w.z) of Eq.(32),

and RTF2,f of Eq.(34) and of Eq.(30). as functions of V1, V2 and _{for j= 1,2,.., 6 theoretically or empirically by some method or other.}

The procedure of calculations base on the linear flow theory around the hull developed in this section is named Hull Calculation U, whose flow diagram is shown in Fig. 6. We

consider the case where, as the input data of Hull Calculation U, the hull form and the

incident regular wave w accompanied by çb0 are given, and p,

for j1,2,

6,

R and FR for j = 1, 2,. .. , 6 are known. Assuming V1, V2 and a appropriately, and then estimating RHf, R4e and (wHX, WHY, wHZ) by some method or other, we can calculate We, the converged values of ¿

for 1=1,2,..., 6, and further F°, F° and F.

Then, if

the ship is in the complete self.propulsion state, Eq. (39) holds, that is, we can obtain

V1, V2

and a

uniquely. Thus, we have V1, V2, a, We,

for j= 1,2,..., 6, q

and

(wHy, WHY, wHZ) as the output data.

[Note 1]

When the hull only was towed independently in the regular wave, Maruo8 derived F21 in Eq. (142) as

/ 'öo

P,n/2 _{r3or12\ k1k0cosçboJH1(ki, 19+ß+jr)2}

Fi=-2rp(J

+ I

_-J

\ or/2

io

or/2 J _1-4fl0cos19

P2Jr-10 k2kØcosØ0H1(k2,19+/9+7r)2 th9

2.irp /

'1-4flocos19

Therefore, the added resistance of the hull in the regular wave is expressed as

=

2P(f0+f2_f3Z)

002 o o02 ki{kicos(19 +ß)- k0cos çbo}IH1(ki, 19 + + jr)I_{1 - 411 0cos 19} th9

+2

_Jo

(2Jn90 k2{k2cos(19+j3)k0cosØ0}1H1(k2, 19+ß+,r)2 1-4A0cos8

However, we can not adopt the above-obtained expression in the case of this paper,

because the equation of energy balance around the hull does not hold under the condition d8

of self-propulsion.

7.

Procedure to Estimate the Propulsive Performance

When we give a ship composed of a hull, an infinitely bladed propeller and a thin

rectangular rudder, an incident regular wave q5 w a mean angle çbo between the direction

of this wave and that of the hull center line, and the mean angular velocity Q of the propeller as the input data, the following quantities are known:

Input: Geometrical Positions and Forms of Hull, Propeller and Rudder; Incident Regular Waves w, Q Assumed w,, w,, u for J=1,2,-Hull Calculation U V1, V2,a; Ç for j=1,2,--,6 cbH; Wax, WHY, W11 Q; upx. Upy, Up Propeller Calculation U çbp; Wp, Wpy, Wpz for j=l,2,--'.6; 11Rx- u5, u: a 1. Rudder Calculation U 1 P,

for ¡=12.6

NO

Output: Vi, V2,a,we,Qp,P for j=1,2,-- -,6

o a Q)

Fig. 7 Flow Diagram of Propulsive Performance (

m, X, Z, A, Xp, .[,

I. Ir,. m7-, mL,

w, h, À-2,'r/ko, çb0

In this case, we assume, initially diR and FR

for j=1,2, ' .6 relating to the rudder

and p, (wpx, wp-,,, w) and

_{for j=1, 2,.., 6 concerning with the propeller to be}

known.

Next, applying Hull Calculation U, we can obtain V1, V2, a, We, ¿, for j = 1, 2,

6, expressed by Eqs. (94), (96), (124) and (125) and (w11, WHY, wHZ) of Eq. (32) as the output data. Thus, from Eq. (58) we can obtain up), upe), which are the input data

of Propeller Calculation U. Then, as the output data, we can calculate çb expressed by Eqs. (62) and (63),

(wp, wp, wp) of Eq.(65), F1 for j=1,2,

. 6 and Qp of Eq.(75).

Further, adopting p, Wpy, w) and

for j=1, 2,..., 6 obtained

above anew

instead of the assumed quantities concerning with the propeller, and iterating the hull and propeller calculations according to the same procedure as developed above, we can obtain converged values of V1, V2, a,We,

Ei for 1=1, 2.

. . 6. ÇbH, Ç5p. _{(WHX, WHY, WHZ),}

(wp,

WPY, wp),Fp

for j=1,2,.., 6 and Q

Then, using Eq. (76) we can obtain (URX, URY, URZ) and a as the input data of Rudder

Calculation U. Then, according to this procedure of calculations, we can calculate R

expressed by Eqs. (80) and (81) and FR for j = 1, 2, . . . , 6 of Eq. (89) as the output data.

Further, substituting diR and FRa obtained above into the assumed quantities relating to the rudder, and iterating the hull, propeller and rudder calculations according to the same procedure as developed above, we can obtain the final converged values of Vi, V2. a. We.

¿

for j= 1, 2,...

, 6, Q and P as the final output data.

The above-mentioned procedure to calculate the propulsive performance of the ship

in a regular wave is shown in Fig. 7.

Moreover, for many Q, repeating the above-developed calculations, we can obtain

the values of V1, V2, a, Q and the mean output power P in the complete self-propulsion state. That is, in the case where the ship, the incident regular wave and Øo are given,

we can obtain the relations among Q. V1. V2, a, and Q or P. Therefore, for example.

if the steam turbine engine with the output power P is adopted as the main engine, we

can estimate the speed loss of this ship in the given incident regular wave.

8. Concluding Remarks

When the ship is advanced in the incident regular wave in the given direction, the rudder must be controlled to be at an appropriate helm angle.

In this paper, for the

case where the ship with the propeller and the rudder, the incident regular wave, and the

angle between the direction of this wave and the mean direction of the hull center line

were given, we developed the hydrodynamical theory to estimate the instantaneous forces acting on the hull, the propeller and the rudder and the relations among the mean angular velocity of the propeller, the mean forward velocity, the mean drifting velocity, the helm

angle of the rudder, the mean propeller torque and the mean output power of the main

engine. Then, comparing the results calculated for a given incident wave with the results obtained in still water, we can estimate the speed loss of the ship in this given wave

corresponding to the type of main engine, that is, torque constant for Diesel engines and power constant for steam turbines.

In the future, it is hoped to develop the propulsion theory of a ship in irregular waves by introducing the statistical method or other theory.

At last, the author would like to express his deep appreciation to Miss Keiko Matsuki of Kyushu University for her assistance in arrangement and typing of this manuscript.

References

Tasaki, R., and Fujii, H., "Propulsive Performance of Ships in a SeawayState of the Art," Ship Motions, Wave Laod and Propulsive Performance in a Seaway, First Marine Dynamics Symposium, The Society of Naval Architects of Japan, 1984.

Yamazaki, R., "On the Propulsion Theory of Ships on Still Water Introduction ," Memoirs

of the Faculty of Engineering, Kyushu University, Vol. 27, No. 4, 1968.

Yamazaki, R., 'On the Propulsion Theory of Ships on Still Water Advanced Theoretical Method,' Memoirs of the Faculty of Engineering, Kyushu University, Vol. 45, No. 4, 1985. Yamazaki, R., "On the Theory of Motions and Propulsion of a Thin Ship in Oblique Regular Waves Introduction," Journal of Society of Naval Architects of West Japan, No. 43, 1972.

Salvesen, N.. Tuck, E. O.. and Faltinsen, O., "Ship Motions and Sea Loads," Trans. SNAME, Vol. 78, 1970.

Yamazaki, R., On the Propulsion Theory of Ships on Still Water Improved Theoretical Method ," Memoirs of the Faculty of Engineering, Kyushu University, Vol. 37, No. 1, 1977. Yamazaki, R., "On Line Integral of Wave-Making Theory," Technology Report of Kyushu Univer-sity, Vol. 49, No. 1, 1976.

Maruo, H., "Modern Developments of the Theory of Wave-Making Resistance in the Non-Uniform

Motion," The Society of Naval Architects of Japan, 60th Anniversary Series, Vol. 2, Part I, 1957. Maruo, H., "Resistance in Waves," Researches on Seakeeping Qualities of Ships in Japan, Chapter

5, The Society of Naval Architects of Japan. 60th Anniversary Series, Vol. 8, 1963.

Hanaoka, T., "Hydrodynamical Investigation Concerning Ship Motion in Regular Waves,"

Doc-toral Thesis, Kyushu University, 1957.

Landweber, L., and Yih, C. S., 'Forces, Moments and Added masses for Rankine Bodies," Journal of Fluid Mechanics, Vol. 1, 1956.

Appendix A

The Force Acting on a Body

A body enclosed by the surface S is assumed to be moved in water with a velocity VB. And

if there is no singularity on the trailing surface connected to S, this body is represented hydro-dynamically by the source with strength a and the doublet with strength p distributed on S. Denoting the density of water, the volume bounded by S and the velocity potential by p, c* and 0, respectively, and applying Lagally's theorem2).11) which is derived from Bernoulli's theorem, the force F acting on this body is obtained as

F=pv*

dv8

(Al)

where V represents the gradients (a/ax, a/ay,3/3z), r denotes a point in space, and

indicates the integral containing the singularities over S.

Now, in order to represent hydrodynamically the ship hull, let us distribute the singularities, i.e., the source with strength ae1' and the outward normal doublet with strength pe° on the hull

surface S,. and also the line source with strength LeWt, the x-directional doublet with strength

V1pÍt

and the v-directional doublet with strength V2.i2Le'°- on the intersect C* between the

hull surface and the water plane. Then, denoting the x-component of the velocity of the hull motion. the x.component of the force acting on the singularities, and the velocity potential by V1+1e1"t,

x(We)Piwe[_

_{v*ei+ fi (+u afl)xdSH+j{}

+PL( V1+ V2)}xds]

-[ff(+

) dSH+f {ê+JZL( v1---+

v21-)} aø ds]

, (A.2)

where n,,, denotes the outward normal to S, and ff,()ds, and f(")dc indicate the surface

integral of («) over S, and the line integral of («) over C', respectively. Substituting (CS S

0), (5D. PD, LD' 7LD;(0e) and (m, I-4m' CLm, PLm We) into (5 ,ff., Pi We) in Eq.(A.2), and referring to Eq. (123) of the text, F9 of the text is rewritten as

9=pA,

_p{_iWev*Ei+iWeAHX+AH

}et

(A.3)

This equation is Eq. (133) of the text.

Here, if the hull sheds rearwards the free normal doublet at the aft end line (X=X(Z), y=O) of S, the force acting on the hull is represented as the sum of the force acting on S and the force acting on the free normal doublet on the trailing surface SF. The former is expressed as Eq. (Al), and the latter is equivalent to the force acting on the bound line vortex on the aft end line, whose strength is equal to the difference of bound normal doublets between the both sides at the aft end. However, since the y-component of the velocity at the aft end can be omitted, the latter force is neglected approximately. Thus, in this case, the x-component of the force acting on the hull

is expressed by Eq. (A.3).

Appendix B

Calculation of F° and

Substituting Eq. (137) into and F/ of Eq. (142), we have

jr _{i°k3gcos8(V1cosL9+ V2sin8)Ho(k,8)Ho(k,L9)}

F=iim4pfd8f

_{{k2(Vjcos8+ Vzsin8)2kg}2+2k2(Vcos+ V2sin8)2} dk

F,

1irn2pfd8

xf0

_,dk

[(k( V1cos8 +V2sin8)We}2 kgI2+

p°2{k(Vicose9+V2sin8)We}-Defining V and fi as

V1 = Vcosfi, V2 = Vsinfi.

and letting 8-fi-ir in Eqs. (Bi) and (B.2) be replaced newly by 8, F,,°1 and F, is

p°k3 Vgcos(9+fi)cos8jH0(k, 8+fi+,r)2 (k2 V2cos28kg)2+u°2k2 V2cos2i9 dk where FP= jim 4p I dOf° p'-.+O .1-jr rewritten as k2 Vgcos(L9+5)(kVcosi9+W)Hi(k, 8+fi+ir)2_{dk (B.4)} I(c9)= hm f" p°(kVcos8+W«)k2gcos(8+ß)Hi(k, 8+ß+,r)12 _dk

,+,

um 2p

f'

/'+o I-jr ((kVcos8+We)2 - }2 +p'2(kVcosi9 +We)2 At first, let us treat of Eq. (B.4). We express FJ(,,1 of Eq. (B.4) as

jr

Fi»2pf I(i9)d8

_-jr

k1 _{g-2Vwcos19±/g(gVwcos19) K0} {1 _{2flocos19±1-4fl0coSL9} k21 2V2cos219 - 2cos2i9

Ao=Vwe/g, K0=g/V2, k1>k2

kt1

1/

7L-.J Accordinglyw ave Ig'(k1)I=V2cos2t92kikik2=V2cos2i9(kik2)=g.I1_4nocos19

f(k')}kvc

g (1±%11-4AOcosl9) ¡(k2)

-

e 2Vcos19

Therefore, by using Eq. (B.7), Eq. (B.6) is rewritten as

2 kgcos(19+ß)IHi(kj, 19+ß+,r)2 ¡(k) 1(19) = r E 11 _g11-4Aocosi9 Jf(k1) ircos(8+ß)

jkJHj(k,19+ß+jr)2 f)

/1-4A0cos19 ¡=1

Since ¡(k1) and ¡(k2) of Eq. (B.1) are real, the equation 1-4fl0cos19 >0 holds. Hence, if A0 >1/4, the solution is absent in the range of

19o<19<190

for A0>0,

and yrr90<19<jr+190 for _fl0<0 where

190 =COS

4IAo 0 190

f

(B.13)

and if Jfl0<1/4, we adopt 190=0.

Thus, let us evaluate f(k,)/Lf(k1fl in Eq.(B.12), (a) In case of fl00, we have

i for

f<19<--r90

19o<19<i If( k1)

1 for

_f

1 for f<19<-190, 19o<19<f

f( k2) I

_{1 for}

(B.14)

Thus, substituting Eq. (B.12) into Eq. (B.5), from Eq. (B.14) we have

Now, we use the mathematical formula [vide Note 2 at the end of this appendix], i.e.,

um f'

_{o g(k)2+p2f(k)2}p°f(k) F(k)dk=,rE2 _g'(k1)F(k1) _f(k1)f(k1) (B .7)

where

g'(k)=dg(k)/dk,

g(k,)=O, g'(k1)0

Here, writting F(k), g(k) and f(k) as

F(k)=k2gcos(c9+ß)IHj(k,19+/3+.7r)2 g(k)=(kVcosc9+w)2kg, f(k)=kVCOS19+We , (B.8) we have g(k)= V2cos219(kk1)(kk2), g'(k)= V2cos219(2kk1k2), (B.9) where (B.10)

(Bu)

(B.12)

+ 2.irp / PDo P002

+1

\ 2012 .IDo o9+ß+yrA)12

-1-41 fl01 cos 8

(Zor_Do k2cos(8+j9)lHi(k. 8+ß+ir)l2 _d8

Jo0 _1-4lAolcos8

AO

kft K0

(1 2lAolcos8± 1-4lAolcos8),

=-i(i+

_{1n01 i}

kJ

2cos°o9

Eq. (B.15) is Eq. (145) of the text.

Next, we consider F°0 of Eq. (B.3). In this case, FJ,°' is expressed as

F0 =[2FJ'>]Wo,H(k, D)-'Ho(k, D)

Then, from Eqs. (BiO), (B.13) and Eq. (141) of the text, we have

.4=0,

k1=K0sec29.

k2=O, 8=0 and

Ho(k,o9+7r)=Ho(k,o9) Therefore, from Eqs. (B.15), (B.21) and (B.22) we have

= /

+ j

-

\ kcos(8+ß)lH1(k1, 9+ß+,r)12 d8 F,),' 2irp( r-0o rr/2 3ir12

Jf2

J'

fI2 J 1-4A0cos0 (2r_0o kcos(8+8)lHj(k2,8+ß+,r)l2 _d8 Jo0 I1-4fl0cosL9 (b) In case of A0 < 0, we have i for

-

3,r

lf(k1)l

_{1 for}

_{-< 8< - 8o,}

_{+ 8o< 8<}

2

1 for

jf(k2)J

_{1 for -<8<r-8o,}

,r+8o<8<--Thus, from Eqs. (B.5), (B.12) and (B.16) we have

/ 'or/2 for_Do

f3002)

kcos(8+ß)lHi(k, r9+ß+ir)12

FJ"=2irpl /_\J,r/2

-

_{,r/2 - ,r*Do '1-4A0cos8}

I '002 rorD _{P3002) kcos(8+ß)lHi(k2, o9+ß+ir)12}

d8

2,rp(/

_\or/2

+1

_./002

+1

_{.1or*Do 1-4A0cos8}

Here, using newly 8 for o9 ± r of the above equation, we have

FJ,.'> 2rp( p7r12 f32r12) kcos(8+ß)lHi(k3, o9+ß)12 =

J

002

+1

oo

-

1+4A0cos8 d8

+2irp ('°°

kcos(r9+ß)lH1(k4, 8+8)12 d8 /1+4fl0cos8 where

Then, referring to Eqs. (B.15), (B.17), (BiO) and (B.18), FJ11 is expressed as

+ 2irp

I P,r12 f31r12\

FP1 =4irpK(_\J,r12

_{- J}

flHo(Kosec2c9, +ß)2sec4.9cos(c9+$)dL9 ,r12 /

Çyr/2

= 8irpKJ₇₀₂ Ho(Kosec2l9, 19+ß)2sec4cos(19 +/3)d19

Eq. (B.23) is Eq. (144) of the text.

[Note 2]

We assume g(k), f(k) and F(k) to be continuous and k1 satisfies the following equations:

g(k1)=0, g'(k1)r*oO for 1=1,2,3

k1 > k2> k3>

Then, choosing a1>0 arid b,>0 so as to be 1e1_1 >k1+a1>k1>k1-b1>k1+1 and setting k=k1+E, the integral defined by

1= um f°°_p,*o p°f(k)

o g(k)2+°2f(k)2 F(k)dk is transformed into

tao _f(k1+E)

1= irf g(k)) F(k)dk= ir _{J58(g(ki+E)) f(k1+)I} F(k1+E)dE

where c3(») is Dirac's delta function. In the vicinity of E=0, we have

g(k,+)=g(k1)+g'(k1)+

f(k1+E)1rf(ki),

F(k,+E)F(k1)

Therefore, I of Eq. (B.25) is obtained as

F(k1)d-rr

ií2i

That is, the following equation holds:

F(k,) f(k1) hm

f°°g(k)2+ p02f(k)2p°f(k)

F(k)dk=jrj

t g'(k1)I f(k1)

(B .23)