Prediction of Resistance and Propulsion of a Ship in a Seaway

RAW RT T Q 0m n V a U t w np no C

Prediction of Resistance and Propulsion

of a Ship in

a Seaway

Odd M. Faltinsen

Norwegian institute of Technology; Trondheim-NTH

KnutJ. Minsaas

Norwegian Hydrodynamic Laboratories, Trondheim Nicolas Liapis

Norwegian Hydrodynamic Laboratories, Trondheim. Svein 0. Skjørda}

Kaidnes Mek. Verks ted, Tønsberg

Norway

ABSTRACT

A procedure for calculating added resi-stance, transverse drift force and mean yaw. moment on a ship in regular waves of any wave direction is presented. An asymptotic formula for small wave lengths has also been derived. The influence of wave induced motions on the wake, open water propeller characteristics, thrust deduction and rela-tive rotarela-tive efficiency is discussed. The reason to false values for the propulsion factors are pointed out. Determination of RPM and HP

in

irregular waves is discussed. NOMENCLATURE

added resistance due to waves total resistance

thrust of the propeller torque at the propeller torque at the engine number of revolutions speed of advance Ship's speed

thrust deduction factor (also used as time variable)

wake fraction. Used both as effect-ive wake and radial wake distribu-tioñ

mechanical efficiency of the shaft bearings

relative rotative efficiency propeller efficiency

hull efficiency

propulsive efficiency water-line curve

Bbhetheek van de

ide

Schepsnw- en Sthepvnartkune

Ted'nicr

oc

chzo, Deift

DOCUMENTA1IE :

DATUM

KT KQ D R L B Cb F n a g k A x,y, z x,y, z

V-3

thrust coefficient torque coefficient diameter of propeller radius of propeller

advance coefficient of the propeller ship length between perpendiculars ship beam midships

block coefficient Froude number

-wave direction (a=O is head sea) displacements (j=1,2,...,6 refer to surge, sway, heave, roll, pitch and yaw respectively, see Fig. 1) incident wave amplitude acceleration of gravity circular wave frequency

circular frequency of encounter wave number

wave length

coordinate system as defined in Fig. 1

coordinate system fixed to the s-ip and

coinciding

with the

x,y,z-system when the shipis not

oscil-lating

local coordinate system (see Fig. 4) complex unit

thrust diminuation factor (see eq-uations (73) and (74))

average 8-value over one wave pe-riod (see equation (76))

h distance between water surface and propeller shaft center

boundary layer thickness 62 momentum thickness VmaxlVO see equation (68)

e see Fig. 4

n Generalized normal directions (see equations (12) and (13). Posi-tive normal direction into the fluid

M Mass of the ship

i) _{see equation (19)} I. INTRODUCTION

Estimation of ship speed, machinery power and propeller revolutions is tradi-tionally based on still water performance. But ocean-going vessels often meet sea con-ditions where the seaway influences the ship resistance and propulsive coefficients; and in the case of extreme ship motions, the ship master may reduce the ship speed due to green water on deck, slamming, propeller racing or excessive accelerations.

The increase of fuel cost has increased the importance of the prediction of ship speed and power in a seaway. It is neces-sary to ensure that the ship will not only be economical on a ship trial in calm water but also on a voyage in wind and waves. This is some of the background why we start-ed our study.

We have focused our

attention

on pro-blems connected with involuntary speed reduction and how to find the increase in power required to

maintain

ship speed

in a

seaway. We have

in

particular studied the added resistance and propulsive factors of a ship in regular waves. The change of pro-pulsive factors due to waves is of greatest importance when there is

significant

wave energy at frequencies in the vicinity of the natural pitch and heave frequencies of the ship. This will normally correspond to severe sea conditions. Further, the change in propulsive factors due to waves is most pronounced when the ship is in ballast con-dition. The added resistance on ships may also be of importance for moderate sea con-ditions, in particular for ships with blunt bow-forms.

At the end of the paper, we have dis-cussed how the results from regular waves can be applied to irregular waves. Hopeful-ly our results may later on be useful tools in designing economical hull forms and pro-pulsion systems, or for instance in deter-mining how the trim of a ship can be used to determine an economical ship speed in a seaway.

2. BASIC FORMULATION

If we want to determine the speed, haft horse power or propeller revolutions of a ship in regular waves, we may average forces and moments over one wave period and set up the following two basic equations

V-3-2

Qmhlm = (1)

RT/(lt) = T

(2)

where is the torque delivered by the ma-chinery,

m is the mechanical efficiency be-tween the machinery and the propeller shaft, Q is the hydrodynamic torque on the

propel-ler in open water,

R is the relative rota-tive efficiency, R,, is the resistance of the ship without propeller, T is the propeller thrust in open water and t is the thrust de-duction. The propeller thrust and torque is dependent on the particulars of the

propel-ler, propeller revolution n and the speed of advance

Va = U(l-w) (3)

where w is the effective wake and U is the forward velocity of the ship. The propeller characteristics will also depend on the im-mersion and the wave induced motions of -the propeller shaft. This will be discussed

later on.

We will in the following text first in-troduce a new theory for the added resi-stance of a ship in waves and then discuss how the waves affect the wake, the propeller characteristics, the relative rotative

ef-ficiency and the thrust deduction. Finally we will discuss how we may determine RPM, HP and ship speed in a seaway.

3. ADDED RESISTANCE

Theoretical methods for added resist-ance of a ship in regular waves are either derived by directly integrating pressure

over the wetted ship surface or by using the equations for conservation of momentum and/or energy in-the fluid. The procedures are derived by a perturbation scheme where

- the linear wave induced motions and loads

are a first order approximation to the pro--blern. The added resistance is then found as the mean longitudinal second order force. This implies that the theoretical added re-stance is proportional to the square of the wave amplitude. This is a valid approxima-tion for moderate wave steepnesses. But when the waves are breaking or close to breaking, one may question the procedures.

Maruo (1) has used the equations for conservation of energy and momentum to de-rive a theoretical method for added resist-ance. His procedure is valid for any wave length and wave heading. The disadvantage with the method is that it is difficult to apply from a practical point of view. The accuracy of the method is dependent on how well the linear problem is solved. The far more simpler method by Gerritsma and Beu-kelman (2) is easier to apply. Gerritsma and Beukelman (2) derived their formula by energy considerations, but the rational

ba-sis for their method is not quite clear. Maruo and Ishii(3-) have simplified the original formula by Maruo (1) considerably

by using a high frequency assumption

con-sistent with s-trip theory. The formula re-seinbles Gerritsma and Beukelman's formula,

but the importance of the differences has not been clearly pointed out. Takagi, Hosoda and Higo (4) have derived the same formula as Maruo and Ish-ii (3). They also use the equations of momentum and energy in the fluid, but used other control surfaces than Maruo.

Added resistance in oblique sea has been studied by Hosoda (5) by applying Maruo's method. He concluded that the con-tribution of lateral motions to added resi-stance is relatively small. According to this conclusion, the added resistance in oblique waves can easily be calculated on the basis of the extension of prediction method for head sea waves (Fujii, H. and Takahashi, T. (6)). In a similar way, Ger-ritsrna and Journde (7) generalized Gerritsma and Beukelman's method to oblique sea.

Another way to derive a formula for added resistance is to directly integrate pressure forces over the wetted surface of the ship (Boese (8)). It should -be noted that Boese has neglected the quadratic ye-.Locity term in Bernoullis equation and one term aris-in f

.t

n .ressure o

the iji-stantenous position of the wetted surface .of the ship instead of using pressure on the average position of the ship. These terms are of importance and will be included in this paper. oth procedures are quite simple to apply in practice. The advantage of a method based on directly integrating pres-sure forces compared to a method using the equations for conservation of momentum and/or energy in the fluid, is that it is easier to understand the physical reason for added resistance. Boese derived his formula only for Ilead sea waves, while our procedure is valid for any wave direction and includes the effect of the lateral mo-tions. We also calculate transverse drift force and yaw moment. Further, we do not base the calculation of the diffraction potential of the first order problem on the

relative motion hypothesis.

-All the formulas above are based on some kind of slendership assumption, which makes them questionable to apply for blunt ship forms. This has some practical conse-quences. Fujii and Takahashi (9) have point-ed out the importance of addpoint-ed resistance on blunt ship forms in the small wave length case for speed predictions in moderate sea conditions. They propose to divide the ad-ded resistance into two parts. One part is

-added resistance due to ship motions and another part is added resistance due to wave reflection at the bow (or stern if fol-lowing sea). The added resistance due to ship motions may be calculated by the con-ventional methods while Fujii and Takahashi have derived a quasi-rational method for resistance increase due to the reflection at the bow. Their formula gives a correct asymptotic behaviour for low wave lengths and zero Froude numbers as long as the ship surface is vertical at the water line. But the effect of finite draught and- forward speed is only included in an approximate way. It may be noted that the forward speed

V. 33

effect may be quite important (see Fujii and Takahashi (9)).

We will in this paper present a new asymptotic theory for small wave lengths which takes into account the forward speed effect. From a rational point of view, one can argue against dividing the added resist-ance- into two parts in the way that Fujii and Takahashi (9) did.. Generally speaking, the reflection of the waves and the ship motions may interact in a more complicated way on the added resistance. But their

pro-cedure makes some sense for certain wave lengths regions, i.e. for small wave lengths where the effect of ship motions may be dis-regarded and in vicinity of pitch and heave resonance where the relative motion between the ship and the waves are important. But- it should be noted that is is possible to de-rive one expression valid for all wave lengths if for instance the first order mo-tions and loads are calculated by a three-dimensional sink source technique.

The calculation of mean added resistan-ce in irregular sea is straigthforward if the wave spectrum and the added resistance in regular waves is known. But it should be pointed out that the added resistance will not have a constant value in irregular sea.. It will be slowly varying approximately as the envelope of the wave elevation.

We will in the following text discuss how we calculate the linear wave induced motions and loads on a ship. This is a ne-cessary building brick in the theory of added resistance. We will then present a method for.added resistance based on

direct-ly integrating the pressure over the wetted body surface. At the same time, we will show how other mean force or moment components in waves can be evaluated. As an example, we will study transverse drift force and drift yaw moment. Afterwards we will derive asymp-totic formulas for added resistance, trans-verse drift force and yaw moment in the low wave length case. Finally, we will present numerical results and comparisons with other methods and experiments.

3.1 Linear wave induced motions and loads Let us study a ship advancing at con-stant mean forward speed with arbitrary heading in regular sinusoidal waves. We assume that the first order motions are linear and harmonic. Let (x,y,z) be a righthanded coordinate system fixed with respect to the mean position of the ship with z verticallyupwards through the centre of gravity of the ship, x in the aft direct-tion and the origin in the plane of the un-disturbed free surface. Let the translatory displacements in the x,y and z directions -with respect to the origin be n1,

2 and

respectively, so that n1 is the surge, ₂ is the sway and fl3 is the heave ment. Furthermore, let the angular displace-ment of the rotational motion about the x,y

and z-axis be fl4, r5 and 6 respectively, so that fl4 is the roll, r5 is the pitch and

and the translatory and angular displace-ments are shown in Fig. 1. We will also

ope-rate with a coordinate system (x,y,z) that is fixed in the ship and coincide with the (z,y,z) coordinate system when the ship is not oscillating.

We will use the Salvesen, Tuck and Faltinsen theory (10) to find the wave in-duced motion. This method does not determine the pressure distribution because it is based on a generalized Haskind relation when evaluating the wave exciting forces and moments. When we later shall derive a proce-dure for calculating added resistance, we will need expressions for the linear

pres-sure distribution on the hull. it is also necessary to know the pressure distribution at the mean free surface if we want to find the relative motion and velocity along the ship. This is of interest in evaluating vo-luntary speed reduction due to slamming and green water on deck. We will therefore pre-sent two different ways of calculating the linear pressure distribution on the ship hull. The differences in the method are in the calculation of the diffraction potent-ial. We will write the total velocity po-tential as

-i t

Ux+S(x,y,z)+T(x,y,z)e

e

where Ux+q5(x,y,z) is the steady contribu-tion which determines the wave résistance. We will assume that has no influence on the linear wave induced motions of the ship

(Salvesen, Tuck and Faltinsen (10)). The time dependent pressure will be divided into three parts

i t e

= IFD

where i is the complex unit, i the fre-quency of encounter and t is te time vari-able. Further, is the incident wave po-tehtial, is te velocity potential due to

forced motions in six degrees of freedom and is the diffraction potential of the

restrined ship. We will write the

inci-dent wave potential as

e05Iet

(6)

where g is the acceleration of gravity, Ca is the wave amplitude, k is the wave num-ber, is the heading angle, = i is the wave frequency which is related to the fre-quency of encounter we by

= w0+kUcosc (7)

According to Salvesen, Tuch and Faltin-sen,.we may write the forced motion potent-ial as.

It is understood that real part is to be

taentin all expressions involving

V.3-4

where

= 2fl2+3fl3+4fl4

+(-x+

)3T15+(x-

!__)26 (8)

where satisfies the following boundary value problem

+ - - 0

in the fluid (9)

9z

2j+gj2

- 0 on z=0 (10)

4,i

= enhi on the average wetted surface SB of the ship (11) ere ni is determined by

n=(ni, n2, n3) (12)

txti=(n4, n5,n6) (13)

where is he normal vector to the body

sur-face (positive into the fluid) and r=+y+z.

Further, 4j satisfies a radiation condi.tion of two-dimensional outgoing waves. The j-problems can be solved by the Frank Closefit method.

_J Fig. 1 Coordinate system

The diffraction problem may be diffi-cult to solve, in particular in the low wave length range. We will apply two diffe-rent methods. One method is what we will call a "relative motion method". A second method is due to Skjørdal and Faltinsen

(11) and is only applicable n head sea. This method takes into account that the waves are modified as they propagate along the ship.

The "relative motion method" implies that the wave length is large compared to the c±oss-dimensions of the ship, that the ship is slender and that three-dimensional effects can be neglected. The diffraction potential can then be found as a solution of the two-dimensional Laplace equation (9), the classical free surface condition (10), a radiation condition of outgoing two-dim-ensional waves. The body boundary condition can be written as -

-= -n2u-n3u

on SB kz +ikxcosa-iwt -

m

u _{= WCaSi111 e}

and

- kzm+ikxcosa_i t

ioe

e

Here is an average vertical cross-sectional coordinate, which may be chosen as -T/2, where T is the local draught. We may now write the total time dependent

velo-city potential as

-iwt

_e i

-= I2'2

us,)

+ 3(i3

+ + (X_Ie)2fl6

The dynamic first order pressure at a fixed submerged point can be written as

et

p =

p(.

₊

_U}-)+Te

(15)

and the dynamic wave elevation can be written as

= pg

z=0

(16)

We will now show how we may find added resistance by directly integrating pressure over the hull surface.

3.2 Added resistance by pressure integra-tion method

We start out with the complete BernouL-lis equation

p = -pgz_p4 -

v2+p0+

u2

(17)

where V is the fluid velocity and

p0

is atmospheric pressure. If we neglect the in-fluenôe of the stationary perturbation

p0-tentiàl (see equation 4), we may write

p = -pgZ-p} - PP}

(18)

az

0

Here p is due to the incident waves and its interaction with the ship. Let us proceed by a perturbation scheme, i.e. we write

4,

(1)(2)

(19)

where

41)

is the linear first order pot9

-ial discussed in the last chapter, and is a second order approximation which is proportional to the square of tPie wave amplitude.

Let us now make a Taylor expansion of the pressure about the mean position of the ship. The pressure p5 on the ship correct to second order in wave amplitude may then be written as (14) V-3-5

ap(U

p5 =

-pgz p( 1U ). (2) (2)

ax

in (1) _P(n2+xn6-zn4)r_(.t (20) (1) $1)

-P(n3-xnS+yn4)(

-(1) 2 (1) 2

p_1aq

₊ 2

ax

'ay

(1) 2 'p

az

m

o Here in indicates that the variables should be evaluated on the average position of the wetted ship hull. We will now study separately the contribution to added resist-ance from each pressure term in equation

(20)

The contribution from the "-pgz"-term can be split up into One part which comes from pressure forces over the exact position of the body surface below z=0 (i.e. the sur-face area Sd in Fig. 2), and in another part from integrating over the wetted area above z=0. It is possible to show by Gauss' theorem that the pressure forces over S0 due to the "-pgz"-terrn result in a vertical force. We may therefore write the contribution to the

added resistance from the "-pgz"-term as

2

F1=

pgj

fzn1dzds=-

n1ds (21)

The bar over the expressions indicates time-averaged values. Further, is the wave elevation calculated by linear theory and c

is the water-line curve.

Figure 2 Ship surface

The contribution from the linear pres-a a

(1)

sure term -p ( - IUa cannot be neg-lected. The reason is that we in linear theory integrate over an average wetted sur-face and an average direction. In a second order theory, we have to make corrections for the integration over the correct wetted area of the Ship hull and, for the correct instantaneous fOrce direction.

The correction for the integration of pressure forces over the correct wetted area of the ship hull gave a mean longitudinal force

(22)

(1) (1)

Whenwe integrate -p( i-U..

over the rest of the wetted area of the ship, we have to be careful with what co-ordinate system we use. Since we must inte-grate over the instantaneous position of the wet area of the ship, we

_{sh2t4d_stat}

out with the coordinate system (z,y,z) fixed to the ship. In -a first order theory, it does not really matter if we had used the AXLY,Z) co-ordinate system instead of the x,y,z-systenl. But in a second order problem, we have to be careful. The z-component F of the force on the average wét

sure due to the

pres-sure term p

m is the same as the vertical hydrodynarnic force used in the linear theory. By using the equation of the first order motions, we may therefore write

F-=M

2

1-C333+C355

(23)

dt

where M is the mass of the ship. C33 C35 are hydrostatic restoring coefficients which may be written

pgfbdpgA

C35-pgfbd-p gM

Here b is the sectional beam of the ship and the integration is over the length of the ship. A and M. are the area and

the moment of the watelane.

By decomposing expression (23) along the x-axis (see Fig. 3), we find the fol-lowing mean longitudinal force contribution

x

Figure 3 Coordinate system

In a similar way we can decompose the -component of the force on the average wet-ted surface due to the pressure term

(1) (1)

-p(

) along the x-àxis. We will

find the following mean longitudinal force

FCDM2ZGn4) n6

(25)

Here z is the z-coordinate of the

centre of gavity of the ship.

The contribution 923the force from the pressure term -p( _m can be.

V -3-6

shown to be zero (Faltinsen and Løken

(12)

The contribution from the rest of the pressure terms in equation (20) is straigth-forward and will be presented later. But first we will rewrite the expressioji for the longitudinal force in a more convenient way by introducing the relative wave amplitude along the ship in the express-ions.

We then use the theorem ff5

Vol S

where S is the surface enclosing the volume Vol. Further, the normal vector is positive out of.the volume. We may then show that

-fCn3-xn5+yn4) (3-xn5+yn4) }n1ds

C33 n3 T)5+C35 Ti5Ti5

By using this relation in equation (24), we may write the added resistance as

p1ca

}n1ds e2Me2M2_2GTi4)Ti6 -(1) (1) +pfC(n2+xn6-zii4)(

IU_

lU)

2 (1) 2 (1) 2)}n,ds -) ) -I-where Cr= (n3xn5+yT14) (27)

is the relative wave amplitude along the ship, and c is t-he water line _curve.

Boese (8) has derived a formula fOr ad-ded resistance which resembles equation (26). First of all, it should be noted that his formula neglected the influence of sway, roll, yaw and any flow which is antisymme-tric with respect to the x-z-plane. That means in reality that Boese's formula should be applied for head and -following sea. But even if we limited ourselves to head or fol-lowing sea, Boese's formula would be missing the integral over S in equation (26).

The procedure escribed above may be generalized to calculate other mean force and moment components. For instance, we may write the transverse average force (i.e. force components in the y-direction)

2 and

the average yaw moment F6 abou-t the z-axis as

}nds

(28) pR (T12+xfl -Zn )L.(_(1) (1) 6 4

ay

t -fU

m

(26)

FCDz_(WeM33)

n3n5+C35n52 (24) y y x

a (l) 2 a (1) 2 a (1) 2 +½((a' a )

)}rids

+K. where

2- 2

K2w M13n4_We M(nl_zGnS

and 2 2

K6=(I44We n4+Pv BZG

4+MZGWe fl2 2

+(We I55n5+Pgv(z-z)n

Here 144 and I are moments of inertia in roll and pitch,

46 is the product of inertia between roll and yaw, and ZB is the z-coordinate of the centre of buoyancy.

Similar procedures for mean forces and moments as described above have been used

in the zero-speed case by Pinkster (13) and Faltinsen and Løken (12). The latter consi-dered only two-dimensional bodies.

3.3 Asymptotic low wave length case We will now derive formulas for added resistance, transverse drift force and mean yaw moment in the asymptotic low wave length

case.

Consider a ship in incident regular waves. We assume the ship has vertical sides

at the waterplane, and that the wave length is small compared to the draught of the ship. Due to the small wave length

assumpt-iOns the waveexcitation forces will be

small. If resonance oscillations are not excited, it implies that the influence of

the wave induced motions of the ship can be ne-glected. Due to the small wave length

as-sumption and the rapid exponential decay of the waves down in the fluid, it is only

the part of the ship close to the waterplane that will affect the flow field. This im-plies that we may replace the ship by a sta-tionary, vertical, infinitely long cylinder with cross-section equal to the waterplane area of the ship.

We will nelect all viscous effects,

and base the procedure on potential theory. In the analysis, we neglect diffraction ef-fects from sharp corners of a structure, and assume that the change in the waterplane area is small over a wave length.

We will write the incident potential as Ca _{k0z ikoxcose+ikoysihs=iWet}

(31)

e e

where g is the acceleration of gravity,

the amplitude of the incident waves, w0 tL

circular frequency of the waves, k0 the wave nuitber, the wave propagation directioh with respect to the x-axis (see Fig. 4), is the time variable and We is the circular frequency of encounter. (29)

+w2I466)n5

₍₃₀₎ V-3.7 yo) wove propagation direction

Figure 4 Coordi-nate system in the low wave length case

We will introduce local coOrdinate systems (n,$), along the. water plane area curve, where n is orthogonal to the water-plane area curve and s tangential to the waterplane area curve (see Fig. 4). Further,

(x0,y

) are the x,y-coordinates of the

ori-gin o? the local coordinate system. We may then write the incident wave potential as

a

k0z

ik0nsin(8+c)+ik ScOs(O+c)

e e 0

ikoxocoscL+ikoyosinx-iWet (32)

where e is defined in the figure. Note that n is opposite to the definition of the normal vector in the chapters above.

When the incident waves hit the struct-ure, they will be diffracted. If we use the wave length as a scaling factor and stretch the n,s and z-coordinates with this factor, it becomes evident that the diffraction pro-blem in the low wave length case is equiva-lent to studying incident waves on an in-finitely long vertical plane wall (see Fig. 5). Parallel to the wall there is a hori-zontal steady velocity of magnitude V1 which is a function of x0 and y0. We will limit ourselves to the low speed case. It should be recognized that the steady flow can only be horizontal in the low-speed case. It is evident that the incident waves cannot reach certain domains. This will be shadow regions, i.e. no time dependent flow at all.

Shadow region

Figure 5 Two-dimensional flow situation The total velocity potential in the. non-shadow region will be written as

=Vs +q

where is-the unknown diffraction potential

which stisfies the following body bàundary

condition

kz+ik0scos (e+ct)

=_iwoCaSifl(O+L)

_e

ikoxocosa+ikoyoSinct_iwet

e

We recognize that the body boundary condition varies as

ik0sOos (O+c*)

e-This implies we may write ik0scos (O+L)_iWet

By introducing equation (35-) into three-dimensional Laplace equation, we that i satisfies the Helmhol-tz equation

a2ij 2T 2 2

2 2

k0 cos (e+)*0

n z

the find

The linearized free surface condition can be written as

2

V--) 0 on z=0

which implies that

2

{_iwe+VikoCOS(8+cx)} I+g-O on z0 (38)

In addition,

D has to satisfy a radia-tion condiradia-tion, i.e. it cannot repiesent in-cident waves far from the .body.

It will be evident later that it is not necessary for- us to know the detailed be-haviour of close to the body. Many wave

lengths from the wall we may write k1z ..ik2n+ik0scO5(8+c*)-iwt

$DAe

e

where k2 have to be positive if should

representoutgoing waves. From equation (38),

we find that

(c -Vk0cos(O+c))2 k1- e

- g

From equation (36), we find that

k-2= /k12_k02cos2(o+

It is necessary to require that Ik1l>Ik0cos(e+) I

to have an outgoing wave solution. This is normally the case. If not, we must require that

k2=i 1k2I

V.3-B

so that the diffraction potential is

expo-nentially decaying when n+-.

We will now find A in equation (39) in the case of k2>O. -This will be done by first considering a fictitious problem and apply Green's second identity.

The fictitious problem -we study is inci-dent waves on the same wall as described in Fig. 5. We write the incident fictitious wave potential

where

-satisfies

o on z=0

- (P43)

i.e. the same free surface condition as It is easy to find that the total veloci-ty potential has to be

k1z ik scos(8+a)4wet

*T=1-D_2e

cos(k2n)e °

We will now study the following gene-ralized force on the wall

o

kz

(37) _{F=J(1pI+IPD) e} dz

- n=0

which can be easily calculated by equation

(44). -

-We will now rewrite (45) by Green's se-cond identity. .We introduce the normalized velocity potential 2 defined by - ikscos(e+a)

DOcasin(0)e

(46) ikoxocosa+ikoyôsinaiwet

(see equation (34)). It is easy to -see that

2 Satisfies the same Helmholtz

eqüa-tion as .

Further,

2 has -to satisfy the same

free-surface condition as and when n--k1z. -ik2n

Ae - e -ik0x0cosc-ik0y0sinc

42_iw.sin(e+)a

e (47)

(if sin (e+)*0) (see equation (39) an

(46) ) .

-'41" By using the boundary condition for

'-' we may write equation (45) as

(42)

(44)-(45)

0

F=f

_{(I)I+ D n} (48)

We will now apply Green's second iden-tity with the potentials

_ID

and and to a dOthain enclosed by the wall, the free

sut-face, a vertical control surface far away -from the wall and a horizontal control

sur-face far down in -the fluid. By using that

k1z Ik2n+ikoscos(O+a)iwet

and

2 satisfies the same Helmholtz

equaton, the same free surface condition

to-gether. with the body.condition for we find that (48) may be rewritten

-ik0x0cosc-ik0y0sinct F- e

jkSCOSO-it

k1

(49)

By using that F also can be directly calculated, we find.that

2k1

We have now determined the form of the velocity potential many wave lengths from

the wall. We may write

a k0z

-_w c cos(k scos(6+a)+k nsin(e+c)+_{0 0 0}-w_e

t)

0

cos(kscos (O+a)-k2fl+ci-wt) (51)

g k1z

+Be

+Vs

sin (k0scos (G+a) _k2fl+o_Oet)

.sjn(kscos (e+)+konsin(e+cL)+o_wet) V.3-9 (k0+k1) 1 B(k2-k0sin(O+cz) (sin(8+a)-B

if

k2*k

sin(e+a)

By°using Bernoulli's equation, we may write (54) correct to second order in wave

amplitude as

k

(56) 1

P½pg

-

?

_{z)2_.(*)2}dz (57)}

where

c_0=(h +Vf-)qI0

By using equation (56), we will. now find that equation (57) can be written as

(e+)]

(58)

+½sin(e+)}

where k1and k2 are given by equation (40) and (411. Generally speaking, it will be complicated to find the steady fluid veloci-ty V in equation (40). We will therefore propose to set

V=Ucose ₍₅₉₎

This is consistent with slender body

theory. It gives also the right answer for

e=/2,

i.e. for extreme blunt ships. In case

of small U-values and by using equation (59),

we will find by Taylor expansion that

g0(l cosOcos(e+s))}(60)

When

is found, we may write the mean

drift force components and yaw moment correct

to second order in wave amplitude as

(61)

where

i

is 1,2 or 6 and F1 is

the added

re-sistance, P., is the transverse drift force

and _{F6 is te drift yaw}

_{moment with respect}

to the z-axis, and

fl1s ine

ri2-cose

n3-x0cose-y0si-ne

The integration in (61) is along the non-shadow part L of the waterplane curve

(see Fig 4)

We should note that equation (58) is

different from Fujii and Takahashi's

asymp-tOtic formula for head sea. The differences can be most easily established by using the approximate fOrmula (60) for small U-valu-es. According to Fujii and Tàkahashi's

for-mula, the forward speed term in the. brackets

Of

equation (60) should be twice as large. It should be noted that the forward speed de-pendence in both fOrmulas are significant. where

cz0=k0x0coss+k0y0sincx (52)

and 2k1 k

B=k _+k sin(O+x) (53)

lo2

We are now going to find average force pr. unit length on the wall in Fig. 5. This expression will be the important tool to find added resistance, transverse drift and mean yaw moment on the ship in the asympto-tic low wave length case.

By using the equation for conservation of momentum in the fluid, we may write the normal average force per unit length on the waIl as

F=J [p+ V} dz

2 (54)

n=-where is the wave amplitude at n=- and the integrand has to be evaluated at

n=-By using the equation for conservation of energy, we find that

-

ds=0

(55)

By using equation (51) in equation (55), we find that.

w05 in(O+c)

_eoX000koYO

k2

(50)

wosin(e+cI)a

C

Fujii and Takahashi compared their formula with experimental results for X/L=O.5. The agreement was good. But it should be kept in mind that they only considered one wave length, which may be too high according to the short wave length assumption inherent in the theory. If we study their experimental results for A/L=O.3, it seems to indicate less forward speed effect than their formula predict. But a mOre extensive experimental program is necessary to judge the validity of the different formulas.

Fujii and Takahashi have proposed a practical way to combine the asymptotic for-mula and the conventional forfor-mulas for added resistance. They propose to multiply the asymptotic formula with a quasianalytical formula containing the draught-dependence of the ship and add this expression to the conventional formulas.

3.4 Comparison betweentheoretical and

ex-perimental values for added resistance The "direct pressure integration"-method has been used to calculate added re-sistance on Series 60, C =0.6, 0.7, 0.8 and the container hull frOm -l75, which was used for the comparative hull motion calcu-lation by the member organization of ITTC

14)

The calculated values for the Series

60, C=O.6, 0.7 and 0.8 are presented in

Figs.'7, 8 and 6 together with experimental and theoretical results presented by StrØm-Tejsen, Yeh and Moran (15). Strøm-Tejsen et al. used Gerritsma and Beukelman's formula together with motion calculations by the Salvesen, Tuck and Faltinsen method. We may note the poor agreement between Strøim-Tej-sen et al's theoretical calculations and ex-perimental results for the fine ship forms. It was pointed out in a discussion by Ger-ritsma and Beukelman that it was possible to achieve far better agreement between Ger-ritsma and Beukelman's method and the experi-mental results if their own computer program for motion calculations was used. From the figures, we note the good agreement between our theoretical calöulations of added re- -sistance and the experimental results. We should note the small difference in the re-sults by using either the relative motion hypothesis or by using Skjørdal and Faltin-sen's (11) procedure to calculate the dif-fraction potential. We have also plotted in theoretical values by our asymptotic theory. We note that the theoretical values are significant smaller than the experimental values for Cb=O.6 and 0.7, while there is tendency to agreement between the asympto-tic theory and experimental values for Cb= 0.8. But the great scatter in experimental values indicates great difficulties in per-forming model tests.

Theoretical and exprimental results for

for the ITTC ship S-175 are presented in Figs. 9 - 16. We have Compared the theore-tical results by the "direct integration method" with the theoretical and experiment-al results presented by Gerritsma and

V .3 . 10

Journêe (7). The results are for different Froude numbers, wave headings and wave lengths. Generally speaking, both theoreti-cal methods agree quite good with the ex-perimental results. But there are signif i-Cant differences in the results by the two theories. Further, it is not known how reli-able the experimental results are in obli-que sea. Some of the experimental results for small wave lengths are peculiar.

RAW

pg2B2/L

15

10

Experiment (Strøxn-Tejsen et al)

CALCULATION

GERRItSMA/.

=0.1470 BEUKELMAN ---FALTINSEN Eli AL

F=0.l65o

(RELATIVE MOTION) FALTINSEN Eli AL (DIFFRACTION) $

,

0ASYMPTOTIC THEORY 0.

_.-.

2,

Figure 7 Added resistance Series 60, Cb=O.6..

Experiment (Strqcm-Tej sen et al)

n-966

.283 o 0 0

Li

we/7

Figure 6 Added

resist-ance Series 60, CbO.$. e g

RAW

Exr,eriment (Strøm-Tej sen et al) a , 207o 222* B .

o

Figure 8 Added resistance

Series 60, Cb0.7. 0° 0 c =0.00 Fn= 15 EXPERIMENT CALCULATION 0 FUJII 0 NAKAMURA a'

'0

\ 0

7

- GERRITSMA/

BEUKELMAN ---FALTINSEN El AL (RELATIVE MOTION) FALTINSEN ETAL (DIFFRACTION) --ASYMPTOTIC TREORY

0

1 L/ 2

igure 9 Added resistance S-175 a= 00

a=O0°

Fn=20

0

1 L/A 2

Figure 10 Added resistance S-175, a=0

V -3 - 11

-10

RAW

c = 0.00 _{Fn= 25}

1 L/ ₂

Figure 11 Added resistance S-175, a= 0

0

1 L/A ₂

EXPERIMENT 0 FUJI) ct 3OO _{Fn= 15}

GERRITSMA/

BEUK EL M A N FALTINSEN El AL (RELATIVE MOTION)

/

\

/

\

/

'

1 LI?.. 2

Figure 13 Added resistance 5-175 c*=30°

Figure 14 Added resistance S-175 cz=30°

c=60°

Fn=15

CALCULATION

1 L/X 2

Figure 15 Added resistance S-175 a=60°

V -3 12

Figure 16 Added resistance S-175 4.0 CHANGE OF PROPULSION FACTORS DUE TO

WAVES

4.1 Change of the wake due to waves

Measurements by Nakamura and Naito (16) and Moor and Murdey (17) have shown that the wave induced motions of the ship increase the wake velocities. When the ship model was oscillating in regular waves, Nakamura and Naito showed that the change in wake veloci-ties increased with wave height and that the change in wake was most pronounced around the natural period of pitch. They also performed

forced oscillation tests in pitch and showed that the wake velocities increased with pitch angle and frequency. Their experiments

indicated that the pitch motion of the ship was a major reason to the increase in wake velocities. This can be qualitatively

under-stood by examining the influence of the wave induced motions on the mean pressure along the ship. The change in mean pressure at a fixed point in the fluid due to oscillatory fluid motion can be written as

(l)

2 (U 2 (1) 2

) +( ) +( ) ] (62)

(see equation 20). This pressure causes a change in the mean flow along the hull and therefore also in the wake velocities. If a flat plate of large extent is used as a rough model of the bottom of the ship and the plate is pitching harmonically, a rough approxima-tion of equaapproxima-tion (62) below the bottom of the ship can be written as

p2

22

_x

This indicates that the pitching motion of the ship will cause a drop in mean pres-sure from the middle of the ship towards the stern. This change in pressure gradient will cause the flow to be sucked more towards the

stern of the ship and thereby increase the flow velocities to the propeller.

A consequence of the increased flow velocities is that the boundary layer thick-ness is reduced. Another consequence is that the shear forces on the ship is changed, and that the separation point is delayed. This causes a change in the resistance of the

ship, which may be interpreted as a wave-viscous interaction force. This is not in-cluded in our theoretical procedure. The importance of this interaction force is un-known.

If we use the same simple model to study the influence of forced harmonic heave

mo-tion, we will find that

P'

eII

(64)

Since the associated pressure gradient is zero, we do not anticipate that the heave motion will influence the wake velocities

significantly. It is more difficult to tell in simple terms about the influence of the incident and diffracted waves on the wake velocities. But we may use equation (62) to-gether with velocities obtained from linear strip theory calculations to estimate more

accurately the change in the mean pressure when the ship is oscillating in regular waves. This includes the effect of the

heav-ing and pitchheav-ing motion as well as the inci-dent and diffraOted waves. Fig. 17 shows one example from such calculations for a Series

60 ship with Cb=O.6 in regular head sea waves. The wave length is X/L= 0.913 and the Froude number is 0.2. This is close to the resonance condition of the ship in pitch. The results in Fig. 17 have been obtained by using equation (62) at the mean posit-ion of

the ship. For each cross-section of the ship we have averaged the mean pressure. This

averaged mean pressure is denoted P in the figure. We see that this pressure-distribu-tion -is of a similar nature as- one would expect from equation -(63). If we make simi-lar calculations away from the resonance in pitch we will find a less pronounced

de-crease in the pressure towards the stern of the ship. The discussion above is a theore-tical indication that the pitch motion of the ship is a major reason to the- increase in wake velocities for a ship oscillating' in regular waves.

In order to quantitatively evaluate the effeôt of the pressure distribution (62) on the boundary layer and the wake, we have adopted a rough theoretical model. We would certainly like to stress that our model has short-comings. In the boundary layer calcu-lations we have assumed that the boundary layer is thin, which is highly questionable at the stern of the ship. It is difficult to know how the boundary layer is following the ship in the oscillatory motion, but it is intuitively felt that the ship should not move out of its own boundary layer. We will use the following procedure. We calculate f-irs.t the boundary layer as if the ship was in its mean oscillatory position. But note that we in those calculations will take into account the effect of the oscillatory motion

V -3 - 13

xI L

Figure 17 Average mean pressure due to wave induced motions

of the ship on the mean pressure distribu-tion around the ship. After we have calcula-ted the steady boundary layer due to both the still water pressure distribution and the oscillatory fluid motion, we assume that this boundary layer is attached to the oscil-lating ship. We have used a two-dimensional model for the boundary layer calculations

and assumed that the boundary layer thickness is only a function of the longitudinal co-ordinate of the ship. Physically we should intèrprete this boundary layer as an aver-aged boundary layer for each cross-section. The pressure gradient that we apply is ob-tained from averaging the pressure over each cross-section. Associated with this average pressure distribution p(x), there is a fluid velocity V(x) OutsIde the boundary layer. According to Bernoulli's equation, we may write

V(x)=

/-ai'

(65)

½ p U2

where U is the free stream velocity. The boundary layer was calculated applying the equations Head

A/L=

F,

Sea 0.913 0.20

I

-0.5 -03 -0 1 01 03 05 20 15 10 5 0

4

X4

O(x)=V (x)0.OlO6

fv

()d

and - -L/2

62=O(x)/(V)O666

₍₆₇₎

where 62 is the momentum thickness (see Thwaites (18)) and L is the length between perpendiculars of the ship. We have assumed

that the flow is turbulent from the bow of the ship.

We have applied equations (66) and (67) on a pressure distribution similar to that measured on a Series 60 model (Cb=O.G) in

still water at F-=0.2. The average pressure distribution p(x? that we applied is pre-sented in Table 1.

Table 1

When the boundary' layer thiàkness ob-tained by using equations (66) and (67) was compared to values calculated by Webster and Huang (19) by a more accurate procedure, we found a quite satisfactory agreement. The momentum thickness at x/L=0.45 which is

close to the stern, was found to be

6.,/L=0.00442. If we added the pressure di-stribution presented in Fig. 17, we found that 62/L=0.00403 for ca/LaO.01 and

62/L=0.00332 for /L=0.02 at x/L=0.45. This

confirms the expeced decrease in the

bound-ary layer thickness due to the wave induced motions of the ship..

When the boundary layer along the hull has been calculated, we may evaluate the wake behind the body. We assume that the velocity in the wake behind a two-dimensional body or a body of revolution can be written as

v()=vo_vmax(.l_(_)m)2

(68)

where n is a coordinate defined below. In the case of a body of revolution, n is equal to the radial distance from the rotational sylmnetry axis. In thecase of a vertical plane coinciding with the centre plane of the ship n will be equal to the y-coordinate defined in Figure 1. In the case of the ship the wake distribution will be a mixture of these two cases, and we will define n as the ra-dial distance from the propeller shaft centre. Further, 6 in equation (68) is the

boundary layer thickness, v0 is the velocity at n=6 , v -v is the velocity at n=0, and

m is a emir?1 constant whiôh we have set

equal to 1.5. In a practical case, Vma and v have to be experimentally determine

0

If the boundary layer is chanqed,

-will change. We have

0

V - 3 - 14

(see for instance Tanaka (20)). Here the apostrophe indicates the changed values. This implies that with

v.

(1-w(--))=l maX(1

(fl)

)

in still water, we may write the wake in waves as

v. ₆₂

m2

=1 --

-(l-(,-)

0 2 0

where w is the radial wake fraction. We have assumed that in equation (71). We are now going to use this procedure to compare with the experimental results by Na-kamura and Naito (16) (see Fig. 18).

5

0.65 1.0

1.00 _Q2

1.19 1.0

If we instead use 6/62=0.9, we will

find that (see Table 3), (1 w.)W Container ship F-_ 0.20 model ' I (1-w _w: in regular waves (.1 w.)s (1-w. :in stilt water

I.

IL

I .

-.-oO.5

PropeLLer I

. 0.8

-boss 1.5 -2.0

-

2.5

I

0

0.1 0.2 0.3 0.4 0.5 (I-w) 1.835 1.644 1.51 1.401 1.314 x/L -0.45 -0.35 -0.25 -0.15 -0.05 0.2 -0.1 -0.2 -0.22 -0.2 Pr12 2 0.05 0.15 0.25 0.35 0.45 -0.19 -0.18 -0.17 -0.1 0.15 0.4 0.6 0.8 r/R 1.0

Figure 18 Nakamura and Naito's experiment-al wake results

They measured the radial wake distribution in the propeller plane when the propeller was not there, and the ship was oscillating in waves. As stated earlier in the text, our results are for the same condition. Accord-. ing to Nakamura and Naito's still water re-sults, we find that

VmaX_08

Further, they got the results in regular waves for

Ca't00]

We have earlier calculated 62/6.2 f9r ca(LaO. and 0.02. We have found

62/62=0.91 and 0.75 respectively. If we use 6;/6.=o.75 in our calculations, we will find

te tollowing wake distribution by equation

(70) and (71) (see Table 2) Table 2 (66) ('flX)6 = (69) 1.6 1.4 1.2

Table 3

0.65 0.9

1.091 1.0

In Nakainura and Naito's experiments it seems

like S02-3R. The values in table 2-3 are of

the same order of magnitude as the experi-mental values b Nakamura and Naito

(16)

and show a similar radial distribution as their results. As an extra control we also found that the volumetric mean wake is of the same order of magnitude as Nakamura and Naito's results. We therefore believe that we have clarified important physical reasons to the change of the wake due to wave induced mo-tions of the ship.

4.2 Change of the open water propeller characteristics due to the waves Analysis of propulsion factors based on resistance and propulsion tests in waves is traditionally done by fitting data into open water propeller diagram derived from

tests with the propeller deply immersed.

But it is well known that the open water propeller diagram is a function of sub-mergence, in particular when h/R<l.5, where h is the distance between the water surface and the propeller shaft center. When the propeller is oscillating in waves, it may very well periodically be in instantaneous positions where h/R<l.5. This is particu-larly true when the ship is in ballast con-dition. This implies that it may be physi-cally wrong to use deeply submerged open water diagram in the analysis of a propeller in waves. This may lead to wrong conclusions about effective wake, relative rotative effi-ciency and propeller effieffi-ciency. In order to develop a more physical open water propeller diagram when the propeller is oscillating in waves, we have adopted a simple theore-tical model. Since the wave induced motion occur with a significant lower frequency than the propeller revolution, we have used a quasi-dynamic approach. At each instant-aneous position of the propeller shaft centre relative to the water surface we are using the propeller characteristics that would be valid if the propeller was not heaving and pitching in waves. We will therefore discuss how the open water propeller diagram is a function of propeller submergence.

The propeller sets up a steady Wave motion when it is working close to the free surface. This will lead to a reduction in the propel-ler thrust. This reduction is a function of the propeller loading and the Froude number as originally showed by Dickmann (21). Later on Nakatake (22) and Nowacki and Shatma (23) made similar Oalculations. We have in our example presented below used Nakatakes results.

Corrections due to wave effects are of particular importance when l<h/R<l.5. But when h/R<l, the thrust must be corrected due

to ventilation and the emergence of the pro-peller. The ratio .of the immersed disc area A1 of the propeller to the disc area A0 of

tIe propeller is

AlQh

_{/ h}

2' 72 21T V - 3 - 15 where 0=2 arccos

We have assumed that the thrust is pro-portional to A,.

Another effect we have to take into ac-count is what we call the Wagner-effect. When a propeller blade splashes into the water, there will be a sudden increase in lift. Wagner (24) studied a related problem for the two-dimensional foil, and found that the sudden increase amounted to 50% of the value for the fully developed lift. The socalled Wagner function gives the ratio between the instantaneous lift and the fully developed lift as a function of chordlengths travelled after the sudden change. One procedure would be to use blade element theory and study the distance each blade element travel compared to the chord length after the sudden change, and then apply the Wagner function. In our calculation, we have followed a simpler pro-cedure by examining an "equivalent" section of the propeller. The results are presented in Fig. 19, where x is the ratio between the real thrust of the propeller and the fully developed thrust of the immersed propeller disc and Bj is due to the steady wave motion created by the propeller.

1.0

p. Without v ntilation

Independent of A

0 0.5 10

Figure 19 pdiagrain for a propeller We may now write

KT=BKTO (73)

where KT and K are the thrust coefficients for the eeply immersed propeller and for the propeller at immersion h/R. The thrust. dimi-nuation factor B without the effect of ventil-0

0.1 0.2. - 0.3 0.4 0.5

ation can be written as

where . is defined in equation (72), B1 is

due to he steady wave motion created by the propeller and x is due to the Wagner effect. A comparison between the procedure described above and experimental results by Keinpf (25) is presented in Fig. 19. The agreement is certainly too good. Note that we have not taken into account neither partly nor full ventilation, which is of particular import-. ance for smaller Ja=Va/nD-nunibers, than we

have considered in the example. The effect of ventilation is illustrated in Fig. 20, where the efficiency of a propeller is given

for both the partly and fully developed range as a function of immersion and loading.

1.0 10 100

CT

Figure 20 Change in propeller efficiency due to propeller loading, immersion and ventila-tion (from Gutsche (26)).

If the propeller thrust is changed with immersion, there is also a change in the torque coefficient KQ. But due to the drag force on the propeller, the change in K,. is not the same as the C'hange in. K,. Modeltests by Kexnpf (25) and Gutsche (26) indicate that

where in is a constant between 0.8 and 0.85, and KQO is the propeller torque coefficient for the deeply immersed propeller.

The final part of our procedure is to average B and 8m over one wave cycle. It is

then necessary to know either from model tests or theoretical calculations the rela-tive vertical motion between the propeller shaft centre and the water surface. We may now write the open water diagrams for the propeller operating behind a ship oscil-lating in regular waves as

KT=B KTO (76)

KQ= KQ0 (77)

where the bar indicates average values of B and 8m over one wave period.

From the discussicn above, we may anti-cipate that the change in open water propel-ler characteristics due to waves is most pronounced in the vicinity of the natural periods in heave and pitch and for a ship in ballast condition.'

V - 3 - 16

4.3 False changes in propulsive coefficients due to waves

We have in the last chapter pointed out that it is not physically correct to use the open water diagram for the deeply immersed propeller if the propeller is heaving and pitching and operating close to the free surface. If we use wrong open water diagram it may lead to false changes in the propul-sive coefficients. This is explained below. Denoting in accordance with Fig. 21 the propulsion factors when an open water diagram corrected for immersion is used in. the ana-lysis by:

w,tf, n, n, n,

r,

K, K

and the corre-sponding values with noncorected open

water diagram by:

W, t,

_nP,

_nH,

n0,

KT, KQ, _A we must have

nPR.nH=nn. If thrustidentity is used

in the analysis _(KT=KT) and the torque coef-ficient in the behind condition is' ₁₍₆ we get approximately dK -1

J - (1K

_A

(__12)

dJA K,. A nD

Figure 21 Open water diagram for partly and deeply immersed propeller

hID 0. -0.375 94 14.1 4 0-025 07510 h/00

Fully

ventilated-2 (78) and

m[QO(..L.K)

(32)

+T1RKQ]

dKKdK

-1 (79) R dKTO dKQQ

where and are the slopes of the diagram for the deeply immersed propeller at the advance coefficient For the wake we ii.na tnat

i-w_

'A

(1cT1() (d10

) (80) Further, j.f t=t* l-w .4 .6 8 0.8 P0. 0./. 0.2 0

Equations (78) - (80) have been used in the example presented in Fig. 22. The re-sults are given as functions of .. The open water propeller diagram that was used, was a typical propeller diagrani for a merchant ship. In the example: J4=0.4 H 1.0 KT 0.227 K0 0.029 dKo _-00474 dJ A -0.428 dJ A 0 75 0.80 0.85 0.90 0.95 1.0

Figure 22 False changes in propulsion fact-ors.

The results that one will get if one uses the open water diagram for deeply sub-merged propeller, corresponds to =l. But

for a propeller operating close to the free surface, may differ substantially from 1. From the figure, we note that the propulsive

factors are quite sensitive to L

Obviously we may in principle use any open water diagram in the analysis but we will end up with false values thr w, _{R' TiH} and r if we are not using a diagram corre-spondng to the correct fl-value. With false values, we mean that the result will not

re-flect the physical realities. 4.4 Changes of due to waves

Moor and Murdey (17) and later Nakarnura and Naito (16) showed that sometimes changed in severe wave .condiions compared to still water. Both increasing and decrea-sing values were reported. This change in

can to some extent be explained as false changes due to use of an open water diagram for deeply submerged propeller. But there may be real changes in R due tochanges in

the wake field and the propeller loading. This has been showed theoretically by

Yama-V -3 - 17

zaki (27) and experimentally by Johnsson

(28)

Yaxnazaki calculated the changes in R due to the radial distribution of the mean load and the influence of the different har-monics of the unsteady components of the wake. The latter being the most important factor. The calculations and the experiments showed that _{R fOr a given wake distribution} will increase with decreasing KT

Nakaniura and Naito (16) measured the wake behind the model in waVes and found that the radial wake distribution was more homogen-ous in waves than. in still water. If also Vmax/Vo (see equation (62)) is reduced due

to the decrease in boundary layer thickness, this may result in a reduced TiR-value.

It is impossible to draw general conclu-sions about the influence of waves on TiR from the experimental data we have at the mo-ment. The reason is presence of false effects. But it should be reasonable to use the still water value also in severe conditions if the propeller characteristica is corrected for

immersion.

0 05 2 25

CT

Figure 23 Change in thrust deduction due to propeller loading

1.5

4.5 Changes of thrust deduction due to waves MoOr and Murdey (17) observed that the thrust deduction decreased with increasing wave height and reached a minimum around the

WT =0.6

-WI 0.4

:1:::

2 Wp

-

i.v97t

lwT

w = 0.17 8 It . C- TSf2V.ltRZ, JA *

L

L

0.4 t 0.3 0.2 0.1 0

natural period in pitch. Nakaznura and Naito (16) made similar observations. Neglecting the wave induced wake and assuming an uniform

source distribution over the propel1r disc

Nowacki and Sharma (23) caine to the fol-lowing simplified expression for the potent-ial thrust deduction fraction in still water

t- 2

Wp

l+VT l-w

(81)

(see Fig. 23) where CT=Tp/(RVa2TR2)l T =

Propeller thrust, w = effecive wake, p

potential wake. R = propeller radius and Va defined in equation (3). Equation (81) could probably be applied in waves. If we do that and use information about propeller loading in waves compared to -still water, we will find a decrease mt of the sart order as rreasured

in xtel tests. Overload tests in still water gave si-milar results -to those in Fig. 23 (Bindel (29)). This indicates that overload tests in still water can give information -aut thrustdaduction in waves.

One consequence of the dependence of t on the loading is that there will be no li-nearity between thrust and resistance in-crease. This can be seen for instance by ex-amining equation (2) and by using the

re-sults above.

5. DETERMINATION OF RPM AND HP IN A SEAWAY

We have in the text above discussed how the propulsive coefficients and added resist-ance change in regular waves. We may then use equations (1) and (2) to find RPM andHP

in regular waves.

If the forward speed of the ship is given, we may use equation (2) to find

pro-peller revolution and then use equation (1) to find the horsepower in regular waves. We have done that in a case near the natural period in pitch. The propulsion factors are then influenced by the wave induced motions. As anticipated we found that the added ship power in regular waves did not vary as the square of the wave amplitude, which is the functional dependence of the added resist-ance. This complicates the prediction of added power in irregular sea.

We will here propose a new method to handle the irregular sea case when the pro-pulsion factors are dependent on the waves. This effect is most pronounced when there is significant wave energy for frequencies in the vicinity of the natural frequencies in heave and pitch, and for a ship in ballast condition. We assume that the effect of wind can be added as a constant resistance force. The method resembles a procedure used in cal-culating slowdrift oscillations of a ship in irregular waves (Hsu and Blenkarn (30)) and is based on a narrow banded wave spectrum. The procedure implies.that we create a time-trace realization of the irregular waves and locate the zero-uperossing wave elevation time instants. We then approximate the wave between each zero-uperossing time instants by a regular wave of wave amplitude _a and wave period T. For that -particular regular wave we estimate the added resistance and

V-3-18

the propulsion factors. When that has been done, w solve equations (1) and (2) and find RPM and HP. By repeating this

proCe-dure-w will find a slowly varying time

trace of RPM and HP. We may also find a slow-ly varying time trace of the ship speed if we assume that the machinery characteristics are given. Traditionally one has been inte-rested in average values for ship speed and! or power in a seaway. This is easy to obtain from the procedure above, but one may quest-ion why one should not consider the extreme va1ue of added power.

-If the propulsion factors in waves are the &ame as in still water, for instance in moderate sea conditions, it is easy to gene-ralize equations (1) and (2) to irregular waves by introducing mean added resistance in irregular waves in the left hand side of equation (2). This has been done by Journêe (31). We may note that this procedure do not find the slowly varying time trace of RPM, HP and/or ship speed. It only provides us with mean values.

The procedure described above may be avoided if we carry out propulsion tests of a model with propeller in waves. An obvious disadvantage with the direct procedure is

that it leads to extensive and expensive model testing in order to cover many sea - states. One should also be careful with the

model testing procedure. Tests are usually carried out at model self-propulsion point. This may lead to erroneous answer. In a cal-culation that we carried out, we found that

increases in power and RPM measured on a model working in a seaway at the model self-propulsion point will in severe conditions be much larger than if the model was working at

the self-propulsion of the ship. This indi-cates that one shOuld run the model at ship propulsion point.

6. CONCLUSIONS

A procedure for calculating added re-sistance, transverse drift force and mean yaw moment on a ship in regular waves of any wave direction is derived. An asymptotic for-mula for small wave lengths have also been derived. The agreement between theoretical

and experimental results is generally

good-1 It is pointed out that model tests in the low

wave length range for blunt ships are needed. In severe conditions there will be a negative wake induced by the motions of the ship. This change in wake may be roughly estimated or determined ,experimentally.

Analysis of propulsion tests in waves based on a propeller diagram not corrected for immersion effects will give false W

and values and results, that do not re-flect the physical realities if the model has been tested in severe wave conditions.

Prediction of RPM and HP of a ship in irregular sea, in particular when the ship is close to resonance condition in heave and pitch and/or in ballast condition, is not straigthforward. A procedure to handle the

problem is proposed. ACKNOWLEDGEMENT

The theories for added resistance have been derived by 0. Faltinsen. K. Minsaas and 0. Faltinsen have cooperated on the chapter about "Change of propulsion factors due to waves". K. Minsaas is the originator to most of the ideas presented about propulsion fac-tors in waves. S. 0. Skjørdal has derived the computer programs on added resistance.

N. Liapis has derived the computer program for diffraction of waves in head sea. 0. Ør-ritsiand and Y. Robertsen have participated in the computer calculations.

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