A method for predicting the added resistance of fast cargo ships in head waves

1. Added resist ane in regukr waves.

Up to thepresent, the mos t reliable determitia-tion of added resistance ofa given ship in regular waves is obtained by means of model experiments. Theoretical methods require complicated cal-culations, and don't seem to give results which are quite exact.

In the presentwork an attempt is made to pro-vide an easy method of estimating resistance in-crease in regular head waves. The authors will only consider ships with fine forms, and will aim but at a rather rough evaluation of this added resistance.

The analysis of experimental da ta from several authors, Referencc[1], [2], [3], [4]**), reveals some common features regarding the shape char-acteristics of the curve of resistance increase in headwaves versus YL75, or , (since

T7=

IL

V.

u):

2rrg

) Leningrad Shipbuilding Instibite, visitor of the Office ofTNaval

Architect-ure, State Univcruitv of Chent, 3elgiunì.

Office of Naval Architecte e, Slate L'niveruitv of Ghcnt, Belgiuni.

Numbers in brackets refer to tite list of references mentioned at the

end of this paper.

- Froude numberbeing constant, the curve of ad-ded res is tance, RAW = f(u), shows a maximum (A\V)max' (Figure 1, from Reference. [1]); - for a _{given mode],, (RA\V)rnax increases, and}

the corresponding circular frequency amax de-creases with Froude number, (Figure 2, from Reference [1]).

The main particulars of the models cons idéred here are given in Appendix I.

The curve of added resistance in regular head waves versus u (or

'/7)

_{is assumed here to be} approximated by the following expression

Lb

Ld

rA\V=a(u) exp[c(u)

RAW ieL 015 - 1858Th -Fa 015.181835 where rAV=

pg2 B2/L

where a and e are nondimensional coefficients depending upon the value of the Froude number and upon the form of the ship's hull, and where b and cl are constants for a given model.

OF FAST CARGO SHIPS iN HEAD WAVES

gsE UVTE&T

by _{Laboratorium 'icor}

V. JINKINE and V. FERDINANDE "1. Scheepshydromothaf,

Mek&weg 2,2628 CD De!ftj Sumniary.

An analysis of experimental data from ship model tests, as published by different authors, sug-gests that the resistance increase in regular head waves may be repreented b an anal VUe

expre--sion, which is sufficiently valid for cargo ships with a fine hull form. An attempt is made to deive such an expression suited for the approximate calculation of this added reistartce. The formulae obtained, which may be considered as empiric, contain but one factor, (dependent upon the ship's form) which should be evaluated by means of experiment. The present method makes it ossib1e to calculate the resistance increase ih regular head waves of any length and at any speed, provided that experimental results for merely one speed, or even for one specified wave length and corresponding (synchronous) speed areavailable. Moreover, by adopting an.average value of the above mentioned factor, the added resistance can be calculated without any help of model experiments.

Hence, the resistance increase in irregular head waves can be calculated_{as well, and, as an} ex-ample, the results regarding ah existing contaiher ship are given anti compared with actual full-scale data.

Furthermore, the added resistance in any irregular head sea represented by a standard spectrum can be predicted very quickly by means of the enclosed graphs, which reproduce thé numerical

I5O-Fn 0,15

0,7 i,b

maxis easily derived from (1):

b

lid

(2)

Hence,

b

b/d -b/d

(rA\V)maxsa(-)

e (3)

From (1), a normalized curve of resistance in-crease can be derived, and written in the form

._bexp[b(ld)l

(4)

rA\V c)

where = and

-" max

The expression (4) does not contain the coeffi-òients a and C.

If the h3pothesis concerning a representation like (l)is valid, (Viz. b and d constant for a given

1:2 1:3 o Experimehtat 1,5 1,0 0,5 o

model), the- experimental data points from runs

atvar-ious F, presented in this normalized form, should lie on one and the sathe curve. Figure 3

shows the converted experimental data from

Ref-erence-ji) fora model of a -fast cargo-liner. On-can see, that a single cutve fits the spots. quite

ve1l indeed.

The analysis of experimental data as found in

other reports,

Refereñces [2], [3] and [4], demonstrates the same -rule for another fout

models, Figures-4, 5 and 6.

Moreover, the spots cotrêsponding to the. dif-ferent models, reported on in References [1], [2], [3] and [4], all lie close to the same curve. One analytic fune tion could match all the spots, with-out appreciable discrepancies, Figure 7.

For the sake of a good fit, this approximating function has toconsistof two parts, one for < 1,

and anothetfor-> 1. Hence, the response curve 'added tesis tancè invaves'

= f(), or rA7 =f(),

is split intotwobranches, represented by

differ-Figure 1. Nondimensional resistance increase in regular head waves at constant Froude number för a model f a-fast cargo ship, according to Reference [1].

1,5

entanalvtid functions, having a common point and a horizontil tangent at = 1, or at c =

°ma, rhis is not quite unusual: an analogous rep resenttion of a standard wave spec tru rrì was introduced already in Reference [5].

Fn 0.30 2JD 1.5 r 1,0 0,9 0.8 0,7 0,6 os Ql. -0,3 0,2' 0,1 L)Çjjsec) aD s &o ,s s.o s.s o ,D 1,1 1,2 O Fn 0,15 0: _0,20 0 0,25 : o o 0,8 0,7 0,6- 0,5- 040,3 -0,2 /t75' 0,1 0:7 0$ Og 1:0 11

1,23

Figure 3. Nornalized curve of added Iesistance in reg-ular head waves. according to rcfereñce [1].

r 1,0 0.9 0,8 0.7 0,6 0,5 04-0,3 02-0,1 Model Gerritsmo 61 Model Lewis Fn o 0,128 G - 0,150-- 0,150--0,171 e - 0,193 o - 0,214

D-

0,235

-

0,257

A-

0,278 -C.) 0,B 0.9 W 1,1 1,2 1,3 1,4

Figure?. Variation of noñdimenional resistance in- Figìre 4. Nordalized curve of added Í-esistace in

reg-crease curveswith Froude number, according to Refer- ular head waves, according to Reference (2].

ence [1]. Fn - 0,117 - 0,155 e - 0,194 e- 0233 o - 4272 (a) 0,7 0:8 0,9 1,0- 1.1 1,2 1,3

Figure-5. Normalized cui'v of added resistance in reg-ular head waves-, according to Reference- (3].

1,0

0 0,7 0,6 0.5 04 0.3-Q2 o I

a)Mode[ Numata U-bow

o-

-Fn

W55

o

0,8 0,9 1,0 1,1 1,2 1 lA

Figure 6. Normalized curve of added resistance in regular head waves, according to Reference [4).

1,0 0,9 0,8 0,7 0,6- 05-0,4 0,3-0,2 0, 0.9 1,0 1,1 2 1,3 1,4

(approximately) constant for all the models con-sidered here. According to (5), however, it is noted that b = 11 and d = 14 for < c.) (left

part of the curve), but b = -8,5 and d = -14 for

> rnax (right part of the curve).

The assumption 'b and d constant' makes an approximate calculation of resistance increase in regular head waves feasible However, this assumPtion seems to be valid merely for Froude' numbers in the range F=0.12- 0.30. Forlower

less good agreement with the mean curve was noticed. On the other hand, no experimental data for higher F were available.

Now we need to determine the values of a and ein equation (1). The value of e may be expected to be dependent upon F ad upon some charac-teiistics of the ship like fineness of the hull and load dstribUtiòn. The influence of the loading condition of the vessel is strongly related to the longitudinal radius of gyration \Ve will pre-sent the nondimens i'onal coefficient c in the form

K

c=co(T.)

yy

Fn

(6)

where the value of c0 should depend solely upon the ship's form. c0, K and n ought to be constant

b) Modet Numata V- bow

D 0,1- D Fn G D o - Q155 G - 081

e-

0,207

0-o -

0,233,Q259

p-

0,285 G- 0,181 e-- _0.207

e- 0233

o- 0259

D- 0285'

A convenient approximating function was found:

i=11 exp[-(1 _14)

for < i (5')

and

j _expl (1 14)] for > 1 (5")

These equations (5) prove to be a satisfactory 'representation of the added resistance of all modéls in the range of dimensionless frequencies 0.75< < 1.25, (Figure 7).

There is less good agreementfor longer, and es-pecially for shorterwaves. For > 1.25, a con-siderable scatter of the spots (F, ) concerning

the different models exist.

Neglecting the latte r region of , onecanconclude

that the ship characteristics (form, load

distribu-tion) don't affect considerably the normalized

resistance increase F = f() within the most

in-tereting range of

, viz, in the vicinity of its

maximum value. The influence of these charac-teristics, on the contrary,appears to be important inshortwaves. In the latter condition, wave re-flectión becomes rather predominant, and the ad'ded resistance is not proportional to

_J

any-more.

0.5 0,1 Fn 0,15 - 0,20 Ref. ¿t _0,25 (1) 8 _- 0,30 0,3 e A o

'_

G EXPERIMENTAL DATA o Fn - 0,128 - W50 - W71

-

193 I:, Fn - 0,216 - 0235 Ref. 0.257 (2) - 0,278J o 8

_"il

¿t Approximation exp for

T:'5exp

_[

.i_-")]

for

Yi,o

£ o e U-bow V-bow Fn o 0.155' y

0.181,

W 0.207 Ref. V

-

0.239 V

-

0,259: V

-

0285) ¿t Fn

O -

0,117

O -

0,155 _Ref.

-

0194 (3)

-

0,233

e -

0,272 O G 0 e V V 0,8 0,9 1,0 1,1 ,2 1,3 1,1+ U

Figure 7. Average normalized curve of added resistance in regular head waves for ships with a fine form. V

154

for the given model. From (2) and (G) follows

b

l/d

'maxg

-/L.,K _F n

(k _{/ n}

and, hence, for the given model

n logco

=constant--logF

max d n

An analysis of the experihen.l data of all the models suggests that n/d is constant, and that approximately n/d =

Asd=l4, the respective values of n must be

n=+2, viz.

n = 2 for < Crnax and

n = -2 for o > _max

A further analysis of the experimental data shows that, fòr all the models investigated, the values of ande0 are constant too. For the range

. °max' we can write

14/3

e = co(T-_) F112

where e0 = -0. 09, for all the models.

This means that, according to (2), the value of max' corresponding to the thadmurn of added resistance, would depend solely upön the Froude number and upon the longitudinal radius of gyra-tion, and not on the ship's form. This conclusion is confirmed by a comparison of experimental data from Reference [4] of two similar models, büt having a different foreship, viz. V- and U-bow frames, for which, at a same F11, _max has the same valüe indeed.

Hénce, the approximate value f _max' cor-responding to (RAW)max can be expressed as

1/14 11/14 -14/3 2

0.09(-)

F11-in accordance with (7). (9) will be written as max [ yy _-1/7 =1.170 F11 (7) (8) I.60 1.58 1.56 L 1.52 L. I.50 L48-ExpecimontoL points: O - Gri1smc 72 D Gerrttsma 61 £ - L.wis O -U-b Nurnata Fn

Figure 8. Comparison of the relation (9') with

experi-mental data.

The experimentally determined values Ofc)may z

/are

plotted versus Fn in Figure 8, in order to assess the degree of accuracy when cal-culating Cfl1ax by meañs of the formula (9').

A next problem is to evaluate the maximum resistance increase (rAW)max. According to (3) and (7), for a given model:

(rAW-)flax = constant xaF11' (10)

where 'a' is a factor depending upon F11 and upon the ship's form and load distribution.

Although the actually measured valües of

(rAV)flax increase with increasing F for each model, this (rA\V)max must in fact decline to-wards ze ro for high values of F. Hence, 'a' can-not be represented by a simple expression like aiFam, for example

An anaÏ5rsis of the experimental data suggests that (rA\V)max is approximated quite well by the following formula

(rA\V)max=al F»5exp[_3.5 F11] (li) where a1 is a factor depending upon the load

dis-tribu tion and upon the ship's form.

Figure 9 shows, that the formula (11) indeed represents the values of (rA\V)Inix in a qute satisfactory way.

0.1 01 0.3

(9)

al Exprirnera pQrnts -₀₆ L 0.05 0.0k 0.03 0.02 01 O - GerrIxrna72 O - Grrltma 61 A - Lewis o - U-bow - V-bow Figure 9. Comparison mental data. Numat 0.2 of the

This means that the trend of the peak value of resistance increase with respecttoFris practic-ally the same for all the models, but that the mag-nitude of (rA\V)max depends upon loading condi-tion änd hull form.

The value of the factor a1 which reflects the influence of loading and form, could be determ-ined easily for each investigated model from the available data. These values are given in the fol-lowing Table 1. Table 1 exp t- 3.5 Fn

jb

b x(co.J_)

exp[_3.5F_

0.3

relation (11)with

experi-i .17°d

't'

- For co < co

_{- niax'}

since there b = 11 and d = 14,

3.67 k _r-11 YY

307

IL rA\\,=O.398a1(i-) F

(co./)

k 2 YY xexP[_3.5Fn-0. 09 F

/14

°V

] (12') - Forco>comax, sincethereb=-8.5and d =114, -2,83

fi__s5

rA\V'=G.85al(i-)

_FO29(coV)

-4.67

xexp[_3.5 Fn_53Fn2()

(co.%/) (12")

The coefficient e in (1) is given by (6). It was stated already, that e0 = -0.09 for co< comax, in

accordance with (8). According to (12"), it can be noticed, that for co > co1 thè value of e0,

which has to be introduced in (6) to obtain e for the branch on the right, is e0 =

-5.3.

The formulas (12') and (12") enable us to cal-culate the added resistance of a similar ship in regular head waves, provided that we Iiow the value of the coefficient a1.

The values of a1 in Table i refer to various models of shipswithagiven hull form and a given longitudinal radius ofgyration. The available in-formation is not sufficient to express here a1 as a function of the ship's characteristics. Never-theless, the values of a1, reveal clearly, that the res is tance increase is 1es for a V-bow than for a U-bów, in agreement with former experience. For example, although there are not enough ex-perimental data to derive quite exactly the value of a1 for E.V. Lewis' model with V-bo. it is found tobe about 90 per cent of the a1 for the U-bow model. The same ratio has been found for Numata's models: (a1) V-bow --0.915 (a1) U-how Model a1 1. Gerritsma 1972 41.6 2. Gerritsrna 1961 51.5 3. E.V. Lewis 58.8 4. Nurnata U-bow 57.5 5. Numata V-bow 52. 6

-i-

.(coV) . d/3

¡jd

(12)

According to (4), (9') and (11), the common

expression for both branches, co < '°max and co > co

_{- max'}

of the response

_{curve rAW =}

(rAW) max. F can be written

b/ 3

rA\y=1.i7beh/dal()

n15+b/7

d/7.

156

The, values of a1 in Table i show a difference

between them, up lo abou t 40 Pc r cen t). Howcve r,

we.shoilcÍ take into account the dependence of a1 on the longitudiaal radius of g3'ration.

It could be assumed that

1 X

,'yy

.ai=aO(T-_)

and, fi'om another analysis of the experimental data: x = 2.If this eÑpression and the latter value of x are adopted, the corresponding values of a0 can be calculated. They are given in Table 2.

Table 2

The average of a0 is. 900, and the deviation from it does not exceed 7 per cent. Although the investigated models are few in number, this seems to reveal that the influence of th ship's form on resistance increase inwaves is relative-ly small. No large variation of a0 among ships of this type is expected.

It must be emphasized, that the present analsis refers to models of rather fine, fast ships, with block coefficients between 0.564 and 0.653, and that consequently the results of the present in-vestigation are applicable but to vessels of a similar fineness.

In order to verify the, formulas (12),

calcula-tions were carried out for model 'Gerritsma

1972', at F = 0.25. The. result is shown in Fig-ure 10, and compared ith the experimental values according to Reference [1]. A fair agree-mentbetween the calcula ted curve and the experi-mental data is noticed. The value of a0 used here was derived from the experimental data, (Table

2).

The formulas (12), however, are not quite con-venient for a quick determination of the resist-ance increase. On the other hand, it is easy to

determine rAW = fÇo) by means of the expressions

(11) and (5), provided they are tabulated

(Appen-clix Il), thus ithout going through tedious cal-culations:

- (rAW)lnaX (13)

The procedure and formulas developed in the pre-sent work can be used for:

1. A rough estimation of resistance increase in regular headwaves for fast cargo-ships, without any recourse to tank experimeñts. For that pur-pose one may take an 'average a0 900, and cal-culate rAW = f(co) by means of (9') and (12) or (13). Asanexample, such calculations vere car-riedoutfor the model 'Gerritsma 1972' (but with a0 = 900)at different speeds. The resultant curves are shown in Figure 11. They can be compared with the experimental data, and with the

out-comes of calculations based on theoretical

methods of Havelock and Gerritsma as well, Reference [1]. It appears that the results of the presently proposed procedure are not vorse than

those of the known theoretical methods;

2 the determination of added resistance of a ship atanyspeedin the range of ibterest (F = 0.12 -0.30), when experimental data are available hut for one speed value of this range. Two procedures

can be followed:

- Calculation

o Exp. points Model Gecritsrna 72

w-1.5 1.0-0.5

irir

W el/sec]

Figure 10. Comparison between relation (12) and experi-mental data for a model of a fast cargo ship.

Model a1 a0

(k/L)2

1. Gerritsma 1972 865 2. Gerritsma 1961 960 3. E.V. Lewis 905 4. Numata U-how 920 5, Numata V-bow 840 0.7 0.8 0.9 1O 1.1 1.2 1.3 30 3.5 0 4.5 5.0 5.5

2.0 1.0 Fn 0. 15 0.7 0.8 0.9 1.0 1.1 1.2 13 0.7 0.8 09 1.0 1.1 1.2 1.3 Fn= 0.25

¡

0.7 0.8 09 1.0 1.1 1.2 U -3.5 4.0 45 5.0 5.5 _L)[1/secl MODEL GERRITSMA 72 o o Experim. points. rAW Fn: 0.20 Ca Icutat ions: Gerritsma

-- - Havelock

Proposed method 0.7 Ò.8 0.9 1.b 11 1.2 1.3 r t-3.5 40 4.5 5.0 55 _[1/sec]

Figure 1. Comparison of results of the proposed method without recourse to thnk experiments with 1) experiment-al data and 2) theoreticexperiment-al methods.

3.5 40 4.5 5.0 5.5 _(J[1/sec]

158 2.0 1.5 in-03 Fn 0.15 1.0 ii 12 1.3 C.) (1/sec.) 3D iS i 4.5 5.0

Figure 12. Comparison between results of the proposed

method with recourse to experiments for some other

speed, and experimental data.

a. making use of the cuive i f(u), which 'vi11

be determined by means of the experiments at the one and only speed V, but which is supposed to be the same for any speed V2; ive- derive umax

from (9'):

.

Exp ponts. - Calculation. rAW 2.0-1.5 1.0 OES t400EL GERRITSMA 72 1/7

(u)\7 =

(u max)V

. ()

and (rA\V)flax -from (11):

j (r) maxi

_{'2 = [('AW) max] V1}

i -r

r i..)

(__) .

e35'r

F)

V1

The validity of this method is il1ustratd by Fig-ure 12,. showing the curve conve rted from F11

0.20 to Fn = 0.25, and the plotted experimental data atF=0. 25, for the model 'Gerritsma 1972'; b. making use of formulas (9') and (1) or (13). The resultn of these calculations for F 0. 15,

0.25 and 0.30, based on the curve

determined experimentally- at F 0.20, and on the derived vnlue of a1, are given in Figure 13, and compared with -the actual experimental data. 'rom Reference [3], it is easy to derive u at

which synchí'onous.pi tching occurs. Thisc see ms

tobe very close to umax, and the corresponding resistance increase does not differ a lot fom (rA\V)nlax. 1h faót, regarding model No. 1445 with li-bow of Reference [3], synchronism for pitching in waves /L = i (u = 6.36 1/sec)

oc-AW 2.0 15 - 0.5-1.3 W Li/sec] JE

Figure 13. Comparison between results oftheproposed method with recourse to experiments for one speed in order to derive a1, arid experimental data.

I[7 lb t t i 5 WELec)

0? O9 iO

1. -, 12 Wfl/sec] OES

-3ïb

0.9 0.8 0. 1.0 1.1 1.2 f 10 3.5

..-.

40 4.5 5.0 55 -3b L5 S'O 3O 3.5 1.0 4.5 SD s.s

- Calculation _{Ñodel Gerritsmo 72}

o Exp.points

2.0

IS

in

Fn r 0:3 sd,ronism in pitch (rAw) max(calc.) MODEL LEWIS W (1/sec A/L

curs at F

0.143, and in waves ?i/L =1.25

(o = 5.68 1/sec) at F 0.232. Resistance

in-crease is seen to be rA\V = 1.81 and 3.08

respectively. Using (9'), (11.) and the relevant value in Table 2, one finds

- for F11 = 0.143

= 6.16 and (rAW)m 1.92 - for F11 = 0.232

°hax

= 5.75 and (rAW)max = 2.92

The differences between the calcula ted u_{max and}

u(synchr. pitch) is about 3 and 1.5 per cent at F11 =0.143 and 0.232 respectively, and the dif-ference between (rAW)m

and r\(synchr..

pitch) is about 6 per cent.

So faras can he ciediiced from the data in Rel-erence[3], maximum resistance increase would occutat(ahout)synchronsmin pitch, as one can noticé in Figure 14. Hence, we may in the first instance presume, that mairnurn resistance in-crease occurs atpitch synchronism, and that the diffeences onu and rAW fbund above are due to errors of measurements and calculations. Pro-ided that this assumption is valid, we should

2.0 '

1.0

Ii) max (calci

synchrorsm n pdch (r11) rnax(colc) j rnax(calci W t 1/sec] Fnr 0232. 5,0 0 7.0 8.0 2.9 1. 1.9 _°? AIL

Figure 14. Comparison of calculated _max and (rAW)fl)ax, with experimehtal values ata condition of synchronism

in pitch.

take ca(synòhr. pitch) for umax, at a given speed, and then the following procedure is suggested; - assessmentofu, or A/Lfor which synchronous

pitching occurs at a chosen F11. This u will be considered as _°rnax'

- execution of one model test at this Froude num-ber n a wave train of this specified 'ave length. The measured rAW should be (rAW)m, from whiáh the value of a1 is derived;

- calculation of rAW = f(u) for any speed in the range of interest.

II. Added resistance ii irregular waves.

The mean resistance increase in irregular head waves is

RAW(u)

RAWr8fS(u)

du (14)

o

where , = wave height

S(u)

wave spectral detìsity

Hence, if the wave spectrum (of a unidirectional head sea) is known, the results of the present wo rk in Part I can be applied and the mean resis

t-5.0_ 7.0 8.0

2 1 15 p 0.75

W

160

ance increase in the irregular waves calculated. For instance1 such a calculation has been car-ried out for a containeiship respecting a full-scale measured sea spectrum, Reference 6. The results are presented in Appendix III.

However, to predict the resistance increase, one lias to use wave spec tra as given by analytic

funtions of frequency, viz, standard spectra

(Neumann, Pierson-Moskowjtz, 12th 1. T. T. C., 2nd I.S.S.C.),

As a rule, such a spectruni can be presented

in the following way:

S((C)) = A(cP exp[-DJ

(15)

where p and q are constants, and A and D are dependent on the sea state, _{(functions of wind} velocity or characteristic wave period and/or significant wave height).

Then, in accordance with (12), (14) and (15),

mean resistance increase in an irregular head sea can be expressed as

8pgB2

:RAWL

A o exp[D1 14 -co +C2

_f

(S5+P)[D14

(ùmax

where _{max circular wave frequency} cor-respondiig to the top of the iesponse curve _{rAW =} f(c) for the given Froude number, and

C1 0.398a k 3.67 F_n3.07_(-j)

L5'5

e-3,5Fn -2.83 k _-4.25

C2=6.85a1()

_Fn0'29()

5 4.67 YY _L D1=-0.09(---)

F2()

-4.67

D25.3(.)

3y F _(-j)L (17) co max f

('P)

which are constants for a given ship in

its specified load condition and at a given Froude

numbè r.

Equation (16) can be simplified, ihen the wave

spectrum is expressed in the following normaliz-ed form

S(co)

S(c0v) (18)

max

-

-

_(o

'here = _{is anondimensional wave}

fre-°_rnaxw

quency, if COma,.w corresponds to the maximum

spectral density Sjnx, i.e.

_{if comL} is the

modal circular frequency. From (15) follòws

qD 1./q

co

max'

_p

and S

A(22)/' .-p/q

Then, from (18), (19) and (20):

where, for a given wave, spectrum,

A1 =A(COmaxv)

qDw'

is a constant,

Now we have a normalized response curve rAW () and ano rnialized wave spectrum (j,

But in order to make use of them fOr the

determination of resistance increase _{RA\v, as} given in (14), we need to introduce the ratio:

co

max'

co

maxA (24)

where, for the sake of clearness,

_{0aA will he}

the notation for the former _{°max' viz, the} fre-quency correSponding to (rA,)flX.

-q1 w (21) n S (cou,) = wT) exp[E( and, referring to (18):

S((0)=A(C)mv)P .

_or

exp[-Since _\V

=

and _A

ixA'

(24) is

R8pgC5 C«

û

[e078J1(a)

equivalent tu the relation

= 0w (25) +e0 607 J(a)] (30)

and (22) can be written as

where Cs = (rAW)I1ax. max' a ship's factor,

S1(4=A1 aA exp[-

_-

_{aPA q] ±Sr(A)}

q Le. depending solely onship form, load distribii-(26) tion and Fraude number, which for a given ship

RA\V(0) can easily be evaluated, and

R

=8f S ()

d = qD

AW

o IrL\r\

(\12

fi C A = A(o)nax,)

= A()

P , .a sea state_/

B2 factor, which can be determined after making a

= 8 PÍ S(o) rAW()du

choice of a standard sea spectrum.

o Then, by means of (30), and adopting the

(as-According to (13), and since, on the other hand, sumed) average value a0 = 900, we

approximate-= inA

d, the expression of RA\V can be 13' calculate the mean resistance increase for any

written as ship of the considered type (fine forms), at any

speed many irregular head sea described by one

B2 of the conventional standard formulations. A

RAW =8 pg

_{rA,rnax mA}

X

further detailed treat ment and working graphs

o

are given in Appendix IV.

rAW(cA)do)A (27) Itmay be interesting to note, that the trend of

where, of course, the index A in the integral may change with c of some propulsive characteristics of a ship model in waves seens to he similar to henceforth he omitted.

that of the added resistance, Refereiice [2]. A Treating as formerly each range c < 1 and

procedure, analogous to the one presented here,

c > 1 separately, we obtain, in accordance with

might he used to estimate the change of propeller

(5):

revolutions, torque, thrust and power in regular

B2 and irregulär waves.

RAW = 8 maxA

J1(a) (11-p) exp --Ci)

11_14 P q--q

--a

c ]dc'

.14 q J2(a)=f -(8.5-fp) .

exp[-.5

1 (29) q Ackiioi1edgements.

The authors are indebted to Prof. G. Aertssen, who kindly gave his permission to use sotne full-scale data from the containership 'Dart Europe', and toIr. R. De Lembre (Ceherena) for his help in carrying ut the computer work. Tle full -scale investigation was carried out under the auspices

of Ceberena (Centre Belge de Recherches

Navales) with the financial assistance of

I.R.S. l.A. (Institut pour l'Encouragement de la Recherche Scieñtifiue dans 1 'Industrie et 1 'Agri-culture).

which are functions of a. These functions can be Nomenclature.

tabulated, (Appendix IV). RAW mean resistance increase in waves,

The mean resistance increase in irregular head kp

waves caïì be written in a condensed form: . rMv nondimensional resistance increase

aP[e0785J1(cx) +e°07J2(a)] (28)

r

in waves

-pgç2 B2/L

normalized resistànce increase = rAV

Appendix L

Main particulars of models. Gerritsma

1972

Gerrit'sma

1961

E.V. Lewis E, Numata

L(m) 3.05 2.227 1,525 1,645 B(m) 0.456 0.324 : ,0.204 0.238 T (m) 0.183 ' 0. 0815 0. 0875 V(m3) 0:1434

0.01

0.0151 0.0198 CB 0,564 0.653 0, 60 0.575 0.58 0.668 - -L/B 6.? 6.9 7.5 6.9 L/T 16 8 16.5 18.7 18,8 B/T 2.5 2.4 2.5 2.72 L/Vh/3 particulars 5.82 5.57

617

' 6.07 of form - - U UandV 0.219 0.232 0.255 0.250

longitudinal radius of gyration, m

P' mass density

Cr acceleration due to gravity

Refer'n ces.

Gcrritsma, J. and Beukel man, W. 'Analvs is of the

reistance increase inwavcsof a fast caigo shiiY,

International Shipbuilding Progress, 1972, Vol. 19, No. 217.

Gerritsnia, J., Vanclen Bosch, J.J. and Beûkelñan,

W. , Propulsion in, regular and irregular waves", International Shipbuilding Progress, 961, VOl'. 8,

No.82.

Lewis, E. V. and Numata, E., 'Ship model tests i n

regularand irreguFar seas', Experimental Towing Tank, Stevens Institute of Technolog', 1956, Rep. No. 567.

Numata, E., 'Comparative slamming characteristics of two ship models", S. N. À. M. E. Tehnical find Research Symposium, October 1967.

Mitsuyasu, H., On the growth of the spectrum of

the wind-generated waves' Reports of ReseaiOh

Institute for Applied Mechanics, Kushu

Uni-versity, Fukuoka, Japan, 1968, Vol. XVI, No. 55. Aertssen, G. and \'an Sluvs, M. F., 'Service

perform-ance and seakeeping trials on a large

container-ship'; The Naval Architect, October 1972. (rAW)rnax

(RAW)max rnaximum resistance increase for = constant

(rAW)_{max maxi mu in nondimens ional res is tance} fOr = constant.

ç wave height, m

wave length, m

C,) circular frequency

L length between perpendiculars, m

B beam, m

T draught, m

displacement, metric tons

V dispiacèment volume, m3 CB block coefficient

CP longitudinal prismatic coefficient

V

Froude number =

-V speed of model, m sec'

Appendix Il.

b

Tabulation of the function = e bexp[-b/d 3d]

1

>1

Appendix HE.

Calculation of the resistance increase due to waves in irregular head seas for a large containership.

The method given above was applied for the evaluation of added resistance of the container-ship Dart Europe in irregular head waves.

Full-scale experimental data we re taken from Reference [6], viz, the observations No. 22 and Nô. 23 in the paper referréd to. The principal particulars and the actual (light-load) condition of the ship:

L=218 m , B =30.48 m,

T=7.64m, =30, 250 metric tons, CB = 0.585, 0.25.

Table III-I

The environmental conditions, shiP speed, delivered horsepowe r in the actual seaway and in calm vater are given in Table III-1, (from Ref-erence [6]).

It is noted in Reference [6] that neither pro pelle r racingnor slamming occurred duing these obse rvations.

Figure 15 shows the energy wave spectra, derived

from the full-scale wave records,

(Tucker shipborne wave recordér), besides. the curve of added resfstance in regular waves. The latter was calculated by means of the presently proposed method, taking for a0 the adopted average value 900. According to formula (14),

0.500 0.600 0.700 0.750 0.800

f

0.00] 0.008 0.043 0.091 0.184 0.850 0.900 0.925 0.950 0.975 1.000

f

0.340 0.573 0.715 0.845 0.965 1.Ò00 1.000 1.025 1.050 1.100 1.150 1.200 1.250 1.000 0.980 0.900 0.692 0.517 0.371 0.272 1.300 1.350 1.400 1.500 1.600 1.700 1.900 0.197 0.142 0.105 0.059 0.033 0.02 0.008 Obs. No

Wind \Vavcs Ship

speed (k D (metric h.p.) calm water Power increase A B trüe speed (1m) true dir, (deg.) Sign. height (in) Heading off bow (cleg.) 22 23 - 7 7 28 33

lop

20P 5.3 4.4 15P 20P 19.7 19.5 24,100 24, 000 14,700 14,500 9,400 9,500

164 S (W)(rn'-s.cI 70L 6.0 50 ¿.0 3.0 2.0 1.0 Obs.N.22. Rel. (61 VE SPECTRUM I \Responsecurve

1Ç/

I _\

\

0.2 0.3 0.1. (15 0.6 (17 0.8 OS 10

-

W [1/sed 2.6 2h 2.2 2.0 1.8 li. 1.2 .0

the added resistance due to the irregular head waves are respectively:

- for Obs. No. 22: RAW 30,040 kp

for Obs. No. 23: RAW = 23,960 kp.

The wind resistance was evaluated by making use of the information in Refeience [6]:

-. for Obs, No. 22: = 11,400 kp - for Obs. No. 23: = 13,900 kp,

By means of the model propulsion tests, one could determine the propulsive efficiency in cal rn water = 0.73. According to Reference [6],. a reduction of 15 per cent has to be taken into ac-count for the environmental conditions as met during both observations. So the actual 1D = 0.62. Hence, the absorbed horsepower due to waves, as calculated, is:

- for Obs. No. 22: ÄW = 6,570 hp

- for Obs. Nô. 23: AW = 5,180 hp and the absorbed horsepower due to wind: - for Obs. No. 22: = 2,490 hp

Obs.N. 23,Ref.(61 6w WAVE 3.0- SPECTRUM 2.0- 1.0-0.1 02 03 Oh 0.5 curve

\

OS 0.7 0.8 09 10 -2.6 -2.4 2.2 120 -b.8 jis -L' 1.2 10 0.8 -0.6 0.4 W(1/sec]

- for Obs. No. 23: P =3,010 hp.

In accordance with Table III-1, which gives the measured in the actual seaway an in calm vater respectively, and hence the overall power increase LPD, the added powerdue to waves (and to other factors maybe, except wind) must be:

- for Obs. No. 22: P = 6,900 hp - for Obs. No. 23: P = 6,490 hp.

The agreement bet\veen the latter values and the calculated PAW can't he considered as un-satisfactory. Indeed, other causes of resistance increase may he present. (fouling since the date of measured mile trials, steering effects in bad weather) While the agreement is consequently quite good for Obs. No. 22, the difference re-S mains considerable in the case of Obs. No. 23. However, during the latter observation, a shorter wave period was noted, and the corresponding wave spectrum indicates the presence of an un-usual substantial energy due to shoì-t waves. This factmaybeareason for a less accurate calcula-tien by means of the proposed method, as has been explained formerly.

Figure 15. Full-scale, measured spectra' of seas encountered b' a large containers'hip, and the calculated curve

Appendix V

Approximate prediction of resistance increase iii irregùlar head waves.

The irregular headwaves considered here will be represented by the standard

two-parafrieter-spectra recommended by the 12th I.T.T.C.:

S) Au5

exp[--2

1731/3 ₆₉₁

and D

- T14 i

1/3=significant\va\'e height (m) = 4.0 and

Tj;= the 'characteristic'period = 2Tr-2, with the suggestion that T1 = the visually observed aver-age wave period, (m0 and m1 are resp. 0th and and ist moment of the energy spectrum).

4 85 From (19) and (23) we derive crn

and = = 0.0645 T1

By(29) and the above mentioned basic formula-tion of S(c) we can write

i

expj- 0.7854-1.25a44}d

o

J2(a)f c)35exp[- Ó.60614

i - 1.25 and

I(cx)=cx5. [e°785 ji(s) +e°606 . J2()]

0)

maxw

I(cï)isafunctionofa - only, and has been

rnaxA

calculated for the range of actual interest, from a = 0.55 up to a = 1.7. The results are given in Figure 16.

In the formula (30), which gives RAW, the

ship's factor Cs depends upon the ship's speed size (L), longitudinal radius of gyration (k,/L), and B2/L. It is convenient to present this factor inthe following way

where A = where B2

=C,

_{L s} 1.6 1.5 14 1.3 1.2 11 1.0 0.8 0.7 0.5 0.6 07 0.8 0.9 LO ii 1.2 1.314 1.51.6

Figure 16. Theinfluence of the ratio o

C (used

.2

=ao(T-) 1.5 exp[-3. SFhI X

0.1 0.2 0.3

'igure 17a. Values of ship's factor C'e.

166 L17

FT'1

)rj7

_Vk/L

3 1.053x10 (-)_g

F1357.ep[- 3.5 F]

The values of C have been calculated in the following ranges of parameters

= 110-200 m

F=O.l2-O.36

k/L=o.21_o.27

and are given in the Figures 17a 17g, from which any value of can be derived by inter-polation.

Finally, the resistance increase is calculated by means of the expression

B2

Rv=SP_. C C. I(e)

1.667 Fn 1.2 li 1.0 0.9 0.8 0.23

Figure 17c. Valuesof ship's factor C.

Fn 2.1- 2.0- 1.9-1.8 1.7-1 .6 1 .L 1.3-0.2L 0.2 0.3 0.1 _{0.2 0.3}

C'it1ISed 2.L

2.3

-Ö.26

10

Figure 17f. Valùes of ship's factor C.

OEl 0.2

Fn