Wake of merchant ships

MERCHANT SHIPS

B Y

SVEND AAGE HARVALD

THE DANISH TECHNICALPRESS

COPENHAGEN 1950

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Denne afhand].ing er s.f Daninarks tekniske Hjsko1e antaget tu Í'orsvar for den tekniske doktorgrad.

Danmarks tekniske Højs1o1e,.23. juni1950.

Anker Engelund. Rektor.

-':.

well as. tothe scientist working in the laboratory.

The dissertation princlpally.dea].s with.the problems connected with wake, seen partly from a practical, partly from a theoretical point Of view; but further It, considers questions closely connected tò wake. Thus amongst others the problem of thrust deduction has been taken.up for _a. brief discussion,

The principal part of the, work connected with this treatise has been carried out at the shipbuilding depart-ment at the DANMARXS TEKNIS

HØJSKOLE, Køberthavn.

The.

leader of this department Professor

_{C.. PROHASKA, D.Sc.,}

has during the progress-of the work given the author a great deal ol' sùpport and assistance.

Some of the work had.to be done abroad, partly in Sweden and partly in Holland. In Sweden the author had a

whole series of experiments performed and in Holland stati-stical material was collected.

For five nontha the author was temporarily employed as half-day assistant at the STATENS SKEPPSP,ROVNI.NGSANSTALT, Gtoborg, under Professor. H.F.

_{NORDSTROM, D.So,}

superinten-dent. During this same period the HUGO

_{HAMMRS FOND FR}

IN-TERNATIONELL FORSKNING INOM SJFARTEN supplied the funds enabling the carrying out, of model experiments. Thus the autboi'ha4 good opportunity to study wake. As these experi-monts are described in detail in "Meddelanden frân Statena Skeppsprovningsanstalt' nr 13", only the results are given in this dissertation.

For three month the author was the guest of the 1EDER-LANDSCEE SCHEEPSOTJYJKUNDIG PROEFSTATION, Wageningen, and worked at the scientific department of the tank; work sole-ly connected with investigations of wake. The staff of the tank was of great help and support to the author,

2.

ficaSed,.and the. superintendent, Professor L. TROOST, who wasoxceedingly hospitable and most helpful. The results of

one part. of the work in Holland has been published earlier iñ the Dutch magazine "Schip en Werftt.

Soñe shipyards and. institutIons have put at the authors disposal model experiment data.

ProfessorE. Petersen, Ph.D., Darmmrks tekniske

Højsko-lé, has checked the mathematical expressions in sections D-c and D-cl.

Mr. J, SVÈNSEN and Mr. H.E.GULDHAMMER, M.Se., hâve assisted in preparing the many diagrams, and Mr. GULDÄU1ER

has given much good advice

The translation f ro Danish has been .perforned by'lMr.

T.E.BLUI, B.Sc., and Mr. G. BUCHANAN, B.Sc., hâsveriied

the transiátion.

±ather of' the author Mr. AA. HARVALD has assisted

byreadingthe proofs and hs. been of great encouragement to

the authói' In accomplishing his work.

The expenses of the work in Denmark were met through grant s from AKADEMIET FOR DE TEKNISKE VIDENSKABER, København and. THOMAS B. THRIGES FOND, Odense.

The author tanks most cordially all the above Lnentioned persons and. institutions for their great and valuable 1elp..

Kgs. Lyngby, Deceñiber 1949. Sver4 Aage Harvald.

Definition, measurement and determination of the wake

nd the wake coefficient 7

CHAPTER I : C0MPOENTS OF WA

C. Frictional wake 12

D. Potent1a1 wake 23

a. Potèntial wake of cylindrically 3liaped bodies. with similár fore and after ends (tvro-dimension].

flow) ' 24

ai . The potential wake coefficient 's dependence on the

angle of tnc1ination of the waterline to the cönre

line aft :.. . 39

The variation of the potential wake coeff.iciet

with the propeller diameter 44

The longitudiral variation of the potential.*ake

coeffiòient

-b. Potential wake of cylindrically shaped bodies with different fore and after ends (two-dimensional

flow) 48

-e. Potential wake of ellipsoids (three-dimensional

flow). . . .. . ... . . . ...oo..-. . .-..o là .-. 54

ci. -Comparisonof two- arid three-dirnensiönalflovs(el--liptical cylinders -and eliipsoid-s 64

d, Potentiji wake of' solids of revoluticn (three

dimensional flow) - 69

--e....Potential wake of' ships (three-di nional- f-low)-.-. 76

Wave wake - .77

CHAPTER III : P?EDETERMINATION- OF iA

-Construction of a new diagram for determining the wáké

coefficient of' single screw ships ...,. 85 a. Parameters on which the wake coefuic.eñt dends. 85

b.- Analysis of model experirents

---bi. Variation of wake coefficient with speed 103

b2. Relation between nominal and effective wake

Construction of a new diagram for determIning the wake

4.

H. The wake coefficient of special types of ships 112

Fishing vessels . .. .

112

Tug 'coats .

115

ships in hioh the propellers are fitted in

nozzles or tunnels 116

Ferries with forward and after propellers 118

Patrol and pilot boats 119

Ocean yachts 120

CHAPTER IV : FORMrJIAE OF VJA1OE AND THRUST DEDUCTION

I. Earlier formulae and diagrams for- determining the

coef-ficients of wake and thrust deduction of single and

twin, screw ships 121

J. Testing of the wake-formulae and diagrams for. single

screw ships 147

K. Comments on the wake-formulae and dia'ams for twin

screw ships and for special types of ships 157

CHAPTER V DEPENDENCE OF WA' ON VARIOUS' FACTORS

L. Influence of rudder on wake 159

M. Dependence of wake coefficIent on condition of ship 166

N. Dependence of wake coefficient on depth of water 167

O. The influence of the degree of turbulence on the wake

measured by experiments 169

CHAPTER VI ¡ WA DISTRIBUTION

P. Distribution of wake of single and twin screw ships

171

Q. The optimum coefficient and distribution of wake 178 CHAPTER VII : DEPENDENCE 0F VARIOUS FACTORS ON V/A

R. The Influence of the wake on the nwnber of revolutions

of the propeller 182

S. The influence of the wake on the steering of the ship 185

T. The influence of wake on propeller cavitation and

vibration . -. 189

CHAPTER VIII : THRUST DEDUCTION

U. The relation of the wake coefficient to the coefficient

of thrust deduction in single and twin screw ships 191

V. Summary

-

204

Summary in Danish

207

Synthols and units - . .. 210

A. INTRODUCTIbN,

It is always the hope of the pÌ'oje].ïe±designerto design a propeller such that: Fir'stly it'ill wörk as he

wants lt to, i.e. th.t the prd1lor & êrtairi

ed and

a certain number of revolutioÍis wi1I absóxb a cèrtaln amount of power. Secondly that the ptirnwn intration between ship

and propeller is attained, Still further nowadays there Is a third requirorent, èspecially in the case of fast ships:that the propeller must be free from cavitation. In order to get these wishes fulfilled he must know the velocity or rather the velocities with which the water flows to the propeller disc. What happens outside this region does not concern him. But in the study of the velocities of the flow of water, it is necessary, to examine what takes place outside the pro-peller disc, in order more easy to ascertain the actual con-ditions at the propeller.

In this monograph the conditions of wake from forward to aft, even up to about one ships length abaft the propel-1er, have been examined. The base flow, i.e. the flow present without propeller, as well as the flow present when the pro-peller is working have been studied. The examinations have first of all covered the ordinary types of merchant ships, but in addition many special types of ships have been inclu-ded,

An analysis of the three components of wake: frictional wake, potential wake, arid wave wake has been made. The ana-lysis has principally been theoretical, and only as regards wave wake have experiments been performed.

Further a statistical examination of the measurement of' wake,. of the effective as well as of the nominal wake coefficients, has been Carried out, and the results have been compared in a diagram to be used for quick predeteruii-nation of the wake coefficient.

A comparison of the different approximate fornmlae and diagrams for predetermination of the wake coefficient bas also been made.

At the end of' this treatise are given curves of wake distribution, obtained from numerous model experiments, to

6.

be used by the designer. who, for his designe, employs the theory of circulation.

Finally the relation of the wake coefficient to the coefficient of thrust deduction, and the influence of the wake on steering and on the number of revolutions of the propeller have been dealt with.

CHAPTER I:

DEFINITIONS.

B.

DEFINITION., ASUREIVNT AND DETERMINATION OF T

WAKE AND THE WAKE COEFFICIENT.

By te wake is understood the difference between the

velocity of the ship and the velocity with which the water flows to the propeller.

By dividing this difference either by the velocity, of' the ship or by the velocity of flow òf water, two wake

coefficIents are obtained. The first coéffiient

w

V -Ve

V

is ternied TAYLOR's wake fraction. This is employed in Ame-rica and in the continental part of Eürope, aìid will sole-ly be used in the following. The other coefficient

w

VV0

F _y

e

is termed FRQUDEs wake percentage and is much used in

Great Britain. For the sake of completeness the formulae for transforming or.e coefficient to the other are given:

w = 'F.

and w

=

By potential wake is understood. the wake obtained 1f the ship moved n an.ideal'fluid.without friction, and wave making. In other words the potential wake.. is that wake which mathematically can. be calculated'. This wake is indo-pendent of the direction of propulsion as well as of the

velocity of the ship. .

By wave wake is understood that component of wake originating from the movement, of the water particles in waves.. There is precedence in shipbuilding for employing.

GERSTNER'S theory

fOr

trochoidal

waves,. in

ordertó

explain

phenomena connected with wave.makirig, Although this thèory

is riot fully correct,

frozn'a

_{purely physIcal point of view,}

as It assumes origination

of

rotation

ma

frictIonles

wake phenomena, which will be proved later on.

Finally there is the frictIonal wake which will have to be defined as the difference between the actual wake and. the suri of the potential wake and t}e wave wake, or in other words that change

In

wake which Is caused by the friction,

However, other definitions for the threa wake components are often used, starting with frictional wake, which is the most important of the three. In towing a thin plate through water, the only important wake component will be the

frictio-nal wake. Around the whole of the plate will be a zone in which the water to some degree more or less will follow the

plate. t the front of the plate the thickness of the belt will be equal to zero whereas the thickness will increase

coniming further aft, until at the after end the thickness of the belt will begin to diminish. At a distance from the plate the flow takes place as if a triangular body had been moved in africtionless fluid. To a certain extent there-fore lt is correct to state that the zone of friction is a part of the solid, and therefore the potential- and the

*ave wake can be defined asbefore, but corresponding to an

increased solid. This method of definition is the one most-ly employed, butit is far from good, as it is uncertain how much or how little of the frictional belt which must be taken Into account. In future Investigations of the zone of friction, of the. frictional wake, and of the coefficient of friction, it: will probably be correct to. employ the first method of definition.

Asa rule the wake coefficient of a ship is determined

by the ship propeller actlng.as a wake measurer and a wake integrator, the effective velocity of wake being defined as the difference between the velocity of propulsion and that velocity, which in a homogeneous field would enable the propeller at this definite nuriber of revolutions to create a thrust or to absorb a torque equal to that existing. Then by dividing the two velocities of wake thus found by the velocity of propulsion, two wake coefficients WT and are

ERRATA

Line 31: Pôr "month" read "months".

p. 2. Line 3: Por "has" read "have".

Line 7: Por "Petersen" read

P.. 2. Line 22: Por "tanks" read "thanks".

p. 8. Line 28; After "the"ineert effective". p lo. Line 26: por "size" read "sizes".

p.. 16. Line lo: Por "alearance" read "clearance". P. 17. Line

5: Por ."ha" read "ve.

F. 19. Iie 23: After "frictional" insert "wake".

P. 2o. Line 12: Delete "generally".

P. 22. Line 18: Delete "change with varying speed".

p. 23. Line 15:.Por "different" reed "differently".

P 23. tine 22: Por "the waterlines of ships" read "a

water--

line of

a-ship"--P. 45. Line 5: Por "w" read

"w".-P. 48. Line 19: Por "longitudial" read "longitudinal".

p. 49. Line 8: Por "was" read "were",

'. 54.- Lines32 and 34: Por "ellipseide"- read "ellipsoid". P. 61. Line 7: Por "ellipsoide" reed "ellipsoid".

p. 62. Line 4: Por "ratio" read "ratios". P. 7e. Line 20: Por "exist" aced "existo".

P. 72. Line 7:For "recolutjon" read "revolution".

P. 76. Line 3: Por "ciefficlents" read "coefficients". P.. 77. Line 9 Por "give" read "gives".

-Line 3e: After "in" Insert "each". p, 82. Line _{2: Por "O.005" read "0.010".}

LinS 4: For ."0.005 to 0.älo"reed '0.010 to 0.o2o".

P. 82. Line i8 Por "liminations" read "limitations".

.p _{83. FIgs.41 and 42:. Por ".01" and, ".o2 read ".o2" and}

- respectively.

. 85. Line 12: Por."are" read "is".

Line. 22: Delete "the". -

-Line 17.: After "wbare" insert "a". P.- 91. Liñe 22: For "have" read "has".

p. 94. Liño 3: Por "playa their part" read "play their -parts".

i'. .94. Line 13: Delete "very".

P.99. Line 22: Por "are" read "la"; .

P.1o5 Lthe -19: Delete. "very big". -

:

P.107: Lthe _{7: For."sltght" read "Blightly".}

P.107. Line 21: For "play" read "plays".

F.lo9 Line 8: For "submarins"read "submarines". . .

-p.ïu. LIne

6: For "dimihed" read "diminished". P.114. Line 23: For ."shipe" read "ship".

Line 13: For "nothing is practically known" read "prao-- tic&lly flotiñg is known".

Table 28: Before "(mi)"insert"".

Line 32: Por "therefor" read "therefore". Line 2: For "deminish" read "diminish".

P.126. Line 6: After "TAYLOR" insert n,".

P.128. Line 29: Por "were" read wao".

P.132. Liiie 13: After "Dj±tai» Insert "(24)".

LIne 1: After "and partly" insert "by".

Line 25: Por "PRotm's" read "PBOU.s". --.-

-P.141. Line 21:- For "2/3".read

"3,/2".-P.142-. Line 18: Por "j/l" read "Vt". Line 3: For "draught" read"B/d".

F.l45 Line 8: Por "have" read "baa".

Line 12: For "VXB" rèad "WIPS"

Line 18: Por "wip' n read "KEVIRS'". P.154. Line 12: Por "is" -read "aren.

p;161. Line _{1: After "87" ins±"t ".".}

P.162. Line 15: After "a t

b" Insert "tfig. 86)".

P 166 Line 34 For "conditions read _{trim or draught}

con--

dItjo".

-P.167. Line 13: Por "chip" read"ehip". P.171. Line 11: Por "resultá" read'results".

P.175. Line 8: Por "ail-. designing _{uiposes" read "design}

pur-poses"

-P.179. LIne 16: För "sliÚhtly" road "lightly".

P.183. Line 17: For "were" read "was".

P.187. Line 20: Por "-aeeuma" read _"asaumes"

-Line 16: Por "must" read "can". -Line 14: After "will" iñeet-t "not". P.2o. I,Ine- 15: Por "constructed" road _"composed"

P.208. Line-.22 _{Por "miseren" read"mjsêren".}

-P.208. Line 36: Por "fuldstndjgt»

_{read "fuldotdig"}

P.210. Line _{7: For "will be" reád "have been".}

P.215. Line 19: After "wake" insert ". -.

P.219. Line li: Por "Edinburg" read "Edinburgh"

9

method) or by torque identity. As a rule WT and W aree, different.. Especially In single screw ships the difference is marked and as a rule is somewhat bigger than WQ. In ordinary practice VIT

IS

mostly employed. Sorne experimental

tanks for Instance those In Gteborg and Washington, ue an

average, value of WT and wQ. HORN (47) bas suggested that be used solely, and in the article quoted he gives a some-what illogical proof as the reason why w should always be employed. Other investigators (amongst others TROOST (122)) have suggested that the ca].culation.a of wake be based on CT/CQ(=TD/Q)_Identity. _As

=

this will be an Identity of efficiency and therefore will have the advantage that the Idea of relative rotative éfflel-ency will disappear.

TELFER (113) has suggested lhe emplòyment of a cQ_iden_ tity, not using the eQ-curve itself, but the slopes of the cc_curve. His reasoning being that by comparing the values

'of c for the propeller in open water with the values of for the propeller behind the ship, different REYNOLD's num-bers would be obtained, and therefore also different

curves. Experiments prove, that the shape of the curves is.. practically Independent of EE'Y1OLD's number, and there-fore TELFER is of. the opinion that an Identity of slope

should be used,

Finally .SCHOENIThRR's method must be mentioned (97). He employs a resultant power Identity, using the coefficient

Cr=

Fr/n2d4, where Fr is the resultant of the lift and the

resistance acting on the propeller blade. When the angle of slip varies on account of the heterogeneity of the flow, this coefficIent varies linearly in contrast to CT and CQ. and therefore S'CHOENHERR is of the opinion that this should be used. In practise Cr

T

T + k c should be used, k

being a coefficient depending on the dstribut1on of

pres-sure on the propeller blade and having the value 8.16 to'9.

- But no matter which procedure is used, the wake

ot a. heterogeneòus. When measuring the Wake in the fiture, the àutbor therefóre is of the opinion that the old methods ought to be rejected and heterogeneous field should be

corn-pared to heterogeneouz. This question will be dealt with in more detail in a later section (section L).

If it s still desirable to employ one of' the oid methodS, it would be preferable to determine the wake

coef-ficientbymeans of the cT_identitY, as this is the implest.

For determining the nominal wake cóefficient,

4e.

the

Wake coefficient at the position of the propeller, wen the

prope].lerhae been removed, three ways are avaible: ¶easu ring by means of pitot tubes, measuring by means of 11ade wheels, and measuring by means of resistance rings. Also

the method employed by BRAGG (16), athongst others, mst be mentioned. This method is a type of blade wheel measure-ment, using a free running propeller as blade wheel.

By the three first mentioned methods it is necesary,

in order to obtain the wake coefficient, either to po1rform a volume integration according to the formula w.dA

V

fA

rA or an Impulse integration WI = A ( -w) d

1 (l-w)° dA

As the advantages and disadvantages of the diffeent

xneasurement and. calculation methods have often been. thòrough-ly discussed before (see for instance (87) and

(62))\It

need:

only be mentioned here that in normalpracticebla4e iheels

of different size are used In determining the

nominali

wake

coefficient,, and for integration, a volume inteaton is

used:

-. ,

r° dr

v

frdr

.

the integration being performed from the propeller I to

the propèller tip.

Measurements by means of pitot tubes would bO

prfer-able to measurements by blade wheèls, because they give

11

are very laborious to, perform, they are seldom carried out. Measurement of wake by means of resistance rings is not used very much, as it does not give any better

Informa-tion than measurement by means of bade wheels.

In a latersection the relation between effective and

CHAPTER II:

COMPONENTS OFWAKE..

C. FRICTIONAL WAKE.

In spite of the great importance of the frictional wake. on the action of the propeller as well as on the rela-tion of the ship resistance. as a whOle very few systematic investigations of the frictional wake have been carried out, indeed so few, that they only

in

broad outlines clear up the problems.

When a ship moves through the water, the water les will, as it were, adhere to the ship. The water partic-les, right at the side of the ship, will have the same velo-city as the ship, whereas those at a distance of 0.1 to 0.2 mm from the ship side, according to STANTON, MARSHALL, and

BRYANT (99) will have a velocity only half that of the ship. Inside this thin layer, which will extend over the whole of the surface, provided the surface is smooth and continuous, the flow will be laminar. Regarding the wake conditions this layer is of no importance, but it will have a big influence on the frictional resistance.

Outside this thin layer there will be a region, the friction belt, in which the velocity wil]. diminish slowly until finally becomming equal to the potential velocity. But in this friction belt it is only the average velocity, which will bave a constant value in a point fixed relative to the ship, while the particle velocity in the point will vary from one magnitude to another, and. at the same time the direction of motion also will vary. The flow will be turbulent. As this turbulent friction belt bas a big in-fluence on the wake coefficient, this inin-fluence will be examined a little closer in the following.

The first questions which arise are: what is the thick-ness of the frictional belt, and how is the velocity cilstri-bution Inside it? PRANDTL and VON kRI?AN (51) have answered both of the questions in a partly theoretical way, although only for plane surfaces. In his work VON KRIVPA}4 starts with

13

following assumptions:

I. The distribution of velocity at the plane surface 18 assumed to depend only on the viscosity of the fluid, the density of the fluid, and the frictional forces.

The frictional resistance is assumed to vary with the velocity raised to a certain power n, which in his work bas been taken as 1.75.

The distribution of the velocity is expressed by

S

where y is the velocity at a distance y from the surface, and d the thickness of the friction belt at a distance L from the foremost edge of the surface.

VON R1&N obtains the following expressions for the velocity distribution and the frictional belt thickness respectively for flow over plane surfaces:

= V. ("d)1 (c-l)

and

-

V,

ji

d = O37 L.

( 'V.L)' (C..2)

V being the coefficient of kinematic viscosity.

Since VON 1(ARM&N published his formulae a series of experiments bave been carried out, some of them confirming the formulae, others disproving them.

At the William Froude Laboratory BAKER (5, 6, and 7) performed a long series of Oxperimenta and arrivd at

re-sults, which do not agree with VON E!ARMkN's fòrmulaë.

In the first case BAKER found, that the belt thikhess

was independent of the velocity, and secondly that VON E!kRNN's formula for belt thickness gave too high values,

especially in the'caseof models. BAKER therefore composed

biß own formula:

d2 + 1.5 d = O.o2 L (feet) (C-3)

It must be added, that BAKER defined the belt thick-ness as extending to the position where the velocity is one per cent of the velocity of the surface.

noted, that the difference between the two set of curves is

large.

Fig. I. Thickness of friction belt, determined by VON K'ARMN's and BAKER's formulae ("model")

Fig. 2. Thickness of friction belt, determined

by VON K!kRN's and BAKER'S

formulae ("ship").

Using the results, obtáined by KEMPF (53) with his 68 m long pontoon in Hamburg, for comparison, it will be seen,

that KEMPFtoohas found that the belt thickness varies

with the velocity and roughly speaking the variation in accordancewith VON IRAN's formula,. *hereas the bé1t': thick-.

ness found..b KEMPF is a little less than. that of VON

iARMAN's. Further with the same pontoon KEIJPF has perfóriTied a series of experiments in which the roughness of the sur-face;was varied. It appeared that a change in roughness partly: resulted in another belt thickness and partly

O f5 MODELS --EIP400rn6 57005M

_/

-, v.,- 2/ 2 4 8A000 IÓ 8 /0,,, d vm SH/PS v-5 5,

,

,

250

another distribution of velocities. Expressing _{the. velocity} distributions by

y = V. (7/d)'

n

being varied with the condition of surface, _{the formula} and the experimental results agree very well. The exponent n will assume the following values:

n .= 1/9. VarniShed and polished iron surface _(KEMPF'S

_pontoon

experiments, R = 4.13 .108).

Channel-steamer (s/s Snaefeil), good paint _and smòoth.butts (BAKER). . .

n = 1/7. Merchant steamer (s/s Ashworth) surface _{çiean, but} paint old, overlapped butts, 108 days _siñce:last docking (BAKER, R = 4.77 . 108).

ri =2/11.Model experiments; dealing _{with smooth polished}

surfaces n can be put equal to this _value.

n =1/5. Merchant steamer (S/S Ashworth), 25o _{days since}

last docking, surface covered with

_lóng'ass

(BAKER, R 4.77 .

Fig. 3 gives an idea of the velocity variation with. different values of n. The velocity is _{given as a functión} of the ratio between the distance from the surface and the

thic1ese of the belt.

Fig. 3. VarIation of frictional wake with the ratio between the distance from the surface, and.

the thickness of the friction belt.

15

KEMPF's pontôon experiments _{further.proved,.that as} n increased frörn 1/9 to 1/7, the belt _{thickness Increased} about lo per cent, and also that the distribution of sur-face roughness had a certain influence on the conditions in the belt. Often it is possible _{ona rough surface to}

change taking place in the relation of the friction belt. A niore detailed account is found in the works of 1MPF and

BAR (53, 54, 5, 6, 7,and io).

he formulae mentioned aovo aré only valid for f low alóng plane surfaces. In

dealing

with ship forms, they may wIth góod appr6ximat.on be used for the flow aroúnd the fore body, while their application to flow arround the after

bo-dyls problematical. Nevertheles8 these formulae are

practi-cally always used, when for instance the tip alearan.ce of

the pipellers iia twin srw ship is to be

decided.

BAKÊR (7, and io) is one of the few investigators, who

have attempted to ±ind an epressio for the variation of

te è1

thic egs with form. Unfortunately the material of

invest tÏoiwas very limited and therefore his formulae

a:iid ã.at muât be treated with great reservation. 1n (7) he

pròrts Ì fort1a for determination of the average

thick-ness à cf the friction belt at a distaice (x) from the rar1sv& e plane (ri) at which the run jol the parallel nttddle cody, or from the midship section in vessels with no

aiïUe'lmidale body. The formula is:

d = d 4

(Ç -

d)

(c4)

and G being the wet girth of frame (ni) and Cx)

respec-tively, and d. and d are the belt thickness's found at the respective frames, usiri; BAKER's formula for the. tickness of the friction be2t (0.-3). In th8 case of a sk4p 'having

d/B = 0.5, G may roughly be put equal to 2 G. Thus 1:s ob-taIned:

d=d +d

(c-3)

and as dm as a maximum can rea..1i .í, it should never be

possible to obtain a belt thickness bigger t$n 2

Ç;

bow-¿'ver 1me and ägaln bigger belt

hicknes are metwith.

In order further to elucidate the influence of ship form on friction belt thickness, it should be mentioned, that when considering the flow along a çylinder, shaped

17

bé found., that the belt thickness will be greater or smal-1er than on à plane surface, according to whether the cy-lindór surface is concave or convex.

'ib

fig. I ánd 2 curvos, calculated by means of for-niula (C-5) has also been indicated, dm being made equal to the belt thickness at L/2. As mentioned, the formula only gives the' average thickness, and generally a bigger belt thickness at the waterline, and a smaller at the bottom 'of the ship may be expected. Table I gives a good illustration of how envolved the ielatione are.

Table. 1: Comparison between calculated and measured

thick-.

ness of frictión belt (BAKER).

In table I which has been extracted from one of BAKER's publications (6), a comparison is made between values of the thickness of friction belt obtained by formulae and by model experiments. The two models (length 21.5 feet) have the same principal dimensIons and identical fore bodies. The frame shape in the aftor'body. was a little abnormal; in the one model it was a pronounced U-shape, and in the other the frame shape arid frame area were the

Distance from p.P (feet) 9 0 12.5 15 0 18.5

Wi-dth of frIctionál belt from

formula (C..- 3). 1.33" 1.81" 2.14" 2.58" Width .of frictional belt from

formuïa (C - 4) 1.10" 1.51" 1.91" 2.92" Width of 'frictional belt from

formula (e - 2) . 2,00" 2.75" 3.11" 3.62" Model 1112. Side . 0.96" 1.6" 2.0" 5.0" Eilge 1.44" 1.5" 2.0" 1.77" Bottom 0.74" 1.27" 1.52" 0.99" Model 1113 Side - 1.61" 1.27" 4.87" Bilge - 1.44" 1.60" 2.23" Bottom - 1.14" 1.76" 1.30"

With the exceptioñ of the thicknesses calculatöd by K!AR-WNts formula, al]. the thicknesses given In the table are calculated to the position, at which the velocity Is one per cent of that of the niodel.

From the tables it appears, that the potential field around the model has a big influence on the friction belt, and the probability Is, that it also influences the

eoeffi-oient of friction (compare (18)).

As it is principally the friction belt, which determi-nes the variation in the wake coefficient with the

propel-ler diameter, a wake summation over the propelpropel-ler disc has been carried out for models as well as for ships. As the

un-certainty in the thickness of belt in ship-forms Is very great, calculations have only been carried out for propel-lers positioned behind plates of different length.

In normal model experiments the length of the model varies very little, as models of 5 to '7 m. are generally used in order to be fairly sure of turbulence. Assuming

that Inside this range proportionality between. belt

thick-ness and model length exits, and using VON 1ARMM's values

fox' belt thickness

d = O.37 L

(V/)1/5

and the expression

y

= V.

for the velocity distribution, It is possible to prepare a diagram as indicated in fig. 4 giving the variation of fric-tional wake coefficient with the diameter length ratio

(volume integration).

Fig. 4. Variation of frictional wake coefficient with the diameter-length ratio (paraffin models, plane

surfa-ces).

1$.

u,1.

3,1$

19

For the actual ships a similar.diagram 'is prepared, but

in' this case ,the relations are hardly s simple, it eing necessary to have regard to the ship length as weil as to the propeller diameter0 As the condition of the surface aries

.t Is also necesaryto take this into áccount. The diagram

fig. 5 has been calculated using _VONthRMAN's formulae for belt thickness and velocity distribution.

FIg. 5. The frictional wake coefficient at different lengths

of ship, propeler diameters, speeds, and vaius of surfáce

roughness (plane surfaces).

ow touse the diagram:

Go in wIth ships length, verti-cally upwards unto the curve corresponding t'o the proper D/L and.V, the horizontally to the linó n-= 1/5, and vertically downwards unto the line corre-sponding to the degree of rough-ness (the n in question), from here horizontally to the ordina-te axis, where 'the frictional

:c0t'13 read off .

CorrectÇon lines, for roughness bave been inserted,'-»the

assump-tion being, that the belt thickness is the áarne for all con-. ditlons of roughness.

_:

.

When the wake ooefficients,deteruiined by model

expón-mente are to be related to the ship condition, thát part of the wake., !hiCh is due to the potential. and wave mOtion, will i

be unchanged, while the frictional wake' will be less. By comparing fig. 4 and fig. 5 an idea' of the, scale efféct on the. wake coefficient is obtained. In a case of a ship 120 rn

in' length ánd a'mode]. 6 ni in length, putting n equal to 1/7,

t'heró will be about O.lo difference In the wake coefficients.

The next figure, fig. 6also gives an idea of the sa1e

ef-fect, rpreseriting the wake coefficient as a'function of the

ratio.: b lt. thickness' to propeller radius.

A

Fig. 6. Variation of the frictional wake coeffici-ent with the ratio between thickness of friction belt and. propeller radius (pia-ne surfaces).

In going from model to ship, the belt thickness

will ut

increase proportionally with the radius and thus the wake coefficient will diminish. It should be observed, that in using VON I!PRMPLNtS formulae a bigger scale effect is expe-rienced than normally reckoned with.

In general the scale effect has previously generlly

been determined under the assumption that a certain rátio

exists between the coefficient of frictionanc the wake coefficient. VAN LAMEEN (63) employed in his dissertation

the following procedure in determining w for the ship: Wake measurement by means of blade wheels was performed for a

series of' models of "Simon Bolivar". The models were made in scales 1/So, 1/36, 1/25, 1/21, arid 1/18, and the wake coefficients were plotted over log R, after which the wake coefficlént at the REYNOLD's number, corroéponding to the ship condition was found by extrapòlation. The extrapola-tion was performed from R = lo.1 . 106 _{to R = 7.31}

. io8 and therefore was exceedingly uncertain. According to VAN

LAEIN

the wake coefficient thus fouñd, ad to be correc-ted further, the extrapolation having beén carried out as if' the ship as well as the model bad had smooth surfaces. This last òbrrection was performed by multiplying the wake coefflclént by the ratio of friction coefficient, corre-.

spondthg to sand roughness of ship, to frIction coefficient córresponding to smooth ship. In the case dealt with, VAN LA!VEREN fòund a diminution of the 'waké coeffIcient of about

VA

21

'0.o5 :by going froin model to ship, thus givth asnualier. reduction thanwould be obtained bynieans ofVON'lkR!ANi's

forinu].ae. '.

TELFER, in his publicatian (i1,4Yha'äuaed,a similar prQcedure, extrapolating over 1/a,. where ) is the. ratio:

- Length of ship or model to .a basis. length. .Again in this

ease the uncertainty is great:, partly due to the extra-: polation and partly because, nothing is known concerning; the relation between the frict.onal resistance, and the.

fric-tiorial, ,wae. :,,. ,' ,,,

Investigations, of the scale effect.have also

beenper-forrne4 'by MPF (5?). From KEMF's results CONN.

.,(21,)pro-duce,s a formula giving the ratio a between the wake, coe'f i-.cient,of ship and,model:

a.= - ß(1

-'.. - (C-6)

ß being the -ratio' of frictional resitanc'e't'o total'

reel-stance. ' ' '

Further:- F '='fr'ictional resistance of ship

f = frictional resistance o-f model

- '

" scale- of model- ' '

'''

«' '

-'= specific gravity of salt

water.-It-also -is of great interest to 'know' how far-the- fllic_

tion- belt extends abaf t the surface, whichgener'ates it',and

how- quickly 'its intnsity din1ni"shes'.- BAKE1 1rns

in'-('5)'de-seribed 'some experiments concerning this. -A 8.5"m-Iong pia-- :.te was towed- thìough the tank a-t a'velocity 'of i,s'm/se'Ó. A

p1tot.tube.:1 -placed abaf t of and 'in the centre 'line-äf the plate, registered the wake coefficients:' -0.4'6',-O'.3-31-or

-0.28 as the-distance of the 'pitot tube-from the after edge

of the plate 'amounted to:' 0.6, 3.8-,or 6.6 peri cent of 'the

-length of -the' plate. - '..

L -

-Also with ,,ship'models, experiments have- been perförmed

in order t-o elucidate the longitudinal» variation of'thewake.

In 19-19 SEMP-LE (-in (99)-)- published 'the" results' of-' a-series

"pro-length .abaft the propeller post, the following wak ceffi-,cients were registered: 0.316, 0.300, 0275, 0.251, ábd

0.226. BA1OER (BYas well as TA!LOR (106 or 1o9) have perfor-med similar experiments and obtained siriilar results. On the whole the experiménts prove, that' the wakè coeffictent of ship models, when the propeller is positioned mor than

twice the thickness of the propeller post from the. post, varies very little iti the longitudinal position of the. propeller.. If the propeller is placed closer tó the prc-pelier post, it' comes ithin á region' of' large wake ai4 ma-ny eddies, and it wiÏl work under. very bad conditionso The-z'e is the, possibility thát the relations for thé actuil

ship are is.. for the model, but ' it is not known with

cer-tainty.

Regarding the influence of speéd on the relation In

the frictIon belt at the after end of the ship and th mo del change with varying speed, practically nothing is known

as very few experiments concerning this have boon earied' out. The author. in his Gtoborg experiments (4) found,

that the wake 'at the propeller bub diminished very quickly'

when the speed of the model Inereasód., while the . wake 1at a

distance from the side'of the 'ship'z'einained practically constant,' only oscillattng up and down'on account of ,he influence of the. stern wave. Whether the waké variation at the 1mb was an actual 'ake.v.ariation 'or was only caused. by. variation' in the degreO of turbulence, and Whether th4.sa-.

.me variation-caù be expcted at the actual ship, are

'stions which must be left 'unanswered.

It appears from th previous remarks, that there are

more' unsólvod than. solved problema,. 'when dealing' with

frictional wake. Very iñtèuse and systematic investigation is therofdre needed in this field. To begin with, the re-lations in the frictton belt of plane Surfaces must be thoroughly in,,,vetigte, ánd then the relations Of curved Surface muSt be examined. For curved surfaces a stárt

ight..

be madò by investigating the flow around solids, of Which

23

the potential field can be calculated, otherwise it will be impossible to determine the form effect on the friction belt. For instance the flow around a series of perfectly inmersed ellipsoids, with varying ratio of the main axes, could be studied. Also the flow around solids of revoluti-on, their potential field being easily calculated, could be investigated. Models having a large draught and vertical sides might further be employed, as the flow around such models would nearly be

two-dimensional.

In the last case it would be necessary to make a correction for the wave motion. Possibly these investigations might be connected to an

examination of the influence of form on the frictional coefficient.

D. POTENTIAL WAKE.

In this section the potential fields around different shaped solids are determined and the potential wake calcu-lated at a number of points at the after edge of the solids. In some cases the wake is summed up over the propeller disc, in other cases the calculation is terminated when the distri-bution of wake is settled. Firstly the flow around. Infinite-ly long cylinders is examined. The cross section of each cy-linder is shaped as the waterlines of ships, and by varying the proportions, an idea of' the dependence of the wake ficient on the ships ratio of length to breadth, block coef-ficient, distribution of displacement, and form of entrance can be gained. Secondly for some of the forms the longitu-dinal variation of wake is examined, and finally three-di-mensiona]. flow is studied, but around solids of' simple form only, such as ellipsoids and solids of revolution. As

cal-culations of the fields around cylinders, having elliptical cross section, have also been performed, the two and three dimensional flows can directly be compared.

with similar fore and after ends (two-dimensional f

_Zow).

Calculations of two-dimensional flow can be. &arrted out in different ways. By means of sources and sinks, or by means of conformal transformations, the field around an arbitrarily shaped solid cari be determined.

In this section the first method has been used, an;d the principles will be briefly described. The method was first developed by RAXI}E Ç89) and, later on further sim-plif ted by D.W. TAYLOR (1o3).

Two infinitely large parallel, planes are p1ace. at an infinitely small distance. The space between the planes is filled with an ideal fluid in which motion can take place without friction.

Through

an infinitesimal aperture in one of the planes, fluid is introduced, creating motion in the fluid between the planes. The streamlines extend from the aperture in the form of rays. A so called

source

has been established. If fluid is drawn away from the space between the planes through the aperture, fluid will move from all directions towards the aperture, forming a sink. If a source as well as a sink are present, a flow fom the source to the sink takes place. CombinIng this flow with a uniform

stream of the same direction as the line connecting th source and the sink, a field has been estabished, tri which motion of fluid from source to sink only takes place in a limited area. If the, boundary curve of this area is 'substi-tuted by an inpenetrable shell, the streamline system will

remain the same inside and outside the shell. By having more sources and sinks and by varying their strength ánd the

ve-locity of the parallel stream, tt is possible to vary tiie form of the shell. By trial and error forms sImilar to wa-terlines of a ship can be obtained.

In the field produced by the source the stream

func-tion ata point P will have the value: = k ,

=. . n

-..(rì-i)

where is the ángle between the X-axIs'and the'line còn-necting the source and the point P, 'and q i the' tre'ngth

of the source, whióh cán be expressed by the velocity of

- flow c at the point P as

q=2ITr

c

r being the distance between the source an4- P.: If there are more sources the stream function at P attains, the value

n n

Z k

n = k1, q Gr1 k1 . . '-: (D-2)

where is à f-aotór of the relative strength fthe.'

sources. Correspònding expressions are obtained in the ease of sinks.

In a parallel field having a velocity y0 parallel to the X-axis the stream function at P will be

.=v0y

y being the distance of the point from th X-axis.

Combining the source-sink field with a parallel field of velocity - v0,v becomes

= k1 S - v0y '' (D-4)

As the stream function of the stream line, 'whidh

divi-des the flow into aninnerandouter flow, isquäI tö"zero,

the sìell is determined by

o = s - v,y " '

.('D-5)

It is now desirable to find the velöcityiri'the'direc-tion òf the X-axis at an arbitrary point, añd for this

pur-pose isused

v=-ki

y0

(D-6)

The constant of the strength of flow is determined' by f ixing.the breadth B of the form and using the equation

for zero flow (fl-5):

k

=La.

SM

where SM is the value of S at the point x = O, y Thus is 'obtained ) 25 (D-3) (D-7) D-8).

w

SM '(D-9)

The calculation of the wake could therefore be erfor-.

med as described in the following.

a,A distribution of sou.rces and sinks is chosen. Th, sour-ces and sinks are placed symmetrically on the X-axis on each side of the origin. Their strength is varied accordixg to some simple curve or other. (It proved unnecessary to wó'rk with omplicated distributions in. order to vary the shape of

the boundary curve).. . . . .

In the first series (series I) a triangu.].ar

distributi-on of strength is used. Calculatidistributi-ons'arecárried out .or ten

different distributions of this type, the length of oze of the cathetei (the extension of the distribution of sources)

in the' right angled triangles bei varied. The strength of sources and sinks is reduced tOwardS' the ends; .(sèe 'fl. 7 in which the distribution curveS are indicated). _'L

In series Ii the areas of strength distribution are isosceles triangles,: and the calculations have been carried outfox' five different 'base lengths. .

In the next series, series. III the areas of strength distribution are réctangular, and here, also five. caSe have been calculated.

In series IV triangles have again been 'used _as

_{19 se_}

nés I, but here the strength inóreases towards the .en1s. Five cases were exained..For 'form no. 5 the distributjon however was slightly different,' being trapezoidal (seefig.

7), ' .

'

. . . ' L.

Series' V comprising four forms is a 'sùpplementary11se

ries, and distributions are 'varied _{as' 8hown in fIg.} 'l'

b. After the diStribution of8ouroes and sInks _{'was setléd,}

tho stream function was detérinód at anumber of points in

the area. On account of symmetry, only points in the ,frst quadrant were considered. The stream

function

_was .calcila-ted at the following points: ' '

Ii VA S

VASI

155

L

áPj4r

v'lui

I..

i-

1t1

tA

uv

I-!W4I5

1IpJ

IiU

y

1,Ij4lrn-

!1ÌU

Fig. 7A. S-y-diagrazns for two-dimensional flow.

Souree-8lnkdlstributiozl

18

1rdioated iñ the

4iagrazn, top right.

ÌN6

uMS

!EiA

1h.

1IA

Irw

IIAM

Fig. 7B. 3-y-diagrams for two-dimen9xa1 flow.

Source-sink distribution is indicated in the diagrams, top right.

-Jff2 Jff-3

F4

[il

-'

riir

!JIIU

ari

k41

IA

i

Ì4fÌ

b*

ilI

iir

Í

iNi iviu

ii= ;w

IWA #i

!t4

iiÍ

iì

AUU

i

!I5

v1

J

iUU

.0

AUU

a _'_

11W F'

,*

L'4I

-

u-4Wd

Lia!

i1_

111W

u

L

'

"

.

t_

J

Iii

j$

u

29

In a few cüe's the stream function was further determined at:

x=llo,

y=0

5

lo

. 2o 30

and, as a control, for sorne forms also at:

x=OO,

y=2.

The stream funoton could be determined by means of (D-2),

TAYLOR's table for angle

being used.

Instead of working with a continuous source-sink

di-stribution lt proves siniplest to perform the calculation

with lo sinks and lo sources, placed according to

TCEEBY-CBEFP'S rule inside the areas. The summation then was easy

to carry out, and the distribution could at will be looked

upon as continuous or discontinuous. As TAYLOR's table could

not be uaed directly, when this procedure was eloyed, a

systeniof influence lines were constructed, i.e. curves,

indi-cating the values of the stream function .in certain points

eorrespònd.ing t.o different positions of a source of strength

1. As the work is performed with symmetrical distribution of

sources ánd sinks lt was an advantage to draw lines of

in-fluerice indlcating the mutual action of a source and a sink

of strength i and situated on the X-axis symmetrically with

regard to the origin.

When the, stream function was determined at the specified

points, diagrams as shown in fig. 7 were drawn, the

abscis-sae being the ordinates of the points of the stream system,

and the ordinates representing the stream function:. The

cur-ves indicáte the variation of the stream funotjon with the

distance from the. X-axis for different values of x

The equation (D-5) depicts in the S-y diagrams straight

lines radl.ating from the zero point, and these were inserted

x =

O,

y =

0 5

lo

15 2o. 3o

X =

20,

y = 0

5

lo

2o

3o

x =

40,,

y

0 5

lo

2o 30

x=.60,

y=O.

x =

70,

y =

0:

5, 5

lo

lo.

2o

2o

30 30

x=80,. .y=0

5

lo

2o

30

x=90,..y=O

5

lo

2o

3o

x=100,

y=0

5

lo

15 2o

30

lea II

Series IV

enes V

Table 2: Waterline coefficients for the cy-linders, around which the flow is

calculated. A:

B/L =

B:

B/L = 0.15

C:

B/L=0.20

D: .B/L= 0.25.

No.

A B C D

1

.895

.850

-

-2

.837

.831

.832

.83].

3

.778

.774

.777

4

.728

.719

.723

.729

5

.664

.664

.670

.677

6

.607

.612

.621

.64

7

.549

.557

.573

.505

8

.497

.510

.533

.551

9

.443

.471

.

.499

.524

1.0

.427

.458

.489

.515

No.

A B C D

1

.866

.863

.855

-2

.777

.776

.777

.784

3

.697

.697

.698

.706

4

.609

.615

.625

.637

5

.536

.551

.568

.585

No.

A B C D

1

.823

.819

-

-2

.744

.746

.750

.756

3

.658

.668

.684

.696

4

.586

.615

.632

.654

5

.579

.603

.625

.648

No.

A B C D

1

.811

.809

.809

.

-2

.760

.764

.767

.773

3 4

.705

.690

.716

.700

.

.725

.710

.735

.722

5 .

.642

.657

.672

.690

No.

.A

B C - D

1

.718

.720

.725

.733

2

.598

.611

.626

.63

3

.527

.

.551

.576

.59

4

.352

.390

.427

.158

Fig. 8. Waterlines for cylinders serles I.

Fig. 9. Waterlines for cylinders serles II.

31 ffA o

4iI

9 ¿4ME/ 6 7 6 5 6 7 8 9 w 5 6 7 8 9 ! fO

Fig0 lo. Waterlines for cylinders series Iii.

Fig. 11. Wcterlines for cylinders series IV.

Z34

670

9

-4 5 6 7 4 9 F6A6 J N-A

234567S

y-B

f Th., 6

-C

1V-D 3 7 6 9

33

Fig. 12. waterlines for cylinders series V.

for the following B-values: B =20, 30, 4o, and So, corre-sponding to breadth-length ratios B/L = 0. lo, 0. 15, 0.20, and. 0.25, thus covering norma]. ships ratios. The points of inter-section between these lines and the curves give the ordinates of the boundary curves and thus the ordinates for the water-line shapes. In figs. 8 to 12 the waterwater-lines for the diffe-rent shapes are indicated and in table 2 the waterline coef-ficients are given.

e. Finally the wake coefficient and the variations of wake at the after end of each waterline were deterzaineci. The wáke coefficient was calculated at the points:

x=loo,

7=0

5 lo 15 2o

using the foxmta (D-9)

w_z.

The derivative _{/y was determined graphioa].ly,and 8M} cal-culated as explained above corresponding to selected values af B/2. The results aro recorded in tables 3 to 7.

6 9

9 FME

Table The potential wake coefficient of cylinders seriés

I' at frame 0. L

Series -'No.

Distance from centre line

O

25

5 10 _i5 20

1

.506 .324 .227 .126 .077 .036 .316 !265 .217 .135

.O5

.057 '3 .217 .182 . .154 .117 .086 .060 4 .156 .138 .122 .098 .077. .057 T A 5 .120' .110 .100 .083 .067 .05]. 6 .095 .088 .082 .070 .058 .047

7

.Q80 .075 .071 .060 .050 ... .041 . .068 .064 060 .052 .045 .037 9 .59. .057 .054 .047. .040 .034 IO .055 '.053 .050 .044 .037 .032 1 .788 .514 .357 ' .196 120 . .055 2 .494 .414 .338 .210 .133 .090 .3 .340 .286 .242 .184 .134 .094' 4 .248 .218 _{.192 .153} .120 .090 T 5 .189 .174 . .159 .128 .104 .08]. 6 .152 .141 .130 .110 .094 .074 7 .128 .120 .112 .096 .082 .066 8 . .110 . .104 ' .098 .0134 .072 .060 9 ' .099 .094 .088 .077 .066 .057 10 .094 .od . .083 .072 .063 .053 .1 . . - - - -.689 .576 4.7l " .294 .187 ' .124 ' .474.' .396 V.336 .257. .190 . .130 4 .347 .305 .269 .214 . .169 H .124 '.267

.246

' .224 ' .182 " .148 114 -, 6 , 7 .216 .183 .200. .172 .185 .162: .. .158 .138 .133 ' .116 . .106 .095 '8 .160 ' .151. .142 .Ï23 .104 .088 9 .147 138 ₁₃₀ 3.15 099 083 10 ' .140' ' .132' .124' '.109, .0,95 .080

1.

'' 2'"92

-.

,-

.,..-.746.' .609 .3132 -S .243 ' -f162 .3'. 1b19 . .5l,7 .438. 'p335 '.245

470

T -fl 4, 5 .455 .352 .402 .323 ' .35,5 .296' .283 ' .242 ..223 .194' ' .166 .150 6 .286 .265 .245 .209 .116 . '.141 7 .246 .232 .23.8 .188 .158 .121 8: .217 .205 _.193 '.16.8: .142 '' ' .119 9 .200 . .190 .179. .157 .135 ..Ï14 10 '.3.93 .1139 171 '.149 .131 .110

35

Table 4: The potential wake coefficient of cylinders series

II at frame 0.

It should be noted, that if a source, however small, is positioned at .(loo,o), the wake coefficient will, be equal to 1, even if the graphical determination results in a

diffe-rent value..

Imagining a propeller situated behind the cylindrically shaped body, the average wake coefficient can be found by' performing a volume integration over the prôpeller disc:

21r

_w'

r"Vl

-

(/r)2

a dy

(D-10)

r2 r being the propeller radius.

The integration was carried out graphically for series I, and the result is given in figure 13.

Series No.

Distance from centre line

0 2.5 5 10 15 20 1 .450 .334 .249 .138 .080 .047 2 .214 .187 .160 .116

.06

.061 -II A 4 .141 .103 .126 .093 .113 .084 .091 .072 .069 .058 .053 .047 5 .082 .076 .070 .057 .046 _.038 1 .701 .520 .387 .216 .126 _.073 2 _.335 .292 .252 .182 .135 .095

lIB

3 .222 .200 .179 .143 .109 .083 4 .165 .150 .135 .113 .092 .074 5 .132 .124 .114 .092 .072 .059 1 .971 .721 .537 .298 .172 .101 2 .470 .408 .352 .255 .188

13

II-C 3 .312 .280 .251 .201 .155 .11 4 _.234 .212 .193 .161 .13]. .105 5 .192 .178 .163 .135 .106 .086 I, 2 -.611 -.532 -.457 -.334 -.246 -.173 II-D ₃ .412 .372 .334 .263 _{.203 .155} 4 _.312 .284 .257 .214 .174 .141 5 .257 .239 .221 .180 .141 .115

Table 5: The potential wake coefficient of cylinders

enes

.III at frame 0..

Here the

propeller diameter bas

been chosen as 0o4 L (D is

in relation to L on account of

the

friction belt, see

'sec-tion C)..

From

the diagram it là clearly ieen, that not ónly,

as generally assumed, the fullness. ot the ship, but also the breadth-length ratio, play

a primary part

aa regards

he

wake coefficient.

Tbeaverágeerror of

the wake Coefficient found will be

about 0,003,. For full and broad

forma

it will be aomewht

bigger, while forfine

and narrow forma

it will be

lesa. The

maxImum error of the

wake coefftcleiit is found at the 'sur-face of

the form, as the S-y curve

here bas moat

curvature.

As it would have been

an advantagé in

the

investigation

Series No.

Dlétance from centre line

0 2.5 5 10 15 20 r ,

.2

.286390 .272 .198 .195 .143 .114 .090 .080. .069 51-.L .049 III-A 3' .204 .153 .116 .078 .060 .046 4 .162 128 - .101 .069 .052 .043 5 .157 .123 .096 .064. .050 .040 1 .618 .426 .307 .177 .124 .081 2 .449 .310 .225 .142 .105 .076 III-B 3 .325 .241 .184 '.124 .096 .073 4 .264 .210 . .164 .Ï12 .086 .069 5 .257 .202 .157 .1O4 .081 '. .065 1 - - - -2 .628 .421 .314 .201

:15

III-c . 3 .461 .338 .262 .176 .136. .104 4 .380 .302 .236 .161 : .125 .099 5 .372 .290 .227

.1%

.117 .095

i.- -.. --

-2 .824 .547 .413 .263 . .196 .144 III-D . ' .615 .449 .349. .234 .180 .138 4 .. .512 .402 . .319 .216 .167 .134 5 .501 .359 ' .306 .203 .158 '.131..

37

Table 6: The potential wake coefficient of cylinders 8eriea

IV at frame 0.

rig. 13. Variatión'of potential wake coeffici.

ont with fullness aM

breadth-length ratió of the cylinders (series I).

Distance from centre line

Series No. o .

2.5

5

10

15

20

1

.411.

.280.

.196

.103

.070

.049

2

.35

.227

.157

.093

.067

.047

IV-A

3

.27

.190

.137

.085

.064

.046 4

.254

.182

.133

.08].

.06].

.045

5

.223

.155

.112

.073

.056

.043

1

.641

.434

.305

.160

.109

.076

2

.527

.350

.247

.147

.107

.075

IV-B

3

.424

.301

.218

134

.100

.072

4

.402

.290

.213

.129

.098

.072

5

.350

.239

.174.

p117

.092

.069

1.

.894

.627

.425

.224

.153

.105

2

_.754

.497

0345

.205

.149

.105

IV-C

3

.598

.417

.310

.189

.140

.102

4

.570

.405

.301

.1d

.137

.102

5

.500

.345

.257

.16e

.130

.098

1

-

-

-

-

-

-2

.990

658

.452

.270

.195

.136

IV-D

3

.791

.550

.404

.253

.186

.135

4

.754

.523

.390

.242

.181

.134

5.

.668

.462

.349

.224

.173

.132

_

w

1j

Táble 7: 'The potential wake coeffiient of cylinders series

'V. at frame 0.

to be able to'etart from a 'fixed waterline, it has been'.

examined whether this was possible. Purely theoretically the corresponding source-sink distribution could be found fox' an arbitrary waterline, but in practice the difficulties are

large, as lt Is necessary to solve extensive systems of equa-tions. In such calculations It is necessary to begin with the equation for the boundary curve (D-5), and, by means of

this, find the corresponding values x, y, and S. The positi-. on of the sources and sinks Is settled and their strengths

introduced as unknowns. As the angles between the X-axis and the lines connecting the source and sinks to

pointmon

the boundary curve can be determined by irnp3,egeòmetDy, í'ay-,

stem of equations can be produced. It appears tobe

neeessa-ry to introduce many unknowns in order to secure a fairly certain detèrmination of the velocity, and the resulting calculation, work becomes very complicated.

If a conformal transformation is to be used In the wake Series No.

Distance from centre uñe'

' 2.5 5 10 15 '20 1 .15]. _.153 .128 ' '.092. .07]. .051' 2' .120 .106 .092 .070 '.056 .044 3 .092 .084

.075

.058 .047 .038 4. .042 .040 .038 .034 .030 .027 1 .286' .241 .202 , .146 .112 .082 2 _.193 .170 .145 .113. .090 069 3 .151 .137 .123 .097 .078 .062 4 . .074 .070 .066 .059 '.054 .047 1' _{.40Q .335} .283 .206 .156 .11 'v-c' 2 3 .276 . .220 '. . .242 .199 .210 .178 .162 ' .140. .129 '.115 . .09 .091 4 .112 .106 _{.101 :} .090, .081 .071 1 2 .525' .370 .441 ' .324 .372 .282 .267, .216 ' .204. ' .170 ' .150 .132 3 .300 .272 .243 .191 .154 .122 - _.4 .158 .150 .142 . .127 .115 « . .100

39

determination, the pxloceduxe is as follows. A stream system aroimd a circular cylinder is considered and transformed in-to a stream system around a solid of more ship like form, As transformation formula an expression of the form

z z z

can be used. In the present work this procedure has only been employed in solving special problems.

al The _{otentia]. wake coefficient's dependejç on} the angle of inclination of waterline to centre line aft,

In order t _{examine this question, the results from the} different series were compared, the waterline half breadths on frame 2 being divided by twice the frame distance (tan a1 in fig. 18) and. used as a measure of the angle of inclina-tion. Actually the tangent angle at the A.P. ought to have been Chosen as a parameter, but this angle is difficult _to define, as it only appears to differ from v,2,

The comparison was made for mean wake values detrviix3id by integration of the wake over a propeller disc of 0.04 L

diameter. To reduce the work the wake variation was taken _as being linear. The average value of the wake was determined by the formula

Waverg

=

-

( W05

_Wmi)

which was established in the following way. Denoting the maximum wake coefficient at the centre of the propeller axis

by

= a

and the minimum at the propeller tip by = b

the value of the wake coefficient w at a distance y from the axis became on account of the linear variation

w = a - ( a - b ) y

w

Wa_.

_1j4D

or

a- b

= a-0.42(a-b) which is identical with the formula given above.

In the figures 14, 15, 16,and 17 he wake curves

for

the different series are represented

and in figures 18, 19,

Lo, and .21 the curves for the newly defined

parameter as a

function of the waterline coefficient.

Fig.14. as function of fullness (cyllrideri). Fig. 15. as function of full-nasa (cylinders). Wp .40 j.îo YL..04

N___

.50 .60 7Ò .80 Wp 04.04

IV

40 .30 .20 JV .50 .60 80

Fig. 16. as function of full-ness (cylinders). Fig. 17. w as fune-tian of fu].ness (cylinders). 41

e'

jz/;.

80 wp

4.25

70

I EI

.60

.s.

30 .20 .10 -50 -60 80

Fig. 18. The angle of inclination of waterline to centre

lirio as function of

the fullness (cylin-deve).

Fig. 19. The angle of inclination of water-line to centre water-line as function of the fullness (cylinderS).

Fig. 20. The angle of inclination of water-line to centre water-line as function of the fullness (cylinders). -40 41=-io 30 FEO

-c

2 20

p.

.10 .80 .60 -50

4

30

j,

20 -10 -50 60 -80 .70 .60 .50

1

'0

50 -70 0

Fig. 21. The angle of inolination of water-line to centre water-line as function of the full-ness (cylinders).

Fig. 22. The poten-tial wake coeffici-ent as function of angle of inclination of waterline to cen-tre line (cylinders).

rig. 23. The

poten-tial wake

coeffici-ent as function of angle of

inclination

of waterline to cen-tre line (cylinders).

43 J7fl .60

iii

60 .60 .40 .30 60 °/L..04 o .20

JD

.10 8 .10 .20 30 40

tan ,

.50 IVp .50 ₄ .40 .70 °/L04

AT0

30 .20 8 .10

/4

.10 .20 .30 .1,0 tan _, .50

cated as a function of the angle parameter at

ffle588 of

waterline cz= 0.6o arid 0.7o respectively.

It willbe observed that the percentage variation for all

tha breadth to length ratios is pr

tcftll

tha same, and

that the vriation for the convex forma is far bigger tban

for the concave ones. As the waterlines as a rule are of the concave type in normal shi.ps, it seems probable, that the angle of waterline is only of secondary importance.

Further it should be noted that the wake coefficient is proportional to the angle of' waterline at small waterline

coefficients.

a20

The variation of the otential wake coefficient

with the diameter of the propeller,

For the forms 2, 4, 5, and 8 in the series I.A, I-B, I-C, and I.rD a snmmRtlon of wake over the propeller disc bas been performed for different propeller diameters. The a1mma

tion was made as a volume si,mm'tion (formula (D-10)) graphi-cally by mean5 of a planimeter, and the wake coefficient was calculated for propeller diameters equal to 0.03, 0.04, 0.05, and O,o6 times the length of the forma. The horizontal distri-bution of wake

ae taken

as the same over the whole of the propeller disg

Fig. 24, The variation of potential wake with the propeller diameter, IVp SER/ES Z-A -.02 r'..5O -.04

I-c

-.02 -44 .03 .04 .05 /L

Z-B

-In fig. 24 the results are given as

the deviation of thu

potential wake coefficient from that of this 0.o4 propeller plotted over the ratio D/L. As expected, the biggest wake co-efficients are found at the smallest propeller diameters, but the variation of w with D/L at normal forma is very small.

In the sta,m,tption no regard has been taken to the

propel-1er hub, as its Influence, on account of the comparatively small variation of the wake coefficient, is of minor impor-tance.

a3. The longitudinal variation of

potenl wake,

coeff te tent

For the forma I-3, I5, III-2, and III-4 the wake bas been calculated for all the tour breadth-length ratios at

frame -1, 0, +1, and +2. For these four forms the waterline coefficients are about 0.78, 0.66, 0.75, and 0.62 respec-tively, and the

waterlines are concave (Sshaped) for the

first two forms, while convex for the last two. The

óalcu-lattons have been carried out in the sane way as previously, determining the slope of the y-curves. The results are found

in tables 8 and 9, and for the series A

and B some of the

results are graphically represented in

fig0

25.

Fig. 25. The longitudjn variation of the potential

wake COOffjeent. 45 .15 3ER/E5A

Q5

-, O (FRAME 'Z" wi,, _SEP/ES .25

û

à,.5 .20

\

-, O 1ff-2" (FRAIE 2

Distance from C.L. Distance from C.L. 0 5 10 15 20 I-B-3 0 5 10 15 20 (-.109) (-.004) .007 -.016 -.027 FR. 2 (-.iii) (-.Qo6) .01l'-.025 -.035 ( .288) .176 .100 .060 .038 i ( .452) .276 .157 .093 .057 .217 .154 .117 .086 .060 " 0 .340 .242 .184 .134 .094 .105 .100 .087 .075 .062 " -1 .165 .156 .137 .117 .097 Distance from C.L. Distance from C.L. I-D-3 0 5 10 15 20 0 5 10 15 20 (-.238) (-.008) .015 -.036 -.049 R. 2 (-.311) (-.oii) .020 -.046 -.064 C .631) ( .385) .219 .130 .080 " i ( .824) C .503) .286 .170 .105 .474 .336 .257 .190 .130 " 0 .619 .438 .335 .245 .170 .230 .218 .19). .163 .135 " -1 .300 .285 .249 .214 .177 Distance from C.L. I-B-5 Distance from C.L. 0 5 10 15 20 0 5 10 15 20 (.125) .107 .071 .048 .029 FR. 2 (.197) .168 .112 .076 .046 (.138) .120 .101 .080 .055 1 .218 .189 .159 .125 .086 .120 .100 .083 .067 .051 " 0 .189 .159 .128 .104 .08]. .070 .067 .062 .055 .048 " -1 .110 .106 .097 .087 .075 Distance from C.L. Distance from C.L. I-D-5 0 5 10 15 20 0 5 10 Ï5 20 (.278) (.238) .158 .107 .065 FR. 2 (.367) (.314) .209 .141 .Ó86 (.oB) (.267) .224 i78 .122.. " 1 (.406) (.352) .296 .235 .161 .267 .224 .182 .148 .114

"iO

.352 .296 .242 .197 .150 .155 .149 .37 .123 .106 ." -1 .205 .197 .181 .162 .140