Merchant Ships

WAKEOF MERCHANT SHIPS

B Y SVEND AAGE HARVALD

THE DANISH TECHNICALPRESS COPENHAGEN 1950 ..'M . I

/ ;4 i) .; ,# 3..

- ' f- f .- t ':. -- . , f ç.. - t : . '.- ,..

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. PREFACE.

It is the hope.of the author that this treatise will be a help and guidance to the designer in the shipyard as 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 fora. brief discussion, The principal part of the, work connected with this treatise has been carried out at the shipbuilding department at the DANMARXSTEKNIS HØJSKOLE, Køberthavn.The. leader of this department ProfessorC.. 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 hada whole series of experiments performed and in Holland statistical 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,superintendent. During this same period the HUGOHAMMRS FOND FR IN- TERNATIONELL FORSKNING INOM SJFARTENsupplied 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 solely connected with investigations of wake. The staff of the tank was of great help and support to the author, especially Mr. J.D. VAN MMEN, M.Sc., with whom many probleme were 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. CONTENTS. page

Introduction 5 CHAPTER_I DEFINITIONS 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

coefficient of twin screw ships - 105 4. H. The wake coefficient of special types of ships 112 Fishing vessels 112 . .. .

Tug 'coats . 115 ships inhioh 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 coefficients 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 207 210 Synthols and units - . ..

Bibliography . . .' 213 5.

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 propeller disc, in order more easy to ascertain the actual conditions 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 propeller 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 analysis 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. 70

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 solely 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:

'F. w = and w =

By potential wake is understood. the wake obtained 1f the ship movedn 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 theoryfOrtrochoidalwaves,. inordertóexplain phenomena connected with wave.makirig, Although this thèory is riot fully correct,frozn'apurely physIcal point of view, as It assumes originationofrotationmafrictIonles fluid, it gives a good pr.ctica]. explanatIon of thewave. 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 changeInwake 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 frictional 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 therefore 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 coefficientsWT and are obtained, determined respectively by thrust identity(FRÓUDE's WAKE OP )ERC!ANT.SK±PS BY SVEBD AAGB RVALD ERRATA Line 31: Pôr "month" read "months". p.2. Line3: Por "has" read "have". Line7: Por "Petersen" read P..2. Line 22: Por "tanks" read "thanks". p.8. Line 28; After "the"ineerteffective". plo. Line 26: por "size" read "sizes". p.. 16. Line lo: Por "alearance" read "clearance". P. 17. Line5: 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". P23. tine 22: Por "the waterlines of ships" read "a water- - line of a-ship"- -P. 45. Line5: Por "w" read "w".- P. 48. Line 19: Por "longitudial" read "longitudinal". p. 49. Line8: Por "was" read "were", '. 54.- Lines32 and 34: Por "ellipseide"- read "ellipsoid". P. 61. Line7: Por "ellipsoide" reed "ellipsoid". p. 62. Line4: Por "ratio" read "ratios". P. 7e. Line 20: Por "exist" aced "existo". P. 72. Line 7:For "recolutjon" read "revolution". P. 76. Line3: Por "ciefficlents" read "coefficients". P.. 77. Line 9 Por "give" read "gives". - Line 3e: After "in" Insert "each". p, 82. Line2: Por "O.005" read "0.010". LinS4: For ."0.005 to 0.älo"reed '0.010 to 0.o2o". P. 82. Line i8Por "liminations" read "limitations". .p83. FIgs.41 and 42:. Por ".01" and, ".o2read ".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. Liño3: 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: Lthe7: For."sltght" read "Blightly". P.107. Line 21: For "play" read "plays". Line F.lo9 8: For "submarins"read "submarines". . . - p.ïu. LIne6: 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". Line2: For "deminish" read "diminish". P.126. Line6: After "TAYLOR" insert n,". P.128. Line 29: Por "were" readwao". P.132. Liiie 13: After "Dj±tai» Insert "(24)". LIne1: 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". Line3: For "draught" read"B/d". F.l45 Line8: 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. Line1: After "87" ins±"t ".". P.162. Line 15: After "a tb" Insert "tfig. 86)". P 166Line 34For "conditionsreadtrim or draught con- - dItjo". - P.167. Line 13: Por "chip"read"ehip". P.171. Line 11: Por "resultá" read'results". P.175. Line8: Por "ail-. designinguiposes" read "design poses" - pur- 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-.22Por "miseren" read"mjsêren". - P.208. Line 36: Por "fuldstndjgt»read "fuldotdig" P.210. Line7: For "will be" reád "have been". P.215. Line 19: After "wake" insert ". -. P.219. Line li: Por "Edinburg" read "Edinburgh" P.22o. Line 19: Por "memoirea" read"Bulletin". 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 VITISmostly 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 somewhat illogical proof as the reason why wshould 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 numbers 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 therefore 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, whereFr 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 toCT and CQ. and therefore S'CHOENHERR is of the opinion that this should be used. In practise Cr + k c should be used, k T T being a coefficient depending on the dstribut1on of pressure on the propeller blade and having the value 8.16 to'9.

- But no matter which procedure is used, the wake coefficient found ialways very uncertain, th9 reason being lo the coziditionsof a homogeneous field are compared t. those 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 fieldshould 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 measurement, 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 -w) d or an Impulse integration WI ( 1 (l-w)°A 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))\Itneed: only be mentioned here that in normalpracticebla4e iheels of different size are used In determining thenominaliwake 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 more indication of the distributionf wake. Eut as tIey 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 nominal wake coefficient 'will bè closer examined. 12

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 relation of the ship resistance. as a whOle very few systematic investigations of the frictional wake have been carried out, indeed so few, that they onlyinbroad outlines clear up the problems. When a ship moves through the water, the water particles will, as it were, adhere to the ship. The water particles, right at the side of the ship, will have the same velocity 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 influence on the wake coefficient, this influence will be examined a little closer in the following. The first questions which arise are: what is the thickness 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) Vbeing 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 results, 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 thickness as extending to the position where the velocity is one per cent of the velocity of the surface. 14 In the diagrams fig. I and 2 VON 1ARAN's andAKER's curves are drawn for different velocities, and it will be noted, that the difference between the two set of curves is large.

O MODELS

f5 -- _/v., EIP400rn6 -, - 2/ Fig. I. Thickness of 57005M friction belt, determined by VON K'ARMN's and BAKER's formulae ("model")

8A000

2 4 IÓ 8 /0,,,

d vm SH/PS

v-5

Fig. 2. Thickness of friction belt, determined by VON K!kRN's and BAKER'S formulae ("ship").

5, , , 250

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..bKEMPF 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 surface;was varied. It appeared that a change in roughness partly: resulted in another belt thickness and partly 15

another distribution of velocities. Expressingthe. velocity distributions by y = V. (7/d)' nbeing varied with the condition of surface,the formula and the experimental resultsagree very well. The exponent n will assume the following values: n .= 1/9. VarniShed and polished iron surface(KEMPF'Spontoon experiments, R = 4.13 .108). Channel-steamer (s/s Snaefeil), good paintand

smòoth.butts (BAKER). . . n = 1/7. Merchant steamer (s/s Ashworth) surfaceçiean, but paint old, overlapped butts, 108 dayssiñce:last docking (BAKER, R = 4.77 . 108). ri=2/11.Model experiments; dealingwith smooth polished surfaces n can be put equal to thisvalue. n =1/5. Merchant steamer (S/S Ashworth), 25odays since last docking, surface covered withlóng'ass (BAKER, R 4.77 . Fig. 3 gives an idea of the velocityvariation with. different values of n. The velocity isgiven as a functión of the ratio between the distance from thesurface 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.

KEMPF's pontôon experimentsfurther.proved,.that as n increased frörn 1/9 to 1/7, the beltthickness Increased about lo per cent, and also that thedistribution of surface roughness had a certain influenceon the conditions in the belt. Often it is possibleona rough surface to i 6

make it re or less rough in certain partswitbou any. change taking place in the relation of thefriction belt. A niore detailed account is found in theworks of 1MPF and BAR (53, 54, 5, 6, 7,and io). he formulae mentioned aovo aré only valid for f low alóng plane surfaces. Indealingwith ship forms, they may wIth góod appr6ximat.on be used for the flow aroúnd the fore body, while their application to flow arround theafter bo- dyls problematical. Nevertheles8 these formulae arepracti- cally always used, when for instance the tipalearan.ce of the pipellers iia twin srw ship is to bedecided. BAKÊR (7, and io) is one of the few investigators, who have attempted to ±ind an epressio for the variation of te è1 thicegs 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 thickness 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 Gbeing the wet girth of frame(ni) andCx)respectively, and d.and dare 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, Gmay roughly be put equal to 2 G. Thus1:s ob- taIned: d=d +d (c-3) and as dm as a maximum can rea..1i .í, it should neverbe possible to obtain a belt thickness bigger t$n 2Ç;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 surface, having Itsdirectrix in the direction of flow, it will 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. 'ibfig. 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.

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"

Table. 1: Comparison between calculated and measuredthick-. 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 18 saine, but the frame s were turned through an angle of90 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 determines the variation in the wake coefficient with the propeller diameter, a wake summation over the propeller disc has been carried out for models as well as for ships. As the uncertainty in the thickness of belt in ship-forms Is very great, calculations have only been carried out for propellers 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 thickness 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 frictional wake coefficient with the diameter length ratio (volume integration).

1$.

Fig. 4. Variation of frictional wake coefficient with the diameter-length ratio (paraffin models, plane surfa- u,1. ces). 3,1$

.03 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 usingVONthRMAN'sformulae 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). A ow touse the diagram: Go in wIth ships length, vertically 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 corresponding to the degree of rough- PiT ness (the n in question), from here horizontally to the ordinate axis, where 'the frictional :c0t'13 read off .

CorrectÇon lines, for roughness bave been inserted,'-»theassump- 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 120rn 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 effect, rpreseriting the wake coefficient as a'function of the ratio.: b lt. thickness' to propeller radius. 20

Fig. 6. Variation of the frictional wake coefficient with the ratio between thickness of friction belt and. propeller radius (piane surfaces). IMOD(CS VA

In going from model to ship, the belt thicknesswill 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-

. 106 tion was performed from R = lo.1 to R = 7.31 . io8 and therefore was exceedingly uncertain. According to VAN LAEINthe wake coefficient thus fouñd, ad to be corrected 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!VERENfòund a diminution of the 'waké coeffIcient of about 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,)produce,s a formula giving the ratio a between the wake, coe'f i- .cient,of ship and,model:

- ß(1 '.. - (C-6) a.= -

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 havebeen perförmed in order t-o elucidate the longitudinal» variation of'thewake. In 19-19 SEMP-LE (-in(99)-)- published 'the" results' of-' a-series of experiments-carriedout with' a '5-m mode-1.-with-the "pro- 22 poIler positionedO.5, 1,2, 3, and4 per cent of the model 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 performed similar experiments and obtained siriilar results. On the whole the experiménts prove, that' the wakè coeffictent of ship models, when the propeller is positioned morthan 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 many 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 certainty. 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 Whetherth4.sa-. .me variation-caù be expcted at the actual ship, are 'stions which must be left 'unanswered. It appears from thprevious 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 relations 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 stártight.. 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 revolution, 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 betwo-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 calculated 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 distribution of wake is settled. Firstly the flow around. Infinite- ly long cylinders is examined. The cross section of each cylinder is shaped as the waterlines of ships, and by varying the proportions, an idea of' the dependence of the wake coefficient on the ships ratio of length to breadth, block coefficient, distribution of displacement, and form of entrance can be gained. Secondly for some of the forms the longitudinal variation of wake is examined, and finally three-dimensiona]. flow is studied, but around solids of' simple form only, such as ellipsoids and solids of revolution. As calculations of the fields around cylinders, having elliptical cross section, have also been performed, the two and three dimensional flows can directly be compared. 2L

a. Potential wake of cylindrically shaped bodies with similar fore and after ends (two-dimensional fZow). 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.Throughan 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 calledsourcehas 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 'substituted by an inpenetrable shell, the streamline system will remain the same inside and outside the shell. By havingmore sources and sinks and by varying their strength ánd the velocity of the parallel stream, tt is possible to vary tiie form of the shell. By trial and error forms sImilar to waterlines of a ship can be obtained. In the field produced by the source the stream function ata point P will have the value:

= k , ..(rì-i) =. . n - 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 25

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

- flow cat the point P as q=2ITr c rbeing the distance between the source an4- P.: If there are more sources the stream function at P attains, the value n n

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

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 velocityy0parallel to the X-axis the stream function at P will be .=v0y (D-3) 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)

Asthe stream function of the stream line, 'whidh divides 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'direction òf the X-axis at an arbitrary point, añd for this purpose 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. (D-7) SM where SM is the value of S at the point x = O, y Thus is 'obtained

) D-8). 26

and. time the wake coefficient becomes

w '(D-9) SM

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, sources 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 distribution of strength is used. Calculations'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 'usedas19 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 formsis a 'sùpplementary11se ries, and distributions are 'variedas' 8hown in fIg. 'l' b. After the diStribution of8ouroes and sInks'was setléd, tho stream function was detérinód at anumberof points in the area. On account of symmetry, only points inthe ,frst quadrant were considered. The streamfunctionwas .calcila-

ted at the following points: ' ' L áPj4r Ii VA S v'lui 1t1 tA uMS!EiAÌN6 1h.IrwIIAM 1IA VASI155 uv y ÀW' i-1IpJ wh I.. !W4I5 1,Ij4lrn- Fig.Souree-8lnkdlstributiozl 7A. S-y-diagrazns for two-dimensional flow. IiU !1ÌU 18 1rdioated iñ the i-. 4iagrazn, top right. I- Jff2 Jff-3- [il ' k41 IA F4 ari ilI riir iUUiir ii=iNi!I5 ;wiviu v1 a _'_ JiIWAÌ4fÌ #i 11W F' -!t4 AUUb* AUUi !JIIU ,,1jpp.iIÍ j$ i1_111W .0 ,* L'4IJ iiÍ . iìLia!Luu- 4Wd Fig. 7B. 3-y-diagrams for two-dimen9xa1u flow. Source- " Iii sink distribution is indicated in the diagrams, top right. ' -t_ 29 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. 5, lo 2o 30 x =70, y =0: 5 lo. 2o 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 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 distribution 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, indicating 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 curves 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 30

Series I

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

lea II 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

Series IV No. A B C D

1 .811 .809 .809 . - 2 .760 .764 .767 .773 3 .705 .716 .725 .735 4 .690 .700 . .710 .722 5 . .642 .657 .672 .690

enes V 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 Table 2: Waterline coefficients for the cylinders, around which the flow is calculated. A: B/L = B:B/L = 0.15 C: B/L=0.20D: .B/L= 0.25. 31

Fig. 8. Waterlines for cylinders serles I.

ffA o 9 ¿4ME/

4iI 6 7 6

5 6 7 8 9 w

5 6 7 8 9 ! fO

Fig. 9. Waterlines for cylinders serles II. 32

Z34 670 9 -

4 5 6 7 4 9 F6A6 J Fig0 lo. Waterlines for cylinders series Iii.

234567S N-A

y-B f 6

Th., -C

1V-D

3 7 6 9

Fig. 11. Wcterlines for cylinders series IV. 33

6 9

9 FME

5 .6 7 6 9 6

Fig. 12. waterlines for cylinders series V. for the following B-values: B =20, 30, 4o, and So,corresponding to breadth-length ratios B/L = 0. lo, 0. 15, 0.20,and. 0.25, thus covering norma]. ships ratios. The points of intersection between these lines and the curves give the ordinates of the boundary curves and thus the ordinates forthe waterline shapes. In figs. 8 to 12 the waterlines for the different shapes are indicated and in table 2 the waterlinecoef- 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,and8M calculated as explained above corresponding to selected values af B/2. The results aro recorded in tables 3 to 7. 34

Distance from centre line Series -'No. 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

.788 ' 1 .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 , .216 .200. .185 .. .158 .133 .106

7 .183 .172 .162: .138 ' .116 .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. ,- .,..- -S ' - '' 2'"92 .746.' .609 .3132 .243 f162

.3'. 1b19 . .5l,7 .438. 'p335 '.245 470

4, .455 .402 .35,5 .283 ..223 ' .166

' T-fl 5 .352 .323 .296' ' .242 .194' .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

Table The potential wake coefficient of cylinders seriés I' at frame 0. L 35

Distance from centre line Series No. 0 2.5 5 10 15 20

1 .450 .334 .249 .138 .080 .047 2 .214 .187 .160 .116 .06 .061 II A .141 .126 .113 .091 .069 .053 - 4 .103 .093 .084 .072 .058 .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 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 different 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)2 a dy r"Vl (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. 36

Dlétance from centre line Series No. 0 2.5 5 10 15 20

r , 390 .272 .195 .114 .080. 51- .2 .286 .198 .143 .090 .069 .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 .136. III-c . 3 .461 .338 .262 .176 .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 ' .180 . .615 .449 .349. .234 .138

4 .. .512 .402 . .319 .216 .167 .134

5 .501 .359 ' .306 .203 .158 '.131..

Table 5: The potential wake coefficient of cylinders enes .III at frame 0..

Here thepropeller diameter basbeen chosen as 0o4 L (D is in relation to L on account ofthefriction belt, see 'section C)..Fromthe diagram it là clearly ieen, that not ónly, as generally assumed, the fullness. ot the ship, but also the breadth-length ratio, playa primary partaa regardshe wake coefficient. Tbeaverágeerror ofthe wake Coefficient found will be about 0,003,. For full and broadformait will be aomewht bigger, while forfineand narrow formait will belesa. The maxImum error of thewake coefftcleiit is found at the 'surface ofthe form, as the S-y curvehere bas moatcurvature. As it would have beenan advantagé intheinvestigation 37

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

Table 6: The potential wake coefficient of cylinders 8eriea IV at frame 0.

rig. 13. Variatión'of _ w potential wake coeffici. ont with fullness aM breadth-length ratió of the cylinders (series I). 1j 38

Distance from centre uñe'

Series No. ' 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

' 2 .276 '. . .242 .210 .162 .129 .09 'v-c' 3 . .220 .199 .178 .140. '.115 . .091 4 .112 .106 .101 : .090, .081 .071

1 .525' .441 ' .372 .267, .204. ' .150

2 .370 .324 .282 .216 ' ' .170 .132 3 .300 .272 .243 .191 .154 .122

- . .4 .158 .150 .142 . .127 .115 « .100

Táble 7: 'The potential wake coeffiient of cylinders series 'V. at frame 0.

to be able to'etart from a 'fixed waterline, it hasbeen'. 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 equations. 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 theirstrengths introduced as unknowns. As the angles between the X-axis and the lines connecting the source and sinks topointmonthe 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 afairly certain detèrmination of the velocity, and the resulting calculation, work becomes very complicated. If a conformal transformation is to be used In the wake 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 waterlineto centre line aft, In order t examine this question, the resultsfrom the different series were compared, thewaterline half breadths on frame 2 being divided by twice theframe distance (tan a1 in fig. 18) and. used asa measure of the angle of inclination. Actually the tangent angle at the A.P.ought to have been Chosen as a parameter, but this angle is difficultto 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 discof 0.04 L diameter. To reduce the work the wake variation was takenas 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 distancey from the axis became on account of the linear variation w = a -( a - b ) y Assuming no vertical variation of the wake, theaverage wake 40

yalta could be expressed as:

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 17he wake curvesfor the different series are representedand in figures 18, 19, Lo, and .21 the curves for the newly definedparameter as a function of the waterline coefficient.

Wp j.îo YL..04 N___.40 Fig.14. as function of fullness (cyllrideri).

.50 .60 7Ò .80

Wp 04.04 IV Fig. 15. as 40 function of full- nasa (cylinders). .30

.20

.50 .60 80 41

e' jz/;.

Fig. 16. as function of fullness (cylinders).

wp 4.25

70 I EI .60 .s. 30

.20

.10

-50 -60 80

Fig. 17. w as fune- tian of fu].ness (cylinders). 42

41=-io -40

FEO-c2 Fig. 18. The angle 30 of inclination of waterline to centre lirio as function of 20 p. the fullness (cylin- deve). .10

.80

.60 -50 4

Fig. 19. The angle of inclination of waterline to centre line 30 as function of the fullness (cylinderS). 20 j, -10

-50 60 -80

.70

.60

.50 Fig. 20. The angle of 1 inclination of water- '0 line to centre line as function of the fullness (cylinders).

50 -70 0 43

J7fl

.60 Fig. 21. The angle of inolination of waterline to centre line as function of the fullness (cylinders). iii

60 .60

.40 Fig. 22. The poten- 60 tial wake coeffici- .30 °/L..04 o ent as function of JD angle of inclination of waterline to cen- .20 tre line (cylinders). 8 .10 tan , .10 .20 30 40 .50

IVp

.50 4 .70 AT0 .40 °/L04 rig. 23. The potential wake coefficient as function of 30 angle ofinclination 8 of waterline to cen- .20 tre line (cylinders). /4 .10

tan, .10 .20 .30 .1,0 .50 44

Pirially in figures 22 and 23 the wake coefficient is indicated as a function of the angle parameter atffle588 of waterline cz= 0.6o arid 0.7o respectively. It willbe observed that the percentage variation for all tha breadth to length ratios is prtcftlltha 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. Thea1mma tion was made as a volume si,mm'tion (formula (D-10)) graphically 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 distribution of wakeae takenas the same over the whole of the propeller disg

IVp SER/ES Z-A Z-B - r'..5O -.02

Fig. 24, The variation of -.04 potential wake with the propeller diameter, I-c

-.02

-44 .03 .04 .05 /L 45

In fig. 24 the results are given asthe deviation of thu potential wake coefficient from that of this 0.o4 propeller plotted over the ratio D/L. As expected, the biggest wake coefficients 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 importance.

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 respectively, and thewaterlines 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 Aand B some of the results are graphically represented infig025.

3ER/E5A Q5 .15

-, O (FRAME 'Z"

Fig. 25. The longitudjn wi,,SEP/ESû à,.5 variation of thepotential .25 wake COOffjeent. \ .20

1ff-2"

-, O (FRAIE 2 I-A-3 0 5 Distance from C.L. 10 15 20 I-B-3 0 5 Distance from C.L. 10 15 20 FR.2 "tI -1 0 ((-.109) .105.217.288) (-.004) .100.154.176 .007.087.117.100 -.016 .060.075.086 -.027 .062.060.038 FR. 2 " -1 0i ((-.iii) .165.340.452) (-.Qo6) .156.242.276 .137.184.157.01l'-.025 -.035 .117.134.093 .097.094.057 I-C-3 0 (-.008) 5 Distance from C.L. 10 15 20 I-D-3 0 Distance5 from C.L. 10 15 20 FR. 2 " -i 0i C(-.238) .230.474.631) ( .218.336.385) .19)..257.219.015 -.036 -.049.163.190.130 .135.130.080 "R. -1 2 0i ((-.311) .300.619.824) C(-.oii) .285.438.503) .249.335.286.020 -.046 .214.245.170 -.064 .177.170.105 I-A-5 0 5 Distance from C.L. 10 15 20 I-B-5 0 5Distance from C.L. 10 15 20 FR. 2 " O1 (.138)(.125) .120 .100.120.107 .083.101.071 .067.080.048 .051.055.029 FR. 2 " 01 (.197) .189.218 .159.189.168 .128.159.112 .104.125.076 .08]..086.046 I-C-5 " -1 .070 .067 Distance from C.L. .062 .055 .048 I-D-5 " -1 .110 .106 Distance from C.L. .097 .087 .075 FR i2 (.oB)(.278) 0 (.267)(.238) 5 10.224.158 15.107 i78 20.122...065 FR. 2 " 1 (.406)(.367)0 (.352)(.314) 5 .296.209 10 Ï5.235.141 20.161.Ó86 Ta].e 82 The""- -1longitudinal 0 variation of the potential wake at some cylinder shaped solida. .267.155 .149.224 .37.182 .123.148 .106.114 ." -1 "iO .205.352 .197.296 .181.242 .162.197 .140.150 III-A-2 0 Distance from C.L. 5 10 15 20 III-B-2 Distance from C.L. 5 FR. 2 " i (.o6)(.003) .069.005 .003.059 -.001.039 -.003.028 FR. 2 " i (.o88)(.005) 0 (.008) .109 .093.005 10 -.002.06]. 15 -.005.044 20 I'" -1 0 .3.286 .098.143 .081.090 .065.069 .054.049 "I' -1 0 .161.449 .153.225 .127.142 .102.105 .086.076 L. III-C-2 o Distance from C.L. 10 15 20 III-D-2 0 Distance from C.!....5 10 15 20 FR. 2 " -1 o1 (.123)(.008) (.012).226.6281 .007 (.153) .218.314 .178.201.130 -.003.143.151.085 -.008.120.108.061 FR. 2 " -1 o1 (.161)(.010) .296.824 (.2òl)(.015) (.009).281.413 -.004 -.010 .233.263.172 .188.196.112 .157.144.08]. III-A-4 O Distance from C.L. 5 10 15 20 III-B-4 0 Distance from C.L. 5 10 1 20 FR. 2 " 1 (.085)(.072) .162. .066.046 .047.024 .035.014 0?7.009 FR. 2 " 1 (.139)(.117) .075.108 .076.039 .057.023 .Ò44.0i5 " -1 0 .073 . .069.101 .057.069 .048.052 .04]..043 ." -1 O .118.264 .112.164 ,093.112 .078.086 .066.069 I t I-C-4 Distance from C.L. III-D-4 Distance from C.L. FR. 2 " (.167) (.1ô8) 5 .055 10 .033 15 . ô2220 FR. 2 (.226) 0 (.146) 5 .075 10 v.045 15 20 " 0i (.2oo) .380 .155.236 .161 109 .125.082 .099.063 ' O1 (.269) .209 .148 .16.7:.110 .085.030 Table 9: The longitudinal" -1 variation .170 .162 .134 .111 .995of the potential wake at some cylinder ehape " -1 .229.512 .218.319 .181.216 .150 solids. .128.134 lT

Ths absi.ssa i the position andthe. ordinate Isthøwa?e oefftiit at the distance 5 units(Oo25 L) from the centra line ofhe solids. It iill be noted that there is a rather bigdifference in the ca:ves for the twoserles, thevariation at frame O in the seriof cove form beine considerably lss than that of the series of convex form, also that only in the latter the maximt.u.wake coefficient is to be found in the vicinity of fra_ae O Fis, 26, serves as a supplement, showthgthe longitudi.. nal variation ofwake distribution for erlinders having ellip.. tical cross section. The calculation of flow around these cylInders was carried out without employing sourcesand sinks (see section cl). The investigation showsthat a longitudinal displacement of the propeller of the magnitude, which would be possIble in practice,does not alter the potential wake anything worth mentioning, whén the waterlines are of normal form.

-toco

7,-fo Fig. 26. The longl.tudial and :ia atwart-sKtp variation of the po- tentiJ. wake coefficient atcy- lindes having elliptical cross section. xÍÍ

.05

X f5,2

o 25f' WO 70f' t

b. Potential wake of e ca117 shaped bj_es with differentfore and after ends (two-dim___ i flow). Model experiments show that there is a change ofthe wake 49

coefficient at the propeller when the lines of the entrance are altered. Such a change may originate, partly from an alteration of the potential wake, partly from an alteration of the friction belt, partly front achange inthe wave wake. The observed change may be strongly affected by uncertainty in the measurement. In this section, only the influence of the entrance ozi the potential wake will be examined. The calculations was performed in the normal way by means of the method of sources and sinks. Four arbitrary unsymmetrical distributions of sources and sinks were considered, and the results compared with the results from formshavingsimilar entrance and run. In determining the value of the stream function 'tp TAYLOR's table (1o3)hasbeen used directly, for which reason the calculation is based on sources and sinks positioned at points spaced L/2o. In the S-y-diagrams fig. 27 the source and sink distributions employed are indicated. In fig. 28 areshownthe waterlines of the after body and in fig. 29 the waterlines of the fore body.

Pz-2a I

Fig.27. S-y- diagrams for two-dimensional flow (flow around cylinders having dissimilar fore and after bodies). / 21-3a Z'-3b / FT-4a The source-sink / distribution is indicated in the ;\ diagrams, top right. 50

J .1 4 5 6 7 8 9 FRAMEI .-a I 5 6 7 8 iííiil23456769

L/ 6 7 8

Fig. 28. Water].ines for after part of cylinders (series VI).

D /1 /2 /3 /4 15 16 f7 78 /9 R4/. 20

384

I /2 /3 /4 /5 /6 /7 19 FRAME20

'O Il f2 /3 f4 /5 /6 /7 18 /9FRAME2O

5,3

2FOb 'J íiÍì Fig. 29. Waterlines for fore part of cylinders (seriesVI). 51.

The waterline and wake coefficients (frame 0). are found in tables lo and Il, and table 12 respectively.The firattable contains the waterline coefficient for the part abaft frame lo, the second table the waterline coeifioient and the ratio B/If for the part abaft frame 8. In thefo'llowing tables a comparison bas been made between the wake coefficient for these forms, and the wake coefficient of forms each having same entrance and run. Sorné of these fo'ms of comparison were derived by interpolation and are marked in the tables with an X. In table 13, in which two very slender forms t= 0.4E) are compared, the biggest differencé in w for the two forma

is O.006. - In table 14 where three forms, haVtng a wateriie coefficient aft of 0.64, save been compared, the difference is a little bigger. The biggest difference 0.015 is found at the broadest form(B/L =0.25) near the centre line. The difference of th entrance of the three. forms is cc,nside-

SeriesVÏ A B C D

- 1 .404 .434 .455 .479 2 .634 .641 .649 3 .660 .673 L .687 .711 4 .630 .639 .646 .656

Table 10: The waterline coefficient of the after enof cyinders havingdiffe-.

rent foré and after endS. -

.VÌ 3 A B :. D oLi .551'. .5841 .605 B/L .129. .192 .257 .3?1..

Table 11: The watr1ine coéffÌciënand B/L for the part of the. cylinder sitiated abaf t frame (förm VI-3). 52

from centre line Series No. 0 5 ' 10 15 ' 20

1 .053 .048 .043 .038 .032

2 .099 ' .0b9 .0Th .063 .050

3 . .103 .089 .076 .062 .049 4 .097 .087 .075 .062 .049 J 1 .058 .080 .072 .063 .053 2 .157 .141 .12]. .100 .079

3 ' .169 .146 .123 .100 .079

4 . .155 .137 .118 .099 .078 1 .128 .117 .l0 .092 .078

,. ,. 2 .223 .196 .368 .140 .111 3 .243 .211 .179 .145 .115 4 .219 .194 .168 .140 .111

1' .174 .159 .143 .125 ' .106 2 .294 .25-9 .222 .184 .146

3 .332 .286 .241 ' .196 .155 4 .291 .259 .222 .14 .141

Table 12: The potential wake coefficient at fraxne O of cylinders with different, fore and after ends.

Distance from eeitre line Series No. 0 5 15

VI-A 1 .053 .048 . .q43 .038 . .032 - x. .051 .046 .041 .035 .030

VI-B 1 ' ' .088 .080 .072 .063 .053 - x .087 .077 .068 .060 .Q5].

VI-C 1 .128 .111 .105 .092 .078 - , x .125 .111 .100 .087 .075

VI-D 1 .174 .159 .143 .125 .106 - x .171 .153 .137 .120 .104

Table 13: Comparison between the potential. wake coeff i- cients of form VI-3 and of a form, produced by interpolation between I-iO and V-4 (the influence of the färe body on w). 53

Distance from centre line

Series No. - 0 5 10 15 20.

VI - A 2 .099 .089 .076 .063 .050 VI - A 4 .097 .b87 .075 .062 .049 I - A x .104 .088 .075 .062 .C9

VI - B 2 .157 .141 .121 .100 .079 VI - B 4 .155 .137 .ii8 .099 .078 I - B x .168 .142 .118 .098 .077

VI - C 2 .223 .196 .168 .140 .111 VI - C 4 .219 .194 ,-168 .14.0 .111 I - C x .237 .201 .168 .139 .109

VI - D 2 .294 .259 .222 .184 .146 VI - D 4 .291 .259 .222 .184 .147 I - D X .309 .263 .221 .182 .144

Table 14: Comparison between the potential wakecóef- ficients of forms having somewhat similar after bodies and fore bodies of different form. I-X is an interpolation form.

Distance from centre line Series No. 0 5 10 15 20

VI - A 3 .103 o89 .076 .062 .049 I x .108 .091 .074 .059 -

VI - B 3 .169 .146 .123 .100 .079 I x .174 .148 .120 .095 -

VI C 3 .243 .211 .179 .145 .115 I x .255 .218 .179 .139 -

Table 15: Comparison between thepoterìtial wake coefficients of form VI-3 and an interpolation form from series I. 54

ràbly' lArger, than would be metwithin practice fo.r normal ships. Finally a compartson!between form VI-3 with a fôrm' from series I has' been made in table 15. As fori VI-3 has its maximum breadth situated a àonsiderable disl3ance f rein frame Io, the coefficients of the form were determinedwithframø 8 as midship section, and the comparisonwas made at the now breadth- length ratio of. the after body.. Thé4iffe_ renco in w is of the same degree of magnitude as inthe other comparisons. Onthe Whole it can bsaid that thei±ifluenoe,ofHthe form of thé fore body on the potential wake at the poition ofthe propeller lá emaIl, so small even that in pracIee

it can be noglectéd. . . .

e. Potential wake of elli soids thxe-dimensional flow). So far only twO-dimensional flow has been consideredl but, in thé following, three-dimensional f iów wl.11 be ìealt with. Unfortunately it is considerably more difficult calculate three-dimensionally than two-dimensionally, and it will be necessary toconsider only flows around solids of simple form, such aé flow around ellipsoids. By means of these calculations it ought to be possible to get some lmthvledge of. the Influence of the draught on the wake ,eoeffio1èr9. Further it .Îs of interest to see, if the influénce of the breadth-length ratio Is the sane as in the ease f two.dImen- atonal flow. As all elllpsólds bave the same block co,efti- cient, =0.524,, there wilibe no possibility for i'nvetIga-

tion of the Influence of the fullness on w. I Imagine an ellipsoid movingIiian ideal fluid of Thf i- nito extension in all'direotione. The coordinate aystemis arranged with its origin in the middle of the eìiIpso!4 and the axes in the directiòns of the main axes. ot the e].lip- solde move, in. the direction'otthe X-axis with a velocity U. As the flow ábove and below the XY plane is the saine, this plane may be looked upon as a water surface n. which hoever rio waves can rise. 55

The principle forthe basis ofcalculations willbe briefly outlined in the fo1lowing, tor the rest referto (26, 61, or 86). The equation of an ellipsoid is;

= I +4+.4 (D-11 ) a, b, and o being as usual the lengths of the semi axes. Now it is' an advantagepartly tochange over toellipticalcoor- dinates instead 'of cartesian coordinatès. 'Astransitlónfor- mula was used: 2' 2 - (D-12) + + ;: + Por determining the velocity potentIal functionthere Is the surface condibIori:

U. for A = O . (D-13) By Introducing

= a b c and L=/(a2,+X)(b2+)(c2X)(D-14) JA(a+x)L añd fuxther

x =a.b.c..J__.dX -

thevelocityfunction can be found: 2__U_ç xcz = xc (D-15)

As it is the velociti component in the direction of the X-axis, which Is wanted, nuist be differentiated withregard to x:

=c1. + C1x'=CO( +

'it can be proved that dA4I 2 dx + r. { (a2+A)2 + (c+ A) X) (D-17) By substitution the velocity is found equal to: 56 2° Cî' x2a' b' o = cl. 2. - 4, X y z (a + + (b2+)?+ x)2] (D18)

y(aI 2 +x)(b2. +7j(c.2 +) In order to determine thisquantity, it is necessaryto, make f'iirther transforrnatlofls

: ::.fl(2d , [Fkk,J )°L-

. '22 I k5ifl21a2b2 x=sinç= where F(k,cp)andE(k,p) are normal elliptical integrals. Subsoquéntly the calculations wòre car'ted out inthe following way: The semI axes of theellipsoIds were chosen0 The length of all the ellIpsoidswas't'aken as'2o, wit1the breadth as 2.o, 3,0, 4,o, and 5o, and thedraught wavaried

from 0.2 to 2,5.(saé further table16). i As at present, only the wakedistribution at,.t'$L ends of. the ellipsoids is required, .x wasput equal to a = lo, and in this plané some twenty points wereselected,., at wh.oh the wake was determined. T14a was doriOby choosing z = 0, z = 0.25 z = 0.5 o, and z =0.75 c. After this a )s was ass.unedaiid y wa8 detórninied by means.of (Dl2), To begin with,an attempt was made to f md y =about O, 025 b, 0.5 b and 0,75 b. Often Itbecame 'nece,ss.a1y to try'a few times..with different1ValUeS ofA,before the required 'value Of y was'found." Whore 'necessary

extra pointS were In8erted. . L When oorre8po13dIg va1ues ofX, y, z and A vei etermI ned, the wake'waS found byusirig(D-18)and (D-19).TGva- lues of the elliptical integrals neroderived by meana of LEGENDRE'S tables (72) and curves for thedifference E. drawn as. a function of p for difrerentvaluesoÍ° k = áinc (8incalculated 'for consecutive complotedegrees). B means of cross òurves supplementary curvescorresponding to the k values of theòlflpóIdawere:insertede'' N E-1 E-2 S-4 E-5 E-6 E-7 E I E-9 S-10 E-11 E-12 E-13 S-14 E-15 E-16 b .1.0 - 1.5 2.0 2.5 b/a o .2 .3 .4 .10 .5 .8 1.0 .6 .15 1.5 .4 .6 .8 .20 1.0 1.6 2.0 1.0 .25 2.5 o/2b Table 16: The semi axes a, b, and o of the u4.lipsoid.s, .101around. .15 which the flow is calculated (s-i to E-16).. .20 .25 .40 .50 .20 .50 .10 .15 .20 I .25 j .40 .50 .20 .50 (1.. (b-10c..Z) - -.E2.. (b-/.0-c.3) E3(b-f.0, c-.) E4 (b10, c 5) - E5 'b./.o, c-.8) E6 (bI0, c-10) 7WE WAKE O/ST/81JT/ONATT//EEND :9J) E-7 (6i5.c.6) E-8 (b.t5,c15) rl)) b E15 (b-2-5,c.t0.J &16 (b2.5,c.24) E9 (b-Oc.û4) jjE10 (b2O,cO6.' E11 (b20,cOà) E-12 (ô2.0,ci0) &13 (b20,cI.6) ÌJJ Et (b-2.0,c2O1 1411):jJj jjJ endeFig. 30. of Curvesdifferent for ellipsoids. constant potential wake at the j 'JJ Ii, 59

Y Y E-1 E-2 O b/4 b/2 3b/4 O b/4 b/2 3b/4

0 1.000 .043 .018 .011 0 1.000 .065 .027 .016 .018 .011 .058 .016 z 0/4 .114 .042 z 0/4 .130 .027 c/2 .068 .037 .017 .011 0/2 .073 .048 .025 .016 3c/4 .048 .032 .017 .010 30/4 .050 .039 .023 .015

Y Y E-3 E-4 O b/4 b/2 3b/4 O b/4 b/2 3b/4

0 1.000 .085 .036 .022 0 1.000 .104 .045 .025 .072 .021 .085 .042 .026 o/4 .142 .034 z c/4 .150 0/2 .076 .055 .032 .020 0/2 .078 .061 .037 .024 30/4 .052 .043 .028 .019 30/4 .052 .045 .032 .022

Y Y E-5 E-6 O b/4 b/2 3b/4 O b/4 b/2 3b/4

0 1.000 .153 .069 .041 0 1.000 .181 .083 .051 .111 .062 .121 .046 c/4 .170 .039 z c/4 .179 .073 0/2 .082 .070 .050 .035 c/2 .086 .072 .054 .039 3o/4 .052 .047 .038 .030 3c/4 .050 .046 .040 .034

Y Y E-7 E-8 O b/4 b/2 3b/4 O b/4 b/2 3b/4

0 1.000 .129 .053 .030 0 1.900 .259 .118 .070 c/4 .191 .110 .050 .030 c/4 .258 .175 .102 .065 Z z c/2 .103 .080 .046 .028 c/2 .i18 .102 .076 .054 30/4 .070 .059 .040 .026 30/4 .070 .065 .054 .043

Table 17: Potential wake coefficients of ellipsoids at frame 0. 60

Y, Y E-9 E - 10 O b/A b/2 3b/4 O b/4 b/2 3b/4 "0 1.000 .093 .035 .019 0 1.000 .140 .052 .029 .175 .86 .035 .019 c/4 .207 .118 .051 .028 Z 'c/4 Z 'c/2' .106 .b72 .033 .018 c/2 .119 .089 .047 .027 3c/4 .Ö77 .Ö59 .031 .018 30/4 .082 .068 .042 .026

Y. Y, E - 11 E - 12

O b/4. b/2 3b/4 O b/4 b/2 3b/4

O l.00Ó .185 .069 .038 0 1.000 .222 .084 .047 c/4 .232 .144 .066 .037 0/4 .253 .163 .080 .046 Z Z c/2 .127 .1Ö1 .059 .035 c/2 .133 .108 .068 .042 3c/4 .085 .073 .050 .032 3o/4 .086 .076 .055 .037

Y, Y E - 13 E - 14 O b/4 b/2 3b/4 O b/4 b/2 3b/4

0 1.000 .293 .126 .072 0 1.000 .336 .148 .086 .304 .208 .114 .068 c/4 .332 .224 .129 .080 z c/4 Z c/2 .144 .123 .086 .058 0/2 .148 .128 .094 .065 30/4 .087 .080 .063 .048 3o/4 .087 .080 .066 .051

Y Y E - 15 E - 16 O b/4 b/2 3b/4 O b/4 b/2 3b/4

0 0, 1.000 .228 .086 .145 1.000 .404 .180 .101 .081 .044 c/4 .403 .273 .155 .093 Z:,c/4 .269 .177 Z 0/2 .147 .120 .070 .042 c/2 .179 .153 .110 .075 3e/4 .098 .086 .058 .038 3c/4 .101 .092 .075 .057

Table 18: Potential wake coefficients ofellipsoids atframe 0. 61

The calculatIons were carried out to suáh anaccuracy that the poselbia error in w was lImited toO0ool. in table 17 and 18 the wake eoefflc2.enta thusfound are. recorded, and in fIg, 3o curves representing constant wake for the different elliisoids are indicated. These curves picture the ake in the tangential piane at the one end of thÒ elllpsolde and in3ide the area of half breadth times draught0 FIgS 31 gIves a cozparison between the infiuencs of the breadhlengtth ratio on the wáke ¿oefficient by two- and three-dImensional flows,

2-DIM. ELLIPSE SEE/ESI .25 S-DIM. d,5- --

Fi:g. 31. Coiuparisonof the influence of B/L on wat two- and tbree-dîiensional flow. ;zJ 84 .10 -/5 -20 -25

The two curves for the two-dimensional flow represent the variatIon of the wake coefficient abaf t cylinders, having elliptical cross section (for calculation see later), and the variation of the wake coefficient abaft cylinders having sharp edged cross section (series I. and = 0.524). The two curves for th? three-dimensional flow indicate the wakecoefficient at the poInt x = a, y = Ö and z= c/2 forel1ipsoIdssome of which have d/B equalto 0.5, and some ith d/B = 0.2. The elliptical cylinders have the sanie cross section as a section through the ellipsoids at z = c/2. Qualitatively there is a good accordance between two- and three-dimensional flow, while qùantltatively there isa difference, Thus it Is seen from the figure, that the wake variation with B/L at theellipsoids havingd/D 0.5 and :0.2 will be re8pectively about 4o and 55 per cent less than at the corresponding two-dInensional flows. Fig. 32 gives an idea of the influence of the draught ori the wake. Curves are drawn for the wake variation with the draught-breadth ratio for the points (a,0,-c/4),(a,0,-e/2), (a,b/4,c.c/2) and a,O,-3c/4) at the foui' breadth-length ratio, A, B, C,and D. .Froz the diagrame it is séen, that onlfor the forms with. great breádth there Is any marked váriat ton. I is therefore probable, that at normal ship forms te in-. r f luence of the breadth-draught ratio on the wake can

neglected. r

: .10 A. .4011U1 .05 35lUPI O '/ 2 -3 -4 unu--c D Fig. 32. The dependence u,.. .10 of potential wake on draught (el].ipsöids). iUÍrI -/5 I.2 3I4-

Ir yO 'D 41111 .10 z-c

.05 .05 '.' d,5

o -t .2 -3 -4 O .1 2 .3 .4

It is alo'interesting to notice the small influe ce of the'proe1ler diameter on the potential wake coefficiént. For the ellipsoids E6 and E-14 curves, over the periphri- cal variation of' the wake have been prepared, the curves being drawn over the wake at the distances 0.01 L,0.0?, 0.03 L, O.o4 L, 0.05 L and O.o6 L from thò centre of te propeller (see figs. 33 and 34). Iñ these figures sonieiof the average values 'of the wake. are also indicated. Thèse have been determined by integration over thin rings hv1ng the diameters stated. The values correspond tó .thôse*Ich would be registered by ideal blade wheels As the variation 63

from ring to ring Is very small, no radial integration over the whole of the propeller disc baa been performed. It is -rather peculiar, that the wake coefficient increases aa the propeller diameter becomes larger.

Wp E-5 4t-.o, 5 --.02 .20 -"03--096 Fig.33. The peripherio. --.-o ---f09 cal heterogeneity in \ --- -.- --ff3 the potent iaj. wake abaft an ellipsoid. .10

450 0 900 1350 f8O

E-14 i

:: : Fig.34. The periphe- j::: rical heterogeneity /62 In the potential wake abaft an ellipsoid. ----

.10

e 90 /350 o 45 A5 64

cl. Comparison of two- and three-dimensiona± f low3 (elliptical 3ylthders and ellipsóids). In order to facilitate the coruparison of two- and three- dithensional flows, the silipsoids were divided into Infinite- ly thinellipticalcylinders, for which the waica coefficient Was ca].culated.under the assumption of two dimensióziel flow. Arounla cylinder ofellipticalcross soction (semi axes a and b) the stream function and the potential velocity function may. beexpressed (27) respectively as equal to

= tJb.j + e j (D-20) and

U being the velocity of' the elliptical cylinders.' In the follo'ing U is put equal to 1, andi are elliptical coor- mates, defined by x=ccosh°cos r y=esirth. sinn c=ya!2 -b2 The velocity at an ,arbitray point is found either by dirfe- rentiatingywith'regard to y,or p with regard tox. 'By introducing

=c, (D-22)

lp=c,. e' sin ii is obtained,and:

cil1' - - Tj + Id ='- C,. O 3iflT s C0STdy

(b-23) Differentiation of (D-21) gives O = o' sinht' cos r - o. cosh'sin dy

(D-24) I = o' cosh sin rj. + 0 sinh SCB .d1 y dy 65

and

d71 .sinh t cos 1 dt coáh . sin ' dy - i + csh) c3_co32 +cosh2t) (D-25) By substitution in (D-23):

e4 2 2 (. coshi' sin TI + sinht.soci-j) (-033r + cosh ) (D-26) which by using C1 b 2 2 TI attd etnht+ cosht= et T'= a- b sin TI = I- 008 can be transformed into

-r-- (coaj-6tcoaht) (D-27) a- b oosh2- The calculations are carried out for four ellipticalcylin- ders having breadth-length ratios equal to : 0.10, 0.15, 0.20 arid 0.25, The wake coefficient is determined partly for x = a and partly for different values of x corresponding to the distances from the tangential plane of' the ellipsoid x = a to the horizontal sections in the ellipsoid The procedure then is:

to ssume a value of et, find et,cosh , slnh , òos r (D-21), r, sin r, y (D-21), and finally dV/dy can be determined, being directly equal to the wake coefficient, because U has been put equal to 1, by means of (D-27), In fig. 35 curves of constant wake are drawn, assuming that the flow around the ellipsoids is two-dimensional. By this method of calculation, the draught of the ellipsoids will have rio influence on the wake curves, and therefore the difference in the wake coefficient will be &eatest at twò--azid. three-dimensional flows for the ellipsoids having small draughts, As seen frontable 19 the differences between two-and tbreé- dimensional flows will be considerable even at the large dra-ughts, The comparison covers the whole field b x e, and particular attention should be paid to the columns in thevicinity of y 0, and z = - c/2. Further lt should benòted that the 66

wake coefficient at two-diménsional flow is not always bIgger than that corresponding to three-dimensional flow.

Fig.35. Curves for constant potential wake abaft é].].ipsoid8 in hypothetical two-dimensional flow.

'E-f.

L EWP5ES A b 10 X- 11.5 a1O j B: aF0b- 1. - Fig.36. Wake C: a-0b-2.o distribution - D a-10b- ¿5 curves (wy) ELL1PJO/D L-16 X-10Z- O a-fob 25 abaft son el- c.s liptIcal cy- lindera- and an ellipsoid. i-

D / 2 3 4 5 6 7 8 9 io4

k èll sub8idiary rerlt of the investigation of fw aro md elliptical cylinders is in f ig.36 shown in the form of òurva indicäting the athwart ship wake variation a small distanc& abaft the cylinders: (z=11 .5) . In the same, figure the corresponding curve for ellipsoidE-16 for z = loand . z E-G E-8 Y=o E-16 E-6 E-8 Y=b/4 E-6 Yb/2 Y='3b/4 Y=b w2w3 i.0001.000 i.0001.000 i.0001.000 E-14 i.0001.000 .275.151 .341.259 E-14.404.336 E-16.457.404 .159.083 .185 118E-8 E-14.206.148 E-16.235.18o .109.05]. E-6 .121.070 E-8 E-14.129.086 .139E-16.101 .082.035 E-6 .047.086 -8 E-14.087.056 E-16.083.063 o w3/w2 1.00.179 1.00.258 1.00.332 1.00.403 .66.121 .175.76 .224.83 .273.88 .073.52 .102.64 .129.72 .155.77 .47.046 .065.58 .080.67 .73 .43 .55 .64 .060.76 c/4 w3W2 .59.303 .60.433 .61.541 .65.625 .48.251 .54.324 .378.59 .65.423 .41'.178 .50.206 .56.229 .64.244 .36.129 .46.141 .54.147 .62.150.093 .034.34.099 .44.099.044 .099.54.053 .63.096 w3/w2 w3 .086 : .148 .179 .072 .102 .128 .153 .054 .076 .094 .110 .039 .054 .065 .075 .034 .041 .047 .053 c/2 W2 .80.108 .72'.165 .66.223 .64.278 .68.106 .158 .61.208 .60.255 .100 .140 .173 .198 .091 .118 ' .136 .146 .o8i . .098 .04 .106 3c/4 w.3/w2 w2w3 .036.050 .057 .078.087, .101.101 .036.046 .056.065.65 .077.o80 .097.092 .036.040.54 .054.54 .073.066.54 .090.075.56 .035.034.43 .052.043.46 .066.051.48 .079.057.51 .033.032.42 .049.038.42 .058.044.45 .066.048.50 w3/w2 Table 19: 1.39 Comparison between w 1.23 1.12. 1.00 for two- and three-dimensional1.28 flow around ellipsoids (at frame 1.16 1.04 .95 "1.11 1.00 0.90 .83 .97 .83 .77 o). .72 .97 .78 .76 69

z = O is shown. The figure gives an idea of the rate at which, the potential field diminishes, and also how large the model cari be made in a model experiment without having to fear. wall effect on the wake coefficient. From the figure is seen, that in experiments with.E-16 no wall effect, worth speaking ofIsobtained, if the breadth and the draught of the model is one tenth of the breadth and the depth of the tank respectively, which are normal ratios: at model experiments.

d. Potential wake of solidi of revolution (three- dimensional flow). ±n order to throw further light on the difference of two- and three-dimensional flows the wàke distribution of so solids of revolutiOn with 'similar fOre md after ends was cálculated. These systematical cálcnlattons of the correlation were not very elaborate as the wake distribution only was determined for 24 forms, of which six had the breadth-léngth ratio O.lo, sx 0.15, six 0.20 and finally six 0.25. For these calculations sources and sinks &iso were employed and the procedure, suggestedby D.W.TAYLOR.in 1895 (1o4),'has been used. In the following a brief account is given of the calculations and the theory on which they are based. As regards questtons of detail reference is made to (1o4 anß. 28). It should however, be mentioned that FUHRMANN (39) éarller determined the potential fields ofbodiesha- ving balloon form. FUHRf1ANN also employêdsources andsInks, but these calculations can in general justas well be carried. out using doublets, which are source-sink couples. Again the conditions .f.an ideal flui4, this time of in-. finite extension in all directions, are considered. A solid of revolution is plaeed somevherc n the f-lu.d, End as the flow around thesolid imisttake place in the same way for al planes passing thróug thé áxis of the solid, "it is only'necessary to carry out calculations thons single

semi-plane. - 70

Fundamentally the calculations are performed in the sanie way as for two-dimensiònal flow. To begin with a coordinate system (X,Y,Z) is established in the fluid, then a source- sink distribution is chosen andthe sources and sinks positioned on the X-axrs, each on their oiside of the origin. The stream function at an. arbitrary point P is

foundby n . = k .cOS9n k, S (D28) being the angle between the. X-axis and the line connecting the source (sink)tothe point P, and k isa factor proportional to the strength of the source (sink), being positive for sources and negative for sinks. It is noted, that the expression Is analogous to the expression for two- dimensional flow, only that has been replaced with cos In a parallel fiòld in spacé the stream function at the point P distanced r +z2 from the X-axis, will

attain the value: .

= 1/2 V0. (y +z2) (D-29)

y0.being the velocity, which is assumed parallel to the X-axis. As symmetry exist about this axis, it can be imagined that P is positIoned in the XY-plane ar..d thus:

t= 1/2 e I (D-3o) By combining the source-sink field with the parallel field 2 'jV= k, ' S - 1/2 y0. (D-31) is obtatn.d,The equation, of the boundary curve then becomes:, 2 O = k, S 1/2 y0 (D-32) If the breadth of the solid of revolution be B, the cónstant of source strength k is determined by: 2 k= .vV'=,(B/2)2.v (D-33) 2S. o 2S SM being the value of S at the point x= O, y = B/2.Tbe distributión of velocity in the field surrounding the solid

of revölution can be determined by: V 71

(B/2)2 y=i. o.Y'=(k -y . ) = y (1 X y dy ' dy o y O .2 SM°7 i and the wake coefficient found as: (B/2)2 d s W 2.SMy dy As it is difficult, for small values of y, to determine vexactly, using the formula (D-34), the following method has also been employed. For a flow starting from a source of strength S the radial velocity will be: s - and therefore the velocity parallel to the X-axis:

;2)3/ (D..36) X2 2 2 = (z2 + This quantity is calculated for all the sources and sInks and then asunm1ation Isperformed. The sum multiplied by the constant of source strength gives directly the wake coefficient, provided the velo3ity of motIon has been made equalto 1. In practice the calculations were carried òut In this way, so that S was first determined at a series of points by means of (D-28): x= 0, 7=0 5 Io 15 2o 30 x= 20, y=0 5 lo 2o 3o x= 4o, y=0 5 lo 2o 30 x= 6o, y=0 5 lo 2o 3o X= So, y=0 5 lo 2o 30 z = 9o, y = O 5 lo 2o 30 x=loo, y=O 5 lo 15 2o 30. As the sources and sinks were placed on the X-axis for every tenth unitrangingfromX = -loo to z = loo, TAYLOR's tables (104) cOuld directly be employed. The distribution of sources and sinks may be taken eitheras cOntinual oras point shaped, all depending on whether the suation is re- 72

garded as performed by the trapezoidal x'ule or as an ordinary smtmntiOn. Having determined S, a diagram of S as a function of y was prepared. Fig. 37 indicates such diagrams for different distributions of sources and sinks (also shown in the figure), and the curves correspond to different values of x. In the

3 O .x

20 biiiri.. 11 /5 IJibi 'IL I-ill. II-/01. V1

20 S'JuuuuuuI!Jl---"-I

I, / '5 i // i/f P / ,r I /1 'aunilii --.-- _/-, y i- Fig. 37. S-y-diagrazna for flow aroundsolids of recolútion. The source-Sink distributions areindicated in the diagrams, top right. 73 same diagram curves (parabo].as) corresponding to the equa.. tion (D-30) have been drawn. The points of intersection of these parabo].as with the S-y-curves give the coordinates of the boundary curve, and thus the contour of the solid of revolution. The wakò was only determined in the plane x=loo, which was done by using (b-35) and' (D-36). In the case Where the first mentioned fox'mila was used, dS/dy was de-

termined graphically. ' In fig. 38 thò contours of the 3ölids 'of revolution are indicated. The curves may be looked upon as the waterline of the solids.-Iù table 2o are given Ithe waterline coefficients for these waterlines and the block còeffioi- ente. Finally table '21 gives the wake coefficient At diff e- rent points all situated in the plane x = loo. In the figures 39 and 4o a comparison between two and three dimensional flow is indicated. In the first figure

PIg. 38. Contours of solids of revolution around whiori

the flow is determined0 ' ' 74

w, - -

1' -

______2-.4ND3-.'MEN5/ONA FLOW -

-I--

MP/EWD

.05

.30 45 50 .55 .6

..Ftg. 39. Comparison betweenthewake coefficient (w ) foi' solids of revolution and for infinitely long cylInders having corresponding breadth-length ratios and block coefficients.

.6 Series VII A .B C D A B C D No. 1 .771.766.768..771.564.552.547.543 2 .747.747.749..754.514.511.510.511 .637.645.655.668.398.404.412.422 'I. 4 .548.565.584.602.309.325.342.359 " 5 .643.656.669.683.391.404.416.429 6 .633.641.650.660.415.419.424 .432

Table 20: The wáterline and the block coefficients for solids of revolution, for which the potential flow has been calcuJ.ated..

the comparison Is based on block coefficient. Iñthecase of two-dimensional flow the wake coefficient for propellers having D/L = 0.04 is employed, while the wake coefficient at the point x = loo, y = O è.nd z = - d/2 (= - B/4) bas been used for three-dimensional flow, asthevariation of 75

Series VII

Y Y i - 2 5 10 15 20 30 5 10 15 20 30

A .035 .028 .022 .016 .Ò09 A .038 .026 .019 .014 .008 B .080 .064 050 .038 .022 B .087 .060 .043 .031 .019 C .145 .116 .090 .068 .038 C .157 .109 .077 .056 .034

D .23]. .186 .143 .109 .061 D .253 .174 .124 .090 .053

Y. Y 4' 5 10 15 20 30' '.' 5 10 '15 20 30

.A .021 .016 .012 .0Q9 .006 A .014 .010 .008 .007 .0Ó4 B .049 .037 .028 .021 .013 B .034 .024 .019 .016 .10

C .092 .069 .051 .039' .025 C .064 .046 .037 ,030 .019

D .151 .113 .083 .064 .041, D .106 .018 .063 .05]. .032

Y Y

5 . 6

10 15 . 20 30 . 10 15 20 5 5 . .30

A .026 .018 .013 .009 .005 A .018 .015 .012 .010 .006 B .062 .044 .030 .023 .013 B .040 .032 .027 .022. 014 C .116 .081 .056 .042 .025 C .072 .059 .047 .039 .026

D .189 .134 .093 .070 .04]. D .116 .095 .077 .064 .04?,

Table 21: The potential wake coefficient of solids of revolution (at frame o). the wakecoefficientwiththe propeller dîaeter ISso small, that It may be neglected in a qualitativeeompari- ou. In the figure th numbers of the solids of revolution are written at the points, which are spread somewhat on account of the unsystematicchangesIn the contours of the forms. Just as found for two-dimensional flow the forma with. S-shaped contour have the smallestwake (no. I and no. 6). 76

'.' Wp - / / ,5(E5ID" - i'

.40 '

2-NID 3-' MINSIO 4 FLOW 'Z ' / - / .....I-8/.

--, - ,',, .- lD_/

T z,'. ------.

Fig. 40. Comparison between the wake coefficient (w .) for solids of revolution and for cylinders having corresponding waterline ciefficieñts.

In the figure 4o the comparison is based on the .wáter- line coefficient. For the solids of revolution the fullness coefficient of the contours bas been looked upon as waterline coefficient. For the two-dimensional as weti ai for the three-dimensional flow the wake value at a point 5 units (0.025 ,L) distanced from thé, centre line has been used for .00mpartsóñ. 'It will be seen, that qualitativel' there is. no difference in the variation of the wake coefficient for two- and three-dimensional flow, whereas theré is a big quanti- tativo difference. It is therefore not unlikely, tt for ships of normal form a similar relation exists between wake coefficient, block eoeffiòient and breadth-length ratio.

''e. Pötential wake òf ihips (threé-dimensional f lw). It would be of great interest to Imow thé 'potential flow aroufldolid8 more ship shaped, and of espeial interest to'coinpai'e the flow around ships having U- and V-

j shaped frsmes reapeotiv8ly Theorètically it should be possible to carry out cal- 77

culatiozis of the field surrounding ship sha3d solids, but the work of calculations is very tedious. The calculations may bé perfrmed by eloylng sources andinks positioned indlÍferent ways in space The field produced b* the sources and 5inks is combined with a parallel- flow, and then it is possible to determine the boundary curve. But no doubt lt will be necéssary to try a considerable. number of tinas, beoré a dstributton of surces and sinks is found, which give a suitable form. The sUrface of the ship form, of which the wake is desired, coúld also be Imagined covered withmany small sources and sinks. An advantage in this method would be. that it is unneces.sáry to determine the boundary, withbe exception óf the preliminary calculation, by which the necessary nuinberof source-sink couples are determined. This procedure has previously been used In determining the wave resistance of certain ship forms.

-. E. WAVE WAKE. When a ship or a model moves through water it will generate a system of divergent waves and a system of transverse waves. The energy of the divergent waves is lost to

the ship, while there is a possibility of regainingsoi of the enexgy expended in the transverse waves. To make this

- possible té propeller of the ship ist be situated in a position,such that the movement of particleé in the wavé liii the direction of motion of the ship (i.e. a positiv wave wake). Or expressed in another way, there must be a favourable interference between the wave aystemgenorated by thé ship and that produced by the propeller, It is assumed that the wave above the propelleris a tróchoidal wave, the wave wake coefficient is determined

-, by:

-. 2zi- (E-1)

where X is the wave length determined by: 78

= 2 ir V/g= 0.6405 V2 (E-2) and where r is the radius of the particlemovement, depen ding onthe wave height, the velocity and the distance from the water surface. It rst be observedthat the expression. for wis only valid directly below the wavecrest, below a trough of tha wave the formu.la bas theopposite sign. By investigating the shape of the stern wave of different ships, it is found, that the shape does not resemble trochoidal waves very closely, and the wave length does not agree with (E2), but as the stern waveis pioducod by In- terferirig waves, generated along the sides of the ship, there is a possibility, that it. has beenbuilt up by trochoidal waves of wave lengths determined by(E-2). Therefore the wave wake coefficient can be determinedat an arbitrary point by meansof (E1), when for half wave height, the wave ordinate abovethe point is used. In hi publiat1on (42) the auther trIed to prove, that in this way, a good approximation for the wake coeff i- oient could bé obtaine4, The wave wake experlilents, which have been performed by the author, can briefly be described s.s follows. Abaft the run of each model, avertical plate was placed longitudInallyin order to record the wve pro- f Ile, end by means of blade wheels the wake was determined abaft the models. To make surs that the influence of the frictionbelt was a minimuxa, the measurements were performed at a distance slightly more than B/4 froii the centre line of the mode 13. Measurements wore taken at different speeds, partly abreast,the propeller and partly at frames - 1/2, -2, -4, -6,and -S. The results of the experiments were compared tO wake coefficients calculated. accordingto the troohoidal theory. In all cases there was good agreement between the two set of values. Subsequently the wake was determined at the position of the propeller, partlyby means of blade wheels of different Sizes and partly by using the propeller as wake integrator. The blade wheel experiments were very interesting, as the variation of the 79

wake coefficient with speed dependedon the size of the blade wheel. With the small blade wheelsthe wake coefficient dIminished considerably1th Increasing speed,.whule the wake coefficient measured by the b±ggerblade wheels remained fairly constant, with humpsand hollows corresponding to the variation of theexpected wave wake. The prò- peller also registered a diminishingwake and humps and hollows in the same places. Themagnitude of the htunps and hollows found in bothcases was In full agreement with the trooboidal theory, but on the otherhand the considerable diminution of the Wake coefficientmust be due to hango In the, friction wake, In order to throw further lighton the question, a short reference to earlier publishedexperimental results is useful. In 1932 HO (48) publIshed the result ofwake measure- monts performed by blade wheels,and at the samé timegave the profile of the stern wave atcorrèeponding speeds. The experiments were carried out witha 4nilong model ofa twin screw merchant ship. The draught ofthe model was 0.223niand the measurementi ere takenat 0.083 and 0.140 nibelow the water surface, At the firstdepth the measurements were recorded at 0.1oo, 0,200, and0.400 ni forward of the A.P. At the second depth the same positionsWere used and aleo 0.600 m forward ofA.P. In all the positions the tip clearance of the propellerwas the same The model was run at speeds 1q25, 1,4o, and 1.55rn/sorreepondjng to 14.5, 16,o, and 17,75 knots for theactual ship (the high speeds chosen In order better to elucidatethe nfluènco of the stern wave). In table 22 themeasured wake values bayO been compared to the wave wake,obtained by using the tro- chold.al theory starting from thewave contour. The desig natIon "a" in the tables gives theresults from the upper positions of the propellers, and thedesignation I to IV denotes the positions from aft toforward. It is seen, that part of the várlation. can be explainedby the variation of the wavewake, butIn many places the observed,variation is 80

a b

1.25 1.40 1.55 1.25 1.40 1.55

W .280 .308 .277 .128 .139 .132 .029 .024 Ww .035 .039 .030 .025 .266 W - .254 .283 .144 .164 .151 .024 .026 .025 .020 w .037 .023 w .118 .139 .088 .108 .129 .109 ç .019 .029 0 .013 .021 0 w .065 .054 .074 iv O -.009-.011

Table 22: Comparison between observed wake and wave wake expected (HORN's experiments). bigger than the calculated, which perhaps is due to uncertainty of the experiments. The experiments of YAMAGATA (133) are alsowell known. For a wooden model 4.626nilong (B = 0.694, d = 0.268, and 8 = 0.694) he determined the longitudinalvariation of the wake at speeds ranging from 0.4 to 2.8rn/s. He recorded measurements extending from the model up to 6 ni abaft.His wake curves plotted over the distance from A.P. show afine wave form having decreasing amplitude withincreasing distance. The distance between humps and hollows agreed very well with those determined theoretically. YAMAGATA measured wave wake amplitudes of more than 0.05. Determination of the wave wake of actual ships bas also been carried out on a numberof.occasions. For these re- cordings the Hamburg tank (see (57)) used a resistance log, which was towed behind the ship at different distances. Previously the resistance curve of the log had been determined by tank trials, and thus the speed could be easily obtained. As the speed, registered by the log, depended on Its position in the transverse wave system, a profile of speed could be recorded using the results from repeated experiments with the log at difCerent distances from the ship. The deviations froi the average speeds represented 81

the wave wake, and in the article quotedit amounted to about 0.o2. The Berlin tankbasinmany pub1icatins (see for instance (loi) and (127)) employed the trochoidaltheory, when corrections had to be made for wave wake.Therefore it can be assumed, though nothing has been publishedto this effect, that the Berlin tank has tested thetheory experimentally. But after- all, too few experimentshave been. carried-out for the determination ofthe wave wake, It is often important in wake investigationsto.. know the degree of magnitude of thewave wake As a wave profile rarely is available, an effort hasbeen made to prepare a diagram for predetermination of the height ofthe stern wave. The magnitude of the stern wave must firstlydepend on the. speed and the length of the ship, andasthe ratiov/i/L is of great importance with regard tothe interference of the wave system from entrance andrun, it is natural to use this -ratio as a parameter. Further it isprobable that the ratio of the breadth to length of theship playos a considerable part,'as this ratio alsoinfluences distribution of pressure around the ship and thereforealso the wave 'formation, To begin With the availabledata for this invo- 'stigation, have .been.ari'anged accordingtoV/VTand B/L. In the diagram fig. 41 are indicatedcurves forhA.as a function of V/(Tfor different valuesof B/L. As the scattering of the points in sucha diagram is very exten-. sive, an attempt has been mad to find one more paranieter. It is a fact, that the draught, blockcoeffi-ient, and waterline coefficient all playa part, ás an increase in one of them will generally result inan increase in wave height'. Different combinations of the coefficientshave been tried as parameter, but none of them proved to beparticularly effeo.. tive,. Theblockcoefficient was used in fig.

42 for the sake - of simplicitybeing not lesssuitable than the others. The - auxiliary' curves have only been inserted at small ra- 'tios, as the experimental materia], at large B/Lratios wa too limited for closer analysis.As practically all the 82

points were inside the dotted lines infig. 42, an uncertainty ofh/X less than O,005 can ingeneral be expected, Which means that the uncertainty of the wavewake coeff i- cient will be about O.005 to 0.010(8ee fig. 43 where the wave wake coefficient taindicated as a Vunetion of h». ford.fferent propeller positions. zbeing the distance f rom,the 'ater,surface to thepropeller axis). Table 23' is u80d for determining the wavelength X at different

speeds. - . It,ia not surprising that the. uncertainty in the predetermination, of wake, can be of the above mentioned magnitude, as only some o the factors, on which the' wave system depends, have been taken into account. Theangles of waterlines and' their sbape including the pösition of 'shoulders, have for instance been neglected. Further in 'many' ships the. hull is notparticularly suitable for" the speed, because probably the choice of' principaldimensions has :bad" certain liminations. If in one öf the diag'ams'for hA, a curve Was inserte,d, corresponding to theratio h/X 'for a certain ship, this curve, for small valuösof Would' be practically parallel to the abscissa, and then suddenly for a certain value ofV/tfLrise, and later again fall, thus indicating a marked 'inf1unce of speed onwavé height. As for the majority of the models the waveheight was known at one spèed only, the diagrams cannot.be expected to give more than very approximate values of h/A

8 10 11 12 V() 5. 6. 7 '9 lO. 13.8 17.0 20.6 -24.4{ è (rn) 4.3 6.1 .3 20 ' ' 17 18 19' V () 13 14 ' 15 16 (m) 28'7 33.3. 38.3 43.5 49.1 55.b ' 61.3 ' 67.8

26 . 27 28 V 21 22 '' 23 24 25 90 2 97 8 106 2 115 124 132 , (rn) 74 9 82.3

Table 23: The variation° ofw'ae length with eed. FIg. 41. The height of the stern wave as function of the speed X'atjo and the breadth- length ratio of the ship.

-. °2ïu: .

.60

Fig. 42. Thevariation of the stern wave WIth .0/ - - 54 N. the speed ratio0 -' N

60 .80 £20 4.J5 02

.0/

.60 80 000 020

FIg.43. The variation of .00 wave wake withwave height and position of propeller. If the wave wake problems, are tobe examine4 more in detail, it will be necessary to perform-several special experiments. Thts it would be of. greatinterest to obtain measurements with pitot tubes over thewhole of the propeller disc at different speeds. By theseexperiments the wave profile should of' cOursebé.re&istered for all the speeds. It might 'then be possible toasóertain how nich the hull is "shadowing" the wave motion atthe position of the propeller. The experiments ought tobe carried out for normal as' well, as for exaggerated ship forms. As 'the speed of the ships Is Increased,the wave wake becomes more and more important. Itwill,therefo2'ß soon be necesSary to investigate the posibilitiesof utilizing it to the utmost extent. For investlgatlofl8 of cavitation thediagrams fig. 41 and fig.' 42 maybe of sonte help, when thestatic pressure is to be determined. 85

CH APTE RI I I PREDETERMINATION OF WA. F. CONSTRUCTION. OF.. A NV., DIAGRAM FOR DrERMIwING THE WA COEFFICIENT OF.. SINGLE SCREW SHIPS. Ai attempt has been made, on the basis of the considera- tionsin sections C, and E regarding the influence of the different factors on the wake coefficient, and on the. basis of tie results of model experiments perfornied at "Nederlandsch Scheepsbouwkundi.g Proefstation" in Wageningen, to construct a diagram for quick predetermir,at'ion of the wakecoefficient of single screw ships. Part of the results from thIs investiga- .tio has earlier been published (43), but are. repeatedin the following for the. sake of clearness. - Firstly. will be discussed the parrrOters to be used for such a predetermination, and later an account is given of the use of the experimental data available.

a. Parameters on which the wake coeffIcient depends.

No doubt onebrthe most' important factors for the 'wake 'coefficient is the fullness of the ship. The calculatedpo tential wake coefficients for two- arid tbTee- .dimensona1 flows indicate clearly that the lake coefficient inc'reáses with the fullness coefficient, and further 'indicates tht there i no proportionality 'betle'en the two' cOefficIente. The question now arises as to which of the fullnesscoeffi&iènts of the ship should be chosen as paraméter,' and further if a coefficient for the whole oftheship should b'e 'used or pre- ferably one for the after body or the run aloneIt is possible to use:the blockcoefficient 8 , the horizontal prismatic coefficient pór the vertical pr'ÎthatIc coefficientp. It would bemostnatural to use the block coefficient8, as this;. isthe coeffIc.Ient the designer knows 'ir'st, B i1ng i'nstead of 8tbere isnot nich gained, as in most öf 'the modern ships the idship section coefficient ', geiièzal'Ïy corres very close to1.00 If ti's ships in question have a la:rge varIation 'in 'ß válues, it rníy be ari advartage tóuse. 86

p, because p then better than 8 Indicates the distribution of displacement. Employment of the vertical prismatic coef- f iclent is not very gopd either, as this coefficient takes Into consideration the shape of the frames rather than the fullness, and still it does not give a real indication as to the shape of the frames. Many investigators are of theopl.nion,that to ascertain the wake coefficient only the shape of' the run need be considered. This also is correct to a certain extent, as the influence of the entrance on the potential field at the after end of the ship Is exceedingly small. However experimente prove that the entrance plays a certain part, but the wayin which it act8 'has not yet been made clear. In fig. 44 the wake coefficient is indicated as afunctionof speed Ío-r three different coasters, all of them tested with three different propellers (the experiments were carried out by TROOST and published (123), where more 'detailed description can he found). Thethree coasters all had identical runs and a displacement of about 620m3. One of them hada norml entrance, while the two othérs had a maier-entrance and an 'entrance with' a yourkevitch-bülb1 stem respectively. It is interesting to see how differently the nine curves run. Their shape probably to some extent reflecte the influence o1 the entrance on the frictionalbelt, but.a great part of the difference'between the curves might be due to expert- mental 'uneertainty. In any 'case it seems difficult at present to explain these differences in a satisfactory 'manner. Also a brief mention st be made, of SCHAFFRAN's experiments (93') with 2e feet models of' merchant ships400 feet long. They all had the same principal dimensions,coefficients, and shape of frames, only the distributionof the displacement differed. The block coefficient was 0.778, whilé the block coefficients of the after body varied from 0.67 to 0.84. This variation in the'distribution ofdis- placement only caused a change in 'the wake from 0.29 to o.36 (apparently too small valuee'). SCHAFFRAN'Sexperiments prove, that a small shift of the certre ofbuoyancy only has 87 a very slight influence on the wake coefficient.

5V/MuSI 56 6 99 rq - , 'r os-

I os HIUAJ%Jj769 ' M

Fig.44.The tuf luenco of shape of före body on wake coeffiò lent 9'. forthreecoasters having identical after bodies .(TROOSTs experiments).

A similar series of experiments has been performed 'by NORDSTRÖM (84). The wake coefficient was dèt.ned'for four models, all having the same principal dimensions, saxne block cöefficlont (0.625')and'satheshape of aires. Only the position òf the centre' of buoyancy was varied, The re- suits at 18 i't were: t -3 per cent -2 per cent -1 per cent O per cent of L

w: Ö.246 0.238 0.226 ' 0.217 Thus 'the wake coefficient increased by 0.òl forevery one per centthe centre of buoyancywas movedaft. Be all thl as it may, the probability isthat by u- 'sing the blóckcoeffiólentfôr the whole' ship,only a 'small error is introduced, Fromthewake' calculation ofcylinders itwas found, that the angle oflnclinatloti of, the:waterlineaft had á certain Influence on thewakecoefficient, butt.as 'this in- f luence wasvery athall it is' neglected' 'heí'e, On the other hiid 'lt is necessar to takeinto accöunt sa the breadth ofthe ship. This was proved by the calculations of flow around cylinders aswell as around solids of revolutions and ellipsoids. Therefore the ratio B/L is introduced as the second parameter, L being the length on the waterline of the ship. Actually some regard to the draught of the ship ought to be taken, but as all the ships considered in this section are sea going ships (sailing on infinite depth of water), the ratio draught to breádtb aliays being abotttOL.4, nô account of draught is taken. A small change in this ratio only alters the wake coefficient vry little (compare fIg. 32). From the saine figure it can be seen, that an Increase In this ratio gives an increased wake coefficient, and that this increase will be dependant uponthe size of propeller and on the ratio of the ships breadth to length. However the conclusion, that loading of a ship also will increase the wake coefficient, must not been drawn. The contrary will in general be the case (see section M). As the difference In two-dimensional and three-dimen-. siorial flow is large, It is necessary to consIder the shape ofthe frames of the ship. The fact is, that the flow around a ship having pronounced U-shaped frames will take place re two-dimensionally than the flow around ships having pronounced V-shaped frames. In the construction of the new diagram therefore a discrimination between ships with. normal- shape of frames, ships with U-shaped frames, and ships having V-shaped frames, will be made. Later on will be defined, what is understoodby U-shaped and V-.shaped frames. As previously proved, the propeller diameter, or better still the propeller diameter in proportion to the length of the ship has a great lafluence on the wake coefficient registered by the propeller. In del experiments it may, as previously mentioned (section C), be shown that the thic'kness of the friótion belt Is proportional to the length of the ndel, thus thè ratio D/L expresses to what degree the propeller is working In the friction belt. In 89 considering the actual ship, it is no longer sufficient to consider"solely the ratio D/L, but the otualvalue of L ist be taken into account. For simplicity L is talceì'i 'as the length between the perpendiculars, even if it would have been more correct to employ the length from the propeller tó thé' stein. A considerable number of model experiments have, in the course of time, been carried out in order to clear tip the variation, partly o the wake and parti-y of the thrust deduction with the propeller diameter, Mention should be made of the experiments carried out by TAYLOR in 930 (112). The same mode], was run with 12 different propellers of varying sizes. The result of the investigation is recorded in fig. 45 in which the wáke coefficient is indicated as a function of D/L. In the same diagram are shown results from experiments performed by KEMPF (56) and YAMJLGATÄ (131). Inthe forwer series of experiments a model was run with two different propellers, and in the latter a model was tested with 17 different propellers. Comparing the figure with figure 4 in' section'. C tt Is, seen,. that there is a very good agreement, only the wake variations meesured by model experiments are a little bigger than the ones determined according to VON KTRN's ;forìmi'].a.It' is also of interest t'o examine, if the shape othe frames has any influence on the variation of the wake coefficient with the propeller diameter, axid if so, how big this tri- f luence is. To throw some light on this' question the experiments ofYAMAGATAdescribed in (130) may be used. Four models having identical fore bodies but different after bodies were run and the wáke was measured by meáns o'f blade wheels and determined by intégration over the propeller; dsc. The shape of; the frames of the four"aft'er bodies varied from 'V-shape to very pronounced U-shape Froth the' diagram in figure 46, which gives the waké. coeffioléntas a function of D/L for the four nio'dels, it l's seen tht; the variation of, wake with D/L is fair].' indépendent of frame thape. Besides, the variations in fig. 46 'are a -little7 90 bigger than in fig. 45. Perhaps this is Just accidental, or perhaps it is due the fact tbat,in the one case the nominal wake coefficient is used and in the other the effective wake coefficient.

YAMAG.4TA .------' \ IMPP \\ TAYLOR .-

Fig.45.The variation of wake coefficient with propeller \\ diameter.

m!u-ro 195 -.U-FR

/97 --V-FR. .45 \\

-40 Fig.46. The variation of wake \\ \\

coefficient with D/L for \ .- different shapes of after \ \ body frames (YAMAGATA's .35 \ '. experiments). \ .' \ 's .30 "S N.

.03 .04 .05

That.the rake of the propeller, and the fitting of the propeller in the aperture and its shape, havesome mf luence on the wake coefficIent is certain, but probably this influenceis very small, as normally it is ensured, that the propeller aperture is suitable for the size of propeller. In section D-a3 it has been shown, that the potential wake varies very Slowly longitudinally, and in section C, that also the friction wake only varies a very little, when the propeller is displaced longitudinally within a range, which is practicable, when it is to be kept at a proper distance 91

from the propeller post (about 'twice its thickness). It Is therefore the opinion of the author that in preparing a wake diagram or formula it is not necessary to take Into consideration the rake Öf the propeller, the posItion of' the propellern the apérture., nor the shape of the áper- ture, unless the aperture is of extreme form, for instance of the type suggested byHOGNER(46). It Is a fact, that in comparing wake coefficients determinéd for the same ship and with identical propellers, only the propeller rake beiùg diffOrent, that propeller which has thé least rake

wli register the biggest , In the same way lt is not considered necésìary to introduea te skew bé.ck òf the propeller, iteblade number, blade shape, bladé aròa or number of revolutions as parameters in the new formulae. Probábly the fuùct ions mentioned may have sorné thf'luenca on the way in which the pópel±er ihtèg±'ates the wake, but thewill never be able t play any priary parta As acomparison some experlmntscarried out by BAR (12) using model 1119 may be mentoned; the iodòl was run with a series of different propellers, and the experiments proved, that variation in the folio'ving quantities have noInfluence on the wake registered: 1. xuyaber of blades1 2. coitour of blades, .3. pitch variation, a.nd 4, cross sections òf blades ODD ha's perThrmed many expo'rimentsproving, that a change innumberof' revolutions only haáverylittle effeòton the wake öef fietént. TIIUé exarriining the results 'obtátned for the ndels 1945 A and i959A (120), a variation in theiurnberofe volutionsinsidethe rangeN =90 -17 mans in this case, that themaxinnim variationin thé wake'ceff1c1en 1.s 0.015 and 0ó45 respectively. There. is no trend In thé variation. In zrst ships the propeller shaft ispositioned about 0,4 times thè draught above th& keel. If it is puta little higher orlower, is of very littleimpoztancefroniá wake point of view,becausegenerally th lines of the run are 'shapedaccording tothe position of the propellr. In ships In which the propeller ispoeLtioned either unusuilly high or unusually 92

low, lt is necessary to consider the ratio E/d (the height of' the 8baft'dIV'ided by thedraught),as the wake coeff i.. dent in the first case is. relatively big,. In the second caserelativelyarai1.An idea as to.bow big euch a correction should be, canbe obtained by studying souc experl- nte performed by TAYLOR (lof) and bTODD (120). With a nd.el about2o feet. long, havirg a block coeTicient of 0.75 andan ellipticel counter, TAYLOR ce.x'ried out some ex- periint8 with the propeller, placed at different heights above, the keel and with different distances f rothe propeller post. Three different propellers were 'Used In the experlr'nti, and in table 24 the results correspondIng to twoof the longitudInal posit lo-s are recorded. Fron the tablee it; appears,, that a of 0.1 in E/d. at p sition II. caue achange Inwake of .0.04, while the sa alteration at I causes a change of' 0.06. Asthe propel'er norcelly will be at some place between position I and II, lt 'may be.reckoned that a change (E/a)1 In E/d causes a change of wake of' 112.(E/d)1. T0D's experiments wlt.h the 'dela 1945.and 1959are of asómewbatdifferent kind, as in addition toalteringthe pqsItion of the propcllers, the propeller diameter,ar.d, thelines andcharactsr of the counter were changad.With the biggest propeller diameter, the prOpeller was highest above the ,keel..and the counter was elliptical, whiiE with the smallest propeller, it was In the lowest posit lori and the stern was a very ful). and. deep cruiserstern. In order,hettcr,,to be able to conpare thé wake coefficientsneasired, these have beencorrected so that theall are valid for the ideal P/L = 0.o4. The result is represented in figure 47, the abscissa being E/d and the ordinate the wake coeffi.lent. Inthe same figure is indicated the result of TODD's .experimen.t. with. 1445 (119).,If a mear. curve is to be drawn in, lt must be a straight iIe making an angleof.450 with the axés. Thus the .ròsult is identical with that found by TAYLOR's ex-

.perlent. . .. ' ' 93

Position II Position

Prop No 570 569 571 570 569 571

Wd=O.6 39 37 38 .53 46 43 W EVd=0.2 24 21 21 24 25 22

Table 24 The variation of vake coefficient with height of propeller axis above keel (TAYLOR's experiments).

0L f945 1 1959 f---. / 1445 / f445(d.13S + 05

The variation of wake Flg.47coefficiet with thé heIght of the propeller axle above keel (TODD'S experiments).

.30 140

Finally ltiàtbe eritiòned, thatw's varlticn with E/d. is conaiderable bigger at ships having V-shapòd framea than at shipswitliU-shaped. That the rudderplays aconsiderablepart with regard tothewake coefficient,will be dealt with in détail in section L, but aemoet rudders Inmodern ships are of the saineshape this àffect will beneglected at thebeginning. When the speed of the ship is altered, thé stern wave andthereforethe wákecoefficient will change, and it is also likelythatthefrictional wake changes. with speed. The questlòn will be more closely dealt with in, a later sec.- tiori, and it should ònly be mBntloned. hero, that atpresent it Is riot poùlblé to introduce any speed parameter. Naturally the conditionof theshIp is of aat importance for the wake coefficient. The influence of change in draught bas justbeen mentioned, and as regards thèInfluen- 94

ce of thé trim of the ship, this wil]. be dealt with in section M. Also the condition of the surface of the ship and the degree of fouling plays their part (see sectionM)0 By a change, of the thrust load of the'propei].er the wake coefficient.1 determined by FROUDE's method, will change, but the alteration wUl not be big. Comparing exper. mental reults, determinedby the continental method, with resultsdetermined bythe Britlh method, it will generally be found, that the wake coefficIent deternired by the fIrst method isabout 0,01larger than that obtained by the seoord method, the reason being thedifference inthe race- contraction. The higher the thrust, the bigger the race- contractIon will beA very thorou investigatIon of the influ.énce of the degreeof the propeller thrust loading on 'the registered. wake has beencarrIedout'by VAN LAEREN

(65), ,. '. In an attenpt to produce a forla or 'a diagram for the predeterminat!.on of the wake coefficient for ode,ls it therefore seemslogicalto Introduce the following pararne-

tars: ' i. The block coefficient (the total displacement divided by the length of, the load waterline., thé waterline

breadth aridthe, average draught).. ' The breadth-lengthratioB,/L breadth of waterline di- viöed by length ofwaterline. The dImot.er-length ratioD/L, propeller dtaeterdI,

videdby the lengthbetween perpendiculars. ' The ratio betweenthe: height of the propeller axis, abovethe keeland the draught aft, Eid. This parameter is only used when E/d deviatesvery much from 0,4 (ses later)0 A parameter taking into consideration. the ±rame shape (will later be defined). 6,, A' parameter for the hape of the rudder.; only to be used in special easeS. A parameter for the distribution of the displacement. 'A parameter for the conditIon of the ship. b.. Analysis of_model experiments. The results frOm about200xwdel experiments, ail of them from single screw ships and ail carried,outat the Wageningen tank,have been employed forthis wOmit. By solely using data from ne tank,the followingadvantages should be obtained: IAll the wake coefficientshave beendetermined in. the sanié way. 2. The sanie artificial turbulerc.e generator baa been used in all the experiments (this however is only ofsecon- dary impoit.arice), 3, All themodels have been of the sanie material and their surf'ace8 treated in the sanie way. One disadvantage in ònly employing one set of resulte is the however rather unlikely possibility of the pÑsence of one-sided errors. The experiments used, have been carried outoniidels of normal cargo ships, tankersand passenger ships as well as for tug boatsar.d coasters. Ships having nozzles and tunnelshave been excluded., and ónly results ofmeasurements, carried, out in deep water, have.been used. All theexperi- sientahavebeen performed according to the contipental method,andTAYLOR's wake fraction has been uaed in all the diEg1'eui8. In most of the experiments a trip wirewa3 fixed at 5 per cent of thelength between perpendicu1aa from F.P. tostimulate the turbulence. In the analysis the ships firstly werò arranged in 'oups according to their breadth-length ratió, and inside each group thewake coefficient wasshown as a function of the block coefficient, but the scatteringof the points was so extensive, that it was extremely difficult to draw a mean curve. It was now assumed, that the.cause of this scattering must be the large variation in thera- tló propeller diameter to length of zrodel0Therefore to begin with a D/L correction diaam was prepared (indicated in.. fig. 52.) by using the curves offig045. The correction curve was chosen as theaverage of the curves shown, 96 Then ail experimental data were corrected to D/t equal to 0.04 and again plotted in the same way as before. Now the sc&ttering was less, and mean curves could more easily be drawn. The result, is given in figure 48. Most of the pointa; mark the effective wake values determined by CT_identity, while the remainder represent the nominal wake coefficients (i). The nominal wake coefficients are average volume values, determined by blade wheel measurements. Many points in the diagram are connected by vertical lines. Points so connected are fromthe same shipand denote the limits within which the wake coefficient varies,whenthe speed varies within the normal rexìge of speeds (for a 13 1/2 knot ship the range is lo - 15 knots).

:t 4i-.I3 -m, L. . .5 5 b ma Jâ -+ r 5OjIIIIIII.uuIIIIII!1 . - A: i .1

IIi'r R 4..f7 I. I !L_ L. '.

Fig. 48. Wake coefficients found by experiments (single screw ships (D/L = 0.04), Wageningen experiments). 97

For the determination of the mean curves In, the different groups some of the results from section D wereuebd. By correoting the wake values so they are valid for models having D/L = O.o4, the part of the variation caused by the frictionalwake badbeen practically removed, and therefore the rest of the wake variation snict principally be due to the potential wake. As the potential wake variation with B/L and8 had previously been determined for flow around cylinder shaped solida, it was only natural to examine, if the theoretically found curves (fig. 13), after displacement in the vertical direction, could be used as mean curves for the values found by experiment. It was very soon proved, that by displacing the whole set of curves vertically as one unit, a quite good agreement with moat of the B/L values was obtained. This might leadone to believe that the frictional wake was independent of the shape of theship, butas the ships, arranged according to increasing breadth, got more and more V-shaped frames (see later), it seems not unlikely that adrop in thepotential wake coefficient, which reallyshouldbe expected for the more V-trained models, is counterbalanced by an increase in the frictional wake coefficient. This means that the frictional wake should increase by increasing breadth of ship. Having thus determined a set of mean curves, the next step was to find the reason, why all the points did not lie on the mean curves or in the vicinity of them.It was necessary first ofair toascertain which shape of ships, and which propeller and rudder arrangementswere normal, and which were abnormal. Therefore the po8itiOn of the propeller relative to rudder and aperture, the shape of the rudder, and finally the shape of the frames inthe after body were examined for all the models. Certaincoeffi cienta charaoterising the property concernedwere established, and finally the normalarrangementsettled by finding the average values for the different coefficients.

In fIg. 49 is indicated a propeller- and rudderax'- rangement, and in this figure are given themeasurements used in cbaraoterising the arrangement shown. Thecoeff i- 98

cienta t/l, ]./D, a/D, b/D, and c/D, where D Is the propel- 1er diameter, were calculated, and their average values recorded In table 25. In the same table are given average values for D/L, d/B and E/d together with the value of the tangent to the rake-angle of the propeller. However only a small part of the scattering can be explained by deviations from the average values of the tables (D/L and E/a being exempted),

Pig.49. The characteristic dimensions of the propel- br-rudder arrangement.

Rake D/L a/B E/a t/i 1/D a/D b/D c/D tan

.404 .405 .394 .162 .705 .106 .192 .099 .178

Table 25: Coefficients of the "normal form".

Next there was the frame shape to be investigated. It was examined, whether It was possible,by means of the vertical prismatic coefficient, to characterise the frame shape, but It was soon discovered that thiscoefficient was not very well adapted for this purpose. Then a new coeff i- oient f, as indicated in figure 50, was computed. This coefficlent was calculated both on frame I and on frame 2

(L is equal to the distance from frame O to frame 20). 99

Fig.50. Definition of the frame shape coefficient (f).

As the coefficient on frame 2 gave a considerably better characterisation of the frame shape, than that on frame 1, the one froa frame 2 was preferred and was used in the following calculations. In the diagrams in fig. 48 the points having very big and very small f values clearly lay on each side of the curves, while the pointshaving moderate f values lay grouped around the mean curves. In order to study the main tendency in the position of the points a little closer, for each B/L the average valua of f for all the points and for the points above and below the curves respectively, were found. In fig. 51 the result is shown. The curve y denotes the variation of theaverage value of f with B/L for points situated belöw theaverage wake curves; thecurve wdenotes the same but for all the pointa, and finally the curve u indicates theaverage valuesof f for point' situated above the wake curves at the different ratios B/L. The two dotted curves indicate the f-values for ships with pronounced V-shaped frames and for ships having pronounced U-shaped framesThe figure shows that generally speaking the ship with U-shaped framesare lying above the average wake curves, while ships with V-shaped frames are lying below. In other words the result of the investigation is, that for ships having eithermar- ked U- or V-shaped frames, the shape of the frame mist be considered,while this Is not necessary for ships havîng loo slightly U- or V-shaped frames. It is the authors opinion that for f values between 0.30 and 0.60 at B,"L = 0.13, and between 0.40 and 0.90 at B/L = 0.16, it is notnSceS3aPyto consider the frame shape.

w wo r f V, // ,' - iT1ls,zJ! '1

Fig.51. f's variation with B/L and with wake coefficient. A I hW - 5 .40 __"

.17

An investigation of the influenceof E/d proved, that ships having a big E/d have a bigger wake thannormal, and fòr ships having a small Eid the opposite is the case. On the basis of these results a diagram for the pro- determination of the wake coefficient was prepared. The diagram is shwn in fig. 52 and gives besides abasis coefficient, corrections for the propeller dialTtetr,the frame shape, and the height ofhe propeller axle. The diameter correction can be used for all diameters,where D/L differs from O.o4. The diagram for the frame shape correction is employed for f-values outside the rangegi- ven above,and the E/d diagran is used, when E/d is outside the rango 0.35 0.45 (no correction in the case of ships having marked U-shaped frames) It may probably seem a little peculiar, that correctionsfor f and E/d should only be carried out in special cases,butit appears, that the degree of accuracyin the determination of w does nàt increase by making the correctionuniversal. Unfortunately however the corrections often are very large and the w-values therefore influenced by theuncertainty Of the corrections. Anidea of the uncertainty in w can be obtained by looking either at fig. 53 or at fig. 54. In fig. 53the abscissa is the value of the wake measured,while the 101

"POPELLEPDIAMETER CO2ECr/ON

.05 Fig.52.,Diagram for predeterminati.. o onof the wake coefficient of aingle .05 screw ships.

.5, 020

I I FPAMECOQECT/ON

.05 11-FRAME :.. 7 . T7.

V-FRAME

.15 50 .55 .60 .65 .05 75 6 L6' COECT/ON

.60 .05

ordinate Is the wake, foundby using.the diagram.s. It is only the results, which have been employe& in theconstruc- tion of the diagram, which have beei tested here.Thebig scattering18In certain cases due to experimental uncer- r

102 tainty; in some of the experiments'rrwas foundequal to 1.10, which in all probability Iserratic.Inother cases the scattering is due to inacoux'acyin the diagrams, or due to the fact that nominal andeffective wake coefficients bave been used indiscriminately. Also thevariation of the wake with the speed is a cause of uncertainty.The next fi... gurefig. 54, indicates how often one can expect by using the diagramsto find a wake coefficient,thatdeviates from the correct wake coefficient by less thant w Thus it seems possible In nearly 90 outof loo cases to determine a w which differs less than± 0.o4 from the "correct" value (the experimental value).

- w,

G G. o 40 .G : :. Fig.53. Compari- s :. son between observed and cal- 30 culated wake (Wageningen experiments).

.10 /.

.10 .,0 . w

Fig. 54. The probability of determining w with accuracy Ê w by means of the diarani fig. 52. 103

At times it will be proper to correct for the distz'i- button of displacement, and this can be done by using NORDSTROM's results. The author is of the opinion that when the centre of buoyancy is situated thus:

1/6= 3,0 4,o 5,0 t = +1 per cent O pez' cent -2 per cent of the correction will be equal to O. Further, as previously mentioned, a displacement of the centre of buoyancy of -i per cent will cause an increase in the wake coefficient of 0.01. In connection with this analysis of model experiments, an analysis of variation of wake with speed, and of the re- latton between nozflinal and effective wake coefficient was carried out. In the two following sections the results of these investigations will be described.

bi. Variation of wake coefficient with eieod. As previously mentioned the wake coefficient of a model varies due to a variation in both the wave wake and the frictional wake. It has been attempted troni the experimental results, used above, to find the lines on which this variation takes place. Unfortunately it is difficult, from the experiments carried out foz' the ship yards, to draw any exaot conclusion regarding the variation of the wake coeff t- oient with the speed, as the experiments have only beencarried out at big intervals of speed. An investigation of the main tendency of the variation gave the following result. In about 65 per cent (group A) of the experimente examined w was falling at rising speed, in 25 per cent (group B) w was constant, and finally only in io per cent (group C) w was rising. The variations were not very big; thus in group A the wake coefficient was falling about O.004 for every knot increase of speed. Takinggroup A plus group B the average diminution was 0.003, and finally taking the sum of the groups A, B, and C the diminution was about 0.002 for every knot increase of speed. As a rule ships with relatively small propellers belong togroup A, 104

while shipswith big propellers willbelong to group B or C. Thus tug boats, pilot- and patrol boats will as a rule be in one of the two last groups. With very small propellers the largest variationwi].l befound, which indicates big variations in the friction belt. The blade wheel measurements (mentioned in sectiân E) also gave the result that wake regi- stared by small blade wheelsdecreased farmore than that registered by big blade wheels. If the small oscillations of the wake are to be studied, measurements must be taken at very small speedintervals, and than it will be of interest to. compare the wake variation with the variation of the shape of the 8tern wave (42). One big question le, whether the variation found experimentally will be confirmed in the actual ship. Variati- ons due to the stern iave are sure to be repeated,but on the other hand It is not known, whatwillbecome of the frictioncontribution.

b2. Relation between nominal and effective wake. As mentlone4 earlier, the nominal as well as the of'- fectivewake coefficients wereemployed inthe preparation of the wake diagrams, it being expected, that there would not be any big dIfference in the two coefficients. It also appeared that the nominal wake coefficientsInthe diagrams fig. 48 were in much the same position, withregard to the mean curves, as the effective wake coefficients. Later ou the measurednominal wakecoefficients were compared to the coefficientsobtained from the diagrams(f ig. 52), and it appeared,that in45 per cent of the investigated cases the nominal cOefficient was leus than, in 4o per cent bigger than, and in 15 percent equal to, the diagram wake. The deviations from the diagram values were bigger than by the effective values. The curve in fig54 had to be lowered about lo per cent at w -= 0.04 in order to be valid for nominal wake coefficients. In a few of' the experiments the wake has been determined partly by FROUDE's method andpartly by blade wheels, snd here lt appeared, that the effective wake 105

sometimes was bigger (5e per cent of the cases), sometimes smaller (35 per cent of the cases) and sometimes equal to (15 per cent of the cases)thenominalwake.But as the experimental material was somewhat limited, the figures must be taken with the greatest -reservation. Investigators have often dealt with the relation between nominal and effective wake,, but as yet the question is far front solved. In (132) YAMA()ATA gives a formula fox' determination of the effective wake coeffIcient, when the nominal Is known. But the formulapresunesexact knowledge of the se- parate propeller blade sections, and therefore isnot ge- neraUy applIcable. Also PROHASKA and VAN LA1REN have studied the question (87 and 62),especiallywith re.gaTd to an analysis of the mode of acticn of the blade wheelsand the relation between the nominal wake coefficient measured by blade wheel and thatmeasured bypitot tubes. One o! the mainresults is,that by wakeIntegrationperformed with blade wheels, their mass moment of inertia together with the degree of heterogeneity of the field will play a very big part In the wake registered. In the publicationsmentioned is explained how to take account of these relations, but the uncertainty in thecorrectionsIntroduced is great. When the propeller designer in daily practice usesthe nominal wake coefficients, it is, a3 a rule, for settling the wake distribution. If also he bas an effective wake coefficient at his disposal, he generally raises or lowers thewake dIstribution curve, until agreement between the two coefficients is attained. If there is a big difference In these coefficients, It may indicate that one of them Is wrong, and in such a case It is the authors opinion that it would be safer to employ the value, determined by SCHOENHERR's. formula (see later), or by the authors diagram.

G. CONSTRUCTION OF A NEW DIAGR8M FOR DETERMINING THE WAKE COEFPIC lENT OF TWIN SCREW SHIPS. The amount of information available for twiñscrew ships was not nearly -so big as for the single screw ships. It has 106

therefore not been possible to carry out a thorough Investi- gation, but certain guiding principles for estimation of wake can be stated. As the potential wake is the moat essential wake com-. ponent of twin screw ships, it is natural to examine this first.. The calculation of the potential wake of cylinder shaped solids and of solids of revolution has proved that the wake coefficient Increases wher, the block coefficient and the breadth-lengthratioincreases. Fig. 55 Indicates the wake coefricient at a distance B/4fromthe centre line and at a distance d/2fromthe watersurfaceas a function of the two parameters for the two kind of solids. From the figure it can be seen that an increase In the block coefficient from 0.50 to 0.75 will cause an increase in wake of 0.o8. Thus the variation Is not as big as for single screw ships. From the calculation of the potential wake it also apprears that the propeller diameter is of secondary impoz'- tance for the registered potential wake.

Fig.55. The potential wake coefficIent at the .20 // I r / afterperpendIular of S401E5 z1_- r the "ship" at a distan-

e B/4 from the centre b line and d/2 below the water surface.

PIT- I -40 .50 60 .70 Leoô

The friction wake has not nearly as big an influence as for single screw ships, the tip clearance In twin screw ships being usually large enough for the propellers to work clear of the friction belt. By model experiments the propeller will work partly in the friction belt, on account of Its reÏatfvely greater thickness. But also in this case the influence of friction on the wake coefficient Is small, which has been confirmed, among others, by HUGHES' experiments (49). HUGHES has performed a long series of experiments with 107

a model having a block coefficient of 0.66, and with propel- 1ers of different sizes. Some of the propellers were placed a constant distance front the centre line of the ship,. and 80mOplaced so that the tip clearance was constant In each case the variations of wake were only small; when the pro- poller diameter increased, and when the tip clearance was reduced, the wake had a slight rising tendency. It is therefore probable that for twin screw ships the wake coefficient can be considered independent of the propeller diameter. HUGHES' experiments also show that a displacement of the propeller longitudinally, and a change in the height of the propeLler axis above the keel, do not cause any big changea in the wake either. A. series of experiments in which the propeller axis was supported by shaft brackets proved, that the wake coefficient was increased by 0.03, whenthe ratio of the height of propeller axis above the keel to draught (Eid) increased by 0.2. In the case of a ship ha- vi.ng bossings instead of shaft brackOts, the variation was still less. Just as in single screw ships, the shape of the frames play a certain part, but no reports of experiments dealing with this question can be foun&. Most of the twin screw ships bave V-shaped frames, and thus it will rarely be necessary to correct or the frame shape. mJGHES.also perfornied eperimènts comparing the influence of shaft brackets versus bossings on the propulsion. His experiments proved that, in general, when changing from shaft brackets to bosses the wake coefficient increase5 by 0.03. In some of the experiments (proellere placed very high and propellers with. small immersion) HUGEES, however, found the oppo5ite variation. Also the angle which the boss makes with the horizontal plane and the direction of revolution of the propeller play a part. This relation was f ir8t provedby LUKE (75), and his resülte with a model having a block coefficient of 0.65 are represented in fig. 56. Later on SCHOENHERR (96) has taken up the question, and his results are found in fig. 57. It is to be noted, that only in ita broad features is there agree- 108

- Fig.56. The varaion of the wake 1PN/W,.....- coefficient with. the aug18 be- tweenthebossings and the non- zontal plane at in- and outwanti turningpropellers (according to LUXE).

A¿ES' 50 NG 0 20' 40' 60'

W jb4PD flVIN6PFOPELLERS-- .30 uIIuI-I- Fig. 57. The variation of the wake coefficient with the angle / between the bossings and the ',/ ,/' I horizontal plane at in- and out- s %j// ward turning pr3pellers (accor- .. ding to SCHOENEERR). 0I.IiiU

nfent between the results of the two investigators, In practice the disagreement is of no Importance, asthe angle between the bosing and the horizontal plar.e usually will be in the vicinity of30Besides the position of the bossings, their shape also plays a part with regard to the effective wake coefficient. If the bosses are given a practically horizontal, plane underside, it will produce an inward turning flow, which will give an increase to the regIstered wake coefficient, with propellers turning outwards. In the opposite case, the bosses having a plane top and the propellers turn outwards, the wake coefficient re.stered will be smaller, than with symmetrical bossings. It should, by the way, be noted that experiments prove that the highest degree of' propulsion efficIency is attained by having synetrical bosses (67). Many other factors, such as the breadth-draught ratio of the ship, the shape of the propellers, their rake and position relatIve to the bossings alsò play a part, but no systematic experiments are to. be found forthrowLnglight on these relations, 109

1i -i

). (ß) a_ffi) il H, J::f.

-50 6

Fig. 58. The variation of the wake coefficient with block coefficient (twin screw ships): - Top: models run at Wageningen. Bottom: models run at Berlin (B), G8teborg (G), Hamburg (H), Michigan (M), Parts (P), Ted.dington (T), and Washington (W). A: morchant ships with A-brackets, C:cruisers, D: destroy- ers, S: submarine, and 2R:merchant shipswith two rudders. The wake is registered by outward turning propellers.

A formula or a dtaam for the predetermination of the wake coefficient can at present only be built up so that it takes account of the most, important factors. The material available, has been arranged in the diagrams fig. 58 in which thewake coefficient isgivenas a function ofthe block coefficient only. An attempt was also made to introduce the breadth-length ratio, but lt was found impossible onaccountof the insufficient information. The investigation has been divided into two sections, the experiments performed In Wageningen being separated from expert- niente carriedoutat other tanks, becausefarmore infor- 110 mation was available for the Wagen.tngen- experiments, than for the other experiments. From fig. 58 it can be 8aen, that the same mean curve can be used in both of the diagrams. As mean curve a curve of the type found by calculating the potentia]. fields has been used. The slope of the curve is not quite as big as that of curves corresponding to the wake at the after perpendicular, at the centre line, and at half draught; but still theslope isbigger than that of curves corresponding to the.wake at the after perpendicular, at one fourth of breadth from he centre line and at half draught (compare fig. 55). It has notbeenpossible to dis- cover the reason for all the deviations from the mean curve. Only in cases of the wake coefficient of older war ships (C, D, S) or of merchant ships fitted with two rudders just abaft the propellers (2 R), or in cases of the shafts being supported by shaft brackets (A)couldthe devtations be easily understood. Frora the Wageningen-experiments the average fig'.u'es of certain characteristic quanttes were found in order bhat possibly in this way the deviations could be explained. The average values found were: Breadth-length ratio B/L = 0.127 Diameter-length ratio D/L = 0.030 Draught-breadth ratio d/B = 0.344 Shaft height ratio E/d = 0.382 Propeller distance from A.P. 0.044 L Angle of bossing with the horizontal 31° Tip clearance 0.005 L, and for the great majority of the models the frames aft were of V-shape. But only a small amount of the scattering could be explained bydeviationsfrom these coefficients. Therefore it must be due to olnciding circumstancesandto inaccuracy in the experiments. The mean curve of the wake which has been established here agrees quite well with TAYLOR's and SCHOENBERR's formulae (8ee section I). In designing it is suggested that the curve indicated in fig. 58 be employed, and corrections carried out in the follow- 111 ing cases: Deviation of B/L from 0.13. For correction, fig. 55 eón be used. In case of pronounced U-shaped frames,w is increased by O.o4. Incase of shaft brackets, w is dinilahed by O.o3. In case of bos8ing angle deviating from30,diagram fig. 57canbe used for correction. In case of inward turning propellers, diagram fig.57 can be used for correction. In case of unsymmetrical bossings,a correction ought to be made, but the magnitude of tbl.3 correction cannot at present be given.

- 4.J2D -- -.13 .

Fig.59. The nominal wake coefficient of ,win screw ships as a function of the block coefficient (Wage- ningen experiment). :

50 .55 .60 .65 3

In fig. 59 the nominal wake coefficient, determined by blade wheels, Is indicated as a funct.on of the block coefficient. The wake coefficients, are average volume values and determined as mean values of the wake regIstered by right- and left-handed blade wheels. In twin screw ships It Is ab8o- lutely necessary to carry out experiments with both kinds of blade whee].s, as a single series of experiments, using either right- or left-handed blade wheels, will only give, on account of the inclined inflow, a wake coefficient which Is dependant upon the pitch of the blade wheels. Often conclusions are drawn from the experimental results with regard to the rotation of the mf lowing water, but this is erroneous. In fig. 59 the in- sertion of curves for different B/L ratios has been attempted. It Is seen that the deviations In this diagram are far smaller than in the diagrams of effective wake coefficients. It is interesting to note, that when SCHOENHERR's curve (see later) 112

for the effective wake coefficient, determined asthe average of coefficients found byinward andoutward turning propellers, is added, the curve will go right through the groupof points. This indicates, that for twin screw ships at least,the effective wake coefficient is equal to thenominal. When calculating the dimensions of propellers,according to the circulation theory, lt is conon in thecalculation to use the effective longitudinalvelocity of the mf lowing water. Accordingly this velocity can now be determined by using the curves in fig. 59. The variation of the wake coefficient with speed was clearly different f ron that for single screw ships. A small statistical investigation gave the following result: In about 40 per cent of the cases examined, the wake coefficient was rising with the increasing speed, in 30 per cent it was constant and in 3o per bent it was falling. In general the variations were only small.

H. THE WA COEFFICIENT OF SPECIAL TYPES OF SHIPS. As certain ships, such as fishing vessels, tug boats, ships having the propellers arranged in nozzles or tunnels, ferries having forward and. after propellers, patrol and pilot boats, and ocean yachts are difficult to arrange anng the other typea of ships, they will in the following section be dealt with separately. Together with. the wake coefficients also the coefficients of thrust deduction will be given.

a. Fishipgvessels. The smallest fishing boats, of which the wake coefficients were available, were three fishing boats having a displacement of 112n3 (82). Of these three boats, the one was of an older design (riumberl in table 26), while the other' two were modern (no. 2 and 3) In the two latter, it had been ensured that the conditions of the flow of water to the propeller were good. In table 26 the most important dimensions and coefficients are recorded together with the wake coeff i- L (ni) y (in3) 18.9 112 1 20.5 112 2 21.9 112 3 26.2 183 4 26.2 177 5 27.6 228 6 33.5 350 7 38.0 402 8 .D/L B/dmB/L 0.0930.3660.320 2.26 0.0860.3220.294 2.14 0.0800.3280.254 1.99 0.0780.5100.216 2.33 0.0640.5200.216 2.46 0.0600.5290.236 2.71 0.0720.4830.204 2.0]. 0.0740.4960.184 2.30 WQwpEId .23.290.44 .17.200.43 .15.200.44 .15.190.38 .18.190.38 0.39 .22 .14.190.34 0.41 .19 v(RN) t Table 26:6.5-9 Wakeand.35-.40.19-.24 thrust deduction data 6.5-9.5.25-.28.15-.17 .21-.24.14-.16 7-10 .27-.31 7-9.5 - .19-.20 7-9.5 -of fishing vessels. 7.5.15. - .24-.29 2-12 - .13-.14 9-11 1,14 cienta and the coefficients of thrust deduction. No, 4 and 5 in the same table give the same quantities for vessels a little bigger (3o). No. 4 is of older design,while no. 5 is of new design. No. 6, 7, and 8 are still bigger fishingvessels; a more detailed description of 6 and8 can be found in (91), while no. 7 refers to an unpublished Wageningenexperiment. From the table it appears that the wake coefficient re.. mains relatively fixed, while the coefficient of thrustd- duction varies considerably.. From these data the diagram fig0 6o, t be used for design, has been prepared. The wake coef- f iclent is here given as function ofB/L and 8 If .D/L dif.. fers -ery much from 0.07 correction may be carriedout. in the usual manner (fig. 52 and 104).

i Fig. 60. The wake and thrust deduction coefficient of fishing .20 - ._ll. vessel3 (6 froni 0.30 to 0,50 and D/L = 0.o7). .15

.18 20.22.24.26.28. 3O

Often lt is of importance to have some knowledgeof the propulsion, characteristics for vessels in the trawling condition. Firstly the ships is considered sailingfree at its greatjst possible speed. Inthis condition there Will be a certain w and. t. If the resistance la increased by putting out a trawl, and if the engine constantlydevelops the same horsepower, the speed will diminish and at the sametime 'the coefficients of propulsion will change. Generally thewake coefílcient will increase-and the coefficient of thrust de.. ductióii will decrease. If the trawl resistance is gradually increased, the wake coefficient to begin with only changes very little, but later risessteeply. It may be estimated, 115,

that a change of speed troni lo to 4 knots will give an increase of the wake coefficient of 0.03. On the other han4 the coefficient of thru8t deduction will vary nearly linearly, and it may be assumed, that at zero epeed lt will bave the value 0.03 - 0.04. In ships having a badly shaped propeller post t however can be as high as about O.lo at zero speed0 As the wave wake in this type of ship, on account of the big V,[L is of great importance, it will be found, that the wake coefficient, when the ship is 5ailing free,will rise considerably with increasing speed.

The wake coefficient and the coefficiènt of thrustdeduction for bigger trawlers can be determined in thenormal way, by employing the diagrame fig. 52, and fig. 104 (section U).

b. Tug boats. In table 27 data for five single screw tugboats tested in Wageningen are given. Poux' of them have normalpropeller

i 2 3 4 5

y (ni3) 14 54 476 555 17C

L (ni) 11.3 15.8 42.4 39.4 22.6 B/L 0.276 0.260 0.197 0.208 0.266 B/dm 2.60 2.57 3.00 2.38 2.18

0.35.2 0.523 0.485 0.509 0.457 D/L 0.074 o.1o8 0.066 0.059 o.o80 Wda 0.41 0.41 0.44 0.40 - w .20 .24 .16 .3-.26-.19--.21 t .20-.25 .17 .28 .21-.26 -.22--.28

v() 5-9 9 11 . 10-13 5.5-9.5

Table 27:Wake and thrust deduction data of tug boats. 116

arrangement, while the fifth has thepropeller fitted in a nozzle. In. designing tug boatsof normal form and withnormal propeller arrangements, w and t canbe determined by using fig. 61. wt61F - .25 .-- Fig. 61. The wakeand thrust deduction coefficient of tug boats(D/L = o.o7). .20 - -

15 1V

.20 .22 .24 .26 -28

During towing, the wakecoefficient and the coefficient of thrust deduction arechanged as indicated in fig. 62. The curves bold good for tugboat no. 2, which bas been tested with two different propellers,and the curves correspond to constant horse power.

t .50

-40 Fig. 62. Thé variation of wake and thrust deduction coefficient when towing, utilizing the full horse power.

Regarding no. 5,ships fitted withnozzles will be dealt with in detail in section e.

e. Ships-in which the propellers are f ttted innozzles or tunnels. When a propeller Is surrounded by anozzle, the wake coefficieùt alters partly due to a change in thebasic flow, 117

and partly because the mode of action of the propeller is changed on acòount of diminished race contraction... There is however the probability, thai the cImge in .th as.ic - flow is only of minor importance.. Some blade wheel easiirements con- firm this assumption. On the other hand the nozzle causes a big change in the effective wake, determined in the normal way according to FROUDE's method. In tug boat no. 5 (table 27) the nozzle has had the effect that the wake coefficient as well as the coefficient-of thrust deduction have become negative. Unfortunately infallible gudanee , fir. usó in designing this type of ship, cannot be given, Also in the case et ships having the propellers placed in tunnels, nothing is practically known. The fact, that shipwith n zzlea and tunnels, as a rule are 4esigned, so as toTbe most economical when-sailing In shallow-water, does not improve matters. A slight-idea of- the. intrieácy.ofwake

-probleme. In these types . of ships may be got. bylookinE -at fIgures 63 and.64. Both-the figures are tho'resuit.ôfWage ningen- experiments with a tri-pie screwed tug bóat- (l3oo'H.P.) fit-ted with tunnels. The first figure: indicates the.: variation of-the wake-- coefficient-with: Speed, at normai- propulsion exper-iments, for the side propellers as well as for.the centre propeller. Results from three different depth of water: In-. finite, 3.5, and 2.o n, are given. The draught of the tug boat was 1.40 m. In the second figure (fig. 64) the resulti from some towing experiments are given. At all speeds the full horse power was developed. It can be seen that at some

J- Fig. 63. The variation .50 of wake coefficient with 40 speed of a tug boat having three propellers in / tunnels (three different depths of water, C = 2m centre screw, S = side screws), o -

. -8 '2 118

w - - .40 'C ------Fig.64. The variationof wake coef- .20 w.12- " ficient with speed of a tug boat having three propellers in tumiels, - duriñg towing, utilizing the full horse power, on three different depths of water (C = centre screw, S = side WAU screws). .80 VAUU rauu. speeds the wake coefficient assumes big negative values. For similar ships even still bigger negative values, up to about 1,50, are met with. The negative wake coef2icients zwist be due to the deficiencyof PROUDE's method.

d. Ferries with forward_and after propellers. Only data from two ferries of this type were available. The one was tested in G8teborg (83) and the other in Hamburg. Table 28 gives data and coefficients for the two ferries and in fig. 65 variation of the wakø and thrust deduction with speed ha3 been indicated for ferry no.1. For ferry no. 2 w and t of the after propeller varied from 0.35 to 0.39 and from 0.35 to0.38.respecttvely; w diminished with speed, while t was fairly constant. miring the test the

1 .2 (m3) 365 388 L (rn) 35.4 39.0 B/L 0.237 0.218

B/d . 343 375 6 0.501 0.518 D/L 0.045 0.045 0.35 0.56

Table 28: Data for ferries having after- as well as forward propeller. 119

forward propeller was running free. - The difference in the two sets of eperimenta1 results ntxst be due to the difference in the placing of the propel- 1ers ox' to the difference in. the rudder arrangements. In designing, the normal formulae and diagrams (fig. 52 and 1o4) can be used in. fixing w and t for the after propel.. 1er, when. it is working alone. FrOm fig. 65 w and t can be estimated for the forward propeller and .forthe after propeller when both propellers are working,

w .35 35 IF

.30

.25 25

A .20 ---: M .20 _-__.__

.15 .15

8 84 9 9.5 10/CN 8 85 9 95 lOHN

Fig. 65. The variation of w and t with speed for a ferry having forward and after propellers. A : after propeller working alone. F : forward propeller working alone, AP: after propeller both propellers working. FA: forward propeller )

e. Patrol and pilot boats. Table 29 contains data and coefficients for some vessels of this type tested at Wageningen, and in fig. 66 is given a design diagram derived from these data.

Fig.66. The wake and the thrust deduction Coefficient for patrol and pilot, boats : (D/L = O.o5) .15

o .40 42 .44 .66 .50 2Ö

1 2 3 4 5 P P P L L

V(ni3) 143 252 385 24.2 479 L (ni) 30.0 34.0 42.1 14.6 41.8 B/L 0.187 0.171 0.181 0.250 0.191 B/do 2.95 3.11 3.17 3.33 2.71 0.448 0.492 0.502 0.414 0.487 D/L 0.045 0.048 0.042 0.062 0.043 0.40 - 0.40 0.41 0.40 w .16-.19 .20 .23-.24.13-.18.22-.24 t .11-.17.21-.24 .25-.26.15-.19.18-.20 v(KN) 9-13.5 9-13 10-13 5-9 9.5-14

Table 29: Wake and thrust deduction data for patrol and pilot boats.

f. Ocean yachts. Only for a single boat of this type wake and thrust deduotiQn data were at dispoeal: V =56ni3 L=15,8m B/L=0.3o8 B/d=1.95 6 = 0.29 D/L = O.o67 E/d 0.44 At 9,5 Iaiots the nominal wake coefficient was 0.20 and at 7 and 1imots the thrust deduction coefficient wa0.24 and O.2o respectively. (Wagen.tngen experiment). 121

CHAPTER IV: FORMULAE OF WAKE AND THRUST DEDUCTION.

I EARLIER_FORMULAE AND DIAGRAMS FOR DETERMINING THE COEFFICIENTS OF WAKE AND THRUST DEDUCTION OF SINGLE AND TWIN SCREW SHIPS. In the course of time many investigators havetried to find a relation between the form of the shipand its condition on the one hand and the wake and thrust deduction coefficients on the other. In this sectionsome different, older formulae and diagrams for the determinationof the two coefficients wILL be mentioned and briefly discussed.As the methods and principles, employed by the differentinvestiga- torsInpreparing the formulae and diagrams, have ineach case been the same forsingleand twin screw ships, the two types of ships wi].]. be dealt with at thesame time0 The for- rnulae and diagrams for the coefficientof thrust deduction have been includedinorder to render the treatise more applicable.

1. MC. DERMOTT. By application of the wake data published byFROUDE in 1883 and 1886 (37and 38) MC. DERMOTTooxaposed the following fornia].ae (published 1896 (80), the formulaehere being taken from (81)),

Single screw ship: w = 0.16 . (L6- 0.6)

Twin screw ship: w = 0.13 . (L116- 1.1) (I-la) p beIng the prismatic coefficient, the midship section coefficient, and L the length of theship in feet. That the wake coefficient is proportional to thefullness Is correct inside a limited p-range. Whenp Is used asaparameter, it is correct to employa e-correction, as in comparing two ships, having the same p, butdifferent , the ship having the bigger will be more slender at theends, and therefor probably have a smaller wake coefficient The length of the 122

ship ntors into' the formulae inthe wrong way, as the wake coefficient will demi.nish withincreasing length of ship on account of 'the rèlative diminutionof the friction belt. Sometimes MC. DERMOTT'S formulae arefoundstated in a different way (73):. Single screw ship: w, =0.195 L1!6 -0.5) (I-lb)

L1'6- 0.9) (I-le) Twin c'ew ship: WF= 0.158 midship section area. 'y bezg the displacement, and O the

2 LLH0US In a contributionto a discussionin 1910 (45) HILL- HOUSE gives a formula determinedfrom FROUDE's and MC.DEEMOPT's wake 'values:

O,ol (3o q'- 0.75L/B - 5.5) ' (I-2). but döse notstatewhether it applies to single' or to twin screwships. It will be seen that HILLHOUSEhas introduced a correction f or.length-breadth ratioofthe ship, but tiais correction is not big.enough.

30 TAYLOR l9loi. TAYLOR 'in 1910 published the formulae forwake and, thrtisdeduction, which hitherto bave been zst used. By applying LUKE's data (75) TAYLOR(loB) found that with a good approximation 'w for twin screw shipcOuld be expressed as: w .2 ,+ 0.55 8 = t (I-5) Thus TAYLOR disrégars the position of thepropellers, but presumes, that the propellers arepo3itionod inthe region abreast oftheafter perpendicular,andwith the centres distanced 1.2 propeller radius from the centre line., In (loi) TAYLOR gives a formula for the wakecoefficient of singlo screw ship: w - 0.05 + 0.5 8 (I-3a) He doesnoti gis anformula for t, but only remarks 123

that LU's experiments söem to prove, that the coefficient of thrust deduction for single screw ships is a little bigger than for twin scréw ships (formula (I-3)).

4. TAYLOR 1923. By application of LUKE's new experiments (76 and 77) TAYLOR in 1923 (Ib) gave two new foriiulae, which should correspond. to average shIp forms In a better way:

Single screw ship: w = - 0.1 + 0.5 5 (I...4) Twin screw ship: w = - 0.16+ 0.5 6 (I-4a) TAYLOR présumes that for twin screw ships the angle between the bossings and the horizontal plane is such that the rotation of the in.f lowing water to the propellersmay be disregarded.

Single screw Twin screw

ship . ship

L w Wf0 w

.50 .230 (.200) -.038 (.075) .55 .234 (.225) -.021 (.101)

.60 .243 (.250) . .007 (.130) .65 .260 (.275) .045 (.157) .70 .283 (.300) .091' (.185) .75 .314 (.325) .143 (.213) .80 .354 (.350) .200 (.240) .400 (.375) .90 .477 (.400)

Table 30: TAYLOR's wake. table.

In the new edition of "The Speed and.Power of Ships" (1933) (1o7).TAYLOR. gives aseries ofwake coeffIcients determined for ships wider way (also reported (111)).From the horse power of the engine and thenumber of revolutions, the torque of the propeller was determined.Further as the speed of the ship was. recorded, and thecharacteristic cur 124 ves of the propeller wereknown, the wake coefficient was easily calculated. In table 30 TAYLOR'svalues are stated, and, in the parenthesist, the wakecoefficients, which would be found by employing (I-3)and (I-3a), are recorded. As the uncertaii.ty as regards themeasurement of the borse power is considerable, theresults must be accepted with reservation. For the thrust deductioncoefficient of single screw ships TAYLOR in 1923 gave theformula: t = 0.3 6 (I-4b)

ROBERTSON According to "Shipbuilding and ShippingRecord" (81) ROBERTSON in "The Shipbuilder" 1913published the following formulae: Single screw ship: w = - 0.o5 + 0.45 ç (I-5) Twin screw ship: w = - 0.20 + 0.5 ç (I-5a) Thus formulae corresponding closely to TAYLORt formulae.

GILL. In the article in "Shipbuilding andShipping Record" (81) is also recorded a formula of GILL, the formulahaving been published In the saine magazine injanuary 1927. Single screw ships: I - w = 1,15 - 6/1.5 (I-6) Twin screw ships: I - w = 1.27 -b/1.5 (I-6a) In another form these formulae can berewritten: Single screw ships: w = - 0.15 + 0.67 8 (I-6b) Twin screw ships: w = - 0.27 + 0.67 6 (I-6e) It is seen that these formulae ingeneral give a somewhat bigger wake, than those of TAYLOR's.

7. SHIPBUILDING AND SHIPPING RECORD. In (81) another formula is also found,theorigin of which is not stated: w=ç3 or (I-7)

'r (I -7a) I The formula Is valid forsingle screw ships. 125

8. LUKE. In a publication in 1917. LUKE (77) gives à diagram for determination of the wake coefficient of single and twin screw ships. This diagram is repx'oduoed.in fig. 67 after transformation from FROUDE's to TAYLOR's wake coefficient.

5//iGL5CE' MODE .30 TIv7N 52EW'DEL r ,' 84S$ING

5L0 D

.20 Fig. 67.. LUKE's diagram - for determination of '' wake.

.10

InLUKE's original diagram the curves were straight lines having the followingequation: WF= a + m8

6 being, as úsual., the block coefficient, a and in aie con- stants,which ror twin screw ships have the values: The angle of the boasings a m to the horizontal plane 00 -0.246 0.820 22.5° -0.229 0.712 450 -0.2o9 0.597 67.5e .0.186 0.483 no bossinga -0.232 . 0.828 LU1 original diagram curves foi the thrat deduction oeffioient were also given. For twin screw ships LU found,, that this coefficient is practically Indepen- dent of the angle between the bossings and the horizontal plane, and that it has the following values: 6 = 0.55 t = 0.103 0.60 0.138 126 0.65 0.166 0.70 O.190 0.75 0.200

9. THOMSON. In the contribution to the discussion of a paper by TAYIR J. THOON (116) gives two diagrams, one för deter. mination of the wake coefficient of single screw ships, the other for thé determination of the wake coefficient of twin screw ships0 The diagrams are shown in fig. 68 and 69. In the diagram for single screw ships THOMSON employs the prismatic coOfficlent ç as primary parameter. As secondary parameter he uses the propeller diameter in feet divided by the cube root of the displacement in English tone. For twin screw shipS THOMSON further uses the tip clearance (a) as parameter. Also a is measured in feet and

L Fig. 68THOMSON's diagram for determination öf the Wake coefficient of single screw ships.

O1O2O3O 40W

P-i-L - O/JTWADTU 80MGSL'ED 70H0r' 'A'" Fig. 69. THOMSON's diagram for determi- - I nation of the wake coefficient of twin screw ships. mu

.50 60 0 127

the cube root of the displacement in Eng].läh tons isuséd as a divisor,

lo, POINCET. In LEF3L's publication (73) is given a formula origi.' nating from POINCET: W= 0055 6 0.21 (I-io) tFr = 0.50 8 - 0.19 (I..1ôa) t =%ip/(l+ tpr)

In other publications, fr instance in (25), theexpressi- ons for w and t, have been interchanged. tpr is the French coefficient of thrust deduction. The formnlá forthrust deduction applies to single as well as twinscrew ships, while the wake coefficient is only valid for twinscrew ships; for. single screw ships w has to be increased byabout 5o pOr cent, The formulae have been developed fromthò same experimental data as TAYLOR'S, therefore thesimilarity is not surprising.

il, SCHAFFRAN. SCHAFFRAN himself has not composed any formulaOr diagram for predetermination of wake and thrustdeductiòn1 But many of his results are presented in sucha maimer that they can be directly employed in destgnThis applies to the results of a sertes of experiments, inwhich SCHAFFRAN (G2) kept the frame shape and theprincipal dizTnáions of

10/CN .70 12K/V- /

Fig. 70. SCHAFFRANB wake and thrust de- ductionCUI'V08for models having U- zhaped frames (sing.' le screwships)0 128 the models constant while the fullness wasvaried.The frame shapewas pronouncedU-shaped,and as in so many of SCHAFFRAN's experiments, the models were small,only about 2.8 m in length. His curves for the wake andthrust deduction coefficient of these models having abreadth to length ratio of 0.13 and a propeller diameterof 4 per cent of the model length are reproduced in fig. '73.

12 SCHIFFBAXJKALENDER., Also in "Schiffbaukalefldel" the wake and thrustdeduction coefficients are given solely asfunctions of the fullness of the8hip: Single screw ships: w = - 0.24 + 0.75 8 (I-12) t = 2/3 w + 0.01 (I-12a) Twin screw ships: w = - 0.35 + 0.83 8 (I...12b) t = 2/3 w + 0.02 (I-12e) Theseformulae should appl7 to conditions at the ship in opposition to most other frmu].ae which apply tothe model conditions. In a case, where the wake and the thrustdeduction coefficients are to be determined moreexactly, "Schiffbaukalerider'" suggests, thatBRAGG'swake diagrams andWEINGART'sthrust deduction diagrams be employed.

13. BRAGG. In 1922 BRAGG (16) published the results of a big experimental series, the aim of which was to find the varia.. tion of wake with the ship form. For a series of models BRAGGdetermined the wake, by placing, abaft the model and clear of it, a propeller, which worked In the saine way as a blade wheel. The number ofrevolutions of the propeller were registered, arid from the calibration curve the wake coefficient could be determined. The experiments were car- x'ied out with propellers of different sizes, and their position in relation to the model was varied in different ways. UnfortunatelyBRAGG'smodels were rather small, only about lo' long, thus the measured frictional wake was disproportionately big. But the error thus introduced is partly compon- .aated for by the position of the propeller, which was a 129

little abaf t the rudder post. From his experiments RGG concludes that the wake coefficient st depend upon Propeller diameter relative to draught of ahip. Draught of ship relative to its breadth. Longitudinal position of the propeller. Athwart ships positio.n of the propeller. 5 Vertical position of the propeller. 6. The vertical pri to coeffIcient of the ship. In 1924 BRAGG (17) published wake dIagrams (seo fig. 71) derIvedfròmthe data published in 1922, using as para-i meter the following ratios: 1, The breadth divídød by the draught. The vertical prismatic coefficient. The propeller diameter divided by the draught. 4, The height of the propeller axle above the keel, divided by the draught. In the same publlcttòn BRAGG bas given a sertes of effective wake coefficients obtaiied by experIments usIng 2o' models. These còefficiénbs a'eod approxlmately with the nOi.nal coefficients from earlier experiments. From this BRAGG concluded that the nominal wake was bigger than the òffective yaks. This conclusion seems however rather daring,taking into account' that the length of the models in the two sets ofexperiments differed as much as 1¼) fest. The paramsers employèd by 'BRAGG are not completely valid. Asmentloned earlier ('section F) the length of the model ought to be taken into account, some way or other, and further zp ought n to be employed ascoefficIent of fineness. when dealing with a twi.n screw ship, BRAGG is of the opinIon, that hIs curves can still be used, when 2/3 of the reiste'ed value is takeñ. BRAGGprswnê's howeJer, that the bosIngs do not have an extrême shape or position. BRAGG also determIned the thrust deduction coefficient by his self propelled models, but he did not analyse hisex. peri.neral results. This was done later by' WEINGART (126), who co tri ed a diagram (see fig. 72)iflwhich t/w Is given as function of the vertical prisaatic coefficlnt. and the r 0/,0.76 /d O30 V r 0i'd U20 2.4 26 2.8 2.0 i28! 2.4 26 r 0/d /d, 0.48 088. Fige 71. BRAGG's diagram for wake determination. !ePpii$ 131

:Fig.:72. WEINGART's .8

diágram for de- .7 termination of the 6 thrust deduction .6 coefficient. -5

.84 ¿e 88 90 .92 .94 -96 .96

block coefficient for single as well as for twin. screw sh.ips.

14. BAKER, In his book "Ship Design, Resistance and Screw Propul-. siori" (8) BAKER givesdiagrams for determination of wake and thrust deduction for single as well as for twinscrew ships. BAKERgives two diagrams for wake determinationfor 3tngle screwships. In the one, only the prismatIccoeffi- cient and the traino shape have to be known; Inthe other the prismatic coefficient of the after body isused as a parameter, and here also the frame shape mist be known. - Fig. 73.Indicate3 the two diagrams in whicha transforma- tiOn tOTAYLOR'swakècoefficient has been made. The diagrams are valid fOr ships having a breadtho'draught ratio of 2.o to 2.5, and at lesathan 1. For an increase of the.breadth-draught ratio from 2.o to.3.o, the wakecoot- fie lent shou].d Increase by 6 to 9per cent, lo to 15 per cent, or 15 to 2o per cent, when the ratio of thehelht

ML ¿LS

- Fis. 73. BAR's diagram for determinationof w for sigle. screw ships.;

.60 70 132

of propeller centre abovekeel to draught at propeller post, is 0.30, 0.39, or 0.48respectively. Fig. 74 gives a diagram for w for twin screw ships, and fig. 75 represents BAKER'S diagram for thedetermination of the coefficient ot thrustdeduction orsinglescrew ships.

TOP t4RY,E 5MIPS E MI/TI? FWPTICMI STEPNS,41(E TI/E L/PP LI_IT 4'VEN TOPI/ME (MOOED CPLII.S(R 5TEEW5. (41(1 TI/E LOWER

w (ARGOVC55E15 I/7I 5OsOMI$ Fig. 74. BMR's diagram for determination of w .10 for twin screw ships. N,

IA .55- .60 65 70

Fig. 75. BAKER'S thrust deduc- '5 tion diagram. GOOl

15 I

15. VAN DIEREN. On the basis of wake data published by LUKE, BRAGGand others VAN DIEREN has conpo8ed the wake diagramsshown in fig. 76. VAN DIEREN divides the ships into three groups:

1 Single screw ships. 2. Twin screw ships having bossings. 3. TwIn screw shipswithout bossings.In each group he arranges the experimental results according totheprismatic coefficients of the after bodies andfurther employsthe ratio of midship section area to square of ship length (loo. as a parameter. Theuse of this parameter is excellent as the wakecoefficientincreases with B/L as well as with d/L3 VAN DIEREN's curves are constructed as series of hy- perbolae. 133 S1NeWR j

.50

.40

w 7W/N C.QEW WITHB /N0 Fig. 76. VAN DIEREN's wake diagrams, .30

W TWIN .40

.20

16. SODEELUND. By employing a series of trial txps and tank results SDERLUND (102) has prepared a set of diagrams for determination of the wake coefficient and the hull efficienc7of single and twin screw ships. Completely empirically, without theoretical considerations, SDERLUND bas found that by de- picting the functions: I 3L.

frs and

over an approximative thrust loading

the best coordination of the points from the trial trips was obtained, In fig. 77 and 78 SDERLUND's curves are represented. In the diagram for twin screw ships the curve of hull efficiency is valid for ships with bossings as well as without. On the other hand the wake curve is only valid for ships with bossings. For ships without bossings the (1-w)-curve will lie about 0.002 units higher. For single screw ships the uncertainty is considerable.

5H/PS .020

Fig. 77. SDSRLUND'S curves for the wake coefficientand the hull efficiency(single screw ship). .012

.0/0 o a 80 100 120 1W

TWIN.CREW/1/PS WIT/I:055/NS

Fig. 78. SDERLUND'S curves for the 'u. wake coefficient and the hull ef- .018 ficiency (twin screw ships).

.010 a 40 60 80 100 /20 135

The diagram for twin screw ships is valid for ships having a block coefficient ranging from 0.63 to 0.8o and a from 0.54 to 0,75, whereas the diagram for single screw ships is valid for ships with block coefficient between 0.7o and 0.Bo and V/Irbetween 0.52 and 0.83. It appears that the parameters chosen are more complicated than necessary. Thus the number of revolutions of the propeller N will have very little influence on w and t. The application of the towing resistance R alzo is a little peculiar. Surely it would be better and easier to use some form coefficient or other, thereby avoiding an estimation of the resistance, In order to arrive at the resistance,S3DERLUBD suggeststhe formula: o,64 R=145.8 y2 e V being in knots, in metric tons in salt water, R in is a coefficient (see (129)). kg, and0e

17. TELFER. Byreanalyzing BRAGG's experimentá (16 and 17) TELFER (113) found that the wake coefficient of single screw ships could be expressed in following way:

w = f(p) (1 - ) (I-17) Here f(ç) is a functIon taking account of the fullness of the ship, E is the height of the propeller axis above the keel, d is the draught, D the propeller diameter, and B is the breadth of the ship. As practically all of BRAGG'S models had the same breadth to length ratio,TELFER, in order to find the influence of this ratio on w, had to ana- lysehe results of TODD's coaster series (118).TELFER then found that the wake coefficient was directly propor tional to the ratioB/L:

w = f(q) C 1 - ) (I-17a) Now f(ç) had to be determined. For this purpose TELFER employed as many results of self propulsion, as ho 136 couldobtain. Partly in an imperical way and partly theoretical consideration befoundthat the function: = L f(cp) l-F where F (1-ITh) was suitable. The wake formula thus became: 3 BE 3D w - / (1.17e) - Fra TELFER emphasized in the publication as well as tri the discussion which followed, tha the formula was only valid inside the normal range of ships. Thus the formula cannot be employed for propellers fitted abaft of plane surfaces or in the case where E is equal to zero. In this sanie publication TELFER goes one step, further by amending the formula to take account of the tip rake and skew back of the propeller. Thus: 3 BE, 3D+2R I17d) W= p - 2 R being the sum of the tip rake and the skew back. Ingoing from model to ship TELFER suggeststhit the wake coefficient found be reduced by 0.33 to 0.04. Even during thedi8cussionof the publication it was pointed out that the formula did not give the correct values, especially for fine ships. TELFER therefore later (115) corrected his formula to:

3ß BE, 3D + 2R (I-17e) w - - - 2 3

But also this formula was met with criticism (125). In the authors opinion, it is unnecessary to introduce R in a formula of this kind, as this quantity is only of secondary importance. If R is introduced, account ought also to be taken of the shape of propeller aperture and the shape and the position of the rudder. In the same publication (113), TELFER, by using RANKINE's theories, prepared formulae, for determination of the coefficient of thrust deduction:

t =, w (I-17f) 2 (1 -w)(1 tv-) e 137 u/v0 being determined by u -1 +aT Vi (I-17g) "o= 2 u being velocity caused by the propeller, and Ve theinflow velocity of water to theprojel1erdisc. aT is thethrust loading of the propeller. TELFER has compared the t values, obtained by the formula, with observed values '(TODD's 3O- ries), and found,thaton account of'the rudder, t ought to be increased by SD (Ve +2u )2 2T (I-17h) s being the width of the rudder post, D the propeller diameter and T the totalthrustof the propcller. Regardingthe accuracy of TELFER'S wake formula it could be 'mentioned,that SCHOENHERR(96) statesthatthe average deviation by formula (I-17d) in 65 cases tested amounted 0.047.

18, K.&RX. In 1941KARl (5o)published a very elaborateformula for determination of the wake coefficient.Theformula is the result of a statistical investigation of certain in- formations, the quantity and origin of which KARl doesnot ment ion.KARl'sformula is valid for single screw ships, thus: p3/4 N i WF - 2'5 I'S (1-18) loD '(DSR) ' Cw+A B In this formula: =FROUD's wake coefficient, N=number of revolutions per minute, P =theaverage pitch in feet, D=thepropeller diameter in feet, DSR= the expanded blade area ratio (the blades only),and A, B,and are quantities determinad thus: 2.75 A V 2.29 118 (loC --0.50)) (DSR) ' 138 where 8 Is the block coefficient, V the speed in knots and L the length of the ship In feet.

B - The value of K depends upon the blade area distribution of the propeller. Denoting the distance of the centreof gravity of the blade area from the hub by CG, the radius of the hub by r, and the propeller radius by R, K will be equal to 14.5 if CG/(R-r) = 0.455. In the case where CG/(R-r) is bigger or smaller than 0.455, K is determined by: K =j45 - ( 35 ( CG 0.455))9/2 CG K = 14.5 - ( 35 (0.455 respectively. Finally C,, is a constant based on experience arid taking into account the shape of the hull, rudder,and position of propeller. KARl suggests that C shoûld be fixed by comparing similar ship forms. If a comparison is impossible, the following valuesofC may be employed& C,, = 173 8- 60.40for ships having double plate rudder and a fin on the rudder post. = 173 8- 64.15for ships having oertz rudder, = 173 6- 63.15for ships having streamlined semlba- lanced rudder with a thickness of rudder less than 1/5 of the rudder length. = 173 8- 66.7o for ships having semibalanoed double plate rudder with unsymmetrical leading edge. = 173 8- 77.lo or ships having balanced reaction rudder. Whether KARl's formula gives correct or incorrect values of the wake coefficient is difficult to say, but it is evIdent that it is laborius to employ. One big draw- back in the formula is thefact,thattheinfluence of the hull, which most certainly is primary, is left to judge- mont, while quantities suchas the number of revolutions of 139

the propeller, the average pitch. of the propeller,the blade area, the blade area centre of gravity, and thespeed of the ship are introduced exactly in the formula. Nueroua model experiments prove that all the last mentiored quantities are only of minor importance, in fact mostlyso insig- nificant that the uncertainty of experiment completelyob- literatse their inflvence, In the saine paper (50) KART also givesa formula for the coefficient of thrust deductIon. Thus: t = 4 (1 - 5) (1 -.s) where S is the true slip,Sa "inflowsupratio", P the propeller pitch, and D the propeller diameter.Sa can be found from: _1mveo.e. - S, R m being constant for given hull and propeller shape,Ve being the speed of advance, S,a "diameter surface factor", and R a "revolution factor", In KARl's paper is described how these factors are determined.

19. SCHOENHERRI. On the basis of 49 dIfferent selfpropulsion expe.ri- mente performed in U.S. Experimental ModelBasi,t, SCEOENRR (95) has prepared a diagramfor determination of the wake coefficient of single screw ships1 This diagramis shown in fig. 79. The abscissa X isa function of the volunie of displacement sy ,the breadth B, thedraught d, and the propeller diameter D. 2/3 =(B+2d)D The ordinate is the wakecoefficient, and in the figure, curves are indicated for different valuesof Y, which is a function of the length of the ship L, themidship section area O, and the prismaticlIo coefficient q': 1140

Al]. the more important data of the shipand the propeller, except the frame shape, havethus been taken into account.SCH0ENHRRstates that average deviation in w for the 49 models is 0.04.

Fig. 79.SCH0E1HERR's diagram fordeter- miriation of the wake coefficient of 20 single screw ships. .10 .

1.6 26 32 36 4.0 44

2e. SCHOENHEERII. SCHOENHERR has later extended his investigations,and the results are published in (96). 65 experimentswith. 61 single screw models and 60 experiments with 52 twin screw models are analysed. From the result of thisanalysis SCHOENHERRcomposed the following formula for w of single

screw shiDs:- B/ - ,i, 1/2(E/d- D/B- w =0.10 + (7 - 6y)2.8 1.8cp) (I .'2o) Here; L = the length of the ship, B = the breadth of the ship, d = the draught of the ship, D = the diameter of the propeller, E = the height of the propeller axis above thekeel, = the vertical prismatic coefficient, p = the horizontal prismatic coefficient, = the angle of rake of' the propellerin radians,and a coefficient havlrg the value of 0.3 fornormal types of stern and the value 0.5 to 0.6 for sterna having the deadwood cut away. SCHOENHERR states that the average difference between experimental and formula values is± 0.027. It is seen that this formula 1 composed of all the quantities which generally are considered to have most in- 1141

fluence on the wake coefficient. It is only a question whether enough regard is taken of the ship form by the suggested employment of.' and p. Further the f ornnila might. have been constructed in such a way that,izi general,it would be unnecessary to correct for the rake of the propeller. Another question is, whether a number of the figure con- stants are due to the comparatively limited experimental material. SCHOENHERR relates the thrust deduction coefficient to the wake coefficient: tkw (I-2oa) k being = 0.50 to 0.7o for ships having streamlined rudder or contra propeller rudder, k = 0.7e to 0.90 for ships having double-plate rud- dors attached to square rudder posts, ar k = 0.90 to 1.05 for ships having oldfashioned plate rudder. For twin screw ships the wake coefficient can be determined by: 5( w = 2 I - 3) + 0.2cos2 . G - 0.02 (I-2b) for ships having bossixigs and outward turning propellers. Here 6 is the block coefficient and Q the angle of the bossings to the horizontal plane. (Average deviation 0.o23, 38 caSes). w = 2 (I - 6) + 0.2cos2 (90 - Q) + 0.02 (I-2oc) for ships havingbossings and inward turning propel- 1ers. (Average deviation 0.o12, 7 cases).

(1 e. w = 2 - ) + 0.04 (I-2od) for ships without bossings. (Average deviation 0.o24, 15 cases). For twin screw ships the thrust deduction can be do- ternhirLed by: t = 0.2F w + 0.14 (I-20e) for ships having bossings. (Average diviationI0.oIB, 45 cases). t = 0.7o w + 0.o6 i L2

for ships without bossings. (Average deviation. 0.014, 15 oases).

21. LEFOL. One of the newest diagrams (fig. 8o) for determination of the wake and thrú.st deduction has been constructed by LEFOL (73). In the óonstruotion of bis diagram, LEFOL used earlier published experimental data. Aa parameters ha employs the following quantities: For single screw ships: takes care of the displacement distribution, and a coefficient Y, defined in fig. 80 as the ratio of the areas NPQR to NPM on frame I (L, = distance from frame O to 20), is used for fixing the frame shape. A third parameter the pro- piller diameter divided by the length of the ship is also used. It is presumed that the rake of the propeller is 8. For twin screw ships X and D/L are again the basis parameters, but Y is substituted by3/1,j being the tip clearance of the propeller. LEPOL gives the wake and thrust deduction for standard ships defined in relation to the parameters and to the screw rake. Corrections for deviations from the standard are performed by means of supplementary diagrams. The procedure in.determining the wake is as follows: The wake coefficient is read off from the main diagram, it is thenmultiplied by the factors k1 and k2 from the correction diagrams. If in single screw ships the rake of the propeller deviates from80,w is to be reduced by 40 O.ol and t by O.o2 for an increase in rake of For twin crew ships, the standard rake (propeller + bossing) being 160,w and t will change 0.ol for a change in.rake of 8. LEFOL's diagrams are very logically constructed, but unfortunately he has chosen a standard form having rar too much V-shape (Y = 1/2), necessitating practically always correction for frame shape. Hereby unnecessary uncertainty w /NG1E-SCREW SNIPS 7M4-3CPf hi 5//Pi 40- y.04.0048.45/C NPO03NVFQN.O5 VAWE$ PA/lE 8 -30 phoN)? A for;.' V PA/IIBAhO OSCPEW84.003 /4WE3- /I4KEQFJ5.j15 -o,s,6 5: .30 TEA/lE -20 - - - .00 -20 -30 .40 ' - SS C,k, , 1Ajk, rs- k, TE CONNECTION Lo - IN TER/ISEO ' w Y .9 IN '5' Y -9 .9 - INTENSarCOPPE 7/ON k A .6 .7 2 Y k, 4 SS Y -0? 7$ hr q03 Cl COPPE I.2 '" - CONNE. rs CONE CT/ON /N7EP . 00j, OPTEN ' i - IN TE o -8 03 035 .5/2 - "- ". -05/ : ductionFig.8oA.for (singlewàke determinationLEFOL's and screw thrust diagram ships). de- wakeFig.termination 8oB.and thru2tLEFOL's (twin deduction diagram screw ships).for'de-lì'!! 144 is introduced. The D/L-correction diagram for single screw ships seems a little peculiar, as according to the diagram the smallest wake occurs at biggest draught in case of small propellers, whereas the opposite is shown for ships having big propellers.

22. VeLKER. For use in estimation calculations OSBURE(85) in a hand book gives H.VOLKER'S table, which is represented in table 31.

l:4Q

kG, a,k o - C) k cri u, 0E-1 kE-1 Cf) b , o k r-4 Ci rl U) 4.) CI) 4' o o

.8 .35 .23 1.07 1.15 .7 .29 .19 1.03 1.10 .6 .23 .15 1.00 1.05 .4to .5 .14 .09 0.96 1.00

Pable 31: \k5LKER's w and table.

23. HEWfl4S. For more accurate determination of thewakecoefficient OSBOUR11E in bis handbook (85) gives E.F. KEWINS'diagrams. These are, as far ascan be seen, basedupon BRJ1GG's and LUKS's experiments. BEWINS' diagrams (see fig. 81) for single screw ships are rather surprising, as he, for main parameter, uses cor- 145

tain quantities defined by the curve of Intersection between the hull of the ship and the circular cylinder,having the propeller circumference as. directrixcurve (seo the sketch fig. 82). Further he employs theratios breadth to draught aft and, tip clearance to pitch ofpropeller as parameters.

It cannot be asserted, that BEWINS!choice of .para. meters have been specially fortunate.For single screw ships be considers solely the conditionsIn the imnediate. vicinity of the propeller, thusdisregarding the length and the fullness of the ship. On theother band be gives.

5Ck 5/IVGLE-5CeEw3hq '5 IvWou CURVE

- -- ..1A' IP

.04 .08

.03 .06

.02 .04

4/ .02

30 4.o .110 .20 30

TWIN-SCREW 5wiP

,iiirw ;#. W7A'VFAVIJMWM

- .

4 .5 .7 . 70 .78 2 6 9O '3'

Fi.g.S1. KWIN' wake diagrams. 146

a corréetton for the variationin the pitch of the prou peller a òorreotion which isgenerally considered super- f1uousThe fact that the diagrams are difficult to use is also a great draw back. His diagrams fortwin screw ships are also shówn in fig. 81 and are used in thefol- lowingznanner: From the proper value of the vertical prîsma1iccoè'fftcient on the abscissa proceed vertically upwards unto the proper curve of breadth-draughtratio, then horizontally to the left part of the diagramwhere w is givenas a function ofj/B, which Is the ratio of tip-clearance to the.heght of the propeller axisabove the keelIn thése diagrams the choice of parameters ta not, so striking as in the diagram for single screwships, but instead of using the vertical prismatic coefficient he, surelyoughtto have chosen the block coefficient or the hoizonta1 prismatic coefficient.

Fg,82. tefinition of HEWIN.! paranaters.

24 VAN AIEN i (2) VAN AiN'has given the following wake .forila: Lw =. 0.1 8 + a(B/d LIB + 18.bo ,) (I-24) a i a faàtor .tiklng regard ofthe. frame shape and taking the valuea:

' . . a = Oo15 'for ships'with V-rrantea, a O.19 fr ships witháltitly V- and U-shaped frames, a 0.à23for ships withU-shaped.frameé, a .0,o25 to 00o27 for ships with pronounced U-shapedframes. For tug' boats iü which Sis about 0.50, a canbó taken as 0.010. 147 it is remarkable, that according to VAN AIN, the block coefficient should onlyhave alight influence on the wake coefficient, at&d the propeller 'diameter none at ail.

J. TESTING 0F. THE_WAX.FORMUIIAE AND DIAGRAMS FOR SINGLE SCREW SHIPS. In order to get some idea of the accuracy of thewake formulae méntionéd in the preceding sections, the formu].ae have been tested on 17 different modelexperintS, 'thé calculated wake coefficient beingcompared to those found by experiment. In this investigation the authors Wageningendia'am (43), Identical with the d1aarn fig. 52, but Without correction for E/a, has also been tested. in- addit.1ón the. best of the formulae have beennior.e 0103017 examined, being tested on a long- seriesof ships... The models f Irat employed were no. 1 BAKER's model no. 1119 (12), no. 2-5YAMAGATA!S models no. 195,196,197 and 19('130), no.-II TODD's models no. 1945 A, B, Cand 1959A, B, C (120), no. 12-14 three models tested in the Pai'±s tank (BI,' UI, Vi) (128), nt. 15a fniit carrier tested in the G8teborg tank, no. 16-1? two cargo ships tested in the G8tebor, tank, no. 266 ard 266 a (42). .I:n all the experIments, except YAMAGATA'S the wake' found byformiila was cónipared to the effective wake òoef- ficlent found by experiment, determined by cT_identity. For YAM&GATA'S experiments the comparison was made with nominal wake coefficient detérmined by blade wheel mea- sureients. Iñ table 32 the principal dimensions and the coefficients of all the nioo.els are recorded, together with certain further quantities, which have been used in determining the wake coefficients. In thé same table theake cöeffieient found by experiments is also.ven. 148

The re:;ults are shown in fig.£3, cóntainirg 25 diagrams corresprding to the 25formulae teìted. The is çlotted over ti-te wa- calculated wake,WFOTmI observed ke valueo, w. In each diagramall the pointstheröfore ouaht to be on theline x = y, with eceptiorL ofthe diagrams calculated by formulae givingship wake coef- f teient, for instsnce MC.DERMOTT'Sar4SCHIF)AUKALEN- DPR's.In these csses the points ought to havebeen fròm 0.05 to 0.lcbelowthe line X y. the case of some of. theforrmilae, calculations were nt carriedoùtforall of the models. Thus SCHAFFRA'8 curveswer8 only used onthree ships with pronounced U- .ahapedfraine8, the curves beingvalid exclusively for this type of ship. KARl 's formula wasapplied to four models only, because in usingthisformulae it appeared: tha the calculations wereextremely laborious, that certain. important coefficientShad to be guessed,and that other coefficients had toobig an influence on the result. Thus the blade Shape of thepropeller had so big an effctthat with speciál blade shapes there wasthe risk ofobtainingcompletely fictitious results. From the figureit isseenthat SCH0EKERR II is undoubtedly the best formula. 0fthesimple formulae TAYLOR I (1916), GILL and. SCHIF'BAUKLENDER arebest,indeed considrably better than most of the morecomplicated forilae. For the further investigationtherefore, these for- mulas together wIth BRAGG'$,,TELFER'S,LEFOL's and the author's Wageningon diagram were employed.The models used

.; for this investigation were: no. 18-2oYAMAGATA'S model 310 with three different sizes of propellers (131), no. 21.22modøl'tested by KEMPF (56), two different propellers, ° TAYLORV5 model no. .2933, five different sizes of propellers (112), no.. 28.33 TODD'Smodela1223, 1210, 1169, 1266, 1242,' and. 1256 (identical with 1o98, 1106, '1101,1o69 A, 1111, and 1118) (117, 118, and119), e SCHI/TB4L/ML. 78eAGG e eeo_ O // .30 0 .ÍlÍ:Y000 j*;: .20 CC. O eQ 46w 20 .30 40w Wpo,M.: 1 : TELff24 : SCHOFNHPP 21 ¿(FOL .40 I 0 .30 G 0G 30 .30 e G Oe O :: .20 °'°.30 40w .20 20 .30 .40 w .20 4 .40w .20 .20 40w determinationFIg. 84. Test Ing of'w of (singlethe beat screw formulae ships). and diagrams for 154

no. 34-42 EMERSON' s modelsAw;AM, AN, By 3N'Cw,CM, and 0N (34), no. 43-44EMERSON's models D and F from the R-sonos (31), no. 45.-47 EMERSON's modelS B, F, arid H.from the S-series (32), . 48-49 EMERSON's modelS G and M from theU-zeniòa (33), no. 50 ÄCRSON's model 2933 (1). The principal dimensions, troni coefficients and measured wake coefficients are given n table 32 and the results from the wake calculations in fIg. 84. The repre. aentation is on the same basis as for fig. 83. The resú1tof the investigation Is;

By all the method8 the uncertainty iri the predetØri.-- nation of thé wake coefficient is considerable. This is

obviously due to inaccuracy partly in the method of cal.- - culation, and partly in the wake coefficient expentinen- tall? obtained. To this imiat be added possible errore. Leastscattening is obtained by SCHOENEER'a forimila, but for thé ships tested, the form la generally give a too small values. Probably the fo la could be iroved by jisiñg the constant 0.12 instead of O.lø. !,Iearlj the sS effeàt would be obtained if instead of using the factor 4.8 was employòd.In this way a bigger correction Wotld be obtained át the bigger wake coòfftcents here. the deviations aro moat pronounced. .he author's diagram gives the best grouping arotind the line x y, butmaniimber of caseS the differònù between the measured and .the calculated wake is óonaide.. Ñbly. 4 The simple approximation forzmilao do not give bigger iflaximum differences between the measured and the calUiaa edvalues than the more. eomplicatedformulae, but be: probability of obtaining the correctvalueislfBBe 5,Whether TAY0R's, GILL'S or SCIFFBAUKALENDER!Sfornfll- ]ia la applied, the result is practically the same,butin going f rom model to Ship différent ways of correcting have to be used. 6. in BRAGG's diagrams it appeared, that 'V had too iieh 155

WFODM

.40 I : .11 ....

.30 30

.20 .204 1 20 40w .20 .30 :

.20

.30 40w

Fig. 85. Testing of the diagram fig. 52: part I: WFàRM = basis w with correction fox' D/L andf, part II: WFOBM = basis w without corrections, part III:wF0 = basis w with correction for D/L, part IV: WF0 = basis N with correction for D/L, f, and EId. influence. For ships having U-shaped frames and small block coefficients ws obtained too high awake, the value in these cases being disproportionately big. As in the Wageningen diagram no correction has boén givenfordeviation from the normal ratio (E/d = 0.4e), height of propeller axis abovekeelto draught, thé new diagram, fig. 52, was also tested.The result is indjòa- ted in part IVfig. 85. Fig. 85 also gives an Idea of the influence of the different corrections.Inpart I has beeii.determined directly by means of the Wageningen diagram,i0.."bAsis w" with corrections for D/L and f. In part iI'ì "basis. w" without corrections, in part T-IIcórÌ'êctiónfòr D/Lonly ismade, andfinally inpart IV coi r'eótions bave 156

been made for the. propeller diameter as well as for the frame shape and the height of the propeller axis. In part IV however, the correction for frame shape was only .per formed for f-values within the following limits: at B/L = 0.13 f < 0.30 or .f > O.6o and at B/L = 0.16: f<0.4o orr>0.90. For other values of B/L the limits were obtained by linear interpo].át ton. In part I o,n the other hand the coef- Íictents were corrected for all f values. In the same way the coefficieùts were corrected for E/a in IV only when E/d 0.35 or E/d > 0.45. and when the framès were of V-shape.

- The result of this examination is: If no correction is applied, the uncertainty is the same as by the simple formnlae. By applying a correction for the propeller diameter a eomparativoly good agreement between the measured and the óai.c'ulated values is obtained. By further ¡applying corróotions ,f or height of propeller axtá and' frame' shape, still better agreement is obtained. In general as a resulto the whole investigation it can 'be stated: For the first estimation, TAYLOR's formula: w =, 0.5 6 0.05 ought to be used, as it, gives fairly good results and x'sover js so simple that it can be remembered easily. . For, a nreaccurate calculation SCR0ENRR' s forila: w=O.10 + 0.5 (E/d-B/D-k,r)is

preferab]e.. . ,' ... ..

3,. Instead of $CHOEERR's f orxmila, the author's diagram can be, used. By doing sotimsWillbe saved. .4.,. T4YLOR',s n4SCHOENIRR.!s for4ae as well as the author's diagramive te wake of the del. Therefoe the wáke coefficient found will, have to. be reduced in trans- ferrtng from n.del to ship condition. This, reduction is rather considerable and may even be bigger than 0.10. As 157

mentioned earlier some idea o't this i'eductjo can be ob- tainedby studying the diagramsfig04 and fig. 5, but as fortute] nothing la knowñ regarding the influenceof dhipform on this reduction, the valuederived from the diagrams is very approximate. 5. When a designer gets wakemeasurements performed by a model tank, it is in order to securea better basis of calculation for future designe. Fornormal types of ships auch measurements will no longergive him any new information as be, in general, will obtaina wake coefficient agreeing with SCHOENHERR's f orniilaor the author's diagram. If he obtains a coefficient deviatingconsiderably from the one calculated, he, will notbe able to apply it to future designs, as in al]. probabilityit will be wrong or be the result of accidentally coincidingcircumstances, which cannot be expected toreoccur frequently, 6, For the model t anks thecase Is somewhat different. As their material is very big, theyought continually to check the formulae and attemptto improvò them. In order to be able to studymore accurtoly the influence of the ship form on the wake, theexperimental method suggested in section L ought to be Introduced,If possible a sy- steniatical series of experimentsfor the determination of the variation of the wakecoefficient with the ship form ought soon. to be performed,as the éarlier serles of experiments ofthis kind are obsolete.

K. COMMENTS ON TEE WA-FQfl1tJI AND DIAGRAM5 FÓR TWIN SCREW SHIPS AND FOR SPECIAL TYPES OFSHIPS, It would have been of great interest toperform an examination of the wake-form1ae and diagramsalso for twin screw ships, but as the material availablewas very 3imited, it was found, that süchan Investigation would be too uncertain, Moreoverfor a great numberof the model experiments parts of the necessary informationwere mis- sing. 158

The uncertainty of the *-va].ues found by formulae is somewhat less than for single screw ships. In most cases it is accurate enough in the w estimation to employ only 8 as parameter. If it is decided to useSGHOENKERR's for- inulae, calculation8 will be avoided by means of fig. 57, which is a graphical representation of the formula. Also regards the special type of ships, the material was too limited for an effective examination. 159

C H A P T E R. .V

DEPENDENCE OF WAKE OÑ VARIÔUS FACTORS.

L. INFLUENCE OF RUDDER ON WAKE. Inexáminiug the influence of the rudder on the wake, the Influenceonboth the nominal and the effective wake' must be considered. In the first case Its effect must nearly exclusively be due to its daing of the flow of water, while its effect in the second case must alsO be due to its changing of the rotation in the propeller race. As the dimn1ng is caused by the potential field around the rudder, the problem is to determine this field. When the flow Is considered as being two-dimensional, theso- lution la quite simple. It iS possible to transform firstly a parallel field into the field around a circular cylinder (see the sketch infig. 86) and then the 'lattèr Into the field surrou.ndlng a rudder shaped solid.

Fig. 86. Sketch showing the stream line systems found by transformations. d

But thereby only the flow arounda rudder, situated in.,a parallel field is obtained, It alsoIs of interest to examine the relations when the. rudderis situated In a field, In which the streamlines converge towards the rudder. Such a stream line systemmay be obtained by employing. the same transformationon the field e, as that which converted field a intO b (see fig0 86),chosing as centre for the transformation a point positioned 160

slightly to the right of the rudder. Then by applying to the field à the transformation. whichconverted b into c, a field as shown in e is obtained,in which a rudder-aba- I pod solid is placed abaft a ship-likesolid. As the last -: transformation will produce nothing new, it bas not been performed in this examination. In the following, the mathematical expressionsfor the transformations and the fields will bebriefly mentioned, referring for further information to(29). The two-dimensional flow is, as usual,presumed to take place in an incompressible fluid andwithout friction and rotation. Thu8 for a rectilinearparallel flow: 'the velocity u=-U v0 the potential function P = -UX the stream function = -Uy and w=p+i=_U(x+iy)=U.z (here w denotes a complex function in the two real variab- lea x andy ) For the flow around a circular cylinder is obtaine4: p=-Ux(1+a2/r2)

v=_Vy(1-a2/r2) (L-1) :w-U (z+a2/z By employing the transformation: Z=z+a2/z (L-2) 'on the field around the cylinder, theparallel field can be established. By means of the reversetransformation it is thus possible to work the opposite way, even if it, 'from a mathematically point of view., is nere difficult. By applying thetransformation: Z=z+a2/z on the field b, having displaced theorigin of the trams- förmation a short distance in the direction of the X-axis., the flow around a symmetrical profile, resembling a stx'eein- line rudder, is obtained. By varying e,, the position of the origin, different shapes or rudder can be produced. Some of the 3tream line systeme thus determined are represented 161

in fig. 87 In other words thestream lines around the cir- culai' cylinder have been transformed byieans of the ex- pressloas:

X =(1+- )(x+e ) (L-3) (X

=( 1- (z )

e0.08

Fig.87. Flow e-0/5 around three rudder shaped cy].inders (parallel flow). / /// // /1/0DER

e-023

------.- / //f/':///..7. //// / /

Finally in order to obtain a convergentfléld, the transformation (a-b) was employed once re, but this timeonthe field c,the centre for thetransfoe.natjon being chosen at a point to the right of ths profile, In i orderto limitthe work of calculation, the last transformation was performed 'aphically, Firstly was constructed a complete diagram ofthe flow around a ciraÏarcy- linder(fundamentally thesaine as b in fig. 86), in which 1b2 diagralli curves for thev].00ity potential function as well as for the stream function wereclosely spaced. The transformation was then carried out asfollows: The cooP- ,d.inat$s of arbitrary pointsin the c plane, for instance points on contour and streamlinea., were read off. By using these vaucs. as. $p and '4Vvalues in the transforma. tion dia'am the correspnding xandy values In the d. plane were found, and thenthe new streamline system drawn. By using two differentpositions for the origin of the transformation the two streamlinesystems fig. 88 were produced.

a:.

Fig. 88... Flow around two rudder shaped cylinderswhen placed, each of them, behind a circular cylinder.

It is evident that the diagram canbe employed in

transforming from a to b. - From the streamline sy3temthe wake coefficient for a propeller, having a.radius R, is determined by meansof the formula:

. (L.4) w ( a1.- b1)/a1 . being equal to 0.7 R and b1 thedistance from the 163

X-axis to the corresponding streamline ata point mf i- nitely remote. By varying the propeller diameter and the 'distance of the propeller from the rudder the diagram fig. 89 was produced. This diagram is valid for parallel inflow, and the example (tu = 0.24, D/]. = 1.40, a/].=O.4 giving w = 0.04) indicates how it should be employed.

Fig.89 .Diagram t 0.70 0 .1 .2 .3 .4 .5 W showing the de- / .12 pendonee, of the aLf/ potentIal wake J .10 coefficient on the diniens ions Aa IIiU .08 and position of the rudder (pa- .06 .06 ralle], flow). .04 Ii . .02 EX. IiL. .20 i .30

In the diagrams of flow for convergentfields, the dotted lines Indicate the flow,, whenthe rudder is rene- ved, thus the change in wake caused by therudder can be determinéd by observing the changeof' the stréamlines. The change of the wake is determined for a/i= 0.2, 0.3, and 0.4, and the result is represented infig. 9o. The thickness ratio of the. rudder tu beIng0.2, curves for the change, in a :.paraflel field have alsobeenhown, and the hatched regions give an impression ofthe difference between the two kinds of fields. From the small devi.atfân of the two setsof curves it may be concluded that, In general, therudder behind a shipïill.change the potential flow In thesame way as a free rudder in a parallel flow. On theother hand It is not certain, whether the In.fiuence of therudder on the rotation In the propeller race, 'and herebyon the effective wake,. does not dependupon the Inflow. In order to get the question of the influenceof the i 164

1;; a. b. - cow OrRE

a4 .08

¿'4 -

r .o 4. l.a ¡.5 2.0 14

Fig. 90. Diagramgi7ingthe difference between the potential wake caused by the rudder in a parallel field. and in a convergent field. rudder on thé wake more closely elucidated, the author had son experimental serles carried out at Statens Skeppe.. provningsanstalt in Gteborg. A detailed description of the experiments is found in (42). Two models, the original and the elongated, the latter produced by inserting a parallel middle body, were used in this investigation, and blade wheels were employed in determining the nominal wake coefficient. w was determined for the models with and without rudder, 'but as only a few runs were carried out for the models without rudder, the re3ults ist bé treated with a certain amount of reserve. In the case of the f ix'at model, the rudder meant an increase in the wake coefficient óf about 2 per cent, while the increase in case of the other model was only about 0.8 per cent. According to the dlagrant. fig. 89 the Increase should have been 3 to 4 per cent. Perhaps the differencö between theory and practice is caused. by the blade wheels, not registering the correct speed owIng to the heterogeneity of the flow. That the rudder changes the rotatIon in the propel- 1er race,andhereby the effective wake coefficient registered by the propeller, was proved by self-propulsion experiments of the models fitted with different rudders, and of the models without rudders. It appeared, that the 165

wake coefficient, broadly speaking, increasedlinearly with the thickness ratto of the rudder. Pora model run with a rudder (t/l = 0.2) the wakecoefficient will be 0,07 to 0.o8 bigger than for model withoutrudder. As the increase of the potential wake is at themost 0.03 to 0.04, the rest must be due to the effectof the rudder on the rotation of the propeller race. In the past, examinations of the influenceof the rudder on the wake have been carried outseveral times, amongst others by VAN LAMMEREN (64). As hisresulta with the models of "Simon Bolivar"are rather interesting, some of them will be represented and used in thefollowing. First his blade wheel experimentsmust be mentioned. They were carried out in the sanieway as employed by the author, and using models made to differentscales. For all the models, the change in the nominalwake, caused by the rudder, was found to be within 0.005 and0.o2o, in good agreement with the theoretical values of theauthor. The propeller experiment in open waterwas performed by VAN LAMMEREN both in the normalway and by having a rudder fitted abaf t the propeller. As he,for both the experimental series, determined thecharacteristic curves of the propeller in the usualmanner, it is possible to determine the change in effective wakewhich is caused by a rudder in a parallel field. By using the principleof thrust identity it is found, that hisrudder (t/l - 0.13, D/l - 2.1, and a/i 0.25) caused an effective wakecoefficient of about 0.05 (cQ-identity giveswQ - 0.o35). A wake of I to 2 per cent was to be expected, and theremainder of the wake therforernu beapparent wake, caused by the changed mode of action of the propeller. By employing thetwo sets Z propeller characteristiccurves in determining the wake in the self propulsion experiment,VAN LAREN obtained the following results: (Two differentpropellers were used, and crn-identity has been emp].oyed inthe wake determination), 166

Prop.no.168 169 1. Model with rudder, propeller in open water with rudder w = 0.2940.298 2. Mode]. with rudder, propeller in open water without rudder 0.3190.322 3. Model withput rudder, propellerin operi water without rudder 0.2870.286 Thu8 even if the propeller arid rudderis looked upon as a unit in the wake determination, the wakecoefficient will not have a fixed value, independentof this unit, but the range of variation of the wakewill be diminished. In the author's opinion a new method oughtto re- place VAN LAMEREN' s method in futureinvestigations of the variation of wake with ship forni. The propeller experiments in the open waterthus should be substituted by an experiment in whichthe characteristic curves of the propeller weredetermined by ruming the propellers behind astandard form. Defi- ning some wake value or other forthis form, a surer basis of comparison would. be obtained..

M. DEPENDENCE 0F WAKE C0EFFICIENT[. CONDITIONOF SHIP.. The wake coefficient of a single screwship will be least, when the ship is new andwhen its bottom is clean. By axid by as the surface of theship's bottom becomes rough ori account of corrosion, erosion ox'old paint, and gradually more and more marine animals andplants tiok to the bottom, the wake coefficientincreases, because the conditions of flow change. On accountof this, the increase of wake may be as much as 0.03(compai'e section C). In twin screw ships this question does notexist, the influence of tb's friction being withoutimportance. As ships during trial trips are oftentested under conditions differing from that of themodel experiment, it is 1nteresting to know bow big aninfluence such a change. of condition may have upon the wakecoefficient. Furthex' 167

aa only very few experiments to elucidate. thiá change of 'wake, 'were found, the author baLtwo experiòntal. series

of this kind carried 'outin.Gäteborg. ., The investigation was carried outin two stages. Firßtly a pair of ndels were run at different draughts md on.an even keel, and then at varying trtms;,.one enea with Constant draught aft, and one with constant draught arnidsh.tpa (the experiments are .further. described in (42)). Similar experiments have been published, amongst Others1 by KENT and CUTLA1D (58), TODD (12o) andBPJR (9), 'ànd on basis of these and the'aüthorta.sxpenjments the following rttles may be 'established: 'In the case of a chip 2oo-400 feet ln.length and having â propeller cóÏpletely submerged, any further loading of the 'ship 'will Cause a:diminutton of the ake coefficient òù áboüt 0.005 for each foot ot increased draught (the re the ornièer stern is submerged, the better the relations of flow 'beoome)..,In the case where the ship baa a trim by the 'stern, the draught amidships being kept constant, the wake coefficient will decrease by O.004 for every additio- Ml'foot of trima On the. other hand, when the draught aft 'is kept constant, every foot of trim will onlycause a decreaSe inwaké' coefficient of':Oeoo2. It is presed' that the propellerLsperfectly submerged for all the conditiona. In the.. casO where the propeller is on» partly sub. merged, it appòars that au increase of draught will cause a cOnsiderable increase of the wake coefficient. It is .howóver rather problematic, whether wake coefficients do- terminated, by. FROUDE's method for partly. aubmrged propel-. 1ers are Of any importance,' the fields oomparéd having

'completély different characters. . ' .

N. DEPENDCE OF WAi COEFFICNT. ON DEH OF WATER. In' calculating in 'section. D the potential, fields,'. around difförent solida it appeared that the fields wOre praotcally free of wall. effect' when the di3tance of...the solid's from the wall, was about ten times their half.. breadths. 168

In other words the depth orthe water will have nomf luence on the wakeeoeffieieñt, if the depth'of the water'ts depth of ten times the draught,but below this limit the water begins to have aneffect. As only few experiments toelucidate this question fixed li- have been published, itis difficult to lay down nes of guidance. In (59) KONING publishedresults for a series of ex- perinielits with models ofsmall coasters and canal boats (V = 1o7 to 127ni3). The models were tested partly in deep 'ailing water and partly underconditions corresponding to ratio of on the Wilhelmiflacanal. In the first case the depth of water to draught was'19.5, whereas in the other 0.53 were case it was 1.6.Wake coefficients from 0.33 to observed in the first ease,while in the latter they ran- ged from 0.43 to 0.83. Theaverage increase wasabout 55 per cent. Here it alsómust be taken into consideration, that the increase 'ias not dueonly to the decreased depth of water, but also to thesides of the canal. The ratio of canal width to. breadth of ship wasfrom 5.4 to 6.4. In an earlier section (sectionH) experiments have' been mentioned with a model of atug boat tested at three different depths of water, and trifig. 63 and 64 is indicated how intricate the questionis. In another connection, GAWNtSexperiments with a mo- dei of a cruiser havebeen mentioned ((41), also compare sectOfl 3). Further it is very difficult toobtain exact values for the variation of wake withdepth of water, 'as the width of the tank ought to beincreased at the same time as the depth of waterin the tank is decreased, in order that the results may not beaffected by errors caused by- wall effect. An idea of the influence ofdepth of water can be obtained by theoretical deductions.It is possible mathematically to determine the flowarroimd profiles in cascade, For a cascade consistingof'syxnmetriel elements and placed in a flow parallel to the axesof the profie, on 169

each side of :an..element. prof-ile- thére will- O - rSCtili ..near-atream.:1j The -f low thòrOfoeaù be conáidred-as .taçing place in -a :canaiand..by comparingwith the flow ,arouzdthe aame solid positjoned:jna. fluidóf intimité øZten8ion.tbe Lwaileffect can-be determinòd. A peculiar form of wall effect ismetwthin sib- marines sailing at a distance below thewater surface but otherwise in deep water. Herea bigger wake will be foutd on the top of the boat than at the bottom, thewater surface acting as a wall (comparo theexperiment with a submerged solid of revolution mentioned In (127)).

O. TEE INFLUENCE OF TRE DEGREE OF TURBULENCE ON THE WAKE MEASURED BY EXPERIMENTS.

In wake determination according toFROUDE' method the action of the propeller in open water iscompared to that behind the model. In the firstcase the propeller will work partly in a laminar flow and partly ina turbulent flow, whereas in the lattercase the propeller is practi.. cally always working in a turbulent flow.In other words if nothing is done to produce turbulence inthe propeller experiments in open water, anerror is introduced, in the wake determination. This error will dependupon which type of propeller is being employed, whichexperimental scale is being used, and by which'revolution nuitber the- caracteristic curves are determined,as in the first place the circulation around certain blade sections,espe- cIally thick sections, willvary very considerably at certain REYNOLDs numbers, and secondlythe position of the transition f rout turbulent to laminarflow will vary, On the other hand the de'ee ofturbulence with the ship model plays practicallyno part. Experiments prove that, regarding the. wake coefficient, itis allthe same whether an artificial turbulencégeneration is used or not or what kInd of turbulence generatoris employed (comparo ALLAN'S experiments (3)). 170

With blade wheel measureiìents the author bas expexi- eneod that the degree of turbulence around the del playa no part, as measurements with bladewheel have been performed several times in the merning without apre1imizary du run and od agreement was found with measurements taken after the dwimiy run. 171

C.H.A PT E R VI WAKE DISTRIBUTION.

P. DISTRIBUTION OF WAKE OF SINGLE AND TWIN SCREW SHIPS.: For the.design of propellers it is.often of importance to know the wake distribution. Experiments can be oar- ned out for determining the radiál distribution of the wake by means of blade wheels or resistance rings; if al- só the penipherical variation is wanted, pitot tube measurements are necessary. Instead of being based on experimental resulta the wake distribution is often estimated for instance by means

of VAN LAMMEREN'S diagrams (69) represented in fig. 9f. :The. diagrams are based on a loñg .series of expertments, r As abücissa the vertical prismatic coefficient is used1 while the ördinate is the ratio of the wake coefficient of One ring element to wake coefficient of the whole of tho.propeller. Curves are given, corresponding to the different ratios of radius of ring element to propeller radius. It is a little peculiar,,. thát no regard to the size of the propeller has been taken in the diagrams for sing- acrew.ships. In order to throw solve, light on this question, the radial wake. distribution just abaft a plane suifaoè can be considered. it WIiI' be solely the friction causing the variations of. wake, and the size of these variations can thròfore be determined by employing the theories of the fniötlonal belt0 By integration of the curves for the periphenical wake variation (indicated in fig. 92) at differexit radii, the curve of the wake distribution In rig. 93wasdetermined. nwas taknas 0,18 (see section C). Under the assumption that the plate was 6 m long and its speed 2 m/s, the diagram in fig. 94 was constructed. Here the abscissa is the, ratio r/R, and the ordinate is the ratio of w1 for the ring element tow for the whole 'of the propeller. It appears that thereis a marked differerìcq in the wake distribution for big and 172 small propeller diainetezs. VAN LPJMEREN's diagrams therefore ought to bave been defined for one certain size of propeller only. For other sizes of propeller, the distribution curves might then be extented or shortened.

u 'i'' 30 - I Fig.91. VAN LAMIEIN'S curves for determining the wake 2.0 distribution. 511 .2

1.0 1.0

.9.- .8- .5- V 'J.82 .84 .88 .90 .74 .78 .82 -.86 Single Screw Ships Twin Screw Ships

Fig.92. The peripherical variation of the fric- ional wake coefficient abaf t a plane surface. 173

Fig.93. The radial variation of the frictional wake abaf t a plane surface.

T2 W. MM

qi,

.06 5m

Fig.94. The radial variation of the frictional wake abaft a plane surface for different sizes of propeller diameter, .10 'Ç,.

O .2 .4 .6 .8

The calculations of the potential wake distribution of cylinders as well as of. solida of' revolutionand ellipsoids prove that the potential wake vartes relatively little over the propeller disc, regions very close. to the ship being exempted. An impression of the magnitude of the variations is obtained by looking at the figures 33 'and 34 in section D. The figures give partly theperip}ierical and partly the radial wake variation abaft two ellipsoids. 174

However, the influence of the potential wakecannot be disregarded altogether, as It will cause a differene3in the distrIbution Curves for ships having U-and V-shAped frames. An idea of the degree of magnitude ofthis dir-. fereÍóó is obtained by lookingat fig. 95, which repre- senta results from some ofYAMAGATA'S experi'ments (130). Thefour. models all bad the sameprinpaldimefl5iOfl,8i block coefficients ai4 ¡orebodies, while the fiáme shApe of-the after bodies Wasvaried. Two of the models (no. 195 And -198) hAdpronounced U-shaped framer, whereas one bad V-shaped (no. 196) andthe fourth pronunced V-shaped frames (no. 197). It Isnoted, that the raialvarlatiön is far smaller for the modelshaving U-shaped frames in the after body, thanor the models havingV-shaped fra-

w YAitIAGAkA N. /95- .80 /96.-

- /97 .m 198 -- .60 ç Fig.95.The radial va- - - - - nation of the wake coefficient of models .-, .40 having V..shped and N\;. - U-shAped frames In the URQAMEJ after body (YAMA.GATA'S .30 experiments). t.qA MES .20. -

.10

.0/ .02 .03 .04 r4oD

been taken to L In VALAMMEHEN'S diagram regard has .the frame shape bemplOying p as parameter, As in the author's opinion it isinsufficient only to einploy'p, an attempt has been-mdò to construOt a newdiagram. The sa- in. section F was Um materia]. u3sd In thern investigations: emplòyed.AllthedistributiOfl curves were oorrectedto - D/L equal to0.04 and - then arranged accordingto block -- eóeffiòieÌt and breadth-length ratio.The resulting mean urves ax'é' indicated-in fig.96, and-in fig. 97.correòt-ion curves, tO. be-used i-n,.caseof special -frame shape, i'e'&.- 175

ven. It should be noted, that the deviations from the nean curves given are often big (± 0.05). Part of the -difference is due to the shape of the after body,and part due to the dependence of the blade wheel Integrations on the heterogeneity of the field. But as very, little is Imown concerning the scale effect on the distribution curves, it is the author's opinion that the curves are appileable for all designing purposes.

w .50 4L .13

.45 I 40IiLIU us& .25

Fig.96. The radial varia- .2 .3 .4 .5-.6 .7 .8 tion of the wake coefficient of single screw w ships (D/L = O.o4). .50 - 17

.35 30.

.25

.2 J

Unfortunately the ecale effect plays a big part In single screw ships. To get an impression of this influence, the wake distribution abaft a plate 6 n long, and having a roughness corresponding to n = 0.18 Is compared to that abaft a 15o n long plate,havingn = 1/9. In fig. 98 the two distribution curves are indicated, and the difference i 76

+ Fig.97. Correction for frame shape (distribution of w for o ìI I., singLe screw ships). r

.04 .35 \ M L0I50m,51

Fig.98. Comparison between wake distributionabaft a 6 m long plate("nadel") and abaft a 150 n longpla- te ("ship").

between them is very considerable. How the relations are, when dealing with curved surfaces, is not mown, but in all probability, they will resemble the relations at plane surfaces. From diagrams of the wake distribution around twin screw ships, it appears, that the friction belt extends past the boasings, just as if no bossings existed. Besides the primary friction belt a secondary belt is formed around the bosaings. In order to fix the radial wake variation of this type of abip, the author used the same material employed in theconstructionof diagram fig. 59. The radiál wake curves were first arranged in groups according to block :cQeffieients and. then extended or shortened so as tobe; valid, all of them for D/L = 0.03. Meancurves were drawn a4 fal.red.as indicated in fig.99. In thisfigure the ab- 177

Fig.99. The radial variation of the wake coefficient for models having twin screws (D/L = 0.03).

r/ .2 .3 .4 .5 4 .7 .8 .9

Fig.loo. The radial variati.. on of the wake coefficient for twin screw ships (D/L = 0.03).

.2 .3 .4 .5 .6 .7 .8 .9

soissa is the ratio of the ring element radius to the radius of the propeller, the ordinate being the averageWa- Ice coefficient of the ring element. It is of interest to observe that the wake coefficient when the radius of the ring elements increases will firstly decrease and then increase in the region near the tip. The initial decrease is due to the velocity variations in the secondary friction belt around the boesinge, whereas the wake increase at the tip is due to the fact that the propeller is partly working in- the prinry friction belt. But as the friction belt In model experiments has about twice the thickness it ought to have, there is reason to believe that the increasing wake towards the tip is only found in the model experiments. It can also be assumed that the secondaxy friction belt is stronger in model experiments than in reality. Fig. 99 therefore has .178 been anflded as shownInf i.100 and thò author advoes- tós the use of this latter diagraminthe design of propel].ers..

Q,' THE OPTIMErM COEFFICIENT AND DISTRIBUTION OF WAKE. By, the optimum wake coefficient' and wake distriutiGfl may be understood that wake coefficientankiwake distrI... 'btion'by Which is obtained thebestpropulsive coefficient. As it ispossible,during theeonstruction of thelines of a ship, to influence upon the wake.nd its distri- but ion, for instance by usIng 'frames more or less U.. or 'V-shaped,' the problem will be briefly di3oussed' in the following paragraphs. The efficiency 'of propulsion of a ship is. expressed

by: ' ' -EHP _1-t '1'h p'rr - I- w p rr EHP and DEPbeingrespectively the effective horsepower and the de'livéred horsépower, his hull efficiency, the propeller òfficiency,-'andfinally1rris the relative rotative efficiency. The problem therefore is to determine a formandherebyaw, where'

'REP . DEP

La as small as possible. Roughly speaking It might be said, .it inchos.nga form giying a big wake value, a small : value fEP is obtained, provided the forxchosen doest 'require too big an REP. Now firter 'î is dependant upon w, aa the Ideal propeller efficiency'

= I

being the thrust loading: ' ' '

' R I' ' T" :'T' 1' F (1-t)(1.w)2

Normally the influence ofwOfl will bconsiderably big- gerthanon Îlj. Moreover the relative rotative efficiency -179

'*ill 'dépend upon the wake. distrtbutibñ, the irréglarity öf whi'ch can cause'amarked:reductlon inpropeller effi- cièncy. As a modirication.in the shape of the hull, beai- deschangng w, also changes t, it is obvious. that the problemis véryintricate0 Different investigators (RANKfl (9o),FRESENIU$(36), HOR1 (47),HELMOLD (44), DICKMANN(23) and severalothers) have, with certain. sithplifying assumptions, attempted. t solve the question theoretically. DICKAN) in orderto

carry outis calôulations assumes: . . I That the propeller bas infinitely many blades,and thus' the condîtio±is become státionary.

.2. That the- rotation in the propeller race be neglected (inthecase of a propeller with a very anali pitch and -big ev6lut1on number, the rotation will be a _____ 3.That the propóller is only slightly loaded.'. Further hé assumes,. in most of his reasonings, thatthe wake is constant ovex the whole of the propeller disc. DÏCKMANN finds that in generai, the potential flow will have an unfarourable effect on the propulsive .eff i.. ciency, when the potentlaiwake is poéttive. Intho same way a negative potential wake coefficient will havean favóupabIé effèct, which Is utilized in nozzle-propellrs. Regarding the frictional wake as well as thewave wake DIG1MAI finds, that itIs anadvantage to have bigposi- tIve wake coefficients. Although by means of DICKMM4Na theotes mani problems ioOnécttønwith-theinteraOttón between ship and propeller, can bè explained' '(refer (68)'), itappears a little peculiar that better propulsion is, attáinedbya ñegat I.vè. than by a' positive potential. wakO. Comparing a ship, haying pronounced wîth a ship having pronounced V-shaped frámea, the wake óoef- ficient f the tirét ship willbefar bigger than thatof the latter, probably first of all because the potential waké coefficient is bigger for U-shaped than for V-shaped ships. However, experiments prOve that the big wake Is not'ac- coznpániedbya big thrustdeduction, whichmeàns that the hull efficiency Is Improved' by changing over from V-frames 180

to-U-frames. It will be remembered.,that in this treatise no. direct proofto.the effect that the potential wakeis bigger at U-shapelthafl at V-shapedframes, baa been given. This was only assumed as aresult of the comparison between two and three-dinißnaiorlalflow. DICKMANN'S assumption of constant wake over the wholeof the propeller disc is perhaps the reason whyaccording to his theory ships with V-shaped frames havebigger propulsive eff i- ciency than ships with U-shapedframes, contrary to general experience. In order to obtain the highestefficiency of the propeller, the pitch is adapted to theradial variation of.the wake. It is however impossiblein the calculation of the propeller to take into accountthe peripherical wake variation, which nnist be as small aspossible. This can be attained by using U-shapedframes. If for some reason or other V-shaped frames arepreferred, the propeller ought to be chosen as small aspossible, in order to avoid that parts of the propeller areworking in a strongLy heterogeneous field. HOGNER(46) has attempted to obtain a minimum wakevariation either by fitting a big bulbaround the propeller axis or by makingthe propeller aperture very big. For a twin screwship, having bulb-shaped bossings, HOGNER states that the effect saved will be from 4 to 5 per cent (his nadel was runwith propellers- designed for the originalbossings). HOGNER is of the opinion that his design bas thefollowing advantages: Gain in efficiency by suitable variation ofthe pro- pefler pitch. Gain in efficiency by better utilization of the frio- tional wake. Less thrust deduction. Less strain on the propeller blades, which therefore may be made thinner. Less danger of vibration. Less danger of cavitation and erosion. ROGNER however bas performed only relatively few experiments. As his construction will be expensive to carry I 81

out in practice, lt is problematièa]. if itwill, ever be employed. Very few experiments concerning theaction of the propeller In a heterogeneáus field have beencarried out. SCH11IERSCEjSKI (94) has examined the propulsionof two submerged forms having the same principal dimensionsand coefficients. (L = 1.6 in). One of the formswas a sölld of revolùtion, the otherwas also forthe greaterpart shaped like a solid of revolution, and the extrêmeafter end bad a frame shape resembling V-shaped framesThe same pro- pefler (D = 0.163inand P/D = 1.o) was used for both of the módele, Thé experiments proved, thatthe efficiency of propulsion was lo tó 11 per cent betterfor the solid Of revolution than for tlie V-ended solid,butit Is doubtful whether the same big difference Wouldhave been obtained If for each form an especially adaptedpropeller had been used. The question of the action Of thepÑpellerbehj,nd U- and V-shaped after bodies ought to bethx'oroughly in- veatigated theoretically as wellas experimentally, and the Investigation subsequòñtlyextended to án examination of the over all propulsive efficiencyin servicé. 182

CHAPTER VII: DEPENDENCE' 0F VARIOUS FACTORS. WA.

R. T_ INFLÚENCE 0F T WAIT N T NUER OF EEVOLUTIONSOF TEE'PROPELLER. In design it may often be ofimportaxce to estimate the influeñee of possible inaccuracyhere and there on the final result. In this section theinfluence of the uncertainty in the wake ooeffic.entwill be moze closely examined. As mentioned eáriier thisuncertainty originates either from the actual rodelexperiment or from the tzlans_ formation of model data to ship data. As many desgrrs calculate theirpropel1e for the dèl condition, an6 then take accountof the scale ef- feet by correcting the number ofrevolutions of the propellers, it would hé natural bere toexamine the effect of a wakevariation onrevolutions. This procedure has been usedby VAN LAEREN,amongst others; in (63) and (lo). he states the uumbr of revolution8 asI to 2 per cent bigger, at the ihip than átthe modél, dependant on the scale of ndel. For a comparison the followingsmall investigation bas béen made. For the propellerof the Wageningen-Beries E 4.40 having a diameterof 3.5 m and a pitch of 3.5, the propeller thrust

H TcTD4I12 a well as the necfssary horsepower

P 2 Q. n 1/75 = D5c n3

was deterrdned for differentwake coefficients (w = O.2ö, and. for different numbers of 0.25, 0o3Ó, 0.35,and 0.40 ) revolutions,assumiñgthat the ship was sailing at a apead of 15 knots. tú fig. lól curves areindicated for the variation of T and HP with number of revolutiön..Under the assumptionthat the coefficient of thrust deduction was constant, or in other' words, that for all the wake coef- 183

AQOPE4L&E W&EN/IV4(N B44N +0- P.3.5,

- Vf50v

Fig.lol. The va- nat ion of the N propeller thrust OW and the delivered. horse power with number of revo].u- 6000 - -tions at diffe.. P.P(L1ER - f

WAG/WINN 8-4-40 - rent wake coef- D.3.5,,p.3.5,,, ficients0 50e-

4000 -I - -. -

3X0 ..- YA:

- /00 - - 200 N

ficienta the same propei.ler thxiist was noedd inorder to produce the -peed of 15 ots, the relation between number of revolutIons aEd wake1 and between number of revo lutions and horse-power, at constant propeller thrust, wa3 detenxined (-see fig. 1à2).. Supposing proportionality be tween the numbòr of revolutions of the -propulsion machine- ny and the horsepower, the characteristIc of the engine were -Insertòd on fig. 1o2, in which the characteristic curves for different sizes of engine are Indicated as dotted.lines. 1t Is seenthat, -if the wake coefficient bas

been. estImated to high, -the propeller must rotate ata 184 bigger number of revolutions inorder to produce the necessary propeller thrust,and unless the nan pressure in the cylinders of lareciprocating engine is increased, 'the desired speed of' propulsionwill not be attained. If, for instance, the wakecòefficieut has been estimated O.o5 too high. it causesa necessaryincrease in horse- powei and nuinbòr ofrevolutiorks of respectively 4 and 2 per cent. I On .rópeating the calculatiofl5for anoher propeller, the relatioùs will be found tobe perfectly analogous with those stated above. In going. from model to ship,the wake coefficient is.. di.m1zisbed by 0.05 to O.1o., which means,that in the cal-. cu].ations for the model conditionit is necessary to use a iumber of revolutions, which in thepresent example is :. 'rom 4 to 8 revoluttons pet'minute less than that of the propulsion machinery. It is better fromthe beginning to work with the wake distributionvalid, for the ship.. In order also to. get animpression of the influence Öf an error in the estimation ofthe coefficient of thrust seduction, curves corresponding to. different propeller thrusts have been indicate,d in fig.1o2.

relatiOn Fig, 102. Top:'The ... tetween Wake coefficientand thunber of revoÏutiofls for. '55 constant proeller thrust. Bot.tot: Rélation between thrse power and number of. revolutions for constant propeller thrust. u.

"--Ecli ACTE TICS'."' 185

TAYLOR (1o7) in his book "The Speed and Power of' Ship&' states, that 1f the wake coefficient cannot be exactly estimated, it is better tO use a figure too high than too low, and TAYLOR gives the following reason: the wake coefficienti8estimated too high, during the trial trip the engine will revolve more quickly than calculated,:which as a rule is permissible.n the other hand rif' the 'estimation of wake coeffiôient has been too lôw, it may be impossible to achieve the desired revolutions of the ezigine without reducing either the pitch or the dia-. meter of the propeller. The author is Of the opiniòn that the oppositeought to be done, as there may be a risk, that the horse power 'réqulred will iot be available. Further lt Is simpler to reduce thø size of the propeller, than to increase 'It. Finally a fairly con method of correctionfoie scale effect must be mentioned. By this method thepower of 'the model multiplied by a con8tant determined by:

K= I +Oo12\J

1WL being thé length of the waterline.n m. At the same time the number of revolutions of 'the propeller is In- creased by 2 to3 per cent according to the scale of the model, and then a trial'trip òurve Is cónstructed. As it is still 'difficult to determine the horsepower exactly on the trial trip, the method can only be checked approxima- tely.

S. THE INFLUENCE OF THE WAKE ON T S RING OF.

THE SHIP. ' Besides influenàlng the mode of actiOn of the propeller, the wake also has an 'influence on the manoeuvre- ability of the ship. Often'a marked change in the field of flow around a ship has been the caüsë of Its disaster. A 'ship may get 'into shallow water or very close to anó-' ther ship, and thus the potential field around the: j1 186

may be changed,, Which in its turnmayreduce the ability of the ship to manoeuvre. If the wake' isincreased' at the rudder of the ship, less water will flow tothe rudder,, thereby reducing the steering qualities. In thissection these relations will be' studied more closely,using the results of BAKER's and BOTTOMLEY'S experiments(14). These two investigators carried out eperiments ith single screw aè well as twin screw mools,the models being fitted with different rudders. Their basismodel Was that of a single screw shiphaving the following, data: 4&o fést, B 52 fest, d = 23 fest, =lo400tous, 0.75, so 41e of del 1/25. '1Popeller data: D = '16.2 feet,' H/D= 1.49, FU/F = 0.40. 'In the experiments the normal force on the rudderand the 'udder moment were determined and compared to the eorre.. 'sponding quantities obtained when only a fin was placed .n,front of the rud4er. BAKER and BOTTOMLEY deternined the ratio (r) of thé forces on the rudder behind themodel (without propeller) to the forces on the rudder in open watei', and thus obtained an idea of the Iìfluenco of the wake on the forces of the rudder. För a speed corresponding to lo knots the ratio r will have the following ave-

age values: - Mdder angle rudder moment ratio normal force ratio, 3O 0050 0.43 150 0.41 0.37' Assuming that the normalpressure P on the rudder is equal to: P = kAV2 being a constant depending on the size of the rudder angle, A the area of the ruddér and V the velocity of' the flow of water, the ratio just defined must be equal to' the ratio of the square of the vòloóit.ies of flow. Ii' it further is assumed that the forni of the model tested bas been quite ordinary, thwake coefficient may be put e-, qual to say 0.34,. and the ratio r will then be about .0.43,

agrees well with the experimental results. -' In order to find the Influence of the degree of fine- 187

ness on. r, an experimental serles was further carried out for three models having the sanie principal dimensions as before, but with different lines. Only the rudder moment was measured, and the results were as follows: f5o a3o° 0.8o 0.26 0.34 0.75 0.4o 0.50

0.70 0.49 0.55 V = lo knots, If it is now assumed, that the wake coefficient for the fullest forni is about 0.40 and for the finest form 0.29, in the same way as before it is found that r = 0.36 and 0.50respectively. WIth a model of a twin screw ship having the same principal dimensions as the single screw ship BAKER and BOTTOMLEY carried out corresponding experiments and obtained the following results: rudder moment ratio normal force ratio a 3 0.58 a = 15 0.42 0.43 V= lo knots. It will be seen that r assume bigger values than for the single. screw ship on account othe differently shaped after body. The results of the experiments with twin screw modela of different fulnesses are a little surprising, as the, reduction of rudder forces is biggest for the finest model, which results probably are due to experimental errors: rudder moment ratio normal force ratio a = 15° 30 50 0.75 0.42 0.62 0.43 0.58 C.7o 0.42 0,58 0,41 0,53 0,65 O.4'. 0.57 0.40 0.49V=12. knots, For the sake of completeness it should be mentioned, that on account of' thepropeller race the change of the steering qualities caused by the wake, Is compensated. Further, according to BAKER arid .BOPTOMLEY the ratio between the rudder forces on the rudder behind the model with working propeller (single screw model) and the md- '188 der forces on a rud.er behida plate, having an aperture but nô pròpeller, i: rudder montent rationormal forcé ratio =3oo 1.13 1008 =15° 0.96 0.93.V = lo knots. Fortwin screw ships the ratio between the rudder forces on the rudder behind the model with working propellers (outùard turring) and the rudder frces on the rudder behind a plate (no propellers) was found to be: rudder montent rationòrmal force ratiO

=300 . O.7o 0.63 z= l5 0.43 0.41 V = lo 1ots. other words it I'S necessary to provide twin, screw ships 'vIth bigger rudders than single screw ships on accóunt of the differencé in. "effective" wake. ' In single screw ships care must be ta1en when stop- ping the propeller,s the effect of the rudder then is very much reduced.

» Then a ship is being'overtaken by another ship, the. field of fÏow wilinöt change, if thedisteincebetween'the hips l's greater than ábout 4 times the .bredths of the ships (compare section D-cl), If the distance is less,, will be a pòssibility, that the two shipá will inter- act whereby steering beàomes difficult. Inpeciallun- fa'ourable cases collision maybe unavoidable. Any'fixed lines of guidànc.e cannot be laid down, as to many,factòrs playa' part, sCh as the shape of the shipsL the Sizeß, and tite speeds. On.lya fewinvestigatorsha'jJe examined the question(amongst 'others TAYLOR (lo5)1 BAKER (11)and: PROHASKA (8e)). When a ship gets into shallow water, tie wakeòef. ficient will rise (Óo'mpare 8ection N), and thereby the. s'ering'quality'of the ship wlfldiniinish. Maxy expertments (13) have been performed witsteering in' shallow water',' but only in a few casés has the wake béen. determi- néd.GAWN (41) has ca'ried' Out some experiments with a dei 'of 'HMS"Ñelson", in which the' initial-shlp tui'- fling' rnömeñt and the wake' coefficient were determined in I 89

deep water as well as in shallow water (52 feet deep). The draught of. the cruiser was abòut 30 feet. At the same time as the propeller registered an increase in wake of O.o95 (from 0.195. tó 0.290), the turning moment for a beim of 10 diminished about 41 per cent, for 20 about 3o per cent, and for 3á° about 16 per cent. In single screw ships the same problems are found, but no reports of experiments are found coñcernlng the change in wake and ruddertorque. Generally itwill bediffi- cult to predict what will happen, as the velocity of water in the propeller race will here play a big part. Finally it must be mentioned, that for ships sailing in shallow water over sloping ground, suction and pressure forces on the ship will arise and may at times be so large, that the ship will not answer the helm at all (compare (41)). The forces arise on account of the difference in the poten- .tial on starboard and port side of the ship. Also in this case the experimental material is limited, and general principles cannot be established.

T. THE INFLUENCE OF WAKE ON PROPELLER CAVITATION AND VIBRATION.

For the sake of completeness it.might also be mentioned that the wake or rather the wake distribution plays a big part with regard tó propeller caviat ion and vibration. Most propellers are designed so that in a poripheri- cally homogeneous field they will be cavitation-free. But if studying the action of these propellers in a radially as well as peripherically heterogeneous field, it will be seen, that every time the blades pass through a region of considerable wake, the distribution of pressure will be such, that possibility of cavitation arises. The amount of inortance attached to the question, is not easily seen, as the problem has only been dealt with a few times so far, as for instance in (78). As the forces acting oñ the propeller blades vary with the wake, the reaction forces of the hull will also 190

vary with the wake, and thus there will be1 the possibility of vibration in the propeller as well as in the hull.. Pro- peller vibrations have previously been dealt with in a number of publications, amongst others by LEWIS (74). At times this vibration mabe the cause of singi-ng propellers0 It cannot with any certainty be foreseen if the propeller will sing, but fortunately itis generally quite easy to change the osciUation figure of the propeller andthus avoid the noise. 191

CHAPTER VIII: THRUST DEDUCTION. U. THE RELATION 0F THE WAKE COEFFICIENT TO THE COEF-. FICIENT OF THRUST DEDUCTION IN SINGLE AND TWIN SCREW SHIPS.

During the preced5.ng years this question.has. been dealt with several times theoretically as well as experimentally, but still there are many problems unsolved. In order to deal with the question theoretIcally it is neoessary to Idealise; thus most of the theories ars based on the assumption that the velocity of the inflow is the saine over the whole of the propeller disc, and thus many faults are introduced. Of older theories, RANKINE's (9o) and FRESENIUS'S (36) are the best 1iown. Amongstore recent treatnts of the subject, the investigations òf HORN (47), HELÌOLD (44), TELFER (1.13), and DICKMANN (23)- should be mentioned. DICKMÁNN, whose examination mist be said tobe the most thorough, found that the relation of the potential wake coefftcent to the pOtential thrust deduction coef fi- dent was equal to:

t = 2 p (U-1)

aT being the thrust loading T. a (u-2) T= 1/29v.F2 The friction- as well as the wave thrust deduction coefficient may according to DICKIEANN, be pút equal to O. It is of interest to compare withHORN's and TELFER'S formulae. Thus according to HORN

t 1-(1-w) l+1+aT. p. (U-3) i +Ii.+(1 _w) aT and to TELFER wF (U-4) 2 (1+ U/'Ve where + 4 a - I U/Ve and a,= irD.v8 192

vt frc2lnthe- -f at that ththree fôrùu1ae all have been formed of the sane quantities, the 5imilaritis not striking. A very thorough experimental invsstigation of the re- letton of ake to thrust deduction has beecarried out by VA1 LAIVREN (66). In the following VA1 .LAvREL'smain results will be brief» mentioned. i. The thrust deduction coefficient of a ship when sailing depends first of all on the periphorical 'htergeneity. of propeller thrust and wake, and only varies slightly with the -:thrust loadIng of the propeller. 2. If the p±oe1ler thrust is periphericaly constant, the thrust deduction coefficient will depend. on the ratto of di3c area tò 1dship section area( F/0 ), 3 For constant thitist loading the thrust deduction 'will epend upon.the frctional and potefltial W.ko. 4. A f.ori1a for thrust déduction should be build up of'two karts,, the first part taking account 'of th variation of the 'hrust deduction with p/o, the thrust .load1rig, and the shape the after body (the influence -of the last two factors is small), and the second part taking account of the variatIon of' the thrust deduct.on wth.the- periphorteal heterogeneity of wake and thrust lòading. 3. The existing forilae for thrust deduction ars thus not correct. 6. The sealS effect on thrtist deduction is very small. It is seen that VAN tAEN'3 experimental results do iot quito agree wit1ithetheoretical.re,suItkOf the other investigators. In order to elidate the relation between thrust dedüc- tlon and'wake, the author exa.1ned the same material which hd. been used for the, wake examination. As it 'was not considered that there wa any simple relation between thrust deduction and Wake, but tht probably the two coefficients dependod to a lrge éx'ent on the same factors, a start was made by finding so of the -factors which bad an influe'ico on the thrustdeductton. In the following, single and twIn' scréw shipswill 'be dealt with se- 193

parately, starting withthesingle screw ships. Ït.was ratural to assuma, and experiments with similar bodlòs also prove,thatthe thrust deduction is dependant on the fullness and the breadth-length ratto of the.ship. The experimental material therefore was arranged in groups according to B/L, and in each group it was arranged according to block coefficient. The result is represented in fig 103. Mean curves were- drawn and in f ig. 1o4 all the. mean curves are put together. This last diagram gives, in other words, the thrust deduction coefficient for ndeIs having normal form and normal rudder and propeller arrangements (seetab- le 25).

r '

mIL 1 !IPI 4 I ¿

V - - 45 - - 6 . O 45 Lt'! ,, -d 6 i

1 Ir - I L L 1

. I.. . t,... , 155

-. .

It I . . V. L 8,j..I7

L,s ô ô f, 4 .5

Fig. 103. Thrust deduction coefficients found experimentally (single screW ships,- Wageningen experiments). 194

:': .30 -- H -

,/3 TT luaU

Fig. 104. Diagram for predetermination ofthe thrust deduction coefficient(single screw ships).

As in the wake investigation an attempt was now made to find the reason why the points did not lie on the meancurves. In the preparation of fig. 1o3 no correction hasbeen ade for the propeller diameter, and thereThre, from the re- lathe small scattering of the points, it may be concluded that the dia eterdoe3nothavethe sa large effect as in the case of the wake. In order more closely to study the èffect of changing the propeller diameter, the same expert- ents as for thecorre spondtngwakeinvestigation (112,56, and 131) wereemployed, andthe result is indicated infigo o5. TAYLOR'S model had a normal frame shapé in the aftér body, whereas thetwothers had V-shapedframes.FX'O the dia'aItappears that thetlu'tistdeduction coefficient +aries less with the pròeller diameter thn does the wake ¿oefficient, and thai the frameshape is likelyto play a ¿onsiderablepartinthe variation of t. Further, to throw some more light on this question, the result of YAM GATA':s experiments with themodels195, 196, 197, and 198. (13o) were arrangedas shown In fig. 106.As all the models have the same principal densions anddisplac6ment,ndonly 195

differ as regards shape of after body, the diagramsprove that the shape of the after body plays a part in the.variation of thrust deduction with D/L, and also that the shape of after body plays a part in the absolute value of the thrust deduction.

t . YA N'0A TA o KEMPF .25 TAYLO2 . o

Fig. 1o5. The variation of the thrust deduction coefficient with the propeller diameter (sing. le screw ships).

- . e

.03 -04 -05

t N2198oU-çQ. .25 /950--u-FP /96 ----w.e,/4L /97 e--- -p _0 Fig.1o6. The variation of the thrust deduction coefficient 20 with D/L for different frame shapes aft (single screw ships, YAMAGATA'sexperiments).

-03 .06 .05

For ships, having V-shaped frames aft, it may be reckoned that an increase in D/L from 0.03 to O,o5 will cause an increaSe In t of abouj 0.04. In ships having U-3haped frames aft or frames of normal shape the variation of t with vryIng D/L may be neglected. Actually D might have been made non-dimensional, dividing D by the distance from the propeller to the aftermost frame in the parallel middle body,but as this distancè ta generally not known at the design stage, it has been prefered to use L, In order further to find the cause of the scattering of the points in fig. 1o3, the diagram was compared to the 196

corresponding diagram for wake. They were examined in order to find out ifitwas the same ships which, in both diagrams, fell outside the main curves, andif the points were situated in the same way in the two diagrams. Thus it appeared that in about 45 per cent of the cases, the points which either were above or below the wake curve, would be positi3ned on the opposite side of the thrust deduction curve; in 3o per cent they were on the sameside, and in 25 per cent the points were on the curves eitherin the one or in the other of' the diagrams, butonly in one of them. Thi.is thero was a certain probability, that the thrustdeduction coefficients of the ships having V-shaped frames aft, would be above the curîsz, while ships withU-shaped frames would be below. That this was the case for shipsha- vthg pronounced U- and V-shaped frames could be seenisne- diately, whereas it was necessary, for normal forms, to proceed in the same way as for the wake analysis. The frame coefficient f was derived and its average value for each group of ships found. Then the results werearranged as indieted in fig. 107. As the difference in the position of the three curves is seen to be small, it is necessary to be careful when correcting for the frame shape. It should be noted, that the curves do riot give a perfectly correct

- image of the relations, because in the case of the V-shaped ships the correction for propeller diameter has been omit-

t ed.

r t

.80

Fig. 1o7. f's variation I80Y( 11 60 with B/L and with thrust INr58E.k/H5WAxEc- s deduction coefficient.

4f 1.,g .13 '6 .15 ./7 197

VAN LAREN is of the opinion that the p.eripherica]. heterogeneity of the wake plays an important part in the tlirust deductIon, but as the perphercal heterogeneity in the wake is due to the shape of the after body, it is the after body and not the wake variation which determines the thrust deduction. As for the.wake coefficient, the thrust deduction coefficient depends upon the height of the propeller axis abo7a the keel, On examining TAYLOR'S experiments (106), it will be noted that a change in E/d from 0.2 to 0.6 cau- sesan increase in thrust deduction of O.oO. In otherwords a change: (E/d)1 will cause a change in the thrust deduction of: 0.2 (E/d)1. T0D's experiments (12o), however, show a somewhat bigger variation (about 0.4 .(E/d)1). The results of hIs experiments are represented in fig. 1o8,and In this case no correction for the propeller diameter bas been made (compare figure 47). Therefors it seems likely that the t-values for the 'small propeller diameters are a little too small, and the author is of the opiniofl, that it must generally be concidered that a change: (E/d)1 will cause a change In thrust deduction of: 0.3 (E/d)1. It has not been possible to establish arule, neither for the influence of the draught-breadth ratio, nor fox' the influence of the fors body. A number of experiments (compare 31, 32, 33, and 123)) prove, peculiarly enough, that the influence of the fore body on thrust deduettonat times i. conderable, One oE1RSON's experiments shows that a

t o --/959 o 045 (off5.2') - /445(0113M -05 Fig.1o6. The vari.e.torL of thrust deduction coeff!cierot with héight of propeller aìis above keel for. single screw ships. (TODD'S exper.- n;ents).

-.05

30 198

change in the fore body causes a change in the thrust deduction coefficient of 0.o5, but no doubt this le due to the lack of turbulence in the case of one of the dels in question. According to NORDSTROM's experi'nts (84) the distri- butIon of displacement is of no Importance regarding t. The rake of the propeller has a small influence on the thrust 'deduction coefficient. BAKER (12) bas proved, that a j50 30 rake change from to wi].]. cause a change in t from c.o2 to 0.03. But, as in the case of the wake, the propeller - rake may be neglectedif the aperture is of normal form. Likewise. the position of the propeller longitudinally is of secondary Importance. The skew back, the number of blades, the shapeof blades, the blade area, and the pitch of thepropeller may be neglected. In spite of the fact, that the rudderChanges the mede of action of the propeller, its influence onthe thrust deduòtion is very small (compare VAN LAMREN'S experiment (66)). VAN LA1'UREN also has proved, thatsmall changes In the 'thrust loading will not affect the thrust deduction coefficient. Neither does the condition of the ship have any special effect. Thus a change of draut means practically nothing '(compare TODD's and KEMPF's experiments (119,120, and 56)) to the thrust deduction coefficient, providedthe propeller does not come in the immediate vicinity of the water surface. VAN LA!MEREN'S experiment (66) with rough models proved, that the thrust deduction was also independent of the degree of roughness. Finally there is the dependence of the thrust deduction on the speed of propulsion. Just as for the wakein- vestigations the available material from the Wageningen experiments was employed. It appeared that in 42 per cent of the cases examined the thrust deductioncoefficient diminished when the speed increased, In 21 per cent' of the casez lt was constant, and finally in 37 'per centof the cases it increased, which means that, In advance, no rule can be laid down regarding t's variation with y. The va- 199

nation was bigger than for the wake, and the variations occurred absolutely independently of the wake variations. In 35 per cent of the cases the curves for thrust deduction and wake were, simultaneously, either rising, constant or falling, in 34 per cent of the cases one of the coefficients was constant, while the other was either rising or falling, and in 31 per cent of the oases the curves had opposite tendencì'. Concerning the causes of the variation of the thrust deductIon coefficient with speed, at present nothing is known, but perhaps this question too is connected with the question of turbulence. However, it is interesting to see, how big the difference is betweenthe thrust ded,.icton coefficient found by experiments and that derived from the diagram in fig. 1o4, applying the just mentioned correctIons. Thus as parameters for the predetermination are used: 1. The block coefficient (e), 2. B/L, L being the length on the waterline. 3. D/L; only in the case of ships havIng pronounced V-shaped frames aft, is the correction made, using the following correctIon formula:

t1 = 2 (D/L - 0.04) 4. E/di thé correctIon is only made when E/d is bigger than .0.45'or smaller than 0.35, using the correction for- niula; t2 = 0.3 (E/d- 0.4o) 5. The frame shape. The correction is only made' when the f ranas in the after body are pronouncéd U-shaped or V-shaped. The following formulae,are .xsed: V-shape: t3 = + 0.02

U-shape; t3 =. - 0.o2 Forthe examination of the exactness of the diagram and the corrections, the material from Wageningen wasagain used, and in fig. 19 the result of thisInvestigation is shown. The value experimentally found is selectedas abscissa, while the ordinate is the value determined by the diagram. Bycomparingwith the corresponding diagram for the wake it will be seen that the indefIniteness is 200

Fig. 1o9. Comparison between observed and calculated thrust deduction coefficient.

- 30 t

the seme, except that in the case of the thrust deduction :diaani a couple of points áre situated sOme distance from the line x = y. These points originate from very broad ships having small block coefficients, and the thrust de- duet ion coefficient therefore had to be estimated by extra-

polatIon. . . It is only possible to find a: relatively simple re- lat ion between thrust deduction and wake when using the normal forms as defthedn table 25. In th diagram fig. Ib the wake coefficient has been. used as abscis3a., whilst he thrust deduòtion coefficient is the ordinate.Three sots of curves are shown, the first et indIcating the relation between t and w for constant block coefficient, the secänd indicating the same relation but foi, constant breadth-length ratio, and the third the re3ation between t andy.for constant hull efficiency, h = ____ . Even if exclusively considering ships-of normal form ad with D/L o.o4, it is seen that. ño proportionality between t and w can be f' ourid. As t añd w also vary in dif f e- !.rent ways with franO shape, propeller diater, aid speed, .:lt..Ià the authòrts opiñion that in any futureinvéstigation the two coefficients ist be. òonsidered. as bei±ig completely

ndeendent of eah other. . The relation between t and w in twin screw ships then was investigátedbeginning by examining the. factors which 201.

V ' /7 //

- ,' // -,/ // // / .24// /

-V<' /7 ,// / - // Z , ,/ / r , / / -" / ' / / j' - / 7 .7 / , ,- / f / I, / f / / / r ,' / 1< / / -/3 '-f -f f / / / / j/- 55 , j j' / /___ o ,J/ / / / 7/ r - / / / /60 I / .5. j' . j' 1' / -/6 7 7 j / z i / /7/ / f/f /7/ /7 / jSa »2 .24--t'-26- .28' 30 /.3 j35" 38,'w

Fig. Ib. Relation between wake coefficientand thrust deduction coefficient for single screw ships, havingnormal form arid D/L = 0.04. were supposed to have aonia Influence on the thrust deduction, For this Investigation the Wageningen matérial-was used, and, as before, the results were arranged according to theblock coefficient ofthe dels. Arrangement ccoi'd- Ing to breadth-length ratIo is riot possibleon account or -the limited. material. Fig. III Indicates the result, and it appears that the devIations from the mean curve are ut so big afor the wake coefficient, It ipractIcaÏ1y'for the same ndeis as inthe wake 'exminat ion,thatthelarge de- viat Ions are found.

Fig. IIi.Thrustdeduction coefficientsdetermined by i T:. experiments (twin screw ships, Ji Wageni:ignexpei imante). 202

8053IG5 ,-' ABe4 ,,,' .20 /

Fig. 112. The variation of thrust deduction coefficient with propeller diameter for twin screw .05 ships (HUGHES' experiments). 025 .03 .035 04

.05 .05 -o - .025 .03 .035 .04 .025 .03 .035 L04

As for the wake, HUGHES'experiments (49) explain to some extent the problemsof twin screwh1ps. In fig. 112 some of HUGHES'experimental results have been graphically shown, the thrust deductioncoefficient being plotted as a function of the ratio of the propellerdiameter to length of ship. The different curvescorrespond to different positions of the propellers and to differentconditions of loadthg of theship. Fromthe figure it appears that there is only a slight difference in variationbetween ships ha- virig bossings and ships having shaftbrackets. From the same figure it is. evident,that by changing from shaft brackets to bossingsthe thrust deductIon increased by about O.o2. Further fromHUGHES' experiments it is seen that an average increase in E/d of 0.2, causes an increase in the thrust deduction coefficient of 0.035. It is peculiar that the experimentsindicate that the thrust deduction only varies verylittle when the position of the propeller is changedlongitudinally. On the other hand a small athwart-ship displacementalways affects the thrust deduction coefficient. Thus anincrease of the ratio of the tip clearance to shiplength of 0.005 will cause a diminution in the thrust deduction coefficientof O.o3. Further the form and posItion of thebossings do not play the big part in the thrust deduction, asin the case of wake. Also the direction of rotationof the propeller 203

t f

.25 7/,

.20 Fig. 113. Relation between w aridt of twin screw ships having normal form and D/L = 0.03.

- .05 .10 .15 .20 .25k

is without influence. The Wageningen material was also employed in an in-; vestigation of the speed variation of the thrust deduoti-. on coefficient. It appeared that in40per cent of the cases, the thrust deduction coefficient increased when the speed increased, in 25 per cent it was -constant, and in 35 per cent it decreased. This variation is practically the same as for the wake coefficient, but the thrust deduction and the wake vary, as for single screw ships, absolutely independently of each other. From the above it undoubtedly is evident that it is extremely difficult, not to say impossible, to find a general relation between the thrust deductiOn and the effective wake coefficient, but, as in the case of singlé screw ships, a relation can be found between the two coefficients for the normal ship loris. Tile result is represented in fig. 113. This diagram might be useful for a first estimate. If an attxnpt is made to insert all the experimental results in such a diagram, it will be found that the scattering is very extensive. ]2o

V.. SUMMARY.

The main results of this investigation are:. 1.- The frictional wake is practically the only important actor governing the variation of the wake ?oefficient with the propeller diameter, and the radial varation of the wake. By employingVAN lMv!AN'stheory of flow a.ong.plane'surfa- oes, the majority of these variations may' 1e explained. ?. Calculations of the potential wake of ¶olids with different shapes have proved, that the potential wake primarily depends upon the fullness of the solids and their breadth- length ratio. The after body has a big influence on the Po- tential flow, whereas the Influence of the fore body is Small. A comparison between two and three-dimensional flows proves, that only the qualitative relation for thx'ee.-di.- ensional flow can be inferred from the results foi two- dinsional flow. 3. Most of the wave wake pbenomenae can be explained by the

trochoidal wave theory. . In employing 6,. B/L, D/L, i'(frame shape coefficient) 'and E/d of -a ship as paraìeters, the wake coefficient, of screw ships can be preestirriated w1tI fairly good couracy. The probability is90per cent that the wake is determined with discr8pancy less thanO.04JThe new w-diagram (fig. 52)..ls .constru ted onhe basis, of mathematical calculational of the potential wake,. on the basis 'of. YA ARÌvMI's theories andof along serles of model experiments. Of older forilae and diagrams for the termination. of forsinglescrew ships, SCHOENEERR's f orzi].a is the best (same standard of accuracy as by the author's diagram). If nly a simple fori1a is preferred, it is suggested.that

. . - TAYLOR's forila be used. . Predetermination of, W in the case of, tWin screw ships be effected by using the same parameters. Account: must also be, taken of the bossings or shaft b'eckets and.of.the direction of rotation of' the propellers. 205

7., For special. type.s ofshipa, special diagrams should be .employd for. thedeterminationof w, in order that .thecor- rectionwill nothe too big. . In single screw ships the ratio between the nominal and. the effective wake coefficient varies in a rather unexpect- ed way. In twin screw ships on the other band, the two ccf ficients have the same value. In the first case t is the rudder, most probably, which causes all the trouble, as lt has. a big influence on the effective wake, but only a small influence on the nominal wake, The scale effect on w is considerable, but it can not be said with certainty, how big lt is, as very little is known regarding the friction belt of curved surfaces. lo, Earlier curves of the radial wake distribution have been Illogically constructed. New curves have therefore been devised. li. For ordinary types of ships lt is no longer necessary for the designer to carry out model experiments In order to detsrmine. the w and the w-distribution, as the formu.lae and diagrams gIve the results with the same certainty as obtained by present experiments still affected by scale effect. 12, No relation exists between t and w. The two coefficients vary absolutely independently of each other. The author's investigatIons bave borne out that It would be of considerableinterest to have performeda complete mathematical calculation of the potential wake fora solid having pronounced U-shaped frames, and fora solid having pronounced V-shaped frames. As regards experiments, attention ought to beconcen- trated on the relations ft the friction belt at planeas well as at curved surfaces. Research In this fi.eld would clarify many problems still obscure, amongst whichthe scale effect on the wake. 15, An experimental series, determiningw and t for models, the shape and dimensions of whichare systematically varied, would be of great Interest. In suchan investigation It 206

would be of advantage to substitute FROUDE?3 method for the deter1nation of w by a direct comparison between the relations of a standard model arid those of the nodel,which is being tested. 207

X. SUv1ARY IN DANISH.

DSTR.ØM 0G }.DSTRØMSF0EDELING VED HANDELSSKIBE. Efter en kort omta].e af definitions- og mâlingsm.der for medstrø'm og medstrmskoefficient analyseres de tre mod- strmskomponenter: friktionsmedstrrn, potentialmedstrm og blgemnedstrm. Potentialmedstrømmen beregnes for en lang rkke forskelligt formedo legemer: cylindre, e].lipsoider og omx1re jningslegemer. I de føigende kapitler konstrueres ved benyttelse af de teoretisk fund.ne re].ationer og en .rgde modelforsøgs resultater (Wageningen-forsøg) et nyt diagram tuforudbe- stesmielse af medstrmskoefficienten. En samuienligning med og meflem deldre tilnmrmalsesformnler og diagrammer fore.. tages. Ide sidate kapitler behandles medstrnsfordelingen over propellardieken, niedstrmmens indflydelse .pstyrin- gen og propellerens. omdrejningstal og medstrmmens relation tusugningen.

Hovedresultatox'ne af undersge1sen er: - Friktionsinec1strsninen er sà godt soin onebeetemmende. for medstriskoeffic1entens variation med propellerdiauieteren og for medstrmmens radielle variation. Ved benyttelse af VON K!ARAN's teori for strønniingen lange plane flader kan største delen af disse variationer forklares. Beregningen gr potentialxnedstrnunen ved de forskefligt formode legemer bar viet, at denne først og fremmest af- nger af legemernes fyldighed og af dares lmngde-bredde forhold. Agterskibet hai' ator indflydelee pâ potentialmed- strØen, hvortrnod forskìbete indílydelse er ringe, En sammenligning melleni to- og tre-dimenstonale strne vi- ser, at man ud fra resultaterne ved de to-dimensionale str- me kun kan slutte sig tude kvalitative forhold ved de tre- dimensionale. 5.De fleste blgemedstrsfmnumener kan forkiares ved tro- choideblgeteorien. 4.. Benyttes skibets ,B/L, D/L, f .(spanteformskoofficjent) O8

og E/d som parametro, kan medstrmskoeftictenten vederi- keltskruesklbe forudansmttes med ret god nZjagtiged. Der er 9ö% 'sandsynlighed for, at w buyer bestemt med større 'nøjagtighed end 0.o4, nâr det ny w-diagram (fig. 52)

:flyttes.

5'. At ldre formier og diagranmier. tuw-bestemmeiso vd enkelt3krueskibe er SÖH0ENRR's formol den bedate (samnie sikkerhod sorn ved benyttelse al' det ny dtagì'am). Vil man kun benytteen simpel formel, foreslàs dot altid atbei. 'nytte TAYLOR's formel. Ved dobbeitkrueskibe kan w forudbeatenunes ved ben3't- tolse at de same ledetal. ensyn má ogsà tages ti]. bosser- ne eher akseitrerne og ti]. propollernes omirejningsròt- 'ninger. Ved speciehle skibstyper bør speoialdiagrarner ánvêndos ved w-beateminelsen, for at korrektionerne ikke skai b].Xve

for store. , Ved enkeltslOEueskibe varierei' torholdet 'rnel'lem den nomine]le og der effektive medstrømskoofficient pá ret t'i].- fmidig mâde, Ve4dobbeltskruesklbe bar de 'to koofficienter derimod samme vmrd. I første tilfide er'dèt efter al'sand- synhighed roret,der er ârsag tumisaren, idet dot bar ator .ndfiydelse pâ. den effektive medstrøm medens det' kun bAr lilie: ind.flydelse pà den nominelle. Skalaoffekten pá w er ator, men endnu vides ikkò id sikkerhed, hvor ator den er, idot meget' lidt kendea orn friktionebmitet -ved kruirnie fiador.

10 . Pid].igere kurver for den radlelle 1nedstrmsfordelng hal' vret ulogisk opbygget. Nye er derfor konstrusret. ii. For konstruktren or det ved almindelige skibstyper ikke ]mngere nzdvendigt at foretage modelforsøg för 'at f â. og W..fordelingen bestomt, idet 'formiamo og dtagi'ammamne giverdisse mod lige sâ ator sikkerhed somforsøg, sâ].Enge skaiael'fekten ikke er ful4t bestemt.

'12. Der ekaisterer ingen relation me].iem t og w. De to koefficienter varierei' fuldstndigt uafbmngigt af binaiden. 13. Undersgelsen bar vi3t, at det er at interesse at be- 209 regno potentia1medstrmmen matematisk ved.1egeiier med ud- prgede U- 0g V-formede spanter. Ekeperlinezitelt bør man først og fremrciest intereesere sig for forholdene i frikttonsbltet ved sve1p1ane sóin krumme flader. Ved at studero forholdene i friìctionsb1tet vil man ogsà have mulighed for at bostemme skalaeffektenp. medstrmmen. En forsøgsserie, hvor w og t bestemme8 ved modeller, hvïs form; og dimensioner er systematisk varieret, br ud- føres. Ved en sdan undersgelse br FROUDE's metòde ti]. w-bestexmneise tonados. I stedet bør w besten1mos Ved sani- menhigning at forholdene ved en standardmodel og den model, man prøver. :210

Y. SYMBOLS AND UNITS.

So far as possible, the symbolsemployed in the English literature have also been usedin the present wox' Sone of the symbols have a double meaning,but their signification will be evident from the context. In thefollowing lIst only the most importantsymbols are mentioned, while symols only occasionally used will beexplained when employed

Generally :Symbol Juit Explanation 'x,(or x) scale. m sec acceleration of gravity kg. nieec density 2 -1 V m sec kinematic viscosity R=VL/v - REYNOLD's number Shin and model L m or feet length m or feet length between perpendiculars ¡n or feet length on waterline B m or feet breadth ¡n or feet draught midship section area O . ¿or feet2 volume of displacement y m . t or te. weight of displacement - waterline coefficienti - midship section coeff,icint block coefficient - horizontal prismatic coefficient - vertical prismatic coefficient entre t ¡n or feet the distance of the (or per cent) of buoyancy forward öt L/2 SUffiXa and denote after body and fore body respecti1vely denote that the coefficient concerned is suffix andWL obtaIned with regard to and 'L respectivel denote that the quantity concerned is valid suffix andM for ship and model respectively 211

Propeller D m ór feet pÑpeller diameter

R . m or feet Tip radius - r m or feet distance f ròr the axis to sectioni!1propeller blade

P ,. . ' m or feòt pitch. -

P/D - -. . pitch rat E m or feet height of propeller axis above keel0 Rudder

i m or feet length .

, ..

t m or feet thiòkness . Speeds y or V msec1or knots speed of propulsion

- - ,

_i' . -1 n og N sec or min nuxiber of revolutions

Ve ' msec1 speed of advance

A =v/nD or Ve/flD ' speed coeíficient. .

w = (v_v0)/v - . -. wake coefficient

wp=. (v._ve)/ve. . - wake percentage (FROUDE.)

. wake coefficient (thrust identity) WT ' - (torque identity)

. (frictional) WFR . -

- . '(potential) (wave).

WV - ' . (nominal, vOlume- mean) - w1 ' (nominal, impulse- mean) Thrust, torque, and effect

T ' kg thrust of propeller

Q. kgm torque of propeller - c =T/D4 n2 . thrust coefficient n - torque coefficient

EHP ' HK (metric) effective horsepower DR?, 11K delivered horsepower n - (quasi-)propulsive efficiency

- hili efficiency

t = (T- R.)/ - ' thrust deduction côefficient 212

Potential field x, y, and z ni co-ordinates p m2sec1 velocity potential function m2sec stream function

s = Z ' m2sec1 stream function Elliosoids and ellipses

a, b, and e ni semi axes

ni elliptical ordinate E and F ni elliptical integrals Waves ni wave length b ni wave heig)at z ni depth below water surface

r ni orbits radius (trochoidal waves)

Friction belt d. m tbicmess n - exponent depending on condition of surface 213

Z. BIBLIOGRAPHY.

ACKERSON, L,, "Test restlts of a series of fifteen mo- deis", ASNA 193o, p. 289. AIOEN, J.A. VAN,en STOLNEYER, H.,"Bepaling van het nia- chineverniogezi van zeegaande e,s, en d.s. vracht-. en passagiersachepen en van kustboten ten beboeve van het voorontwerf", Schip en Werf 1949, No. 18, biz. 378. ALLAN, J.F.,and CONK, J.F.C.,"Effect of laminar flow on ship mode1s', TINA, read in Copenhagen onAugist 31, 1949. t.' Vt J AMTSBERG, H., "Untersuchungen über dio Fomniabhngigkeit d s Reibungswiderstandes", JdSG, Bd. 38, 1937, S. 177. BAKER, G,S., "Ship wake and the frictional belt", NEC, vol. XLVI,1929-3o, p. 83. BAKER, G.., "Wake", NEC, vol. LI, 1934-35,p. 303. BAKER, G.S., "Ship Desi, Resistance and Screw Pro- pulsion", 1933, vol, I, p. 23 (7), vol. II,p. 74 ànd 113. BAKER, G.S,, "Experiments with low pitch ratio screWs behind a single-screw hull", TINA 1942,p. 17. (lo) BAKER, G.S., "Ship Efficiency and Economy", 1942, p.2.' (li) (Io), p. 1o6. BAKER, GB., "The desii of screw propellers, with special reference to sIngle-screw ship", TINA 1934,

0 3370 BAKER, G.S., "Steerin& of ships in sl*llow water and canals", TINA 1.24, p. 319. BAKER, G.S., and BOTTOMLEY, G.E., "Manoeuring of ships", National PhysIca.l Laboratory, Collected Re$earches, vol. XXIII, 1932, p. 149, (part I: BAKER and BÖTTO}EY, part Il-III-IV: BOTTOIltY). BOTTOMLEY, G.E., s. (44). BRAGG, E.L, "A study 8f the wake of certain models by means of a current meter", ASNA 1922, p. 93, BRAGG, E.M., "Wake and thrust deduction of selfpropei- led models", ASNA 1924,p. 93. (IB) BRETTING, Á.E., "Nyere Udvikìing indenfor Hydraulik- ken0gRosultaternes Anvendelse pá forskellige tekn- ske Problemer", Ingeniren 1949, nr. 6, s.129.

(19) BRYANT, C.N.,S. (99). .(2o) CONN, J.FSC., s. (3). CONE, 1T.F.C., NEC 1935-36, p. D 106(disc). IJTLAND, R.S.,s. (58). DICKMANN, H.E., "Wechselwirkung zwischenPropeller und Schiff unter besonderer BerVcksichtigUflgdes Wollen- einflusses", JdSG 1939, S. 234.

(24) DIEREN, E. VAN, "Volgstroomkwestiesbiischepen en modelJen?, De Ingenieur 1932, No. 33, blz. 119.

(25) DOYRE, C., "Théorie du Navire", 1927, p. 583. DURAND, W.F., "Aerodynamic Theory",1934, vol. I, p. 293, (26), p. 2oo. (26), p. 2o4. (26), p. 175. (o).EDWARD, J., and TODD, F.H., "Steam drifters: tankand sea test", lESS 1938-39, p.51. (31) EMERSON, A., "The effect of shape of bow onship resistance", part I, TINA 1937, p. 188. EMERSON, A., "The effect of shape of bow on shipresi- stance", part II, TINA 1939, p. 132. EMERSON, A., "The effect of shape of bow on shipresi- stance", part III, TINA 1941, p. 40. EMERSON, A., "Resistance of hulls of varyingbeam", TINA 1943, p. 172. FREIMANIS, E., s.(83). FRESENIUS, R., "Das gruridstliche Wesen der Wechsel- wirkung zwischen SchiffskJrper und Propeller", Schiff- bau 1921, S. 257. FROJJDE, ILE., "A description of a method of investigation of screw propeller efficiency", TINA 1883, p. 231. FROUDE, R.E., "The determinatión of the most suitable dimensions for screw propellers", TINA 1886, p. 25o. FURIVIANN, G., "Theoretische und experimentelle Unter- 215

suchungen an Ballonrodelleri", Jahrbuch der Motorluft- sohií'f-Studiengesellschaft, V' Band, 1911-1912. GAUTIER, J., s.(128). GAWN, R.W.L., "Steering and propulsion of H.M.S. Nil- son in a restricted channel", TINA, read in Copen- hagen on September 2, 1949. HARVALD, SV.AA. ,"Medstrm9koeffiCiefltefl8 afhrngighed at rorform, trim og hrkbøle", SSPA nr 13', 1949. HARVALD, SV.AA., "Gegeveris ter bepalinvan het volgstroom- en zoggetal bij het voororitwerp van zeegaande enkelschroefschepen", Schlp en Werf 1949, No. 19, blz. '404. HELMBOLD, H.B., "Schraubenzog und Nachstrorn", WRH 1938, Heft 22, S. 354. HILLHOUSE, P.A., TINA 191e, p.88 '(disc;). HOGNER, E.,ttNogra försÖk att âstadkornmaaxelsyniirie- t.risk medstrmsfördeling kririg fartygsproÏlrar", TTS, Dec. 1932, s. 85. (4 HORN, F., "Measurement of wake, 'NEC 193738, p. 251. HORN, F., "Ungleicbíörmlgkeitseinfliìsse bei 'Schiffs- schrauben", Hydromechanische Froblemé 'des 'Schiffsan- trieb 1932, S. 371. HUGHES, G., "Model experiments on twin-screw propulsion",

part I, TINA 1936, p. 145, '

part II,' TINA 1939, p. '149, ' '. '' , ,

part III, TINA 1941, p. 11. ' ' KARl, A., "Components of propulsive efficiency and

' their variation", TINA 1941, p. 62. ' KARMN, TE. VON, "1!ber laminare 'und turbulente Rei- bung", Zeitschrift für angewandt Mathematik und Mechanik 1921, S. 242. KEMPF, G., "Vergleich des Nacbstromes am Schif f und am Modell", WEH 1925, S. 115. ' ' KEMPF, G., "Weitere Reibungsergebnisse'an ebenen glatten und rauhen Flthhen", Eyromechan1sche Prob- lerne des Schiffaantrieb, 1932, s,, 7. 216

KEMPF, G., "Ontheeffect of roughness onresistahce of ships", TINA.1937, p. 1o9. IMPF, G., "eber denEinfluss derRauhigkeit auf den 'Widerstand vonSchiffen", JdSG 1937, S. 150. KEMP?, G,, "Further xde1 test on îxunersion of propel 1ers, effect of' wake and v1cosity",NEC 193738,p,349, KEMP', G.., "Measuréments ofthe propulsîve and.sruc- turai charact'eristic3 of ships", ASNA 1932, p. 4, KENT., J.L., and ÖUTLAiD, R.S.,"Effect on the pxpu1 sion of a single screw cargo vessel of changes In the shapé anddiùiensionsof the propeller", lESS 1939-40, p.137. (59)KONING, J.Q., "Eenonderzoek naar vorm er voortstuwing van binnenbeurtschepen", .Einnenschepvaart, 78..i97, blz. 66, (6o) KONING., J.G., s. (67),. 'LA!, H., "HydrodynaDlic5", 1895, p'. 461. LAIVIEHEN, W.P.A. VAN "Analyse der voorstuwingsebnì- ponentenin verband methet schaaleffeot bij scheps.. modó1proeven", 1938, biz,58,

(62), b].z.83. - (62), biz. 122. (6?), blz. 80, '(66)(62),blz. 1o7. (67) LAMMEREN, V.P.A, VAN, TROOST, L.,. en KQNING, J.G., "Weerstaxd en voortstuwir.g vanschepen", 1942,' biz.. 86, (68) (67), biz. 141.: (69) (67), biz. j67. (7o). (67), blz0135. '(71)LAEN, é.P.A., s. (87). LEGENDEE, A.M., "Tralt4 dés Fonct1on&Elliptique s, Paris 1826. LEFOL, J., "Les interactionsentre la carena et ] proa

puiseur", ATMA 197, p. 221, : LEWIS, F.M., "Prope11er vibration", ASNA 1935, p. 252, and ASNA1936, p. 5o1. LUKE, W0J., "Experimental. investigations on wakeand thr. déduction values", T.fl'A 1910, p.43 21 7

(76..) LUKE, WJ., "Furtherexperirnerrto.upon wake arid thrust deduction",. TINA 1914, p. 33. (77) LUKE, W.J., "Further experiments upon wake andthrust deduction", TINA. 197, p. 1o7.. (78). MANEN, D. VAN, "Oncierzoek naar de mogelijkheid. in tun.. nel eonjuistbee].d te verkrijgen van de in werkelîjk- heid aan de sobroef optredende cavitat-ievòrscijzselen", De IngenIeur 1949, No. 238

(79) MARSHALL, D,, s. (99). .. (8o) MC. DERMOTT, G.R,, "Sörow propellers", ASNA 1896,, p.159. .X., "Wake", Shipbuilding and, Shipping Record,Oct. 1927, p. 444. NoRDSTROM, H.F., "FÖrsk med fiskebâtsmôdellsr", SSPA nr. .2, 13. NORDSTROM, H.F0., och FREIMANIS, E., "Modellf8rsk med en fMrja", SSPA nr7, 1947. NORDSTROM, H.F., "Further test with medèls of fast cargo vessels", SSPA nr.14, 1949. OSBOUBNE,A., "Modern Marine Enginier's Manuel", 1943, vol, II,. p. 231.1. PEARSON, K.., "On tbe motion òf pherical and ellipsoi- dal..bodies in fluid. media", The Quarterly Journal of Pure and AppliedMathematics, vol. ?o,, 1885., p. 6o and p. 184. PROHASKA, C W., und LAREN, W,P.A. VAN, "Mitetrom- messung an.Schiffsmodellen", Schiffbau 1937, Heft 16, S.. 257. De Ingenieur 1937, No, 35, biz, W, 83,.. PROHASKA, C.W., "Sugning melJ.em hirianden overhalende Skibe", TT 21. dec. 1946, s. 1333, RA1NE, M., "One plane water-lines in two dimensions", Philosophical Transactions1864, p. 369. (9o) RANKINE, L, "On the mechanical princp1es of the action of propellers", TINA 1865,p. 13. SCHAFFRAN, K.,"Modelversuòhe fr.Fischere1fhrzeúge", Schiffbau .2/9 Marz und 23 Marz 1921, S,. 78.. SCHAFFRAÑ, K., "SystematischeVersuche mit Frachtdamp- .fermodellen", JdSG 1921, S, 2o2. SCHAFFAN, K., "SystematischeVersuche mit Han4els- 21

schiffamodellen", Schiffbau 13-8-1949, S. 583. SCHMIERSCHALSKI, H., "Zwei Versuche zurFeststellung dea Einflusses der Gleichforuhigkeit desNachstromes auf die Propulsion", HydrodynamischeProble des Schiffsantrlebs, 1932, S. 397. SCHOENHERR,K.E.,"Recent development in propeller design", ASNA 1934, p. 90. SCHOENBERR, K.E.,"Propulsion and propellers", Prin- ciples of Naval Architecture, 1939,vol. II; p. 149. SCHOENHEIR, K.E., NEC1937-38, p. D 152 (dIsc.). SEMPLE, J., "Some experiments on full cargoship models", TINA 1919, p. 320. STANTON, T.E., MARSHALL, D., and BRYANT, C.N.,"On the conditions at the boundary of afluid in turbulent motion", Proceedings of Royal Society, A vol. 97,192o, p. 413. (loo)STOL!!EYER, H., s.(2). (loi) STROEBUSCU, E.,"Untersuchungen fiber den Mitstrom von Schiffsmodellefl in Propellerberelch (auf Grund von Versuchen in der Preussischen Versuchsanstalt fflr Was- serbauurid Schiffbau, Berlin)", Schiffbau, 1932, S.247. (102) SDERLUND, K., "Standardiserede modellf8rs8ksresUltats praktiska anvndningt', TTS dec. 1935, S. 85. (1o3) TAYLOR, D.W., "Onship-shaped stream forms", TINA 1894, p. 385. (1o4)TA!LOR, D.W.,"On solId stream forns, and the depth of water necessary to avoid abnormal resistanceof ships", TINA 1895, p. 234. TAYLOR, D.W.,"Some model experiments on suct1on of vessels", ASNA 19o9, p. 1. TAYLOR, D.W.,"The Speed and Power of Ships", 1933 (191o), p. 121. (lo?) (1o6),p. 117. (loB) TAYLOR, D.W., TINA 191o,p. 83 (diSc.). (1o9) TAYLOR, D.V., "Some experiments on propeller position and pulstVe efficiency", ASNA 1922, . 1à3. (Ib) TAYLOR, D.W.,"Propeller design baseduponmodel ex- 219

periments", ASNA 1923, p. 57. TAYLOR, D.W., "Propeller desii developments",Proceed- inge of the World Engineering Congress Tokyo, vol. DCX, 1929, p. 145. TAYLOR, D.W., "The variation of efficiency of propulsion with variation of propeller diameter and revolution", NEC 1930 -31, p 317. (11) TELFER, E.V., "The wake and thrust deduction of single screw-ships", NEC 1935-36, p. 179. TELFER, E.V., "Frictional resistance and ship resistance'!, TINA, read in Edinbur June 28, 1949. TELFER, E.V.,TINA 1949, p. 175 (disc.). THOMSON, J., TINA 1925, p. 88 (disc.). TODD, F.H., "Further mode]. experiments on the resistance of mercantile, ship forms - coaster vessels", TINA 1931, p. 139. TODD, F.H.,"Screw propeller experiments with models of coasters" TINA 1933, p. 122.

( ' TODD, F,H., "Further resistance and propeller experiments with models of coasters", TINA 1938, p. 136. 2o) TODD, F.H., "Further experiments with modèls of cargo- carrying coasters", NEC 1941-42, p. 81. TODD, F.}I., s. (30). TROOST, L., NEC 1937-38, p. D 147 (disc.). TROOST, L., "The effect of shape of entrance on ship propulsi.on", TINA 1949, p. 159. TROOST, L., s. (67). VEDELER, G., TINA1949, p. 179 (disc.). WEINGART, W., "Nachstrom und Sogziffern far Schiffe mittlerer Geschwindigkeit", Schiffbau 1-8-1934, S. 237.. WEITBEECHT, H.M., "lTher Mitstrom und Mitstromschrauben", JdSG 1931, S. 117. VERDIRE, G. DE,et GAUTIER, J., "Formes de carènes pour cargos rapides", ATMA 1948,.p. 481. VOLKER,H., "Eine Methode zur angenharten Bestizmriung des Sehiffewiderstandes auf Grund von Schleppverauchen", WRH 22-1-193o, S. 35. 220

combined et- (13o)YAMAG?TÂ, L,Modo1 exporinients of the feet of aft-body formsand jxropeller revolutions upon the propulsive sco1ornyof single-screw ships", TINA 1934, p. 386o YAMAGATA, M., 1'Modelexperiments. on the optimum diameter of the propellerof a single-screw ship", NEC 1937-38, p. 381e YAAGATA, L,, "Furthertnodei experinent8 of the con- bined effect of aft-bodyforma and propeller revolutions upon the propulsion economyof single-Screw ships", Jouri'.al of' the Society,of Naval Architecte in Japan, No. 57, 1935, p.149. (33)YAMAGATA, M., "Wake measurementby a wo'king propel- 1er", Internationale Tagungder Leiterder Schlapp- versuehsanStaitfl, Berlin 1937, S.67.

Abbreviations.,

ASNA Transactions of the Society ofNaval Arthiteets and Marine Eng.narS, New York. Technique Maritime et ATMA Memoires de l'AsoctatiOñ AnauciqUo, Paxis. lESS Transactions of the Ingtitutionof Engine.ers and S&aipbui]-dex's in Scotland, Glasgow. Gesellschaft, Berlin. JdSG Jahrbuch der Schiffhautechflischefl NEC Transactions of the North-EastCoast Institution of. Engineers and Shipbuilder,Newcastlei SSPA Meddalanden f r.n .StatensSkeppsprOvflthgsaflstalt (Publications of the Swedish StateShipbuildingEx- perimental Tank, G6teborg. TINA Tran8aetOfl5 of the Inst.tutG ofNaval Architects, London, TT Tknisk Tidskrift,, NorrkBpiflg. TTS Teknisk Tidskrift, Skeppsbyggfladskoflst,Norrkping. WRB Werft, Reederei, Hafen, Hamburg. £00

r 0 91 61 PO 1 0 C P C 9 6 01 To as 01 61 11 11 61 00 10 00 90 00 00 LO 00 60 OC 10 00 00 60 CO 90 00 J IO CE 60 10 OP Ob PO CO 90 LP pP GP 01

10 to io to p w io w o O Ii II ti o o p p p ii 10 0 0 11 0 pi ii p p ,g 9 0 9 Ci P t! O t'O pii toot t O O 10 O t! 0 O 000 000 POi 000 000 00? P'OOI 1'06 c'poI O'pOl 001' OOP OIP 001 001 001 001 9'9C1 9'9OT 0111 O'6L P'I.CT l'OCT c'LOS 000 oSP OIP 000 000 000 000 000 000 009 000 000 POP OIP 000 OOP 009 006 000 000 OLP 006 000 000 00G 000

. - . 60001 G'LOJ L'96 - ,'opi C'6Ci p'6Oo C'PII 6'GOi 00601 PoCo 000Cl - CII TIP '110 016 ro III 111 160 TIP OIP 019 100 pop pOP 600 600 -

Il O'OC 66 91 91 90 00CC 'CO I P'CC 00Cl 0000 9'CC 0P'LI P0'61 60061 91'91 6101 0601 09001 C9'pT O9'pI T6'91 1690 O'9C C'9C C'PC 6'9C 6gO 0050 OS 6G 00 060 OLP 0CC O'60 O'IP 01 CC 05 09 CC 00 OC CI PC CO OC CC 60 C'PC 0000 P O'OO Pl'L, 011 00000 0061 6101 CP'OI C6'CI 'CI 50 10'! CL'). 01'! cIL 6CC U'C CC'p CC00 C00p 09'!. 09'L 900 9'?? 0000 0000 Ç 9? 91 90 00 0T'L O6'L 0'O) C'PI C9'9C 01'CT 0C'C 19'61 90 00 9? 90 90 PO to 00 60 00 60 P? 9GO 0301 /,091 1561 POrI 0CC). PLOP LOOP 06061 65061 00061 00001 0-rl OILS 00101 00001 60161 00101 COOL 09cl SOLIO 009Cl 00501 00001 0096 0090 0096 0096 0096 0000 0001 9960 L'pIO 601,1 PlOT 05601 00005 01001 01601 IC OLI OILOT 05600 06/11 01101 Ii OjO 09911 00011 00011 00011 00901 00001 0096 O OIL ¿11 P11 016 000 (PP 6to' 060 0GO ¿00 001 901 60! LOI O'C9 0009 OCT 061 CCI 1.01 LOI 1001 TOOT TOOT TOOT COL OPP 060 L011 0601 000 009 PLO 0001 0001 01,01 0061 CoPi 61,01 0661 0000 CLOT 6001 poOl LoOp OCT I 1001 9601 9601 SCOt 001' 011' LOI' LOI' 000' 091' LOT' lOI' 101' 1.01' LOi' obi' COo' GOT' tOT' 601' pOT' 001' OPT' 0000 100' 191' I/P 001' 001' CCI' oIr opt' coo' coo' OPT' 191' 101' COT' 901' POl' PCI' 901' 001' cpI' gli' bOl' 0000 601' 0000 601' 001' 101' COr PET' 'COI' 0000 60'? 60'? Cr0 0000 6000 60,0 0 PP 90'? 0000 1190 ¿0'? L6'G 00000 LOG 00'? 9O'O P000 6000 Pca oro 0000 0000 050 060 0C'O 0000 COO P0? 0000 1O'O Pro OC'0 P/P 0o'e PT'? 06 0000 oro OG'I i00o o'"7 0000 60'? pro 6c'c Pc'? 600 60'? 60'c oro 0cL' POL' POL' 019' PLO' 019' POL' POL' 009 PoL' 001' 00000±000' 109' 009' 009' GoL'CG9'JOLG' 099' 069' Cpa' p09' 669' p09' 069' p09' 609' 600' POL' GIL' COL' 016' TaL' TOI.' 0CL' 1cL' 101' 101' SOL' TOI,' 100' PLL' OU' POL' COL' POL' LOL' 609' p69' - CGO' 999' ¿69' 090' 009' L69' OLP 009' 110' CIl' -- « 9 '019' COL' POL' COL' 000' 001' COL' COL' 001' COL' POL' COL' POL' POL' 000' 009' PLO' pLP' OLP' 906' 996' gPO' 616' GpO' PeG' PoI' 066' PpP' 066' OLP' LIP. LL6' COO. pLG' 696' 006' 006' 066' 066' p6. p 66' 06' 06' 66' 66' pG. 06' p 66' 196' TPO. 006' 166' 166 006' TpO' ipP 506' 066' 066' 066' 066' 006' 066' 066' 66' e 561.' COL' GOL' 99' 90G' 0Cc' POP' Lt)' PLO' 601' PPL' OLI' Pop' LOO' LaO' 1,00' 000' 001' 9Op' POI.' LIP' LOp' ?Cp' p COL' COL' COL' OSL' Cod' Cap' LPO' 096' 086' OGL' 66L' C6L' 0GO' 66L' 04L' 06L' OSI' 5] COP' POP' OIp' 600' OTO' cGt' 090' OOL' OPI.' 011.' PCI.' POL' 009' tOp' PC' OiL' OPI' OPI' 019' PLc' 9 619' 009' 6OI 069' 069' 069' cOt' POL' 119' 1L9' tL9' 1L9' OLP' 609' 91L' bol' 69L' 660' LOP' 601.' COL' (CO' COL' dc p COL' 001' OEL' COL' 000' TPL' TPL' 000' 000' OPL 0OL 0Ol TLP a 0 Pl9 POL. - J -- p. 0LL 099W O9p 0O apg Oci6 ppp. 9W llO l90 £90 COO ¿Op tP9 092 6Og POp 016' oT6 OiP 0T5 0Op 01p 09p 006 l0 060rn OOY 016. 105 LaY LO6 l05 106 ¿06 JOY 106 l.O5 l21 P5 opp 05P l.p OL9 100 t06 - -J : :- : J -. P 106 6P0 6°0 600 600 l00 000 P LI LOO P10 LPP co0 01C 0TC CYO ora OT? £YC 000 000 PL6 0LG PL6 tL6 0L6 9Lp OOp 50l. a99 £00 OcC 6 01 0L6 0l6 CL6 (l6 0L6 Cl6 Cl6 CLG CP6 OO6 CL6 £L5 0L6 C16 006 006 O CrOl 9901 O 999 9901 09 999 000 O0C 0 00 090 96? 960 oOb 50C ¿C9 0iC 900 OPT opt 09t 0O0 OOT 0o t o-co T10 l091 it0 000 000 010 p66 901 911 91T 91T 9L1 9LT PLI 91r pc Cl 0090 COOT OL'9L O/y OLOC 0191 0 00 Ct/O 0U O (t 00 O 911 3L. .rr. ptO 0P6 009 tl 001 0Ot 000 oc ot o-t oc oc 0-r 0a Ç6 P6 ---- Py 06 06 OY 06 EP pJ GL L 90 PL 9L 00.-c OL 9th 909 9L9 06O £6P £6L 06Y 061 199 00 00O 000 9LY 609 TbL 000 000 6T9 OL 96L hO OL P/a 919 O6L 99o9,9o° L6 OL hL LOO iL9 LL9 ¿19 ¿09' 009 JL9 iL9 ¿19 £69 059 069 59 O69 O9 £69. 0L9 01rO OVO OFO 0OO 000 O09 £iO O0O T00 LOO c0o oto Oto 0bO 000 000 oOO OOO o/ci 000 0l0 (O0 t0 0O0 00 L0 O0O cbo 000 690 O9O 10o 000 t00 ccc t00 110 too cco bco* coo cOo' 110 000 t0 0cc, oOo oco cco coo cOo 1O 80 900 901' 900 OTt toO P/h OO I9i tOI'. thO b91 OhO OO 011 tiGO ¿0° 060 000 000 060 OG 000 oOc Oh oOb 000 c(b Oc bc tc 010 cc 0L0 cI0 OLY ÇIh 010 OL( 1L0 010 cc0 060 060 90Y 901 900 900' 0O9Oc J - t 9 9 9 9 01 £1 0 00 OT 00 £9 9 gg__ c 09 00 O? cytT 091 OOt 091 2T OT 6900 tT t0 091 OI tI tot coo 001 L90 Obi L9T L'gO ¿91 jI d91 oc h -' 0 £ 11 Ic 60 6 11 0 F O 90 i tO 90 O 00 00 6 00 cc 1? t e 0 LO ? ( Ot Lt 9t 6 £0 LO 0 0? ° ±L 6 LO O__ J6 J--_ 1;-

0100 POC 00 eCO ocrrococjeoe to 00 erce rette ic orç 3Tt naj atirt tr J 151

,,// WF OEJKE eTNO5O/ 7YLQ /// 1LALOP/I O8ET5Ot/ // .47 ., .S//P8 ,// // F/NCt r /// ,/ / // // / 7/ 30 ./ 30/ .30 .30 30 /7/7/ 3,Z1: ._/ __ _ 30 7/4 30 30 4 //// e / ,/r: /3$ / 20 T 0_/__2_ : . L : 3 .30 20 40w Ît1 L7 L0 .2oJoLo o . ., d .3C m---4T ---r jr 1:r.ßA KEk' 4SdD[PL') TEL FE? - // /1 2TELFE/JZ //7 // 4

/ , / ;../ , / : 3o / 3o + :) .30 .30 //Z1 .30 / /. // ,, , / r / / // /// / 20 / 20 / . .29 r2 / .20 / // ,// :4 w 4 w oo .30 .40 t ¿ .30 40w 20 .41)wJ L.3O /: L30 30 t30 /20 .30 ¿ ¿0

k/pQ1 tffSOEN//E/ 27 .:.1VcKE,Q / 7 30 .30 30 o t

// 20 .2)u .20/4 V z, .30 / i:'2 P.40. 30 .40 ¿í..__.0 30 t4uí.L' .40 w

Fig1B. Teing of the 1et irnown forciuiae and diagrams for wake determinat5ori (single scriwh5.ps) ECT ON

CRE ENO SIDES END BKTS 70 SEAMS STFFE NOR LOWER END SEE CONT IN 005 SIDES OF STIFFENERS 81<1 TO SEAMS WELDED UPON SKIS TO SEAMS DOCK LOWER END SN, PEO PATI0 i 111111 SSmm -j S5mm STIFFENERS SEE LAN SEE PLAN L 100w 65 7 (n 9) PLATING 1OQnnr,, g o Orrnn

STIFFENERS 6o 13O 55 a L 1OO&5 7 P 180k 9) (r '32658) PLATING Or'nni o nnm

. STFFENERs I l60 P 78oa) L l3O65 LlOO6s? 1 PLATING 150mm SSmm 9mm STIFENERS L 130 os 8 L 102 75 r 135 5S 8) I PLATINO 10.0 rn g mm SSmm

STIFFEER5 160 8:8)8 :5S SiL 10O5 NG 73D55) lüCmm SSmm IR 200 IO L 120 PLATING 75 g J IR22C 11 - 15 Tmm POS 51FFENERS I PS 200.15 F35 SOIF ENEP SCANTLINGS w o 105 SISEN IN PARENINESIS NEF i 'O ER 115 -o STIFE NERS 9 0010ES 120 WINDOWS FRONT STIFFENERS R 200'7 OSER I DECK HITE (MIN UNDER LONOL DECK GIRDERENDS.

.±j5

WHEEL HOUSE TOP INICKNESS OF DECK PLATINGSO mm SuFFERERS L 75 SPACING I 655mm

In AÇfl;JJ

I -H------I-if ¶4O T7------_ - LL50I2O5155 + ---L_ -- TL- T

. H - - - . L L t .- -- ±r- _ I-p 160 9 'ACING 540

-p 140 S

CONSTRUCTION PLAN BRIDGE DECKHOUSE

450 0 5015 R E MAR K G.

o PILLAR UP .: PILLAR UNDER FRAMESPACING ?GSmm

STALE