Report of Resistance Committee

5/14/2018 Report of Resistance Committee

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RESISTANCE SESSIONChairman: Prof. C. W. PROHASKAReporter: Vice-Adm. R. BRARDSecretary: Prof. L. LANDWEBER

REPORT OF RESISTANCE COl\iMITTEECommittee Members

Vice-Adm, R. BRARD(Chairman)Prof. T. INUIProf. L.LANDwEBERIr. A. J. W. LAP

INTRODUCTIONThe terms of reference recommended by the 10th

LT.T.C. were "To study the fundamentals of ship re-sistance with attention to wavemaking resistance andthe relationship between the various components ofresistance". \The Resistance Committee has held five formal meet-ings each lasting two days: inParis in November 1963,

in Wageningen inMay 1964, in Hamburg in September1964, in Paris in June 1965 and in Feltham in January1966.A questionnaire was circulated to all member organi-

zations in order to discover the extent of the use ofblockage corrections and of turbulence detection tech-niques, and to ascertain the frequency of occurrence oftank "storms". The results arc presented in this report.With the direct encouragement and support of the

Committee an English translation of a Russian text book"Theory of waves and wave-making resistance" by A.A. Kostiukov has been prepared by Professor Oppen-heimer of the University of Iowa, and edited by Dr. J.N. Newman and Professor L. Landweber.In its final presentation the Committee has divided

the subject into five topics:I Definition of termsII Viscous resistanceIII Wave resistanceIV Low resistance hull formsV Restricted water

Prof. J. K. LUNDEMr. J. R. SHEARER(Secretary)Prof. G. WEINBLUMProf. K. WIEGHARDT

DEFINITION OF TERMSThe Committee considers that it is essential to the

proper understanding of its subject that the importantquantities Viscous Resistance and Wave Resistanceshould be defined in a way that has a clear physicalmeaning. The most logical basis for doing this is to re-late the components of resistance to the method ofenergy dissipation involved, and the Committee there-fore recommends as follows:Viscous resistance is the component of resistance as-

sociated with the expenditure of energy in generat-ing vorticity, vortices and turbulence.Wave resistance is the component of resistance as-sociated with the expenditure of energy in generat-ing gravity waves.

These two quantities together represent the total dis-sipation of energy from a normal displacement vesseland they should, therefore, be additive to give the totalresistance. At high speeds energy may, however" bedissipated in spray and a third term, Spray Resistance,is required:Spra resistance is the component of resistance asso-

ciated with the expenditure of energy in generatingspray.Viscous resistance can be regarded as having twocomponents:Skin friction resistance, the component of resistanceobtained by integrating tangential forces over thehull surface.


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10 RESISTANCE SESSION

Viscous pressure resistance, the component of re-sistance obtained by integrating pressures due tothe thickness of the boundary layer and the wake.

VISCOUS RESISTANCEThe proposed definitions of viscous and wave resist-

ances do not involve any assumptions regarding theirindependence and a rigorous approach must assumethat both are functions of Fn and Rn. Thus the totalresistance coefficient CT for a smooth ship may berepresented as follows:

CT=Cv(Rn> Fn)+Cw(Fn, Rn} (A)To perform this sub-division experimentally it is

necessary to evaluate the components independently.The viscous resistance can be obtained directly from awake survey as long as the energy of eventual longi-tudinal vortices is negligible. A limited number of suchsurveys have been carried out and the results ale illumi.-nating although rather inconsistent, probably due to theuse of different techniques of analysis. The Committeerecommends that further surveys should be carried outin as many cases as possible, the surveys being so de-signed that alternative analysis techniques can be ap-plied and compared.Good approximations to the viscous resistance forships operating at low Fn are the resistance derived froma submerged double model, or the resistance of a sur-face model at very low Fn' Provided that turbulencestimulation is effective, these two methods generally giveresults which are in close agreement but which differfrom the results from wake surveys at higher speeds,implying a significant dependence of viscous resistanceon wavemaking. Consideration should also be given toa proposal by Horn that an average speed increase maybe estimated from measurements of the sinkage ofmodels.Since the processes for the direct evaluation of vis-cous resistance are too complex and two time-consum-

ing for day to day use, reliance must be placed on ana-lytical or empirical estimates. These are generallybased on the concept of a correlation line and an as-sociated form factor and require the assumption thatviscous and wave resistance are independent thus:

CT=(1 +k)C1(Rn) +C2(Fn) (B)Active research is in progress in an attempt to estab-

lish a relationship between k and suitable parametersdefining the hull form. The aim of geosim research isto determine whether equation (B) is a sufficiently goodengineering approximation to equation (A). Geosimseries tend to be limited in range, however, by turbu-lence stimulation problems at the lower end, and bytank boundary interference at the upper end, and pro-

gress is therefore closely linked to progress in the de-velopment of blockage corrections.In relation to the correlation problem itself the Com-mittee strongly recommends that a three dimensiona.extrapolation should be adopted. Five formulae havebeen proposed for the evaluation of the form factor, oneby Lap, one by Granville, and three by Hughes. Inaddition considerable work has been done in Japan, anda new basis is being evolved by Hughes. The Commit-tee does not therefore consider that at this stage it canrecommend a specific procedure but suggests that tanksshould try the various methods and compare the resu.tswith the eventual aim of standardisation of a procedure.Of the two components of viscous resistance, neitherviscous pressure resistance nor skin friction resistance

can yet be directly evaluated. The total pressure re-sistance, which has been measured in a few cases, con-tains both wave resistance and viscous pressure resist-ance. Skin friction can then be derived by subtractingpressure resistance from total resistance and results ob-tained in this way suggest a significant relationship be-tween skin friction and wavemaking. Viscous pressureresistance can then be regarded as the difference be-tween skin friction resistance and viscous resistance butas a small difference between large derived components:accurate determination is unlikely. Direct measurementof skin friction by determining local shear stress wouldbe invaluable and research in this direction should bepursued. The use of Preston tubes for this purpose isbeing developed and there are indications that hot filmprobes can be adapted to this work. Both these tech-niques are critically dependent on calibration of the de-vices in pipe flow conditions where local friction can bedetermined from measurements of pressure loss.Separation is a function of boundary layer flowwhich

is increasing in importance with the increase in fullnessof large bulk carriers. Hot film probes have been suc-cessfully used to detect separation on model scale.Factors which can cause changes in viscous resistanceare of importance for three reasons; firstly as a potentialmeans of reducing viscous resistance, secondly as acause of irregularities in model experiments, and thirdlyas a possible means of controlling scale effects. Fivefactors have been considered by the Committee:(a) Boundary layer control Boundary layer thick-ness can be reduced by the application of continuoussuction over the afterbody. The quantity of water to bepumped is, however, very large, and the small suctionopenings are liable to choke. The method has beenused in an attempt to reduce the scale effects on pro-pulsion experiments but the suction introduces second-ary effects which make analysis difficult.(b) Effect oi Additives It has been demonstratedthat the addition of dilute solutions of certain long mole-


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COMMITTEE REPORT 11cule polymers either to the water in a towing tank or byejection through openings in the sides of a model hullcan reduce viscous resistance by up to 60%. Full scaleapplication to torpedoes has been tried but the quantityof solutioninvolved in furl scale application to ships ap-pears to be prohibitive. It has been suggested that theeffect could be used to reduce scale effects in resistanceand propulsion experiments, although it is doubtful ifthe flowin contaminated water is similar in detail to thatin pure water at higher Rn. Considerable research isin progress in an effort to understand how these sub-stances operate and it is currently assumed that theeffectis a visco-elastic one, depending mainly on the di-mensional and mechanical properties of the molecules.(c) Flexible skins Experiments have not givenany clear indication that skin friction can be reduced by

the so-called Kramer skin. Theoretical studies suggestthat a material with the right elastic and damping prop-erties might reduce turbulent motion, but it is con-sidered unlikely that a material with these properties inthe proper proportion could be produced. Further pro-gress in this direction is doubtful.(d) Special surface treatment All materials ofcomparable roughness have the same skin friction andthe only properties to be sought in a new material arethe permanence of the surface condition, and its resist-ance to biological contamination.(e) Gas lubrication Little work has been done in

this field. The general indications are that frictionwould be reduced by a continuous air film and probablyslightly increased by a mixture of air bubbles and water.Three of these factors have a significant bearing onmodel experiments. A propeller introduces a form ofboundary layer control by suction which has beenshown to delay or eliminate separation on full after-bodies. Long chain molecules can occur naturally as abi-product of the decay of algae, and the formation ofair bubbles on model hulls can occur in certain circum-stances. An enquiry circulated to 55 member organi-zations yielded 32 replies of which 9 referred to changesin model resistance which were attributed to changes inthe water, 3 quoting severe "storms". One of these"storms" resulted from algae and analysis of the waterproduced direct evidence of the presence of the poly-mers known to cause such changes. The second wasdue to uncontrolled bubble formation associated withthe method of treatment of the main public water sup-ply, and the third followed refilling of the tank aftercleaning. A water board expert reporting on one ofthese incidents recommended in very strong terms thattowing tanks were insufficiently aware of the need tomaintain a high standard of water and he recommendedthe universal adoption of biological filtration.

Practice in regard to turbulence stimulation varieswidely. Of the 32 replies received from members, 1, Ltanks reported having equipment for turbulence detec-tion, generally hot film probes, but none reported astandard practice of checking that stimulation is ade-quate. Two researches on the efficiency of turbulencestimulators indicate that studs of suitable dimensionsare more reliable and effective than trip wires or sandstrips.

WAVE RESISTANCEIn spite of considerable effort no significant improve-

ment has been achieved on the Michell integral, whichrequires the ship to be thin and wave slopes to be small,The application of slender body theory as distinct fromthin ship theory has given results which are in worseagreement with experimental results. If it is acceptedthat practical ships are not thin, but that wave slopesare reasonably small, then it is possible to programmesolutions. A direct approach in this direction has beenattempted in the US but the computation of the line-arised free surface condition has proved to take too longto allow useful application. A further attempt in whichthe singularity distribution was derived by satisfying theboundary condition at the hull surface and at a rigidwater plane, and thence calculating the wave resistancedue to this singularity distribution, gave a result whichexaggerated the humps and hollows of the resistancecurve more than does the Michell solution. On theother hand an approach on these lines has been success-fully used as a basis for the derivation of forms of lowwave resistance, a problem which is discussed in thenext section.Considerable development has taken place in the

technique and theory of measuring wave resistance. Thebasis of this is the computation of wave energy frommeasurements of the surface disturbance. Although themethod is approximate, in so far as the energy calcu-lations are based on the application of linearised theoryto the real disturbance, the errors are unlikely to besignificant except in the wake, where the relationship ofwave height to energy is suspect. Nevertheless a num-ber of applications of the approach have been made andsome very interesting results have been obtained. Inone case of a thin parabolic hull, summation of viscousresistance measured by a wake survey, and wave resist-ance derived from the surface disturbance has given agood approximation to the total resistance. Two mainmethods have been developed depending on whether thewave profiles used are measured transversely (transversecut method) or longitudinally (longitudinal cut method),and a linking theory exists between these two methods.In towing tank work both have an application, the


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transverse cut because it can be applied directly on atowing carriage, and the longitudinal cut because it ismade at fixed points in the water and could be appliednot only to models independent of the towing carriage,but also to full scale ships. The Committee considersthat the direct measurement of wave resistance in thisway is a very powerful technique which should be de-veloped fully.

Low RESISTANCE HULL FORMSThere are two possible approaches to the problem ofdeveloping low resistance hull forms:(i) to select optimum characteristics from a large

store of data, either by inspection or by statisti-cal methods(ii) to try to optimise the associated hydrodynamic

phenomena.The first of these is adopted by the majority of shipdesigners but a systematic statistical approach is only ofrecent development. This has been applied successfullyin the case of trawlers and current indications are thatsignificant improvements can be obtained.The use of wave resistance theory as a guide to opti-mum design is of also recent origin and appreciableprogress has been made. The direct approach to theproblem is to minimise the resistance integrals andthence derive a corresponding singularity distribution,but this is limited by the overall limitation of the line-

arised theory. The alternative is to try to minimise thewave pattern and this has practical advantages, firstlythat it can be applied to specific regions, for examplethe forebody, where boundary layer effects are small,and secondly that results can be verified bv visualinspection of the wave system, or by wave patternmeasurements.The development of bulbous bows is closely linked

with the theoretical problem of minimising wave resist-ance, and optimised forms frequently have bulbs. Thesuccess of bulb and ram bows in reducing the resistanceof large tankers, particularly in the ballast condition,cannot be entirely attributed to wave cancellation sincethe reduction in resistance is of the order of the totalwave resistance, and the effect on viscous resistancetherefore requires to be investigated. There is evidencefrom wind tunnel experiments that trailing vortices aregenerated from the bilges of very full forms, and it hasbeen suggested that the bulb may reduce such secondaryflows.

RESTRICTED WATERA theoretical approach to the problem of restrictedwater is subject to the same limitations as the wave re-sistance calculation. The problem is to derive an aug-

mented velocity distribution and then to use it to calcu-late a resistance correction. No new developments havetaken place in this direction, but the use of availabletheoretical solutions should be investigated.The Committee does not consider that blockage cor-rections are important in the larger tanks for experi-

ments with models of reasonable dimensions. Smallertanks have a more difficult problem because laminarflow prevents small models being used. Of the 33replies received in response to the questionnaire, 14mentioned some use of blockage corrections, 6 used ablockage correction as standard practice either for allwork or for specific classes. Where corrections werereported the following methods were used: Hughes (5)Schuster (3) Taniguchi (3) Emerson (1) Landweber/Schlichting (1).The Committee is unable at this stage to make a spe-

cific recommendation but has made arrangements for acomparison of these methods to be carried out. Block-age corrections assume considerable importance in re-lation to geosim experiments and it is important todevise a test procedure for studying geosim resultswhich is not dependent on any assumptions regardingcorrelation methods.When it is necessary to simulate full scale conditionsof restricted water additional problems arise both ingeneralising the data to take account of variations in thegeometry of the channel, and in carrying out the experi-ments and analysis. In particular, additional precau-tions are necessary to ensure full turbulence, and extrastimulators are frequently required. In extremely re-stricted conditions the interaction of viscous and waveresistance becomes more serious and the validity ofstandard extrapolation procedures becomes suspect. Asin the case of the basic correlation problem, directmeasurements of components of resistance by wakesurvey and wave pattern measurements are necessary inorder to establish a test for approximate methods.

SUMMARY OF RECOMMENDATIONS(a) Experimental research to determine viscous re-

sistance by wake survey methods should be carried outin as many cases as possible, in order to assist in de-veloping a datum for the comparison of correlationmethods. The study of resistance and flow on doublemodels should be extended.

(b) Experimental research to determine wave re-sistance from the wave pattern should also be carriedout in as many cases as possible in order to establish thevalidity or otherwise of summations of this and wakesurvey results, because it provides a comparison forcalculated wave resistance, because it offers the possi-bility of full scale determination which would allow the


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COMMITTEE REPORTcorrelation datum to be extended to high Rn, and be-cause it shows immediately the influence of designchanges on wave resistance.(c) Detailed flow studies should be pursued active-ly because of the possibility of assessing the effects oflocal hull features such as bulbs, and the effects of sepa-ration, and because it may provide information on thepossibility of optimising hull forms in relation to skinfriction.(d) Analytical approaches to the evaluation ofwave resistance should be pursued actively because oftheir relation to the design of optimum hull forms.(e) A three-dimensional correlation method shouldbe adopted as soon as possible but a geneal test for thevalidity of specific formulations will not exist until moredata for (a) and (b) have been obtained and the Com-mittee cannot at this stage recommend a particularformulation.(f) Blockage corrections are only important forsmaller tanks using relatively large models, and ingeosim studies. The most widely used correction is thatof Hughes. The Committee has arranged for a comparison of formulations to be made and cannot yet makea specific recommendation. The effect of restrictedwater on the wave resistance derived by linear theoryshould be checked experimentally in as many cases aspossible.(g) Of the factors which can influence viscous re-

sistance only two are of general importance to towingtanks, additives and air bubble effects, in both casesbecause of their relation to the reliability of modelexperiments.(h) The Committee strongly recommends that alltanks should aim to maintain a high standard of waterquality, and that all deviations or storms should bethoroughly investigated by a water expert.

BIBLIOGRAPHY1) G. HUHES: Correlation of Model Resistance and Ap-plication to Ship, R.l.N.A., 1963.2) G. HUGHES: Ship Model Viscous Resistance Coefficients,

NPL Ship T.M. 80, March 1965.3) S. TAMIYA: On Resistance Experiments on Ship Mod.els, International Shipbuilding Progress, Vol. 1, No.2,1954.

4) K. YOKOO: An Investigation of Ship Model Correla-tion, Transportation Technical Research Inst. of Japan,Report 45, Oct. 1961.

5) H. MARVO: Scale Effect on Ship Resistance, 10thI.T.T.C., London 1963.6) H. SASAJIMAand I. TANAKA: Form Effects on Viscous

Resistance and their Estimation for Full Ships, 10thI.T.T.C., London 1963.7) K. TANIGUCHI: Measurement of Wavemaking Resist-

ance, Shipbuilding Association Report (in Japanese)1965.

138) R. BRARD: Consideration of the Experimental Deter-

mination of the Form Effect from Results about Transi-tion, 10th I.T.T.C., London 1963.

9) G. HUGHES: Results of Resistance Tests with a StandardModel, R.I.N.A. 1963.

10) K. TANIGUCHI: The Resistance Tests on the LT.T.C.Standard Model, Mitsubishi Technical Bulletin, Feb.1964.

11) G. HUGHES: Results of Resistance Tests with a Stand-ard Glass Plate, R.I.N.A. January 1966.

12) K. C. BARNABYand A. L. DOREY: A Towing TankStorm, R.I.N.A. 1964.

13) J. R. SCOTT: On Ship Model Resistance MeasurementErrors, R.I.N.A. 1964.

14) J. Wu and L. LANDWEBER: Variation of Viscous Dragwith Froude Number, 10th I.T.T.C. London 1963.

15) K. KEY: Verification of Method of Determining theViscous Drag of a Ship Model, University of Iowa M.S.Thesis January 1965.

l6) J. R. SHEARERand J. J. CROSS: The Experimental De-termination of the Components of Ship Resistance for aMathematical Model, R.I.N.A. 1965.

17) H. LACKENBY: An Investigation into the Nature andInterdependence of the Components of Ship Resistance,R.I.N.A. 1965.

18) S. D. SHARMA: Untersuchungen tiber den Zahigkeitsund Wellenwiderstand mit besonderer Berucksichtigungihrer Wechselwirkung, Ifs-Bericht Nr. 138, Hamburg.December 1964.

19) R. L. TOWNSIN: Frictional and Pressure Resistance ofa Victory Model, I.S.P. Vol. 10, No. 104, April 1963.20) S. K. CHOW: Free Surface Effects on Turbulent Boun-

dary Layer Separation, University of Iowa, Inst. ofHydraulic Research, March 1965.

21) J. D. LIN: A study of the Boundary Layer on theAfter-Body of a Ship, Hydronautics Inc. Progress Report132-1, 1962.

22) W. WEBSTER: Study of the Boundary Layer on ShipForms, Hydronautics Contract Proposal to S.N.A.M.E.,Feb. 1965.

23) R. L. TOWNSIN: Boundary Layer Separation from ShipModels, R.I.N.A. 1965.

24) International Seminar on Theoretical Wave Resistance,Vols. I, II and III, University of Michigan, August 1963.

25) D. K. AI: A Class of Multiple Integrals of the 'Station-ary Phase Type occurring in the Theory of Water Waves,Hydrodynamics Laboratory Report NO'..:1 .J .L4J C.LT.Pasadena, 1965.

26) H. MARVa: Problems relating to the Ship Form ofMinimum Wave Resistance, 5th Symposium on NavalHydrodynamics, 1964.

27) P. C. PlEN: The Application of Wave making ResistanceTheory to the Design of Ship Hulls of Low Total Re-sistance, 5th Symposium on Naval Hydrodynamics, 1964.

28) E. O. TUCK: The Effect of Non-Linearity at the FreeSurface on Flow past a Submerged Cylinder, Journal ofFluid Mechanics (1965) Vol. 22, Part 2.

"29) B. YIM: Some Recent Developments in the Theory ofBulbous Bows, Hydronautics Inc. Tech. Report 117-6,August 1964.

30) F. GREENSHIELDS: Unpublished Report to Messrs. JohnBrown Ltd. on the Treatment of Water in Towing Tanks.

31) See (12).32) M. KmSCH: Flow around Bodies in Shallow Water and


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14 RESI sTANCE SESS IONin Channels, Forschungsh Schiffstech 12, 1965.

33) M. IKEHATA: The Second Order Theory of Wave Mak-ing Resistance, Journal Soc. Naval Architects of Japan,117, 1965.

34) R. GUILLOTON: Procedure for Calculating Isobars for aLinearised Hull, Assoc. Tech. Maritime et Aeronautique,1965.

35) C. ELATA and J. TIROSH: Frictional Drag Reduction,Israel Journal of Technology 3, 1965.

36) G. E. GADD: Turbulence Damping and Drag Reductionproduced by Certain Additives, NPL Ship Rep. 66, June1965.

37) G. WEINBLUM: Ships of Least Wave Resistance, Forst-chungsh Schiffstech 12, 1965.

38) S. SATO and S. OKADA: Effect of Bulbous Bow uponthe Resistance of Ships with small LIB and large CE,Journ. Soc. Naval Architects of Japan, 118, 1965.

39) A. M. FERGUSON: An Investigation into the Effects ofTemperature Distribntion on Water Movement, R.l.N.A.1965.

40) S. C. ROSE and P. GRIFFITHS: Flow Properties of Bub-bly Mixtures, AS.M.E. 1965.41) H. MARUOand M. BESSHO: Ships of Minimum Wave

Resistance, Journ. Soc. Naval Architects of Japan, 1963.42) J. KOTIK and P. THOMSON: Various Wave Resistance

Theories for Slender Ships.43) T. S. RAYBURNand J. P. GROSE: A Note on the Form

Resistance of Ships, R.I.N.A 1964.44) R. GUll.LOTON: A Theoretical Study of Ships in a Per-fect Fluid, AT.M.A. 1964.45) J. V. WEHAUSEN: Effect of Initial Acceleration on the

Wave Resistance of Ship Models, Journal of Ship Re-search, 7, 1964.

46) H. GATZER: Investigations into the Effects of Turbu-lence Stimulators on a Ship Model, Schiffbautechniskc14, 1964.

47) M. BESSHO: Study of Full Ship Forms in View of the

Wavemaking Resistance Theory, Journ. Soc. Naval Archi-tects of Japan, 1964.

'18) N. HOGBEN: The Computation of Wave Resistance froma Wave Pattern by a Matrix Method, NPL Ship Report56, 1964.

A9) N. HOGBEN: Record of a Boundary Layer Explorationon a Mathematical Ship Model, NPL Ship Report 52,1964.

50) A. EMERSON: Model Experiments using Dilute PolymerSolutions instead of Water, North East Coast Inst. ofEng. and Ship, 1965.

51) H. GROTRIUS-SPARK: Geosim Test, Schiff. und Hafn,17, 1965.

52) V. G. Stsov: On the Theory of Wavemaking of a Shipin Still Water, Iswestijis Akad Nauk U.S.S.R., Otd. techNauk Mech iMaschinostrojenie, 1961, (German trans-lation by Dr. Maria Kirsch, English Translation by Dr.Hogben), NPL Ship Tech Memo 66, 1964.

53) A. A. KOSTIUKOW: Theory of Waves and WavemakingResistance, Text book in Russian, published 1959. Englishtranslation by Professor Oppenheimer, University ofIowa.

54) L.W. WARD: Experimental Determination of Ship WaveResistance from the Wave Pattern, Webb Institute, No-vember, 1964.

55) L.W. WARD: Wave Resistance Surveys on a Ship Modelof Minimum Resistance, Webb Institute, August 1965.

5.6) J. N. Newman: State of Art of Wave Resistance,ATTC, 1965.57) P. C. PIEN: Some Experimental Results of Hull Form

Research, TMB Report 2144, November 1965.58) G. L. MELLOR: The Effects of Pressure Gradients on

Turbulent Flow Near a Smooth Wall, J. Fluid Mech.(1966) Vol. 24, Part 2.

59) G. L. MELLORand D. M. GIBSON: Equilibrium Turbu-lent Boundary Layers, J. Fluid Mech. (1966) Vol. 24,Part 2.

APPENDIX IVISCOUS RESISTANCE

by L. LANDWEBER (Inst. oj Hydraulic Research)The following note represents a "state-of-the-art"report on certain specific aspects of viscous resistance,and its relation to a fuller understanding of ship resist-ance. The following topics are discussed in detail:(1) Correlation Lines and Form Factors.(2) Measurement Errors.(3) The Betz-Tulin Method for Obtaining ViscousDrag.(4) Variation of Viscous Drag with Froude Number.

CORRELATION LINES AND FORM FACTORSTwo new methods of estimating the viscous resistance

of a ship form have been proposed by Hughes. In one'?he recommends for the ship-model correlation formulaCv=O.0620r (lOgloRN-2.18)-2

where RN is the Reynolds number and r is a form factor,to be determined at a sufficientlylow Froude number F Nso that the wave resistance coefficient Cw is negligible.This formula is derived as a mean from the analysis ofseveral geosim series. However, because of the widedispersion of the data about their mean values, and indi-cations of variation of the form factor with Froude num-ber, the case for the above correlation fomuIa does not


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seem to be strong.Dr. Hughes may agree with the above opinion, since

he has proposed an alternative correlation formula in amore recent paper'".

v= viscous drag. canst. =X(IOglORN-2)-2V2/3V2in which V is the displacement, v the velocity of themodel, and x is a form factor, given in terms of a curveand formula as a function of the block coefficient andthe ratio S/V2/3 where S is the wetted surface area. Hisanalysis shows that, for most models, this frankly em-pirical procedure gives form factors within 3 percentof the mean.Geosim data have also been recently reanalyzed by

Professor Tamiya'" on the basis of the formulaCt--C 10=KC 10 +Cw

in which the resistance coefficients are based on wettedsurface area, and C10 is taken as Schoenherr's flat platefriction coefficient. If the form factor K is a constant,graphs of C t - C10 versus C10 at constant Froude num-ber would plot as a straight line of slope K and interceptC w o However, his results show slopes varying withFroude number, and sometimes negative.

Itmay be recalled from the 1962 report of the Resist-ance Committee that Yokoov had also found a vari-ation of the form factor with Froude number. Sincethis variation follows the humps and hollows of thewave resistance, Maruo'" has suggested that this resultis due to scale effect on wave resistance, rather thanwave effect on viscous drag.Another formula for the form factor, based on an

approximate but rational theoretical analysis, taking intoaccount the effect of the increased potential-flow ve-locities on the frictional resistance, the growth of theboundary layer on the viscous pressure resistance, andan empirical formula for airfoil pressure drag, is givenby Sasajima and Tanaka'" in the form

where C, is the block coefficient and the quantity P isgiven by a curve as a function of the beam-length ratio.A remarkable paper by Taniguchi?', which reports

on the results with a geosim series of a 45,000 tontanker, for which the viscous drag was also determinedby wake surveys, and the wave drag by surface-profilemeasurements, will also be mentioned here. He foundthat the form factor varied from about 1.34 at FN=0.15to 1.2 at F N=0.21, and that in the ballast condition,the wave resistance coefficient is not zero at FN=O.1 as

APPENDIX 15

is assumed in a method of Hughes for determining theform factor.In studies of form effect, some method of turbulence

stimulation is usually applied to assure an early transi-tion to a turbulent boundary layer. Hot-wire studies ofthe boundary layer of models of two are carriers, re-ported by Brard'", indicate that a helicoidal wire issuperior to studs as a turbulence stimulator. Since thesurface area over which the laminar boundary layer ex-tends can be determined by means of the hot wire, itwas recommended to test without turbulence stimu-lation, and to determine the resistance with a fullyturbulent boundary layer by calculating a correction forthe known extent of the laminar boundary layer. Aninteresting feature of this proposal to allow a larger areaof laminar boundary in model tests is that the bound-ary-layer thickness and the width of the wake would bereduced at the stern, so that the flow in this regionwould be more nearly similar to that of the ship, andthe scale effect on the wave resistance would be reduced.These various investigations of the concept of a vis-

cous drag given by a correlation line and an associatedform factor have not yielded a strong confirmation ofthe validity of the procedure. Geosim studies indicatethat one can select an average form factor for a shipform with which one could probably predict prototyperesistance with more realistic "roughness allowances"than with the fixed I.T.T.C. or A.T.T.C. correlationlines. Since the form factors obtained have varied sig-nificantly with Froude number, one may well question,as Maruo has done, whether the term (1 +K) C10 de-rived from geosim tests gives the viscous resistance orsome mixture of viscous and wave effects.

MEASUREMENT ERRORSIt is well understood that many pitfalls must be

avoided to obtain a meaningful value for the resistanceof a ship model. Even if the dynamometer and thespeed-measuring system are accurate and reliable, thelength of run is long enough so that a steady pattern ofboundary layer, wake and wave system is established,and the model has been verified to be sufficiently trueand smooth, there still remains the uncertainty of theextent of the laminar boundary layer, the error in ve-locity due to residual currents and surges in the towingtank, the important effect of minute quantities of cer-tain contaminants in the water, and the error due to thetank boundaries.An overall measure of these errors is given by the

results of tests of the I.T.T.C. 15.8-foot standard model.It was stated in the 1962 report of the A.T.T.C. Resist-ance Committee that a report of tests in four Britishtanks "presents a discouraging picture of dispersion of


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16 RESISTANCE SESSIONdata, not only among different tanks, but also in re-peated tests at the same tank." Two additional reportson tests of this model indicate a much improved level ofconsistency. Hughes" finds, after 200 tests of the stand-ard model at NPL over a period of four years, that theresistance remains within 1 percent of its mean value.Taniguchi'?' found that his results in the Mitsubishitank at Nagasaki agreed well with the NPL mean curve,after applying the blockage correction.

ilvjv= 1.4m(L/B)3/4Comparison with the results by other Japanese tanksshows a maximum discrepancy of about 3 percent. Inthe Mitsubishi tank the yearly standard deviation variedfrom 1 to 1.7 percent; no "storm" phenomenon was ob-served. Nor was there a significant correlation of iletwith the temperature distribution in the water.A subsequent study by Hughes-!' of the resistance ofof a glass plate, about 44 inches long, resulted in valueswhich fluctuated within 1percent of the mean at 4 and

5 fps, and 1-1/2 percent at 3 fps, with extreme vari-ations of about twice these values. Since the resultstended to be in phase, i.e. all high, medium, or low atthe same time, and in qualitative agreement with thetendency of the results with the standard model, it wasconcluded that the variation was a genuine hydrody-namic one. Use of the glass plate as a daily check forcorrecting resistance to a standard has been suggested.A large reduction in the resistance of ship models in

the Fort Steyne tank in December 1960 was reported byBarnaby and Dorey'>'. For a standard model the vis-cous resistance was about 20% less than its usual value.After a thorough investigation it was concluded that thephenomenon was due to contaminants, either a poly-saccharide material or certain algae. It was found thatthe effect could be eliminated by chlorination, or byemptying the tank and introducing fresh water.The importance of the residual currents in a towingtank was pointed out by Scott''" on the basis of ananalysis of the resistance measurements of the I.T.T.C.standard model in four tanks, the Teddington Nos. 1and 2, the St. Albans, and the Mitsubishi tank. He con-cluded that the principal source of both random andsystematic error is current drift between runs, evenwhen anti-drift curtains are used, and suggested that anaccurate current meter or an improved system for driftdamping is required.

THE BETZ-TuLIN METHOD FOR OBTAININGVISCOUS DRAG

1. An important step in the derivation of the Betz-Tulin formula for the viscous drag of a ship model re-quires that the potential flow outside the wake be con-

tinued analytically into the wake region in such a waythat the boundary conditions at the free surface over thewake region be satisfied. It should be borne in mindthat the Betz method does not require that the Betzsources be found, nor that there be a unique set of them;all that matters is that it be possible to find a set of suchsources.The problem may be formulated mathematically as

follows: It is required to find a potential function(x,Y,z) in a region R bounded by a submerged sur-face S, the assumed border of the wake, and above bythe free surface So on which the conditions _ o ~ . = O anda npressure p=O are satisfied. On the submerged surfaceS both and}j_ are assumed to be known. Here na ndenotes distance increasing in the direction of the out-ward normals to S and So. The Betz source distributionwithin R would be given by

p2=4rrq(x, Y, z) ( 1)where q(x,Y,z) is the volume density of the sourcedistribution.Application of Green's third formula, modified to suitthe present problem, gives the relation41!(x,y,z)+ H ~(! +H)p2(~,r;,()

R

where r is the distance between a pair of points of R,r= [(x_g)Z + (Y-r;)2+ (z- , ) 2 ] 1 / 2

and H is a harmonic function such that _ ! _ +H satisfiesrthe free-surface boundary conditions. Then (2) indi-cates that the potential function (x,Y,z) also satisfiesthe free-surface boundary conditions. By expressingp2 in terms of second differences and replacing theintegrals by quadrature formulas, (2) assumes the formof a set of linear equations for an array of values of on So and in the interior of R, with the right member of(2) giving a set of known values. Thus, although theexecution of the indicated numerical solution of (2)may be extremely difficult, the indicated procedurewould yield an approximate solution of the stated prob-lem, and hence, by (1), an approximate determinationof the Betz source distribution.2. A refinement of the Betz-Tulin formula for the


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viscous drag of a ship model will now be presented.According to the Lagally theorem, the force on a wakesource is proportional to the potential flow at the loca-tion of the source. Both Betz and Tulin have assumedthat, to a sufficient accuracy, this velocity may be takento be that of the free stream, and have expressed theforce on the sources in the form

DSB= -,of U(ul-u)dSwhere ,0 is the mass density of the fluid, w the area inter-sected by a transverse section of the wake, U the free-stream velocity, u the actual longitudinal velocity com-ponent, Ul the corresponding velocity component of theanalytically-continued potential flow.By Gauss' flux theorem, the Betz source distributionq satisfies

4/r f qdV= fCUl-U)dS (4)the volume integral extending over the volume of thewake from the body to the control section w. Applyingthe Lagally theorem, we obtain for the total force on thesources

Hence, applying the mean-value theorem and (4), wefind

where ul is a mean value of U1 over the region boundedby the control surfaces.If a, is replaced by the local value Ul> the expression

for the viscous drag becomesDv= f[po-p+,oU(U-U)

+ J,o(U - u1)(U-2u + Ul)JdS (7)instead of the formula of Reference 1),

Dv= f[po-p+ ,ou(U-u)+ ~ p(U-U1)2JdS (8). ,

VARIATION OF VISCOUS DRAG WITHFROUDE NUMBER

Determinations of viscous drag of ship models bymeans of wake surveys have been made at several tow-ing tanks since the last I.T.T.C. Conference. For a 10-foot model of a Series 60, 0.60-block model it wasfound by Wu and Landweber'


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18 RESISTANCE SESSIONvaries strongly with Froude number, showing humpsand hollows in opposite phase to the curve of wave re-sistance, and does not follow the trend of either theA.T.T.C. or I.T.T.C. extrapolators. On the other hand,the results with a double model in a wind tunnel, ob-tained in the same manner, are practically coincidentwith the 1957 I.T.T.C. line.Studies of the effect of the free surface on flow sepa-ration by Chow20) at Iowa and Lin21) and Webster-" atHydronautics indicate the probable mechanism of thevariation of the viscous drag with Froude number. Theprimary effect of a free surface is to introduce zones ofstrong adverse piezometric-pressure gradients, wherethe surface-wave profile along a hull rises from a troughto a crest. This causes the boundary layer to thickenrapidly, the shear stresses to diminish, and the viscouspressure drag to increase, by amounts which depend onthe shape of the wave profile, and consequently on theFroude number. The possibility of the occurrence ofseparation is also increased.Chow's study of the flow about a vertical strut pierc-ing the free surface showed that separation occurred

farthest upstream at the Froude number FN=0.25, andthen moved downstream with further increases in theFroude number. It was observed that the presence of afree surface affects the occurrence of separation in twoways:(a) At Froude numbers less than about 0.40, theadverse piezometric pressure gradient at a free surfaceexceeds that on the same form when deeply submerged,and hence the occurrence of a zone of separation neara free surface is enhanced.(b) Because of the curvature of the streamlines atthe crests and troughs of a free surface, secondaryflows may occur in the boundary layer. This has beensuggested as the explanation of isolated zones of sepa-ration below the free surface. It appears to be neces-sary to take these secondary flows into account in anytheory for the prediction of separation and viscous drag

of shiplike forms at a free surface.In the Hydronautics study a procedure for computingthe line of separation on a ship form, employing line-arized gravity-wave potential-flow theory to computethe pressure gradients, and Cooke's method (assumingsmall deviations between directions of flow inside andoutside the boundary layer) for computing the three-dimensional boundary layer, has been applied to theSeries 60, 0.60 block form. In remarkable agreementwith Chow's observations on an entirely different form,it was also found that the earliest separation occurred atFN=0.25. Observations at the Davidson Laboratory,however, failed to show any separation, as one mightexpect for so fine a form. This indicates that the calcu-lation procedure used is as yet unsuited for makingabsolute predictions, although the agreement withChow's observations on a fuller form suggests that thepredicted trend is probably correct.Townsin'"" has also found that the extent of the sepa-ration zone varies with Froude number. Hot-film sur-face probes were used to detect separation on a 100-inch Taylor Model of 0.875 block coefficient, an 8-foot"Victory" ship model, and an 8-foot BSRA model of0.80 block coefficient. The Taylor model showed muchmore separation at FN=0.122 than at FN=0.183. No

separation was detected on the Victory model. TheBSRA model showed a larger separation zone at FN=0.119 than atFN=0.219. Although these results indi-cate that the viscous drag varies with Froude number,the trend of the extent of the separation zone is oppositeto that observed by Chow and predicted by Webster.It appears, from the aforementioned research, thatboth the frictional resistance and the viscous drag of aship model are functions which vary sensitively with theship form and the Froude number. The refinement ofmethods of obtaining the viscous drag, and its routineuse by towing tanks, seems to be a necessary step in thedevelopment of a more rational procedure for predictingthe resistance of a ship from model tests.

APPENDIX IIREPORT ON WAVE RESISTANCE

by J. K. LUNDE tSkipsmodelltanken, Trondheim)

1. Since the design of ships always involves com-promise between various desired characteristics thetheory of wave resistance should, from the point ofpractical application, solve the following two main

problems:a) the determination of wave resistance of a givenhullform,b) reduced wave resistance development of hull


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forms for which the reduced wave resistance isincorporated in the compromise of the desiredhuH characteristics.

These two main problems have produced differenttrends in research although an adequate theory is ableto handle them both.In order to restrict the synopsis we consider only shipmoving uniformly and rectilinearly at or under a smooth

surface of deep water.2. Looking back through the years a big step for-

ward in this field of naval hydrodynamics was takenwhen Kelvin published his linearized theory of shipwaves. True, in this theory the most drastic approxi-mation of all was made, i.e. the entire moving hull wasassumed to be collapsed into a moving point singularitythus leaving one parameter, namely its strength to de-scribe the ship. At first sight one would perhaps notexpect such a theory to produce much of practical andtheoretical value. However, that is not so becauseKelvin's theory yields, even when it is treated approxi-mately by the means of stationary phase, importantaspects of the wave pattern created by the ship, aspectswhich are in good accord with observations. For ex-ample, the existence of two different wave trains, theconfinement of the disturbance within a sector with anangle fixed independent of the speed and the speedwave-length relation are all correctly given by thistheory. It is quite remarkable that such a drastic ap-proximation gives these results; the only phenomenonleft out is the wave interference.Kelvin's theory was superseded by the more advancedlinearized theory by Michell. This theory was however

overlooked and forgotten for many years. In his theoryMichell assumed that the entire hun of the ship was col-lapsed into a slender vertical disc-a so-called thin ship.Thus, apart from assuming the well known linearizedconditions, the boundary condition at the wetted surfaceof the ship was satisfied on the vertical centre line planeof the form rather than on its wetted surface.Later Havelock developed his linearized theory for asource moving under the free surface and suggested gen-

erating bodies by locating a continuous distribution ofsources and sinks or directed dipoles on the verticalcentre line plane of the body or on a surface on or in-side its wetted surface. This theory is in many waysmore descriptive than Michell's approach and hence ap-peals more to engineers. By placing the continuoussource and sink distribution over the vertical centre lineplane of the body and by determining the source andsink strengths for a thin ship and neglecting the disturb-ance of the free surface, i.e. assuming unbounded wateror extremely low Froude numbers, it will be found thatnot only are Michell 's and Havelock's wave resistance

APPENDIX 19

formulae identical but their velocity potentials are com-pletely identical too.3. The search for a better approximation to thedetermination of the source strengths in the linearized

problem has continued in later years. Assuming asource distribution on the wetted surface the determi-nation of the source strengths involves the solution of aFredholm integral equation of the second kind over thewetted surface. Hess and Smith have indicated how thisproblem can be solved when assuming a finite numberof sources over the wetted surface of the ship and neg-lecting the disturbance of the free surface. Thus theirmethod has, from the view point of the wave resistancetheory, some grave limitations but is, on the other hand,valid for arbitrary bodies.

It is understood that a computing program is underway in connection with the solution of the Fredholmintegral equation for the determination of the sourcestrengths in which the disturbance of the free surface istaken account of.H the distribution is over the vertical centre line planethe determination of the source strengths, when the free

surface is taken into account, involves the solution of aFredholm integral equation of the first kind.4. In the modern derivation of Michell thin-shipapproximation of the velocity potential it is common tobegin with the exact boundary conditions for the prob-

lem, assuming irrotational flowof an inviscid fluid. Thevarious unknown functions are then assumed to be ana-lytic functions of some parameter describing the thin-ness of the ship. After various manipulations the origi-nal boundary-value problem is replaced by a new line-arized one whose solution is taken to be an approxi-mation to the desired one. However, there is someuncertainty as to the proper restriction for this approxi-mation to be a good one. . Renewed research intoMichell thin-ship theory has made it clear that the re-sistance computed by this theory is the most importantterm in the more exact wave resistance expression, as-suming the beam/length ratio to be small but with norestriction on the beam/draught ratio or the slopes ofthe wetted surface.5. In Michell's theory the thin ship is, in effect, heldfixed in space with the water streaming past. Attemptshave been made to generalize Michell's theory in allow-

ing the ship to be a floating rigid body performing smalloscillations relative to a motion of a translation withconstant velocity. It then turns out that the rolling,yawing, and swaying oscillations are all damped whilethe surging, pitching and heaving oscillations are notdamped to the first order. These results are what onewould expect and comes about because of the way inwhich the slenderness parameter is defined for the gen-


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20 RESISTANCE SESSIONeralized Michell theory, i.e. from the assumption thatthe ship is a thin disc, the plane of symmetry of which isthe vertical centre line plane of the ship. In the absenceof friction, the only mechanism available for damping isthe creating of surface waves which carry off energy toinfinity. For a Michell ship an oscillation of infinitesi-mal amplitude of first order in pitching, heaving andsurging would create surface waves having amplitudesof second order at least, while the other possible oscil-lations of a Michell ship, having amplitudes of firstorder, can cause surface waves of first order. Since foractual ships, however, the pitching and heaving modesare damped to about the same extent as the other modesone is again reminded of the slogan that we should beconcerned with what type of theory fits the ship and notwhat type of ship fits the theory. One obvious methodof overcoming the difficulty is to retain the theory andsimply add damping terms with coefficients to be fixedempirically. An other and more satisfactory possibilityis to explore other ways of linearization, for example byassuming the draught/length ratio to be small. Theanalysis in the latter case is however still incomplete andsuffers from drawbacks similar to the thin ship, but withthe vertical and transverse modes reversed.6. There has been a good deal of research into theapplication of the linearized slender body theory, whichhave been well established in aerodynamics, to the pre-diction of wave resistance and ship motions in general.This is a linearized theory based on the assumption thatthe hull is slender with respect to both beam anddraught. Hence the hull is regarded as being collapsedinto a line segment and the effect of the hull is generallyreplaced by appropriate distributions of singularitiesalong this line segment. Alternatively, use is made ofMichell thin-ship theory allowing the draught to becomesmall, i.e. by expanding Michell's integral for smalldraughts. It is optimistically suggested by some authorsthat the slender body theory will eventually lead to arational and successful theory for predicting ship motionin waves. It should be noted however that the waveresistances predicted by the different linearized slender

body theories are not in agreement with each other overthe range of Froude numbers.7. The problem of finding a hull form which pre-sents the least resistance under conditions imposed bypractical requirements has always been a goal for thenaval architects. Indeed the vast efforts at the experi-mental towing tanks aims at fulfillment of this goal. Onthe other hand, mathematical methods have been triedto the determination of the ship form of minimum resist-ance under simplifying conditions.Separating the total resistance of a ship into a com-ponent due to viscosity and a component due to wave-

making both these components will depend, amongother things, on the ship's form. The relation betweenthe viscous resistance and the ship form is however notsufficiently understood to make an analysis feasible.Under extremely simplifying conditions the wave resist-ance is given by the celebrated Michell's integral involv-ing either the functions which define the shape of thehull or, as shown by Havelock, the functions which de-fine the distribution over the vertical centre line planeof sources and sinks or directed dipoles by which thehull is assumed to be generated. The problem to mini-mizing Michell's integral is a mathematical problem inthe calculus of variations and has stimulated the interestof hydrodynamicists for a number of years. In anypractical applications of the theoretical results whichmay be obtained one is forced to accept the assumptionthat the linearized theory of wave motion is an accurateenough approximation to the actual fluid motion aboutthe hull and that the exclusion of viscosity will have noserious effect.The problem of minimum wave resistance can betackled only when some side conditions are imposed.This is due to the fact that without any restrictionsthere is a solution which creates no waves. In fact, weknow of a class of singularities which does not createwaves in the linearized case. Even with the side con-dition of fixed displacement and draught the minimumwave resistance problem has no solution. Thus theminimum problem which has been tackled has usuallybeen less general in scope, having been simplified to findthe curve of sectional area for a form whose transversesections are given by a prescribed equation. The sim-plest case is the elementary wall-sided ship of infinitedraught. The condition of constant displacement is nowinterpreted as constant water plane area. It has beenfound however that the problem has really no solutionin the strict sense since the end conditions are violated.A similar situation appears in the case of finite draughtbecause of the logarithmetic singularity in the kernel ofintegral equation which has to be solved. In these casesMichell's integral has been interpreted as involving the

functions defining the shape of the hull.The situation is however somewhat different whenMichell's integral is interpreted as involving functionswhich define the distribution of dipoles over the verticalcentre line plane. In this case the problem of minimiz-ing the wave resistance of infinite deep struts under asingle side condition of constant sectional area has asolution. The same is true in the case of elementaryships of finite draught. The minimizing problem withthe side conditions of constant displacement, draughtand beam, i.e. constant block coefficient is of greatpractical interest but has no solution. If, instead of con-


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stant beam, constant moment of inertia of the waterplane with respect to the transverse axis is assumed aunique solution exists. The minimizing problem withthe side conditions of constant displacement and con-stant wetted surface has also been investigated. Thesolution of this problem is, approximately, equivalent tothe ship form for which the sum of wave resistance andskin friction is a minimum.8. Attempts have been made to find a slender shipform of minimum wave resistance but the solution in-

volves only ship forms of a restricted class and is by nomeans the best among all admissible forms. Alterna-tively, the aysmptotic form of the minimum solution ofMichell's integral for vanishing draught has been in-vestigated but the results are not yet quite conclusive.9. The use of bulbous bows in order to obtain re-

duced wave resistance for ships has become very popu-lar with ship owners in later years. The waves left farbehind any body is, in the linearized case, an integralsummation of elementary sine and cosine waves. Theseelementary waves travel in all directions () satisfying-1 ': /2 ~ ()~ 1': /2, where () is the angle between the nor-mal to a straight line through the ridge of each ele-mentary wave and the direction of motion of the body.The amplitude function of each elementary wave is afunction of (). Analysed in this manner it is not difficultto determine which waves are due to the bow, which aredue to the stern and so on for an elementary form.If the strength of the source distribution is a cosine

function in the horizontal direction and, for example, aconstant in the vertical direction, i.e. the ship has ap-proximately U-shaped sections, the two wave systems faroff of the ship can be represented by an integral sum-mation of elementary sine waves focused at the bow andstern respectively all with positive amplitude functions.1t is well known that the waves created by a pointdoublet, (i.e. a sphere) can be represented by an inte-gral summation of elementary sine waves focused in thefree surface vertically over the centre of the doublet, allwith a negative amplitude function. This suggests thatby a suitable choice of both the strength and the verti-cal position of doublet at the bow it will be possible toobtain a considerable reduction of the bow waves forthis type of model at a chosen speed. It is however notpossible to obtain a waveless bow by this device. Indeedno finite ship can have zero wave resistance.Similarly a suitable doublet at the stern would reduce

the stern waves at a chosen speed when assuming in-viscid fluid and when the effect of propellers and otherattachments are neglected. However, the influence ofthe viscosity and the wake near the stern is so importantthat the stern problem should be considered separatelyin connection with practical application.

APPENDIX 21The bow waves are purely positive sine waves if thestrength of the source distribution is an even powerseries and are purely cosine waves if an odd powerseries is used. It has been found that these positive sine

bow waves can be completely eliminated by a concen-trated doublet distribution (i.e. a cylinder) of suitablestrength along the forward end of the distribution andextending vertically to infinite depth. As the infinitedeep doublet distribution can not be realized in anypractical application, it must be truncated at a finitedepth. The truncation invalidates the perfect cancel-lation of the waves generated by each systems. How-ever the deeply submerged part does not influence toomuch the surface waves.Although a doublet is a good device for cancellingpositive sine bow waves it has no application in con-

nection with cosine bow waves. A singularity of onestep higher order than a doublet is a quadrupole. Thewaves created by a quadrupole consists of cosine ele-mentary waves focused in the free surface verticallyover the centre of the quadrupole, all with a negativeamplitude function.The cosine bow waves formed by a source distri-bution which is an odd power series can be completelyeliminated by a quadrupole distribution of suitablestrength along the forward end of the distribution andextending vertically to infinite depth. A quadrupole it-self in a uniform stream does not produce a closed body,

but it may, when combined with a doublet line. Thissuggests that these quadrupoles can be used to improvethe form of the bulb and hence its effect when bothcosine and sine waves are present.Negative cosine bow waves can also be eliminated bya source distribution of suitable strength along the for-

ward end of the distribution and extending vertically toinfinite depth. Of course, a sink distribution at theafterbody of the ship has to be employed in this case inorder to have a closed body.

In general, the linearized bow waves consists of ele-mentary waves which have different characteristics ineach direction of advance. Therefore it is not possibleto match in all direction of advance the three maincharacteristics of elementary waves, i.e. focus point ofwave, phase and amplitude of elementary waves frombulb's with those from a general ship bow so that allwaves are cancelled everywhere. Indeed, the shipshapes and hence the source distribution for ships withbulb has to be chosen with care because:a) positive sine bow waves require a doublet bulb,b) negative cosine bow waves require a source bulb,c) strong positive cosine waves with weak sine bowwaves require a doublet bulb combined with

either a source (sink) or a quadrupole bulb.


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In this type of research the source distribution has,in general, been taken as given and the hull form(stream lines of a double model) determined under theassumption that the disturbance at the free surface isnegligible. This hull form is then used in the modelexperiments.For a model assumed to create only positive sine bowand stern waves a forward shift of the wave phase has

been observed. This necessitated fitting a bulb whichprojected quite a bit forward of the bow in order to ob-tain a favourable wave resistance instead of locating thecentre of the bulb at the stem, which is the position thelinearized first order theory suggested.Various explanations for this shift have been put for-

ward, the most recent being based on higher order wavetheory. In the expression of Green's function in theclassical wave theory a particular term, which is sup-posed to be small, is neglected in the first order theory.For a particular simple source distribution this term wasincluded in Green's function and it was found then thatthis led to a forward shift of the wave phase comparedwith the results from the first order wave theory. Thissuggests that the discrepancy between experiments andthe first order wave theory as regards the wave phasemay be due to this term. It indicates also why the useof ram bow is favoured in many cases.A sizable bulb can also be used for the purpose ofguiding the water flow toward the flat bottom from the

beginning rather than letting the water spill over thebilges to reach the bottom at a later stage. Used in thismanner a bulb prevents the formation of eddies andhence reduces the form drag. If in addition the combi-nation hull and bulb produce low wave resistance thetotal resistance is reduced. This has been put forwardas an explanation why a large bulb on tanker modelscan result in a great reduction in total resistance at lowFroude numbers.10. The method of steep descent has lately been put

forward as a possible method for the purpose of modify-ing given hull forms in order to obtain reduced waveresistance.Note is taken of the close relationship between theEuler-Lagrange equations of the classical calculus of

variations and the gradient of a function in a finite-dimensional vector space. Indeed, the Euler-Lagrangeequations may be regarded analogously as the gradientof a functional in function space. This suggests thatthe wave resistance as well as the hull form should bedescribed in terms of a finite number of the same varia-bles. Considering each of these variables to represent acoordiuate in a finite-dimensional vector space, thegradient of the wave resistance in that vector space isdetermined. If these defining variables are changed in

a direction as nearly parallel to this gradient vector asthe necessary side conditions permit we decrease thewave resistance as rapidly as possible for a givenamount of motion through the vector space. In thismanner we produced an improved hull form. Follow-ing the trajectory far enough we may approach a sta-tionary value. This description leads to the employ-ment of a method of steep descent for the calculation ofimproved hull froms.11. The Froude scaling method of analyzing ship

model resistance data presuppose that the total resist-ance experienced by a ship may be decomposed intotwo parts, i.e, the viscous component which depends onthe Reynolds number, and the residual part which de-pends on the Froude number. Although this was aningenious suggestion made years ago for solving a trou-blesome scaling problem, grave difficulties have beenencountered in the practical application of this scalingJaw. One of these difficulties is the interaction betweenship waves and boundary layer. Although this problemhas been recognized for years no one has actuallylooked deeper into its nature until recently.The total vectorial force experienced by a body mov-

ing in a real fluid consists of two terms, namely thevectorial pressure resistance drag which consists of thesurface integral (taken over the wetted surface) of thenormal fluid stress, and the vectorial skin friction dragwhich consists of the surface integral (taken over thewetted surface) of the fluid shear stress. For flows ofinfinite extent, the pressure drag, which vanishes in in-viscid flow, is due to the pressure distribution beingmodified from the potential flow by the presence of theboundary layer, and is usually in this case called theform drag. In the presence of a free surface, however,gravity and capillary surface waves now exists to furthermodify the pressure distribution. In this case the pres-sure drag includes not only the form drag of the viscousorigin but also the resistance which is intrinsically due towave-making. The concept of wave resistance arisesonly when the far-field is brought into the picture.Otherwise both viscous form drag and wave resistanceare contained in the surface integral of the normal fluidstress. A feature of the surface wave and boundarylayer interaction is that it occurs as a whole entirety.The existence of the boundary layer effects the externalpotential flow and its wave system; the waves in turninfluence the pressure distribution in the boundary layerand hence also skin friction. It follows that the shapeand structure of the boundary layer in the presence ofwaves must be determined with the interaction takeninto account.A complete calculation of the boundary layer for a

given body which moves in or beneath a free surface


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and produces gravity waves may, at present, be toocomplicated. Since we are mainly interested in deter-mining with adequate accuracy overall characteristics,e.g. skin friction of momentum thickness, rather thandetails of the flow the momentum-integral equation fortwo-dimensional incompressible boundary layers ismade use of together with certain more recent develop-ments in the boundary layer theory. With the boundarylayer determined in this manner the skin friction is de-duced as well as an equivalent ship form, this beingequal to the original form plus the displacement thick-ness. The wave resistance, which is determined in theusual manner but for the equivalent ship form ratherthan for the original form, consists now of the followingterms:a) The wave resistance experienced by the shipwhen moving in an inviscid fluid.b) The wave resistance experienced by a formcorresponding to the boundary layer represented byits displacement thickness and moving in an inviscidfluid.c) The body-boundary layer interaction.One should not expect too much of such a research atpresent because too many simplifying assumptions havebeen made. The important part is however that a starthas been made towards an investigation of the real prob-lem which concerns three-dimensional flow of a turbu-lent boundary layer near the condition of separation.12. Although we are concerned with the three-

dimensional ship wave problem the research carried outin connection with two-dimensional waves should notbe overlooked.The question of determining the plane potential flowpast a circular cylinder beneath a free surface under

APPENDIX 23gravity is an old problem which recently, however, hasundergone renewed investigations.The first approximation is to replace the cylinder bythe dipole potential, modified so as to satisfy the line-arized free-surface condition. However, no dosed bodyis generated by the first approximation. In particularthe front and rear stagnation points are on differentstreamlines. Moreover it appears likely that at no finiteorder of approximation is a closed body generated. Thisfinding may have a bearing on the related linearizedproblem of determining ship-like bodies from givensource distributions.As to the second approximation, circumstances areknown in which the effect of the second approximationto the Bernoulli equation at the free surface is more im-portant than to include modifications to the flow dueto the fact that the singularity distribution which gener-ates the body in an infinite fluid no longer does so exact-ly in a fluid with a linearized free surface. This findingalso may have a bearing on the related ship wave prob-lem where consideration has been given to whether innumerical terms some improvement in accuracy mightbe achieved by going back to the exact boundary con-dition on the actual wetted surface.Second-order effects are present even with a linearfree surface condition. Large as the corresponding cor-rections to the first-order linearized expressions for thewave-induced forces on the cylinder are, the non-linearsecond order corrections are considerably larger. Thisclearly indicates the importance of being quite clearunder what circumstances the linearization process hasa rational justification. It can not be said that we havereached this stage yet in the theory of ship waves.

APPENDIX IIISHIPS OF MINIMUM WAVE RESISTANCE

by G. WEINBLUM (Univ. Hamburg)

1. The development of forms of least resistance is aproblem of utmost practical importance; it represents acentral problem in ship-building science, in analyticalship hydrodynamics, as well as in model work. Thisstatement applies particularly to wave resistance be-cause of its strong variability with the ship form. Theo-retical research on ships of minimum wave resistance isadvancing now on a broad front and is beginning to

inspire experimental investigations. Notwithstandingmeritorious attempts the application of this research todesign is still in a rudimentary state because (1) theo-retical results are based on the ideal fluid concept, and(2) even under this idealized assumption linearized so-lutions have only been obtained for restricted classes ofhull forms. As an example of (2), we mention that noreasonable expression exists so far for the determination


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of the wave resistance of ships with a transom stern.It is our purpose to summarize briefly what so far has

been achieved, and to indicate what should be done inthe future to meet urgent needs of naval architecture.Theoretical work must finally lead to the determinationof ships of minimum total resistance or rather of opti-mum performance including propulsion effects* , latereven considering seaway conditions. From the point ofview of technical application the theoretical methods sofar available rarely represent more than a heuristic ap-proach. It must be emphasized, however, that findingforms of low resistance rather than "exact" optimumforms represents the technical problem, thus mitigatingthe rigor of analytic work.2. Assuming ideal fluid, linearized theory, and sim-

plified ship forms (or singularity distributions), theoptimization is based oni) finding the minimum of certain resistance inte-

grals orii) on the extinction of the wave pattern (especiallyof bow waves).

The first approach is theoretically the straightforwardand appropriate procedure. The second, however, hassome advantages in so far as it can be applied to regionswhere the ideal fluid concept is valid to a reasonableapproximation, i.e. to the bow.or, more generally, to theregion of the forebody.To 2 i) : Minimum solutions or, more generally, so-lutions yielding low values have beena) investigated primarily for Michell's integral as themost important and classical wave resistance

formula.b) Similar investigations refer to the wave resistanceof wholly submerged bodies of revolution andc) to Pienoids (bodies generated by singularities dis-tributed over a skeleton surface).

Further integrals which can, and should be handledin a similar way are those given byd) Havelock for systems of continuous and discrete

distributions (but not for those distributed over afixed surface!)

e) Hagner's integral for pressure systems,f) Maruo's expression for semi-submerged ships

(can be subsumed under d),g) pertinent resistance integrals for motion on shal-low (finite depth) water.

The mathematics involved have caused a lot of con-cern. It has been shown that in many cases no so-lutions exist of the underlying problem of the calculus ofvariation. By generalizing the class of functions ad-mitted, and proper interpretation of results, most diffi-* Unfortunately, a paper by SISOV on the optimization ofthrust deductions so far has not been available.

culties can be avoided. These painstaking investigationshave led e.g. to the interesting results: keeping the dis-placement constant changes of hull forms (distributions)are possible such that the wave resistance is not influ-enced by these changes. This indicates the fact thatequivalent resistance properties can be attained bywidely different hulls. On the other hand other calcu-lations show that, to slight changes of form undercertain conditions, large variations in resistance maycorrespond.Although the "exact" method of calculus of vari-

ations has yielded valuable results, the direct method(Ritz' method) appears to be more fruitful from apractical point of view. The influence of vertical distri-bution of displacement, in addition to the dominantlongitudinal distribution, on the minimum resistance hasrecently been clarified. The wave drag becomes almostnegligible for theoretical optimum forms below aFroude number of say 0.27 and similarly for optimumfull forms (high prismatic forms) at usual cargo shipspeeds. The importance of bulbous forms is supportedby this analysis.To 2 ii): The same result concerning the bulb is

obtained by the inspection of the wave pattern, andmethods designed for wave extinction. Because of itsevidence and advantages from the point of view ofphysics investigations in this field appear to be promis-ing (started by Inui, his school and followers).Notwithstanding the physical shortcomings of the

ideal fluid concept efforts should be concentrated tostudy the following problems.1. Development of hydrodynamic (singularity)

models for all important families of ship forms (e.g.with transom stern) and determination of pertinentvelocity potentials under simplified conditions.We mention as a solution of more general interest

(not applicable immediately to ships) as found re-cently by Bessho for optimum dipole distributionsover lines or planes normal to the direction of trans-lation.2. The determination of body forms generated bysingularities in presence of a free surface (Tuck,Kajitani) and2a. the inverse problem-determination of gen-

erating singularities for a given body moving close toa free surface.3. Second order theory (in first line for theMichell theory) including critical survey of non

orthodox attempts to improve the linear theory e.g.such as made by Guilloton.Results of second order investigations should greatly

further endeavours to find optimum forms.3. The mentioned steps in developing second order


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ideal fluid theory, important in itself, are perhaps stillmore decisive as a necessary prerequisite for derivingoptimum forms with respect to wave resistance and laterto total resistance in the real fluid.The theoretical and experimental work carried outwithin the last years admits a favorable outlook (Inuiand his school, Wahausen and his school, Eggers,Sharma, Shearer, Hogben, Gadd, Ward, etc. for the de-termination of wave resistance, as well as T. Y. Wu'sfundamental study on the influence of waves on fric-tional resistance). It should be emphasised that evenour knowledge of the viscous drag of ship-like bodies

APPENDIX 2 5

moving in unbounded fluid ("double model technique")is absolutely inadequate. These investigations shouldbe extended on a broad front including the problemspresented by the influence of the frictional wake and ofseparation on wave generation. There is no qeustionthat the determination of optimum (favorable) forms oftotal resistance and the possibility to convert model datato the full size ship will be decisively promoted by thisresearch. Already at present the impact of analyticalmethods of optimization on model technique is extreme-ly beneficial.

APPENDIX IV

EFFECTS OF SHALLOW AND RESTRICTED WATER

by A. 1. W. LAP (Netherlands Ship Model Basin)

In principle most of the problems dealt with in thereport apply also the resistance of models or ships inshallow and restricted water. In general it can be saidthat the greater the restriction in dimensions of thechannel, the more intense the phenomena and unfortu-nately also the greater the difficulties in both conductingmodel tests and judging and extrapolating their results.A few items, however, need maybe some furtherconsideration.

1. INTERACTION OF WAVE AND VISCOUSRESISTANCE

As the distance between ship form and boundaries ofthe channel becomes smaller, the interference of viscousand wave resistance becomes much more serious. Espe-cially in the case of long and comparatively slowmodelsit may well occur that the gap between ship bottom andtank bottom is completely filled up by boundary layerflow, in which case the question must be put, in how farthe general equation for the total resistance may besplit up as:

c.s-c.i; F n) + Cw (Rn, F n)For very small water depths this will probably not bethe case and a solution must be found to determine howfar under these circumstances the viscous resistance de-pends on Froude number and how far the serious dis-turbances in the potential flowdue to viscosity and theireffect on viscous resistance may be taken into accountas a form correction factor.

Part of these interaction problems can in principle besolved by conducting geosim tests in very shallow waterin order to check whether or not lines of constantFroude number are still parallel, when plotted to a baseof R; or f(Rn). Much useful information can also beobtained by determining the viscous resistance by meansof a wake traverse method or by pressure measurementson the hull. The interference effects further form anadditional difficulty in calculating the wave resistance inshallow and restricted waters in a non-ideal fluid. Aneffective general solution suitable for practical appli-cation does not yet exist.

2. FORM EFFECT CORRECTION TOCORRELATION LINE

For deep water it is generally agreed that form effectcorrections must be applied to the correlation line, al-though there are differences of opinion about type andmagnitude of these corrections. In particular there isuncertainty about the behaviour of the viscous pressureresistance and as a consequence the question must beput, whether the correct correlation line is representedby the line of zero Froude number of a model family.This problem, not even solved for deep water, becomesof extreme importance for the extrapolation of test re-sults for shallow and restricted water, since the differ-ences between the line Fn=O and the basic correlationline may quite well become four or five times as large asthey are in deep water. Consequently the form effectcorrection to the correlation line, if determined from


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2 6 RESISTANCE SESSIONsingle model tests on base of the assumption that thelow speed part of the C t curve coincides with the re-quired correlation line, may become extremely large.This results in a much steeper extrapolation and a muchlower smooth ship resistance prediction. On the otherhand a roughness correction has to be applied to thefrictional resistance, that is maybe much greater thanthat for the same ship in deep water. This is also dueto the form effect.

3. SHALLOW WATER AND CANAL EFFECT ONTHE COMPONENTS OF RESISTANCE

One of the most important problems in the evaluationof tests in shallow and restricted waters is the conver-sion of the results obtained into those for other waterdepths and widths. This is necessary because not al-ways the full size channel can be reproduced to scale bythe available basin. It is also necessary sometimes inorder to be able to limit the number of tests.As far as resistance is concerned, several closely re-

lated conversion methods were developed, all based ontwo basic assumptions:

1) the total resistance of a ship form can be splitup into viscous and wave resistance, the first of whichcan be calculated for any required condition.2) Schlichting's assumption, that the wave re-

sistance per ton is the same in first approximationwhen the generated waves have the same length.It is known that these methods give unsatisfactory

results in many cases, which is not so surprising, since:

a) they do not take into account any form effectin viscous resistance, as a result of which considera-ble errors are maybe made in splitting up the totalresistance into its components.b) it was recently suggested that Schlichting's as-sumption is an oversimplification. By assuming that

the resistance per ton is only a function of ).IL *Schlichting introduced the parameter _ _ ! ! _ _ *, which. i " 1 V2QIS not a constant.A better assumption is therefore maybe that the spe-

cific resistance C, is dependent on AIL only.The physical meaning of Schlichting's oversimplifi-

cation is that he neglected the wave height or, in otherwords, that he assumed the wave height automaticallyto be equal as soon as the wave length was equal.

Quite recently it was shown that on base of three-dimensional extrapolation and the modified Schlichtingassumption good correlation could be obtained betweenresults of model tests at various water depths as long ascomparatively slender ship forms are considered. Forbluff bodies, such as barge tows, that are big relative tothe channel, the method still yields unsatisfactory re-sults. There is a possibility that this is caused by lami-nar flow effects or by incorrect treatment of the viscouspressure resistance, which may be relatively great forthis type of bodies.

* A=length of generated waves L=ship lengthV=ship speed n=wetted surface.

APPENDIX VRESULTS OF QUESTIONNAIRE ON BLOCKAGE CORRECTIONS, TANK STORMS,

AND USE OF TURBULENCE DETECTION TECHNIQUES

In August 1964 a letter was circulated to 55 memberorganizations and individuals. The text was as follows:

"The Resistance Committee was asked by the 10thI.T.T.C. to make recommendations regarding certainaspects of towing tank work, and to do this needs infor-mation on current practice and experience in these mat-ters. It would therefore be most helpful if you couldsupply any relevant information on the questions listedbelow.1. Blockage: Is it your standard practice to use a

blockage correction, or have you had experience ofusing such corrections? If so, has the method been de-scribed in published work; if not, are you willing to offerinformation for circulation to other members?

2. Water Contamination: Have you ever experi-enced a change of resistance directly associated with achange in the chemical or biological state of the water,and if so, was a cause detected or an analysis of thewater made?3. Turbulence Detection: Have you a standardtechnique for turbulence detection or for the measure-ment of turbulence level, and if so, is this commerciallyavailable, or has it been described in published work?

If you have unpublished experience in any of thesefields and are willing to let the Committee have infor-mation, it would be useful if you could name the officerin your laboratory who may be contacted for further de-tails. All information should be sent to the Secretary


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ApPENDIX

OrganisationTable 1. Use of blockage corrections

Method & ApplicationountrySchiffbau technische Versuchsanstalt,ViennaNational Research Council, OttawaVersuchsanstalt fur Wasserbau undSchiffbau, BerlinS. P. Div. Transport. Technical Re-search Inst., TokyoMitsubishi S. & E. Ltd., NagasakiStaten Skeppsprovningsanstalt, Gothen-bergSaunders Roe Div., Westland AircraftLtd., CowesJohn Brown Ltd., ClydebankNPL Ship Division, FelthamUniversity of NewcastleUniversity of MichiganWebb Institute, New York

AustriaCanadaFed. Rep. of GermanyJapan"Sweden

United Kingdom"""U.S.A.

"

Schuster method for models over 20 ft long.Hughes method for large, full models.Standard blockage correction by Schuster method.Taniguchi method used in No.2 Tank only.Taniguchi method used in scientific work only.Schuster method included in correlation factor.Standard blockage correction by modified Hughes method.Hughes method has been used for special problems.Hughes method used for special problems.Emerson method used for tank comparison.Standard correction based in part on Hughes.LandweberjSchlichting method has been used for specialproblems.

OrganizationTable 2. Towing tank storms

InformationountryFranceJapanU.K.

""""

Bassin d'Essais des Carenes, ParisTrans. Tech. Res. Inst., TokyoSaunders Roe Div., Westland Ltd.,CowesJohn Brown Ltd., ClydebankVickers Ltd., St. AlbansAdmiralty Experiment Works, HaslarMessrs. Thornycrofts Ltd., Fort Steyne

Storms have occurred e.g. in the Spring and after refilling.Water now chlorinated.Occasional variations in resistance.Occasional variations in resistance.Severe storm reported, associated with formation of gas

bubbles. (See Ref. 30)Occasional variations in resistance. Water now treatedwith sodium hypochlorite and algicide.Occasional storms, and random variations.Severe storm reported, associated with algal growth Curedby chlorination. (See Ref. 31)

OrganisationTable 3. Use of turbulence detection techniques

Method & ApplicationountryOccasional use of hot film probe.Occasional use of colour probes and hot film probes.Hot film probe used in research work.Benzoic acid film used when required.Hot film probe used in research work.Soluble film method sometimes used.Hot film or inkstream used for special applications.Hydroquinone diacetate, or hot film sometimes used.Hot film probe used to check stimulation on full models,and in research work.Hot film probe sometimes used.Hot film probes used.Hot film probes sometimes used.

CanadaFed. Rep. of GermanyFranceJapanNorwaySwedenU.K."""".S.A.

National Reseach Council, OttawaVersuchsanstalt fur Wasserbau undSchiffbau, BerlinBassin d'Essais des Carenes, ParisS. P. Div. Trans. Tech. Res. Inst., TokyoSkipsmodeltanken, TrondheimStaten Skeppsprovningsanstalt, Gothen-bergSaunders Roe Div., Westland Ltd.,CowesVickers Ltd., St. AlbansNPL Ship Division, FeltharnAdmiralty Experiment Works, HaslarUniversity of NewcastleWebb Institute, New York


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28 R ES IS TA N CE SE SS IO Nof the Resistance Committee.Your co-operation in this work will be greatly appre-ciated."32 replies were received and in the following tables

reference is made to those which contained informationof a positive nature. The Committee would like toacknowledge the assistance given by member organi-zations in this matter.

APPENDIX VICONTRIBUTIONS TO THE 11TH LT.T.C.

During the course of its deliberations the Committeereviewed and discussed a very large amount of pub-lished and unpublished matter. In addition to this agreat deal of valuable new material has already beenpresented in the form of formal contributions to the11th LT.T.C. These were not available when the Com-mittee prepared its report, but they do much to advanceour knowledge in certain of the fields covered i

Report of Resistance Committee

Documents

components of resistance

spray resistance

resistance sessionchairman

spra resistance

viscous resistance forships

report of resistance

component of resistance

resistance derived froma