ae = SF i ae eee {Sle Sass Se ee Seri eee ede =< a RTT ed i Ninth Sympasiun / NAVAL HYDRODYNAMICS a SSS cs es SSS SST SS Soe pe SS chee as = ee rae eS SSS SSS Fs —s SSS ce a ea ee = FR ie Rp DA FEE EE Sa ee eS VOLUME 2 on ek gern eee Se Sas Soe <== TST SIT Se PLES Sele onl’ ee FRONTIER PROBLEMS ow ts = At Ree te ere bi on cc EAS RE a ei ia q iy mara ete? i 9 RAL TD UCU MARE AE SRA A AME SEATS i iseienannenenninneninnnsinannnteasenssnesisnssinnenivensen-ensiaba ACR 203 CHtice of Naval Research Dopartmnant of the Navy =— SS ee ee = A eet Wee Mile Ps. A iy ee A ~ sa! q oe, be Pa ; = Hy ; yar w E ¢222000 TOEO g UC 000000 0 a 1IOHM/TEIN ARUN SUE Ba aie tite on iat ii a te My Uy ae i A le i NY , hy ali ny ah . \ iP A et i We co . 7 Rte re airy an uh GUE) ba ae Tyce Pen if te j i ee bs i i\ i 7m 4 7 f ; i, y ul 1 My if } wy, Ninth Symposium NAVAL HYDRODYNAMICS VOLUME 2 FRONTIER PROBLEMS Ae ge, WANK. ' va a WUUS HOLE Mace 4 { — i sponsored by the J: wWwHo j R | OFFICE OF NAVAL RESEARCH the MINISTERE D’ETAT CHARGE DE LA DEFENSE NATIONALE and the ASSOCIATION TECHNIQUE MARITIME ET AERONAUTIQUE August 20-25, 1972 Paris, France R. BRARD A. CASTERA Editors ACR-203 OFFICE OF NAVAL RESEARCH— DEPARTMENT OF THE NAVY Arlington, .Va. For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price $13.55 Stock Number 0851-00063 PREVIOUS BOOKS IN THE NAVAL HYDRODYNAMICS SERIES “First Symposium on Naval Hydrodynamics,” National Academy of Sciences—National Research Council, Publication 515, 1957, Washington, D.C.; PB133732, paper copy $6.00, 335-mm microfilm $1.45. “Second Symposium on Naval Hydrodynamics: Hydrodynamic Noise and Cavity Flow,” Office of Naval Reserach, Department of the Navy, ACR-38, 1958; PB157668, paper copy $10.00, 35-mm microfilm $1.45. “Third Symposium on Naval Hydrodynamics: High-Performance Ships,” Office of Naval Research, Department of the Navy, ACR-65, 1960; AD430729, paper copy $6.00, 35-mm microfilm $1.45. “Fourth Symposium on Naval Hydrodynamics: Propulsion and Hydroelasticity,’’ Office of Naval Research, Department of the Navy, ACR-92, 1962; AD447732, paper copy $9.00, 35-mm microfilm $1.45. “The Collected Papers of Sir Thomas Havelock on Hydrodynamics,”’ Office of Naval Research, Department of the Navy, ACR-103, 1963; AD623589, paper copy $6.00, microfiche $1.45. “Fifth Symposium on Naval Hydrodynamics: Ship Motions and Drag Reduction,” Office of Naval Research, Department of the Navy, ACR-112, 1964; AD640539, paper copy $15.00, microfiche $1.45. “Sixth Symposium on Naval Hydrodynamics: Physics of Fluids, Maneuverability and Ocean Platforms, Ocean Waves, and Ship-Generated Waves and Wave Resistance,’’ Office of Naval Research, Department of the Navy, ACR-136, 1966; AD676079, paper copy $10.00, microfiche $1.45. “Seventh Symposium on Naval Hydrodynamics: Unsteady Propeller Forces, Funda- mental Hydrodynamics, Unconventional Propulsion,’”’ Office of Naval Research, Depart- ment of the Navy, DR-148, 1968: AD721180; Available from Superintendent of Docu- ments, U.S. Government Printing Office, Washington, D.C. 20402, Clothbound, 1690 pages, illustrated (Catalog No. D 210.15:DR-148; Stock No. 0851-0049), $13.00; microfiche $1.45. “Eighth Symposium on Naval Hydrodynamics: Hydrodynamics in the Ocean Environ- ment,” Office of Naval Research, Department of the Navy, ACR-179, 1970; AD748721; Available from Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 20404, Clothbound, 1185 pages, illustrated (Catalog No. D 210.15: ACR-179; Stock No. 0851-0056), $10.00; microfiche $1.45. NOTE: The above books are available on microfilm and microfiche from the National Technical Information Service, U.S. Department of Commerce, Springfield, Virginia 22151. Che first six books are also available from NTIS in paper copies. The catalog numbers ind the prices for paper, clothbound, and microform copies are shown for each book. Statements and opinions contained herein are those of the authors and are not to be construed as official or reflecting the views of the Navy Department or of the naval service at large. PREFACE The Ninth Symposium on Naval Hydrodynamics continues in all aspects the precedent, established by previous symposia in this series, of providing an inter- national forum for the presentation and exchange of the most recent research re- sults in selected fields of naval hydrodynamics. The Symposium was held in Paris, France on 20-25 August 1972 under the joint sponsorship of the Office of Naval Research, the Ministere d’Etat chargé de la Défense Nationale and the Association Technique Maritime et Aeroanutique. The technical program of the Symposium was devoted to three subject areas of current naval and maritime interest. These subject areas are covered in the Proceedings in two volumes: Volume 1 — The Hydrodynamics of Unconventional Ships — Hydrodynamic Aspects of Ocean Engineering Volume 2 — Frontier Problems in Hydrodynamics. The planning, organization and management of a Symposium such as this is an undertaking of considerable magnitude, and many people have made invaluable contributions to the resolution of the myriad of large and small problems which invariably arise. The Office of Naval Research is acutely aware of the fact that the success of the Ninth Symposium is directly attributable to these people and wishes to take this opportunity to express its heartfelt gratitude to them. We are particu- larly indebted to Vice Admiral Raymond THIENNOT, Directeur Technique des Constructions Navales, Ministére d’ Etat chargé de la Défense Nationale, to Professor Jean DUBOIS, Directeur des Recherches et Moyens d’ Essais, Ministere d’ Etat chargé de la Défense Nationale, and to Monsieur Jean MARIE, Président de 1’ Association Technique Maritime et Aeronautique, who provided the formal structure which made this joint undertaking possible. The detailed organization and management of the Ninth Symposium lay in the capable and competent hands of Vice Admiral Roger BRARD, President de la Academie des Sciences, and Rear Admiral André CASTERA, Directeur du Bassin d’Essais des Carénes, who were most ably assisted in this endeavor by the charming Madame Jean TATON. Throughout the long days of planning and preparation the experienced and practical counsel of Mr. Stanley DOROFF of the Office of Naval Research provided continuous guidance which contributed in an immeasurable way to the success of the Ninth Symposium on Hydrodynamics. RALPH D. COOPER Fluid Dynamics Program Office of Naval Research ili VOLUME 2 CONTENTS FRONTIER PROBLEMS ©. @. 0 (@ wey ie. (9 Leh. 6.2.0) ah) e). eee 6 eien le mie a aie. a ere Ue) She Je Vener ee ie OPTIMUM SHAPES OF BODIES IN FREE SURFACE FLOWS Th. Y. Wu, California Institute of Technology, Pasadena, California, and Arthur K. Whitney, Palo Alto Research Laboratory, Lockheed Aircraft Corp. Palo Alto, California DISCUSSION, tars taate trv astustind it: Succ ndowal tilde. Ernest O, Tuck, University of Adelaide, Australia REPEY -TrO DISGUSBSION 45. Stee 2. beh ve Arthur K.Whitney, Palo Alto Research Pie Lockheed Corp. , Palo Alto, California DISCUSSION 2:3°7 WA} G9 BRERA VaeeR hy 100) 58 William B. Morgan, NEAT Ship Research ana Development Center, Bethesda, Maryland lv 1i27 LeZg 26 REPLY TO DISCUSSION Theodore Y. Wu, California Institute of Technology, Pasadena, California PSE WOO MON cog eG wee es. 5 SEM eine ale. open eet BOS 1129 Vsevolod V. Rogdestvensky, Shipbuilding Institute, Leningrad, U.S.S.R. REPLY TO DISCUSSION. ...nctapsl ets eae Sere Ms 1130 Arthur K. Whitney, Palo Alto Research Lab., Lockheed Aircraft Corp., Palo Alto, California HYDRODYNAMIC CAVITATION AND SOME CONSIDERA TIONS OF .THE INFLUENCE OF FREE GAS CONTENT #3... sre 1131 Frank B, Peterson, Naval Ship Research and Develop- ment Center, Bethesda, Maryland WES CHS STON, sie as coe ot ee Joy ot o. Sy cone OES LPet Edward Silberman, St. Anthony Falls Hydrau- lic Laboratory, Minneapolis, Minnesota DISCUSSION © i aw 2 ee © #8 aACMR ATS woe a ee 1182 Serge Bindel, Délégation Générale a la Re- cherche Scientifique et Technique, Paris, France DISCUSSION cia eos a ae ee A ee ae a 1184 Luis Mazarredo, Escuela T.S. de Ingenieros Navales, Associacion de Investigacion de la Construccion Naval, Madrid, Spain REPELS PONDISGUSSION: ) ovate fle et Bates © es 1184 Frank B. Peterson, Naval Ship pesedveh and Development Center, Bethesda, Maryland VORTEX THEORY FOR BODIES MOVING IN WATER... LBS i: Roger Brard, Bassin d'Essais des Carénes, Paris, France DISCUSSION fF 3 ak 2h See iate Bie. oro of eR on Paee aa 1278 Peter T. Fink, University of New South Wales, Kensington, Australia REPLY, TO. DISCUSSION 4.écerleus Se one ge ee ne 1280 Roger Brard, Bassin d'Essais des Carénes, Paris, France DES We ROI eae eens we, st mos iegans om SLi Eee 1281 V.N. Treshchevsky, Kryloff Research Ins- titute, Leningrad, U.S.S.R. Vv REPLY ‘TO DISCUSSIONS... i Hie ee tea ear oe Roger Brard, Bassin d'Essais des Carénes, Paris, France DISCUSSION 3. 0ctct PP ehh tet et ena et esas ; John P, Breslin, Stevens Institute of Techno- logy, Hoboken, New Jersey REPLY ‘TO DISC USSIONUA Pirie ce ae Roger Brard, Bassin d'Essais des Carénes, Paris, France Page 1282 1283 1284 ON THE VALIDITY OF A GENERAL SIMILARITY HYPOTHESIS PORT Ie. 0 OND) WARP EEO Wise tn a Ce cite aint (sree animes ea are ieo Fink and Eduard Naudascher, University of Karlsruhe, Germany DISGUSSTON? 4 AALS HK CRP Pale (9. COR Paul Lieber, University of California, Berkeley, California REPLY *2©@ DISCUBSION? t 2% 27 ha" st ee ears Leo Fink, University of Karlsruhe, Germany VELOCITY DISTRIBUTION AND FRICTION FACTORS IN FLOWS. WITH, DRAG -REDUGIAON oon oc omar sete eke Michael Poreh and Yona Dimant, Technion Israel Institute of Technology, Haifa, Israel DISGUSS LOW a0: 4. 6 xy w, fay oe) ACT ARI SOS OP Thomas T, Huang, Naval Ship Research and Development Center, Bethesda, Maryland DISCUSSION oie. 6 oie se ects ie iol ob a nue aenerie oueomd Jaroslav J, Voitkounsky, Shipbuilding Ins- titute, Leningrad, U.S.S.R. | pal 3 REPLY, LO DISCUSSION. ook Bak sage we ceal Michael Poreh, Technion Israel Institute of Technology, Haifa, Israel OCEAN WAVE SPECTRA AND SHIP APPLICATIONS .. Ming-Shun Chang, Naval Ship Research and De- velopment Center, Bethesda, Maryland vi 1285 1304 1304 1305 1324 1325 1325 1326 1331 DISGUSSION ve csaus eee s. cc ee FART EES, GLa eV William E. Cummins, Naval Ship Research and Development Center, Bethesda, Maryland DISGUSSION. «iiss ‘gutivan atitmeracaue Tan a e64e Sao Michel Huther, Bureau Veritas, Paris, France DISS Ue OUON majmyy sxcecaviia es) 6 «a ele eh ja ee 8 es Manley Saint-Denis, University of Hawai, Honolulu REPLY ck OG. DISCUSSION. 5.5 soe a-e ce oa 2 8 ee Ming-Shun Chang, Naval Ship Research and Development Center, Bethesda, Maryland THE ROLE OF THE DOMINANT WAVE IN THE SPECTRUM OF WIND-GENERATED WATER SURFACE WAVES... E.J. Plate, University of Karlsruhe, Germany DIS GUSSION) 04s. Aseyueslerve! « at dh airs te Yo, yah ah eens M,. Huther, Bureau Veritas, Paris, France DISGUSOLOM o7 eS cee eee ee ae eee aero eran R, Tournan, Bureau Veritas, Paris, France REPLY TO DISCUSSION 422.253 Be ee eee E.J. Plate, University of Karlsruhe, Germany INTERNAL WAVES IN CHANNELS OF VARIABLE DEPTH Chia-Shun Yih, University of Michigan, Ann Arbor DISGUSSTONM TT soe lett ts aoe ee a ene eee a eens a Michael N, Yachnis, Naval Facilities Engineer - ing Command, Washington D.C. REPSY PO DISCUSSION Tl sh. sere ee eee tec C.S. Yih, University of Michigan, Ann Arbor MODELING AND MEASUREMENTS OF MICROSCOPIC STRUCTURES OF. WIND WAVES. 155. . Cho 2) cine eda o Jin Wu, Hydronautics Inc, Laurel, Maryland THE WAVE GENERATED BY A FINE SHIP BOW)..... T. Francis Ogilvie, University of Michigan, Ann Arbor DISCUSSION s gesvsogy aber Hecate sree den ChG pay st * eal Renier Timman, Delft Institute of Technology, Netherlands vii Page 1365 1366 1367 1368 Laat 1394 1394 L395 139% 1432 1432 1435 1483 L523 REPLY TO DISCUSSION ¢ = 222 e225 2 a aan T. Francis Ogilvie, University of Michigan, Ann Arbor DISC USBION 3 2 o:¢:2e 4 2 eae sie ee wo am Ernest O, Tuck, University of Adelaide, Australia REPLY TO DISCUSSION . Seth el eee eee : T. Francis Ogilvie, Baivereity of Michigan: Ann Arbor DISC USSION® 52472 SING APY art AROS! Py aie, F Gedeon Dagan, Technion and Hydronautics Ltd., Haifa, Israel REPLY TO DISCUSSION (2S sd: c's or. ee a ‘ T. Francis Ogilvie, University of Michigan, Ann Arbor TRANSCRITICAL FLOW PAST SLENDER SHIPS .... G.K. Lea, National Science Foundation, Washington D.C. and J.P. Feldman, Naval Ship Research and Development Center, Washington D.C, DISCUSSION 33.5 6 BS S POTEET TEN EY ic Ore Ernest O, Tuck, University of Raode Australia REPLY TO DisGUnslOm 0 Xe. se Sa ae ea George K, Lea, National Science Foundation, Washington D.C, DISCUSSION 2 ite 2's Stare ee Take ee ene Ian W, Dand, National Physical Laboratory, Feltham, «U UK. REPLY TO DISC USS ton tut. eae eee eaode George K, Lea, Natosal Science Réuiidatilen: Washington D.C, COMPUTATION OF SHALLOW WATER SHIP MOTIONS R.F,. Beck and E,O, Tuck, University of Adelaide, Australia DISCUSSION \, 0: 4. sible ol dice tela ena Bose ee eon . Valter Kostilainen, Ship Hydrodynamics Lab., University of Technology, Helsinki, Finland Viii Page 1523 1524 1524 1525 1525 152 1540 1540 1540 1541 1543 1582 Bee EO DIG USO U ON earn ucemiaars musicady pa ue Robert F. Beck, University of Michigan, Ann Arbor, Michigan DISGUSSION cious: oa ye ace es 0 BRAUN AAR Sie Cheung-Hun C, Kim, Stevens Institute of Technology, Hoboken, New-Jersey REPEY FO DISCUSSION: «. 62:2 «(5 test. APs Is Robert F. Beck, University of Michigan, Ann Arbor, Michigan DISC USSEON Fe g.8 «paloma sats TO. Got RST RL, PIS Cheung-Hun C, Kim, Stevens Institute of Technology, Hoboken, New Jersey REPLY TODISCUSSIONBIUS wre emelsuicie. Robert F. Beck, University of Michigan, Ann Arbor, Michigan SEAKEEPING CONSIDERATIONS IN A TOTAL DESIGN in. gl SC DLC 1, OGD epilemempeamche rt age tei aia Re A telat 2 Chryssostomos Chryssostomidis, Massachusetts Institute of Technology, Cambridge, Massachusetts DISC USSi OI xe cwrae ws ae aan: hi, SER RRM? Geol aN CPIM toy Gots Manley Saint-Denis, University of Hawai, Honolulu DISCUSSION, yc xveca duce tn ekow cu cy sae esaluice Mie a tatene as Raymond Wermter, Naval Ship Research and Development Center, Bethesda, Maryland REPEL Y TO DipC Usain. occ, cpannesa swe tesedeote ne oe Chryssostomos Chryssostomidis, Massachu- setts Institute of Technology, Cambridge, Massachusetts DTS. CUSSION ciel cis coped stalin eke zetellee feAigthe: thet 's Reuven Leopold, U.S. Navy Naval Ship Engi- neering Center, Hyattsville, Maryland REP EY OV ils OUSaI ON ia. ccna innit auhe' Ie ae, aubern Chryssostomos Chryssostomidis, Massachu- setts Institute of Technology, Cambridge, Massachusetts DERE Le! = 21, © J RAR RA EER a a RS ars eM ae eee an Michel K. Ochi, Naval Ship Research and Development Center, Bethesda, Maryland 1x Page 1582 1583 1583 1587 1587 1589 1619 1620 1621 1622 1624 1624 REPLY TO DISCUSSION, ... aver 9 cicolttes smelt ielee ts Chryssostomos Chryssostomidis, Massachu- setts Institute of Technology, Cambridge, Massachusetts DISCUSSION cs. wis ait eucal nish dilening us Guts . Fre Surrey, U.K. REPLY, VO DISGUSS TOM: 6. uc padiecd ul tcl iietds Chryssostomos Chryssostomidis, Massachu- setts Institute of Technology, Cambridge, Massachusetts DISCUSSION elite wis vaccine, A cale eens Sr heute Meet wa Edmund Lover, Admiralty Experiment Works, Hastax, ‘Gosport; Hants;: \U ,Ky REPLY iO DISCUSSION) a oan on eae Chryssostomos Chryssostomidis, Massachu- setts Institute of Technology, Cambridge, Massachusetts THE APPLICATION OF SYSTEM IDENTIFICATION TO DYNAMICS OF NAVAL CRAE'T .. 0 we« vic ce. Se ieeus< Paul Kaplan, Theodore P, Sargent and Theodore R. Goodman, Oceanics Inc,, Plainview, New York DISCUSSION wus ven cncie snsaay cued s4.cm owen wep -eet Rote Re D.S. Blacklock, Hydrofin, West Chiltington, Sussex, U.K. DISCUSSION), 3. 04.chs<. ecu cess ners tata ner et aka aes Peter T. Fink, University of New South Wales, Australia REPLY TO DISCUSSION jes 1 oma oad are oa Rais ee Paul Kaplan, Oceanics Inc,, Plainview, New York NON LINEAR SHEP WAY E: WHE OR Y, so sesiccivee sa ae cece Gedeon Dagan, Technion Haifa and Hydronautics Ltd. Rechovoth, Israel DISCUSSION .. pis bee we ss BROS Ernest O, Tuck, University of Adelaide, Australia Page 625 1625 1626 1626 1627 1629 1690 1692 1693 1697 1736 DISCUSSION) 5, 5 c-tv nae or eo epee ae oh a eee Edmund V. Telfer, R.I.N.A., Ewell, Surrey, U.K. REPLY ..T.© DISCUSSION: oi.cu con ae cata ale ee Soe Gedeon Dagan, Technion Haifa aa4 Hydro- nautics Ltd., Rechovoth, Israel ON THE UNIFORMLY VALID APPROXIMATE SOLUTIONS OF LAPLACE EQUATION FOR AN INVISCID FLUID FLOW PAST A THREE-DIMENSIONAL THIN BODY .......-. J.S. Darrozes, Ecole Nationale Supérieure de Techniques Avancées, Paris, France WAVE FORCES ON A RESTRAINED SHIP IN HEAD-SEA WE WIGS: 6. oes Ga Pee eee ee ee ee ee eee Odd Faltinsen, Det Norske Veritas, Oslo, Norway DESCIUSEEON Pans fe sce.) OBE a orig ek Se ee Michel Huther, Bureau Veritas, Paris, France PIS CSS OI hs. o cakie. Sone aged hs Porro ua ged oe Aes ao Cheung-Hun C, Kim, Stevens Institute of Technology, Hoboken, New Jersey REP By VO DISCUSSION che Se tee aon AS Odd Faltinsen, Det Norske Veritas, Oslo, Norway DISC UBC eect oie: fo sengen selina cn tenes a aire ote Cheung-Hun C, Kim, Stevens Institute of Technology, Hoboken, New Jersey REPLY TO DISCUSSION oa wo scien cae i a ee Odd Faltinsen, Det Norske Veritas, Oslo, Norway FREE-SURFACE EFFECTS IN HULL PROPELLER INTERAC TION? $ «..2 2.0589. ein. Peta a WER Rae ie Nemes ° Horst Nowacki, University of Michigan, Ann Pe eats Michigan and Som D, Sharma, Hamburg Ship Model Basin, Hamburg, Germany DES GWE SEG ee clk i bla Sets ay alee hes UN Edmund V. Telfer, R.I.N.A. Ewell, Surrey, U.K. Xl Page 1736 i 4 1739 1763 1842 1842 1843 1843 1844 1845 1945 DISCUSSION, 155 of ./0p ta. ae arce” entcnr net ealoee Meee ; Georg P, Weinblum, Institut fur Schiffbau der Universitat Hamburg, Hamburg, Germany DISCUSSION, oo. oi, ous iss, a oeumbereae ante Sauk Klaus W. Beacret Institut fur Schiffbau fen Universitat Hamburg, Hamburg, Germany DISCUSSION oo ie eu be dipin ec a cael’ ashi, ceo aera p Carl-Anders Johnsson, Statens Skeppsprov- ningsanstalt, Goteborg, Sweden DISCUSSION, jr seatet Alea ers Bah oP S Soe a Gilbert Dyne, Statens Skeppsprovningsanstalt Goteborg, Sweden REPLY £O DISCUSSION, «. «weno s/iaue su elmamdee Som D, Sharma, Hamburg Ship Model Basin, Hamburg, Germany SHEAR STRESS AND PRESSURE DISTRIBUTION ON A SURFACE SHIP MODEL : THEORY AND EXPERIMENT T.T. Huang and C.H. von Kerczek, Naval Ship Research and Development Center, Bethesda, Maryland TONS CON a oo fe acc. an fa, selena ae Gl ei L. Landweber, University of Iowa, Iowa City DISCUSSION 00.) a. in 6: wipe i asineiiey: 4: ates cay 2 emma John V. Wehausen, University of California, Berkeley, California REPLY TO DISCUSSION sc cecemumihee neh geen age Thomas T, Huang, Naval Ship Research and Development Center, Bethesda, Maryland DISGUSSION se.ie: %) Gia) cptenlole, owe aia se eee Oa arte: Jean-Francois Roy, Bassin d'Essais des Careénes, Paris, France REPLY. TO DESC RS PON) os copay chim chee reign ele Thomas T, Huang, Naval Ship Research and Development Center, Bethesda, Maryland Xli Page 1948 1949 1950 1950 1953 1963 2008 2009 2009 2010 2010 FRONTIER PROBLEMS Thursday, August 24, 1972 Morning Session Chairman : Pr.).J.V.-Weéhausen University of California, Berkeley, U.S.A. Page Optimum Shape of Bodies in Free Surface Flows. i T. Y. Wu (California Institute of Technology). A. Whitney (Lockheed Aircraft Corporation, Research Laboratory, (U.S.A.). Hydrodynamic Cavitation and Some Considerations of the Influence of Free Gas Content. 1131 F.B. Peterson (Naval Ship Research and De- velopment Center, U.S.A.). Vortex Theory for Bodies Moving in Water. 1187 R. Brard (Bassin d'Essais des Carénes, France). On the Validity of a General Similarity Hypothesis 1285 for Jet and Wake Flows. L. Fink, E. Naudascher (Universitat Karlsruhe, Fed. Rep. Germany). Velocity Distribution and Friction Factors in Flows with Drag Reduction. 1305 M. Poreh, Y. Dimant (Technion Israel Institute of Technology. 1109 was ey 6 : Me : UBCUISIN 6 i) wk ess tak a 1 Onorg P. Wotsblien, ine iewe ting § * Cadvpretens {Awritineey, a Din. WSSiON Kiaue 6 2A PSO ot Ba ea ver ohrat Haaaibn PAPO S, Geacenay USI OM... agent be : Cipin Adele b BE Rta Ae Beige Vie 4emeLes Nay aint gi Lad Io cay a an > / e441 % ‘ . 5 - t Dyer,” Bieter Skhppaprovelndynnateite ETS OVEN cm uedow Vo, 12 + wenesiedee ete shetax LAG: LRG Yo eax : Grow ds rye, | ree bay rig Bt:o Made! Seen, ogee lAinters,: Carinae =yye . HERA STRESS ANEMONE Rae ae Kt erie ae pqade t PUAF ACE | pittortiogh) hem tes om) mache. Erie ne BOM aOR Op) DESI, DS tga: Revenrcn ant Devevomrc dated VIMAR. Moi soueh vps rena oem ebimuo. ames boa aoliatived oir at ler rat ‘oo? saietaw? af op? Teba esac i ge apt Ses Jeune taba eleoean dened oe LSC USSION pe 8.0 ots imemqolsy yj "iE ohm We tem, Univer aity. of Cat a! Oct < Tei) orleley, CaliteteWl oi aanivoM agiboa 7oY ‘yroadT t a re ate poy sal oleae — ein brexra ff 5 « he a won el | g 4 he _lerneas tt « shomaa 2. pean, fh ip Ra shapes and eas) Dew atest tainting yivbilaY OISt- (SSaroe wingr’d naw bas be Jran- rene gu). xsdemshasd eh att dh Caracas, Pass 701) .qo bel ,odutefead | = ee apeaiakaits | aosin Tbe melendiniedc sort Laine Husk, Na P Hi Os oi" pepe a gerd YES Oy bee Gi aed” YOO Mate Ie ¢golcadveT Yu etiniveg! ont! OPTIMUM SHAPES OF BODIES IN FREE SURFACE FLOWS Thee WE Caltfornta Instttute of Technology Pasadena, Caltfornta, U.S.A. and Arthur K. Whitney Palo Alto Research Laboratory, Lockheed Atreraft Corp. Pato’ Alto, “Calt fornia, “UsS.A: ABSTRACT The general problem of optimum shapes arising in a wide variety of free-surface flows can be charac- terized mathematically byanew class of variation- al problems in which the Euler equation is a set of dual integral equations which are generally nonli- near, and singular, of the Cauchy type. Several ap- proximate methods are discussed, including linear- ization of the integral equations, the Rayleigh-Ritz method, and the thin-wing type theory. These me- thods are applied here to consider the following physical problems : (i) The optimum shape of a two-dimensional plate planing on the water surface, producing the maxi- mum hydrodynamic lift ; (ii) The two-dimensional body profile of minimum pressure drag in symmetric cavity flows ; (iii) The cavitating hydrofoil having the minimum drag for prescribed lift. Approximate solutions of these problems are dis- cussedundera set ofadditional isoperimetric cons- traints and some physically desirable end condi- tions. i Be Wu and Whttney I. INTRODUCTION The general problem of optimum shapes of bodies in free- surface flows is of practical as well as theoretical interest, In ap- plications of naval hydrodynamics these problems often arise when attempts are made to improve the hydromechanical efficiency and performance of lifting and propulsive devices, or to achieve higher speeds of operation of certain vehicles. Some examples of problems that fall under this general class are illustrated in Figure 1. The first example is to evaluate the optimum profile of a two-dimensional plate planing on a water surface without spray formation, and produc- ing the maximum hydrodynamic lift under the isoperimetric cons- traints of fixed chord length € and fixed wetted arc-length S of the plate. The second example depicts the problem of determining the shape of a symmetric two-dimensional plate so that the pressure drag of this plate in an infinite cavity flow is a minimum, again with fixed base-chord & and wetted arc-length S. The third is an example concerning the general lifting cavity flow past an optimum hydrofoil having the minimum drag for prescribed lift, incidence angle a , chord length f and the wetted arc-length S. In these problems the gravitational and viscas effects may be neglected as a first appro- ximation for operations at high Froude numbers. Physically, there is no definite rule for choosing the side constraints and isoperimetric conditions, but the existence and the characteristic behavior of the solution can depend decisively on what constraints and conditions are chosen, Mathematically, it has been observed in a series of recent studies that the determination of the optimum hydromechanical shape of a body in these free-surface flows invariably results in a new class of variational problems, Only a very few special cases from this general class of problems have been solved, the optimum -lifting-line solution of Prandtl being an outstanding example. There are several essential differences between the classic- al theory and this new class of variational problems. First of all, the unknown argument functions of the functional under extremization are related, not by differential equations as in the classical calculus of variations, but by a singular integral equation of the Cauchy type. Consequently, the ''Euler equation'' which results from the consider- ation of the first variation of the functional in this new class is also a singular integral equation which is, in general, nonlinear, This is in sharp contrast to the Euler differential equation in classical theory. Another characteristic feature of these new problems is that while regular behavior of solution at the limits of the integral equation may be required on physical grounds, the mathematical conditions which insure such behavior generally involve functional equations which are Fib2 Optimum Shapes of Bodtes in Free Surface Flows difficult, and sometimes just impossible, to satisfy. Because of these difficulties and the fact that no general techniques are known for solving nonlinear singular integral equations, development of this new class of variational problems seems to require a strong effort, Attempts are made here to present some general re- sults of the current study. Some necessary conditions for the existence of an optimum solution are derived from a consideration of the first and second variations of the functional in question. To solve the re- sulting nonlinear, singular integral equation several approximate methods are discussed. One method is by linearization of the integral equation, giving a final set of dual singular integral equations of the Cauchy type. When the variable coefficients of this system of integral equations satisfy a certain relationship, this set of dual integral equations can be solved analytically in a closed form ; the results of this special case provide analytical expressions which can be exten- sively investigated to determine the behavior of a solution near the end points, Another approximate method is the Rayleigh-Ritz expans- ion ; it has the advantages of retaining the nonlinear effects to a certain extent, of incorporating the required behavior of the solution near the end points into the discretized expansion of the solution, but the method is generally not convergent, A third approach depends on a thin wing type theory to describe the flow at the very beginning, a variational calculation is then made on an approximate expression of the physic- al quantities of interest. These mathematical methods will be discuss- ed and then applied to three problems described earlier. While the results to be presented should be considered as still preliminary, since exact solutions to these problems have not yet been found, itis hoped that this paper will succeed in stimulating further interest in the development of the general theory, and, in turn, aid in the resolu- tion of many hydromechanic problems of great importance, II. GENERAL MATHEMATICAL THEORY To present a unified discussion of the general class of op- timum hydromechanical shapes of bodies in plane free-surface flows, including the three examples (i) - (iii) depicted in Figure 1, we as- sume the flow to be inviscid, irrotational, and incompressible, taking as known that the physical plane z=x+iy and the potential plane f=¢ +ipf correspond conformally to the upper half of the para- metric ¢= & +in plane by the mapping that can be signified sym- bolically as fee way ase re.. plete, (1) PEES Wu and Whttney where v is an analytic function of ¢ and may involve geometric parameters Cy a eee Cys so that the wetted body surface corresponds to 7=.07, || <1, and the free surface, to = ot, ts Ceca ae Specific forms of the function v ({) will be given later, but our pur- pose at this time is merely to illustrate the type of nonlinear varia - tional problem that arises, Description of the flow is effected by giving the parametric expressions. f= £(6)ieand ciaf=.e 15) Go) n=l osilon(di/da)up= maedrad (2) being the logarithmic hodograph, The boundary conditions for w may be specified either as a Dirichlet problem, by giving Yr iven Haier, REMeACEh HAO ) (g ease , (3) or as a Riemann-Hilbert problem, 6 = Im w(f +10) = 6 (é) (1 E|-n 1). (4b) The formulation of the w problem is completed by specifying a con- dition at the point of infinity, say SLO; ( |z] > ©), (5) and by prescribing a set of end conditions, which are generally on (ey enais palit 1) iT] io) (6) 1114 Opttmum Shapes of Bodtes in Free Surface Flows or similar ones. The end conditions are usually required on physical grounds in order that the fluid pressure is well behaved at the end points £ = + 1, at which the free boundary meets the wetted body surface. The solution to the Dirichlet problem (3), (5), (6), ive. and the solution to the Riemann-Hilbert problem (4), (5), (6), given by ly a7 : 1 2 t)dt OU te aee - 1) | (8a) i at (ee) with 1 l ne aon (8b) t (l-¢,) are A ae baa to each other, as can be readily shown. Here, the func- tion (cee - 1 2 is one-valued in the ¢ =plane cut from f£ = =1 to 5 = 1, On the body surface, we deduce from (7), by applying the Plemelj formulas, that 1 i i f Tet su] (lel<1) (9) rats Wu and Whttney where the integral with symbol C signifies its Cauchy principal value, and also defines the finite Hilbert transform of I(t) , as denoted by Hy [r] From this parametric description of the flow we derive the physical plane by quadrature Z(¢ ) = i et) ae dg (10) With the solution (7) - (10) in hand, we see that the chord 3 wetted arc-length S , angle of attack a , as wellas the drag D, lift L, etc. can all be expressed as integral functionals with argument functions [(&) and £8(£&), which are further related by (9). III. THE VARIATIONAL CALCULATION The general optimum problem considered here is the mini- mization of a physical quantity which may be expressed as a function- al of the form under M isoperimetric constraints i tg [T, Bic, ... o] = i Fg(T,8, &3c),... e) dé =Ap (12) where si are constants, L i ee eee Optimum Shapes of Bodtes in Free Surface Flows The original problem is equivalent to the minimization of a new func- tional I [Y, es), teas ae = i Ie = Ay ) , (13) where dg 's are undetermined Lagrange multipliers. We next seek the necessary conditions of optimality. Let [(&) denote the required optimal function which, together with its conjugate function B6(£) given by (9), minimizes I [T, B |. We further let 6 I'(&) denote an admissible variation of [ (£), which is Holder continuous, satisfies the isoperimetric constraints (12) and the end conditions (6). The corresponding variation in B(&) is found from (9) as 6p(&) = -H,[6P] eet oes (14) The variation of the functional I due to the variations 6T and 6B is AW. = pleat sta dle poset 8.5 Seat OC. | - bl RoR ce (15) where 6c,,'s are variations of parameters c,. For sufficiently small [6 r| ; |6 B | and [aca] , expansion of the above integrand in Taylors'series yields 1 z ARE Ve OYE ea 6 I + at 6°I FURS (16) 2 1 : eal Rebs =a 2 where the first variation 61 andthe second variation 6 I are 1 l hee [ ot a 66 Jatt sc [tars separ. (17) -l Lit? Wu and Whttney 2 2 2 i ie. LI ag at lei oe, (a8 d= + 6c Sc_ (2 dF [ Jc vc a (38) n m + cross product term between bc. and, 6T_or. .668., in which the subindices denote partial differentiation. The variations GE; sof depend on 6. as wellas on I’. For I to be mini- mum, we must have tied eck all Saar (19) yh 6 Pee SS we (20) in which 8B and 6 8B are understood to be related to [ and 6F by (9) and (14). Relation (19) assures I to be extremal, and with the inequality (20), I is therefore a minimum. Now, substituting (14) in (17) reduces it to 1 i ee | fe +H, [F, ]for( as + 6c. [ord ecaat (1 7)" =] I after inter-changing the order of integration, which is permissible under certain integrability conditions (see Tricomi 1957, § 4.3) which will be tacitly assumed to hold. Since the variations 6T (£&) 1118 Optimum Shapes of Bodtes tn Free Surface Flows and oc. are independent and arbitrary, the last integral in (17)! and the factor in the parenthesis of the first integrand must all vanish, hence (22) The nonlinear integral equation (22) combines with (9) to give a pair of singular integral equations for the extremal solutions, This is one necessary condition for I ia to be extremal ; it is analogous to the Euler differential equation in the classical theory. Presumably, cal- culation of the extremal solution ['(¢&) from (22) and (9) can be carried out with h,, ... Ayqy regarded as parameters, which are determined in turn by applying the M constraint equations (12). While we recognize the lack of a general technique for solving the system of nonlinear integral equations (9) and (22), we also notice the difficulty of satisfying the end conditions (6), as has been expe- rienced in many different problems investigated recently. The last difficulty may be attributed to the known behavior of a Cauchy integral near its end points which severely limits the type of analytic proper- ties that can be possessed by an admissible function I ( & ) and its conjugate function 6 (é&). Supposing that these equations can be solved for [I (£; Cy > Cor +e C,), we proceed to ascertain the condition under which this extremal solution actually provides a minimum of I [r] . From the second variation 6°1 we find it is necessary to have 1 [' a°F } ac;) dE>0" 5 (23) =i 1119 Wu and Whitney l 2 2 [ be 9) F2E pee Re * eet trey J dé > 0...) Gey By substituting (14) in (24a), interchanging the order of integration according to the Poincaré-Bertrand formula (Muskhelishvili, 1953) wherever applicable, it can be shown that (24a) can also be written as (24b) 1 1 | Foe ry, ae ‘: AOE) s(t) or(eyat E: B aaa -1 -1 di > 0 where g(é) = a aay soe vats CER)? eae Fig + HLF, I If we suppose that F s+ eee Pees are Holder continuous, and consider a special choice of 2 6T which vanishes for |e - EG | 76 9 tg bounded (orl B) and is of one sign for |é ~ Es <€, where §& ° is any interior point of (-1,1), then it can be shown that the first term on the left side of (24b) predominates, hence a necessary con- dition for (24 b) to hold true is the inequality g(&) > 0, or E espa Fag >0 ( || ra Yoo (24c) This condition is analogous to the Legendre condition in the classical theory. The preceding illustrates the method of solution of the ex- tremum problem by singular integral equations. We should reiterate that the integral equations are nonlinear unless F is quadratic in I 1120 Optimum Shapes of Bodtes tn Free Surface Flows and £6 . No general methods have been developed for the exact so- lution of nonlinear singular integral equations. Further, it may not always be possible to satisfy the condition IT ( + 1) = 0, which are required on physical grounds, With these difficulties in mind, we pro- ceed to discuss some approximate methods of solution. IV. LINEARIZED SINGULAR INTEGRAL EQUATION The least difficult case of the extremal problems in this general class is when the fundamental function F [r j B | is quadratic im ol vand 6. thatis F(T, 8, § sc.) = ar “;2pTe ees Zr Wage Gs in which the coefficients a,b, ... q are known functions of § and may depend on the parameters c,, ... Cy. It should be stressed that the above quadratic form of F can generally be used as a first ap- proximation of an originally nonlinear problem in which F is trans- cendental or contains higher order terms than the quadratic. With this approximation the integral equation (22) isthenlinearin IT and Bp , and reads aD be ep Se Be br ce rg) t ei< Ph (26) which combines with (9) to provide a set ot two linear integral equations, both of the Cauchy type. The necessary condition (24c), obtained from the consideration of the second variation, now becomes ae jos ee) S- <0 (ss ll eat als) oe (27) For the present linear problem (regarding the integral equations) two powerful analytical methods become immediately use- ful. First, the coupled linear integral equations (9) and (26) can always be reduced toa single Fredholm integral equation of the second kind. When the coefficients a(t), b( &) and c(é ) of the quadratic terms satisfy a certain relationship, the method of singular integral equations can be effected to yield an analytical solution in a closed form. (i) Fredholm integral equation RiAI Wu and Whttney By substituting (9) in (26), we readily obtain a(é) P(e) - b(&)Hy [T]+ H, [br] - , [c()H,[r]] = - -H, [al] - p(é) Upon using the Poincaré-Bertrand formula (with appropriate assump- tions) for the last term on the left side of the above equation, there results where il gi) = ae me This is a Fredholm integral equation of the second kind, with a regul- ar symmetric kernel, for which a well-developed theory is available. (ii) Singular integral equation method When the coefficients a,b,c, satisfy the following relation- ship a( &)otine(te) o>it) Dian bl &)S bo Sane bo = const , (29) the system of equations (26) and (9) can be reduced in succession to a single integral equation, each time for a single variable, and these equations are of the Carleman type, which can be solved by known methods (see Muskhelishvili 1953), yielding the final solution in a closed form. In the first step we multiply (9) by bo, and subtracting it L122 Optimum Shapes of Bodtes tn Free Surface Flows from (26), giving (30a) where 22) = ae ree et! , v(t) = -H, [a] er on) ae u (3 0b) After this Carleman equation for + is solved, a second Carleman equation results immediately upon elimination of B between the ex- pression for + and (9). The details of this analysis are given by Wu and Whitney (1971). These analytical solutions are of great interest, since in their construction there are definite, but generally very li- mited degrees of freedom for choosing the strength of the singularity, or the order of zero, of the solution [I[(é) and £6(£) at the end points §& = * 1. It is in this manner that the analytical behavior of the solution [(&) and £6(£) can be explicitly and thoroughly examined. This procedure will be demonstrated later by examples. V. THE RAYLEIGH-RITZ METHOD The central idea of this method, as in classical theory, con- sists in expansion of I(&) and £(£&) ina finite Fourier series B's ) = es Y 4, ©O8 a (31b) This expansion is noted to satisfy (9) automatically. The functional FIRS Wu and Whttney I [ ar 2 ] is now an ordinary function of the Fourier coefficients Te : Tv Roa(ele Boke. e~) = fee 8.5 cos Osc c )sin6 dé 1 n m 1 fo) SC sb ea Sy SEY (32) For I to be extremum, we require that d1/ O Fee = 0) (lz eyok, ma) ; (33) and a1 / dc, =(7*¢ Giger ie LOPE (34) These (m+n) equations together with M constraint equations (12) determine the m coefficients Yor Ym? " parameters c,... Cy, and M multipliers \, ,... M.- It should be pointed out, however, that the coefficients y;'s and parameters cj's generally appear in the expression for I(y,,c;) in a nonlinear or transcendent- al form, making their determination, by algebraic, numerical means or otherwise, extremely difficult even when their number is moderate- ly small, such as three or more. The preceding general theory will be further discussed and clarified with several specific examples in the presentation of this study. ACKNOW LEDGMENT This work was sponsored in an early stage by the Naval Ship System Command General Hydrodynamics Research Program, ad- ministered by the Naval Ship Research and Development Center under Contract Nonr-220(51), and partially by the Office of Naval Research, under Contract N00014-67-A-0094-0012. 1124 [1] [3] Optimum Shapes of Bodtes tn Free Surface Flows REFERENCES MUSKHELISHVILI,N.1., 1953, "Singular Integral Equations", Groningen, Holland, Noordhoff, TRICOMI, F.G., 1957, "Integral Equation", New York, Interscience Publ. WU, Th. Y. & WHITNEY, A.K., 1971, "Theory of optimum Shapes in Free-Surface Flows", Rept. No E 132 F. 1, Calif. Inst. Tech. Pasadena, California. 1125 Wu and Whttney (1‘o‘s ‘og paxts) Q wnwiuiw (6ui33!7) MO} AjiAngd (jeuolsuewlig-Z) swe[qoig TeostsAyg jo sojdwexy—T[ omsiq (S ‘7 paxis) GQ 6bosp wnwiuiw (6o4p ang) MO|} Aylangd db nN (S‘'g7 pexiy) 7 agi) wowixow Piste : (@ = 44) ao0juns Buludld 1126 Opttmum Shapes of Bodtes tn Free Surface Flows DISCUSSION Ernest O. Tuck Universtty of Adelatde Adelatde, Australta At a meeting like this it might seem strange to ask about existence and uniqueness solutions to mathematical problems but I think it is possibly relevant here. I wondered if the authors know in their examples whether they can expect a solution, on either physical or mathematical grounds ? REPLY TO DISCUSSION Arthur K. Whitney Palo Alto Research Laboratory, Lockheed Atreraft Corp. Pato Alto, Californias U.S.A. That is one of the unresolved questions in this minimisation technique, we simply do not know the answer at this point. If it turns out that solutions to the exact nonlinear equations do not exist, this still does not invalidate solutions by the approximate methods. It may mean, however, that as we take more and more terms in the appro- ximate solutions, these solutions do not converge. An existence proof, especially a constructive existence proof, would be very much desi- rable. t127 Wu and Whttney DISCUSSION William B. Morgan Naval Shtp Research and Development Center Bethesda, Maryland, U.S.A. In connection with your third examples, how do you plan to determine the point of separation on the suction side of the foil when the leading edge is blunt ? REPLY TO DISCUSSION Arthur K. Whitney Palo Alto Research Laboratory, Lockheed Atreraft Corp. Pato Alto, California, U.S.A. These problems are formulated as inverse problems, so you know the point of separation in all cases. Thatis, if [(é) is givenas the dependent variable, the hodograph variable is then determined and the physical plane, including the shape of the wetted foil surface, is given by a quadrature. REPLY TO DISCUSSION : Jtheodore Yi4Wu Caltfornta Instttute of Technology Pasadena, Caltfornta, U.S.A. May I add a few comments to Dr. Morgan's question ? For the lifting problem, I think we can also impose two conditions as iso- perimetric constraints for a fixed chord and a fixed arc-length. These constraints may provide a good method to overcome the difficulty due 1128 Opttmum Shapes of Bodtes tn Free Surface Flows to unknown position of the separation points. Suppose we start with K = 0 (K being defined as the ratio of the wetted arc-length to the chord length minus 1, or K = s/£ -1), then we know that at K=0, the two end points of the cavity boundary would be of the type of fixed de- tachment, at which point the curvature of the free streamline is sin- gular. As K is increased by giving more arc-length to the body pro- file, we hope the profile can be expanded in such a way as to arrive at the required optimum shape. When K reaches a certain positive value, one of the end points of the optimum profile would firts reach the state of smooth detachment (in the sense that the local curvature of the free streamline will be equal to that of the body at the detach- ment). Near this critical point (K = K, say) and from then on (K>K,) I think other physical quantities such as the viscous effects and the physical condition,that the pressure on the body remains now- here less than the cavity pressure, must be thoroughly examined from the final results predicted by the theory. This proposed procedure is to be carried out in the future study. Would this answer Dr. Morgan's question ? DISCUSSION Vsevolod V. Rogdestvensky Shtpbutldtng Instttute Lentngrad, U.S.S.R. I think you have done very interesting work, but there are many questions in this problem, In order to decide the problem in ge- neral, it is necessary to make good tests. I should like to ask, have you any comparison with experience ? Have you any experimental data ? EV29 Wu and Whttney REPLY TO DISCUSSION Arthur K. Whitney Palo Alto Research Laboratory, Lockheed Atreraft Corp. Pato. Alto, Californias U.S.A, No, we have not tested any of these optimum shapes experi- mentally. 1130 HYDRODYNAMIC CAVITATION AND SOME CONSIDERATIONS OF THE INFLUENCE OF FREE GAS CONTENT Frank B. Peterson Naval Ship Research and Development Center Bethesda, Maryland, U.S.A. ABSTRACT Hydrodynamic cavitation inception on an axisymme- tric body with a 5-cm diameter was measured ina standard water tunnel. Particulate matter and free- gas bubble size distributions were directly measur- ed immediately upstream of the bodies with a high- speed holographic technique and related to calculat- ed bubble trajectories. Discrimination between par- ticulate matter andgas bubbles was possible for dia- meters larger than 0.0025 cm. Inception was mea- sured acoustically and high-speed movies at 10,000 frames per second were taken to verify the type of cavitation present. The influence of headform sur- face chemistry was studied using plastic, copper, and gold-plated bodies with and without various types of colloidal silica coatings. Physical surface charac- teristics were checked with scanning electron mi- croscopy. All cavities observed in the water tunnel tests were approximately hemispherical in shapeand translated along the headform surface. When the results were compared with previously reported tests in a high- speed towing basin, it was concluded that the mea- suredfree stream gas bubbles in these standardtest facilities did not significantly contribute to the nu- cleation of cavitation when acoustic detection was used, Other recent research is summarized that de- scribes the de novo production of stable hydrophobic particulate in water through the mechanism of aera- HSI Peterson tion. These particulate are felt to have a major role in the cavity nucleation process. INTRODUCTION Over the years there has been continued discussion about the role of the air content in water on the cavitation inception process. Typical recent surveys on the subject have been by Eisenberg Big Holl [2] , Knapp [3] , Plesset [4] , and vander Walle [5] . Much of the discussion has been concerned with the determination of the nature of the "nuclei'' that are attributed to the onset of a vapor cavity. At the present time, no conclusive results have been reported that fully explain the relative importance of the free stream gas bubbles, the unwetted (hydrophobic) solid particles and the gas trap- ped in crevices on the test body. Presumably, each of these postu- lated nuclei sources will contribute to the formation of cavitation, with various degrees of relative importance. The actual importance of each during any given test will be dependent on the fluid and body characteristics and the pressure and velocity fields. What is presently needed is a series of definitive tests that would elucidate the role of the various types of nuclei as a function of the various controlling parameters. Before this series of tests can be performed, an adequate physical understanding must be deve- loped to recognize and plan a definitive experiment. It is the aim of this paper to assist in extending the presently available knowledge on the cavity nucleation process in hydrodynamic cavitation. Since the available literature on cavity nucleation has been surveyed by many writers, the contribution of the present paper will be concer- ned principally with the recent work performed at the Naval Ship Research and Development Center (NSRDC). The results of other investigators will be considered and compared to the extent that their work has a bearing on the interpretation of the observed phenomena, The recent cavitation research at NSRDC has been concerned with developing a better understanding of the role of the free and dis- solved gas content on hydrodynamic cavitation. The emphasis of the work reported here will be concerned with the importance of gas bub- bles in the free stream for the type of cavitation occurring on one headform shape at one velocity in a water tunnel. By restricting the test conditions in this manner, changes in parameters such as the * References are listed on page 1156. 1 Bs 74 Cavitation (Influence of Free Gas Content) free stream turbulence, body boundary layer characteristics, and body pressure distribution are minimized. Specific studies were performed to evaluate the importance of headform surface nuclea- tion sources. Surface treatment procedures were designed to reduce the surface nucleation capability of the body. In this way the role of free stream nuclei could be more clearly defined. The actual gas bub- bles and solid particles just ahead of the headform were recorded by high speed holography. The path and stability of the bubbles as they passed over the body were determined analytically. These cal- culations were necessary to determine through what cross-sectional area upstream of the body all bubbles must pass if they are to con- tribute to the visually observed cavities on the body. The inception condition was measured acoustically and high speed movies were ta- ken to verify the type of cavitation present. The headforms tested consisted of several bodies each of plastic (Delrin)* and metal (copper and gold plated copper) materials, Cavitation tests on several bodies of the same material gives a check on the surface machining accura- cy and the material variety assists in evaluating the role of surface originating nuclei. The results of these water tunnel tests were com- pared with previous tests of the same bodies in the high speed towing basin. The significant aspects of the towing basin are that essential- ly no free gas bubbles are present and the turbulence levels are very low. From all of these studies the importance of free gas bubbles on acoustically measured cavitation inception can be evaluated for at least the headforms tested. In the interest of introducing into the cavitation literature re- cent research results from other disciplines pertinent to cavity nu- cleation, a review of this work will also be given. Specifically, these results demonstrate a mechanism by which stable hydrophobic parti- cles can be produced in water by the process of aeration, EXPERIMENTAL FACILITIES, INSTRUMENTATION, AND PROCE- DURE All of the experimental studies were carried out in the standard test facilities at NSRDC [6]. The cavitation inception studies to be reported here were performed in the 12-inch water tunnel using normal tap water filtered to remove particles larger than25 wm, Deaeration was accomplished by passing the water through a standard design pac- * Delrin, Acetal Resin, manufactured by E.I. Dupont de Nemours and Co., Wilmington, Delaware. 1133 Peterson ked column desorber. The test body used throughout the work was from the series tested by Rouse and McNown igah This is the same series from which the headform used for the ITTC comparative tests was selected [8]. The bodies tested had a minimum pressure coeffi- cient, ss Fe » equal to 0. 82, a diameter of 5 cm, and were installed in the water tunnel as shown in Figure 1. An axisymmetric headform was selected as the test body for several reasons, First, inception measurements are relatively straight-forward, because the body is stationary and it is easy to manufacture with a high degree of accu- racy. Secondly, vortex cavitation is not present. Cavitation inception was detected acoustically for all of the results presented here. The measurements were made by locating a hydrophone inside the headform on its axis. Details of the equip- ment and operational characteristics can be found in references [9] and fi O}. The noise level of the facility was determined for a tunnel pressure slightly above that corresponding to a cavitation inception number, g; , equal to “oe, . The associated electronics were then adjusted so that all cavitation noise exceeding the tunnel background level would be indicated. The cavitation inception number gj, is defined as Po a CER Ta -P g| = where P,, and Vg, are the upstream pressure and velocity respecti- vely, p is the density, and P,, the water vapor pressure. The cri- teria for the actual inception was selected to be one acoustic event per second. The technique of acoustic detection of inception will be considered in more detail later in this paper. The test procedure used in the water tunnel studies was to in- stall the headform and then deaerate the water to the desired total dissolved gas content. This dissolved gas content was measured with a standard van Slyke apparatus. All tests were run ata free stream velocity of 9.1 meters per second. The test section pressure was adjusted in stages to produce a range of cavitation conditions as mea- sured acoustically. When the cavitation inception pressure was rea- ched, a series of holograms were made of the bubbles and solid particles in the water just upstream of the body. High speed photo- graphy at 10,000 frames per second and 20 microsecond exposure time were also taken for selected runs at the inception conditions. 1134 Cavitation (Influence of Free Gas Content) BUBBLE TRAJECTORY ANALYSIS Implicit in all discussions on the role of free stream gas bub- bles in the cavity nucleation process is that they will be transported into a sufficiently low pressure region to be available to nucleate the cavity. The mere existence of gas bubbles in water is not in itself sufficient to conclude a knowledge of their importance. For this reason the bubble trajectory and its radial dynamics must be evalua- ted. When this information is combined with the bubble size and population information, then a better understanding of the importance of these bubbles can be developed. The trajectory of a bubble in a flow field with large pressure gradients has been considered by Johnson and Hsieh [1 1] ,» Hsieh [12] , and Schrage and Perkins i 3] . The governing equations de- rived by these authors can be reduced to essentially the same form and contain the following assumptions: 1. The flow field is axisymmetric. 2. The bubble remains a sphere throughout its trajectory. 3. The fluid is assumed to be inviscid for the purpose of the flow velocities and pressures. 4. The bubble is assumed to be sufficiently small so that the flow field is not affected by the presence of the bubble. 5. The fluid is not taken to be inviscid with respect to the bubble, i.e., the bubble experiences a drag dependent on the relative velocity between bubble and fluid (see ref. 14). 6. Diffusion of gas and heat transfer through the bubble wall are negligible. The equation for the dynamics of bubble radius, r shown to be 2 3 2 r 4 *b - ci a % 12. P_+(P -P po acne ns Bes - P(x,y)} (1) b V s TAY 3 : dt p Tho Th Th eit can be t is time, p is the fluid density, Py is the vapor pressure of the fluid, b135 re) 6 Peterson a subscript denoting a value at the initial bubble position, is the surface tension of the fluid, P(x, y) is the external pressure of the fluid at (x, y), (x,y) refers to the location of the bubble in Cartesian coordi- nates. The vector equation of motion for the bubble moving in an axisymme- tric flow field, can be written as where —)> u 2 3 du_ il a eee Be bei Sage = Bids Ch wr (v-u) | v-u | (2) 4 dr cle B. 3 py lege LE Sie 2 he 2nrr3 VP, = mV + 21p rr (v-u) are unit vectors in the x, y directions, respectively, is the fluid velocity vector = vi + =a 5 is the drag coefficient, (see reference [1 4] im is the pressure due to flow, and is the pressure due to gravity. Using these equations, Schrage and Perkins 13] compared their analytical prediction of the bubble path with experiments in both rotating water and glycerin and obtained excellent agreement. A numerical study was carried out at NSRDC where the potential flow field around the headform was combined with equations I and 2 to determine the trajectory and radial dynamics of a free stream gas bubble. The description of the pressure and velocity field around the body was determined through the use of a computer program for po- tential flow around an axisymmetric body {1 5] : The most important aspect of these calculations was the deter- 1136 Cavttatton (Influence of Free Gas Content) mination of the region upstream of the body in which the bubbles would have to be located in order to produce cavitation. The results of the numerical calculations are shown in Figures 2 - 4 as the local pressure coefficient, Cp , experienced by the bubbles along the bubble trajectory. Figures 2 and 3 show the situation for a typical cavitation inception condition experienced in the 12 inch water tunnel with a metallic body. The bubble screening effect is easily seen, The 25 wm diameter bubble does not experience as lowa local pressure as the 50 um diameter bubble when they both start at the same point upstream. Correspondingly, the 25 wm _ bubble does not pass as close to the body as the 50 wm bubble and does not strike the body as soon. Figures 5 and 6_ show the variation in bubble diameter for some of the bubble sizes considered. None of the bubbles experienced ex- tremely rapid growth rates. For the range of bubble trajectories considered, all the bubble wall velocities were less than 0.1 meter per second. Once a bubble touched the body, the numerical method is of course not valid. However, it appears reasonable to assume that when the bubble touches the body, its translational velocity may decrease sufficiently to permit further volume increase. On this basis it was concluded from Figures 2, 3, 5, and 6, that all bubbles would have to be initially within the cross-sectional area of radius 3.75 mm upstream from the 5cm diameter headform for them to produce cavitation. For the purposes of further discussion, the bubbles out- side this area are assumed not to contribute to the cavitation on the body. The question still to be resolved is whether the bubbles that strike the body will in fact actually produce a vaporous cavity. Be- fore discussing this aspect of the problem, the numerical calcula- tions of the bubble trajectories over the same body with the same in- ception coefficient, 0, , but ata pressure approximately 3.4 times higher, should be considered. Figures 4 and 7 represent a typical inception condition when the same headform was tested in the high speed towing basin 16] . The experimental results from the basin were essentially the same as those obtained in the water tunnel and therefore the same og; was used in the calculations. The interesting result is that for the higher speeds in the basin, the bubble trajector- ies are slightly further from the headform and therefore the bubble diameters are correspondingly smaller. If the bubble strikes the headform, it strikes further downstream, From this result it can be concluded that if given identical bubble size distributions, then the rate at which bubbles produced cavities should be directly propor- tional to the velocity of the body in the basin or conversely, the up- 1137 Peterson stream velocity in the water tunnel. This conclusion assumes that the viscous effects, such as boundary layer separation, do not in- fluence inception. Further discussion on boundary layer separation will be deferred until later. Following the same reasoning, if the body size had been changed, then the number of freestream bubbles cavita- ting per unit time would vary directly with the square of the ratio of the body diameters. The key to all of the preceding discussion on bubble screening is whether in fact the free stream gas bubbles are responsible for cavitation. The remainder of this paper will be concerned with expe- riments specifically planned to extend our understanding of the role of gas content in water. VARIATION OF FREE GAS CONTENT AND BODY SURFACE QUALITY A. Criteria for Cavitation inception The commonly accepted criteria for the onset of vaporous ca- vitation is when a cavity grows "explosively", with the local pressu- re less than or equal to the vapor pressure (e.g. er and is ge- nerally considered the only true cavitation. On the other hand, gase- ous cavitation can occur at pressures either greater or lower than vapor pressure, with gas diffusion into the bubble possibly important and the growth rate of the bubble considered something less than "explosive''. However, these definitions are not specific enough for the purposes of the discussion here to delineate when in fact a cavity is growing "explosively", This problem was apparent to Hsieh [12] when he calculated bubble dynamics in the bubble trajectory. None of the bubbles he considered had what could be considered '"'explosive" growth, but in fact had a bounded maximum size. Therefore Hsieh arbitrarily defined a bubble to be cavitating when its diameter reach- ed a certain minimum ''visible''size. On the basis of the trajectory and bubble diameter calculations in the previous section for the NSRDC headform, it is felt that only bubbles actually striking the body could nucleate a vaporous cavity. This assumption is based on the observation through high speed pho- tography that the cavities appeared hemispherical in shape from the time of their first observation and translated along the body surface during both the growth and collapse. The "'visible'' size criteria is not applicable here since once a bubble strikes a body, the calcula- tions are no longer valid and in fact the visually observed cavity growth could correspond to gaseous cavitation. Since all of the cavi- ties observed in these studies were observed to be translating along 1138 Cavttatton (Influence of Free Gas Content) the surface, it is particularly important to have some means of di- scriminating between vaporous and gaseous cavities. If the cavity is truly vaporous, and grows "'explosively"', then the collapse should also be far from equilibrium with the local pressure field and should produce noise. This definition has been used by innumerable investi- gators. For headform studies, Saint Anthony Falls Hydraulic Labora- tory (SAFHL) [18] , [19] and NSRDC have regularly been using the acoustic radiation as the criteria for inception, It can be shown that the velocity of cavity collapse for nominal- ly hemispherical cavities scales approximately as the square root of the pressure difference across the cavity wall [20] ; [2 1] eins, if one assumes that the shape of the cavity during collapse is essentially the same for slight changes in this pressure difference, then the col- lapse velocities also would only experience slight changes. On this basis it will be assumed in this paper that the noise produced by the collapsing cavities will not be significantly affected by small changes in the pressure of the water tunnel, When the acoustic impedence between the water and the head- form material is changed, then the amplitude of the noise detected by the hydrophone will be affected. As one would expect, the peak noise amplitude will vary over a finite range. This has been experimentally shown by Brockett fi 0] for a headform made of Delrin which is a good impedence match to water. When the headform material is a metal, such as copper, then there is a poor acoustic impedence match and one would expect to detect a lower peak noise amplitude from the col- lapsing cavity. This material influence cannot be entirely cancelled out by adjusting the detection threshold of the electronics on the basis of background noise. Therefore, it is expected that the DELRIN head- forms will indicate cavitation at a somewhat higher water tunnel pres- sure than for a metal headform. This aspect is not of significant con- cern here because direct comparison of the cavitation inception num- ber, o;, for the two types of materials is not intended. The most im- portant concern is to determine how variations in the free gas bubble distributions affect the inception on a headform of the same material. B. Free Stream Bubble Size and Distribution Measurements The microscopic gas bubbles immediately upstream of the headform at inception conditions have been measured with a high speed holographic technique. This technique was selected because it appeared to be unique in its ability to (1) make direct measure- ments with no calibration required, (2) discriminate between bubbles 1139 Peterson and solid particles, and (3) record both bubble size and spatial di- stribution in a large liquid volume instantaneously. A mathematical analysis of the complete holographic process for bubbles and solid particles is given in Appendix A. The schematic representation of the holographic equipment at the water tunnel is shown in Figure 8. The resulting holograms obtained in these studies recorded the bubbles and solid particles contained within a liquid volume 5 cm in diameter and 15 cm long. A small magnified view of a hologram for 2-25 wm diameter wires and many bubbles and solid particles, is shown in Figure 9. As shown in Figure 10, typical exposure du- ration was 10 nanoseconds, The hologram is then used to produce the 3-dimensional image of the contents of the original volume. This volume was scanned with a traveling microscope and the size and location of the bubbles and solid particles recorded. Figures 11 and 12 show the appearance of a bubble and a solid particle as the micro- scope is moved away from the focussed position. For the optics u- sed in these studies, it was determined both analytically and experi- mentally that 25 um diameter was the smallest bubble size that could be reliably distinguished from a solid particle for the optical configuration used. Smaller bubbles could have been distinguished if different optics had been used but a sacrifice in the fluid volume recorded would have been necessary. The smallest size possible would have been approximately 10 um diameter because of the na- ture of this type of holographic process. Conclusions to be made later in this paper will show that the additional effort to measure smaller sizes was not merited. Typical bubble and solid particle size distributions are shown in Figure 13. A comparison can be made between the number of measured bubbles calculated to strike the body and the total dissolved gas con- tent of the water. This is shown in Figure 14 along with the corre- sponding go, and the headform material. High speed photography has shown that for a dissolved gas content referred to test section pres- sure, afar. , of 1.45 and o; = 0.61, approximately 1000 tran- sient hemispherical cavities per second were visible on the headform. However, when a/a-.= 0.22 and go, = 0.48 and there were only on the order of 10 visible hemispherical cavities per second on the headform. These observations are in general agreement with the cal- culated number of bubbles that would strike the body. The most si- gnificant result apparent in Figure 14 is that for changesin a/g TS around the saturation condition, very large changes in free gas con- tent will occur with very little change in oj . It appears that al- though 9; is less than \Spmin , the visible cavities do not produ- ce a significant amount of noise when they collapse. The small diffe- rence noted between metallic and plastic bodies is attributed to the 1140 Cavitation (Influence of Free Gas Content) large difference in acoustic impedence between the materials as pre- viously discussed in Section A, Brockett [9] ; [10] has also obser- ved, during studies with the same shape headform, that not all visible cavities produce noise. It should again be noted that the estimated rate at which bubbles struck the headform did not include bubbles less than 25 mum in diameter. Therefore, the estimates are to be considered low. In any event, it certainly appears that there were a sufficient number of bubbles available to account for the number of visible cavi- ties observed photographically. When the dissolved gas content of the water was reduced, some filtering of the water also occurred. Howe- ver, it can be seen from the particle size distributions in Figure 13 that when the gas bubble content of the water was reduced by over a factor of 10, little change occurred in the number of solid particles present. At this point no conclusions can be made concerning the effect of the presence of the solid particles in the water on the nucle- ation process. If they have a density greater than that of water, then they may have a trajectory which tends to direct them away from the low pressure region of the body. If their density is approximately that of water, then one would expect that these solid particles would have trajectories corresponding to the streamlines. In any event, it is clear that a large number of solid particles were always present and for the sake of completeness their size distributions are presented here, When the dissolved gas content, a/a Ts » was reduced below approximately 0.6, the number of gas bubbles were so few it became impractical to manually scan the image volume with a microscope. However, inception measurements were made and these are given in Table 1. The results indicate that the addition of new water into the tunnel may have had some effect, but the statistics are inadequate to verify this point. From the data one can also see that the typical de- crease in cavitation inception number, go, , occurs as the dissolved gas content is reduced. Based on the previous discussion, however, it is not readily apparent what type of nuclei are most affected by this change and further discussion of this result must also be deferred. C. Modifications to the Surface Characteristics of the Headforms Nuclei originating from gas trapped in crevices on the body surface have been postulated as one source of cavitation nuclei. There is sufficient experimental data available to show that under certain circumstances this type of nucleus can be a significant factor in cavity formation on a body [1] : [2] ; [22] . Therefore, the possibility exists that this nucleus source may have been a factor in the experi- ments reported here. Studies were carried out to evaluate this factor in two ways. First, headforms were plated with gold to minimize sur- 1141 Peterson face irregularities and corrosion, Second, an attempt was made to increase the wettability of the solid in order to promote wetting of microscopic crevices by the water. Electron probe microanalysis of the gold plated copper head- forms was carried out. In the secondary electron mode of operation no copper x-rays were found that would have indicated a pore in the plating. Although resolution was limited to lum, experience with gold plated materials of this type leads us to suspect that no pores of smaller size were present, A scanning electron microscope was used to study the surface features of both the plastic (DELRIN) and metallic (gold plated) bo- dies. The most significant surface feature on the gold surface was the scratch shown in Figure 15, All surface scratches were less than 0.5 wm across and shallow. All protuberances appeared to be less than 0.2 yum. The plastic surface shown in Figure 16 can best be characterized as consisting of a series of shallow scratches, the width of which are larger than the typical surface roughness element. High speed photography gave no macroscopic indication that cavities repetitively occurred from any single location. The gold plated bodies were coated with 1% colloidal silica and the plastic headform surface coated with a positive sol also of 1% concentration. This procedure is described in detail in Appendix B. The results of studies on various surface treatments can be summarized as follows : a. Gold plating a copper headform to give a smoother surface did not change a, , b. Colloidal silica coating on the gold plated surface did not change gs c. Use of a positive sol on the plastic headform did not change og It is concluded from these studies that cavity nucleation was not significantly affected by roughness elements or from gas trapped in hydrophobic crevices on the solid surface of the headforms. D. Boundary Layer Separation Separation of the boundary layer could have a strong influence on the local velocity and pressure distribution of the headform. As is 1142 Cavitation (Influence of Free Gas Content) well known, separation can occur in both laminar and turbulent boundary layers. In both cases, the separation will take place down - stream of the minimum pressure point on the surface in the region of a positive pressure gradient. A relatively crude experiment was per- formed using a fluorescent oil film on the headform surface [9] ; [16] . The result indicated that separation occurred at an X/D = 0.5, for velocities below 4.2 meters per second and a water temperature of 10°C. This is in agreement with tests performed at the California Institute of Technology with a hemispherical nose headform [23] : There, laminar boundary layer separation was also found to occur downstream from the minimum pressure point. From the high speed movies of the headform in the 12 inch water tunnel it was found that many bubbles were already visible at the minimum pressure point, This same result was apparent in the experiments run in the high speed basin and reported earlier [1 6] . On the basis of this discus- sion itis felt that boundary layer separation, if present, occurred sufficiently downstream to be of negligible influence on the inception observed in the experiments. E. Comparison Between Water Tunnel and High Speed Towing Basin Cavitation Studies In order to clarify the role of the free stream gas bubble in the cavitation nucleation process occurring in the water tunnel tests, it is worth while to compare results with those obtained in the high speed towing basin at NSRDC. As previously reported {1 6} , these same headforms were mounted on a strut and tested in the towing basin, The prodedure was to wait at the end of the basin for 45 mi- nutes prior to each run, In this period of time the basin water became very smooth and high speed photography was possible through its sur- face. It was found that the incipient cavitation number varied between 0.6 and 0.8. The higher values were again typical of the plastic (DELRIN) headforms and this is attributed to the difference in acou- stic impedence between the metal and plastic. In the towing basin the inception velocity was also determined with a hydrophone inside the body. Unlike the water tunnel tests, these acoustic results were found to agree with the high speed photography. From the data on the buoyant rise of bubbles in water [14] 3 it can be estimated that a bubble 4 wm in diameter will rise 270 mm in the 45 minute period. Larger bubbles will rise correspondingly faster. By any one of a number of theories for gas bubbles in water (e.g., [24] , [25] ) it can be shown that bubbles smaller than 4 um in diameter should have dissolved completely in a matter of minutes. This is supported by the experimental evidence of Liebermann [26] . 1143 Peterson Thus it appears that the probability of bubbles existing in the towing basin water immediately prior to a run, is extremely remote. Now, if in fact free stream bubbles are necessary for cavita- tion inception, then a dichotomy exists between the basin and the wa- ter tunnel studies. The measured free stream bubbles in the water tunnel canaccain for the visually observed cavities but not the acou= stic determination of cavitation inception. The towing basin acoustic determination of inception agreed with the basin high speed movies and the water tunnel acoustic inception determination. As already pointed out, the probability of free gas bubbles existing in the basin water is extremely remote although the dissolved gas content was approximat- ely 100 percent saturated. On the basis of these studies, it appears that the free stream bubbles contributed to the production of the visibly observed cavities on the headforms but were not necessary for the generation of those cavities that produced acoustic radiation during collapse. Justas numerable investigators have concluded before, the results of the studies reported here can also best be explained by the existence of a hydrophobic particle with gas trapped within a crevice ( [27] - [29]). There has been a considerable amount of research performed by in- vestigators in other research disciplinesthat has significantly increas- ed the plausibility of this postulated nucleation mechanism. Within the cavitation literature available to this writer, it appears that these new related research results have not been discussed. Therefore, the next section will deal specifically with this related research. STABLE HYDROPHOBIC PARTICLES IN THE WATER The concept of cavity nucleation by a hydrophobic particle in water has long been the subject of considerable discussion. The basic hypothesis is that a small quantity of gas is trapped in a crevice of a particle and stabilized by the surface tension of the water because the particle it- self is hydrophobic. This theory was first advanced by Harvey, etal [27, 28, 29] and has most recently been reviewed by Apfel [30] .A number of recent experiments have been carried out that indicate the hydrophobic particle may play an important role in the cavity nucle- ation process in water [30 - 33] . In keeping with the nature of this paper, a survey of the literature on this subject will not be attempted but rather only those references most pertinent to the discussion will be considered. One of the most detailed experiments was carried out by Greenspan and Tschiegg [32] with an acoustically excited cylindric- al resonator. They found that the cavitation threshold for unfiltered 1144 Cavttatton (Influence of Free Gas Content) water increased significantly as the dissolved gas content was reduced. However, after filtering the water through an 0.2 um filter, the threshold was then essentially independent of the air content for un- dersaturated water. For organic liquids, the threshold was high and was not affected by filtering. Hayward [3 1] used a ''tension mano- meter'' to produce a tension in the liquid of 0.15 bar. Various li- quids were tested by measuring the number of bars prepressuriza- tion a sample would have to be subjected before it could withstand the 0.15 bar tension in the device. Nine organic fluids, including a water-in-oil emulsion, were tested and all were found to withstand the test tension with no prepressurization required. Of the liquids tested, only water was affected by the prepressurization and Hayward concluded that only water contained cavitation nuclei capable of sta- bilization. A further result was that distilled water (of unstated qua- lity) and polluted river water both required approximately the same level of prepressurization. These experimental results are consider- ed typical of the efforts directed toward understanding the role of the particulate in the cavity nucleation process. In the case of hydrodynamic cavitation where the body is mo- ving in a stationary fluid or conversely, a fluid is moving pasta stationary body, an important consideration is how these hydrophobic particles are produced and why they remain suspended in the water. As has been pointed out by Plesset [4] , if the solid particles have densities in the range of 2-3 gm/cm? , then their radius must be on the order of 0.01 wm to remain suspended in quiescent water. On the other hand, unwetted particles of this size would require a tension on the order of approximately 100 bar to nucleate cavities. Before this subject of the Harvey model of cavitation nuclei is pursued further, some recent oceanographic research pertinent to this subject should be considered. Sutcliffe, et al [34] have found that aeration of filtered sea water will produce a suspension of inso- luble organic particles. Some of these particles eventually settled out after aeration but most always remained in suspension. A signifi- cant amount of this particulate was larger than the 0.43 mum pore size of the filter. It was found that large surface-active organic mo- lecules adsorb at the air/water interface of the bubble to produce a monomolecular layer. This layer can be aggregated into insoluble organic particles by folding into polymolecular layers to form colloi- dal micellae or by collapsing into fibers. Coalescence of these colloi- dal particles then produce a semistable suspension of organic mate- rial. Riley [35] has confirmed the Sutcliffe, et al, work by also producing through aeration insoluble particulates from the dissolved organic matter in the sea. He also found that the aggregates will in - 1145 Peterson crease in size by coalescence or further adsorption and eventually become indistinguishable from natural aggregates. The longest di- mension of typical newly formed aggregates was on the order of 2 Drapes Wallace and Wilson [36] have shown the effectiveness of concentrating dissolved organic compounds as particulates through aeration. They found that for their test protein solution of 5 parts per billion, aeration gave almost 100 percent recovery of the compound in the form of particulate. This is typical of the concentration of dissolved organic com- pounds in seawater: The sum of these compounds, however, can reach the part per million range, Similar results have been found by other investigators not specifically studying the de novo particulate production in water. For example, Liebermann [26] found in the course of studies on the so- lubility of air bubbles in water that the contamination at the interfac- ial boundary between the air and water had no effect on the diffusion of air into the water. After the addition of many organic compounds and surfactants to the water, he stated that ... ''no laboratory con- dition could be found in which the rate of bubble diffusion was signifi- cantly altered.'' Lieberman also showed that when a bubble in multi- ple distilled water collapsed on a chemically clean surface, a micro- scopic amount of residue remained. When the pressure was reduced to 1/4 bar, the residue quite frequently nucleated another bubble. In another series of experiments on the diffusion of gas out of a bubble, Manley [37] found results similar to Liebermann, In this work, also, bubbles collapsing in distilled water left a small deposit of impurity. From the above discussion, it is apparent that in the typical cavitation test facilities, there should be no difficulty in producing particulate capable of cavitation nucleation. These can remain suspen- ded in quiescent water and can readily be produced whenever at least some degree of aeration of the water takes place. CONCLUSIONS The general objective of this work was to develop a better un- derstanding of the role of the free and dissolved gas content in water on the nucleation of hydrodynamic cavitation. The means by which this was accomplished was to use only one type of body, a headform, 1146 Cavttatton (Influence of Free Gas Content) in the natural existing environment of standard test facilities with emphasis on the measurement of flow conditions and the control of headform surface condition. This simple body produced only discrete cavities translating along the surface. From these results and a com- parison with the pertinent literature certain conclusions can be in- ferred. The results substantiate what other investigators have found in that a very precise definition of inception is necessary. When noi- se is used as an inception criteria, then it was shown that free gas bubbles were not specifically needed for the nucleation of the noise producing transient cavities under discussion in this paper. Hydro- phobic particles can function as an adequate source of nuclei. The dissolved gas is important because it can affect, through the mechanism of diffusion, the amount of gas trapped in hydrophobic particles for a given pressure history of the water. Conversely fora given dissolved gas content, changes in the normal pressure history of the water will also affect the ability of these hydrophobic particles to nucleate cavities. For the experiments reported here, body surface nucleation of cavitation was not considered a significant influence. If a material such as teflon is used which is hydrophobic and known to be porous on a microscopic scale, then surface nuclei could in fact be the controlling source. New stable nuclei can be generated in the typical test facility water whenever a gas/water interface is produced because of local adsorption at the interface of dissolved organic material. In a water tunnel this could occur during the filling process, by the introduction of locally supersaturated water or even during the deaeration process. In both water tunnels and towing basins this could occur during the actual tests where bubbles of one form or another are produced. In any event, the persistance of these hydrophobic particles can be ex- pected unless very special water treatment procedures are followed. If either 2- or 3- dimensional boundary layer separation occurs, g; may be affected but the type of cavity nucleus initially responsible may not be important if an attached cavity eventually forms. For flow fields to cavitate when nuclei mobility across stream- lines is required - suchas a vortex - then the free gas content of the water can be expected to be of prime importance. But here again, care must be taken to specifically define whether cavitation is based on visual or acoustic observations. 1147 Peterson If scaling of cavitation inception from a model to a prototype is required, then the detailed properties of the flow field must be considered in conjunction with a consideration of the type of nuclei controlling the inception process. The essential aspect of these conclusions are of course not original in this paper, but it was the attempt of this paper to add addi- tional physical basis for their validity. ACKNOWLEDGMENTS Many individuals at the Naval Ship Research and Development Center have assisted in obtaining the results reported in this paper. Specific recognition should be given to Dr. H. Wang and Mr. C, Dawson for their contribution in developing the computer program used in the bubble trajectory analysis. This work was authorized and funded by the Naval Ship Systems Command under its General Hydromechanics Research Program, Task SR 023 0101. APPENDIX A ANALYTICAL EVALUATION OF THE HOLOGRAPHIC PROCESS FOR A BUBBLE When a light beam is incident on a bubble, some of the light is reflected at the first surface. However, a significant amount of light is refracted at the first surface and eventually passes out through the bubble. In Figure Al, several rays are shown. As shown by Davis [3 8] , ray 2 gives the largest contribution to the transmitted energy for 0° <0< 40°. This information will be used to represent the light passing through the bubble in the following calculations. In order to differentiate between a bubble image and an opaque spherical particle image, the light transmitted through the bubble must be observed. Thus, the holographic process must be evaluated to determine how the transmitted light can be expected to influence the holographic reconstruction of the bubble image. The general method will be an extension of the method used by DeVelis et al., for solid particles [39] ; The wave equation in vector form 1148 Cavttatton (Influence of Free Gas Content) 1 oo? Vii et) 1A favient = describes the propagation of optical monochromatic radiation, where Vtdbdetes. Sle ee ~— and W(x) is the complex amplitude. Physically, the wave amplitude will vary as Re \w(x)e =e ny If the operations on V(x, t) are linear and only the long time average is required, then the physical quantity is the real part of the final expression. Thus, for our application, the wave equation can be transformed to the Helmoltz equation lege athe Ee eclonladen aur00 (2A) 27 w : where the wave number k =—>— =-@-. Here we have applied the restriction that the radiation is essentially monochromatic. The so- lution of equation (2A) can be written in integral form with G(£é :x) the appropriate Green's function, — where w(é) = g(&) | onthe surface s, and SCS ay ee ord, Ey with i and j unit normal vectors. S isa plane perpendicular to the Z axis and its outer normal is in the direction of negative Z. The Green's function for this case is ikr cree a). e d 4 mr where PS hf Ze +|&- -x | 2 : 1149 Peterson Using the paraxial approximations, Z>> )F-x | , and the Fraunhofer (i.e., far field) approximation klé |? ~ 0, then it can be shown that 2Z ; - oe ee eS Ss ve) =Giy tte oF se g() ez **aF (44) The boundary conditions, e(£) , selected to represent the bubble are eH =O +g + 5,08 sagpye! 2p CRY le BEEP Heer ae (5A) nes 1; Pel?! where Dy) o, Ele The first 2 terms of (5A) represent an opaque object andthe3rd term is an equivalent to a lens with a negative focal length f, attenuation factor a , and Gaussian transmittance distribution. The use of a negative lens is an approximation to ray 2 of Figure 1A, It is assu- med here that a uniform plane wave is incident on the object plane (€). Substituting g, ) in equation 4A gives i.e., a plane wave. . . 7—* —? . . The second integral gives W2 (x) when g,(&) is substitut- ed into 4A. Fora circular disc of radius [f|=fo ifo _ikz ikp* ¥, &) = Sues ela ved [225 4 pbo/Z) (7A) where p= |x| —, The thirdintegral W, (x) can be shown to be s ee Des eae As ree v, @) z Ee ic elke JtnZ a7 715 git,P? (ga) 1150 Cavitation (Influence of Free Gas Content) By combining terms, ‘ - , > vs) oe te aikz u Cop ic3 eicsP (9A) where ak i MEN wae at clef 2e)® Co oes (22)? (10A) 2 a2 + (k/2f)2 E eu tan” (k /2fa) i k 2 rae ot ee + (k/2ay] */22) = ee + wh See “4 It should be noted that equation (8A) was evaluated with the assi- stance of reference [40] . Thus the complete wave amplitude di- stribution in the xX plane, a distance Z away from the object plane, cr is 2 ik ikp2 /2 -c. p*-ic. ic] (11A w (x) Se aes pau cae = J, (kp Lo/Z) -ic,e ee teas it ) Since both photographic emulsions and the human eye are square law detectors, the quantity actually measuredis w (x) w (x) * the intensi- 2 oe ae a sin(kp?/2Z) J, (kpbo/Z) + (Lo/p)* I” (kpLo/Z) - ela ie sin(c, - Coa} (12A) cls c, J, (kpLo/Z)e 2 Pcos(k p2/2Z ie, = c, P*) +2 prtcsee For photographic emulsions it has been found that after proper ex- posure and processing the emulsion density will vary linearly as fol- lows, 1151 Peterson D = 6 log 10 I A BD) =) leee a Ss where 6 = constant I = incident intensity I = transmitted intensity As Now, since wave amplitude varies as v.(1)° us iT then pam 0 A a a i i t isp -t : : Mein - 6/2 and for a unit amplitude incident wave, D. = I. If 6 is positive a photographic ''negative'' is produced. Conversely, if 6 is negative a photographic ''positive'’ is produced. The holographic process has been analytically described for the production of the hologram, If on the hologram a plane wave is incident, then we have the same analytical situation previously de- scribed, The new boundary condition is the variation in amplitude of the incident radiation, a. [vay ey ae where YW™* is given by equation (12A) and A is the plane wave amplitude outside the hologram diffraction pattern. (yw *)-4/2 has the form (1-X)Vandif |X| < 1 and sufficiently small, taking only the first 2 terms of a series expansion gives Ae ees = x (14A) Thus, the use of the integral solution to the Helmholtz equation, (4A) with the boundary condition given in (14A), will result in an image of the original object. 152 Cavitation (Influence of Free Gas Content) From equation (4A) it is apparent that the wave amplitude in the hologram plane is essentially a Fourier transform of the wave amplitude distribution in the object plane. With appropriate change of sign, the process of going from the hologram plane to the image plane is just an inverse Fourier transform. The evaluation of the image intensity distribution has been performed numerically. Thus, by the use of a computer program, the influence of hologram size, emulsion signal to noise ratio, and many other factors can be studied analytically. The effects of either the holographic process or the test facility can be estimated prior to the actual physical measurement. The following constants were used in the calculations for Figures A2 - A4, Z 1 100.00 mm k 9045/mm u 0.03 mm ho a oe yee) p = 7 (i.e., the limit of integration) f = -0.3 mm a = 1080 6 =" \ 4 A = 4 As the image focusing distance, Z,5 , is changed from 99.7 mm to 100.0 mm and then to 100.3 mm, it can be seen in Figures A2 through A4 respectively, the focusing property of the hologram. When Zs equals 100.0 mm, then the bubble and opaque sphere shape are in focus, The light passing through the bubble produces an in- terference pattern within the bubble outline. For Z> equal to 100.3 mm, the bubble shape is no longer in focus, but the apparent point source of the light passing through the bubble is in focus. This is the distinguishing property of a bubble image in contrast to that of an opaque sphere. 1153 Peterson APPENDIX 5B APPLICATION OF COATINGS TO THE HEADFORM As a result of a previous study at NSRDC, [22] » it was concluded that through the use of the principles of surface chemistry, the number of cavity nucleation sites on the solid surface could be significantly reduced. In order to assist in evaluating the role of free stream originating nuclei, it was necessary to determine whe- ther or not surface nucleation sites were contributing to acousti- cally measured cavitation inception on the headform, The underlying objective of the surface coating procedure was to enhance the wettability of the solid on a microscopic scale. The coating selected was one formed by the application of colloidal silica, As is well known, [4 1] » amorphous silica has a very low interfa- cial surface energy in contact with water. This is basically because the atomic structure of water is quite similar to silica. When silica dissolves in water, the process involves simultaneous hydration of the Si0, surface and depolymerization, This leads to the formation of monosilicic acid, (SiO, ) ts 2m (H,0) = m Si(0H), From monosilicic acid colloidal particles of silica can be produced. On the surface of each particle a monolayer of water is chemisorbed that can only be removed.by heating a dried surface to 600°C. Itis also known that these particles will have a negative charge in an alkaline medium. The concept of the coating process is to utilize this negative charge and the colloidal particle dimensions to microscopically coat a solid surface. The surface selected was gold that was recently plated on the headform. The headform was flushed with spectro- scopically pure acetone and then cleaned in a special chamber with steam produced from a potassium permanganate solution, Following cleaning, a 1 percent solution of colloidal silica* was put in the chamber and a positive potential applied to the headform. The head- * Ludox SM, Colloidal Silica, manufactured by E.I. Dupont de Nemours and Co., Wilmington, Delaware. 1154 Cavitation (Influence of Free Gas Content) form was dried in the chamber and then removed. When a small quantity of water was applied to the surface, it immediately spread and then appeared to dry as a film. Later applications of water also spread over the surface. Plastic bodies made of Delrin were also coated with colloidal silica. However, since the plastic has a weak negative charge na- turally occuring on the surface, a special positively charged colloi- dal silica** was used. After cleaning the surface and applying the positive sol, the water also spread over the solid surface and appear- ed to dry as a film. Both the plastic and the gold plated bodies were then immersed in a container of water distilled from a potassium permanganate solution to minimize organic surfactant material from contaminating the coatings. The bodies were installed underwater while still in the pure water of the container. The test results comparing the coated with the uncoated bodies are given in Table l. ** Positive Sol 130M, manufactured by E.I, Dupont de Nemours and Co., Wilmington, Delaware. Li55 LJ [5] [6] [io] Peterson REFERENCES EISENBERG, P., ''Environmental and Body Conditions Governing the Inception and Development of Natural and Ventilated Cavities." Appendix I, Report of Cavitation Committee, Proceedings, 12th International Towing Tank Conference, Rome, 1969, pp. 346-350. HOLL, J.W., ''Nuclei and Cavitation", Journal of Basic Enginee- ring, Trans. ASME, Series D, vol. 92, No 4, 1970, pp. 681-688. KNAPP, R. T.; “Cavitation and Nucter", Trans, ASME, Vol. ge, 1958, pp. 1315-1324, PLESSET, M.S., "The Tensile Strength of Liquids", pp. 15-25, Cavitation State of Knowledge, ed. J.M. ROBERTSON and G.F. WISLICENUS, ASME, 1969, New-York. van der WALLE, F., "On the Growth of Nuclei and the Related Scaling Factors in Cavitation Inception", Netherlands Ship Model Basin, Laboratory Memorandum No. V2, Wageningen (1963). See also International Shipbuilding Progress, Vol. 10, No. 106, 1963, pp. 195-204. VINCENT, da C., ''The Naval Ship Research and Development Center'', Naval Ship Research and Development Center Report 3039, June 1969. ROUSE, H. and McNOWN, J.S., ''Cavitation and Pressure Distribution, Headforms at Zero Angle of Yaw'', State University of Iowa Studies in Engineering, Bulletin 32, 1948. JOHNSSON, C-A., ''Cavitation Inception on Head Forms. Further Tests", Appendix V, Report of Cavitation Committee, Procee- dings, 12th International Towing Tank Conference, Rome, 1969, pp. 381-392. BROCKETT, T., ''Some Environmental Effects on Headform Cavitation Inception", Naval Ship Research and Development Center Report 3974 (in review). BROCKETT, T., ''Cavitation Occurrence Counting - Comparison of Photographic and Recorded Data"', ASME Cavitation Forum, Evanston, Illinois, June 1969, pp. 22-23. 1156 [11] [12] [13] [14] [15] [16] Liz] [is] [19] [20] Cavitation (Influence of Free Gas Content) JOHNSON, V.E., Jr and HSIEH, T., ''The Influence of the Trajectories of Gas Nuclei on Cavitation Inception", 6th Sym- posium on Naval Hydrodynamics, October 1966, Washington, D;.G.5.. pp. 4.63.-182. HSIEH, T., ''The influence of the Trajectories and Radial Dynamics of Entrained Gas Bubbles on Cavitation Inception", Hydronautics, Inc. Report 707-1, October 1967. SCHRAGE, D. L., and PERKINS, H.C., Jr, " Isothermal Bubble Motion Through a Rotating Liquid", Journal of Basic Engineering, Trans. ASME, Series D, Vol. 94, no. 1, 1972, pp. 187-192. HABERMAN, W.L. and MORTON, R.K., ''An Experimental Investigation of the Drag and Shape of Air Bubbles Rising in Various Liquids'', David Taylor Model Basin Report 802, September 1953. HESS, J.L., and SMITH, A.M.O., ''Calculation of Potential Flow About Artitrary Bodies", Progress in Aeronautical Sciences, Vol. 8, 1967. PETERSON, F.B., ''Water Tunnel, High Speed Basin Cavita- tion Inception Comparisons", 12th International Towing Tank Conference, Proceedings, Rome, 1969, pp. 519-523. EISENBERG, P., ''Cavitation Dictionary"', Appendix VI, Report of Cavitation Committee, 13 th International Towing Tank Con- ference, Berlin/Hamburg, September 1972. SCHIEBE, F.R., ''The influence of Gas Nuclei Size Distribu- tion on Transient Cavitation Near Inception", Project Report No. 107, St. Anthony Fall Hydraulic Laboratory, University of Minnesota, May 1969. SCHIEBE, F.R., ''Cavitation Occurrence Counting - A New Technique in Inception Research", ASME Cavitation Forum, New York, November 1966. CHAPMAN, R.B., and PLESSET, M.S., ''Nonlinear Effects in the Collapse of a Nearly Spherical Cavity in a Liquid", Journal of Basic Engineering, Trans. ASME, Series D, Vol. 94, No. 1, March 1972, pp. 142-146. P157 [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] Peterson NAUDE, C.F. and ELLIS, A.T., ''On the Mechanism of Cavi- tation Damage by Nonhemispherical Cavities Collapsing in Contact with a Solid Body", Journal of Basic Engineering, Trans, ASME, Series D, Vol. 83, No. 4, December 1961, pp. 648-656. PETERSON, F.B., ''Cavitation Originating at Liquid-Solid Interfaces'', Naval Ship Research and Development Center Report 2799, September 1968. ARAKERI, V., ''Water Tunnel Observations of Laminar Bound- ary Layer Separation Using Schlieren Technique", ASME Cavitation Forum, San Francisco 1972, pp. 29-30. EPSTEIN, P.S., and PLESSET, M.S., ''On the Stability of Gas Bubbles in Liquid-Gas Solutions", Journal of Chemical Physics, Vol. 18, 1950, pp. 1505-1509. WARD, C.A., 'On the Stability of Gas-Vapour Bubbles in Liquid-Gas Solutions", University of Toronto, Department of Mechanical Engineering Report TP-7105, May 1971. LIEBERMANN, L., ''Air Bubbles in Water", Journal of Applied Physics, vol. 2o, No. 2, 1991, pps 2Oo-20 1 HARVEY, E.N. etal., "On cavity Formation in Water", Jour- nal of Applied Physics, Vol. 18, 1947, pp. 162-172. HARVEY, E.N., etal., ''TRemoval of Gas Nuclei from Liquids and Surfaces", Journal fo American Chemical Society, Vol. 67, No. 1,. 1945, pp. 156-157. HARVEY, E.N., etal., "Bubble Formation in Animals; Physical Factors", Journal of Cellular and Comparative Phy- siology, Vol. 24, 1944, pp. 1-22. APFEL, R.E., ''The Role of Impurities in Cavitation-Threshold Determination", Journal of Acoustical Society of America, Vol. 48, No. 5, (part 2), 1970, pp. 1179-1186. HAYWARD, A.T.J., ''The Role of Stabilized Gas Nuclei in Hydrodynamic Cavitation Inception", National Engineering Laboratory Report 452, May 1970. 1158 Cavttatton (Influence of Free Gas Content) [ 32 ] GREENSPAN, M, and TSCHIEGG, C., ''Radiation-Induced Acoustic Cavitation; Apparatus and Some Results", Journal of Research National Bureau of Standards, 71C, No. 4, 1967, pp. 299-312. [33] | MESSINO, C.D., SETTE, D., and WANDERLINGH, F., "Effects of Solid Impurities on Cavitation Nuclei in Water", Journal of Acoustical Society of America, Vol. 41, No. 3, 1967, pp. 573-583. [ 34 | SUT CLIE FE, Wi... Jz, BAYLOR, EB Ro, and MENZEL. D. W. "Sea Surface Chemistry and Langmuir Circulation", Deep-Sea Research, Vol. 10, no, 3, 1963, pp.r2a3-243. [35 | RILEY, G.A., "Organic Aggregates in Sea Water and the Dynamics of Their Formation and Utilization", Limnology and Oceanography, Vol. 8, No. 4, 1963, pp. 372-381. [36] “WALLACE, G.T., Jr., and WILSON, D,F., "Foam Separa- tion as a Tool in Chemical Oceanography", Naval Research Laboratory Report 6958, November 1969, lee val MANLEY, D.M.J.P., ''Change of Size of Air Bubbles in Water Containing a Small Dissolved Air Content", British Journal of Applied Physics, Vol. 11, January 1960, pp. 38- 42, [3 8 | DANVIS;’ GYEet, "Scattering of Light by an Air Bubble in Water", Journal of the Optical Society of America, Vol. 45, No. 7, 1955; pp"5722581 . [39] DeVELIS, J.B., PARRENT, G.B., and THOMPSON, B.J., "Image Reconstruction with Fraunhofer Holograms", Journal of the Optical Society of America, Vol. 56, No. 4, 1966, pp. 423-427. [40 | GRADSHTEYN, I.S. and RYZHIK, I.M., Table of Integrals, Series and Products, Academic Press, New-York, 1965, p. 485. [41] ILER, R.K., The Colloidal Chemistry of Silica and Silicates, Cornell University Press, Ithaca, 1955, 1159 o On CO] PW DN = NY NY NIN @ &@ @ ale |= 2 =a | oe WON |]=/0 O86 On OI fF WN || Peterson TABLE 1. ACOUSTIC INCEPTION ON HEADFORMS (V = 9.1 meters per second; Acoustic event rate = 1 per second) HEADFORM COATED ey DELRIN DELRIN DELRIN DELRIN DELRIN DELRIN DELRIN DELRIN DELRIN DELRIN DELRIN NO YES YES YES YES 149 149 220 49 145 159 166 88 162 21 44 152 81 91 42 34 1160 REMARKS NEW TUNNEL WATER SAME TUNNEL WATER AS RUN 1 SAME TUNNEL WATER AS RUN 1 SAME TUNNEL WATER AS RUN 1 SAME TUNNEL WATER AS RUN 1 NEW TUNNEL WATER NEW TUNNEL WATER SAME TUNNEL WATER AS RUN 7 NEW TUNNEL WATER SAME TUNNEL WATER AS RUN 9 SAME TUNNEL WATER AS RUN 9 NEW TUNNEL WATER NEW TUNNEL WATER SAME TUNNEL WATER AS RUN 13 SAME TUNNEL WATER AS RUN 13 NEW TUNNEL WATER SAME TUNNEL WATER AS RUN 15 SAME TUNNEL WATER AS RUN 15 SAME TUNNEL WATER AS RUN 15 SAME TUNNEL WATER AS RUN 15 SAME TUNNEL WATER AS RUN 15 SAME TUNNEL WATER AS RUN 15 SAME TUNNEL WATER AS RUN 15 Cavitation (Influence of Free Gas Content) INANAARAR ANN HEADFORM , ae. a: j oA * OP ZZZZZZZIL — QO ee oil a es a ” | LL oT of raagsati' Li SY Pee SELIEAA PITA tt ey PLLLILELEBEPELEEOL EEE EE, ‘a ¢ “i AS) ml ZZ772 zt i; CIRCULAR TO RECTANGULAR LONGITUDINAL SECTION CONTRACTION OF TEST CHAMBER FLAT OPTICAL PORT SCALE: 3/4” = 1'-0" FIGURE 1. MODIFIED 12” WATER TUNNEL 1161 Peterson INITIAL BUBBLE DIAMETER ————— 25 im 50 um O = TOUCH BODY INITIAL UPSTREAM LOCATION = 3.75 mm OFF AXIS P upstream = 0-296 bar Vupstream ~ 2:1 m/sec BODY DIAMETER = 50 mm ON BODY SURFACE PRESSURE COEFFICIENT (C,) AT BUBBLE LOCATION AXIAL DISTANCE FROM FACE OF HEADFORM (mm) FIGURE 2. BUBBLE DIAMETER ALONG BUBBLE TRAJECTORY: INITIAL UPSTREAM LOCATION OF 3.75 MM OFF AXIS; LuG2 Cavitation (Influence of Free Gas content) O =TOUCH BODY oa INITIAL UPSTREAM LOCATION = 5 mm OFF AXIS — — INITIAL UPSTREAM LOCATION = 6.25 mm OFF AXIS P UPSTRE aM= 0.296 bar Vurstream ~ 2-1 m/sec BODY DIAMETER = 50 mm ON BODY SURFACE PRESSURE COEFFICIENT (C,) AT BUBBLE LOCATION “2.5 0 -25 -5.0 AXIAL DISTANCE FROM FACE OF HEADFORM (mm) FIGURE 3. PRESSURE COEFFICIENT VARIATION ALONG BUBBLE TRAJECTORY: INITIAL UPSTREAM LOCATION OF 5.0 MM, 6.25 MM: VUPSTREAM = 91 M/SEC 1163 PRESSURE COEFFICIENT (C,) AT BUBBLE LOCATION Peterson © =TOUCH BODY —— INITIAL UPSTREAM LOCATION = 5 mm OFF AXIS — — INITIAL UPSTREAM LOCATION = 6.25 mm OFF AXIS PupsTREAM ~ 1-06 bar Vv BODY DIAMETER = 50 mm upstream ~ '8 m/sec ON BODY SURFACE “2.5 0 -2.5 -5.0 AXIAL DISTANCE FROM FACE OF HEADFORM (mm) FIGURE 4. PRESSURE COEFFICIENT VARIATION ALONG BUBBLE TRAJECTORY: INITIAL UPSTREAM LOCATION OF 5.0 MM, 6.25 MM; Vypstpeam = 18 M/SEC 1164 Cavitation (Influence of Free Gas Content) 10! © = TOUCH BODY —— INITIAL UPSTREAM LOCATION 5.0 mm OFF AXIS —-- INITIAL UPSTREAM LOCATION 6.25 mm OFF AXIS 19° E 4 INITIAL DIAMETER 250 um Ww = Ww = 1071 a INITIAL DIAMETER 50 pm a a INITIAL DIAMETER 25 um > = a 10-2 s VupsTREAM = 9.1 m/sec 0; = 0.65 Comin = 0-82 10-3 10 75 5 2.5 0 -2.5 -5 -75 AXIAL DISTANCE FROM FACE OF HEADFORM (mm) FIGURE 5. BUBBLE DIAMETER ALONG BUBBLE TRAJECTORY: INITIAL UPSTREAM LOCATION OF 5.0 MM, 6.25 MM OFF AXIS; V ieee ad WU SEC 1165 BUBBLE DIAMETER (mm) 10-2 10°3 10 Peterson O =TOUCH BODY —— INITIAL UPSTREAM LOCATION = 3.75 mm OFF AXIS INITIAL DIAMETER 250 um INITIAL DIAMETER 25 um / Vupstream ~ 9-1 m/sec 0; = 0.65 Comin Cc = -0.82 Pmin 7.5 5 2.5 0 -2.5 -5 -7.5 AXIAL DISTANCE FROM FACE OF HEADFORM (mm) FIGURE 6. PRESSURE COEFFICIENT VARIATION ALONG BUBBLE TRAJECTORY: INITIAL UPSTREAM LOCATION 1166 Cavitation (Influence of Free Gas Content) O =TOUCH BODY —— INITIAL UPSTREAM LOCATION = 5 mm OFF AXIS —— INITIAL UPSTREAM LOCATION = 6,25 mm OFF AXIS INITIAL DIAMETER 250 um INITIAL DIAMETER 50 um BUBBLE DIAMETER (mm) Vv upstTREaM ~ '8 m/sec G, = 0.65 Comin = -0.82 10 75 5.0 2.5 0 -2.5 -5.0 75 AXIAL DISTANCE FROM FACE OF HEADFORM (mm) FIGURE 7. BUBBLE DIAMETER ALONG BUBBLE TRAJECTORY: INITIAL UPSTREAM LOCATION OF 5.0 MM, 6.25 MM OFF AXIS; V upstream ~ '8 M/SEC 1167 Peterson WATER FLOW TEST SECTION WALL GLASS Y RUE PORT LASER A phtecby BEAM EXPANDER 15 CM PHOTOGRAPHIC PLATE HEADFORM FIGURE 8. BASIC SCHEMATIC FOR BUBBLE MEASUREMENT WITH FRAUNHOFER HOLOGRAPHY 1168 Cavttatton (Influence of Free Gas Content) FIGURE 9 MAGNIFIED VIEW OF A HOLOGRAM FOR TWO 25m WIRES, BUBBLES, AND PARTICULATE NSRDC FIGURE 10 RUBY LASER PULSE SHAPE NSRDC 1169 Peterson % om, A bar 60 um Diameter Defocussed 0.3 mm, Focussed FIGURE 11 - BUBBLE IMAGE ly A 70 u»m Diameter Defocussed 0.3 mm Focussed FIGURE 12 - SOLID PARTICLE IMAGE 1170 NUMBER OF BUBBLES AND Cavitation (Influence of Free Gas Content) @ SOLID PARTICLES O BUBBLES a/a,, = 1.62 g, = 0.71 VOL MEAS = 67.0 cm* DELRIN a/a,., = 1.49 G; = 0.65 VOL MEAS = 48.9 cm? a/ay = 1.32 0; = 0.66 VOL MEAS = 77.7 cm? SOLID PARTICLES N —) ala, = 0.88 0; = 0.69 VOL MEAS = 102.8 cm? DELRIN a/@.7, = 0.63 o; = 0.62 10 cu VOL MEAS = 58.7 cm3 0 0 25 50 75 100 125 150 175 £200 225 «250 «275 DIAMETER OF BUBBLES OR SOLID PARTICLES (um) FIGURE 13. BUBBLE AND SOLID PARTICLE DISTRIBUTIONS ETL Peterson fon (DELRIN) A 0.65 (AU) 0.66 (CU) yY via w 4 o, (MATERIAL) Ao 8 i] 0.62 (CU) TT £ 0.69 (DELRIN) BUBBLES PER SECOND STRIKING HEADFORM _ =) ro) 10 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 TOTAL AIR CONTENT REFERRED TO TEST SECTION PRESSURE (a/a,.) FIGURE 14. NUMBER OF BUBBLES > 25 #M DIAMETER STRIKING HEADFORM FOR VARIOUS (ox rs ) BZ Cavitation (Influence of Free Gas Content) FIGURE 15 TYPICAL SEM PHOTOGRAPH OF GOLD PLATED BODY bE Peterson FIGURE 16 TYPICAL SEM PHOTOGRAPH OF DELRIN HEADFORM 1174 Cavitatton (Influence of Free Gas Content) FIGURE A-1. LIGHT THROUGH A BUBBLE — TWO MOST IMPORTANT CASES PDS IMAGE INTENSITY (UU*) Peterson —— OPAQUE SPHERE —— BUBBLE OBJECT RADIUS = 0.03 mm 0.01 0.02 0.03 0.04 IMAGE RADIUS (2) mm FIGURE A-2. LIGHT INTENSITY VS IMAGE RADIUS Z. = 997 MM 1176 IMAGE INTENSITY (UU*) Cavitation (Influence of Free Gas Content) —-— OPAQUE SPHERE -—— BUBBLE OBJECT RADIUS = 0.03 mm 0 0.01 0.02 0.03 0.04 0.05 IMAGE RADIUS (2) mm FIGURE A-3. LIGHT INTENSITY VS IMAGE RADIUS Z » = 100.00 MM Rt77 Peterson —— OPAQUE SPHERE —— BUBBLE OBJECT RADIUS = 0.03 mm IMAGE INTENSITY (UU*) 0 0.01 0.02 0.03 0.04 0.05 IMAGE RADIUS (2) mm FIGURE A-4. LIGHT INTENSITY VS IMAGE RADIUS Z = 100.3 MM 1178 Cavttatton (Influence of Free Gas Content) DISCUSSION Carl-Anders Johnsson Statens Skeppsprovnitngsanstalt Goteborg, Sweden In this very interesting paper the author discusses the diffe- rent mechanisms proposed throughout the years as responsible for cavitation inception. One of his conclusions seems to be that, for the type of experiments referred to, nuclei trapped in crevices of hydro- phobic particles could play the main role in the nucleation process leading to cavitation inception, A consequence of this reasoning would be that the role of free stream nuclei is not so important as has been assumed during the last years. The dissolved gas content is important however as it can af- fect the amount of gas trapped in the hydrophobic particles. I will show a slide, which may give some support to this con- clusion. The slide shows photographs of the free bubbles in the test sections of the two tunnels at SSPA during tests similar to those de- scribed in the paper. The same body was tested in the two cases and can be used for estimating the bubbles sizes. The test conditions are the same in the two cases: water speed 7.5 m/s, G'S ues. a/a,= 0.1 and the water taken from the same storage tank. The upper photograph shows -the test section of the new large tunnel ; the lower that of the smaller tunnel. The difference in bubble spectrum is quite large and is of course due to the large difference in the pressure history for the water entering the test section in the two cases. The point is that the same inception cavitation number of 6, = 0.45 was observed visually in the two cases. This might indicate that the influence of the free bubbles is not important. 1179 Peterson b) test in the small tunnel Comparative tests of the same body at the same a in the two tunnels of SSPA 1180 Cavttatton (Influence of Free Gas Content) DISCUSSION Edward Silberman St. Anthony Falls Hydraulte Laboratory Minneapolts, Minnesota, U.S.A. I want to comment on two points. I would agree that the free stream nuclei, which are very important, need not be gas bubbles in the usual sense that we may think of them as having uncontaminated interfaces between gas and water. I made observations about 25 years ago (1), using a microscope, of bubbles being dissolved under pressure in a rotating apparatus. This was in connection with resorption pro- blems that we were working on at that time to get rid of bubbles in water tunnels. I was measuring time rate of change of diameter of single bubbles generated by cavitation. In watching these bubbles I found that a relatively small number of those generated would not disappear but instead would collapse ona thin, wrinkled, opaque skin. I recorded this fact in the reference paper, although at that time it did not seem important. When the pressure was reduced again these bubbles would expand just like normal gas bubbles. I think this is per- tinent to what Peterson said, and enables us to treat nuclei as though they are gas bubbles. I believe that what is left in the water could be these bubbles with skins on them. It should be mentioned that these bubbles in the rotating appa- ratus, when they appeared to be pure gas bubbles would be near the center of rotation, but when they collapsed ona skin, their diameters remained constant, and they would wander around in the centrifugal pressure field without regard to the pressures indicating mean bubble densities near that of water. Such bubbles could sustain themselves in a towing tank for a long time - maybe indefinitely. My second point refers to the implications in the paper that it is necessary for the nuclei to touch the body in order to produce ca- vitation. I do not believe that this is correct. If you think about what I (1) SILBERMAN, E., ''Air Bubble Resorption", Tech. Paper No. 1, Series B, St. Anthony Falls Hydraulic Laboratory, 1949. 1181 Peterson said avout the possibility of the bubbles being of almost neutral densi- ty, perhaps it is then very likely that such nuclei follow the stream- lines or very close to them. It is our belief from work that we have done more recently at our laboratory that bubbles following these streamlines merely have to enter the low-pressure field of the head- form in order to produce cavitation (1). Of course, more bubbles will cavitate near the body than farther from it. DISCUSSION Serge Bindel Délégatton Générale ad la Recherche Setenttfique et Techntque Parts, France ( Translated from French) I read with a great interest the paper presented by Dr. Peterson and I must say firstly that I do not agree completely with its last conclusion, When Dr. Peterson tells us that his paper is a contribution among others about cavitation and that there is nothing original in its conclusion, I frankly believe that he is too modest. In fact this contribution seems to be very important in several respects. First due to the nature itself of program, and to the quality of the measurements. In particular, it seems to me that for the first time the bubble spectrum in front of the test body has been measured, not only the spectrum of free bubbles but also that of the solid particles by means of an unquestionable method, here the holographic one. That leads to estimate that the author's conclusions are based on serious data and consequently that this paper is by no means negligible. This point being reminded, I would like to make for my part two series of remarks ; first on the analytical calculation of the tra- jectories. It has to be pointed out what are the limits of such a cal- culation ; it is based on some simplifying hypotheses, which are not original, but these hypotheses are not completely valid, even far from the body. As was shown for example by Foissey in a recent paper (1) SCHIEBE, F.R., ''Measurement of the Cavitation Susceptibility of Water Using Standard Bodies", Proj. Rpt. No. 118, St. Anthony Falls Hydraulic Laboratory, 1971. 1182 Cavitatton (Influence of Free Gas Content) presented to Association Technique Maritime et Aeronautique (x), using a singular perturbation method, the solution of the problem cannot be reduced at not order to Stokes solution. Foissey showed in particular that some terms, which are currently put in the equations are not in fact correct, even far from the walls ; a fortiori close to the body, in particular inside the boundary layer, since it would be necessary to take into account the rotation of the bubbles and their deformation. But the analytical calculation may have a qualitative interest, namely to show the screening effect. Concerning this effect it seems that there exists some discrepancy between the present re- sults and those obtained by Johnson and Hsieh from the same hypo- theses. For Johnssonand Hsieh, if my memory is good, the bubbles are kept away from the solid body and only the smallest ones starting from the axis of the body can reach it. On the contrary, Dr. Peterson tells us that bubbles starting away from the axis can reach the body and cavitate on it. I would like this point to be cleared. My second series of remarks are relative to the respective influence of the stream free bubbles and of the hydrophobic particles as cavitation nuclei. When the air content in the water of the tunnel is decreased, it is noted that the free bubbles are decreased in num- ber and diameter but that the noise remains constant, and from this it is concluded that hydrophobic particles are responsible for cavi- tation, at least for cavitation noise. I believe that this conclusion may be true but that it is perhaps premature. It is indeed possible that the bubbles which can be observed when the air content is high are not cavitation bubbles but gaseous bubbles or pseudo-cavitation bubbles, that is bubbles filled with a great quantity of air, and leading to visual but not noisy phenomena. The noise could be perhaps produced by smaller free bubbles i.e. by true cavitation bubbles and not necessa- rily by hydrophobic particles. I believe that these considerations bring us back, as ever, to that difficulty of defining incipient cavitation. This can be only defined by its effects, either bubble growth, that is a visible phenomenon, with the difficulty of making the distinction between cavitation and gaseous bubbles, or an acoustical phenomenon that presents difficulty for an analytical treatment. In the ITTC Cavitation Committee, we have had serious discussion before reaching an agreement on the definition of (x) FOISSEY, C., "Application d'une méthode de perturbation singu- liére a 1'étude de la cavitation naissante.'' Association Technique Maritime et Aeronautique, session 1972. 1183 Peterson the term ''cavitation" itself, and now Dr. Peterson seems to question it in his paper. That means that there is yet much work to be done. For my part Iam convinced that studies like that which was presented here can bring a new light on this problem, and I hope that they will pursued, DISCUSSION Luis Mazarredo Escuela T.S. de Ingenteros Navales Assoctacton de Investtgacton de la Construccton Naval Madrtd, Spatn I want to thank Dr. Peterson for showing the results of his very accurate experiments. Tests like these may give a real expe- rimental basis and avoid the contradictory results we sometimes found. This is for instance the case with the no influence of the sur- face state which has been shown. Though this might be predicted, since conditions in its crevices are permanent and there is no input (which would not be the case in a central propeller or in boiling) such a confirmation is wellcomed. Since positive results are still more interesting, may Iask Mr. Peterson wether he intends to perform tests, in the future, to check the magnitude of the influence of speed on this transient phenomenon ? REPLY TO DISCUSSION Frank B. Peterson Naval Ship Research and Development Center Bethesda, Maryland, U.S.A. In response to Mr. Johnsson's comments about his photogra- phic bubble measurements, all I can say is that in our work we tried to differentiate between the visual and the acoustic measurements. If we had in fact used the observations taken in high-speed photography then we would have said that inception had occurred earlier when there were more bubbles in the water. I think that might have been shown on one of the slides I presented. But in spite of that we still, appa- 1184 Cavitation (Influence of Free Gas Content) rently, came up with the same general conclusion as to the bubble di- stribution importance. I should like to thank Professor Silberman for pointing out a paper of his which was unknown to me. It is unfortunate since the Navy Department apparently funded that work and I was remiss in not seeing it in the literature. I would like to say that all of this work can be considered a conservative estimate in that I emphasised free gas bubbles must touch the surface. If I gave looser bounds to the calculations and said that the bubbles did not have to touch the surface but only had to reach within a certain distance from the surface, in fact it would make the case I was presenting even stronger. As you may have noticed, the actual bubble distri bution was not even used in this work. All we took were the total number of bubbles observed. We did not even discrimi- nate between bubbles that had an order of magnitude difference in size because we felt it was not necessary to make the point. On Mr. Bindel's comments, I should like to point out that we could only discriminate between bubbles and solid particles larger than 25 micrometres and the conclusion that bubbles were not impor- tant at all was drawn by inference from the studies that we made in the high-speed basin. The analytical calculations of the trajectory by Foissey are not familiar to me but I think a comparison would be in- teresting between the methods of calculation. I might say that the paper pointed out that the calculations that I have used were compar- ed with experimental measurements of bubbles in pressure gradients in water and in glycerine and there was quite good agreement, sol suspect that there may be a close agreement between the work of Foissey and the calculation methods that we used. This work does agree with the work of Hsieh. The equations used were essentially the same. Perhaps it is unfortunate, but in the slide that I showed I did not show that some bubbles would be drawn away from the body, but given a pressure distribution on the surface - and in my case it was different from that of Hsieh and Johnson - and given different bubble sizes, some bubbles would by the pressure gradients be. forced away from the surface. I should like to reference the work of Dr. Brockett at the Naval Ship Research and Development Center. He has ina report correlated the noise produced by the collapse of the cavity and the visual observation through high-speed photography of the collapse of the cavity on the surface. It has been shown that ba- sically in the work on this body the cavitation that occurred on the surface produced the noise and to the best of our information the cavities that were further from the surface and did not touch the sur- face, did not produce noise within the significant range of this work. 1185 Peterson It is very possible that for the pressures that we had in these tests, the bubbles that were off the surface never really reached, or very few of them reached, the region at the vapour pressure of the water. Iam not sure of the extent of the work that will be carried out, Professor Mazarredo, as far as the influence of speed is con- cerned. The work requires a lot of effort and making detailed mea- surements at various speeds in water tunnels could be significant, but it is possible that the towing basin work which was done ata speed twice as high may, at least for the time being, serve our pur- poses of analysis. I appreciate all of discussion and I'd like to thank all of you for giving the pertinent and valuable comments. Bese 3E 1186 VORTEX THEORY FOR BODIES MOVING IN WATER Roger BRARD Basstn d'Essats des Carénes Parts, France ABSTRACT This paper presents in a synthesized form old and new results about the vortex theory for bodies mo- ving in water. It is shown that the hydrodynamic forces exerted on such a body can be derived from the knowledge of a vortex distribution kinematical- ly equivalent to the body. A method is proposed for determining both the bound vortices and the free vortices when the bound vortex distribution is cho- sen to be adherent to the hull surface. The total vortex distribution is divided into two parts. The first one consists of a volume distribution inside the hull and of a vortex sheet on the hull surface. The volume distribution is identical with the vor- tex distribution due to the angular velocity of the body. The sheet is determined by the condition that this first part of the total distribution induces out- side the hull a velocity null everywhere at every time. This first part may be calculated once for all. The second part consists of the free vortex sheets shed by the body and ofa bound vortex sheet on the hull. It is equivalent to a normal dipole dis- tribution whose density is the solution ofa Volterra equation. The determination of the hydrodynamic forces exerted on the body is derived from the dy- namical equilibrium of the fluid outside the body, of the fluid inside the body and of the total bound vortex on the hull. This system can be subdivided into three systems: the quasi-steady system, the system due tothe added masses and the system de- pending on the history of the motion. 1187 Brard This paper has been written to suscitate new re- searches in the field of the maneuverability and control of marine vehicles. INTRODUCTION Fhe vortex theory in incompressible, inviscid and homogene- ous fluids plays a role of importance in many chapters of Ship Hydro- dynamics. However it is not systematically applied to all the problems where it should be especially useful. This is the case of those relat- ed to maneuverability and control of bodies which behave as rather poor lifting surfaces because of their large displacement/length ratios. The research reported in the present paper has thus been un- dertaken with the purpose of determining how much help one can ex- pect from the theory when dealing with such bodies in any given steady or unsteady motion, Indeed the question was not to draw up a new vortex theory, but rather to extend known results relevant to fluid kinematics and dynamics and to increase their generality and effectiveness. The joined table of contents suffices to show the writer's line of thought. The startpoint is Poincaré's formula which permits to deter- mine the velocity in a closed domain when the vorticity inside that domain and the velocity on its boundary are known. This leads toa mathematical model where the hull surface is replaced by a fluid surface moving without alteration of its shape. There exists an in- finite class of vortex distributions kinematically equivalent to the body. They only depend on the choice of the vorticity distribution in- side the hull. The most interesting one is that which permits the ex- terior fluid to be adherent to the hull. Inside the hull the absolute fictitious fluid motion coincides with that of the body. From the point of view of kinematics, one of the features of the theory is that the total vortex distribution can be divided into two families almost inde- pendent of each other. One consists of a volume distribution inside the hull and of a vortex sheet over the hull, it is so calculated as to induce outside the body a velocity which is null everywhere. It only depends on the angular velocity of the body and can be determined once for all for any given hull. The second family is the union of a vortex sheet distributed over the hull and of the free vortices shed by the hull. It is determined by the condition that the fluid inside the 1188 Vortex Theory for Bodtes Moving tn Water hull be at rest with respect to the body and by a complementary con- dition expressing that the pressure is continuous through the line of shedding of the free vortices. Only the second vortex family has a physical meaning. But both are necessary for determining the hydrodynamic pressure on the hull. This is not really surprising since both the fluid inside the hull and the bound vortex sheet over the hull must be in dynamical equilibrium. A consequence is that the classical expression for the force exerted by the flow on an arc of vortex filament which does not move with the fluid cannot be readily extended to the case when this arc belongs to the bound vortex sheet adherent to the hull. The hydrodynamic pressure on the hull is expressed in terms which only depend on the total vortex distribution. The dynamical problem is thus completely solved for any given hull in any given motion what- ever the incident flow may be. The theory developed in the present paper is quite general. Its application to practical ends does not seem to lead to insuperable difficulties provided that reasonable assumptions can be made con- cerning the position of the free vortex sheets with respect to the body and the possible variation of that position with time. In any case, it is shown in the last section that an older and less complete vortex theory is still useful in maneuvering. Thus it is hoped that the pre- sent one can guide the experimental and theoretical researches which are to-day urgently needed. I. A BRIEF SURVEY ON VORTEX THEORY The vortex theory can be divided into four parts. (i) The first part is, in fact, a chapter of Vectorial Analysis. The vector V is the velocity of the fluid points in a certain fluid motion at a fixed instant t and the vorticity » is defined as @= curl V = VAV. Cra) The starting point is the Stokes Theorem, according to which the flux of @ through an open surface is equal to the circulation of V in the closed circuit consisting of the edge of the surface. A con- sequence is that no vortex filament can begin or end in the fluid. A vortex filament is therefore a closed ring or its ends are located on the boundary of the fluid domain, or at infinity. A consequence is that the intensity of a vortex tube is a constant along the tube. The 1189 Brard intensity of a flat tube or of a tube whose all transverse dimensions are null can be finite. This is the case of the vortex tubes on a vortex sheet and of the isolated concentrated vortex filaments. Equation (1.1) can be solved with respect to V. Poincaré's formula gives V_ when the vorticity is known. A particular case is the Biot and Savart formula which expresses the velocity ''induced" by an isolated vortex filament. A consequence of Poincaré's formula is that the perturbation flow due to a body moving in an inviscid fluid can be regarded as generated by a vortex sheet distributed over the hull of the body and fulfilling the condition that the fluid adheres to the hull. There is a kinematical equivalence between the body and the vortex sheet. (ii) The second part deals with the evolution of the vorticity with time under the assumptions that the fluid is inviscid and that the exterior force per unit mass is the gradient of a certain potential. The basic theorems are due to Cauchy and Helmholtz. The intensity of every vortex filament is independent of time and the vortex fila- ments move with the fluid. This means that every vortex filament is composed of an invariable set of fluid points. Lagrange's theorem follows according to which the fluid motion is irrotational if its starts from rest under the effect of forces continuous with respect to time (shock-free motions). This theorem seems to be contradict- ed by the possible existence of vorticity in the motion of an inviscid fluid, but the difficulty can be overcome by considering such a motion as the limit of the motion of a real fluid when the viscosity goes to zero. Although the second part of the theory is based on the Euler equation, it only deals with fluid kinematics. (iii) The third part of the theory concerns the dynamical interaction between flow and vorticity. If the set of fluid points belonging to an arc of vortex filament does not move with the fluid, this interaction cannot be null. The concept of force exerted by the flow on every bound arc of a vortex filament is now classical. Conversely, the set of fluid points belonging to this arc exerts a force equal and opposite on the adjacent sets of fluid points which proceed with the general flow. As it has been shown by Maurice Roy [1] , the system of forces exerted by a steady flow on a body in a uniform motion can be obtained in this way. This led to an important generalization of the Kutta-Joukowski theorem. Later von Karman and Sears have success- fully solved the problem for wing profiles in a quasi-rectilinear non uniform motion [2] . The pressure distribution on such profiles has been calculated by the present writer [3] . There exist now 11990 Vortex Theory for Bodtes Moving tn Water powerful methods of computing the pressure distribution on a wing of finite aspect ratio in the same kind of motion (see, for instance [4] The case of bodies of high displacement/length ratio in an unsteady motion is sensibly more intricate than the case of the usual lifting surfaces and there was a need for a general theory. Poincaré's formula gives means for determining such a vortex distribution on the hull and inside the hull that the fluid adhere to the hull. This vortex distribution is kinematically equivalent to the body. But it is not the only vortex distribution with this property. Furthermore if the motion of the fluid about the body is unsteady, any vortex distri- bution kinematically equivalent to the body varies with time. Lastly the theory would be without practical interest if it were not capable to take into account the effect of the free vortices shed by the body and that of an arbitrarily given incident flow. This paper gives an answer to the problems arising from the afore-mentioned needs. (iv) The fourth part of the theory concerns vorticity in viscous fluid motions, but it does not fall within the scope of the paper. Il, POINCARE'S FORMULA VORTEX SHEET Let > be a part of a certain surface. The two sides Ds ’ Dre of » are distinguished from each other. The unit vector n normal to a is uniquely defined at every point Pot Se and in the direction from Mie towards Dos . We put PP, = np(0+) +¥ BP emis Blt): .oPsoand~P4jbeimg the ue eas a of np with deawandia Diawmespectivelys (Bigure t,and 2), Let Dee - year! denote ae surfaces described by the points Pi, Pj such that PP! = = $= Tp, PPE -— Tip , respectively. We pepe that a Toten w is Se isine 2 G1 aiepribapad between a and De . @(M)is normalto n,, at every point M of each rien P; P,. When e{o , €(M) has a finite limit T (P) tangent to. a . Let 6 p » Tp denote two unit vectors tangent to)’ at P, such that the directions (np. Op? Tp) make a right- handed system, with 7p in the direction of Tp. Aline ¥ tan- gent to 7 at each of its points is a vortex filament of the limiting distribution. Let oe L, be two lines Y close to each other. Let eC be a line orthogonal i the £ 's and containing P. It intersects 12 at P, . We may put PP, = =do6_. For the limiting distribution, ak Pp ut o is the flux of the vortex through the area of the infinitely 1191 Brard flat rectangle P; P, P,, P;, andis therefore equal to the circula- tion dT of,the velocity v in the closed contour of this rectangle. Consequently (2 ga) Conversely, if the velocity V is discontinuous through a surface > Saas and if the jump is nee to Das oe then the_ above formula afeaes a vortex sheet. (, )) .s,1_«)), with tp = =Np VP: ) - = V(P. )| € The expression (2.1) is the local intensity of the ''vortex ribbon'' located benyesm & and L, . Itis a constant along the rib- bon if no nea sae @ ’ coming toed the regions outside x joins the distribution _1_ — over nae We will see in the next Section that the opposite case is frequent. POINCARE'S FORMULA We consider in the fluid a closed surface S witha tangent plane at every point. Let D; D, be the interior and the exterior of S. The unit vector n hs eae to S is in the inward direction. The interior side Sj of S is considered as included in D;, and the exterior side S, as included in D, The time t is fixed. The velocity V is supposed continuous and twice continuously differentiable within D;. Let A bea vector function of its origin M and defined by i192 Vortex Theory for Bodtes Moving tn Water We use the classical formula curl curl A = Vdivin © AA, a2 where A is the Laplace operator: A = v- = Lasinage a and V the operator ne Os , the system of rectangular axes Me ee SG O(x, » X55 xX being right- hace By Poisson's formula : i) -V(M) if M e D.. fr if MeD e Furthermore it is easily seen that n AV a 1, fff et) psf ae MM! ase) + fff erp a = | ; e (nV) yor. div A if at dS(M') + | curl A iH curl Brard This is the Poincaré formula. Let us suppose that the fluid is incompressible. The triple integral in the second term is not necessarily null for sources with a density o = div V could be distributed through D;. The double integral in the second term represents a source distribution over S with the density o =(n. Ve . There is a jump of V through S. Its normal component n (V, - is -@V)y, . According to (2.2) its tangential jump from ie e to Ded is that due to a vortex sheet over S, with the vorticity =-1 (m7 A Via € Poincaré's formula solves the equation o= WAV with respect to V, when the vorticity @ and div V are known inside Dj, if, furthermore, V is known on De ts Biot's and Savart's formula - (Figure 3) Let us suppose that all the points of S are at infinity, that div V=0 andthat @ is null everywhere except in a very thin tube with a transverse area 5 vie . Let us suppose that the measure of 6 ys goes to zero, while ai eso . The vorticity reduces to a vor- tex filament with the intensity [ tangent to ee. Let ds denote the element of arc of Then Poincaré's formula gives (Mm) = Leupp ML). sea hay r ds(M! V (M) = curl — ds(M') = curl re [os , (2. 4) This is the Biot and Savart formula which gives the velocity induced at M by the vortex filament Fhe intensity of whichis [ As: uw, Si. excepton oe there is a velocity potential @ except on pices > be an open surface whose edge coincides with and n the unit vector normal to ye , Oriented in the posi- tive direction with respect to the arc ds of . One has Vv =< VO wine = ff =o — sar du (Mt), Gee 7 M'! x 1194 Vortex Theory for Bodtes Moving tn Water ® is therefore generated by a distribution of normal dipoles over ). ; with the constant density [.. Of course, ® is not single-valued. If C is a closed circuit intersecting a, at P and surrounding one time, then the circulation of V in that circuit is [Fe A Adrabeae wiTe ths Feet the pete, pies G Application to vortex sheets Let us consider the partys ye of a vortex sheet. The contri- — bution of ‘aa to the velocity V is given by av =— curl ee |] Sa Neer dd (M (2:2) III. VORTEX DISTRIBUTIONS KINEMATICALLY EQUIVALENT TO A HULL SURFACE Let be denote the hull surface of a solid body completely submerged in an unbounded, incompressible fluid at rest at infinity. Let Vy denote the velocity of the body and Q E its angular velocity. One has ay = we = On curl E 7K): E Let D- - resp, D, - be the intervor.- reap. the exterior — of ay , and Se ; o. the Nie sides of yu: For convenience, i ‘S D; and de @ De.) The-unit- vector normal to > is in the Saeed By eeemar et V ener the velocity of the fluid inside D,. One has nV_ = a V -om yi 2 Gu Poincaré's formula applied to VE inside D, gives : EES Brard a oe --. (M') scifg Ses aan le D. ' : F —, [= ie } Vals aZow’)] I Ven if Me D. ; BT maleate > MM! “ehh Oe ae he The same formula applied to V inside De gives: (SOAV Jn 2, n Vv) I woffa ddu(M »| F grad = ee ee oo a 2M") = Adding these two equations and taking into account the boundary con- dition (3.1), we get ee 7 (nat o- (M') Vee onl eff E00 sf SE) | = ae (3%) Vig (M) if MED, " V(M) if MED,, where (3.3) T(m') = -#,, s/o) : Vt) | on Se This shows that the vorticity distribution 1196 Vortex Theory for Bodies Moving tn Water (She O, 4B) eyes 04), (3. 4) is kinematically equivalent to the body inside D, and generates inside Dj a fictitious ‘motion identical with the Guedes of the body. The relative velocity Ve = V-V' fulfils the condition V, =\0 jon pay (3.5) Therefore, the vortex sheet ones allows the fluid to be adherent to the solid wall pee of the body. ‘This gives the image of a very thin boundary layer which the real boundary layer would reduce to if the fluid viscosity m were going to zero. It is easily seen that curl V = 0 inside D, » so that WIEV . @) inside Dos @ being the unique solution of the Neumann problem with the boun- dary condition For the sake of brevity, let us put: Bee te T(M') se E foo =f ne MM' ddu(M'), i. ze [fF ~MM'~ =, (3. 6) The components of Jr are continuous and continuously dif- ferentiable inside 0 oe 1D (and harmonic within i) Those of J. and of curl J are benaien tad ai through Pe because 1197 Brard Equation (3.2) expresses that curl J (M) = V_, (M) - curl J. (M) when M describes D. ; (3. 8) and therefore entails Aa). oe ff», 209 a 0(M") = V_(M)-curl J,(M), MED yD (3. 9) curl ie (M) - curl J (no | = 0 within D. , ee < <} fy iS ' Q far as (a ta iS SI | =/ 0: within D. ; the difference VE - curl J, is within D; the gradient of a harmonic potential, and, for (3.8) to be satisfied everywhere inside D,, it suffices that it be satisfied on ee ~ Cres: (3x 8) Aare 3), 08) 1198 Vortex Theory for Bodtes Moving tn Water Consequently, T has to be determined by (3.9) and the complementary condition nT =0 o>. (39") Equation (3.9) is a vectorial Fredholm equation of the 2nd kind. It is singular since T= An, X_ being a constant, is a solution of the homogeneous equation LD ae a T(M') we “ = (n A Dia faa lf oo TRRE™ ALD ABA" ia 40 For (3.9) to have solutions, it is necessary and sufficient that its right side - say B (M) - be orthogonal to H on)> [f= ads bokovg (3. 10) my This requirement is fulfilled because Vi and curl Jp are divergen- celess inside Dj. Hence, if T' isa particular solution of the com- plete equation (3.9), the general solution is die — ead Uh Dhak Same ea But nT' = Gonist.i°= amt) ion: Ye , and therefore PPS) Bion a (3891) f : : : 1 is the only solution which fulfils both (3.9) and (3. 9') (1) (1) The proofthat (3.10) is sufficient and that nT’ = const. on 2» has been omitted because it is possible to substitute for (3.9) a scalar equation which does give rise to no difficulty (see Ch. VI, eq. (6. 6'). An equation similar to (3.9) has been considered by J. Delsarte [5] in the case of a fluid motion inside a closed vessel. In Ch. V, we will deal with an equation (5.7) analogous to (3.9) and (6. 6'). 1199 Brard T being determined in that manner, the velocity V' gene- rated by the vortex distribution aa on )> and te Wp inside Dj; is equal to VE within Dj Therefore, as the jump of v' eaegn ‘Ss is purely tangential, one has n V' = n Ve on os and V' evidently coincides with the velocity V_ of the irrotational fluid mo- tion inside D, The above results can be extended to the case when there exists inside D, some incident flow of velocity Vig . It suffices to replace VE oy TS - Ve into the right side of (3. ‘9). The resulting velocity is i= inside D. ; V inside D e If De were bounded by solid walls and (or) a free surface, one would have to add singularities distributed beyond the boundaries. These singularities would be linear functionals of T and therefore T would be found on the right side 6f (3.9) too. Furthermore, it is seen that the incident flow V_ could be due totally or partly to free vortices shed by the hull itself. In such cases, the right side of (3.9) would depend upon the history of the motion of the body. Various remarks (i) It is to be noted that the vorticity inside Dj; could be chosen 1 2a 1° 49 p; MM! oe The possibility condition (3. 10) would still be fulfilled. And, for the same reasons as above, T would still be determined uniquely by (3.9) -and.(3..9"),, But, inside D; , the resulting velocity could no longer be identical with VE ; arbitrarily. Te should be replaced by a In all the cases, the resulting velocity Vv on), = 1. ey between )); and >, - is given by V (M) => | Faw) " ¥ (a) (3. 12) 1200 Vortex Theory for Bodtes Moving tn Water (ii) _When T is determined so that V(M; Vz ve (M;), the jump of V - VE through >, is equal to Vr(M, Jor and V_. (M) == V, (M,) 8 is perpendicular to VR on »S and the edges of the vortex-ribbons on oy are orthogonal to the streamlines € of the relative motion over ), é If, furthermore, no free vortices are shed by the hull, these edges make closed rings ~ on yy . (Figure 4). Let do be the element of arc of a particular streamline . The intensity of the vortex ribbon between two rings L , ©” close to each other is dI(M) = V (M_) do(M_). (3. 13) (iii) If G=0 inside D,; then, dI’ is a constant when M describes LY , and the fluid cation inside D, andinside D, depends on the velocity potential = Hee = a(M'), (R = MM’) & (M This potential is generated by a normal dipole distributions. [I is determined up to an additive constant. If w @ £0 inside Dj, then, dI is no longer a constant between Y and ww’; ds being the element of arc of A » one has Ot dio des 26 idicw da dads The same aba Cee See in the case of the vortex distributions (D, 7 2 ® p) al Par I) - see Section V, Figure 5.1. 1201 Brard IV. FREE AND BOUND PARTS OF A VORTEX SHEET EULER'S EQUATION IN A MOVING SYSTEM OF AXES The fluid is assumed to be inviscid, incompressible and homogeneous. Consequently, its mass density p is independent of position and time. Let De be the hull surface of a body moving in the fluid. Let S', S denote two right-handed systems of axes. S' is at rest and the fluid motion with respect to S' is said the "'abso- lute'' motion of the fluid. S moves with the body and the motion with respect to S is said the ''relative'' motion. The subscript R refers to the relative snot F is the absolute exterior force : Va Ss -eurk V and ¥ =-G_ are the absolute velocity, vorticity and acceleration. The instantaneous motion of S with respect to S' is termed the''entrainment' motion, It consists of the addition of a rotation about a certain axis A and ofa translation parallel to. a If O and M are two points of S, the entrainment velocity Ve(M) of M can be expressed as V(M) ="V_{0) + QessAsgoOMas where Q E is the angular entrainment velocity. The relative velocity of a fluid point P located at M at the instant.t.. 1s R E One has Y(P aa a Pe Ae OP). saa Vena Vigne t = RP OR =i foybla lly Whe Td Pas © a ahs and the exterior force Eg per unit mass in the relative motion is 1202 Vortex Theory for Bodtes Moving tn Water given by Let p be the pressure. The Euler equation in the system S can be written in the form : 1 — R —>— = = ie AW = = = = 9/ 5 A = ae ee wade CE eee et av Re => = i, = + @ AV_ + V(—v = ke R R ( a) ) and the relative velocity Vp of every fluid set belonging to this bound part is null. On the contrary if dE belongs to the free part > 2 ¢ of the vortex sheet, it moves with the relative velocity Vp inside the sheet. Let De fie Wiese denote the two sides of pat » 1 be the unit vector normal to f in the direction from ys fe towards yy fi » and let ¢« be the infinitely small thickness of the sheet. The vorticity inside the sheet 1S) Gete=s Lieqvath T ho. ner Nar by Peat T, (M) a R (M,) Ve mi) Far ay dy oe € —— > € MEd). » MM, = ,,.—-iMM-=m,.s , Since there is no exchange of matter between the sheet and the adjacent fluid sets av. a> R Fale MGR) =? edd, +p iy FM). Vv. (M,) s EvaAwe) Vip(M,) a pe, (4. 5) 1205 Brard Furthermore, as the pressure does not depend on the axes, we have, by the momentum theorem Se Hy (OE) = Hy, [pg (ME) = wy Om) ] AE - m0 This quantity is null because of (4.4) and ¢« =+0. Consequently p, (M_) = p, (M,). (4. 6) Hence the pressure is continuous through the free vortex sheet. For the sake of simplicity, let us suppose that we deal with only one moving body. We also assume that we deal with one vortex sheet only. The vorticity generated by the body is thus concentrated inside that vortex sheet. The absolute velocity V in the domain D' really occupied by the fluid can be divided into three components: - the velocity V4 induced by the vortex distribution (Dis 1 2.0 ett (>, 4T ) which permits the fluid located between) ; and Sis to be at rest with respect to the hull; > - the velocity V, of some incident flow, - the velocity Ve due to the free vortex sheet shed by the body. According to Section III, we have 1206 Vortex Theory for Bodtes Moving tn Water with the boundary conditions Va (M.) = ¥V 2 (M,) - Vi (mM) - ¥ z (M,) on ES a+ 345 -{M) =0on). Vo is due to causes located outside 2; e and one may consider that there exists a velocity potential ®, such that at least in the region of Dj close to De evandinside D,, + > Va + V- is irrotational outside D + Dix; i thereiore; Mz. and Mj being given on f, and f; » respectively - with MeM;j = ny (0+), one can find a path starting from Mj, and arriving at M, sothat V be irrotational everywhere along this path. Putting > V= Vd F) (4. 9) we obtain, by integrating (4.1) along the path : ‘ se pyeee (M,)-p,(M,) = - » @%) Hence (4.6) gives: (4. 10) One has 1207 Brard Consequently Baas 2 2 Sia My | v2 (M.) - VQ om) | (4. 11) Thus, when the relative motion is steady, T;(M) and VR(M) are colinear, and both are in the direction of the bissectrix of the angle (VR (Me), VR (Mi)). _ are no longer equal to each other and therefore the direction of T(M is no longer that of the bissectrix. If the relative motion is unsteady, bine (M,) | and lvR im) Equation (4.10) expresses the dynamical equilibrium of any part of a free vortex sheet. DEFINITION OF THE FORCE EXERTED BY THE FLOW ON AN ELEMENT OF THE BOUND VORTEX SHEET ADHERING TOA MOVING BODY Let us consider nowa set dE of fluid points belonging to the element d ‘ of the bound vortex sheet. We have V.. @ty-= 0, and —> , for, because of the adherence of the fluid t yi , a force = at y is exerted by the element d ys of the hull surface on dE. The equilibrium of dE requires: 1208 Vortex Theory for Bodtes Moving tn Water Conversely, the equation y oe = EB (M.) =P mt | i, 42 (M) (4. 13) ees shows that the fluid sets adjacent to dE exert the force aX, on the set dE. V. THE STRUCTURE OF THE VORTEX SHEETS GENERATED BY A MOVING BODY Two cases will be studied in the present Section. (i) One free vortex sheet only is shed by the body, (ii) Two vortex sheets are shed by the body. In the first case the fluid motion is irrotational everywhere outside the body, except through the free vortex sheet pa . In the second case, if the fluid motion is unsteady, vorticity is necessarily distributed in a certain volume downstream from the body. A - STEADY MOTION IN THE CASE OF A UNIQUE FREE VORTEX SHEET _ We can define on ba such a vortex -I’ that each of the two vortex distributions . Gi = 28 5)+(2,2) 6.1) Dr = (eS) (LTS .2) be complete and that they sum up to the total vortex distribution M = i rms 24) 135 y 2) 9), 0a) 1209 Brard —} ! The vortex family (D. , 2.) + (2 ,—) i The new vortex distribution (5.1) will be chosen so that the velocity = maneaiet 20 (a V"(M) =*eurl ff B00 500 + curl =f — =r aD, (M') D 2 : (5. 4) be identically null inside D, Inside Dj; We is the velocity of a fictitious fluid motion due td a certain free Thc dl Son per unit mass (determined in Section VI). One has : eur! Vi" Somng inside D.. (5.5) E at MUR eS) igh eee D. (5. 6) is obviously equivalent to the condition yy (Ma) = 0 on Ee (5. 6") The fictitious fluid motion defined by V' is one of the fluid motions which could exist inside the vessel bounded by by’ ; if the body were at rest. (5. 6') means that 1 T'(M') es | pM’) curl ee ED aaE )= - curl a M. eri he dD.(M )on PDN 1210 Vortex Theory for Bodtes Moving tn Water Because of the discontinuity of the left-hand side through dy , this equation can be written in the form : i, AT'(M) “ cont fron ddu(M') = where M is located on > This is a Fredholm vectorial equation of the 2nd kind which is quite analogous to (3.9) in _spite of the fact that the condition to be verified concerns the side AGE instead of the side ie First one observes that this equation is singular, since the left hand side vanishes when T'(M) = Any, A_ being a constant. The right-hand side must therefore satisfy a possibility condition. MM' Gof Gera on m) dD (M) = 0, II. sie oa e one must have M') I mim.) | ate fff, aM ap,()| adi(M,) = 0. (5.8) a This condition is obviously fulfilled since the latter expression is equal to Zl Brard 29 _(M') 2 1 1D Se div curl |— a ee ee ED on | aD (M)<29o ; ip s/f, MM : It follows, as in the case of (3.9), that the solutions of (5.7) are T Ber + \n, (M) = TY (M) +3, a ith — é wane n T) = constant ys and therefore that there exists one, and only one, vortex os which is tangent to > and satisfies (5.7). E One has 7 ahacin. Vy Ake =n AV" : y T' (M) = -n,, af (M_) -V a) ny, AVM) on »2 (5.10) At each point M on oe é (n}, : Ou ‘ tna) are three unit vec- tors making a right-handed system ; n is normal to in the inward direction, and 7 is in the direction of T!'. The lines @' tangent to @' and the lines’ tangent to 7' determine on 3 two systems of orthogonal curvilinear coordinates, | the arcs o' on €@' and s' on being oriented in the direction of @ and r' respectively. ! Let us consider (Figure 5.1) two lines € (o ) and Sa (0',) close to each othey and two lines Co (sh) Nees (s', + ds'). The flux dT” of the vortex 1 through the area Mg, Mg, Mj, Mi, is equal to V'(M) 8 a4 do' (where do! =o,' -¢,'). Through the area Me Me, — Le , it is equal © V'(M}) 6,41 do! = dT -wn de ds' 1 ape > 4 : VecACte eo Fit a a. ak - with ds' = s', - s';. Hence Sar r (s,@) o.n 29. .n,) This results from the fact that the vortex ribbon whose edges are ‘(o') and & '(c',) loses, between € (s')) and © (s',) , vortex filaments entering D;. The . are closed rings, since the ends of any segment of L2iz2 Vortex Theory for Bodtes Moving tn Water a vortex filament 292 ~ interior to Dj necessarily belong to the same line £ '. — ae The vortex family (ee a oa oe a ai ) € € The-velocity V'' induced by this vortex distribution is given m (rT). my, | T.(M') V''(M) = oof ff adu(M') aed, aarti 2,009] , f (5,13) V" is irrotational everywhere except on oz and on ae Re On, =e ; one has V"(M,) = (V - Vo) Me and, on Qj, V"(M,) = (V,-V'-Vo) us, Accordingly : V'"'(M_)-V"(M.) = V2 (M,) 4 WiUM. (5. 12) Let's, -0 _denote two unit vectors on oy , t (M) being in the di- rection of (T - T')y , so that the three directions (Se Sar} make a right-handed system. It follows that @ (M) is in the direc- tion of Vr(Me) + V'(M;). The curves @ tangent everywhere at 6 and the vortex filaments LY on be define a system of orthogonal co- ordinates (o, s) increasing in the directions of @ and 7 ,» respec- tively. We may consider V" as gene'rated by a normal dipole distri- bution on >3 + yoy . Let # and yw ¢ denote the density of the distri- butions on) and a respectively. We have: © (M) ff bw (M"') a ae aos ia) (5. 13) 2, lg1s Brard Let pare ; ae: , denote the two sides of y hy. » Nf on pa f being in the direction From > a towards dt, One has (M,) + wg.) = O(M,) - oM,) ; gu Ol ieee dir -|¥ (m,) + Vem) . O(M)doa ; Let Q@ denote (Figure 5.2) the intersection of ~ with e Two parts )\, and pe of }> are adjacent to each other along The part psi corresponds to )\¢ and the part dU, t Wee - Hs be the determination of uw on and He Ee Let Ls ; cL %: VY be three vortex filaments intersecting at the same point, By, and located on De fas yy ; Day , respectively. Similarly, L f » ae ; 5 intersect RR at the same point B and NAG oan nes intersect R at B'. The points B and B' are chosen in opposite directions with respect to B°. Thus A f belongs to the vortex ribbon Ly, the edges of which are L rae ae, . Simi- larly Br belongs to the vortex ribbon L, and L3 to the vortex ribbon L, , the edges of L, being ~,, sf, while those of L, are WH, , ae . Let Mg, Mf be the intersection points with hers se of the curve intersecting ae at Mf. Letus define ina similar manner the points M, and M; on L, and Ly ,» and the points M, oF Mi! Aon L, and £ 2 2 We now suppose that Mf, M* and M5 are chosen so that . to i O = S to i oO = 5 td iN Ory (5. 15) n being a small length that we shall finally equate to zero. 1214 Vortex Theory for Bodtes Moving tn Water Let [ '(C) be the circulation of V" in some closed circuit We consider the following circuits : rei ee Se M*%. Me. oe M? M; M: ; eo rh ey we 2 2 C) M M'! M; M, M P “1 cai 1 l e} (S M' M M M! M! F 2 e e5 f, f, €2 G M M! M! M M F al 1) it 7) 19 1) GC M M! M! M ; f as roe & Ty G M M' M' M M (= ©) e} €2 eo €) We have Tr" CE, = 0, | (C,) = Qd; T!! (C,) = 0, T"' (C.) = Q, The first equality gives M3) - #,(Mf) + O(7 ) (5. 17) T215 Brard The other three equalities (5.16) give: v"(M2 ) MjM, + V"(M ) MM, = taal © a bc A) 2 Zz WN ° ! = ! ° ! = Vv ae De ay ae M}M, Oye AG Euv | . MMi + [FaiMy y+ ¥ (se MIM, = - ta (oo ae O(7) The left member is equalto [ "(C.) + O(7); we have thus: Pies) = r(C,) + -O (ar). (5. 18) But IT '(C,) is equal to the flux of Z - Es through any open surface S, the edge of which is C,. We may take S = ‘rectangle M VES Vi Mi + S e Qe) ey 1, 1, l + rectangle M' M M. M: (5. 19) Sa Sh tae where S, is any open surface the edge of which is the contour 1216 Vortex Theory for Bodtes Moving tn Water We can take for S, a surface entirely located within Dj . Therefore the flux of 2-2" through S, isnull. Let % be the unit vector normal to the first sacle so that the pees directions M, ; Me ; nj}, vy make a right-handed system and let v5 be the unit aegior norinal to the second rectangle such that the three directions — Mi Ms n, 5 also make a right-handed system. The vectors , , ro ¢ are tangent to x, VO at M‘ and to Se at M*, , respectively, and both are in the aietind towards B°. hokee ; _7 rec ). = jares Mj Me) Mh Mi), (A) pe : LS eeD A, (LT TRL l TT! + [area M'! M, M, M; V5 ( z ) Me * Z Z Z 2 2 The intensities dQ of the vortex ribbon L, and dr, of the vortex ribbon L, are given by r - = = dT = hice M M' M! M. #(- ae ’ 1 1 1 1 1 dr, = lie M' M M. M! 74. tae ’ zZ 2 2 2 respectively. Hence " = ie < - r (C.) qr, ae | vis 7) + dt ( ’ > T 5) Similarly, vs being the unit vector normal to the rectangle Mg, ME M}, Mr, and such that the three directions —_ Mr =, nf, v-¢ makea nieke- -handed system, we have : - 1217 Brard Py = dr, .( Vv Pee Bs where dQ, is the intensity of the vortex ribbon Lr. According to (5.18), it is seen that iia —_ cc gh: (5-20) Let Bf , By? be the points derived from B, B' by the trans- lation "+ = 7; (B) and Vrs se BY those derived from B, B' by the translation - + n, (B). Let ry denote the surface whose edge is the contour By, M;, M}, By and ye the surface whose edge is the contour Bf, By* f it : Siri ay let pals be the surface bounded by the contour e, fit, aya BY, and “177 the surface bounded by the contour M, Mi! By Be , Sy the aa ohare bounded by the contour Cr, and S, the Paeice bounded by the contour es _sthe surtace ¢ , , , S. +2, ay ei se w2e —> is closed and the flux of = through this surface is null. As it is equal to+ (dT, +dI,- Is) if We Fe = +2 andto = (dh + dT oc - dT) if ve tT, =, - 1, we Haye df o£ dF = aE (5.21) To. ZS TS ae ee eo ae 1218 Vortex Theory for Bodies Moving tn Water This shows that: If L, starts from BB', then both L, and L, end at BB'; If L, ends at BB', then both Ly and L, start from BB'. Consequently, every vortex filament 2 can be considered as the union of two free vortex filaments ¥ f, and & 5 ; Sat and & ty being two complementary arcs of the same vortex filament L, : similarly Z£5° and Le belong to the same vortex filament sho ; SA and x, are clbsed— When 7 goestozero, (5.17) becomes H(B°) = ou (B°) + #)(B*). (5.23) MeGording £0) .(5522), 4f sagen ap) =- 114; then 8, is in the direction from a towards SL, - 65 in the direction from L, towards L,' and Os in the direction from L f towards SP } ay Tit Dip asin ¢ Pegs 1, the directions of 6, , 8, and 6, are opposite to the preceding ones. Thus, from B to B', the variations of Ne I. and l are grep Hp: tly yz a Ecay -*(6)]- ¥ au [topper Tap on aes ae es) -1(L,) } hae are #1 A 4 + Lp f sign if aD oe Ty (stp) a TA) v= K enw (ate) fF - Ab ae iB Me As it was to be expected, (5.21) and (5.23) are therefore equi- valent. Fig. 5.3 sketches the configuration of a vortex ribbon when there exists only one free vortex sheet. This case is that of a wing with a finite aspect ratio. P is the trailing edge of the wing. 1219 Brard Let us suppose that the wing is in a uniform motion of trans- lation and that the relative motion is steady. Figure 5.3 shows the general configuration of the vortex distribution. The free parts of the vortices are closed at infinity downstream from the wing. (In the figure the concentration of the free vortices around the edges of the free vortex sheet is neglected), The second frontier between), and >» consists of the bound part of the vortex filament shed at the junc- tion of the leading edge with the trailing edge. The sum )/, oe does not cover entirely the wing surface) > . The vortices located on ee eae ay Pas 5) contain no free arc ; they make closed rings entirely located on 3° Let som denote the intersection of dr f, With >) a and 43. that of >> f, With y e, : These two curves are infinitively close to ‘ The relative velocity Vp is tangent to PB, e, at every point Ba. end to e> at every point Be, . These two points can be 2 e2 ; considered as belonging to Deice and to a fe respectively. Thus VR (Be,) - Viz (B*) and Vig (B*)\- VR (Be,) are equal and orthogonal to pee Balt |VR (Be) | = | Vp (Be_) | because the pressure is con- : 2 : : tinuous through the free vortex sheet as shown in Section 4. Con- sequently Vp (B°) = 0. Hence Lf is orthogonal to ‘om ° vend also are orthogonal to since they are orthogonal to ve on and on Do. respectively. Furthermore the intensities dI, = OVER (Bé,) . BB' and di, =.- Vp (Be,) BB' are equal. Hence, as shown by Maurice Roy. df) V2) adr -— ar (5.24) f Of course, this relationship does no longer hold when the motion is unsteady. B - STEADY MOTION IN THE CASE OF SEVERAL SHEDDING VORTEX LINES One case of several free vortex sheets is sketched in Fig. 5.4 which represents the lower half of a double model the two halves of which are the images of each other with respect to the (X, Y) - plane. The (Z, X) - plane is the longitudinal plane of symmetry of the hull. The fluid is unbounded. The body is in a uniform motion of translation in the positive x-direction and the fluid motion is sup- posed to be steady. The drift angle a = (Ox, OX) is positive, so that the starboard side is the pressure side and the port side is the 1220 Vortex Theory for Bodtes Moving tn Water suction side. The transverse cuts of the hullare V or U - shaped, the radius of curvature of the U's is small. Experiments show that, if a is not too great, then the re- lative streamlines on the port side), are less inclined on the (X, Y) plane than those located on the starboard side ae - Let E,, EGF Ah fs. be a sequence of points on the lower half-stem. On a5 ; the relative streamlines coming from the E,'s leave the hull at points S;, located either on the lower half-stern-post or on the keel, in the (Z, X) - plane. 2p ae , the relative streamlines leave the hull at points Sy also located either on the lower half-stern-post or on the keel. But the points S} located on the stern-post are below the points S, , and the points S, located on the keel are upstream of the points S;,. Consequently, the two streamlines arriving ata point B on the keel (or on the stern-post) come from two different points E. The relative velocities on these two streamlines are equal at B, but their directions differ. This entails the shedding of a free vortex filament from B. Thus there exist three surfaces oF . One, denoted De. Rae: generated by the free vortices shed along the keel ; the second one, denoted Ee is the mirror image of Ye ; the third surface, denoted oa is generated by the free vortices shed along the stern-post et is its own image (Figure 5.5). Let PD ' denote the complete stern-post S, SS, and B 5 . the keel and its image, respectively. Every free vortex filament ¢ starts either from the upper half of CP or from BR and goes at infinity downstream from the body. Because of the steadiness of the motion, Y f coincides with a relative streamline. Consequently, ae is nearly parallel to the (X, Y) - plane. So does the upper edge of aan . The said vortex filament Y ¢ comes back from infinity towards the body and reaches it either on the lower half of oF on Ro: The start point B, of £ 5 and its end B, are mirror images, Let S, be the bound vortex BS B, on abet and Ce the bound vortex B, By on ‘eo >: As the relative velocity on we is greater than on ee the intensity dIy of the free vortex a eee starting from the body between two points B, , B} and arriving on B,By is equal to the difference dI, -dI, , dI, being the intensity of the bound vortex ribbon L, on)), starting from B,By and arriving at B, Bi » while ar, is that of the bound vortex ribbon L, on))5 starting from B,)By and arriving at B, Bi at es dt al (5. 25) Brard Hence (5.24) does not hold. 1 Let C denote a closed contour around the stern post. Let rae denote the port side of y and pare its starboard side. C intersects ae at Ms Z De, at M, and Ds at M, . The circula- tion of the fluid velocity V in that circuit ig null. Hence, if all the points of C are close to the stern-post 0 =[orng, ) - "iM, | + [eras ) = ora, ) Teor, ) - Orta 1 a ] ] 2 z where 1 sac ! re) l ! + =f MM!) —9— ad (M') (5, 26) ! ! If the normal to rae is in the direction from bes ¥ to Dine » we obtain 2 1 ui(B) = 45 (B) = (B), (5.27) H, being the determination of uw on a ,» while Uo is its determi- nation on Similarly, by taking 7, on 2, , inthe positive z -di- 1 1222 Vortex Theory for Bodtes Moving in Water rection and, on >» fo in the negative z-direction, it is easily found that for B on B PE = Bia) (5.28) : e ; fox) on Equation (5.23) holds in all the cases. Let us consider two values a , a. Of a , with ada The dissymmetry of the flow is more strongly marked forasa ’ The part of free vortices coming from BR and arriving on? & is greater in the second case than in the first one, while the part of those coming from the upper part of cP and arriving on the lower partis smaller. This entails a rapid variation with a of the posi- tion of the lift, that is of the y-component of the hydrodynamic force exerted on the body. It has been assumed that the line “Bh is in the (Z, X) plane, In fact, if the bottom is flat, the line WR becomes a curve with posi- tive values of Y. However this phenomenon cannot alter seriously the velocity induced on the hull by the vortex distribution. One among the advantages due to the substitution of a normal dipole distribution for the vortex distribution is that one needs not know exactly the direction of the free vortex filaments. When a is too large, the relative streamlines on Poe tend to pass from ae to oF and separation occurs on the suction side. In such case, the above considerations do not hold. C - UNSTEADY MOTIONS IN THE CASE OF A UNIQUE FREE VORTEX SHEET Let us consider a point Mg on me ¢ attime t, and let a t? denote the position of Py at t'< t. Let P be the fluid point located at Mr at t. The condition B. (t') M -| ¥. (pyr) a (5.29) L223 Brard with By, (t') infinitely close toa point B Pass) , determines the time t' .at*which “Phase lett . It may happen that the position of B on R depends on t'. In that case, B can be defined by its abscissa 0B on . During the time interval (t', t'+dt') a free vortex filament of density d,:d, is lying on a closed contour By Br Cr Cr By , with Br Be =d, (B) and B,C; = VR (Bey t') dt'. Ate this vortex filament is lying on the closed contour Mg My; N¢ N¢ Mg : where ! ! = ae ! = ! Be M; { bee) eared) ahs tae M,N; Va(My > t) dt t! We have therefore t z e) dQ. (M, ? t) = eee d To, (7), 7) dr (5. 30) t! or equivalently, t rf fe) B du, (M, , t) i ai dd. (ae Hy) a(1), 797 (a 3a t! If B is independent of t', then we obtain by integrating from one of the edges of Dis Bpa(M.. pot) =uhbe (= po) -(#, -#,) ; pa alan Ey EE ae t (5. 32) with B(t')M, = x: 1) dr t' In the latter case, the support of the vortex sheet is generated by the relative trajectories of the fluid points leaving rh ae Su a 2° 1224 Vortex Theory for Bodtes Moving tn Water D - UNSTEADY MOTIONS WITH SEVERAL VORTEX SHEETS As shown in Subsection B, a double model in a uniform motion of translation in a direction V, parallel tothe (X, Y) - plane of symmetry, sheds three free vortex sheets starting from the keel YY, ,» from its image @, , and from the stern-post, respec- tively. We consider below a body which could not be necessarily a double model. Its stern-post is denoted S,S S, ; its stem is the arc E,EE,; the segments E,S, and ES, are parts of straight-lines. The origin O of the axes OX Y Z moving with the body is in the middle section, the X-axis containing the points S and E, and the Z-axis being in the longitudinal plane of symmetry of the body (See Fig. 5.6). At a given instant t', there are between two points B,)Bo on the keel ES, (or on the lower stern-post), two bound yortex ribbons, one on the port side ),, , with the intensity dX se (B and the other one on the starboard side), , with the intensity dX 012 (B , , t'), These two vortex ribbons end along the same arc B,; ceo , on the upper keel E 4 4 (or on the stern). The BY es tion of B, ee depends on B, neces on the abscissa X of B, and sSeeibly on, G'. ot!) In the interval (t', t' + dt'), the increments of the intensities of the two Ves vortex ribbons are dt' dX (2F1) 5. dt' dX ae Bt’ . If they are not equal to each other a free X vortex ribbon is shed. It begins on the segment Bje +! Bie, t! and ends at By, s Pea the points with subscript e are infinitely close to B, ,th BY ths Bist Py, . For the sake of brevity, we con- sider this free vortex ribbon ie an arc of free vortex filament Bie, t! Boe! with the intensity dt' dX woah Deodt, =F} at’ idx l 2) BR. tt The fluid point P which was at Bie, t! at time -ot' os «sha time t ata position Jy ;4:. In the system of axes moving with the body 3 — Vy W Ww . 5. 33 Bie, t' Ix, t! sf YR Pillidett ( ) 1225 Brard Similarly, the fluid point P which was at Bo. at t' is at t ata point ly 41, and " = W Bo I , ee (Pee) dee (5. 34) The fluid points which were at t' onthearc Bo, By ,t! infinitely close to B oP 1,t 1 areat t onanarc Jy tly 4. Let ty be the time of the beginning on the unsteady motion, e suppose that the y-component of ie has on the arc ie j Piss , a constant sign (po- sitive) during the time interval [t,, t) (Fig. a aan But it might occur that this sign changes at times t, , tz,... We also suppose that this sign does not depend on the abscissa X of B,. The most general case is still more complicated and will not be examined in this paper. The points Ty describe a line starting from IX to and ending at Bo. . Similarly the points Jy describe a line starting from Jx, to and ending at Bi, re These lines generate a surface doe and a surface de, ; u smerd ci The arc Jx, t! yy t1 gene- rates a surface Sy ‘beginning at Jy, ie Ty ts and ending on the arc Bo By . There Ei lay: So. is one of the edges of Dt and the e ayy IE 2? fe) arc Ito So. is one of the wm ea8 of oe . The fluid points which wereat t! on So, Si. describe a sea when t' varies from t= to) os. On oy» , at t' , we had on the port side pate a bound vortex fo) filament of intensity dy I, (X,t') on the arc BoB, 41, and on the starboard side jogs a bound vortex filament of intensity dy J (X, t') with the support B, | 4: By . During the interval (t')..ot' + de')etthe variations of these two intensities are dj: dyT, (X,t') and dy: dy r, (X,t'). It is because they are not equal to each other that a free vortex is shed. At t' the support of this free vortex is Bie ! Bo e and its intensity in the direction from By, to Bo. is equal e; t! to dir uly (x5 t') = r, Cx; ")] . Because the intensity of a free vortex is time-invariant, the intensity of the free vortex filament whose sup- portis Jy 4: ly 4, is at t also equal to dy: dy i ea r2(x, | : The closed contour Bo, aba Jy t! Ty t! Bog is thus the support of a free vortex filament of intensity dj: dy E (33,/¢") -T, (x; ) ‘ 1226 Vortex Theory for Bodtes Moving tn Water Let L(t’) denote this free vortex filament. On the arc By t! Jx ¢1 e; ? we find the union of the vortex filaments, (7) shed during the in- terval E pt!) ae Thettotal intensity ofthat union rt Jx, to sah e-y ; thus varies with Jy t eet Jy tt it is equal to ee dit dy E (Xt") -T, (xX, | = dy [ry (5) tara x, | and it is in the direction towards Jy to Similarly we have a union of vortex filaments on the arc Ix, to Bo; its intensity at Iy 4: is equal to dy P, (x,t') - Ty (Xx, ‘| and it is in the direction towards Bo. Let us consider a point Mr on dug, . It is one of the points Iy 4: It belongs to the trajectory of a fluid point P which was at an anterior time t' ata certain point Bo. lence, Mf, being given, Bo, and t' may be considered as determined functions of My and t. The same is true for any point Mg, on), f, @ndany point M/ on oF , and also for any point M whichis at t ona vortex filament ! hia le pa We have supposed that the arc Bo. Biest is infinitely close to the arc BoB t on port side bag and nevertheless that Vp on the arc Bo, Bi, is not tangent to Le This implies a contradiction of the same nature as that encountered in the scheme relative to a steady flow about a wing of finite thickness. We have seen in the latter case that the relative velocity on the wing is tangent to its trailing edge and we however assumed that the free vortices leave the wing in a direc- tion orthogonal to this edge. In the present case, the distance of the arc Bo, Bi. from the arc B,B; 4 is not really null, for the boun- dary layer is not infinitely thin. The contradiction seems to be an in- eluctable consequence from the assumption that the fluid is almost inviscid, We now drop the subscript e and consider that the support of the vortex filament YQ (t') at time t is the closed contour BoB, ¢ Jx,t'Ix,t1 Bo. Thearc BoB t is bound while the three others are free. The intensity of this vortex filament previously determined is dis dy [T] (X, t') - I, (X,t')]. The quantity dy[T, (x,t') -T, (x,t!')] is positive for the free vortex to be shed from port, but its variation with t!' may be positive or negative. To make easier the drawing, the angles between By Jx and the longitudinal plane of symmetry has been considerably magnifi- edin Fig. 5.6. Furthermore, one observes that, if the minimum value of X onthe keel is necessarily equal to == (for SoS) plays the role of the ordinary trailing edge of a wing), its maximum value TZ27 Brard It, can be less than ee At time t, the surface Sy generates a volume W, when X increases from -=>- to its maximum value. We therefore deal with a three-dimensional wake W,, which was not the case in the preceding sections A,Bor C. Within W, a vorticity OF is continuously distri- buted. The vector w(M) is tangent to the arc Jy ¢: Ix ,! which pas- ses through M. oma The total velocity Vs induced by the vortex Slama cabana te) at a point M located inside or outside W;, is given by Poincaré's or Biot-Savart's formula: > 1 a fs does V (M) = curl) — era cat dX, (P) 2] 2 Bite t=) hyphae or +H, sy 2%, ()+ Te ito aw, ay (5. 35) pa; t = The velocity V; is irrotational outside (W, + its boundaries). It is, in particular, irrotational within D; . The other vortex distributions to be considered are the distributions —> QG, = (Since 4 (D; » 28), Lote Ora ' (5. 36) The velocity V' induced ryY is null outside the hull, and curl V' = 22, within D . (5. 37) The vortex =e See! consists of ring vortices on at Each ring is made of two arcs Bolas on 235 and Bi, t Bo on Disk The intensity of the ring is a constant_dy Y5(X, t) along the ring. Onan arc B,B, 4 located onthe part2,, of 2, , there exists the vortex filament of intensity dy T; (X, t) already considered in formula (5. 35) - see integral extended to ‘Be . The vortex distribution 2 is equiva- lent to a normal dipole distribution (2, Mon). It generates a velocity V5 » and one has Mics ee (V,, iN 4) -V, a“ vo within D, (5. 38) where Bia is due to some incident flow on the body. Outside the body, the total velocity is — —> = = V(M) = V,(M) + V,(M) + V (M) . (5. 39) Let us put Vv, =2 P. (outside W, + its boundaries) (5. 40) £228 Vortex Theory for Bodtes Moving tn Water To calculate Dr » one can observe that a vortex filament Keer) is equivalent to a uniform dipole distribution on any open sur- face the edge of which coincides with fYy(t'). For that surface we may take the surface Sy. t! consisting of the five following parts : i (i) The part dy (X) of a behind the arc BOB : sie (ii) The part of ys interior to the contour S I I Br See: i O 2 4h se Ga Oe 2 ? ? (iii) The part of Pe interior to the contour s\J L J B, Ss, ; E Set Meet ae (5.4: (iv) The part of pas interior to the contour I s Sd I : ot! o 1 -4, t! = t! (v) The part of the surface S! generated = the arc Jy tly ¢ when X varies from - 5 to the value X defining the vortex Z filament Fy (t")s = this surface let us select a unit vector ? normal to it ; for instance, Vison 2 1, 2m the outward direction with respect to the hull. On the fifth part of Sx t vy is thus directed toward Be ete!’ Det). M*, ‘M7 be two points cnheitely close to each other, M* being on one side of the surface and M™ on the other one ’so that M~M’ is in the direction of Y. The circulation of the velocity induced b x(t') ina circuit starting from Mt, turning around x(t') and ending at M is equal to the intensity of X, t! and equal to the density of the normal dipole distribution on the surface. Hence the velocity potential due to the dipole distribution is did ?y 4 (M) = = 4,4 d x [A r (Xt!) -2,(x, >) Aff ees, ~ ana &5(M"). (5.42 X, x The total potential outside W, and its boundary is max - smo =f 1 fe x, tM) : (5. 43) ! According to (5.41) the contribution from Bis 1 - 1 Feuate (X, t) = eee t) xp aia (M') ; 5. 44 4 In 2 lhe Onan M'M 1 ( ) Brard 1 this notation recalls that any point M' on »s belongs to an arc BoB) 4. the abscissa X of By, being at t afunctionof M. ‘ Let us consider a point My, Mz, or Mj on jo een or as . For instance M, determines a pair(X, t') and it belongs only to fe) the vortex disse pte (7) for 7» Us, Uy denote the components on the moving axes S of the velocity Vp (0) of their origin O and u, , us , Ug those of their angular velocity Q ,. According to (6.6) and (6.6'), we may write : 6 (iM, t). = 2 @ (Myra (ee (6. 7) In this formula, M is the point moving with the body which coincides witht Mi! tatiti= t,he DRheg @ lj 's only depend on the hull geometry and on the position of the system of axes S with respect to the body. Furthermore, it follows from the second equation (6. 1) that (ap) ! z= ! S, ! bi ' q (M t) ® (M t) © (M', t)) .(M', t), with M'cQo and M'M! = mie (0)... (6. 8) Consequently, the density uw of the normal dipole distribution on Pi which generates ao. the solution of the regular integral equation : 1232 Vortex Theory for Bodtes Moving in Water (6. 9) Se Me pS ae ME te eM! Let us consider the following three particular cases: Casea - Vv. =0,; no free vortex is shed by the body. According to (6.7) and (6.9) we obviously obtain wim. t) = 4% v(m) u(t). (6. 10), Case b - Me = 0; o (t) aad | ae aes (0) = constant ; the fluid motion is steady with respect to the body. Hence, the support of the free vortex sheet is close to the surface generated by half straight lines starting from the curve in the direction of -V 7 (0). At P fixed with respect to the body and located on the generatrix starting from B, we have Bee ee 3 es (6. 10), M, and uw, being the values of u in the vicinity of OF on the parts oe ? = of > adjacent to each other along FB: To determine Mon bm a complementary condition is needed. Since the pressure p is necessarily continuous through the free vortex sheet > ¢ this condition expresses that p is continuous on > eS through@e : EA33 Brard Casec - a = 0; the motion of the body consists ofa translation nearly uniform with small variations of V,,(0) and o, In that case, the support of the free vortex sheet x f is practically the same as in case b. This case will be studied in some details in the next Section. VII - HYDRODYNAMIC FORCES EXERTED ON THE BODY The case a previously defined (Section VI) can be subdivided into two cases, a, , a, according as the motion of the body is uni- form or not. Case a7 The body is in uniform motion Since Vie is nullinside D, and no free vortex is shed by the body, the fluid,motion is steady with respect to the system of axes S. Hence C*R =0 inside D,, and in this domain, Euler's equation ot dV SN anal = p ais dt reduces to l ae 1 2 — V Se a ee -V (— Pn ashe Vag Be Ca AR 1 2 a 1 Z - V(— Q - = i i (3 me? 5 Vp) inside D, We have therefore tak. i _lbsietsn uF bay? ? P4(M.) =F aS (M.) a VRiM,) + constant on 2d. CH. 1) The system oA of hydrodynamic forces exerted on the body consists 1234 Vortex Theory for Bodtes Moving tn Water of the elementary forces [palme) ny d i ono . This may be writ- ten in the form : iG & Q? r“(M_) m7 ep T(M) AV,(M) ) aX am) 2.2)e3 where Va(M) = + yim) + 7 (| = = FM) ; (7. 3) This relative velocity is calculated inside the bound vortex sheet by using the vortex distribution ; it can be considered as the incident velocity on the element of bound vortex filament T( € a>.) which is at rest with respect to the body. The equilibrium of the set E of fluid points located inside D; requires ly > 1 72 Ma a = ax y, = V a Q Aes E BW MEANS: day" Sele This gives 1 2 Zz a ee) ! p4(M,) 5 =e (M,) + constant on ag (7.4 The system of forces exerted on the set E' of fluid points belonging to the bound vortex sheet is thus —_> ie [(oar,) Hyg - P(M,) Fy) 2X09 | ; |- pT(M) AV, (M) aX] a 5) 4 Finally we have : L235 Brard where Fa Gaies (7.7). 1 m_ being the mass of E, and r(C) the vector orthogonal to the axis A of the helicoidal motion of the body with its origin on A and its end at the center C of the domain Dj. Fo is thus the centrifugal force exerted on the set E of fluid points, i.e. the centrifugal force acting on the fluid mass ''displaced"' by the body. Case as This case differs from case a, in that V (0) and (or) Q E may depend on time. a Q 724).” “does ao lodges aopky te ean ey Unetae ay ger apply At eS dQ y E curl Ye = eer Se : (7-8), and - VE cannot be identified with +—Vpq . The equilibrium of the set E of fluid points therefore requires that the expressions for the absolute force per unit mass be different inside Dg and inside Dj. iF = V denotes the expression of that force inside D, , its expression inside Dj is F + F‘, and we must have ee Woe (UG: es ty i NE oes p d ae ; dt E ene ty ag av. ay I ! =2 = il ' = i I curl F'(M »t) ae curl— (M »t) curl +> (M »t) inside D, (7. 9), i Since V'(M',t) - V.(M', t) = - V © (M',t), h —> Sacre Fe gee ee Vw (M',t_) ot oO : 7 8 1236 Vortex Theory for Bodtes Moving tn Water Ss. ¥, ' a ! _4 Vv ' ' pg(M', t.) pM Hye te ay Yat t) + Vw Oe £ ] (ts oe or i (M!,t_) = -9_ @ (M!,t_) +> a@ r? to INA | ond, LO Lk AO sae (Fi 10") a2 The unknown function W, has to be determined by the condition that ue Woo eee ais: Ss Bg (M: : t) reduce to 5 | es (7 bas, when case as reduces to case ay This implies that W, is harmonic inside D.. Since u j the derivative -—~—® (M;,t,)) contains two kinds of terms. The terms of the first kind are those due to the variations of the u; 's when the rotation of the system S of axes is ignored. Let us denote a(t.) &,; (mM) the sum of the terms ofthe first kind. The terms of the second kind are due to the fact that M; is fixed on ),;, while M! is fixed in the space referred to the fixed axes S'. Consequently they sum up to -V (M: , t) Vd, (M:, Le L237 Brard We thus obtain : oy (Re) be r= ee) vb, (Mi, t), (TNE ate This expression takes the same values in both cases aj and a, provided that in the uniform motion of the body corresponding to case a,, Vp(0) and Q ¢ be identical at tg with those of the real non-uniform motion. Since %, is harmonic within Dj , it is uniquely determined in that domain by (7.12). We have Hpsiaie) =~ Bale) & 0m) + 82) 22(4,,4,) on Dy (7.13), 2 Now, let us consider the velocity potential ® inside D,. Euler's equation gives, after (4. 1'), 1 ODi a 51 1 92 aks lace Pas = ? oP ? ->V J p ee at . 7 ‘i z Eto) Z Ce tS) 2 Pe ts) on ds, (7. 14) at By comparing with (7.1), , we obtain 1 OD (m',t ) = 0 on >» ~=incase a,. (7. 15) ot eo e 1 ay In cases a, and a,» we have D= 2 inside D, ‘ (7. 16) and according to (6,7) and (6.10), 1238 Vortex Theory for Bodtes Moving in Water ® (Mi, t.) =@ 4 (M:, t) + u(M',t ) = Zou (t,) Jey + M; wy) on Lu. d (7.17), Hence, with the above notation, we have : (M,) + ¥ (0%) | yr (M!,t.).v,(M!, t,) t fe) The second term on the right side only depends on the instan- taneous velocities. It is the same in cases a,and a, for equal velo- cities Vp(0) and QE . But, in case ay the first term on the right side is null ; the term on the left side is null also by virtue of (7. 15), Hence we obtain : 3 ! = = if a ® (Mi, t.) = pede fo, + H (00) (ire 18). Let us put dvi {* = a dM AN (M) dv (M) (7. 19) The system of forces exerted on the bound vortex sheet is that of the elementary forces [F| | — “e) = x es | ue) ee s o =} s Qu Kd | 4 i] > =) . oa s iS BY uM iS umes + 1 a. we Lge a The system of forces exerted on the set E of fluid points is that of the elementary forces 1239 Brard ut) #, .204')] -|-» (Pato (mM) Hy, — 0% r “(M) oy dy my) (7.22) - -FUt je 3 (6) ® 5(M,) a2] Putting es oe [a s roe eee. - » T eta! > Ka we see that the final expression for the system of hydrodynamic forces exerted on the body is as follows a ee is the ''quasi-steady''system of forces and BA is that due to (7.23). qs. the so cailed ''added masses". vase b- This case is a limiting one, involving the assumption that the motion of translation of the body became uniform ata time t consi- derably anterior to the present time t,. Since 2 = 0, one has (7. 24) b q cannot reduce to a torque because of the effect of But the free vortex sheet. 1240 Vortex Theory for Bodies Moving tn Water Casec - The x'-axis is taken in the direction opposite to the mean velocity Vir of the origin of the system of axes moving with the body. Let uy be the absolute value of the x'-component of V},. The sup- port of the vortex sheet may be considered as generated by half straight lines starting in the positive x'-direction from a certain line on a . Let Py; and B; denote two points on the same gene- ratrix, By being infinitely close to G. We put & = |B} P}| . Let 0 = - © be the time of the beginning of the motion of the body and t, the present time. If the fluid motion were steady as in case b, one would have M-(Py,t.) = u,(By,t,). Because it is unsteady, one has For reasons which will be elucidated later, we put (By ’ > = ’ a 6M He ; pw AP? t 30+) = (Bet) eg ' of ATS ! 0 ! emer 2 i (Pp fist t') uw (Bi t.) u(P rt st -t ), with (7.26) | One sees that, if u(Be, t) is a constant in the time interval (t',t ), then ys(P},t,; t,-t') is equal to that constant at t, , provided E up(t,-t'), then pw ¢ at P} still depends at time t, on the variations of w,(Bj,t) for t< t'. In particular, the 1241 Brard fluid motion would be steady at t, if mw ¢(By,t) were constant during the infinitely large interval (- » , tj). One may write ®, (M',t. 3) 04) «= aff u (Ph, to ; 0+) om ae ad, (Pj). (7.27). f This expression is the velocity potential due to the normal dipole distribution on )/,. The expression ! é Sel Bela ye ! , at 3 1 ! JOBS RG aa Ss | Ae a a) np, M'P} ad (Ps) ‘ f (7.28), is the difference between the true expression of D(M', t, ce +) given by (7.27), and that which would be reached at t, if the motion of the body had been uniform in the interval (t', to),the velocities Vz (0) and E— during this interval being constant and equal to those of the real motion at Bee Obviously : +O; (7.29) @,(M',t, ; t+ oo ) is the limit reached when t' ——~- © . This limit defines the steady case b when Vp(0) and Qf are constant (since t 0! The difference 5@,(M', t 3) foo prey Z piM'.t, 3 +0 )- @(M',t. 5 0+) (7. 30). 1242 Vortex Theory for Bodies Moving in Water is the deficiency of Py, that is the difference between the ® ¢ deter- mined in the steady case for the present velocities VE(0, t,) and la 2 (to) at t, andthe true @ in the real motion at t, Let Be, and at denote two points infinitely close to B}, but located on 3 and 53 ’ respectively. We have ! = ! = ! . u (Bi, t.) HAP ans ts) Big t) (7. 31). Mis the solution of the integral equation -H(M', t.) Nie a _ d Lu (P!) Sy = @ (MM! it ) <4@ ! . ! My a (Mit, 30+), M! being on pide (7232) We have u(M',t )= u(t) u,(M) -G o (Ms, t 3 0+) ) (7.33). Ae a linear and wee eneous functional the argument of which is the function ® ¢ ( (M', ; O+) : it may be written in the form GC (M' me KUM, P) OP eG +) dd (P').) ees Let us substitute (7. 27), into (7.33), and take into account (7.26) and also the condition (1) K is the resolvent kernel of equation (7. 32). 1243 Brard s = ! Pp eh i eh es , t). (7. 34). Then we obtain for determining M(B} ; t, 10+) a_Volterraintegral equation. This shows that the fluid motion time _t,. does depend not only on the present velocities VE(0, t,) and ‘-@ Q R(t), but also on the history on the motion during the whole interval t < ty Let us consider now the system aoe of hydrodynamic forces exerted on the body. We readily obtain Ey ee cae WA +Z, (7.350 where Fe = [ar | 3 - pT(M) AV, (M) aX (on 2 E 1 eal Q i I > =) by © H ro¥ 'o) pay iI ae) nN] es (t) r (M, t) #422000], EF. . fs pia )(® 45) + ou a )a,, ads (vo |. A ; AS D(M', t 5 0+) ve an ax’ | ; (7. 36) | The definition of "| and of ee are exactly the same asin formulas (7.20), and (7.22), , respectively. But the system of forces [dF ] does not coincide with that defined in case a. This is due to the effect of potential ®, on potential ®gy. On the con- trary, the systems of forcesY 4 coincide in both cases if the sys- tems of the six accelerations u;(t,) are identical. Let Sf op denote the system of forcesH evaluated at t, under the assumption that Ve (0) and {2 FE coincide with VE (@,t a and f(t.) in the time interval (t',t,), with t'< to. Then the difference 1244 Vortex Theory for Bodtes Moving tn Water = y + G = dF"! a dF'! a dig ek hee ito atk fool oe bee is the deficiency of Yr due to the deficiency noe defined in (7. 30)e:. For the sake of simplicity, the true system 6@q of hydro- dynamic forces acting on the body is very often replaced by the esti- mate i = Baling the Bpeorhunghes (7.38). The system es - Fo is called the quasi-steady system of forces : are -“S% 7 Bhs (#82). The error involved in the substitution of Gea for Mas! is Ra a ee Seal Like A 5 ’, represents an inertial effect, but due to the free vortex sheet only. Let us assume, for example, that a jump of Vx (0) and Q E occur in the infinitely small interval (t. - 0, tN and that these two velocities remain constant for t > ty. It follows from the third equation (7.26). that ! . = bu (Py : to ; O+) 0 for every &— . (7.41). Hence the free vortex sheet is not immediately altered. But, one has : 1245 Brard 3 im tie 3 AR ae = ® (M tii 0+) = 1 ff 2-0; t) sa, MPI a2 (P) (7.42). » Koi at t, has the same components as at t, - 0. ee is null at t,, but the yore of forces necessary for Sesh the jump is infinite. Furthermore, at=to +70 Gs is generally finite. Of course, when t +oco , the deficiency 6 a calculated by taking into consi- deration the new velocities Vz (0) and E tends to zero ; also tends to zero and a ete ee ee (7. 43) Cc The effect of WS A may be considerable. For instance, it has been shown that the jump of the lift of a wing with an infinite aspect ratio is at tg t+ 0 equal to half the difference between its final value and its value at tg - 0 (See [2] ; [3] \). THE ORIGIN OF THE FORCES EXERTED BY THE FLOW ON A BOUND VORTEX DISTRIBUTION The above considerations started from the idea that a dyna- mical relationship necessarily exists between the hull of a moving body and the vortex distribution satisfying the adherence condition on the hull surface. Another viewpoint is that any vortex filament which does not move with the fluid is necessarily submitted to forces exerted by the adjacent sets of fluid points. The proof is classical. It is sufficient to summarize it. Let Foe a vortex filament, Fa its bound part, ds, an arc AY, . Let O denote the middle a the arc ds, >) Oz. fe axis in the eee of ds » ane Fr ,4@", 2 2 system ‘of semi-polar coordinates. Let D' denote the domain 1246 Vortex Theory for Bodtes Moving in Water Hence D' does not include the arc ds, . We consider the relative motion of the fluid with respect to the system of axes just defined. The velocity may be written in the form Wee Va - BV") OVE , ae Viol 6V') where Vo defines some incident flow, V' is the velocity induced by Yana 5 V' that induced by the part ds, of LF Let dE' be the set of fluid points inside D', and dE, the set of fluid points belonging to ds, . One easily sees that the momentum I (dE|) is null, and, therefore, that, I (dE') being the momentum of dE' , d+ 1 spe niGll 1 te a. deere is af 1 GE) =) Tae rae)= ff p V, @V,) as’; (m is in the outward direction , and S' is the boundary of (D'+ds). One readily obtains : Iimeud 1 = ee? (de) = —— Pr ee led : R50 dt Co ae Arata bon Tsk (7. 44) where [ is the intensity Pe ree are aay j ie being the unit vector tangent to ds, and Vj the finite incident velocity on the arc ds 1 On the other hand, by the momentum theorem, one has oer (dE) = dE f] -pu dS", dt ia an 1247 Brard —_> where -dF* is the force exerted on dE' by dE, . From Euler's equation (4.1), we have inside D': ¥ = a5 (curl V_) A(V, + 6V") -V : (V+ v1)? Bo wae at o i a ate : This gives, when R——-0O, Pq =tS4p V; ’ 5V' + constant on S, and é a =>, 1 =F a iim — T(E") = —-dr -— 2 AV Xde (7.45) dt n: SZ i 1 R>0 dF. Set oa whi eli (7.46) dF ip is the force exerted on the arc ds, of the bound part Ly of & by the adjacent sets of fluid points. The force vanishes with Vj, that is when the arc ds, moves with the fluid. Formula (7.46) is of practical interest when the vortex dis- tribution equivalent to the hull is replaced by a unique concentrated vortex and a suitable distribution of sources or normal dipoles on the hull surface. This formula does not imply that the fluid motion around the bound arc of the vortex filament is steady. If [ varies with time, one has to consider that another vortex filament &#’, of intensity dI , appears in the time interval (t, t+ dt). & is distinct from L although their supports have a common part ; the free part of &" does not coin- cide with the free part of ae Let us consider now a flat vortex tube inside the bound vortex sheet over a hull ye . This tube has athickness ¢€ , a width do Let ds, be the element of arc of the tube. Applying (7.46), we have 1248 Yortex Theory for Brdtes Moving tn Water ' = = T Vv =o IT A = dF i pTa ¥, dsias p fF). ae a | Teas ie = -e (TAV,),, 42 (M) (7.47) because V2 (M) wey icika et Wp (M) . dit —_ We know that, if the motion is unsteady, d T must be re- placed by dF. However there exists no contradiction. The term - p-Ck= aS which appears when the vortex tubes belong to a sheet eos from the integration of -P S* through the sheet. When the vortex tube is isolated, there is no igcontinuity of io on the surface S' and the contribution of the term p OV in the integration of pVp on S' is 0(R'ds)), thus negligibly Sthall. This is not the case when one deals with a sheet) CASE OF A HULL EQUIPPED WITH MOVABLE APPENDAGES The treatment of the problem arising from the presence of such appendages obviously depends upon their position with respect to the hull. When the axis of the rudder coincides with the edge of the stern, this rudder may be regarded as a part of the hull. The shape of the hull varies with time. At each instant t there is however a vortex sheet adhering to this hull. The method of Section VI therefore applies in principle. But separation may occur at the leading edge of the rud- der because of lack of continuity. Furthermore the effect of the vis- cous boundary layer is never negligible in this region. When the rudder (or diving plane) is at some distance from the hull, the rudder behaves as a lifting surface with a small aspect ratio. Because of the thinness of the rudder, the concept of the ''in- terior'' of the rudder becomes meaningless and the thin wing theory is to be used. (1) By using (7.46) one can simplify the expression of the fictitious force F' inside D,; when —— #0 ~ See [13] Chapter IlI,B, art. 9. t 1249 Brard VIII - THE APPLICATION FIELD OF VORTEX THEORY IN SHIP HYDRODYNA MICS GENERAL - It has been seen in the preceding Sections that the vortex theory applies to any body moving in water whatever its motion may be. The methods to be used in practice may considerably vary with the shape of the hull, the motion of the body, the boundaries of the fluid domain. To perform the calculations, it may be advantageous to substitute normal doublet distributions for vortex distributions because a regular scalar Fredholm equation then replaces a vectorial singular Fredholm equation. This is why the vortex distribution kinematically equivalent to the body has been divided in Sections V and VI into two parts, one of them being equivalent to a normal dipole distribu- tion. When the motion of the body consists of a pure translation, the vortex theory leads to computations which are not more complicated than those involved when source distributions are used ; they are even simpler when the distribution of the pressure over the hull is needed. This can be of interest when there exists an incident unsteady flow. The theory extends to the case when there exists a free surface, at least when the condition on the free surface is linearized. Neverthe- less, some difficulties are to be expected when the hull pearces the free surface. It is necessary to close the vortex filaments by their mirror images with respect to the plane of the free surface at rest. This can lead to difficulties analogous to those encountered in the case of the Zero-Froude number approximation when the hull is replaced by a normal dipole distribution [6,7 ] One of the main features of the theory developed in the pre- sent paper is that it includes the case of bodies which are neither thin, flat nor slender. However, to the knowledge of the writer, the vortex theory is still used only in cases of thin lifting surfaces. There is thus a need for more general methods and one of the purposes of this paper is to give means to extend the field of applications. In this Section the present field of application is briefly out- lined. Yet the problem of maneuverability and control of marine vehi- cles is examined in a more detailed manner, for progresses in that domain seem to be strongly needed. D'ALEMBERT'S PARADOX There exist many proofs of this theorem. The following one may be of interest,for it clearly explains the physical meaning of the 1250 Vortex Theory for Bodtes Moving tn Water hypothesis required for its validity. The body moves with a constant speed VE in an unbounded, inviscid fluid at rest at infinity. One supposes that no separation,oc- curs and that no vortex sheet is shed by the body. Let ()> , T ) be the vortex sheet which allows the fluid to adhere to the side pate of the hull. The vortex filaments & are closed rings on ss - They are orthogonal to the relative streamlines @ . Let i. be the unit vector tangent toa line Y inthe direction of T, and ig the unit vector tangent to a line € in the direction opposite to the relative velocity VR . A vortex filament © is defined by the curvilinear abscissa ag of its intersection with a line "A, o chosen once for all. The intensity dI of the vortex ribbon located between two vortex filaments ar (wg ) and oy a( o, + do.) isa constant. One has : dT or, A dg =e 6 ) dig: = ¥(o)) age: (8. L) The relative velocity is YR wh a Ve . Thus the hydro- dynamic force on the part (d>, , |) of the vortex distribution is € aF iM at + AF , with CL pT (M) a (- V,) ad (M), 1F. = Prawn ¥ 04) allan 1251 Brard The system of forces dF. is that of the forces exerted by the vortex distribution on itself and is therefore equivalent to zero. Also the systems of forces SY, aS (i of the general theory are equivalent to zero. Consequently the system of the hydrodynamic forces exerted on the body reduces to that of the forces d Fp oe Bis gives the general resultant ‘ oo off B00 6 vy ak (M) 2B and the resulting moment with respect to a given point 0: MH ff OM afFo) Vp (00)] d2U(M) . A —_ Since ds(M) =0, onehas: L (o, = bow a8 Pp cf “of ae (c,) taf ds(M) = 0 (8. 3) Stern L(o,) Furthermore : —> Ta off [Feo iz on | -V. fom. vo] dd (M) ya > ba ty 3 = =p E o,) 6 stern Ye bow of hg Med. (Vv. . OM) ds (M) . s ie tern bow =f > ag (V7, . OM) ds’(M) . (8. 4) (o ) stern This term is not null, at least in general, L252 Vortex Theory for Bodies Moving in Water Equation (8.3) expresses the d'Alembert paradox. If free vortices are shed by the body, then (8.3) is no longer verified. Na has two components ; one of them is a lift and the other one is the "induced resistance". KUTTA-JOUKOWSKI'S THEOREM - The first version of this theorem concerned wing profiles in a uniform motion of translation, with Vo = 0. The wing is an infinite cylinder and its profile € is its intersection by a plane normal to the generatrices. The problem can be considered as the limiting case of that of a wing, when the aspect ratio of which tends to infinity. When the aspect ratio is finite, the relative velocities on the two sides camel P pe of the wing near the trailing edge are equal and opposite. This follows from the continuity of the flow between Ves and and between 2, and va . Hence, in the case of a wing profile, one must have v2 = 0 at the trailing edge B. This is the Kutta condition which determines the density of the vortex sheet on > a taat 1s the ratio “I. on the contour © of the profile. The Kutta condition holds when the motion is unsteady. The theory developed in the preceding Sections applies to wing profiles. But because one deals with two-dimensional motions, the concept of complex velocity potential can be used and leads to considerable simplifications. In particular, one can associate a vor- tex distribution and a source distribution on the skeletton of the hull to obtain the desired profile shape (1) : (1) The determination of the exact distribution of the velocity at the leading edge requires some care [3] : 1253 Brard WING PROFILE IN A QUASI-RECTILINEAR, NON-UNIFORM MOTION This problem has been considered by many authors and fi- nally solved by von Karman and Sears in 1938 eal . These authors gave the correct form of the Volterra integral equation for the total circulation [ around the profile. They showed, in particular, how the circulation behaves following a perturbation in form ofa step function occurring at time t,. The response time is long and appro- ximately corresponds to a path equal to 15 times the chord, but the effect of the total virtual masses is considerable and the lift at time to + 0 is half its final value. In [3] the writer has completed the calculation in order to obtain the pressure distribution on the profile. UNSTEADY THREE-DIMENSIONAL MOTION OF A WING WITH A FINITE ASPECT RATIO There now exist methods for solving the previous problem in the case of a thin wing of finite aspect ratio when the amplitudes of the deviations from a uniform motion of translation are small. Dat and Malfois [4 ] have given a linearized theory using an accele- ration potential py = We ist with 09 F ist = oe + is" P \ £6 =P : ¥ Bin Sie ae e Vr being here in the negative x-direction. The pressure P is given by The component in the vertical direction of the velocity is Wie wie wie S | K(x -§ y-7) 6p(§,n) dédn, 4anpVi wing 1254 Vortex Theory for Bodtes Moving tn Water where 6p is the pressure difference between the two sides of the wing. The difficulties due to the singularity of the kernel have been overcome by the authors who obtained a very good agreement between calculated and measured values of 46 p for harmonic motions. PROPELLER THEORY The determination of the steady and unsteady forces acting on a propeller is a problem of importance (efficiency, risk of cavita- tion, vibrations, noise, etc...). We just mention it for it is beyond the purpose of the present paper. SHIP MANEUVERABILITY THEORY A first step in the mathematical maneuverability theory con- sists of the determination of the bound vortices when the relative mo- tion is steady. This problem has been considered by P. Casal in his Thesis dissertation fora shipata constant drift angle in the horizontal plane and for a ship in a forced turning motion in the same plane(1), Several drastic simplifications were made : (i) The waves generated by the ship are neglected. Thus one deals with the Zero-Froude-number approximation. (ii) The ship is assumed to be infinitely thin ; the heel angle is ignored. (iii) The free vortex sheets are attached to the hull along the keel and its mirror image with respect to the plane of the free surface at rest. (iv) The free vortex filaments start in the direction of the bis- sectrix of the angle between the local velocity of the body on the keel and the keel line. In fact, because of the errors due to the first three assump- tions, the fourth one is essentially used for the determination of the (1) Casal's Thesis was written 20 years ago when the author was staying as scientist at the Bassin d'Essais des Carénes (Report Bas- sin d'Essais des Carénes - 1951) and published much later [9] . 255 Brard behavior of the curves giving the density of the bound vortices in the plane of symmetry of the ship (y = 0). Let §& denote the reduced abscissa 7 , the origin of the moving axes being at the center of the plane y=0. The integral equation of the problem expresses that the velo- city induced by the total vortex distribution is tangent to the plane y = 0. We give below the density f( & ) of the circulation. In the case of an oblique translation, one has : draft/length ratio, a = drift angle, and ree) 6 I & oe I Str + ° nN ; 6 = Dirac function projection on the plane y = 0 is at the abscissa Ey - — SLA ates one has : rVL L 38 = € —— : Sad 5 aa a eae A OU AD st Deen 8 GEASS) en Figures 8.2 and 8.3 show the graphs of ¢ in the first case and in the second case respectively. The force Y and the moment N with respect to the z-axis (vertical upwards) are 1 Zz ie es 1 See oe oy Eee eee = SV re cos ala sina + Boe rox | (5) 1 1 Zz ib Telly Ni =f? == p6¥ he nests (RAL Le ws =| b 7.4 1 cosa | sina By = Cc, an lan | Cc, c(s,) 1256 Vortex Theory for Bedies Moving in Water A, B, A, , B, are constants relatively close to unity, and C, C, are functions of — ;. As Ej is practically invariable, C and C, may also be considered as constants. In fact all the coefficients depend on the hull shape and must be experimentally determined. Let - © , = @ _ bethe y-compo- nent and the moment of the hydrodynamic force exerted on the rudder, M, the added mass for the double model calculated by neglecting the free vortices effect for a non-uniform motion in the y-direction, and I, the similar inertial moment in a non-uniform rotation about the z-axis. We put: 1 L 2 —— T € k ’ ke = ke tk fe) 2 fo) fe) Furthermore, because the circulation can never take immediately its asymptotic value corresponding to the steady motion defined by the present values of a and Baer the present value of Y is not given by the above expression, but by Y - Y, . Similarly the hydro- dynamic momentisnot N, but N-N,_. Y, and ny are the defi- ciencies due to the history of the motion. Finally the equations of the unsteady motion are as follows : b+Y L L da 16; L i | 2 = - — —-=Aa+B —+C — |—|- we on ae Se ae es Last. Guba es eee 18 d 6 L i i; LZ = (ees) Ss A Ye geet ogi (Rae ae Cas av > at ‘aR? iy. else toe | I 2 aoe ae Let = A = S ete Oke Ee Be : : 4 L : : One sees that in a steady motion, the ratio oR 28 given by @ E; ie t A +A,)——_>~———_ = SO (A GC + A —— _ j___}. ( )4 oe Slater! the tee Ci) aR aC 2 L257 Brard If S >0, this equation has one and only one,root. If S< 0, there exist three roots for small values of = phen mt and, in particu- lax, for ,@® v= 0... Seer cae ima\sne hee ate == ), be the three roots with (+=), 4 Ss ae J, . It is to be expected 2R that the steady motion sa )o is unstable. This is confirmed by the study of the equations of the un- steady motion when Y, and N, are neglected. After a perturba- tion, the stable steady motion is reached again without oscillation. When one takes into account Y> and N, , that is the terms depending on the history of the motion, one sees that, if S > 0, the steady motion is still stable, but, after a perturbation, it may occur that the transient motion be oscillating. It may even occur that no straight motion be possible for © equal to zero ; the head is cons- tant in the mean, but it is continuously oscillating. Oscillating motions in calm water are therefore a conse- quence of the delayed circulation around the ship. They appear when the ship has to proceed a long path before the circulation becomes close to its asymptotical value. In spite of the rather rough assumptions involved in Casal's theory, it appears that this theory is qualitatively in good agreement with experiments, except for what concerns the position of the result- ant force in the oblique translations. According to the above expres- sions for Y and N when R= , this force should intersect the plane of symmetry at a point practically invariable and located inside the ship. Experiments on models show, on the contrary, that this point can be located ahead of the bow for very small angles of attack. Then,when the angle of attack increases, the & of this point rapidly decreases and, finally, takes a value rather close to that assigned by the theory. The explanation of that discrepancy seems to be that the free vortices are shed along the stern-post and not along the keel line when a is very small. Because self-sway motions are very undesi- rable, attention is to be paid to this point. That is also for this rea- son that we have indicated above the existence near the bow of a very strong vortex represented, ina first approximation, by a 6 -function. In the past, the wanted maneuvering qualities mainly con- cerned the characteristics of the motions at large rudder angles. 1258 Vortex Theory for Psedies Moving in Vater Some cases of course unstability had been stated, but regarded as rather exceptional and the behavior of the ship in transient motions was not a prior matter of concern. The situation became quite a dif- ferent one with submarines achieving very high speeds. A dynamic stability in the vertical and horizontal planes of motion are required, and an automatic pilot system is used for performing combined ma- neuvers in both planes. Surface vessels such as big tankers also need remarkable maneuvering qualities ant to that end, they are also equipped with automatic steering systems. Nevertheless, the automatic control of the ship does not solve all the difficulties involved in maneuvering. One can even say that, from a certain point of view, it gives rise to new problems. The writer has described some years ago some among the methods used at the Bassin d'Essais des Carénes for studying ex- perimentally the case of submerged bodies, determining the coef- ficients of the equations of the motion and predicting the real motion of the full scale vehicle [10] . Ina recent paper [1 1] , M. Gertler de- veloped analogous views on the purpose of this type of matched ex- perimental and mathematical researches. However powerful this way may be, it leads to the introduc- tion in the equations of motion of much too many coefficients and per- haps in an unappropriate manner. This situation does not favour the prcegress of the knowledge of the fundamentals in maneuverability. In ee) , the writer drew the attention to the time response to a maneu- ver and the risk of erroneous interpretation of experimental results. The writer is of the opinion that new purely theoretical researches are needed. Casal's thesis has been given as an example to base this opinion. The present paper has been inspired by the same line of thought. CONCLUSION Although the Vortex Theory plays an important role in many Chapters of Ship Hydrodynamics, it does not seem to be used in all the cases where it could really be fruitful. It is so when one deals with the Ship herself. Several explanations of the rather reluctant attitude of the Naval Architects with respect to the application of vortex theory to 1259 Brard ship hulls could probably be found. A fact is that an arc of vortex fi- lament does not generate a velocity potential and in practice the re- presentation of the hull by a source distribution can be simpler than its representation by a vortex distribution. However the vortex theory, which allows the fluid to satisfy the physically meaningful condition of adherence to the huil, certainly offers mathematical models closer to the reality than those drawn from the other types of hull representa- tion. Furthermore, it leads more rapidly to the determination of the distribution of the pressure and of the velocity on the hull surface. The present paper is therefore an attempt to explain the fundamentals of the vortex theory and of its application to the kine- matics and dynamics of bodies moving in water. After a brief survey on the various aspects of the vortex theory in inviscid fluids (Section I), one will find in Section II the Poincaré formula which permits the calculation of the velocity ina fluid domain when the vorticity inside the domain and the velocity on its boundary are known, and in Section III the application of Poincaré's formula to the determination of vortex distributions kinematically equivalent to any given ship hull. The class of these distributions is infinite. Each consists of a volume distribution inside the hull and of a surface distribution over the hull. The volume distribution can be chosen arbitrarily. The surface distribution associated with it is de- termined by means of a singular vectorial Fredholm equation of the second kind. Section IV gathers material to be used later to solve the dynamical problem. Section V is devoted to the study of the structure of the vor- tex distribution which permits the fluid to adhere to the hull surface. The surface distribution is the sum of infinitely flat vortex tubes call- ed here ''vortex ribbons". The vorticity inside the hull is twice the angular velocity of the body. Thus, if the angular velocity is not null, the intensity of each vortex ribbon is not a constant along its length. Furthermore, if free vortices are shed by the hull, some of the vor- tex ribbons do not close on the hull. To overcome the difficulties aris- ing from these circumstances, the vortex distribution generated by the body is divided into two distinct families almost independent of each other. One consists of the volume distribution and of the surface dis- tribution associated with it so that the velocity induced by this first family outside the hull be null. The second family consists of a vortex sheet entirely located over the hull when no free vortex-sheet is shed by the hull. In the opposite case, it includes the free vortex-sheets. 1260 Vortex Theory for Bodies Moving tin Water As shown in Subsection D, it may also include a volume vortex dis- tribution in the wake when the motion is unsteady. Section VI deals with the integral equations determining the two families of the total vortex distribution. The singular vectorial integral equation related to the first family can be replaced by a sin- gular scalar Fredholm equation for a Neumann interior problem. It can be solved once for all whatever the body motion may be. The in- tegral equation for the second family reduces to the scalar regular Fredholm equation of the second kind for a certain Dirichlet interior problem when the fluid is unbounded and at rest at infinity. In the most general case it becomes a Volterra equation expressing the so- lution in terms depending on the history of the motion. Section VII is devoted to the study of the system BA of hydrodynamic forces exerted on the body. As stated before the total vortex distribution determines inside the hull a fluid fictitious motion which coincides with the absolute motion of the body. For this kine- matical condition to be compatible with the dynamical equilibrium of the fluid, it is necessary to introduce a certain system of fictitious forces per unit mass inside the hull. The system od, of hydrodyhamic forces exerted on the body at t. can be written in the form where ae s, is the quasi-steady system of forces, that is the sys- tem to which qd would reduce if the motion of the body were uniform in a large interval (t', t,) eo is the system due to the so- called added masses ; it is independent of the free vortices shed by the body. There exists a difference between the structure of the free vortices at tp and at t=+00, the latter being evaluated under the assumption that the motion of the body is uniform to t > t,. This difference affects both the bound vortex distribution on the hull and the incident velocity on it. It entails the term - 1Sfe . The last term Pins ; : ; ¥ AAG ‘a is an inertial effect due to the partial derivative =e at ty. of the bound vortex sheet. Fa reduces at t, + .0 to Za, 5 (t, -0) + YF: (to + 0) if the body motion is uniform for - to +0, but discontinuous between to - 0 and ty. 1261 Brard Section VIII consists of a brief synopsis of the present appli- cations of the vortex theory to ship hydrodynamics with somewhat greater emphasis on ship maneuverability. Attention is drawn to Casal's thesis about this problem. Casal's theory certainly involves too drastic simplifications and some among the conclusions are unac- ceptable. First, a ship cannot be considered as infinitely thin, and even if such an assumption could be accepted in a first approximation, it would be necessary to satisfy the boundary condition on the whole surface of the longitudinal plane of symmetry of the ship. It is still necessary to resort the empirical or semi-empirical methods. But, in the writer's opinion the part devoted to theory is really unsuffi- Crene: The purpose of this paper was to prompt researches in that direction, REFERENCES [ 1] ROY, M., '"Aérodynamique-des ailes sustentatrices et des hélices'', Gauthier Villars, Paris, 1928. [2] Von KARMAN, Th. and SEARS, W.R., ''Airfoil theory for non-uniform motion", Journal of the Aeronautical Sciences, vol. 5, No 10, August 1938. [3] BRARD, R., ''Mouvements plans non permanents d'un profil déformable.'', Association Technique Maritime et Aéronau- tique, No 63, 1963. [4 ] DAT, R., and MALFOIS, J-P., ''Sur le calcul du noyau de l'équation intégrale de la surface portante en écoulement subsonique instationnaire'', La Recherche Aérospatiale, No 1970-5, September-October 1970. [5 | DELSARTE, J., 'Included in ''Legons sur les tourbillons" by H.. VILLAT, Gauthier Villars,..Paris, .19350. [6 | KOTIR. ae and MORGAN, R., 'The uniqueness problem for wave resistance calculated from singularity distributions which are exact at Zero Froude Number", Journal of Ship Research, vol. 13, Nol, march 1969. 1262 L7] [8] [9] [10] [11] [12] [13] Yortex Theory for Bodtes Moving tn Water v BRARD, R., ''The representation of a given ship form by singularity distributions when the boundary condition on the free surface is linearized", Journal of Ship Research, vol. 16, No 1, march 1972. DARROZES, J-S., ''On the uniformly valid approximate so- lutions of Laplace equation for an inviscid fluid flow pasta three-dimensional thin body'', ONERA Report. CASAL, P., ''Théorie de l'aile portante de trés faible enver- gure'', Publications Scientifiques et Techniques du Ministére de l'Air, No 384, Paris, 1962. BRARD, R., BINDEL,\S., LECOEUR, G. and CHAVEREBIERE de SAL, A., ''Le modéle libre de sous-marin du Bassin d'Essais des Carénes'', Association Technique Maritime et Aéronautique, No 68, 1968. GERTLER, M., "Some recent advances in dynamic stability and control of submerged vehicles", International Symposium on Directional Stability and Control fo Bodies moving in Water. The Institution of Mechanical Engineers, London, LORZ. BRARD, Rz, “A vortex theory for the manoeuvring ship with respect to the history of her motion", 5th Symposium on Naval Hydrodynamics, Bergen, 1964. BRARD, R., ''A Mathematical Introduction to Ship Manoeu- verability (2nd Series, David Taylor Lectures , NSRDC - Washington,(in the Press), Sept.1973. 1263 Brard Bigure 2 Vortex Theory for Bodtes Moving tn Water Figure-.3 Figure 4 Figure 5, 1 1265 Brard Figure. 542 1266 Vortex Theory for Bodtes Moving tn Water Figure 5.3 1267 Brard oe 22 $253 53 Eo Figure 5.4 - Half-model in oblique translation (positive drift, permanent motion). The streamlines of the relative motion are more inclined towards starboard than towards port. The lines ending on a same point of the longitudinal contour SS,E, do not have the same direction, hence the shedding of free vortices along SSE. EL streamlines starting points on the stem, si end point of the starboard streamlines, sa end point of the port streamlines. 1268 Vortex Theory for Bodtes Movin tn Water Hipure-5.5. a Ze (< ) part of > on the rear of L 3 (e) (@) Bee eX) z , S a LE, 2 RY z, 4-3 SL) = =e. (ke Pe 2a) fe) il 2 Te »5 L2° 7 ae es ee <= w\ vy Figure 5.5. b 1269 Brard Figure 5.6 P20 Vortex Theory for Bodtes Moving tn Water Figure 5.7 12 Brard Figure 8.1 T272 Vortex Theory for Bodies Moving tn Water o(¥) concentrited VOneex Figure 8.2 = O° ; Se Figure 8.3 1273 Brard NOMENCLATURE l, OPERATORS -(.3 3 3 oe Vv = ome Go ), (the axes X11 X59 X, make a right 1 Zz 3 handed system). 2 ; A =V Laplacian operator A Symbol of vectorial multiplication. 2. VORTEX FILAMENTS AND VORTEX SHEETS 1g Vortex filament d DB Element of area of a vortex sheet eae The two sides of ‘D Infinitely small thickness of the vortex sheet Unit vector normal to the sheet in the direction from pe towards ae 5 Vv Velocity of a fluid point @ Curl of Vv Sea ER — > — T Limittof ¢€@ when e-.0\. T =(T - I") + 1" dm sectia aan Vi, ViEE; > 2 : ; ; 4 T Unit vector tangent to 2 in the direction of TF => = WP) - V(P._), (or V(M - V(M_) : jump of V through > from ee “to e: ise ie M: Z M.), P (or M ), P. (or M.) be- 1 e e i i longing to Bee ae 3 respectively, T = -n Ot 7) ee Line on nS orthogonal to nel Aa —- ip Circulation of V ina closed circuit 1274 Vortex Theory for Bodies Moving tn Water ar Intensity of a flat vortex tube on pan the width of the tube being do. — aT, dT oa? 3. GEOMETRY AND KINEMATICS ‘ys Hull surface D. “ dD: Interior, exterior of ye respectively D, Domain occupied by the real fluide : DLC D. —- V Absolute velocity of a fluid point — Ve Relative velocity of a fluid point : velocity with respect to axes moving with ae Velocity of a point fixed with respect to the moving axes Q, Angular velocity of the moving axes 2, Surface supporting a free vortex sheet = 5 Incident velocity on ya) qd, op Velocity potentials : V =VOif curl V=0, ie 2MP. o = curl Vv ae = curl Ms B (with or without accents) : line on di along which u is at- tached to), —_ wa Velocity induced by the bound vortices a Vi Velocity induced by the free vortices > cs Vector ~I on Ly ar, a on‘2a, m Density of a normal doublet distribution over the hull suniice, L275 Brard Density of a normal doublet distribution over the free vortex sheet pay Vortex filament on yas Velocity potentials generated by the normal doublet distri- bution on >a f respectively —w Velocity i duced inside D. by the vortex family (D. ; ae TT ®) nm (3 Rea in Sections V, VI and VII. > => Velocity potential defined inside D. such that Ve - V' “VP. (Sections VI and VII) 4. DYNAMICS Mass density of the fluid Exterior force per unit mass : F =vll ag, Ya Additional exterior force inside D, if —) 7 08, B= = Hydrodynamic pressure Hydrodynamic force exerted by the flow on an element pes of a bound vortex sheet or on an element of arc of a vortex filament which does not move with the fluid System_of hydrodynamic forces exerted on the vortex sheet (es € System of hydrodynamic forces exerted on the hull System of inertial forces inside D. System of complementary forces pF db. inside D. System of forces - pe) wad (M) on du. M, System of forces -pT (M) ie (M) roe (M) System of forces due to the added masses when there exists no vortex sheet 1276 Vortex Theory for Bodtes Moving tn Water A Estimate of K when the deficiency affecting , is neglect- gq. 8. ed Y. eo 5S Estimate of Umax7u, and ea tae ae are plotted. Note that at center of the pipe dut/dy+ = 0 whereas du,+/dy* + 0. in We have also shown in Fig. 1 the distribution of a for R = 1000 and Rt= 100. We see that the differences between the velocity profiles for R+ = 10.000 and Rt = 1000 are small. Prac- tically the same profile is also obtained in the Newtonian case for Rt = 100; however, the velocity distribution for R* = 100 and At = 300 does not coincide any more with the other profiles which have larger values of Ry A, The velocity distributions according to the various models for Rt = 100 are plotted separately in Fig.3. We see from this figure that the difference between ut and u? for Ass = B00. ds large. Note that the velocity ut near the wall merges with the parabolic equation ut = yt(1- y?/2R+) whereas uj} is tangent to the ut= yt curve and goes above the parabolic profile. We have also plotted in this figure Virk's ultimate profile (Eq. 9). Virk's profile is quite close to u‘* but it also gives at one region slightly larger velocities thanin a laminar pipe flow. Measured velocity distributions are compared with the calcul- ated profiles of ut in Figs. 4 - 7. The values of At were chosen arbitrarily (The data is taken from Virk (1971), Fig.3, using the same symbols to denote the various entries.) The agreement with the data is very good, In particular the velocity profiles in the maximum drag reduction regime, Figs. 6 and 7, describe the measurements much better than the velocity profiles proposed by Virk's elastic sublayer model, : - : : * A computer program for the calculations of u and f is available on request from the authors. ave Flows wtth Drag Reduction (Veloctty and Frietton) FRICTION FACTORS AND RELATION TO POLYMERIC PROPERTIES The dependence of the friction factor f£ V2 on Refl2 asa function of At , has been obtained numerically and plotted in Fig. 8. One sees that at large values of Refl2 the variation of f V2 for constant values of At is described by a logarithmic law. The Newtonian case A+ = 26 coincides wir the line describing the equation f/@= 4.0 log Ref 2-0. 4. Integration of the theoretical limit of Eq. (19) for small values of R*/At gives the laminar friction law. CS SP Y= (23) Several data points appearing in Fig. 1 of Virk (1971), near Ref 2 = 200(R+= 70), are quite close to Eq. (23). However, the avail- able data at larger values of Ref V2 indicate that the values of At obtained so far in dilute polymer solutions are bound by pat 1288 50: At the polymeric regime, as defined by Virk, an approximate relation between A* and the polymeric properties can be found using Virk's correlations, At large values of Ref , where the friction factor curves for different values of A+ are described by parallel lines, Au? is uniquely related to At. From Fig. 8 it was found that at this range + Au /\f2 ~ 40 log (AT/A” + 4) - 28 (24) At small values of Ref 12 the relation between Au* and At depends on the values of Refl2 , however, if Aut(A*) is measured along straight lines originating at Ref l2 >1000 and having slopes which do not exceed the slopes recorded in actual measurements, the deviation from Eq. (24) is less than 5%. The relation between ae and the shear stress can now be obtained from Eq. (3). This equation is composed of two expressions ; for V*< VQ, and for V*>V%,. It is suggested that a better descrip- tion of the variation of Au+ is obtained by the single equation Sas fey Ay ee Le (v/v*_)*] (25) Equation (25) deviates from Eq. (3a) at V"> aiVes by less than 3% and is practically zero for V*< Mi go /2s The values of al according to Eq. (25) should be determined by the intersection of the straight 13h3 Poreh and Dimant line (3a) with the Newtonian profile, which is exactly the procedure used by Virk. It follows from Eqs. (24) and (25) that 4 re ney i. (26) a / ee ee tO eee where ais related to the polymer properties by Eq. (13). We have used Eqs. (26) and (12) to calculate the variation of f-!2 versus Refil2 for solutions of the polymers AP-30 and Guar Gum, (Estimated values of the critical shear and molecular properties are given by Whitistt et al (1968) and Virk (1971), table 5). The calcul- ated curves for the three solutions, and curves for constant values of At are compared with the measurements of Whitistt et al (1968) in Figs. 9 - 12. At small and moderate values of V*/V{, the agreement between the data and the theoretical calculations (solid lines) seems to be satisfactory even for small values of Ref 2 . The agreement is not surprising as it merely reflects the adequacy of Virk's correlat- ions and the slight improvement due to the use of the continuous equation (25) rather than equations (3a) and (3b). The phenomenon of maximum drag reduction, however, appears now in a different light. One sees that when VipW becomes large, the data deviates from Eq. (26) and seem to be correlated with curves of constant At he measurements in the concentrated polyox solutions and the smaller pipe-diameters seem to be bound by the curve At = 350, which is close to Virk's maximum drag reduction asymptote in the range Ref V2 < 1000. However, the deviation from the lines which are calcul- ated using Virk's polymeric regime correlations, and the approach to the maximum value of A* , donot occur only near the maximum drag reduction asymptote. It appears that for each solution, there exists a maximum value of At (or Aut) approximately independent of the pipe diameter. Only when Rt is small the curves coincide in a limited region with Virk's maximum drag reduction asymptote (11). This evidence is not manifested in Virk's model which predicts drag reduc- tion values of the order of 90% for very large shear rates. It is also interesting to note that the measurements of drag reduction with alum- inium distearate in an organic solvent shown in Fig. 12 (McMillan et al, 1971) exceed the maximum drag reduction curve and appear to reach values of At = 600. In the absence of a theoretical model for drag reduction mechanism there is no way at present to determine whether the asymp- totic value of At is determined by properties of the particular poly- mers used, experimental limitations, a dependence of drag reduction on the existence of a minimum level of turbulence necessary to deform the macromolecules in solution, degradation or other causes. 1314 Flows with Drag Reductton (Veloetty and Frictton) CONCLUSIONS It has been shown that the effect of linear macromolecules in dilute solutions on the flow in the wall region, can be described by van Driest's mixing length model with a variable damping parameter At, If the Reynolds number of the flow R* is large, the constant shear approximation used by van Driest can be used. When Rt/At is not large, it is necessary to take into consideration the variation of the shear stress with the distance from the wall. The velocity distri- bution in the outer region is modified in this case using Coles' Wake Function multiplied by a factor. The factor decreases as the damping action of the molecules increases. Although the model does not ex- plain the damping mechanism it suggests a similarity between flows with and without polymers, which is not present in the elastic sublayer model. The model does not explain the nature of the maximum drag reduction asymptote either, however, it is pointed out that the maximum drag reduction curves for a given polymer might be associated with a maximum value of the damping parameter At. REFERENCES [1] COLES, D.J., of Appl. Math. & Physics. (ZAMP), Vol. 5 No. 3,,1954. [2] ELATA, C., LEHRER, J. & KAHANOVITZ, 1966, ISRAEL, J. Tech. 4, 87. [3] McMILLAN, M.L., HERSHEY, H.C. & BAXTER, R.A.,1971, "Drag reduction’, Chem. Eng. Prog. Symposium Series, 111, 67, 27: [4] ME VER OW AS, 1960, Aoi Ch. By t2, S22. [5] POREH, M. & PAZ, U. 1968, Inter. J. Heat Mass Transfer, 11, 805. [6 ] TOMIPA, Y., 1970, Bull, JS, M.E. 13, 995. [7 ] VAN DRIEST, E.R. 1965,’ 5, Aero. Sci. 23,..1007- igi5 Poreh and Dimant [8 ] VIRK, P.S. & MERRILL, E.M., 1969, Viscous Drag reduc- tion (Ed. C.S. Wells) Plenum Press. [9 J VIEK, Pls. / 1971, J. Fluid Mech. 45, 417. fe 0] WHITISTT, N.F., HARRINGTON, L.J. & CRAWFORD,H.R., 1968, Clearing House AD 677467. 1316 Flows wtth Drag Reductton (Veloetty and Frictton) ‘(OO€ =4V ‘92 =4V) STepou SNOTIVA 9Y} 0} BUTPIOOOe suOTINGtT4aysIp AJIOOTIA T[ ‘STA OL Ol Ol L3i7 Poreh and Dtmant ) be! 0.1 0.2 03 04 0506 08 10 Y/R” Fig.2 Defect velocities in the various models (R*+= 10000, AT= 26). VIRK'S ULTIMATE PROFILE Fig.3 Velocity distributions for BY = 100. (A= 26, Ar] 300), 1318 Flows wtth Drag Reduction (Veloetty and Frtetton) 10 10 + Fig.4 Velocity distributions for R = 2300. VIRK'S MODEL + Fig.5 Velocity distributions for R = 1890. P39 Poreh and Dtmant “PL2-5-- 20} SUH NGI e 1p ADOTAPA LSTA TS0OW SYYIA 7% yi, TSQOW LN3S3ud “009 = + Y Aof suotnqtiazjstp AyI0TAPA 9 “BIT see =v JaGOW LN3S3ud 1320 Flows wtth Drag Reductton (Veloetty and Frtctton) / A’ = 350 yt’? =19.0 log,,(Re f'*)-32.4 7 300 40 30 ae f 20 10 y. 1 if *=4.0 log, (Re ¢ 7)-0.4 10? 10° Re f/2 10° 10° 10° : Seg , 9 Fig. 8 Friction factor curves as a function of A . PS2k Poreh and Dtmant = a: a. = Sy & 0.416 ID OM OLIT MO Fig. 9 Measurements of friction coefficients. = a a = oO wn AP-30 ® 0.18 ID ® 1.624 ID Dooo0000w Dam owsion \ \ EG AG)) Fig. 10 Measurements of friction coefficients. L sae Flows with Drag Reductton (Veloctty and Frtctton) UpStale 250 Wee. | e 0.18"ID 4 0.416" ID \ \ \\ \\\\ \\\\ ReVit Fig.11 Measurements of friction coefficients. AP-30 100 W.PP.M. 4 0.416 ID @® 1.624 ID Oe .e) 0 0 ie) 0 6 10 10? 10° 10° Fig.12 Measurements of friction coefficients. 1323 10° Poreh and Dtmant DISCUSSION Thomas T. Huang Naval Shtp Research and Development Center Bethesda, Maryland, U.S.A. The authors are to be congratulated on providing a most detailed velocity-profile model for the turbulent flows with drag re- duction. However, the empirical fundamental equations (equations (3) and (4) are applicable to poor drag reducers, for example guar gum, at certain concentration and shear ranges. It would be more appropriate to put down the limitations of these equations and to indi- cate that the results derived here are valid only within these limita- tions. I have three other comments of a minor nature. The maximum drag reduction asymptote stated in equation (3) and used to compare with the experimental result is not in good agreement with our experimentation. The Virk first formula agrees better with our data and if you use the Virk first formula (JFM 1967) there is very little difference between the Virk model and the present model, i.e. within the accuracy of experiments. Second, the optimal concentration to reach the maximum drag reduction is found by us to be a function of shear stress and thickness of boundary layer divided by kinematic viscosity. The effect of the scale does play an important role in this respect. Once drag reduc- tion reaches this maximum value the results arrived at here are no longer valid. This limitation is suggested to be stated in the paper. The present model does not offer any advantage for predic- ting drag reduction. Nevertheless, it may be more suitable for dif- fusion and heat transfer prediction, as authors stated. In this appli- cation, more definite experimental results would serve the field more than empirical calculation based on bold assumptions, 1324 Flows wtth Drag Reductton (Veloetty and Friction) DISCUSSION Jaroslav J. Voitkounsky Shtpbutldtng Instttute Lentngrad U.S.S.R. The problem being discussed is of great interest, of course, but it is very difficult for us. It is important to describe the pheno- mena in the turbulent flow and in my opinion it is especially useful to try to apply for it Prandtl's ideas about the ''Mischungsweg"' (mixing lengths), Such a method was used by the author and some years ago it was used in the Soviet Union too, In particular the van Driest idea about the construction of formulae for the mixing length was used. I think it is a very useful method and the results obtained by the authors are also very interesting. I should be grateful if the author who presented the paper could answer one question. What, in his opinion, is the prospect of applying that method to the case of the rough surfaces because from the application point of view it is a very interesting problem. Some years ago in England these natural experiments were carried out with a natural ship and it is clear that the roughness of the surface of that ship was of a high degree, and it would be very interesting to apply this theory to a description of that phenomena. DISCUSSION Edmund V. Telfer lia sba Merl Ewell, Surrey, U.K. This is quite a fascinating paper and whilst it has many di- verse applications I would like to ask the author for his opinion on only one point. We have all been attracted by the possibilities of drag reduction, but I would like to suggest that there are also immense ad- vantages to be got by going in the opposite direction so far as ship model experiments are concerned. When one considers the terrific e325 Poreh and Dtmant wealth which has been wasted in all the experiment tanks in the world by the intrusion of laminar flow affecting the accuracy of their results, I wonder whether the question of doping the tank water with something which will have entirely the opposite effect from that of the drag re- ducing polymers should be considered. In other words, is there a dope which could go into the tank water which would make the occu- rence of drag reduction or laminar flow quite impossible ? If that could be found, a lot of the quite arbitrary and, to many people, unac- ceptable devices currently in use to eliminate turbulent flow would be avoided. I offer this thought sincerely to the author of this paper, and would like to congratulate him on the way the paper was delivered and on its content. REPLY TO DISCUSSION Michael Poreh Technion Israel Institute of Technology Hatfa, Israel The concern of Pr. Telfer is well understood. The presence of drag reducing agents in the towing tanks is favorable from the mo- deling point of view. Dr. J. Hoyt has published an excellent review paper in the Journal of Basic Engineering (June 1972) in which he dis- cussed this problem. The uncertainty of the data which has been ac- cumulated in the past is due to the fact that the drag reduction was not controlled or recorded. Pr. Telfer suggests, if I have understood correctly, to eliminate the effect altogether. I believe that this can be done by adding chemicals which will inhibit the growth of algae and drag reducing bacteria as well as creating unfavorable conditions for the stretching of the molecules. However, the more difficult task of maintaining a controled standard level of drag reduction in towing tanks seems more attractive as it enables one to improve the simi- larity in modeling ship motion. The interest of Pr. Voitkounsky in the effect of roughness is natural.I have published a paper in the Journal of Hydronautics in which I proposed an approximate model for describing this effect. The work in the Soviet Union in this area is not known to me and I'll appre- ciate if Pr. Voitkounsky will help me in receiving the papers he has mentioned. * (Jour. of Hydronautics, 4,4,Oct. 1970) 1326 Flows wtth Drag Reduction (Velocity and Friction) As to the remarks of Dr. Huang I would like to stress that the model proposed is not based on Virk's data. In particular it is not based on Virk's maximum drag reduction curves. The opposite, our model suggests that one should not look for a universal maximum drag reduction curve. Equations (3) and (4) are not the starting point of the analysis and have only been used to obtain the correlation bet- ween At and the polymer properties at the end of the paper. If Dr. Huang feels that better correlations exist he can use them in- stead, and the only equation that will have to be modified is Equation 26. Thus his conclusions as to the validity of the model are not ac- cepted. L See a LG aoe ' y 74 ed eg 7. ts f i ! Y Ff rt i +e ‘ r 4 : 1 + iy d fi { a t 7 ‘he . ee i: Pa # Vr , [. e } ~* ' i at t , ¢ *? ' 4 (44 s, hs 4 Cae Fhe. rF ies qe, ° ont ¥ to hast of ce age a3 + nN f eat a ee } gee ‘at Vetus ist ie iss ey) 3 mph? SAP ay ks Epos ea i even: re DE Te CMe ott le yaabie Wishes vide: See the ve fhe te ag Pwihs Ceres eh er iw ov en ew x § 1, tow, Dey ek fats Ta bee Ae a, , r Fy wiveg te Mi « al a . ee Fi i ‘ ay 40) ch hae Donets ae ee t ia $ aot eee Y 7 a £14} Pb paneer ¥ eh ee: Met Ae nae) | whi : rhe growth oF aga aie conga rn dtticwtt ta bh. ei ‘ee * rT as . » ya) i* ane Tere waeer i a My ive wit of Aras comet on 1 VAT ES 1 oe a AT Cte ty Cae ‘ mr, Lar Gaal hs ey nm 4a ee ols tes way ated) 4th ih ti ATCA AS. FM hadwr hy uy Oh Cv the DARA iad . FRONTIER PROBLEMS Thursday, August 24, 1972 Afternoon Session Chairman: J. Dieudonné Institut de Recherches de la Construction Navale, Paris, France Page Ocean Wave Spectra and Ship Applications 1331 W.E. Cummins, M. Chang (Naval Ship Research and Development Center, U.S. A.). The Role of the Dominant Wave in the Spectrum of Wind- Generated Water Surface Waves. 1371 E.J. Plate (Universitat Karlsruhe, Fed. Rep. Germany). Internal Waves in Channels of Variable Depth. 1397 C.S. Yih (University of Michigan, U.S.A.). Modeling and Measurements of Microscopic Structures of Wind Waves. 1435 Jin Wu (Hydronautics Inc., U.S.A.). 1329 SPadONT AUTHOR ved. dS teugo rk “yobarml? monigase doomret? Seachieid .U wslaveyl cabfovitsavl ef ob axd>oredoe sl ob intent aoeaxT ,etreS oye : a nm Ne ‘i _ (eed smHasilagA gid® baw otioegé oval mt ds taneeS aid? love) grat 1M nko? 2 /.A\a.0. 7 andw. Equation (2.4) relates the directional wave spectrum S (k’) to the correlation function R(7,7%) . In the above equation Fourier transform of R(7, £) with respect to 7 and is called the cross spectrum, which is denoted by Co() - iQ(w). Thus, RNG 7) = / [Co(w5¥) - iQ(wi¥)] aT ay (2. 5) where ih o/s = = | S(k 5) oo “dk, = Co(w3t) - iQ(w:t) =" /g (2. 6) Thus, if one represents the directional wave spectrum S(k,,w) ina Fourier series with respect to k,D, where D satisfies _w2h Sm 4; one can obtain the Fourier coefficients directly from equatfon (2. 6) by their definitions. That is 3D S(k), #) = Pann ) : (A, cos nk, D+B_ sin nk D) (2-4) ra ol where A. ==— Co(w3? = (0,0) ) (2. 8) 1341 Ming-Shun Chang AL = 2 Co(w;r = (nD, 0) ) aie i ee aug le 2. (wiz? = (nD, 0) ) (2. 8) By arranging the probes to measure wave elevations such that one can calculate the Co's and Q's upto # = (ND, 0) , the coefficients Ap, Aj, By, .-- Ay» By can be readily obtained from (278). “The relations in equation (2.8) are simple and they form the basis for the experiments described below. The accuracy of the approximation of equation (2.7) is de- pendent on both the order N and the non-dimensional parameter w°D/g . For given N and D the angular resolution, 6 , , is defined as 9, = sin ee 2 B . Physically this is a measure of the width of the angle over hee a narrow band wave-spectrum is spread. @, increases with decreasing w ; that is the angular resolution increases with increasing wave length. On the other hand if N and » are given, then 9, increases with decreasing probe separation, D. Fora spe- cific experimental setup, D can be adjusted for optimum results. As an example consider a narrow band directional wave spec- trum which satisfies k,+ Ak i 2 a e or + — i 2 = e Fr “al I Va Nw < iT] So Otherwise where C(w) is an arbitrary function of frequency w and 2Ak, is the band width of the wave number k, . The directional spectrum can then be represented by a delta function, 6 , and its Fourier represen- tation is given by S(k,;@) = C(#) Dd (k D - kD) co C(#) Dj} 1 en ree a re = = E ) | cos n(k) Kg)? | (2, 20) i ll where n= 1,2, ... THe nth order approximation gives 1342 Ocean Wave Speetra and Shtp Appltcattons N S(k;#) © ad [+ oe cos n(k, -k,) D| (2. 11) ea At k, =kp the approximation S(k,;w) has its maximum value of C() D 1 ilo Lae C(w)D, 1 aaa Seas iN) At kK - Ky) = Wp one bas S(k, 30) = = Nes) It is a factor of -1/(1+2N) smaller than the maximum value. The approximation of equation (2.8) to the narrow band directional spec- trum, equation (2.9) of kg = 0, is shown in Figure 6. From this figure, one sees how a narrow-band directional wave-spectrum is directionally spread out as a function of-“= D in this approach. The loss of accuracy, i.e., increase of spreading, with decreasing-“—D, which has been discussed previously, is clearly shown in the figure. EXPERIMENT The basic approach described in the previous section was ap- plied to the measurement of the directional spectrum of waves gene- rated in NSRDC's seakeeping basin. Wave measurements were taken with sonic probes, using three separate array configurations, which are shown in Figure 7. For the linear arrays, the fundamental distance between the probes was 2.5 feet and the total array length was 32.5 feet . This arrangement enables one to approximate the Fourier series of the directional wave spectrum up to the ninth har- monic. This configuration was suggested by Pierson - “Phe reason this arrangement was used rather than the optimum array sug- gested by eetomaea (FE is that in Barber's configuration the total array length would have been only 22.5 feet ; a greater length was preferred. The linear array was arranged in two orientations relative to the wave generators. In the first case the array was mounted parallel to the West bank of the basin and at a distance of 100 feet from the bank, as shown in Figure 3. In the second case the same array was rotated 45 degrees clock-wise to the North. The third array consisted of a pentagon arrangement, and employed six probes : one in the center and five outside forming an equilateral pen- tagon. The sides were designed to be 10 feet long. The orientation is shown in Figure 3. The seakeeping basin has wave generators along both the West and North banks and the wave generators on the two banks are operated independently. During the study three kinds of directional wave fields were generated. These were : wave coming from West bank, waves 1343 Ming-Shun Chang coming from the North bank and waves coming simultaneously from both banks. Both regular and irregular waves were generated. The periods of the regular waves were 1.6, 2.0, 2.5, and 3.0 seconds. These wave periods correspond to waves with angular resolution of less than 10 degrees upto 90 degrees. The irregular waves were generated from available random seaway tapes. These wave trains had average wave periods ranging from 1.6 to 3.0 seconds. Sonic probes operating at a frequency of 200 KH were mount- ed approximately 20 inches above the still water surface. These devices can measure the instantaneous water surface elevations with great accuracy. From the digitized records, cross spectra were calculated for all irregular waves. Wave amplitudes and wave phases were calculated for the regular waves by means of Fourier trans- forms. The directional wave spectra were then obtained by the use of Equation (2. 8). Some resulting directional wave spectra measured from the linear array are shown in Figures 8 through 16. Figures 8 through 11 are for regular waves and Figures 12 through 16 are for irregular waves. The curves for regular waves represent the directional spec- tra obtained under several different conditions such as different wave-maker dome air pressures, which are indicated in terms of blower rpm, different directions, which are designated by N or W for waves generated at the North and West banks, respectively, and different wave combinations. The coordinates of the figures are the normalized wave vector components in the direction parallel to the array, and the normalized spectrum density S(k 34) S*(k) 34) 156 goat where S, (4) is the average of the one-dimensional spectrum den- sities obtained by the five probes. Since S(k, ;w) is approximated by a ninth order Fourier series, the normalized spectrum density S*(k,;w) should be less than 9,5 as previously discussed, The theo- retical maximum value of S*(k, ;w) depends on the wave conditions. If a wave of a given frequency were generated at only one bank, S*(k, ;w) should have a maximum value of 9.5 in the direction in which it was generated, For other directions, it will be less than 9.5 . The actual value depends on the combination of the wave am- plitudes generated at the two banks. The regular wave results as shown in the figures agree with this theoretical value very well regard- less of the wave frequencies, wave amplitudes and the presence of 1344 Ocean Wave Spectra and Shtp Appltcattons waves coming from other directions. Figure il shows how the an- gular resolutions varied with the frequencies of the waves. As dis- cussed previously, the angular resolution increases with increasing wave period and the theoretical value of S*(k, ,w) is -0.5 at di- rection 0,., (k, -k,) = a ; The measured angular resolution agrees fairly well with the theoretical value. However, away from the peaks, S*(k,;w) oscillated with a much higher value than one would expect, especially for the wave periods of 2.5 and 3 seconds. The irregu- lar wave results do not agree as well with the tehoretical value. Figure 12 shows the resulting directional wave spectra of irregular wave trains generated at the North bank. It indicates that the waves were all coming from the North bank and the peak values of the spec- tra are between 6.5 and 8. In comparison with the regular wave measurements of Figure 7, the peak value is decreased. However, as the tail value also decreases, one concludes that the mixture of the frequencies in the same direction does not affect the angular spread- ing significantly. Figure 13 shows the measured directional spectra under a different wave condition. From it is concluded that there were long waves of period 2.5 to 3 seconds coming from the West and short waves of period 1.6 seconds propagating to the South. The direc- tional distribution of the 2.0 second period wave is meaningless. However, the actual wave field was different from the one pictured above. An irregular wave train was generated at the North bank and a regular wave of period 2.5 seconds was generated at the West bank. The loss of accuracies of the wave directions for the waves with wave frequencies near that of the regular wave is clear, Figure 14 illustrates the same phenomena, For this case the period of the regular wave was 2.0 seconds and the amplitude was smaller in comparison with the previous case. The loss of information on the wave directions in this case was not as serious as in the previous case. The presence of the 2.0 second period wave reduced the peak value of the 2.4 second period wave but increased that of the 1.6 second period wave. Figure 15 shows the measured directional wave spectra for the case of two low-amplitude irregular wave trains pro- pagating at 90 degrees to each other. The directional distribution of the wave is reasonable. By examining the calculated cross-spectra we found that the method used in estimating the cross spectrum was responsible for the errors which appear in Figures 13 and 14. Special care is neces- sary when analyzing the directional wave fields in these cases. By the use of a narrower frequency band-width, the result was improved ; it is shown in Figure 16. 1345 Ming-Shun Chang The directional spectra obtained from the pentagonal arrange- ment were not good and are not shown here. The measured phase lags between the probes had the same accuracies as those obtained by the linear array. However, as previously mentioned, the relations bet- ween the Fourier coefficients and cross spectra are more complicat- ed and thus, the results were not as good as those obtained from the linear array method. DISCUSSION The result of the study on the idealized unidirectional spec- trum indicates the need for improvements in recommended spectral forms in order to obtain a better prediction of long term ship motions, The two-parameter spectral form underestimates the wave energy for both high and low frequencies and overestimates the wave energy over the wave frequency range of 0.1 cycle/sec. to 0.14 cycle/sec. This has been illustrated in Figures 1 and 5, For applications, one has not only to be aware of this limita- tion associated with the idealized spectrum but also the varieties of ocean wave spectra. Fora better representation of the ocean environ- ment, one needs to know not only the averaged wave period and wave height but also other parameters, With more measured spectra a data bank of wave spectra can be established on a digital computer and stored on tapes for direct access. Such a data bank would even- tually make idealized spectra obsolete. It would certainly be more ac- curate than the idealized spectra and would contain samples of all of the various ocean wave spectra. The experiment on the directional waves suggests that the ac- curacy of a measured directional spectrum depends more on the di- rectional compositions of a wave field than on frequency compositions of the waves. This is demonstrated in Figures 11, 12 and 13. In order to accurately measure the spectrum of a swell and wind waves combin- ed sea, one has to use a technique which can estimate a sharp peaked cross-spectrum accurately, such as the narrow band process or the time shift process|'4 . However, it will require much longer re- cords of surface elevation. The linear array is a better probe arrangement than a penta- gon arrangement for the wave fields generated for this paper. How- ever, a linear array does not allow one to separate the waves propa- gating in the direction left of the array from those of the right. Two linear arrays may be needed for the measuring of an actual wave field in which waves propagate in an angle of more than 90 degrees. 1346 Ocean Wave Spectra and Shtp Appltcattons The angular resolution illustrated previously can be improved if one applies a weighting function to the Fourier coefficients, The choice of the weighting function depends on one's taste, as has been discussed by Longuet-Higgins The separation D was 2.5 feet in the present experiment. This separation can be adjusted to improve the long-wave angular resolution. The choice of the D depends on the wave frequency range in which one is interested. If one is only interested in the long waves then one can chose a suitably large D such that—“-D <7 for all frequencies in which one is interested. But in this case, the total array length will certainly be increased, and the increase in the array length can complicate the operations of the detectors. This has to be considered in the choice of an optimum D. By moving the probes opposite to the direction of the waves, one can also improve the an- gular resolution significantly[" | . It is applicable in an open sea where the waves are considerably homogenious over a large area, It may not be suitable for a model basin unless the basin is large and the waves are very homogenious with respect to space. The length of the wave records used for the above irregular wave calculations are approximately one minute long. The accuracy might have been improved if longer wave records had been used. ACKNOW LEDGMENTS The author is indebt to Dr. Wm. E. Cummins for his sugges- tion of the subject and the guidance given during the course of the work. Also, the author wishes to thank Mr. Daniel Huminik for his assistance with the experiments and many others who helped on the manuscript. REFERENCES [1] PIERSON, J.W. and MOSKOWITZ, L., ''A Proposed Spectral Form for Fully Developed Wind Seas Based on the Similarity Theory of S.S. Kitaigorodskii", J. Geophys. Res., Vol. 69, No, 24, 1964. [2] BRETSCHNEIDER, "Wave Variability and Wave Spectra for Wave Generated Gravity Waves", Technical Memorandum, No. 118, Corps of Engineers, Department of the Army, 1959. 1347 B] [4 ] [5] [6] [7] Ming-Shun Chang Proceedings of the 12th Internat. Towing Tank Conference, 1969. CUMMINS,W.E., ''A Proposal on the Use of Multiparameter Stand Spectra'', Proceedings of the 12th ITTC, 1969. PIERSON, W.J. and TICK, L.J., “Wave Spectra Hindeasts and Forecasts and Their Potential Uses in Military Oceano- graphy"', The U.S. Navy Symposium on Military Oceanogra- phy, 1965. MILES, M., ''Wave Spectra Estimated from a Stratified Sample of 323 North Atlantic Wave Records'', NRC Report LTR-SH-118, October 1971. (Unpublished) CHASE, J., etal, "'The Directional Spectrum of a Wind- Generated Sea as Determined from Data Obtained by the Stereo Wave Observation Project'', New York University, College of Engineering Report, July 1957. "Ocean Wave Spectra, Proceedings of a Conference", Prentice Hall, “New York, 1963. BARBER, W.F. and PIERSON, W.J., ''Review of Methods of Finding the Directional Spectra ina Towing Tank'', New York University, College of Engineering Report, 1963. EWING, J.S., ''Some Measurements of the Directional Wave Spectrum", Journal of Marine Research, Vol. 27, No. 2,1969. PIERSON, W.J., ''The Estimation of Vector Wave Number Spectra by Means of Data Obtained from a Rapidly Moving Hydrofoil Vehicle", Techn. Report No. 68-56, Oceanics, 1968. BARBER, N.F., "Optimum Arrays for Direction Finding", NYZes) Science le(N)5 1958: CUMMINS, W.E., ''The Determination of Directional Wave Spectra in the TMB Maneuvering -Seakeeping Basin'', DTMB Report 1362, 1959. PIERSON, W.J. and DALZELL; J. F, , "The Apparent Logs of Coherency in Vector Gaussian Processes Due to Computa- tional Procedures with Application to Ship Motions and Random Seas'', College of Engineering, Research Division, New York University, 1960. 1348 m@-sec E [S,(w)] Ocean Wave Spectra and Shtp Appltcattons 7 > AVERAGE IDEALIZED / \ SPECTRUM / \ | 3.0 | \ | | Nn o AVERAGE HANDCAST SPECTRUM 0 0.157 0.314 0.471 0.628 0.785 wESEGel Figure 1 Average hindcast, wave spectra calculated at station near station India 1349 Ming-Shun Chang Hy3 FT 46 8 10 12 14 16 18 18 = 1614 8 2 D P(10PP 700 RPM 2.0 SEC 6eo- -~@e@@ 1000 RPM 2.0 SEC S*(ky iw) Figure 9 Measured directional spectra of 2.0 sec. period waves only, with array parallel to West bank 1357 Ming-Shun Chang BLOWER RPM DIRECTION” PERION 12 1000 RPM N 25 SEC >>> G00 RPM w 25 SEC ==. (O0;REM N 25 SEC N 600 RPM Ww 25 SEC — — 700 RPM N 25 SEC 10 ee 600 RPM Ww 2.5 SEC | alee 1 | i | 6 3 | iu | * \ ” ' : | ww j ; % I a \ | ) / \ | / \ Os 74 N Ww -4 -1.0 -0.5 0.0 05 1.0 Figure 10 Measured directional spectra of 2.5 sec. period waves only, with array 45° to West bank 1358 Ocean Wave Spectra and Shtp Appltcattons WAVE PERIOD S*(k qi) Figure 11 Dependence of angular resolution on wave period for wave generated at North bank and array parallel to West bank 1359 Ming-Shun Chang CENTER PERIOD OF FREQUENCY BAND 3.0 SEC 2.7 SEC 2.0 SEE 1.6 SEC Measured directional spectrum of irregular waves generated at North bank 1360 Ocean Wave Spectra and Shtp Appltcattons CENTER PERIOD OF FREQUENCY BAND 3.0 SEC 2.5 SEC 2.0 SEC 1.6 SEC Figure 13 Measured directional spectrum of an irregular wave train with a high amplitude regular wave train of 2.5 See.) period 1361 Ming-Shun Chang S* (ki) Figure 14 Measured directional spectrum of an irregular wave train with a low amplitude regular wave train of 2.0 sec. period 1362 Ocean Wave Spectra and Shtp Appltecattons S*(ky;w) Figure 15 Measured directional spectrum of two irregular wave trains 1363 Ming-Shun Chang Figure 16 Directional spectrum of Figure 12 with a narrow frequency bandwidth 1364 Ocean Wave Spectra and Shtp Appltcattons DISCUSSION William E. Cummins Naval Shtp Research and Development Center Bethesda, Maryland, U.S.A. As originally scheduled, I was to be a co-author of this Paper and I should to offer a word of explanation. Because of high pressures in the Navy Department it was not possible for me to contribute to this to the extent that would justify my name being on the cover. All the work and most of the good ideas are Dr. Chang's. I will not dis- sociate myself from any bad ideas, although I do not admit that there are any in the Paper. Fortunately Dr. Chang was able to doan ex- cellent job without me. She has started work in a very difficult field which we have all neglected, and which we must not neglect much longer. I shall say a few words about the first part of her paper on the problem of standard spectra, which is something we have been fight- ing over inthe ITTC now for some six years. I expect there will be a good deal of concern with it next month in Germany as well. The more we learn the more we realise that we are in trouble. We are using sea spectra in the United States Navy and we find that we do not know enough to use them well. I would like to offer a word of explanation on some of the troubles that Dr. Chang showed where there were discrepancies at the two ends of the ''average'"' spectrum. The average which was based on the fully developed spec- trum tended to underestimate the ends, and in the middle it tended to overestimate. I remember Bill Pierson warning us many times that when he and Neumann and Moskovitz and the others who worked on the fully developed spectrum, they based their theories on only 15 percent of the measured spectra available to them, 85 percent could not be considered fully developed. So the naval architects who have been using them, against this advice of the oceanographers, have been concentrating on something that occurs about one time out of seven. If you have been to sea and looked with your eyes open you 1365 Ming-Shun Chang will realise that almost invariably what you see is a local wind sea with a swellunderneath, The wind sea tends to be of relatively short wave length. The swell comes from some distant storm ; it has been through the filtering effect of distance, so itis a narrow band. Even when you cannot see it with your eye the ship usually does see it. The local wind sea is usually doing one of two things : it is either growing or falling. So what you have is really a dumb-bell spectrum. You have a swell here from somewhere else and a wind sea developed locally. These are not taken account of in the fully developed spectra on which we have been basing much of our work, yet they are very important for the naval architect ; we cannot ignore them. I was on a ship just two months ago. It was not a normal ship but nevertheless it is an interesting case, The average wave height on one day was about four feet. There was a swell about 300 feet long for our 250 foot ship. We here heading into it. It was a quiet day. The surface even glassy. The local wind sea was virtually zero - and we slammed about 55 times per hour by count. The next day we had a local wind sea about the same height ; a fewwhite caps, not much ; a lovely day. The waves were much shorter. It was an abso- lutely wonderful day, because the ship was just alive in the water. The ship behaved completely differently. Over and over again we have used the fully developed spec- trum for characterising response when it is just plain wrong. We have to look at all of the seas that the ships encounter. DISCUSSION Michel Huther Bureau Verttas Paris, France I first thank the authors for the very interesting Paper they present. As the authors noted it, for ship behaviour calculations sea states representation by spectra is nowaday commonly used. The main problem for naval architects remains the question of the multi- directionality of the sea. I shall be pleased to know the opinion of the authors upon the two commonly used representations, i.e. the cosine 1366 Ocean Wave Spectra and Shtp Appltcattons of the angle incidence spacial repartition of the energy from the main direction, and the concept of the superposition of different uni- directional spectra coming from different directions. DISCUSSION Manley Saint-Denis Untverstty of Hawat Honolulu, U.S.A. The spectral theory of waves is a difficult subject to discuss with clarity and cohesion because the subject, even today, is in great turmoil and no matter how much or how well is said about it, even more remains in doubt. At the present state of the art, a point can be strongly developed only by disregarding a plethora of other points which, however, remain to haunt one like ghosts demanding to be heard in the night. Miss Chang's Paper is a welcome and well-written exposition on a point of keen interest, namely, do the idealisations made in the effort to obtain a working description of the sea yield disfigurations of reality ? Miss Chang believes that the correspondence between model and reality is unsatisfactory, yet I suspect this is only a ques- tion of standards and that hers are somewhat higher than those of us who, being older, have learnt that nature, no matter how well-behav- ed, cannot be very well fitted by simple formulae, no matter how well they may be conceived. For me, the fit of measured and idealised spectra in Figure 5 of the text seems to be very good. Miss Chang suspects the usefulness of the two-parameter spectrum and suggests a data bank. Such an idea would have been frightening a few years ago but now that computers with abundant memory are available, the suggested solution is feasible, yet for all the merit of the idea I should like to enter a plea for elegance - that is, for minimum effort, that is, for simple formulations instead of quasi-infinities of numbers, the designer will be happier ; for, having been brought up over the past century to depend ona length over 20 wave, he would appreciate something almost quite so simple. Whether the simple formula can be fitted to the vagaries of the sea wave de- pends, in my opinion, on how representative the probability distri- butions of Figures 2 and 4 in the text are, i.e. upon whether the 1367 Ming-Shun Chang probability distributions really represent the sea behaviour, and to this end, Iurge Miss Chang to explain, if possible, the double dip in the curves. After long meditation on the subject I have succeeded only in convincing myself that I do not understand the plot and am in somewhat of a dilemma as to whether the explanation underlying the double dip is a physical or a statistical inaccuracy. At all events, the orderly presentation is a pleasant reward for the reader who seeks insight into this fascinating subject of how to describe the wind-driven sea and leaves him with a stimulating thought for further work. From a single Paper written by a young lady one cannot possibly ask for more, REPLY TO DISCUSSION Ming-Shun Chang Naval Shtp Research and Development Center Bethesda, Maryland, U.S.A. My thanks to Dr. Cummins and Dr, Saint-Denis for their kind comments, On the question of how well the linear supposition of the direc- tional waves can be applied to the ship applications, I do not yet know the answer. This is one of the reasons for doing the directional waves experiment in the basin. The cosine power law spreading in the wave directions has been observed by a few experimental studies suchas SWOP and the floating buoy of N.I.O. However, these data were taken with respect to wind generated waves. For a large percent of the time, the waves in the ocean are a combination of swell and wind waves ; thus, one should not expect this power law to hold in general. The Atlantic hind- cast data could give a preliminary picture on this subject. As to Dr. Saint-Denis' comment on the use of two parameter spectral formulation, I certainly agree that it is dependent on the types of problems one is studying. I do not believe it will be very good for studying optimum design and its associated problems because the shifting period may introduce a large error in probability at the large response end, About Dr. Saint-Denis' question on the probability diagram 1368 Ocean Wave Spectra and Shtp Appltcattons and the dip init, I have as yet not studied it very carefully. However I believe the dip is due to the combination of swell and wind waves. This combination depends on the distribution of the wind field over the area under study. I would say that you will find the answer in the wind statistics. 1369 24 denim 2° te when ‘> Ox pus TASTO" Ube iw het \ agaih hvs ba eroy es ~ ¢ wie intes! (oageykoy. Ag a2 oo ood” rw acer) Wl » ote abe “vowed for the yoatuy is hg Pek eke CE hee S: i opsert od oa jcsecribe the Wind ~~ eh wri 15 «§ ier, ige' a vs MAS yy i, wi ges iw? ti) rther or Se civic iay Fo loeey Sh iay iw ~~ ono ann oo gaits are 3 i - a vw : > 7 % . Fike * REPLY TO OFMCUSSION h (rig os, ’ Chi 4 Ps | i ards ‘ ’ = %e Tas * : y. Ss : oa e © ‘ hehe 7 \ *) ¢ i w tive é y ater cet the ‘f Dy 8 A: th orl £@Tane, Leu wnt you ' P ie ' P opi ° és oe So Qh tha 4% re<‘ioanl ipa 13 by yeizee ? “~ gle eArig in ’ A 4.6 rec oP r ' a f ce ie | , 4 ( m“ weg ct z i ;. « MOWAT, a nila W i.*e Wilt is 1 ‘ aM ey , ce D4 iM tom = Hecht ry its ssw h KE 2 wives < Sue ahecye! 3 uy ive “4 ¢ *) wer 4a er) Bote =, pore a). Whe » oe \ @ j aie f Gur x sie, Waa a sabieet. ie mo. ive Saar tegii ©', Caw’ ctv. tre C26 al twee Ga ee soevtval formaitauus, Lecrteialy agrew et 6015 Keygen t of Si WM prvelawis cid in @tieyat ‘ig sor belleye 14 with, Se Mery eee stadving opticowem Stmgr OP ily Peewee tee yreaems ve cans ap Cty itty period may Wired le lege er nes me prvbab idsy a@ the a _, i pu im Mf a. Siwat O- Setot- Betts question en he prohed bart THE ROLE OF THE DOMINANT WAVE IN THE SPECTRUM OF WIND-GENERATED WATER SURFACE WAVES E.J. Plate Untverstty of Karlsruhe Karlsruhe, Germany ABSTRACT In an appendix to a paper on the structure of wind generated waves Plate et.al. (1969) have shown that the - 5 power law of Phillips is not necessa- rily an indication of a gravitational subrange but could be a consequence of the dominance of a train of significant waves. These waves dre simi- lar in shape but not necessarily sinusoidal. The consequences of these observations were exploited in developing modeling laws for wind generated waves, by Plate and Nath (1970), and interpreting the shape of the spectrum of wind generated waves (Plate. 1971). Although these concepts serve well to explain a number of previously unexplained phe- nomena - such as the "overshoot phenomenon" of Barnett and Sutherland (1968) _ there appear cer- tain features in observed field data which are not in accord with the results. For example, many ocean wave measurements reveal a broader spec- trum than that of the similarity hypothesis, and non-constancy of the coefficient in Phillips - 5 po wer law. It is the purpose of the paper to explain at least qualitatively the development of the ocean wave spectrum in terms of modifications to the similarity concept and to assess the consequences of these modifications for the laboratory simula- tion of ocean waves. I, INTRODUCTION There exist two different methods of describing the state of the sea surface : the method of the dominant wave and the method of the spectrum. The former has recently been replaced in many labora- cura Plate tory and theoretical studies by the latter. In this paper, it will be shown that the two concepts are not mutually exclusive but complement each other. It will be shown that the dominant wave concept may have some important advantages foranunderstanding of the physical proces- ses involved in the energy transfer from the wind to the water surface. In particular, it will be shown that the equilibrium spectrum constant 8 of Phillips can be associated with the slope of the dominant wave. Consequences of this relationship between the dominant wave and the spectrum will be pointed out. Y. THE EQUILIBRIUM SPECTRUM In calculations of forces exerted by wind-generated waves on off-shore structures two different views of the wave field have been held. The engineer of older days has used what is called the ''design wave'' for his structures : starting from the observation that for a long duration wind there arises a wave field in which similar waves of ap- proximately equal lengths and frequency but of variable height domina- te the motion of the water surface, he defined an average wave from the largest third or so of the observed waves and used the correspon- ding height or length for his design. Such concepts lead to the repre- sentation of the design waves in the form of a fetch graph (see Wiegel (1970) for the latest version of this graph). We shall call henceforth this wave the ''dominant wave". This essentially physical view of the ocean surface is con- trasted by the more recent, essentially mathematical representation of the water surface at a point as a random time series as used by many modern writers. This time function has some interesting prop- erties which were determined experimentally. It was found, for exam- ple, that the elevations of the surface at a point constitute, aftera long duration wind, a stationary sample of a Gaussian distributed en- semble (Longuet-Higgins (1953), Hess et.al. (1969)) with Rayleigh distributed extrema, with a variance that can be decomposed into a variance spectrum. Physics enters into this latter concept through the spectrum. As sinusoidal waves satisfy the linear equations gover- ning surface waves, the spectrum was thought to represent a super- position of very many ''component waves", thatis, linear waves of ve- ry small amplitude who by superposition form the large waves. Empi- rically, it was found soon enough that the spectra of the water surface at many different points in many different regions of the oceans had a somewhat similar shape, and attemps were made - which are by now classical - to empirically associate a functional form with the spectrum, whose parameters were empirically correlated with local independent parameters, such as wind speed and fetch. 1a Role of Dominant Wave in Spectrum of Wind-Generated Waves It might well be said that this development found its culmina- tion in Phillips' (1958) derivation of the - 5 - power law for the high frequency end of the spectrum. From reasoning that all component waves shorter than some limiting, longest component are at equili- brium ina state of breaking, he deduced the equation for the high frequency end of the spectrum : S beasties w for pie? th xs (1) where S (w) is the spectral density, g the acceleration of gravity and w» the angular frequency which has a value wm where the maximum = ( wm) in the spectral density occurs. How well the - 5 - power law equation 1 fits the experimental data is illustrated in figure la (from Hess et.al. (1969)). The data are from many different sources and range from short fetch laboratory data to long fetch ocean data obtai- ned at winds near Hurricane conditions. A separate set of data, by Mitsuyasu (1969) is shown in figure 1b. Through both sets, a curve is drawn according to equation 1 with 8 = 1.48. afm ss Unsatisfactory in equation 1] is the fact that 4 is an empiri- cal constant. An analytical model that overcomes this defect by rela- ting @ to the energy input into the wave field from the wind was sug- gested by Longuet-Higgins (1969). Assuming the rate of energy lost to turbulence by wave breaking to be proportional to the wave energy contained in the wave field, and equating this to the work done by the wind on the waves, he was able to show that if all wave components at frequencies above w,_ are in the state of breaking the coefficient f can be related to the drag coefficient C for the wind through : 0,125 legtleas. 6 pd/ee (2) where C = Ty /hg hy , Ly, is the water surface stress exerted by the wind, p is the density, and Uy is the wind velocity at height h, while the subscripts a and w refer to air and water, respectively. A drag coefficient of C= 1.5 X 1073 leads toa value of 8 = 1.3 x 10° in reasonable agreement with observed values, and since it is empi- rically observed that the drag coefficient decreases with fetch, or with duration, the theory is even capable of predicting a slight decrea- se in @ as has indeed been reported. Baz Plate Hl, JHE SIMILARITY SPE CrERUM The model of Phillips and Longuet-Higgins for the high fre- quency end of the spectrum appears to neatly solve the difficult pro- blem of describing the water surface spectrum by means of physical concepts, thus closing the gap between the purely mathematical des- cription of the surface and the physics of the generation process. Yet, there are a number of observed phenomena which do not fit into this model, There is, for example, practically no observed spectrum which does not possess "humps'"', or smalloscillations, of the high frequency end of the spectrum about the best fitting - 5 - power law. These humps appear regularly in the neighbourhood of higher harmo- nics of w, and are more pronounced in the laboratory data (for exam- ple, Hidy and Plate (1966)) than in field data (for example, Moskovitz et. al, (1962), or Liu,(197])). There is, also, no observational evi-= dence of a water surface on which waves of all wave lenghts or fre- quencies are breaking simultaneously. In fact, it appears unreasona- ble to expect that small waves and large waves should be affected by the wind in the same way, because larger waves are always exposed to the wind, while smaller waves either are exposed or sheltered, depending on where they are located with respect to the crests of the large waves. Add to this the strange phenomenon of the ''overshoot'', Both in the laboratory and in the field, if for identical wind conditions a plot is made of the spectral density at one particular frequency as a function of fetch it is observed that the spectral density first increa- ses very rapidly with fetch, then reaches a maximum (for that fetch at which the component coincided with the peak of the local spectrum), and with longer fetches decreased and developed into an oscillatory curve. An example of an overshoot plot is shown in figure 2 which is taken from a paper by Barnett and Sutherland (19). The experimental evidence and the theoretical models can be reconciled through the concept of the similarity spectrum, of which Phillips' law equation 1 is a special case. Similarity spectra are derived on the basis of the idea that by a proper non-dimensiona- lization of the frequency scale and the spectral density scale all ob- served spectra can be made to collapse upon a single curve. In the literature, one finds a number of different representations of a simi- larity spectrum which differ in the functional form of the spectral density distribution, as well as in the parameters by which the mea- sured quantities are non-dimensionalized. Well known is the simila- rity spectrum of Kitaigorodski (1962), but other forms are perhaps 1374 Role of Domtnant Wave tn Spectrum of Wind-Generated Waves more useful. I think that G, Hidy and I were the first (Hidy and Plate (1965)) to suggest a method for avoiding the problem of determining the proper functional form of the scale parameters. To normalize the spectral density we used the fact that the integral over the spec- trum had to equal the variance o* of the sea surface, and we used the frequency “to scale the frequency. In this manner, a similari- ty spectrum of the form : oui )=S(w) — (3) is obtained, where the non-dimensional spectrum Sg is a universal function of w ore We did not specify the functional form of equation 3. But we pointed out that the high frequency end could be represented by the - 5 - power law. Many later writers have adopted the same procedure and represented their spectra in the form of equation 3. Examples are shown in figure 3, in which three different results for the similarity spectrum are given. The solid curve is a ''best fit"' equation through non-dimensional data obtained on Lake Michigan by Liu (1971). - This curve has a maximum at w/w_= 1 of 1.5. Super- imposed are the curves of Hidy and Plate (1965) and of Mitsuyasu (1969). The curve of Hidy and Plate was derived from laboratory da- ta. It has been corrected here for a scale factor of ten by which the vertical scale had been distorted in the original paper. Thus, the maximum of the peak is found to be at about 5 rather than 0.5 (as has been used, for example, by Plate and Nath (1969)). Ata first glance,one mayattribute the difference in the shapes of this and Liu's curve to the difference in the conditions at which the data were ta- ken and one may conclude that there exist different spectral forms for laboratory and field. Older analytical spectra for sea waves are found to have dimensionless peaks close to the one given by Liu. Mitsuyasu (1969) has shown that the maximum values obtained from the spectra of Pierson and Maskovitz (1965) and Neumann (1952 ) are equal to 1.43 and, 1. -15,, respectively. But there is evidence that there must be a different reason for the difference in the peak values of Hidy and Plate (1965) and Liu (1971). There are the spectra for laboratory and field waves, as pre- sented by Mitsuyasu (1969). For both conditions he finds almost iden- tical spectral shapes, with a maximum dimensionless density at w Jw in = | which is equal to 2.74, One may therefore suspect that the difference in the three results may be due to the data analysis 1375 Plate technique employed, and that a similarity spectrum which is the same in field and laboratory is a physical possibility (see also Plate and Nath (1969) ). For the discussion in this and the following section, an equilibrium spectrum of universal shape and in the form given by Mitsuyasu is assumed to exist. The spectral shapes shown in Fig. 3 are averaged in the sense that the 'humps'"' in the experimental spectra have been removed by drawing a smooth curve through them. There is no requirement that a similarity spectrum must be a monotonous function, and the multiple peaks observed in the experimental spectra will, since they occur at multiples of w_, occur again in the similarity spectrum. I have shown in a recent paper (Plate (1971) ) that the concept of such a similarity spectrum leads naturally to an explanation of the overshoot phenomenon, But the significance of the similarity spectrum goes much further. For if the spectra are similar at all frequencies, then the measurement of the properties at one particular frequency suffices to fully specify the spectrum, This is an important conclusion, for it permits to reconcile the spectrum approach with the older dominant wave approach. Ifa single wave component suffices to describe the spectrum, why not use the dominant wave for this component ? This proposition is almost obvious, and yet, there is a very fundamental objection to it. For it is in general not permissible to identify component waves and physical waves. A physical wave isa water surface contour, while the component wave is a Fourier compo- nent. Consequently the former arises from a superposition of many of the latter. Fortunately, the sharp peak in the spectrum indicates that component waves of appreciable magnitudes are clustered around On 3 and as a consequence are only weakly affected by components at higher or lower frequencies. This is of course reflected in the observed wave pattern. Since the dominant waves are waves that belong into a narrow band of components with frequency near w,, it follows that the domi- nant wave, the component wave at wy, and the higher maxima of the time series correspond to very nearly the same thing. In particular, one can assume that the highest n out of m waves below some number n 100 H y ! so i ! r ‘E 1071 ° = ° 10-2 Fi as p oO / Ve ie A fF. e@ FIELD DATA / fl vo © LABORATORY DATA ; 4 x 104 of %8 o Lf ri © ° H / 105 i %. J 106 0.2 1.0 10 100 W (rad-sec™') Figure la Windwave spectra. Data collection of Hess et.al. (1969) 1388 Role of Domtnant Wave tn Spectrum of Wind-Generated Waves 103 27 mh (w) (cm?. sec) “4g? we (wy) = 1.48-10 102 Hakata Bay: Fe4,.5 km \ gF/ Uz = 9.11%105 \e btw) £8.9-107%g? w 10 gF Up 1 1071 —-—-- 1010 107 107° 107 1 10 10? 0 1000 2000 FETCH/ WAVE LENGTH Figure 2 Measured overshoot curves (a) Laboratory, 25 rad/sec component, (b) Ocean 0. 82 rad/sec component. (from Barnett and Sutherland, 1968) P3189 Plate ° Similarity form of Hidy and Plate (1965) x Similarity form of Mitsuyasu (1969) “Best fit “'- spectrum of Liu (1971): = 1.02 (w/wWm)”* OE Nt 4.08 (<-) P Figure 3 Similarity spectra from different sources. #590 Role of Dominant Wave tn Spectrum of Wind-Generated Waves © Mitsuyasu (1969) * Liu (1971) Figure 4 The equilibrium range constant/3vs the dimentionless fetch parameter gF/ue . Data points from Mitsuyasu (1969) and Liu (1971) Elevation (cm) Time, At=0.06 sec Figure 5 Streamline pattern and wind-generated waves. Laboratory data from Chang et. al. (1971) psgt Plate E 5 3 a w ' 107! (LZ6L) * ye ” 1 o - 1072 (LZ6L) ‘Je 38 Opuoy @AeM ueaVs0 paniasqo xX 32 Bueys Aq AsOyesoqge; ul aAem padrsasqo O @AeM SayYOIS JO SotuoWseY sayBIy @ S}UZUOC CD J21INOY JO |pNyIdwD 3AI}D)}24 paspnbs 10°4 10°> 605° 6) 1010 2 relative Frequency o Figure 6 Energy line spectra of real waves 1392 Role of Dominant Wave tn Spectrum of Wind-Generated Waves Figure 7 Figure 8 *300 -250 -200 “150 -100 -050 +0 050 100 150 200 250 300 350 Time delay 7 (sec) Filtered cross correlation for laboratory waves obtained from wave -gages placed 31.5 cm apart along the wind direction at a fetch of 9 m. From Su (1969) / Wind velocity (ems) 620 aE a ar 21.8m } Surface displacement (ors) Average wave and horizontal wind component associated with it. Data by Kondo et. al. (1971) £393 Plate DISCUSSION M. Huther Bureau Verttas Parts, France I first thank the author for the interesting paper he presents today. In the calculation of springing of ships a good knowledge of the energies in the range of high frequencies (i.e. w ~1.50 rad-s~! ) is necessary, solI shall be pleased to know the opinion of the author on the better sea state representation to be used in this case, DISCUSSION Ri.) Lournan Bureau Verttas Parts, France As described at a conference at the "ATMA 1972" "Sollicita- tions externes et internes des navires a la mer'' (J. M. Planeix, M. Huther et R. Dubois), and in a paper in the International Shipbuilding Progress of August 1972. ''Wave Loads - a correlation between cal- culations and measurements at sea'' (J. M. Planeix Ph. D.), Bureau Veritas using classical spectral sea state representation has found a good correlation between calculated and measured ship behaviour. So Iask to the author if such calculation and comparison had been done with dominant wave sea state representation? 1394 Role of Dominant Wave tn Spectrum of Wind-Generated Waves REPLY TO DISCUSSION E.J. Plate Untverstty of Karlsruhe Karlsruhe, Germany I think the two questions amounted to about the same thing. I shall try to answer them indirectly by giving my opinion on how best to represent the sea state in a mathematical or laboratory mo- del. I must speak as a coastal engineer because we are usually concerned with structures that have very few degrees of freedom. In coastal engineering we can usually identify only one or at best a small number of degrees of freedom and of corresponding natural frequencies or eigen-frequencies of our structures. Therefore I would recommend that if one does laboratory studies of the vibrations of a structure one should always set the wave conditions for the laboratory in such a way that the most critical eigen-frequency of the structure corresponds to the peak in the spectrum of the wave, if this is at all possible. Of course, if this natural frequency is of the order of 25 Hz, itis point- less to try to get 25 Hz waves. But if itis, for example, of the order of 0.1 Hz, may be the most important design case arises when the frequencies of the dominant wave and the natural frequency of the structure match. This is somewhat different from present usage among coastal engineers. I should like also to say a word against the necessity for using so-called random waves for modelling the sea state in a laboratory. You have probably all heard that it is becoming more and more fas- hionable to use a random wave generator instead of the older sinusoi- dal wave generator to model the forces on structure. In my opinion this is an ill-considered move for the simple reason that the random wave components you are generating are component waves - that is, each one of these waves is travelling at its own celerity. Therefore, these waves show interference pattern - that is, the small waves modulate the big waves, and vice versa, and you get a breaking of waves owing isi95 Plate to random modulation; you do not get a breaking of waves owing to the fact that they are real waves. Also, you do not find waves which are skewed like the ones I showed in my paper. And I think that you arrive at wrong conclusions, or no better conclusions, from this kind of study than those which you would get by just using a sequence of experiments with sinusoidal waves of increasing frequency and drew your response diagram in the usual manner, because you are basically trying to solve a non-linear wave surface problem by a linear superposition method. If you allow yourself to make this appro- ximation, why not go all the way and just use sinusoidal generators, which would cost you a lot less money? 1396 INTERNAL WAVES IN CHANNELS OF VARIABLE DEPTH Chia-Shun Yih Untverstty of Michtgan Ann Arbor, Mtchtgan, U.S.A. ABS TRAG. Internal wavesin prismatic channels of variable depth propagating along the channel axis are studied. It has been shownthat for whatever stratification of the fluid the frequency of the wave motion increases whereas the wave velocity decreases as the wave number increases. A general method of solution for anarbitrary channel isthenpresentedindetail, which gives the wave velocity andthe fluid motion for a given wave number anda given mode by successive appro- ximations. Finally long waves are studied in some detail,a few specific examples oflong wavesare given, and the connection of the present theory withthecla- ssical shallow-water theory is shown, I, INTRODUCTION Known solutions of gravity waves ina prismatic channel of variable depth which have a degree of general applicability are of three categories. For very long waves (first category) the shallow- water theory (Lamb 1932, p. 273-274) gives (gh) V2. asitheawaverve- locity, where g is the gravitational acceleration and h the average depth. For very short waves (second category) not confined to the edge region the variability of the depth is unimportant, since the mo- tion is confined to the region near the free surface. The third cate- gory is the category of edge waves, which for short waves have an ap- preciable amplitude only near the shores (or the edges), and are there- fore always affected by the geometry, specifically the slopes of the channel near the shore lines, however short the waves compared to D397 Chta-Shun Yth the maximum or average depth. The solution for edge waves (Stokes 1839, or Lamb 1932, p. 447) is exact if the region occupied by the water is semi-infinite, bounded only by the free surface and a plane of constant slope serving as the only solid boundary. If the channel is finite in both depth and width, Stokes'solution is nevertheless valid for each shore if the wave length (in the longitudinal direction) is very short, since the variability of the depth has then an important effect only near the shore lines. Aside from these three categories, and interrelating them, are the exact solutions for water waves in a symmetrically placed triangular channel of vertex angle a/2 (Kelland 1839, or Lamb 1932, p. 447-449) or of vertex angle 22/3 (MacDonald 1894, or Lamb 1932, p. 449-450). These exact solutions are useful because they pro- vide a check for any approximate theory. In this paper internal waves in channels of variable depth are studied. The differential system governing the flow of a system of superposed layers of homogeneous fluids is formulated by first con- sidering a single layer. Then for the layered system it is proved by the use of comparison theorems that the frequency 97 of the waves increases but the wave velocity c decreases as the wave number k increases. Then the differential system governing the flow of a con- tinuously stratified fluid is derived, and the increase of o and de- crease of c as k increases are again proved in general, After giving a few solutions in closed form (under the res- triction of the Boussinesq approximation), a general method of solu- tion for wave motion in stratified fluids is given, In the form given the method is for application to continuously stratified fluids, but it can be adopted to deal with homogeneous fluids, and the manner of adoption is briefly indicated in the last paragraph of Section 7. Finally we study long waves in some detail, and both conti- nuously stratified fluids and layered systems are considered. A few examples are given, and the connection of the theory to the classical shallow-water theory for long waves ina single fluid is shown. Il. THE DIFFERENTIAL SYSTEM FOR THE CASE OF CONSTANT DENSITY If viscous effects are neglected and the motion is supposed to have started from rest, and if the density of water is constant, the motion is irrotational and a velocity potential $ exists, the gradient of which is the velocity vector. We shall use the Cartesian coordinates (x, y, z), with z measured longitudinally, y measured vertically, 1398 Internal Waves tn Channels of Vartable Depth and x measured across the channel. If the velocity components in the directions of increasing x, y and z are denoted by u, vandw, respectively, we have vie esse6r otv o= ics w= : (1) The equation of continuity then becomes 5% F 32 : 34 Sat ee which is the equation to be solved. At solid boundaries the normal velocity component vanishes, so that ag on Suen (3) where n is measured in a direction normal to the solid boundaries. At the free surface the pressure is constant, so that, with the square of the velocity neglected and with 7 denoting the displacement of the water surface from its equilibrium position, the Bernoulli equation is ag a + gn = constant, (4) in which t is the time. Since 1S HOG Vv = A (4) can be written as 1699 Chta-Shun Yth Far Te Be SL es (5) which is the free-surface condition. We shall assume P= £ (53) vy) expi (Kec and (13) by £, , integrating over the do- main occupied by water, utilizing (8), (14), and (15) whenever neces- sary, and applying the well-known Green Theorem, we obtain 2 0 | oh 2 : 24 = I al “+ k aa - (16) ee JI = ll + k ‘ I g m mil 2 PaO! * TH) with I = fot tie CUAY os al = ( ) ; m0 iy m1 fee 2c ly Zy (18) a+b = d dis i fii, Sei a where a and a+b are the values of x atthe shore lines, so that b is the width of the free surface. The difference of (17) and (16) is , (19) so that in the limit (as k; approaches kp) 1402 Internal Waves tn Channels of Vartable Depth 2 gl d 0 5 sae Be 0 ., (20) dk where at+b fz Zz I = ff: aay , J = feiGex (21) a One consequence of (20) is that the group velocity c the same sign as the wave velocity c, since g is always of qe re = eS 2 e = S6 aE (22) so that do ce, (= My es Oi (23) : dk Now we wish to see how c” varies with k*. Rewriting (19) as 2 1 4 isa: ue ee ie = = = a[gis ; (ogi cscs gal (ko - ky) dig : Bro}, (24) and going to the limit, we have 2 2 Lo oe. ac c Le ef bot a who eile ee 25 3 5 ween ed (25) dk 1403 Chta-Shun Yth But from (16), on making kz =k , , we have 2 ge% 2eG (26) dk Since dc Ce = (ap Ko - 0 <4 (ee << Cc ’ (27) which means that the magnitude of c, is always less than the magni- tude of c, whether or not c is positive. For superposed layers, each of which is homogeneous within itself, (23) and (26) are obtained in much the same way. All we need to do is to apply the same procedure to each of the layers, and then apply the interfacial condition (11) at each interface. IV. THE DIFFERENTIAL SYSTEM FOR A CONTINUOUSLY STRATI- FIED FLUID If the fluid is continuously stratified, we shall denote the density of the fluid when it is undisturbed by pg, which is a function of y alone. The density perturbation will be denoted by p , so that the total density is po +p. The mean pressure pg is related to po by the hydrostatic equation 1404 Internal Waves tn Channels of Vartable Depth dpy eat he es (28) Then the linearized equations of motion are 3 ) 3 Po oy (uv w) = laces Oy’ SPA: - gp, 0) , (2.9) where p is the pressure perturbation. The linearized equation of in- compressibility is =—— + wee =4 O =, (30) with the accent indicating differentiation with respect to y. The equa- tion of continuity is Oe: ORNs ge: a are (31) du he! Cross differentiation of the first and third equation in (29) gives (Sse Parse) ees (32) Since dependent variables will all be assumed to have the time factor Baler. (32) gives from which we have 1405 Chta-Shun Yth ae (34) @ being a velocity potential for u and w Let F bea function of x, y, z and t defined by e (35) (36) In view of (34) and the third equation in (29), differentia- tion of (36) gives (37) Assuming for all dependent variables the factor exp i(kz - ot), we can write (37) as (38) Eo °= ohy The subscripts in (37) and (38) indicate partial differentiation. Re- calling that the first and third equations in (29) are and combining (36) and (38) into 1406 re -' as redrrsttei} aomoved (0b) (f%) Nees: , VP hee (3 : : . z Pe: | ; a = \<. e way Th ny) y 6 . 6 oP * eR ; sxe 0 S g%8* of” 5 its is Tesily tha sare Ab che omyuation of cot r s ) 'asy 1 dt sibqaox dile! asixeltiadtptibt wos “seeuly dale piety ying (49) by #4 and vag uper the tha weqretds, ear ‘7 the Green's Theil Bak ving petty erty coowltiloas ins? eel SB for @=#, . + ev, » ‘ww keve LP * Sat eet , tee % : (SE hin 1 ten «Sg Teertey So WA 80 MOITAISAY .V . eat xt { nos lt équsoa edt gaidalidetes bo seoqzeg edt sot ad (85) ostew late Hi wee oe Se: } i ; i i . al A — — ah H > oa 9, GA-, . aed iE * on. : wy0 fiz. ; 4 Te >a ar Es * — ree a _) 7 ine i 7 ‘oo. De)? ian : bat . oe ia 7 2) ¥ , : Chta-Shun Yth With v given by (43), (40) becomes (remembering the factor exp i(kz - ct) ) 2 d Aa" ¢ 2 2 3 bomen ie Sap we Fy Sgt (44) ie 0 oy 5 ee 0 0 The boundary condition at the solid boundary is po ea ONG (45) where u and v are given by (34) and (43), and nn and n, (=0) are the direction cosines of the normal to the solid Soeee The Bernoulli equation is still valid in the free surface (if there is one). In fact, the Bernoulli equation obtained by integrating (29) is simply (39). But we must recall that p is the pressure perturbation only. If we require the total pressure p+ po to be zero on the free surface, and use (28), we have (with the new definition for ¢ ) ails 2 + gn = constant (at the free surface) , (46) PO which gives, upon idfferentiation with respect to t, p BPO %y 2 + ! a BPO 2 2 2 V. VARIATION OF. 5° AND c WITH k For the purpose of establishing the comparison theorems, we shall write (44) as 1408 Internal Waves tn Channels of Vartable Depth I o¢ ee ( for 5-0 7 Ao t 8% rs) > ; (48) and we recognize that this is really the same as the equation of con- tinuity (31). We also recognize that (47) can be written in the much simpler form = OV. \. (47a) We now consider two wave numbers te and ear with the corresponding eigenvalues ax and cea , and the corresponding eigenfunctions $, and ¢, . The velocity components (u, » Vy > Wy ) and (uysVv, + Wy) satisfy Ou Ov ow 1 1 bee x ts Oy i az pits (49) ou Ov ow, Se od cana it SOS ane ee) Multiplying (49) by ¢5 and integrating over the fluid domain, and utilizing the Green's Theorem and the boundary conditions (45) and (47a) for g=90, , b= 1 , V=V,, we have iE +k I + -@. AT Suna fs Le (51) in which 1409 Chta-Shun Yth where a and a+b are the abscissae of the shore lines. Similarly, on multiplying (50) and integrating, we have I Pa ae ee eee: ee ke (52) where Z, 2 2 Z. where g.py. @ pupae // : oe ele 2y dA . (55) + ! ! (ore, Eee, pt EP ) fo aes ae (56) 1410 Internal Waves tn Channels of Vartable Depth where I9,H and K are respectively the value of lone’ Ha wemdak,. when K, = K, and are obviously positive. Hence d a = 0 (57) dk We can write (54) as 2 2 z Le es Zz ea = Ky )| Noam Ga Re pa ee which gives ae I, - (a +R) ae (58) dk k (H+ K) It is a simple matter to show that Ji Kerio vy Oni; where J is the value of J), (or J2,,) when kj = kp and oj =92. From (51), on making kj = ky , we have Hence 2 2 Zz i. = Cry = 6 ee =eU1 Bie Hae ee <= Os and (58) gives 1411 Chta-Shun Yth ne (59) As before (57) and (59) imply, respectively, Cee aO and Pie coe, = SEES I eebenntlaSt 16 ae (74) V2 V2 : ne : ; With A given by (64), given 8B and k we canfind o from (74). Then \* is known and hence m?_ is known, Evidently there is also an antisymmetric mode, given by @= A sinh ax sinh yy cosk (z-ctt+ e) Then (69), (70), and (71) still stand but the dispersion formula is now oc = S— coth — ; (75) or coth see (76) V2- V2n It can be easily shown from (65) that if there is no free surface \* must be negative. The solutions given above all corres - pond to positive A 2 | Hence the presence of the free surface is es- sential for the existence of these solutions. The waves these solutions represent are therefore largely free-surface waves rather than in- ternal waves, and the density stratification has only a minor effect - that of affecting the value of the slope m. For this reason these so- lutions are not very interesting. We note that if B = 0 we have m* = 1, and (73) becomes 1415 Chta-Shun Yth k gt gov Sk digas mirage (77) which is just the solution of Kelland (1839). We shall now proceed to study truly internal waves. VII. A GENERAL METHOD OF SOLUTION Given a density stratification Pg anda channel cross sec- tion, our task is to find the relation between a2 and k* determin- ed by (44) and (45), supplemented by (47) if there is a free surface. Since o?% appears in the denominator of the third term in (44) and as a multiplier of that term, (44) is inconvenient to use as it stands. We shall transform it into an equation in v. Differentiating (44) with respect to y, andusing (43), we have a?" 5 34 Zz “ye ; 5 ye ak v sik vy) + (9,v'). = oie (78) with the accent on pg and on v indicating differentiation with res- PECt tole y. - The boundary condition (45) has to be written in terms of v alone. Since u is given by (34) and v by (43), we can write 3 EP’) Fao ; ( gts 5 ) vdy + £, (x) : (79) where fj(x) is an arbitrary function of x. It will be shown later that the boundary condition (45) demands that f)(x) be a constant. Hence 1416 Internal Waves tn Channels of Vartable Depth bs 3 BP'o Pou = aie ne ( Po a 3 ) vdy . (80) 0 Multiplying (45) by Pp, andusing (80), we have . bk EP 1 3x Leo t 5 ) vdy + n, ew ee ; (81) 0 in which the boundary geometry not only determines n, and no, but will play a role after v has been differentiated with respect to x in the first term. We shall show later that the integral form of (81) can be changed to a differential form. If there is a free surface the condition there is (47) , which again must be expressed in terms of v. By virtue of (43), (47) can be written as a r gP ov. Applying the operator V on this and using (40), we have 24 Wed (3=—=1)/( 4 Equations (78), (81), and (82) constitute the differential system defining the eigenvalue problem, 1417 Chta-Shun Yth We shall now impose two restrictions on our study : (i) we shall assume that the channel is symmetric, and (ii) we shall exclude sloshing modes (with motion in the x-y plane only) from con- sideration. Asymmetric channels can be similarly treated without any substantial additional difficulties, and sloshing modes need a se- parate treatment. We shall describe the boundary of the symmetric channel by and consider only the branch x Ath fleas (83) Then the direction numbers (nj) s 2; 0) of the normal to the boun- dary are Dye So in tes re es oth ae Olan. (84) Restriction (ii) enables us touse the following expansions : Peek (85) and ams = NL # ROR S ereddore opts (86) 1418 Internal Waves tn Channels of Vartable Depth where Z 4 - c = toe? ‘ (87) Substituting (85) and (86) into (78), and extracting the terms of zeroth order in k, we have (P', assumed never to vanish) ! Bas t Xr ! = 2 98P 0%20 gp 0 0 Yoo + bg oa) Dix, (88) which gives v2qg in terms of voo. If terms of zero order in k in (81) are taken, that equa- tion becomes, with n, and ng given by (84), d 1 =e = a= [ of? 9% 20 dy ES iG) Po go 0 0 This equation is valid for all y. Hence we can differentiate it with respect to y and obtain ! s ee = 05 Yaar ee (f'f Pas do) os (90) 1419 Chta-Shun Yth Eliminating er between (88) and (90) , we have ! a OP ee ee 000 Oir\. (91) ! ! (P9¥'o9 0F? 000 or io = (| ! ' ea = ! = =o op) £ (opts ) AB 'o |¥o0 Pl The boundary condition” at y =0 is If the upper surface is fixed, the boundary condition there is Vo 96d) = GE (93) If there is no flat upper surface, and the conduit is full of fluid, (93) can simply be applied at the highest point. On the other hand, if the upper surface is free, the boundary condition (82) can be written as a Ee [2v9(@ . Vo9( | ; (94) Integrating (90) in the Stieltjes sense over a vanishingly thin layer at the free surface, we obtain _ £'(d) 2 B%20(9 = Fa) Yoo(%) Substituting this into (24), we obtain f fi(d) are i - | Yoo) » (95) *If there is a flat rigid upper surface the boundary condition there for v is v5,(d) = 0, which can be satisfied only if 20 20 f'(d)=0. The present analysis for symmetric channels is there- fore valid only if the channel boundary has no flat rigid part. 1420 Internal Waves tn Channels of Vartable Depth which could have been obtained by integrating (91) across the free surface in the Stieltjes sense. If f(y) is a constant, (91) becomes Pg Sigg rie i hgee Brad he 000 which agrees with the equation governing wave motion (Yih 1965, p. 29) ina stratified fluid for k=0. Furthermore, (95) becomes Which is the free-surface boundary condition for a rectangular chan- nel, and which agrees with (82) when v is independent of x and when k = 0 (for which \ = Xo). The conditions (92) and (93) remain unchanged if f(y) is constant. Hence the differential system consist- ing of (91), (92) and (93), or (91), (92) and (95), agrees, as it should, with that for a rectangular channel when f(y) is taken to be a constant. From the system (91), (92) and (93) or (91), (92) and (95) we candetermine Ag and vgo(y). Then vgo(y) is known from (90). We shall describe the next stage of approximation, The pro- cedure of successive paaroximation will then be clear. Taking terms of order k2 in (78), we have 2 ! 1 > i ss e ity oro pegepnOk MABE “20)| on 1421 Chta-Shun Yth The equation corresponding to (89) is now, for terms of order k and free frgm =, vA + ay oe 2 i ( NEP a2? Yok? ovee ) Poranh oY 0 (97) f' (hibeceen da) =afi0 sick Seam il aAK A "20" F where a = d_ gpl il 4f 08? a 405) 0 Differentiation of (97) gives {Po ; : + + \_gp! z ee 2( X98 P'g%a2 * Pov20 28 ?' yao) 6 Yoa) t (eg¥aq) - (98) Elimination of Vo, between (96) and (98) gives, after multiplica- tion by f, Lv = f oe Pahoa) ae enact N28 *'9)o +2 » (99) in which L is the operator on vgg in (9la). We see from the terms. of order x? in (96) that V4g Can be expressed in terms of vag, and is therefore known. Thus I in (98) is known, 2 * Equating terms of 0(x ) in (96) to zero also guarantees the ae tisfaction of the free surface boundary condition for terms of 0(k°x hy as can be seen from a Stieltjes integration of those terms at the free surface and from (82) . 1422 Internal Waves tn Channels of Vartable Depth bt » : : The boundary conditions are, if there is no free surface, ae (() era © Vo2(d) - (100) If we now multiply (91) by voz and (99) by vog, and integrate the resulting equations, by parts if necessary, andusing (92), (93), and (100) whenever possible, we obtain two equations the left-hand sides of which are identical. Taking the difference of these two equations, we have d | ! = =. ot iB f | (s Pavao) ( Pot 8 P Aer I |» =0, (101) 0 which determines A,, since vy and vo are known, Then (99) can be integrated by the method of the variation of parameters to give vg» . Then V 29 is known. Further approximations follow > the same pattern. If the upper surface is free, the free-surface boundary condition can be found by integrating (99) in the Stieltjes sense, and an equation similar to (101) can be found. In fact, to obtain it one need only add the terms Zz = ! ON ff" p o(4) v5 9(4) + dW £8 (a) v 2 0 00 ) 2 d)+4f d (4) d 98g (4) Myo to the left-hand side of (101). It remains to show that the f,(x) in (79) can be taken to be a constant. The argument is as follows. We have obtained succes- sive approximations to the eigenvalue and the eigenfunction, at each stage satisfying all the boundary conditions. If f(x) is not a constant, it is an additional term for the potential @¢ in (44), which gives rise to an additional velocity whose y component is zero, That velocity * Recall that a flat rigid upper surface is ruled out, and at the highest point of the symmetric channel x=0, _ so that V49 (4) does not have to vanish, 1423 Chta-Shun Yth therefore cannot possibly satisfy (45) unless its x-component is zero - or unless £ (x) is a constant. In the next section we shall study long waves. But before we leave this section we shall make two comments: (i) The method of expansion can be slightly modified to deal with unsymmetric chan- nels. (ii) If the fluid is not stratified, Po is constant, and from (90) one deduces that vgg vanishes and with it v290 2lso vanishes, according to (88). However, for the second approximation, in which terms of 0(k? ) are considered, it is found that Vo2 is governed by the equation =O; (102) which is what (99) becomes. The boundary condition on vo9 is identical with (95) : u f'(d) V" 92 (4) = [ays - oa | V2 (4) , (103) ! ee a + mado heey yea cis (104) and 4 fy) 72 105 f 0 Substitution of (104) and (105) into (103) gives 1424 Internal Waves tn Channels of Vartable Depth Ce eB ices ee gh , (106) where h is the average depth. Equation (106) agrees with the result of the classical shallow-water theory for long waves. Thus every comparison we have made indicates the correctness of our procedure. VIII. LONG WAVES We shall give some specific examples of the speeds of long internal waves, and shall consider two special classes of density stratification. (i) Exponential density distribution. If the density distribution is given by (61), we can work directly with (44), which in this case becomes (63), with \*% defined by (64). If we expand ¢ ina power series in k ? > we have 2 4 CA Pag. t Te Graygl¥ ht tids oP ayyl We detmnes (107) : 2 , : , Since we expect A to be negative for internal waves (i.e., waves that do not owe their existence primarily to the free surface), and since *% contains the factor o2 » which contains the factor k 2 we shall write orion ty iF, tg 54.5 (108) in which 1425 Chta-Shun Yth 1 al = - —> (109) 2 ee We shall endeavor to determine y for a given channel cross-section anda given £6 in (61). Substituting (107) and (108) into (63) and taking terms of order k? ; we obtain Zabe,te ssestsethde ads tOsnibak sheng saat (110) Let the channel boundary be given by (62). Then the condi- tion there is, from (34), (43), (45), and the definition of »2 given by (64), md, -, p = 0 Cre} Z The terms containing k in (111) are -2 2m y+ ar et) = OC (112) 2y $ ly) me ay Gyr -S) 0, (113) after substitution of y for mx (it being sufficient to consider one half of the symmetric boundary). Combination of (110) and (113) gives 1 2 M tal Ge. i aeat iT} Oo (114) 1426 Internal Waves tn Channels of Vartable Depth If the inertial effect of the density variation is neglected (Boussinesq approximation), this equation becomes + oN Pie Oy 4 (115) 1 a ! 00 3 * 0 00 the solution of which satisfying the boundary condition at y =0 is Jory), where Jo is the Bessel function of the zeroth order, If the upper surface is free, then (47) gives the condition =- = ! Bey (4) = o'y5(€) » (116) so that (116) is replaced by BJ (vd) = yJ)(vd) . (47) 0 which gives Y. Once y is known, the long-wave speed cg is calculat- ed from ) Ss ciskadt OP4 ce Aces cee (118) It is important to note that the roots of (116) or of (117) are for internal waves only. The speed of waves due predominantly to the presence of the free surface is found in the following way. First of all, differentiation of (46) with respect to t gives directly 2 BPy7 ® i ee (119) op +g! 08? 9 We see that there are no terms free of k in (63) and (119). Hence any solution of (114), which automatically satisfies condition (45) at the channel boundary, is an acceptable solution. But at this stage we cannot determine cy). Proceeding to the second approximation, whe- ther or not the Boussinesq approximation is used, we reach a nonho- mogeneous differential equation in ?>(y) , the solution of which to- gether with the boundary conditions then determines Yor c,. The Co so determined is not proportional to VB, but is much larger, and the corresponding waves are predominantly surface waves, the density stratification merely causing a minor correction if B is small. 1427 Chta-Shun Yth Because of the convenience afforded by the exponential density stratification, we have used the differential equation in @ instead of (78). Remembering (61), (86), and 7) " l RED f 0 aad a Ge one can readily show that (114) is equivalent to (91). In fact, ' poly) is proportional to Pon ‘ (ii) Superposed homogeneous layers. If the fluid is composed of superposed layers, in each layer the governing equation is (2) or (7), The boundary conditions are (8) for the rigid channel boundary, (11) for the interfaces, and (9) for the free surface. Of course, (9) isa special case of (11). Note that the f in (7) is not the f in (83). The solution is now not restricted to symmetric channels. Suppose there are n layers. We shal use the expansion fa ¥) => f tks tc Kk of FP Sees (120) for the mth layern counting from the bottom up. Furthermore, we shall write o = 0 +kao +ko amecweate (121) Substituting (120) and (121) into (7) and taking only terms free from k, we see that the solution is f ek oi reece (122) which satisfy all the boundary conditions if only terms free from k are taken, For the second approximation we have to solve the equations — + — = — ( 7) 5) f 2 Ca for? mle why2,.o.emi ib ae 1428 Internal Waves tn Channels of Vartable Depth together with the boundary conditions. Now (123) is just the equation for potential flow with uniformly distributed sources of strength C If (8) is satisfied (with f now identified with f,,5), andif d, is the height of the mth interface, b) ! = | fleudide =" A eCica 0 (124) ee th N a Qu Q ra ii > @) ob > ine) O! NM ra ies) ¥ Q as, Qu ] I > 5 Q 3 by virtue of continuity. In (124) b,, is the width of the mth inter- face, and A, the cross-sectional area of the mth layer. (See Figure 1), Integrating (11) across b,,, layer by layer, we have 2 er eet Pg eet TER eae 2 hoe 2 AOC Bee foo Nag ee ee) a of) +» = (125) REPRE Pao : ane c. As > Ge “9 n ata rE 20s Tr) tn mee There are n unknowns oe » not all of which are zero. Hence we obtain a determinant which must vanish, Its vanishing gives n values for cg . The last of the equations in (125) corresponds toa free surface. If the upper surface’is rigid, it is to be replaced by & since *Now a rigid flat upper surface is not excluded. 1429 Chta-Shun Yth f',2 must vanish at y=d, being the height of the upper surface measur- ed from the lowest point of the channel | Indeed, the theory given here is a natural generalisation of the classical shallow-water theory for a single fluid of constant den- sity. If there is only one single fluid, and if it has a free surface, the last equation in (125) gives, with n=1, b=b,, and A=A), Ge Gos 404 or ec = gh h being the average depth. This is a classical result, ACKNOWLEDGMENT This work has been sponsored by the Office of Naval Re- search, under Grant NR-062-448 to the University of Michigan. REFERENCES 1 KELLAND, P..,, 1839,""On Waves!’ Trans. RS. Edinz ital 14, 2 LAMB, H., 1932, ''Hydrodynamics", Cambridge Univ. Press and The MacMillan Co. , Sixth Edition. 3 MacDONALD, H.M., 1894, ''Waves in canals", Proc, Lond. Math. Soe.5, Vol. 25, p. 101, 4 STOKES, G.G., 1839, "Report on recent researches in Hydrodynamics, Brit. Assoc, Rep., 1846. 5 YIH, C.-S., 1965, ''Dynamics of nonhomogeneous fluids", The MacMillan Co., New York. 1430 Internal Waves tn Channels of Vartable Depth Figure Caption Figure 1. Definition sketch 1431 Chta-Shun Yth DISCUSSION Michael N, Yachnis Naval Faetltttes Engineering Command Washington D.C. , U.S.A. It seems to me that ''boundary waves" or ''inter-face waves" are more suitable terms for two or three layers of fluid that the term "internal waves''. I have four basic questions associated with the ap- plication of Professor Yih's Paper, especially on actual ocean intern- al waves: 1. Possibility of short internal waves and breaking pheno- mena. 2. Interaction between surface waves and internal waves. 3. Effects of internal waves on fixed or moving structures, 4. Influence of earthquake or disturbances of the bottom of the channel on internal waves. REPLY TO DISCUSSION Ceo... Lan Untverstty of Michtgan, Ann Arbor, Michtgan, U.S.A. I have been asked by Dr. Yachnis to comment on four matters. This is a great compliment. I do not know that much, but I shall try. Iam glad he mentioned short internal waves. I forgot to men- tion short internal waves propagating along a channel of variable depth in the direction of the axis. If you do not have stratification, very short waves are not terribly interesting, for this reason : if the den- 1432 Internal Waves tn Channels of Vartable Depth sity is uniform the motion is concentrated in a region near the free surface and the fluid does not ''feel'' the nonuniformity of the depth. But for internal waves, especially if the modes are high, that is to say if there are many internal zeros, then even for short waves the entire fluid participates in the motion, This has not been studied very much. My expansion scheme is not suitable at all for large wave numbers. One can do an asymptotic study of the differential equation to deal with that case, and I do not think it would be terribly difficult to do so. As to breaking, I should think that breaking is probably more severe for long waves. In fact, we know that long internal waves in a system of two layers indeed very often break, Although there are a lot of other solutions for non breaking waves (cnoidal waves, solitary waves and so on), if you make a laboratory test, pushing a plate against a layer- ed system you will see that indeed the interface breaks, Mathematical studies of the breaking of internal waves are even more difficult than the studies of the breaking of ordinary waves in one single fluid, and I certainly do not know the mathematical theory. You surely remem- ber the last picture shown by Pr. Plate. If a wave goes that far I would keep well away, both physically and intellectually. Secondly, about the interaction between surface waves and internal waves, that too has not been studied a great deal. We all know that after a storm there are internal waves created in the sea. How is the surface disturbance created and how are the messages transmitted from the surface down to the depths of the sea ? Notmuch is known about that. I think, however, that the interaction between surface waves and internal waves can be considered in this way : if the surface waves already created have a frequency very far away from that of any of the internal waves, there is no chance for the re- sonance phenomenon to happen. However, if short surface waves have the same frequency as some longer internal waves, the surface waves can excite internal waves, especially if the amplitude is not small, If the amplitude is small we do not need to worry about excit- ation, As for the effect of internal waves on moving structures, I do not have much to say about that. Naturally there would be an in- ternal-wave drag for moving’ submerged structures. The last question concerns the creation of internal waves by earthquake disturbances, I think that as far as the linear theory goes itis really just a matter of Fourier analysis. If one knows the details fo an earthquake, you can obtain the spectrum of internal waves creat- ed. 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Laurel, Maryland, U.S.A. ABSTRACT The microstructure of the wind-disturbed water surface, characterized by surface-slope and sur- face-curvature distributions, is measured in a labo- ratory tank under various wind and wave conditions, The relative frequencies of occurrence of various slopes generally follow a normal distribution. At low wind velocities, the formation of parasitic waves causes a skewed slope distribution; at high wind velocities, the wave breaking causes a peaked slope distribution. It is also shown that the mean- square slope rises suddenly at about the wind velo- city where the airflow boundary layer becomes turbulent. The curvature distribution of the wind disturbed water surface observed from different angles is generally skewed with greater radius of curvature at steeper viewing angles from the nor- mal to the mean water surface. As the wind velo- city increases, the average radius of curvature decreases rapidly at low wind velocities when waves are effectively excited by wind, and gradu- ally at high wind velocities when waves approach saturated state. The present measurements of surface curvatures are the only set of data of its kind. The mean-square surface slopes are com- pared favorably with those determined in the field; and both sets of data are consistent with the equi- librium wind-wave spectra. 1435 Jin Wu I, INTRODUCTION The wind-disturbed water surface consists of waves of va- rious lengths moving at various speeds. There has been ever-increa_— sing interest in determining the statistical properties of the micro- structure (wavelets) of such a surface. From a fluid-mechanics point of view, these wavelets are involved in the inception of wind waves and are believed to be related to the dissipation of the wave energy. From an oceanographic point of view, the quantitative measurement of the mean square surface slope provides the best determination of the coefficient involved in the equilibrium wind-wave spectrum for describing the directional energy-density distribution of ocean waves. From a meteorological point of view, the microstructure plays a ma- jor role on the radiation of thermal energy from the sea surface. Finally, from the remote-sensing point of view, ripples are impor- tant for interpreting reflection and back scattering of electromagne- tic waves from the ocean surface. A few optical methods have been adopted in the past for de- termining the microstructure of the wind disturbed water surface; these include the photographic method of Schooley (1954, 1955), and of Cox and Munk (1956), and the light refraction method of Cox (1958). However, the photographic method so far developed involves rather tedious data analysis and, moreover, is not completely apt for laboratory application, Owing to the limited fetch, the water surface structure in the laboratory wind-wave tank lacks spatial homogenei- ty, which is required for the photographic method. In the light refrac- tion method, the apparatus consists of submerged parts which offer obstruction to waves and are rather difficult to construct; in addition, the under tank illumination required is inconvenient for a deep wind wave tank which is appropriate to simulate the air-sea interface for more advanced studies, The present instrument, utilizing a light reflection principle, is capable of determining not only the surface slope but also surface curvature with high resolution. In the present study, the microstructure of the wind-distur- bed water surface, characterized by surface-slope and surface-cur- vature distributions, is measured in a laboratory tank under various wind and wave conditions. The features of these distributions are discussed, along with their variations with structures of dominant waves and the growth of slope and curvature statistics with the wind. The present measurement of surface curvatures is the only set of data of its kind. The mean-square sea-surface slope obtained by Cox and Munk (1956) are reanalyzed and compared with the present results. These two sets of data are shown in good agreement and to be complementary to the equilibrium wind-wave spectra, Finally, 1436 Mtcroscopte Structures of Wind Waves some discussion is included on mechanism of wave generation by wind through comparing the cutoff wavelength of the slope data with the calculated neutrally stable wavelength. II. EQUIPMENT AND EXPERIMENTAL PROCEDURE Il, 1 Wind-wave tank and general instruments The wind-wave tank has a 1.5 x 1.55-m cross section and is 14 m long; see Figure la. The top of the tank is covered for 5.5m, up to the test section. Mounted at the upstream end of the tank is an axial-flow fan, and a permeable-type wave absorber is installed at the downstream end. The maximum obtainable wind velocity with a 0. 35-m-deep air passage above 1.2-m-deep water is 14 m/sec. The wind-velocity profile in the tunnel is determined by the vertical traverse of a pitot-static tube. The drift current is measured by timing the motion of floats of various sizes. The surface drift cur- rent is determined by extrapolating the measured current-distribution curve to the water surface. The results of surface drift currents and of wind velocities are used together to obtain the wind velocity rela- tive to the water surface. Two types of instruments have been used simultaneously for wave measurements: a conductivity probe for recording gravity-wave profiles, and an optical instrument for surface-slope and surface- curvature measurements, The conductivity probe, a wave-height gauge, does not provide enough resolution for measuring wavelets that ride on top of gravity waves and have amplitudes only small frac- tions of the latter. A detailed description of the wave tank and its associated instruments has been given elsewhere (Wu 1968, 1971a) II, 2 Optical instrument The optical instrument, shown in figure lb, consists ofa light source, a telescope, and a photomultiplier unit. Supported over the wave tank, the instrument can be set at any desired inclination from the water surface. The photomultiplier receives reflected light only when the water surface is normal to the plane containing the light beam and the telescopic axis. The cross section of the light beam is rectangular, with a length-to-width ratio of 20 tol. The short side of the beam is aligned with the direction of the wind. Therefore, the angular sensitivity of the instrument in the traverse (cross-wind) di- rection is about 1/20 times that in the longitudinal (wind) direction. The angular tolerance of the instrument to the water-surface slope 1437 Jin Wu in the longitudinal direction is about 1°; see figure 2, The focal spot of the telescope on the water surface is circu- lar, with 0.7-mm diameter. This spot is completely bright when the water surface is relatively flat and is partially bright when the curved water surface reflects part of the impinging light away from the teles- cope. Simple calculations have been made along with a calibration test consisting of passing cylinders, with the same kind of reflecting surface but with various radii, under the instrument. The longitudi- nal axis of the cylinder is always parallel with the same axis of the lamp. From the geometry in figure 3a, the following relationships can be obtained for a curved surface with radius of curvature R: a Tog = HK Ccos. a li, - R cos, & 2a = cot a aaa COL SSeS aL ee (1) w/2+R sin a d/2+ Rsin a for a convex surface, and Lp + Rcosa 1 ei COSme =f = t Zo = Ceot + cot 7 (2) w/2+R sina d/2 +Rsin a for a concave surface, where d is the diameter of the pinhole located in front of the photomultiplier, wis the effective width of the plano- convex cylindrical lens for focusing the light, and L,and Ly are the distances from the telescope and the light box lens to the mean water surface, respectively. Only single reflections are considered. By choosing the size of the pinhole to be much smaller than the beamwidth of the light and by putting the instrument away from the surface (for the present setup, w/d = 400 and L/w = L,/w = SO). we can show that the second term on the right of (1) and (2) is much smaller than the first term in each respective equation. In other words, by the proper setting of the distance between the instrument and the water surface ( Lp >> R), both (1) and (2) can be approxima- ted by a = leenwis a 2 t [Le / ( /2)] (3) Hence, the response of the instrument to surface curvature is essen- tially the same for both concave and convex surfaces. 1438 Microscopte Structures of Wind Waves If we designate r as the radius of the focal spot of the teles- cope on the water surface, the degree of saturation of the light signal (s = 1 for saturated signal) can be shown as di 2125 Joos"! = -= oer]? (4) where hla sina is the half-width of the bright portion of the water surface image (see figure 3a). Of course, the radius of the focal spot of the telescope on the water surface, r, depends upon the diameter of the pinhole, d, as well as on the characteristics of the telescope lens. The calibrated response for the present setup, as well as those obtained directly from (3) and (4), is plotted in figure 3b. The scattering of the calibra- tion points is believed to be the result of local deflects on the calibra- tion cylinder. In summary, the light signal is continuous and saturated to a prescribed level, insensitive to the change of curvature, as long as the angular change of the wavy water surface from the downwind face to the upwind face is less than the acceptance width of the instru- ment (about 1°), As the angular change increases beyond that, or if the surface curvature increases further, the signal becomes discon- tinuous. The signal is essentially a light pulse. The intensity of the signal, i.e., the pulse height, is related to the surface curvature; the period of the signal, or the pulse width, is the time required for a detectable slope to pass completely under the instrument. The distribution of the surface slopes is determined by accu- mulating the numbers of light pulses for each inclination of the instru- ment for a constant period (10 min). To determine the distribution of surface curvatures, each series of light pulses is sorted according to their intensity into 50 channels of the pulse-height analyzer, with preset intensity bands. The output from the analyzer, the height dis- tribution of light pulses for a given instrument inclination, is first traced on a x-y plotter and later digitized. The block diagram of the apparatus is shown in figure 4a. The light source is a 1500 W incandescent lamp, approximately 18 cm long. The photomultiplier tube is a nine stage, side-on unit with S-4 spectral response, The high voltage power supply is adjustable so that adequate sensitivity with minimum dark current noise can be obtained. The output of the electrometer is comprised of irregularly 1439 Jin Wu shaped pulses, varying from 0 to 10 V in amplitude and having dura- tions of a few milliseconds. However, the pulse height analyzer (ND-110 128- channel analyzer manufactured by Nuclear Data, Inc. ) requires pulses of much shorter periods and places very stringent requirements on the risetime. The pulse conditioning circuit shown in figure 4a is thus required. The pulse conditioner is capable of analyzing both pulse height and pulse width (pulse period). In both modes, the signal activates the Schmitt trigger. This in turn sets the binary and opens electronic switch N° 1, In the height mode, the track hold amplifier output is compared with the incoming signal. When the input drops, the ampli - fier is put in hold. In other words, the amplifier tracks the signal to its maximum value and holds this value until it is reset. On the other hand, when the signal drops below the threshold of the Schmitt trigger monostablet, is started. This closes electronic switch N° 2 foma sec. During this time the amplitude of the track hold amplifier is ga- ted to the pulse height analyzer as a pulse with a suitable width and risetime. At the end of this time, T> is started again to reset the bi- nary and to close electronic switch N° 1. By doing so the track hold amplifier is reset until a new pulse is received. In the time mode, the operation is identical except that the Schmitt trigger controls a ramp generator, which is tracked to con- vert time to amplitude. Consequently, the amplitude gated to the pul- se height analyzer is proportional to the signal pulse duration. The pulse described above, from the signal to the pulse height analyzer, are shown in figure 4b, The widths of 17, and T, are exaggerated for clarity. Ill EXPERIMENTAL CONDITIONS IiI. 1 Wind conditions The wind-velocity profiles were found to follow essentially the logarithmic law near but not too close to the water surface (Wu 1968). The shear velocities, obtained at different wind velocities, are presented in figure 5. Lines shown in the same figure are the shear velocity for a laminar boundary layer. 1 1s O64.) uy= (F ey Ue (5) and for a turbulent boundary layer in the aerodynamically smooth 1440 Mtecroscopte Structures of Wind Waves flow regime, 1 spt 070: 590 Na" vn( 2 - R % . a (6) n wherein the Reynolds number (RL = ah . L/v) is defined in terms of the free-stream air velocity, U_, the distance between the fan and the test section, L, and the kinematic viscosity of air, V Judging from the trends of the data shown in figure 5, we see that a) the air-flow boundary layer seems to be in the pretransi- tion region for U_< 1.9 m/sec. Because the air was sucked into the wind-wave tank by an axial fan and through guiding vanes, which were arranged to straighten the flow but not to diminish the high tur- bulence level, the latter arrangement was helpful to increase the effec- tive lenght of the wind fetch. b) the effective transition region of the boundary layer from laminar to turbulent is very narrow, at wind velocities between 1.9 and 2.4 m/sec. c) once the boundary layer becomes turbulent, U_ >2.4 m/sec, the transition from smooth to rough boundary layer takes place and this process is completed at Uo = 3.5 m/sec, the aerody- namically smooth flow regime is rather narrow. d) for the aerodynamically rough flow regime, the two groups of data are separated physically by the transfer of the gover- ning mechanism of wind-wave interaction from surface tension to gravity (Wu 1968, 1969); this separation occurs at iy. = 9m/sec. It should be emphasized here that because of the difference of scales (such as wind fetch) between the ocean and laboratory con- ditions and of differences of the wind structures (such as turbulence levels) between various wind-wave tanks, the shear velocity rather than the wind velocity should be adopted to characterize the wind conditions (Wu 1968). By use of the shear velocity, data obtained in the present experiment may readily be correlated with results of other investigations. III.2 Wave conditions From the continuous wave-profile recording, the periods of more than 100 basic waves for each wind velocity are obtained (Wu 1441 Jin Wu 1968). The average values of the wavelength, calculated from the measured period by using the dispersion relation for small-amplitu- de deep-water waves, are shown in figure 6. A general observation is that small-amplitude capillary waves are generated at very low wind velocities (U, < 2 m/sec). Rhombic wave cells are observed before the wind boundary layer becomes fully rough (2 m/sec 9.5 m/sec), the carrier wave is covered rather evenly by ripples. This is the gravity-governing regime. The case with the wind velocity of 9.3 m/sec is in the transition re- gion between the surface-tension regime at low wind velocities and the gravity-governing regime at high wind velocities. The microstructures of disturbed water surface for the three highest wind velocities are very similar to oceanic conditions; nonlinear interaction between short and long waves is believed to be active in this regime. The horizontal contraction of the water surface near the crest of the long wave was stated by Longuet-Higgins and Stewart (1960) to shorten the lengths and to increase the amplitudes of short waves (ripples). When these ripples are saturated, further shortening will cause their breaking. Phillips (1963) then showed analytically that the energy loss by short waves near the crests of long waves is partially supplied by the long wave, and, therefore, causes the attenuation of 1447 Jin Wu long gravity waves. Later, slow moving short waves are considered (Longuet-Higgins 1969) to be swept through by fast-moving long waves and to lose their energy to the long waves ata rate proportional to the orbital velocity in the long waves. More recently, Hasselmann (1972) found that the work done on the long waves through the sweeping is balanced by the loss of potential energy due to taking mass of high potential energy from the crests of long waves and returning it ata lower potential level in the troughs. The dynamics of nonlinear inter- action of oceanic waves is therefore rather confused. However, all of these models are based on one common phenomenon : long waves sweeping through short waves and causing short waves to break on the forward faces of the longwave crests. Some physical evidence substantiating this basic phenomenon has been obtained through photo- graphs of glitter patterns (Wu 197Ib). Figure 13 shows that the minimum radius of curvature is observed at a positive angle for all of the wind velocities. The mini- mum radius of curvature is undoubtedly produced by crests and troughs of the shortest waves. The present results therefore seem to confirm earlier observations and conclusions that waves with the smallest radii of curvature, very likely produced by nonlinear wave- wave interaction, ride on the forward faces of long carrier waves having positive slopes. In other words, the skewed shape of the angu- lar distribution of average radius of curvature is probably due to the nonlinear wave-wave interaction. V.5 Low and high grazing angles for back scattering. The reflection and back scattering of electromagnetic waves impinging on the air-sea.interface depend on the sizes of the specu- lar areas, The latter can be described statistically by the average radius of curvature. The distribution of the size of the specular areas was considered by Schooley (1955) to be substantially the same for all slopes. The results, obtained here and shown in figures 13 and 14 suggest that this earlier consideration may be approximately true for the ocean surface and at limited angles near the normal to the hori- zon. As discussed in the previous section, the skewed angular distri- bution of surface curvature seems to be as expected as a result of parasitic capillaries at medium wind velocities and of nonlinear wave - wave interaction at high wind velocities. The measurements for the three highest wind velocities are most interesting for practical application, because the surface struc- tures for these three cases are believed to be very similar to the air- sea interface. As shown in figures 13 and 14, and stated in the pre- 1448 Mtcroscopte Structures of Wind Waves vious paragraph, the microstructure may be considered nearly iso- tropic for viewing angles very close to the normal to the mean water surface, say, less than the root-mean-square slope. Beyond this region, the sizes of specular areas, represented by the average radius of curvature, increases rather rapidly with the angle from the normal. No data were obtained at very steep angles, where the situation is further complicated by possible shadowing effect. Judging from the data shown in figure 14, the backscattering measurement is ideally made at small angles from the normal, where the sea surface is nearly isotropic. A small error of the angular mea- surement at large angles would introduce serious change of the results because the sea surface in this case is highly nonisotropic. V.6 Growth of high-frequency wind waves. In order to find the over-all average radius of curvature of the disturbed water surface for each wind velocity, the cross product of the smooth data shown in figure 13, is found. One curve shows the angular distribution of average surface curvature and the other curve shows the relative frequency of occurrence of the particular curvature. Consequently, the cross product represents the overall average of surface curvature obtained at a given wind velocity. The overall average radius of surface curvature are shown in figure 15. The data indicate a rapid decrease of the radius of sur- face curvature with increasing wind-shear velocity is observed at low wind velocities and a steady but gradual decrease at high wind velocities. Figure 15 shows that the radius of curvature seems to reach a saturated value of 1/4 cm, when the wave growth with wind ceases, VI. SEA-SURFACE SLOPE AND EQUILIBRIUM WAVE SPECTRA VI. 1 Equilibrium wind-wave spectra. The directional wind-wave spectra W (k) in the equilibrium range was proposed by Phillips (1958a, 1966): — B -4 Gravity waves: VW (k)=— f(0)k , ie oe ale 7 ) - ii (7) Capillary waves: W (k) =qmi t (18 bbknd - ky where k and k are the wave-number vector and scalar, respectively; 1449 Jin Wu k,is the wave-number at the spectral maximum; ky is the maximum wave-number where the influence of surface tension is negligible; k, is the neutrally stable wave-number, B and B' are the spectral coef- ficients for the gravity and the capillary ranges, respectively; finally, f (@)is a dimensionless function specifying the directional distribu- tion of wave components (Schule et al. 1971) where 9 = 0 indicates the wind direction, The wave-number ky can be expressed as | I= deesBgtie Bs shy (8) wherein pis the density of water, gis the gravitational acceleration and T is the surface tension. The neutrally stable wave-number cor- responds to the wavelength at which the energy input from the wind is balanced by the energy dissipation through viscosity. This specific wavelength was expressed by Miles (1962) as a function of the wind shear velocity. It is considered for the equilibrium wave spectra that high- frequency wave components spread isotropically. In this case f (0 ) equals to unity and the one-dimensional spectra, identical in all direc- tions, becomes v¥(k)=(B/2 7 )kK? and W(k)=(B'/27)k? (9) In laboratories, owing to narrowness of the tank the waves propagate predominantly in the direction of the wind. The spectra may be consi- dered to be unidirectional and may be described by (9) in the direction of the wind. The mean square slope of the wind-disturbed water surface, s*, can be obtained from the directional wave-number spectrum WV (k), or =z f [oe) 2 =a A s = j k* w (k) dk (10) The integration should cover the possible range of the wave-number. Substituting (7) into (10), we have (Phillips 1966) s°= B fy (ky/k,) + BI Ln (k,/ky ) (11) The first term on the right-hand side of (11) represents the contribu- tion of gravity waves to the mean-square surface slope, and the se- cond term on the right-hand side represents the contribution of capil- lary waves. 1450 Microscopte Structures of Wind Waves It is noted that the longitudinal (upwind-downwind) component is about one half of the total mean-square sea-surface slope, as short ocean waves propagate nearly isotropically. In the laboratory tank, the longitudinal component is about the same as the total mean-square surface slope, as waves propagate nearly unidirectionally. Needless to say, these ratios can only be approximations, because the propa- gation of wind waves can neither be exactly isotropical nor be exactly unidirectional, VI. 2 Sea-surface slope and spectral coefficients. Cox and Munk (1956) deduced slopes of the sea-surface from the brightness distribution of photographs of sun glitter of the sea- surface. The wind velocity at the time of taking sun glitter pictures was recorded at two heights, 9 and 41 ft. from the mean sea level. The lower height may be too close to the water surface to be free from the wave-induced air motion, especially at higher wind veloci- ties. Therefore, the wind velocity measured at the upper height was used along with the wind-stress coefficient iGr,! formula (Wu 1969), =F 0.5" 'U Cio dos (a 10 (12) to determine from the logarithmic wind profile the corresponding wind velocity U at the standard anemometer height. It may be worthwhile to note, however, that this correction is very small and that the wind velocities U, obtained from both methods are about the-same. The results’ of the mean-square surface slope of a clean surface s¢ versus U_ are plotted in a semilogarithmic form in figu- re 16, Various poumde es layer regimes of the wind (Wu 1968) are shown in the same figure. The results of mean-square slopes obtai- ned from the interior of an artificial slick are not included in this figure. It is very obvious from figure 16 that the data are divided into two groups: one in the hydrodynamically smooth flow regime and the transition region (U,_.< 7 m/sec), and the other in the hydro- dynamically rough flow regime (U > 7m/sec). A straightline can fit the results in each group rather well. The data are scattered at low wind velocities where the wind condition is less stable in the transition region, scattering of the data seems to be inevitable. An excellent correlation of the mean-square slope with the wind veloci- ty is seen in figure 16. Most of the results of Cox and Munk were obtained from a 1451 Jin Wu clean sea surface, where capillary waves were undoubtedly present and should contribute to the mean-square surface slope. On the other hand, some of their observations were taken in the interior of a dense artificial slick, where waves shorter than 30.5 cm (1 ft) were repor- ted by Cox and Munk to be absent. For the latter case, the maximum wavenumber for existing waves, designated as k, (with the minimum wavelength 7, ), is certainly well outside the capillary range, or in other words kg smaller than k,. Therefore, for this portion of the data obtained in the interior of artificial slicks. Equation (11) can be rewritten as 2 BLU, ,?/s) kK. (13) in which the wavenumber kis substituted with e/a 2 1( Phillips 1966). It isSnow obvious, if we replot the mean-square slope data in the semilogarithmic form and fit the replotted data with a straight- line, wecandetermine both B andk, independently. The kg so deter- mined can then be compared with the abserved ke Replotted in figure 17 in the proposed form and fitted with straightlines by means of the least-square principle are the data of three different groups : (a) clean sea surface with the airflow in the hydrodynamically smooth regime and the transition region, (b) clean sea surface with the airflow in the hydrodynamically rough regime, and (c) sea surface covered with slicks. For the last group, the va- lues of B and k, , determined from the slope and the intercept of the fitted straightline, are B=4.6 x 10.7: A, = 38cm (14) The value of B is identical to that obtained earlier by Phillips (1966). Taking into account the scattering of the data and the rather crude visual observations of the minimum wavelength in dense slicks, the A_ can be considered in rather close agreement with the observed value of 1 ft. This agreement also supports the technique used here to determine the spectral constant. For a clean sea surface, the contribution of the capillary waves to the mean-square surface slope, the second term on the right-hand side of (11), cannot be neglected. Relative to the data ob- tained in the slick, those obtained in the hydrodynamically smooth regime of clean water surface are seen in figure 17 to be shifted al- most parallel upward and twoward the left. Referring to (11) and (13) the upward and nearly parallel shift of the data indicates that the 1452 Microscopte Structures of Wind Waves contribution to mean-square slope from wave components having their wavenumbers greater thank, are about the same for all wind velocities in the smooth regime. This trend is along the lines of Phillips concept (Phillips 1958a) of the development of the wave spectrum, in which the higher-frequency waves reach the saturated state earlier. The shift toward the left of the clean-surface data or the shift toward the right of the slick data is also very interesting. This clearly indicates that the wind boundary-layer transition from smooth to rough regime is delayed by the presence of the slick. It is expected that the ocean surface becomes smoother when it is cove- red by a dense oil slick. The difference of mean-square slope between the clean and the contaminated sea surfaces at low wind velocity is about 0.0115. This difference is the contribution to the mean-square surface slope from wave components having their wavenumbers greater than k_. Accepting this argument we can estimate from the slope difference the cutoff wavenumber k, at low wind velocities provided that Kewis smaller than k, . From (13) and (14) we have 0. 0046 Ln (k_ /k_) = 0.0115 k = 2.5¢m_ (15) This value is indeed smaller than k, which is about 3.6 Sen “whe closeness of these two values indicates that the straightline fitted through the clean surface data at low wind velocities may be the upper bound of the contribution from waves in the gravity range. In other words, the contribution to the mean-square slope at higher wind velo- cities shown in figure 17 above the extension of this straight line must come from waves in the capillary range. At first look, this conside- ration may seem rather arbitrary. Actually, since B is very small, the choice of a slightly different cutoff wavenumber k, has an insi- gnificant effect on the results. On the other hand, the coefficient B' to be shown later, is much greater than B, so that wave components in the capillary range contribute much more effectively to the mean- square sea surface slope than those in the gravity range. Consequent- ly, once the integration of the wave spectrum extends into the capil- lary range, a change in the trend of the data, such as that shown in figures 16 and 17, is expected. More studies are needed to see whether it is just a coincidence that this separation of slope behavior coincides with the change of the regimes of the wind boundary layer. An excellent correlation of data between s* and U, g is shown in the hydrodynamically rough regime of wind. This 1928 indicates two possibilities : the contribution to mean-square slope 1453 Jin Wu from waves of capillary range is independent of the wind velocity and the coefficient B has a different value for the hydrodynamically rough flow regime; or k,, is proportional to U, ¢ The former cannot be true, because any downward shift of the Hess data shown in figure 17, in order to get rid of the contribution from waves in the capillary range, would mean that the contribution from gravity wave compo- nents to the slope at the lower portion of high wind velocities is lower than that at the upper portion of low wind velocities. Furthermore, referring to (13), the cutoff wavenumber obtained from the intercept of the straight line fitted to the data at high wind velocities is simply too great to be reasonable. In summary, the data seem to indicate that the mean-square slope is contributed by wave components from only the gravity range at low wind velocities ( Dio < 7 m/sec), and from both gravity and capillary ranges at high wind velocities. In addition, the cutoff wavenumber, or as considered by Phillips (1966), the neutrally stable wavenumber is proportional to U , the square of the wind velocity measured at he standard anemometer height, Following this conside- ration, we now plot U 2 versus the difference between the slope mea- sured at high wind velocities and that contributed by gravity wave components, and rewrite (11) as st°- By (ky U,)°/e)=Biln(k, /ky) (16) Therefore, the straight line fitted through the replotted data shown in figure 18 allows an independent determination of the spectral coef- ficient B' and the dependency of k, onU,_. It is noted that the trend of the data shown here provides a good verification of the form of the Phillips' capillary-wave spectrum, which has not previously been verified by observations. The value of B' is found to be Bi = 3" 15x10 7* which is about the value 1.5 x Lo. offered by Phillips (1966). Com- paring the curves, fitted by the spectral coefficients shown in figures 17 and 18 with those shown in figure 4.17 of Phillips (1966), one is inclined to believe that the former may be more accurate than the latter. Moreover, Phillips adopted Miles (1962) calculation of the neutrally stable wavenumber as the cutoff wavenumber of the slope data in his process to obtain B' , while the present technique allows an independent determination of B' and the cutoff wavenumber. The 1454 Mteroscopte Structures of Wind Waves latter technique is desirable especially when Miles' calculation has not been verified experimentally. VII. COMPARISON OF LABORATORY AND OCEANIC RESULTS The average wavelengths obtained at various wind velocities are replotted in figure 19a. As the wind velocity increases, the wa- ves, as described previously, passing the following stages of develop- ment : (A) infinitesimal capillary waves, (B) rhombic wave cells, (C) long waves accompanied by parasitic capillaries, and (D) brea- king long waves. The mean-square surface slope determined at va- rious wind velocities are replotted in figure 19b. Taking together the results presented in figure 19a, b, we see that capillary waves with infinitesimal amplitudes are the sole contributor to mean-square slope in stage (A), gravity waves are the sole contributor in stage (B), and both contributors in stage (C) and (D) Because of the great difference between wind fetches exis- ting in the wind-wave tank and the field, the shear velocity rather than the wind velocity should provide a basis for comparison of slope data. The upwind-downwind components of Cox and Munk's data and our laboratory results of the same components are replotted in figu- re 20a. This comparison is made possible on the basis (Phillips 1958b) that high-frequency wind waves, the principal contributor to surface slopes, reach equilibrium states at very short fetches. Such a concept is further illustrated by Cox's (1958) measurements of mean-square slopes, which reach equilibrium states, ceasing to grow spatially, at a fetch slightly greater than 3 m. The wind fetch for the present experiment is about 6 m, It has been shown that oceanic slope data are divided into two groups : gravity waves are the sole contributor to sea-surface slope at low wind velocities and both gravity and capillary waves con- tribute to sea-surface slope at high wind velocities. The portion of the oceanic data fitted by a straightline in figure 20a is the second group. A straightline is also drawn to fit the laboratory data in figu- re 20a. It is interesting to see that the fitted line for the laboratory data which contributions from both gravity and capillary waves is parallel with the line fitted through the oceanic data which the same contributors; see figure 19, The same trend of variation between the oceanic and the laboratory data further confirms our earlier discussion : the separa- tion of oceanic slope data into two groups is indeed due to the fact that capillary waves contribute to mean-square sea-surface slope 1455 Jin Wu only at high wind velocities. Moreover, since the slope of the fitted straightline for the oceanic data is related to the spectral coefficients in the equilibrium range, the same slope of the fitted lines shown here for laboratory and oceanic data indicates that the spectral coefficients are universal constants independent of wind fetches. Consequently, the present results also verify the concept on developing the equili - brium wind-wave spectrum : high-frequency wind waves saturate at short fetches. The cross-wind slope has not been measured in the present tank. However, the fetch for the growth of resonance waves is about half the width of the tank, or 0.5 m, because the slope measurement was made at the center of the tank. According to Cox (1958), the mean-square slope should not rise until the fetch is greater than lm. Therefore, the cross-wind slope should be negligible in the present tank. Consequently, the upwind-downwind component of the present measurement should be nearly the total slope. The upwind-downwind components of the laboratory results are plotted along with the mean-square slopes of the sea surface in figure 20b. Excellent agreement is seen between the oceanic data and the laboratory results except at low wind velocities, where the wind boundary layer in the laboratory tank is not even turbulent. The boun- dary layer in the present tank becomes fully turbulent when the wind- shear velocity is greater than 12 cm/sec. The agreement of oceanic and laboratory data further substantiates some previous considerations the spectral coefficients are indeed universal; the spreading of wave- number vectors is nearly isotropic for the sea and is nearly unidirec- tionai in the laboratory. The comparison shown in figure 20b explains the discrepan- cy between the oceanic (Cox and Munk 1956) and the laboratory (Cox 1958, Wu 1971) results. Furthermore, the agreement on one hand im- plies that short waves are directly generated by the wind and the wind- shear velocity is therefore the appropriate parameter for correlating results obtained at different fetches. On the other hand, the agreement indicates the possibility of modeling microstructures at the sea surfa- ce in a laboratory tank. VIII. CONCLUSION In the present study, the microstructure of the wind-distur- bed water surface, characterized by surface-slope and surface-curva- ture distributions, is measured in a laboratory tank under various wind and wave conditions. It is shown that wind waves arise at about the 1456 Microscopte Structures of Wind Waves time when the airflow boundary layer becomes turbulent. The rela- tive frequencies of occurrence of various slopes generally follow a normal distribution. At lower wind velocities, the formation of para- sitic waves causes a skewed slope distribution; at high wind velocities, the wave breaking causes a peaked slope distribution, The skewed slope distribution may produce different back scattering of electro- magnetic waves from the leeward and from windward faces of the basic wave profile. The curvature distribution of the wind-disturbed water sur- face observed from different angles is generally skewed with greater radius of curvature at steeper viewing angles from the normal to the mean water surface. As the wind velocity increases, the average ra- dius of curvature decreases; rapidly at low wind velocities when waves are effectively excited by wind, and gradually at high wind velocities when waves approach saturated staté. The skewness is caused by the presence of parasitic capillaries at low wind velocities and by nonlinear wave-wave interaction at high wind velocities. The back scattering measurement is ideally made at small angles from the normal, where the sea surface is nearly isotropic. The present measurements of surface curvatures are the only set of data of its kind. The mean-square surface slopes are com- pared with those determined in the field, and the difference is explai- ned as a result of various directional distribution of wave components. Equilibrium energy spectra of wind waves was first established for the gravity range and later extended to the capillary range. The veri- fication of the latter extension and the determination of the spectral coefficients for both ranges are discussed on the basis of the experi- mental results for mean-square slopes. Good agreement between present and oceanic results indicate a possible modeling of the micro- structure of the air-sea interface in a laboratory tank. ACKNOWLEDGEMENT This is a summary report of our work on measurements and modeling of microstructures of the air-sea interface. This program is under the direction of Mr. M.P.Tulin, who also suggested the design of the optical instrument, and of Dr. S.G. Reed, Jr., who also reviewed this report. Portions of this report have been published else- where. This work was supported by the Office of Naval Research under Contracts N° N00014-70-C-0345, NR 220-016 and N° N00014-72-C-0509, NR 062-472. 1457 JIn Wu. REFERENCES COX, C.S.¥ and MUNK, W.H., '956, "Slopes of the Sea Surface Deduced from Photographs of Sun Glitter,'' 6, No.9, Scripps Institu- tion of Oceanography. COX, C.S., 1958, ''Measurements of Slopes of High-Frequency Wind Waves,'' J. Mar. Res. '6, 199-225, HASSELMANN, K., !971, 'Onthe Mass and Momentum Transfer Between Short Gravity Waves and Large-Scale Motions,"' J. Fluid Mech. 50, 189-205. LONGUE T-HIGGINS, M.S... 1959, “The Distribution of the Sizesson Images Reflected in a Random Surface, '' Proc. Cambridge Phil. Soc. 555 9 1=100, LONGUET-HIGGINS, M.S., '969, "'A Nonlinear Mechanism for the Generation of Sea Waves, '"' Proc. Roy. Soc. A 311, 371-89. LONGUET-HIGGINS, M.S., and STEWART, R.W., 1960, ''Changes in the Form of Short Gravity Waves on Long Waves and Tidal Currents,'' Jc, Bidid Mech, G. 5b5=5.5. MILES, J.W., 1962, "On the Generation of Surface Waves by Shear Flows,'' J. Fluid Mech. 13, 433-48. PHILLIPS, O.M., 19584, "The Equilibrium Range in the Spececea of Wind-Generated Waves,'' J. Fluid Mech. 4, 426-34. PHILLIPS, "O. M., 1958b, "Commention.a Paper by Dr. Goxm J.. Mar. Res. 16, 226-30. PHILLIPS, O.M., !1963, "On the Attenuation of Long Gravity Waves by Short Breaking Waves,"' J. Fluid Mech. 16, 321-32. PHILLIPS, O.M., 1966, ''The Dynamics of the Upper Ocean," Cambridge University Press. SCHLICHTING, H., 1968, '"Boundary-Layer Theory," 6th Ed., McGraw-Hill Book Company, New York. SCHOOLEY, A.H., 1954, "A Simple Optical Method for Measuring the Statistical Distribution of Water Surface Slopes, '' J. Opt. Soc. Am, 44, 37-40. 1458 Mteroscopte Structures of Wind Waves SCHOOLEY, A.H., 1955, "Curvature Distribution of Wind-Created Water Waves, ''Trans. Am. Geophys. Union 36, 273-78. SCHULLER Ad. ; Jf) a LOD, oS. 5.4. and, DeLHONIBUS, P.S:, 19728, "A Study of Fetch-Limited Wave Spectra with an Airborne Laser," J. Geophys. Res. 76, 4160-71. WU, JIN, 1968, "Laboratory Studies of Wind-Wave Interactions, "' J. Fluid Mech. 34, 91-112. WU, JIN, 1970, ''Wind-Wave Interactions, '' Phys Fluid 13, 1926-1930. WU, JIN, 1971la, ''Slope and Curvature Distributions of Wind-Disturb- ed Water Surface," J. Opt. soc, Am. 61; 852-58. WU, JIN, 1971b, "Observations on Long Waves Sweeping Through Short Waves," Tellus 23, 364-70. WU, JIN, 1972, "Surface Curvature of Wind Waves Observed from Diffefent Angles,;J. Opt: sec. Am. 62, 395-400. WU, JIN, 1972, ''Sea-Surface Slope and Equilibrium Wind-Wave Spectra,'' Phys. Fluids 15, 741-47. WUP ws LAWRENCE SUM, EBAY, rE. S: ;. and. TULEIN, MeP:, 1969, ''A Multiple Purpose Optical Instrument for Studies of Short Steep Water Waves,'' Review of Scientific Instruments 40, 1209-13. 1459 Jtn Wu A =Light Source and Adjustment Assembly; B =Light-Signal Receiver and Adjustment Assembly C =Cross-Beam and Am Unit; D =Hinge-Joint Support; E =Angle Indicator; F = Water Surface (b) Optical Instrument FIGURE 1 - WIND-WAVE TANK AND INSTRUMENTS 1460 Mteroscopte Structures of Wind Waves Oo Oo oO ° > —— _ e =) = =) > ® — x) c — wo -1.0 -0.5 0 0.5 Angle of Inclination, @ (degrees ) FIGURE 2 - ANGULAR RESPONSE OF OPTICAL INSTRUMENT, 1461 Jtn Wu Width of light-box lens opening = w w w — Diameter of pin hole =d Mean water level Mean water level { Concave Surface (a) Geometry Partially-illuminated spot Fully-illuminated Degree of Signal Saturation U.05 0.) O75) i} 5 Radius of Curvature r (cm) (b) Calibration FIGURE 3 - CURVATURE RESPONSE OF OPTICAL INSTRUMENT. 1462 Mteroscopte Structures of Wind Waves Al Signal a Threshold Schmitt trigger of as Q T/H Amplitude "Height" T/H Amplitude "Time" (a) Pulse Shape and Timing Diagram of Pulses Passing Various Stages of Pulse Conditioner TRRIALVIJED DCARHAIIT ~~ | ANALYZER-READOUT Coincidence 7285Ch | G -Channe \ ee ——“ Pulse height | | ' anaiyzer | j A : = Oscilloscope; i | y x-y recorder; | | (when used) or punched tape | ee =— : ee [ SIGNAL | PULSE CONDITIONING | | Photomultiplier tube | CIRCUIT | (S-4} =a | Q Electrometer | Comporator | q p amplifier x | 3 free | | O | Teack/halder Ephone | | omplitten switcn No. | | 2 | | Electronic | | l switch No. I ) ' SSS = | t | | Schmitt | trigger | voltage i power | | I bakes | 5 psec 5 psec { Monostable Monostable ! T T) | 1 ‘ eae ee ED NG Se Pee (b) Block Diagram of Optical Instrument FIGURE 4 - ELECTRONICS OF OPTICAL INSTRUMENT. 1463 Shear Velocity, u, (m/sec) Pre-transition Jtn Wu Turbulent boundary layer Aerodynamically rough flow turbulent ) Transition Surface roughness governed by capillary waves Surface roughness governed by gravity waves — < D > ° pad ' <= _ ° ° [= 7) ~— Cc fe) = “a Cc Oo -_ — _ Transition 1/2 9.0295. - U_ - Turbulent aerodynamically smooth flow iri fc) 6.332 V2 f Se U - Laminar flow iRi/2 ° | 5 19 Wind Velocity, Us (m/sec ) FIGURE 5 - WIND-SHEAR VELOCITY AND BOUNDARY-LAYER REGIMES 1464 Average Wave Length, A (cm) Microscopie Structures of Wind Waves ] 72 4 6 8 10 20 Wind Velocity, U, (m/sec ) FIGURE 6 - GROWTH OF WIND-WAVES. Stages of wave growth with wind velocity: (A) Capillary waves; (B) Rhombic, short gravity; (C) Gravity waves with parasitic capillaries; (D) Breaking gravity waves. 1465 Jin Wu BODjINS 18jDM “SNOILNGILSIG IJOIS JDVAINS - Z INOIS (seaiBap) g ‘uolpuljouy yo ajbuy O€ Slt O SIt- O€- O€ st O SI- O€- Of Sl O SI- O€- oz OL & Ai awiniysu! Port aS x jo21sdo Butuipjyuos \7 BUuDIid 0 Ol- 62°C oe ae \ 90°72 é6l*l 08° | /Agy> \l i | 08° Ll Gy Z es" 16°0 (2as/w) OA ‘AyD0;aA PULA 9 vz 9 vz 9 LZ Frequency of Occurrence, p, (percent) 1466 Standard Deviation of Surface Slope (degree ) Microscoptc Structures of Wind Waves Turbulent boundary layer Cc fe} = = .o] = = nN oO 17.3 ————-Y>Y TT __ Od oO (Q 9 eee 0 2) 10 15 Wind Velocity, Uo (m/sec ) FIGURE 8 - INCEPTION OF HIGH-FREQUENCY COMPONENTS OF WIND WAVES. 1467 Mean Square Water Surface Slope Jin Wu 0.1 0.05, 0 5 10 15 Wind Velocity, U (m/sec) FIGURE 9 - COMPARISON OF PRESENT RESULTS ( @ ) WITH THOSE OBTAINED BY COX (O ) 1468 Skewness ( degrees ) Relative Frequency of Occurrence Microscopie Structures of Wind Waves oO On 0 5 10 15 Wind Velocity, US (m/sec ) FIGURE 10 - SKEWNESS OF WATER-SURFACE SLOPES AT VARIOUS WIND VELOCITIES (a) Skewness of Slope-distribution curves; (b ) Relative Frequency of Occurrence of Capillary Waves at Upwind (O) and Downwind (@) Faces of Gravity Waves 1469 Jin Wu NOILNSILSIG IINLVAYND JDVAINS - LL IYNOIS (wd) 4 ‘aunyoasn> yo snippy ° (28S/w) eS iq) c Frequency of Occurrence, p_ (percent) 1470 Structures of Wind Waves croscoptc ML VLVG IWOldAl - 21 FINO (Wi) B1N}DAINZ 4O sNIpDY £ é l 0 0S OF BINJDAIND 10 snippi abbiaay juUswnijsul jo $4O-4ND 419MO7 Ol Ol Ol Ol saquinyy jauuoy> oe 02 Ol Number of Waves 1471 Jtn Wu SNOILIGNOD GNIM SNOIYVA YIGNN JINLVAYIND JO SNIGVY IOVYIAV JO NOILNGISLSIG YWINONV - El UNI sa|Buy jo uol!uljaq aoDyINs JajDM UDA juawnijsut B2DjINS 1aJOM jo214do uDaW OF JOWON | BuiusojU0> aUudId 4 aNim Gy of Sl. 0 SI-O€- SP- EGE Sy O£ SI ( seaiBap ) ‘ g’uolyouljouy jo ajbuy Sy O€ SI a evel | is ie tA 0 Sl- O€- Sr- Sp0e Sil) 40S 08S Sy- ‘a re 6L°¥ OSI= (08> Sra) ‘Sr 0& SIE €£°6 \ \, 298/W G0°z =f) ONSIS0b=ssr— 0) Average Radius of Curvature, |r| (cm ) 1472 Absolute Average Radius of Curvature, a (cm) Microscopie Structures of Wind Waves 3 ‘ U) = nae 4.79 2.05 m/sec 1.0 9.5 0 T : 6.22 Wath : 9533 “i ° L ° os ° dl. 12.39 13.42 = n 7 oe ee ee ey AVES 22 No Ol. Ze av 4 aia6. 2 -t. Gee Angle of Inclination/Root-Mean-Square Angle FIGURE 14 - ANGULAR DISTRIBUTION OF RADIUS OF CURVATURE NORMALIZED WITH RESPECT TO ROOT-MEAN-SQUARE SLOPE. 1473 Absolute Average Radius of Curvature, cm Jin Wu —_ e o 0.5 0 0.4 0.8 Shear Velocity, u, (m/sec) FIGURE 15 - VARIATION OF RADIUS OF CURVATURE WITH WIND SHEAR VELOCITY 1474 1.2 Microscopie Structures of Wind Waves Hydrodynamically smooth Hydrodynamically rough Transition flow regime flow regime 0.08 52 Mean Square Surface Slope, SZ g 0.5 ] 5 10 50 Wind Velocity, Vio (m/sec) FIGURE 16 - MEAN-SQUARE SEA SURFACE SLOPE IN DIFFERENT BOUNDARY-LAYER REGIMES OF WIND 1475 “$3dO1S JDVIINS JVNOS-NVIW JO SL1NSIY WO SLNVISNOD 1v¥LD3adS JO NOILVNIWY3130 - ZL IYNOI4 (w) B/O'n 201 (ol (0! Ol Ol c0*0 Jin Wu 90°0 }21|s Aq passnoo aonjins Dag @ BIDJINS DES UDS| O 80°O 1476 Microscopie Structures of Wind Waves 0.04 0.02 ° ©) O oO .) O @) O O 5/2 OE 2 10 10 2 2 Yio FIGURE 18 - CONTRIBUTION OF CAPILLARY WAVES TO MEAN-SQUARE SURFACE SLOPE. 1477 Jtn Wu Average Wave Length, A (cm) 2 L Mean Square Slope, s O.8F 1 2 4 6 8 10 20 Wind Velocity, U, (m/sec ) FIGURE 19 - LABORATORY MEASUREMENTS OF DOMINANT-WAVE LENGTH AND MEAN SQUARE SURFACE SLPOE. Stages of wave growth with wind velocity: (A) Capillary waves; (B) Rhombic, short gravity waves; (C) Gravity waves with parasitic capillaries; (D) Breaking gravity waves. 1478 Mean-Square Slope Mteroscopte Structures of Wind Waves 0.10 @ Longitudinal component 0.08 (Wu, 1971) © Longitudinal component (Cox and Munk , 1956) 0.06 0.04 0.02 0 0.10 e@ Longitudinal component 0.08 (Wu, 1971) © Longitudinal component (Cox and Munk 1956) 0.06 0.04 0.02 10° 2 4 6 8 19! 2 4 @ {3 =A 2 Wind-Shear Velocity, uy (cm/sec) FIGURE 20 - COMPARISON OF OCEANIC AND LABORATORY SLOPE MEASUREMENTS 1479 ~-4- oa a Oo meill — sop hom mi cr 7 oer TT i ; rT aan avy : tp (lege nate us e* , yO) 4 Pv Te. ; hia hu eet in Ne nes As Cae Y wo a Ry Vash? +} Saree ae esto hae Le, WA vt caqeimemeM ) cltea ao * : (1TeE ow) . Wren cayrey? lar ari gro go Q (22% jog A binge se) r + | oy ‘ Lins er pe Xe A Q ‘ a : ° “50.0 t o¢ 4 ° * =<) C Ps ° 5 a ' 4 Ce : -- ~ Bt -oeeee— ee F 4) . Ki T tae | ; os i; “Ci rn t ry y ‘ -a8 -* fren for iu! inno, s ) uw 190.0 ° teeenagmos loniikwtigers ° Aco) eM bere an) ? 140.0 0 ~ | iy 1). Ouch ty - : i . 1,04 J r ' Pp ° e — 4 aan | ma” rc * ms eo | nee a : at eieteeies Te ete ae are ne 2 x r rc A we "rts 0. ob « ah} ‘> a! ev 95040 oot = byt WV * AW 42 FT 7s A585 AS a 4 ., “a i. ¥ A bE a5 ht UY RCRL) BOA ISS ty Semen * Oe myo r ANS MEARS, item be Ya " gt eed ArT) 4 Caray A y oar ecity! Ca i arey ws A) Bisahbic Suet tatiaity @cewee i¢ Y thi vi fy wore ; gest GO5)ii' C.A6 *+9@ t: Os Ot preys he ‘ FRONTIER PROBLEMS Friday ,/, August 25,1972 Morning Session Chairman ; Dr, W,E,. Cummins Naval Ship Research and Development Center, Bethesda, U.S.A, Page The Wave Generated by a rine Ship Bow. 1483 T. F. Ogilvie (University of Michigan, U.S.A.). Transcritical Flow Past Slender Ships. ey Ar | G. K. Lea (National Science Foundation, U.S.A.). J.P. Feldman (Naval Ship Research and Develop- ment Center, U.S.A.). Computation of Shallow Water Ship Motions. 1543 R.F. Beck, E.O. Tuck (University of Adelaide, Australia). Seakeeping Considerations ina Total Design Methodology. 1589 C. Chryssostomidis (Massachusetts Institute of Technology, U.S.A.). The Application of System Identification to Dynamics of Naval Craft. 1629 P. Kaplan, T.P. Sargent, T.R. Goodman (Océanies Inc. , U.'S. oer rainaD feeunquievell bee doceeee KR otis lever A,.2.U ,Gbeedtehi spami i wel cid? onal a yd horareaeD | 1A@.U ~hepidold to viierovieD) siviigo x | 7 ge! .eqidd yoboole trad woll Lasiaae {A PU .wevsboedT e3nelo? fasotaevs) esd PP. oP. as plowed bos daveeasA qth LeyeVi) cambiet “See . &.U ..#lmeD ines saottaM qi 2? seta WV wollede bo aosh shbielebA to yitavovis ) AouT © = #008 .4 aS (ilanens> OR golobortieM agiseG laioT #£ a) waebetebtandl ge T? © elutiven! etisavdoeeesM) sibisrotacaeyrd? A 2.U ,vyoloadtout® @ eoioianytl of notisi Raed] mote vé to wolWaoin Ln aeezbooD .h.T cepted “7.7 ,walqasd, 7 (LA 2.U | 267 soheees0y bey t THE WAVE GENERATED BY A FINE SHIP BOW T. Francis Ogilvie Untverstty of Michigan Ann Arbor, Michtgan, U.S.A. ABSTRACT The flow field near the bow of a ship has some cha- racteristics of a high-Froude-number problem, even if ship speed is moderate. Some of these features can be predicted by slender-body theory if the usual assumptions of that theory are modified in the bow region to allow for the occurrence of longitudinal rates of change greater than normally assumed, Analytic- al results are derived for the case of a fine wedge- shaped bow, in which case a universal curve can be drawn for the shape of the bow wave on the hull, re- gardless of speed, draft, or entrance angle (all with- inlimits, of course), The lengths must be nondimen- sionalized by the quantity (HU2/g)' °| where H is the draft, U is ship speed, and g is the gravita- tion constant. Itis shown how this mathematical mo- del matches with the usual slender-body model and how it eliminates certain of the objectionable features of the latter, with only minor complications. Some experimental resultsare shown which generally con- firm the predictions, I. INTRODUCTION The Froude number can be taken as a rough measure of the relative magnitude of inertial forces with respect to gravitational forces in the interior of a fluid region, In the usual problem of ship hydrodynamics, neither of these forces dominates the other in the overall picture, and this fact is recognized in the custom of treating Froude number as a quantity which is O(1) ase -+0, where € is the small parameter that provides the reference for ordering all quantities 1483 Ogtlvte in the problem. If we take as the Froude number F = u/NeL » where U is the forward speed, L is ship length, and g is the gravitation- al acceleration, then the statement that F =O(1) means that there is a characteristic length Ug which is comparable with ship length and which is unrelated to the small parameter, € In a strict sense, this should always be the case. Suppose that ¢€ is a measure of ship thinness or of ship slenderness. As €—0, there is no reason to expect that U?/g should become either very large or very small ; one should certainly be able to specify the forward speed independently of ship thinness or slenderness, and g does not vary significantly in any case, But there are a couple of reasons sometimes not to accept this apparently natural assumption : a) Whe» we develop an asymp- totic analysis, we expect it to be more and more carly valid as the small parameter becomes infinitesimally smail, But we usually ob- tain just one or two terms in our expansions, and we try to use those expansions for computations when the small parameter is quite finite. We may actually obtain more accurate formulas if we assume an un- natural relationship between e¢ and the length u-7e: For example, if the latter is actually comparable to ship beam in the cases of practic- al interest, we may be better off in assuming that U*/g =O(€) when we formulate the boundary value problem. b) The implication about the ratio of inertial and gravitational forces may be locally invalid. That is, in some regions, one of these forces may dominate the other to the extent that the asymptotic solution gives grossly wrong predic- tions in those regions, The first of these two points I have discussed at length ina previous paper [1]”. In fact, the idea was not original there ; it was used many years earlier by Vossers [2] and also by Joosen [3] , for example. The second point is already implicit in slender-ship theory, for one assumes there that rates of change in the transverse direct- ions are very great compared with rates of change in the longitudinal direction, at least in a region near the ship. This means that accele- rations (and thus forces) are greater in one direction than another, and the ratio between them depends on e€ . Thus, to the extent that * Numbers in square brackets denote references listed at the end of the paper. 1484 The Wave Generated by a Fine Shtp Bow we accept slender-ship theory, we have already recognized that the overall Froude number does not characterize the ratio of inertial and gravitational forces uniquely throughout the fluid region, This idea was also discussed in the earlier work [1] already mentioned. There I pointed out that special order-of-magnitude con- sideration should be given to conditions near the ship bow. Because of the presence of the free surface, the fluid particles just a very short distance ahead of the bow are quite unaffected by the oncoming ship, until -suddenly ! - those particles are in the region of highly accelerated flow around the ship bow. The effects of water displace- ment by the moving ship are much greater than the effects of gravity, which normally hold the water surface horizontal, and so the presence of the free surface is momentarily simply equivalent to a pressure- relief surface. All of this can be implied by saying that the flow near the ship bow is a high-Froude-number flow. Thus we come to the concept that the bow flow is a high- Froude-number problem, even if the ship speed is moderate. The previous argument then suggests that we try to relate the Froude- number aspect of the bow flow to the slenderness parameter. In this paper, I have done this in a very pragmatic way : In the usual slender-body theory, we assume, in a symbolic notation, that d/dx = Q{(1) but that d/dy and 0d/dz = O1/e), where x is the longitudinal coordinate, This means that rates of change in the longitudinal direction are smaller than rates of change in the trans- verse direction by an order of magnitude e . (It is this very gradual variation in the longitudinal direction that leads to the typical feature of the slender-ship near field, namely, that the free surface acts as a rigid wall. Rates of change are so gradual that gravity dominates and holds the free surface horizontal.) This intuitive picture is for- malized in the mathematics by stretching coordinates in the transverse directions by a factor 1/e Now we suppose that, near the bow, rates of change of the flow variables should be greater than those usually assumed in slender- body theory. We may expect to introduce such a notion formally by stretching the x coordinate from the bow sternward, But what should be the degree of stretching ? Let us define a new longitudinal coor- dinate, X=x/e™, with x and X both measured from the bow in the downstream direction. If n= 0, we have the usual slender-body theory, and if n= 1 we have the original problem in three dimensions, (In the latter case the stretching is isotropic.) Therefore we seek a value of n such that 0 pC, UP. It is assumed that the shipis ''slender'', which means that there is a small parameter, ¢, characterizing the smallness of beam /length and draft/length ratios. As e« + 0, the ship shrinks down to a line, the part of the x axis between the origin and x=L, L being the ship length at the waterline. But ''slenderness"' means more than this. It implies also that the size and shape of hull cross- sections change gradually in the longitudinal direction. In particular, we shall require that HEAL EE SE OrrGnxt au ds ay Ox even in the bow near field. The ''bow near field'' is defined as the region in which 2 Zl] ee Sis Bhall?snA meh vi ncow ener walls) It is assumed that, in the bow near field, the flow variables are changed in order of magnitude when they are differentiated, according to the following symbolic rules 3 -1/2 3 pes So 5 a hea = O Ox te Or dy Oz eS These effects could be brought about formally through the introduction of new variables, x= Xell2 ,.y=Yeo) z= Ze, after whichiwe would require that differentiation with respect to X, Y, and Z have no effect on orders of magnitude. However, the rules will simply be carried along implicitly, the introduction of such new variables being quite unnecessary. Note that there is one exception to the above procedure : We have already required that b(x,z) =O(€) and db(x,z)/dx =O(e€). This is simply a condition on hull geometry. It has nothing to do with the nature (or existence) of a flow around the ship. 1488 The Wave Generated by a Fine Shtp Bow We assume everything that is necessary for the existence of a velocity potential, which we write in the following form Ux + (x, y, z) As usual, the potential satisfies the Laplace equation in the fluid do- main ; [L] Oat pts teenct oe adage [4/] [o/e*J[o/e 7] The expressions in square brackets give the orders of magnitude in the bow near field of the terms immediately above. Although we do not yet know the order of magnitude of ¢ , it is already clear that we can ignore the term ?¢,. in finding the first approximation to the so- lution in the bow near field. The boundary condition on the hull can be written D's ea < y ZZ [e ] [¢e Ve] [?/e] [¢/¢] Dropping the one term which is clearly of negligible order of magni- tude, we can rewrite this condition + Sn = oer Ub Pete te Ong ae esein Ed oe = 2 “Y= )Gole) Vl + be Ls: be Since the operator 0/dn is similar to, say, 3/dr with respect to its effect on orders of magnitudes, we can now conclude that either ¢=O(e*) or the first approximation to ¢ satisfies a homogeneous boundary condition on the hull. Let us suppose that the former is true. If this is wrong, we shall discover that fact when we consider the other conditions on ¢ 1489 Ogi lvte There are the usual two boundary conditions to be satisfied on the free surface Zz a) ; (S] [3/2] [e3] [e2] [e2] es z= ¢(x,y). rat > c= ° I ieee 2 gf + Ud, fone. +a + ¢ [B] 0 = U q =F Piz ce ap ron) y by eG ? fe eben Ee iree Wes The orders of magnitude involving $ have been noted, but of course we have not yet reached any conclusions, even tentatively, about the order of magnitude of ¢ . In condition [A] we can clearly neglect all of the quadratic terms, and in condition [B] the second and third terms on the right side can be neglected. Thus we have reduced the number of terms to the following [A] 0 [B] 0 Boe Og: I Gq an 1 oe In [A] , the first term cannot be lower order than the second, be- cause we would then have the meaningless result: '=0. Thus, either the two terms are the same order of magnitude or the first term is higher order than the second, If the latter is the case, the first term in [B] is higher order than the second term in [B] , and this leads to an ill-posed potential problem. Therefore we must con- clude that ¢ =O(€3/2 ) , and the two conditions are consistent in orders of magnitude. Note that this order-of-magnitude estimate for § al- lows us to impose the boundary conditions at z= 0 with negligible error. Finally, we can combine the two conditions above into the following [F] OS %e + 7K 78h, on Zea OY 5 1490 The Wave Generated by a Fine Shtp Bow where K= afte i In finding the first approximation to ¢ , we have a boundary- value problem to solve inthe y-z plane. Thatis, we have a partial differential equation involving only the transverse rates of change. The body boundary condition is a simple Neumann condition, but the free-surface condition involves derivatives with respect to x , and soa 3-D aspect is introduced through this condition. The problem in the cross plane is illustrated in Figure l(a). For the moment, we shall confine our attention to a special case of this problem, namely, to narrow bodies which can be generat- ed approximately by a distribution of sources on the centerplane, y = 0. This special case is depicted in Figure 1(b). A modification of our method of solution has been worked out for more general cases, but we shall not consider such cases further in the present paper ; they would only distract us from the simple ideas which are being developed. (a) Zz Figure 1 Problem for the First Approximation (a) General body. (b) Thin body. In both cases, the potential satisfies :¢ t+@ = 0. Vy ZZ 1491 Ogt lute For the thin bodies being considered, we shall suppose that the body boundary condition can be expressed ee ae ~ Ub on y= = O.,) for 2.> - ie The following 2-D potential function satisfies this body boundary condition dé b (x, 5) los (y +e : x) In fact, if we let v and w respectively denote the corresponding velocity components in the y and z directions, we find easily that -H(x) For y= +0, this can be evaluated through use of the Plemelj formula 0 =F - .t U df bx(x, ¢ ) (v 3 ee Ub (x, z) + on aes -H(x) Thus, 7 (capee (ite pecsee Ub (x, yj as required, The above potential function satisfies the partial differential equation and the body boundary condition. To that potential, we can add the potential for any other source distribution which induces no net normal velocity component on y=0, -H(x) = e (5) At first sight, this equation appears rather formidable. But the integral can be considered as a convolution integral, a fact which suggests the use of Fourier transforms to eliminate the y dependence. In what follows, we manipulate some transforms which are nonsense in a classical analysis ; whenever necessary, integrals should be in- terpreted in the sense of generalized functions. We follow Lighthill [7] in such respects. Let the Fourier transform be defined as follows F {ay} = £*(f) = foo eify Egy Ss 1495 Ogt lvte The transform of the right-hand side of Equation (5) can be computed as follows ie,9) 2 2 os) i we ‘ ze fos e ify ——- = ih dye ve [log (y-iH) + log (y+iH) - -0O a ‘ - 2 log l co l . -ily Le AO 6 xo stleghe yf ke Vea yas my - 00 a ee e -HILl] The integral term in Equation (5) can be treated as an ordinary con- volution integral, with the result that asit ¥ ody Py Mache pr ef ee —so - oO - ae [ily* (x :L)] [wi sgn L | = K|L|¥* (x ;L) The integro-differential equation now becomes an ordinary differen- tial equation with respect to x vt (x sf) + Kilt sf) = Pi peti aie The solution of this equation is now readily obtained. A par- ticular solution is the following Tr . - nly 1 - cos K| | ; =" ipa P.O) (by Equation (2) ) . v(x 32) (7) In principle, we should include the complementary solution, and this would be easy enough to do. However, the above solution appears to suffice for all that follows. 1496 The Wave Generated by a Fine Ship Bow There seems to be little point in writing out the correspond- ing expression for @ (x,y,z), which could be done through use of Equation (1), In fact, we shall not even bother at this point to write out the inverse transform of the expression in (7), although we note that the latter can be expressed in terms of Fresnel integrals. Itis worthwhile to write at least the transforms of two related quantities, namely, ¢*, (x; f;0) and $* (x;L;0) o* (x: £0) = a (: - at) cos VK |d| x ; (8) aS VK|L| € - ities) sin VK |Z Kg 2c (9) o% (x; £50) The behavior of $*, at large distance from the bow will be interest- ing to note presently, and e”, is essentially the transform of the wave height, which can be seen from the dynamic free-surface condi- tion [A] ; IV. LIMIT BEHAVIOR OF THE SOLUTION FOR THE WEDGE BOW Behavior as _ |y|-. 00 . Since the potential and its deriva- tives on the plane z=0 are all given in terms of Fourier transforms with respect to y , it is nearly a trivial matter to determine how the inverse transforms act when y— > f°. We need only to examine the behavior of the transforms near their singularities. The only singu- larities occur at £=0. For example, ¢*,(x;2;0) can be expressed $ (x; £50) 2Ua {1 = H”|L| + M3 Kx’ +...) 2UaH E kets Kx’ | + | Treating this transform as a generalized function, we can obtain the limit behavior of its inverse transform by using the methods describ- ed by Lighthill [7] . We find that 2 * (x; y, 0) ——— See as dar al, wy, This shows that, far off to the sides, the disturbance appears to be caused by a vertical dipole distribution. Such a result should not be 1497 Ogtlvte too surprising, since the body boundary condition was satisfied by distributing sources over the underwater part of the centerplane, to which we added a distribution of opposite sinks on the abovewater image of the centerplane. These two distributions alone would certain- ly lead to the dipole-like behavior far off to both sides. Apparently, the third term in the expression for @ , as given in [1] , has negli- gible influence in this sideways limit. Actually, we guaranteed such a result by choosing the com- plementary solution as we did in [7] . Effectively, we have implied that there are no waves upstream of the bow, even in the bow near field. In the final section, we shall return to this point ; it requires much more study in the future. The transform of the wave deformation function can be ex- pressed U Gal) = (== "Go g x and, from (9), this quantity has the following behavior near L=0 % a = L| i (xt) = E l-e Hl sin VK |f£] x (10) Nxt ( ) 5 i zat fi ee 6 et 2 6 The inverse transform then must have the behavior 2 H K (Gy) > = ae E + 7 [o.. as pe It can be shown that the potential itself drops off inversely with y? ,» but this does not seem to provide any special insight into the results. Behavior as x-+e%o , This is an important limit; for it pro- vides the connection to the usual slender-body solution. Let us recall that x =O(e 1/2 ) in the bow near field region. Our solution, when we let x -—co, should match the solution of the usual slender-body pro- blem if we let x —0 in the latter. 1498 The Wave Generated by a Fine Ship Bow In order to obtain these limits, we manipulate the inverse transforms into forms so that the generalized-function procedures can again be used. For the vertical component of velocity, for exam- ple, we go through the following steps > (x, y, 0) bat fete™ a ( - HN cos VK |L|x L — -H ES / al cost y= 5 — cosVKE x 0 — 2 || 4Ue ar ‘ whe Pas /* a mae cos Ax cos K aa ce, Pee : 2 co ; Z, -H> /K ON i oh 2a dite meats = Pe: K || —%0 4UaH wore ey | as x —» co é CET) TwKX The interpretation of this result is of some’interest. The quantities a and H are each of order ¢. In addition, x =O(e'@ ) in the bow near field. Thus, %, =O(e) in the bow near field. Now, we have already commented that the solution in the bow near field must match the solution given by the usual slender-body theory. In fact, the near field of the usual slender-body theory is a far field with respect to the bow region ; x =O(1) in the usual theory. From this point of view, the expressions obtained above for $, represent a one-term inner expansion, and the final formula above is the one- term outer expansion of the one-term inner expansion. In matching it with the corresponding ''far field'', we must reinterpret the va- riables as far-field variables and re-order the expansion. In the pre- sent case, this means only that we revise our estimate by consider- ing x tobe O(1), in which case we observe that ¢, = O(€2) on the plane z=0 as x— ye, This agrees with the well-known result of the usual slender-body theory. We shall say more about this presently. What is most remarkable about the above result is the man- ner in which the flow completely changes its character in the down- stream direction. Very close to the bow, the flow appears to have been caused by a distribution of vertical dipoles, and so the flow at the plane z=0 is almost completely normal to the plane. However, 1499 Ogtlvte as x-—oco , we find that the normal component of velocity on the plane z= 0 vanishes and the flow becomes parallel to the plane. We also examine how the wave elevation varies asymptotical- ly in the downstream direction. We proceed as with ¢, : We write as the inverse transform of the expression in (10) and then mani- pulate it so that it appears formally to be a transform with respect to x . We obtain in this way ras ad = 2a iAx ae l-e ale (Gary ie f= = dre sgn cos K ea eee co 2aH a id oie a idx —_ dd -—— phos ae nik e sgnd E aK DO 4aH 5 + O(1 = pire O(l/x) as x70 : (12) It is worth noting that the y dependence enters only in the term which drops off inversely with x>. We also observe that [ =O(e 3/2 ) in the bow near field, where we assume that x =O(e/2 ) , but when we re- interpret x as being O(1) we must conclude that ¢ =O(e?), This is in agreement with the well-known results of the usual slender-body theory. Finally, we obtain an estimate for (x, y,0) as x40 ., The transform of this quantity was given in Equation (7). It is clear that we cannot follow exactly the same procedure as we did for estimating >, of §, since there is a part of the expression in (7) which does not even depend on x . However, we can proceed in two steps: a) First we consider the part of the transform in (7) that does not depend on x, namely, the quantity We shall find that this is the transform of H Rel Ue i df log (y + iz - oH (13) -H ze=sQ 1500 The Wave Generated by a Fine Shtp Bow The interpretation of this result will be discussed after we prove that it is true. By elementary means, we obtain the following result H U s af 7 -H at tog +12 - 10) z=0 H Ua Ze AZ - ge df log (y +f") -H Oo, ze 2 - =—2 E log (H +y )/y' +Hlogy + 2[y cot ‘y/s)-1]. The last expression is now broken into several pieces, for each of which we obtain the generalized Fourier transform. For the first piece, the transform exists even in the classical sense so 2 2 a 2 z -i H fe sly pogo Ty = | dy cos hy log i y From the point of view of generalized functions, we have the following result One more integral can be computed readily fo evity ly cor Gi) - a | = 2 i dy coshy ly eof (y/H) ~ H| ¥ 0 so 2 yi yH = ro Nae i i) dy sin Ly (coe u Zo) vy S+elt 1501 Ogtlvte is \d| [ : nid These three transforms can now be combined to yield the result stat- ed above, thatis, 9o a ( - 0) — fo oily E log we) + 2[y cot”"(y/#)-H] ; — oo The expression in (13) is the potential for the flow caused by a line distribution of sources on the centerplane and on the above- water image of the centerplane, the potential having been evaluated on z=0. We recall that we started constructing our solution, in Equation (1), by assuming that there was a distribution of sources on the submerged part of the centerplane and a distribution of opposite sinks on the image of the centerplane. We now discover the interest- ing fact that one part of the potential, when evaluated on the plane of the undisturbed free surface, represents a symmetrical distribution of singularities, rather than an antisymmetrical distribution. The symmetrical distribution would have been a logical starting point in the ordinary slender-body theory, in which a rigid-wall free-surface condition must be satisfied. It appears in the present analysis asa natural consequence in the region downstream of the bow region, al- though we started with quite a different picture of the flow around the bow. b) The remaining part of the expression in (7) is oscillato- ry with respect to x, and so we use the procedure that worked well in estimating the downstream behavior of ¢, and £ . We go through the following steps fos Peers (1 - oH) cos VEL] x 27 xe p2 1502 The Wave Generated by a Fine Shtp Bow Zz log Cx + O(1/x ) as SS Me ~ 4UaH oa where C isa constant which cannot be determined from this analysis. From the two-part analysis above, we obtain our desired re- sult, the estimate of the potential on z=0 as x +o: H $(x, y, 0) ~Rel Sf df log yan - -H as xX — © , (14) 4U aH 2 log Cx +O(1/x ) Thus, we see that the potential represents the source distribution al- ready discussed, in addition to which there is a term which becomes infinite logarithmically when x goes to infinity. These results will both appear in a proper perspective when we consider what the usual slender-body theory predicts near the bow. Both of the explicit terms above are O(e€‘loge ) in the bow near field. The appearance of the constant, C, in the above result is an unfortunate consequence of our use of generalized-function theory. In general, the value of the constant may even have to change as the for- mulas are manipulated. Froma strict mathematical point of view, it is quite improper to leave a final formula in sucha shape that it can be interpreted only in terms of generalized functions, especially when it is supposed to have direct physical significance. Fortunately, this is not so much of a problem for us here as might be supposed. The quantities with real physical significance are $, and § , and their estimates are not at all murky. V. THE USUAL SLENDER-BODY SOLUTION If one stretches coordinates near the body in such a way that and then treats derivatives with respect to the new variables as if they had no effect on orders of magnitude, one obtains the usual problem and solution of slender-body theory. Without going through the forma- lism of such changes of variables, we write down directly the boundary- value problem that results for the wedge-like body that we are consi- dering in this paper. The first approximation to the near-field pertur- 1503 Ogt lute bation potential satisfies the following [LJ egies nero [H] Oo = Up! (x) on y=. = ibe; Jy a = © on Za = He 5 [A ] Be UGA =e AKO on z= 0 [B] § Mact® The last condition, [B] » is of course the rigid-wall condition which replaces the free-surface condition. The dynamic boundary condition on the free surface, [A] » serves only for the determination of the free-surface shape, $ , after the potential problem is solved. In the body boundary condition, we have stated a separate condition for the bottom of the wedge, for we do not need or want to restrict ourselves to a ''thin'' slender-body over the entire body length. The above problem can be solved precisely, by mapping, for example. We do not need that complete solution, however. Let @(x,y,z) be the solution of this 2-D problem which has the property 2UHb' 2 &(x, y, z) pe dee se log ts +x us ray a 2 | i as lie . ee Then the perturbation potential, ¢(x,y,z), is given by O(%sVo2)) oS4 0b Yee) to PaGe)ass where F (x) is given by [8] 'F (x) dle ae fg (2) sgn (x- £) log 2|x- | +7 H,(K(x- &) ) + (2 + sgn (x6) )=RY, (K |x- nf , 1504 The Wave Generated by a Fine Shtp Bow where s(x) is the area of the immersed part of the cross-section at : a Yo is the Bessel function of the second kind, and Ho isa Struve function, (Notation is the same as in [9] .) Near the bow, we note that Srey i ZiOl SS tae eae OT ee en where the '"'...'' denotes some smooth function of x. The first and second derivatives of s(x) can then be expressed as follows s' (x) BRON Jong) Oe eee AS SU = 3 Za@kpese)) 4) cee |3 where 6(x) is the Dirac delta function. The function F(x) represents the effects of interactions bet- ween the various cross-sections. For a body in an infinite fluid, we would have just the first term in the integrand, and one can show ea- sily that it represents the flow on the x axis caused by a distribution of sources both upstream and downstream of the point under conside- ration. The other two terms represent the effects of the free-surface, and they combine with the logarithm term in such a way as to cancel any flow upstream of a source. Tuck [8] has shown this explicitly. The integrand of the F(x) expression has a wavelike nature for < x but not for §> x. We are interested in how the above solution behaves as x -_,0. In fact, the easiest procedure for determining this behavior is totreat x as being O(e 2 ) and re-order all quantities accordingly. When we do this (after much algebra, expanding of the Bessel functions, etc. ), we find that 3 4UH K 3 Pc) — [toe x +f log +1, for x = O(ef2) , where Y is Euler's constant. The problem for g@ becomes, for x =O(e1/2), a wedge-flow problem, with a rigid wall in place of the free surface ; its solution is 1505 Ogi lvte H (x,y,z) = ref Se f oF tog (y +12 - 1) (15) When we combine the two results above, we obtain a one-term ex- pansion of the potential, to be matched with the bow-near-field ex- pansion H U 4UaH o(x,y,z) ~ Re al dS log Crary - = log Cz -H 3 1 K 3 —— see ny log C A log z 7 This result should be compared with that in (14) : the matching is perfect, with the previously unknown constant C now fixed. The in- terpretation is, of course, different. In (14) , the potential was ap- proaching infinity logarithmically as x +00; here, the potential is approaching infinity logarithmically as x —0. The kinematic free-surface condition, [B] , does not mean that ¢, is precisely equal to zero on the plane z = 0 ; it means only that oo) z=0 = 0 for the leading-order term in the solution for® . The first approximation to ¢ is O( €) and the first approximation to ¢, is O(e€ ). Thus, the statement that $7 l,=0 = 9 really means that: zl, -~q9 = o(€). This remains true evenas x— 0. Thus, the first-order termin $, automatically has the correct behavior for matching with (11), which gave the behavior of ¢, in the bow near field, under the condition that x +o, Finally, we consider once more the wave-shape function, $(x, y) . From the dynamic boundary condition, [A] , combined with what we have found above concerning the slender-body potential for this problem, we can express $ in the following way [ey = -—-¢ = — F"(x), = =—@ ; eon %2->0 For the wedge-shaped bow, with constant draft, we can find immedia- tely that 6, =>0. (See (15) ) For x very small, we also find easily 1506 The Wave Generated by a Fine Shtp Bow that The final term needed is the one involving @ this quantity being y’ H $ Ua dé P, = Re} i | UP Ouene ? -H the remainder being a quantity which goes to zero at the bow. We need to evaluate this quantity only on z=0, for which we find gue (sgn y) tan ree We now have the following representation for the wave shape Zz 4Ha 2a ARP Sal BE ApS 2 tan Ty] ]- ore The last estimate of order of magnitude is still valid in the usual near field, where x =O(1). In order to match this result with the bow-near-field formula, we must reinterpret the order of magnitude of x, that is, consider that x =O(e'/2), and re-order the expan- sion. When we do this and keep just one term, we have only 4H Gy) ee = O(e7 ) We now observe that this matches precisely with the expression in (12) V. COMPARISON OF RESULTS WITH EXPERIMENTS From Equation (10), we can compute the shape of the free- surface disturbance : E507 Ogilvie pcbe hoget e Je = (abe (16) For y=0, this simplifies further 0 We obtain our simplest form when we make the following changes of variables xi] 2AiK/a Z(X)ote 3 ee GRE) ee (17) oo ae ZAK) = [ #€52) (3) (18) af # Thus, the wave along the side of the wedge can be nondimensionalized in such a way that we have a single universal curve, a function of just one variable, which purports to describe the wave shape for any speed, any draft, and any wedge angle. Of course, we have not yet considered the range of validity of these results, but itis clear that they are very simple results. It is worthwhile to notice the manner in which the length scales are made nondimensional : The reference length is (H/K)'? = HU2/g)'/2 . This is the geometric mean of two lengths, the draft H and characteristic free-surface length U“/g. Also, the wedge half-angle enters in a very simple way: The non-dimensional wave height, Z , must be multiplied by the ratio, 1508 Generated by a Fine Shtp Bow The Wave aspem TeodtajyourutAs uo odeys 9AeM-MOQg P2e1DIpetg ‘7 sandy 3 Cc. } O [xto2-L -2] Xx * n/b= » * MARX =X 0 393 Pee SU 4 Sy 40 ; e TIX NS TV O-} = (OXx)Z 11- (o'x)z ahs =(0')5 1509 Ogt lute a/(r/2) (in addition to being made dimensional on the scale of (H/K) Y2 ). Thus, the theory predicts that wave height along the side of the wedge will be proportional to the wedge angle. Calculation of Z(X) has been carried out, with the results shown in Figure 2. In addition, the integral in (18) has very simple asymptotic approximations which are validas X 40 or Xo, and these are shown by the broken curves in Figure 2, In order to determine whether this result was even approxi- mately valid, we conducted some experiments with a very simple model. The planform of the model was that of an unsymmetrical dia- mond ; at one end, the model was a wedge with a half-angle of 7.5°, and at the other end the half-angle was 15°. Tests were conducted at speeds up to about 15 ft. /sec., with drafts from 4 in. to 16 in. A grid had been inscribed on the model so that wave shapes could be measured from photographs of the bow wave. In Figures 3 and 4, two selected series of tests are shown. In both figures, the model is being tested at a draft of 12 in. There are several qualitative features in these photographs that are worth noting : (i) The model speed in Figure 3(a) is 1.64 ft. /sec., which is only about twice the minimum speed at which waves can travel.ona water/air interface. (Minimum speed is about 23.2 cm./sec.) In fact, capillary waves are quite evident in this picture, as well asin several of the higher-speed test pictures. Whether these ripples can actually be seen apparently depends more on the lighting than on any- thing else. The existence of a sharp edge on the model presumably accentuated the amplitude of the ripples in all of our tests. (ii) In (b) - (e) of Figure 3, the water level at the bow edge is about 1 in. above still-water level. (The white mark at the bow is atthe 18in. draft mark, and the squares are 1 in. ona side.) This rise of water level ahead of the bow is, of course, not predicted in the analysis. We fully expected to observe sucha rise, and we re- cognized that it would represent a source of error in the predictions, What we did not anticipate was that the rise is quite insensitive to forward speed. From a speed of about 5 ft. /sec. (Figure 3(b)) toa speed in excess of 15 ft. /sec. (Figure 3(e)), this rise increases from about 0.8 in. toabout 1.2 in, (iii) The corresponding rise in water level at the bow is greater for the wider-angle bow, but even in this case the level seems to L520 The Wave Generated by a Fine Shtp Bow WU = 95st 7iftle/ sec. Figure 3. Bow wave on a wedge Dratt="12 in. Half angle = 7. 5° EST} Ogtlvte U =7..64 ft../sec. U = 9.80 ft. /sec. WisiED. 46.4t-/ sec. Figure 4. Bow wave on a wedge Draft = 12 in. Half angle = 15° P5112 The Wave Generated by a Fine Shtp Bow stabilize at about 1.7 in. See Figure 4, parts (b) to (e), in which the rise varies between about 1.5 and 1.9 in. while the speed in- creases from 5.0 to 11.5 ft. /sec. (Note : the white mark on the bow here is at the 16 in. draft. ) (iv) The region in which the bow wave dominates the picture in- creases steadily with forward speed. (The analysis predicts that the peak of the bow wave moves aft in proportion to U , the speed.) In the lowest-speed tests, there is a clear wave-trough behind the bow wave. See, for example, Figure 4, parts (a) and (b) : The lowest visible white marks are on the still-water waterline. The trough is not predicted in the present analysis, and so we see that there are non-negligible waves at low speed which simply are not evident under the assumptions which have been made here. We cannot say whether the same kind of troughs occur at the higher speeds, because the model length was not great enough to observe the phenomenon, From Figure 2, it was clear that we have a "universal'' bow-wave curve which is supposed to apply to all wedges at all speeds at all drafts - within some unknown limits. To check this conclusion quantitatively, we measured just the amplitude and longitudinal po- sition of the peak of the bow wave. For the finer wedge, the results for the wave amplitude are shown in dimensional form in Figure 5 ; the corresponding data for the longitudinal position of the peak are shown in Figure 6. These dimensional data are shown only to provide the reader with an impression of the scale of what was observed. The nondimensional wave-peak data are shown in Figure 7 ; according to the analysis, the nondimensional amplitude, Z,,3,, should always have the same value, approximately 1.6. Figure 7 shows clearly that this is only roughly substantiated in the experiments. In fact, there are two ways in which the analysis is obviously deficient : 1) The assumption that made our analysis distinct from the usual slender-body theory was that the bow flow is essentially a "high-Froude-number"' problem, in some sense. The depth-Froude- number is the only reasonable Froude number to consider in the bow region, and one can hardly expect the analysis to give good answers when Fy 0. In fact, it gives terrible answers then ! 2) At the higher Froude numbers, the wave peak occurs at a considerable distance from the bow, ata place where the '"'thin- ship" representation of the body is probably quite invalid. We used the ''thinness"' twice, first in satisfying the body boundary condition approximately, then in evaluating the wave height on the body. (We simply set y= 0 in passing from (16) to (16').) The worse agree- ment for the wider wedge suggests that this ''thinness'' assumption LSLS Ogt lute H = 16 in x A i —aCun thy EXPERIMENTS H = 86 in A H = 4 in vi ie \e Z ~ e@ % a) x ce a a Bile Vins x 4 in H =S ‘Ss e e a = 75° 4 6 8 10 42 14 U (ft%e0c,) Figure 5. Bow wave amplitude on wedge 1514 {6 ANALYSIS The Wave x H = {76 in 20 a H =?2 in. a He om 18 @ H=4in EXPERIMENTS ANALYSIS £ 16 { 4 . ty 14 4 z 12 }— x X Max a 10 Gn) Z e ere 6 4 — ® A 2 x ie, 4 6 8 U_ (#tvsec.) Figure 6. F515 10 Generated by a Fine Shtp Bow Longitudinal position of wave peak 14 Ogi lvte (uIzOj TeUOTSUSUITpUOU UT) OpnzITdure eAeM MOG *) 2ANBIA ‘UID ‘Ul ‘ul 2} ‘ul 9} i li og L=” SLNIN/4 Fd XP SISATWNV 1516 The Wave Generated by a Fine Shtp Bow may well be the cause of the increasing error at high Froude number. If Pay is below some moderate value, it can be seen from Figure 7 that our method of nondimensionalizing the data seems to be still valid even when the Froude number drops below the level at which the analysis is valid. The reason for this is not clear, but the fact may be useful in reducing experimental data, even in cases in which the present analysis is obviously invalid, The inaccuracy of the wave height predictions at high Froude numbers can probably be ameliorated if not completely removed by the introduction of a more precise method of solution of the problem. In principle, it appears to be possible to solve the bow-flow problem without introducing the thinness assumption, and some efforts have already been made to do just this. At the moment, however, we have no results to show for this effort. VI. CRITIQUE OF THE ANALYSIS Intuitively, we visualize a ''slender body" as a body of which the length is much greater that the transverse dimensions. In addi- tion, if we want to be a bit more precise, we require that there be no sudden changes in cross-section size or shape. For such bodies, slender-body theory is likely to lead to reasonable predictions concerning a fluid flow around the body - provided we do not examine too closely what is happening near the ends of the body. The last qualification is necessary because slender-body theory is based on one major assumption which is usually violated near the body ends : It is assumed that the rates of change of all flow variables are much greater in the transverse directions than in the longitudinal direction. For a body with cusped ends, this assumption is valid even in the region near the ends, but the assumption is not valid near the body ends for most bodies of practical interest. The result is that slender-body theory typically predicts some kind of sin- gular flow near the body ends. Such a result is not.necessarily unacceptable. If the singu- larities are integrable in some appropriate fashion and if the solution is approximately correct in most of the flow region, the presence of singularities in the mathematical solution may not even be serious. If one is very careful in obtaining the singularity strengths, one can even make some reasonable calculations concerning the flow around a blunt body in aninfinite fluid. At cross-sections not too near the ends, the presence of singularities in the solution for the body end regions manifests itself as a perturbation of the longitudinal velocity compo- E517 Ogtlvte nent ; this effect is rather minor over most of the body surface, and its precise evaluation is carried out by matching the near-field and far-field solutions. In the free-surface problem, this procedure leads to an es- sential difficulty : In the far-field problem, the disturbance caused by the presence of the body appears actually to be caused by a line distribution of sources along the x axis, this axis lying in the plane of the undisturbed free surface. A concentrated source in the plane of the free surface is completely intolerable, because it causes much more than just local problems. (For example, the wave resis- tance of such a source is infinite.) Therefore we cannot hope to re- present end effects in the simple way that is sometimes so success- ful for bodies in an infinite fluid. In particular, we note the following important fact : No matter how nonlinear the local flow around the bow of the body may be, it cannot appear from afar as if it had been caused by a concentrated source. In fact, an even stronger statement is possible : If, in the far field, the disturbance appears to have been caused by a line dis- tribution of sources, the distribution must have a density which varies continuously. For the wedgelike body considered in this paper, slender-body theory predicts that the source density in the far-field expansion should have a jump at the bow. Actually, there may bea steep rise in the curve of source density, but there can be no jump in value. Otherwise the whole far-field solution has little meaning. The far-field solution must be less singular at the bow than one might expect from infinite-fluid slender-body theory. There is another point of view which also encourages some optimism for treating the free-surface problem. The local behavior at the nose of a body in an infinite fluid appears to be intrinsically a three-dimensional problem. The presence of the body must have a fairly significant upstream influence. However, the additional pre- sence of a free surface should reduce such upstream influences, Moreover, the isobaric property of the free surface may tend to smooth out variations in the longitudinal direction. Thus, one may be greatly encouraged to attempt to analyze the ship problem by slender- body theory. These rationalizations have come, for the most part, after the preceding analysis had been developed and found to compare fair- ly well with experiments. Originally the motivation had been more like that described in the Introduction, In any case, we have found fair agreement between the analysis and our experiments, and so we should proceed to investigate further the internal consistency of the 1518 The Wave Generated by a Ftne Shtp Bow analysis, while we also investigate possible modifications and check them against experiments, In the analysis as presented here, one may observe that the solution in the ''bow near field'' was never matched directly with the usual far-field solution of slender-ship theory. As Hirata [4] dis- covered, this is no small task. I have not yet carried out this match- ing, but Iassume that it would lead only to a modification of the far- field source density in the neighborhood of the ship bow. Presumably, the source density curve would be rounded over ina region of length O(€ V2 ) near the bow. Thus it would be possible to compute the wave resistance of this shape of ship. (We have no assurance that the value computed would be accurate, but it would be a big improvement over ordinary slender-ship theory, which would give an infinite value of wave resistance for the ship with wedgelike bow ! ) We have made only a fewcrude attempts to predict what happens just ahead or the edge of the wedge, and these attempts have not been described. Using a very heuristic mathematical mode, I concluded at one time that the rise in water level ahead of the bow should be independent of forward speed (for a given wedge angle), and it was this tentative conclusion that led us to examine our photographs carefully, after which we came to a conclusion that there must be some truth in the crude analysis, since the water rise is in fact quite insensitive to forward speed. The fact that the analysis is linearis, of course, a great help in obtaining a solution, but the most casual observation of the physical situation (as in Figures 3 and 4) suggests that linearisation may be a great over-simplification. In defense of the linearization in this analysis, I offer just two comments : (i) It always seems reasonable to try a linear analysis of any problem. One must in any case trust experimental evidence for the justification of an analysis. In the present problem, it is evident that the linear analysis is not grossly wrong. (ii) In many mathematical analyses of fluid mechanics problems, apparently unacceptable singular solutions often become very useful when they are properly interpreted. I have already mentioned the ap- pearance in slender-body theory of flow singularities which result from the invalidity of the assumptions in the regions near the body ends. Perhaps an even more interesting situation arises in some pro- blems in which we find that the solution to a linearized problem re- presents approximately the correct flow patterns - but in slightly wrong places. Our bow-flow solution, for example, is not so singular 1519 Ogtlvte as that which results from the usual slender-body theory, but itis still singular. It is interesting to note that the experimental data in Figure 6 would have a more orderly appearance if the ordinate scale had started at about x,,35, ~-2. In other words, the predictions in Figure 6 are considerably improved if we arbitrarily assume that the rise in water level should have been measured from a point about 2 in. ahead of the bow. To this extent, our linearized results follow the pattern mentioned above : They are approximately correct, but in the wrong place.”* The form chosen for the solution in (1) is not an essential part of the analysis presented in this paper. It was an easy way to arrive quickly at a solution for a particular case. It has already been mentioned that this simplification may be at least partly responsible for the discrepancy between analysis and experiments at the higher Froude numbers. Having now determined that we have found some general agreement between analysis and experiments, we shall next try to obtain more precise solutions for these and similar problems. For example, the body cross-section shown in Figure 1 (either (a) or (b)) can be mapped into an auxilliary plane in which body and free surface together make up the horizontal axis. The free-surface con- dition must be transformed, of course, and then an integro-differen- tial equation comparable to (3) can be obtained. This procedure can also be followed for bodies which are not symmetrical or for bodies which have an angle of attack. No solutions have been obtained yet except for that described by Hirata [4] for the case of a plate of zero thickness with an angle of attack. I hope that we shall be able to ob- tain solutions for several more realistic situations - for which com- parisons with experimental data will provide more definitive evalua- tions of the fundamental approach described in the present paper. * A more careful study of Figure 6 shows that the predicted curves have the correct slopes if the origin is placed on a sliding scale, with essentially no shift for the case of small draft, up toa shift of about 5 in. for the maximum draft case. I do not want to try to read too much quantitative significance into this result, however. 1520 The Wave Generated by a Ftne Shtp Bow ACKNOWLEDGEMENTS The analysis described in this paper developed during and out of discussions that I had in the spring of 1971 with several students in the Department of Naval Architecture and Marine Engineering at the University of Michigan : Eiichi Baba, Odd M. Faltinsen, Miguel H. Hirata, Arthur M. Reed, and William S. Vorus. It was a unique expe- rience for me to have these students together, working as a group, analyzing problems without inhibitions even while criticizing one an- other. Each of them contributed in some way to this paper, and I de- dicate it to them, scattered as they now are over four continents. This research was carried out under the Naval Ship Systems Command General Hydromechanics Research Program, Subproject SR 009 01 01, administered by the Naval Ship Research and Develop- ment Center (Contract No. N00014-67-A-0181-0033). NOTATION b(x,z) hull offset (half-width) b(x) special case of b(x,z) (the wedge problem) g gravitation constant H(x) draft of the section at x H special case of H(x) (the wedge problem) L transform variable L ship length r (y* + z2)1/2 U forward speed V's 2 coordinates D4 xVK/H a b'(0) , half-angle of the wedge € slenderness parameter (x,y) free-surface elevation Z(X) nondimensional $(x,0) ; see Equation (17) g/u4 p perturbation velocity potential W(x, y) o (x, y, 0) Fez 1 Ogtlvte REFERENCES [| OGILVIE, T.F., 'Nonlinear High-Froude-Number Free- Surface Problems", Jour. Engin. Math., 1 (1967) 215-235. [2 | VOSSERS, G., "Some Applications of the Slender Body Theory in Ship Hydrodynamics", Ph. D. thesis, 1960, Technical University of Delft. [3 | JOOSEN, W.P.A., "Slender Body Theory for an Oscillat- ing Ship at Forward Speed", Fifth Symposium on Naval Hydrodynamics, ACR-112, 1964, pp. 167-183, Office of Naval Research, Washington, D.C. | 4 | HIRATA, MoH “On*the Steady Turn of 2° Ship) e ae thesis, 1972, The University of Michigan. [5] BABA, E., unpublished manuscript, 1971. [6] MARUO, H., "'High- and Low-Aspect-Ratio Approximation of Planing Surfaces", Schiffstechnik, 14 (1967) 57-64. [7] LIGHTHILL, M.J., ''Fourier Analysis and Generalised Functions", 1959, Cambridge University Press, Cambridge. [8] TUCK, E.O., "A Systematic Asymptotic Expansion Proce- dure for Slender Ships", Jour. Ship Research, 8:1 (1964) 15-23, [9] ABRAMOWITZ, M. and STEGUN, 1.A., Ed. , "Handbook of Mathematical Functions'', Applied Mathematics Series - 55, National Bureau of Standards, 1964, Washington, D.C. Bo22 The Wave Generated by a Fine Shtp Bow DISCUSSION Reinier Timman Delft Instttute of Technology Delft, Netherlands I do not have many questions to ask. The only thing I want to do is congratulate Professor Ogilvie on his paper. Slender body theory for years has been stagnating because we all knew that it did not work well at the bow region. Professor Ogilvie has apparently opened a way to get good results even there. It could be asked, though, what would happen with a blunt body, but that is rather trivial. The only thing you know is that this does not work for a blunt body but it gives a result with this wedge shaped body which is really very good and his way of looking at it opens up a new way of attacking several other problems. So there is hope of making real progress in connection with many ques- tions in this field. Iam very happy that this paper has been presented here and I hope that it will open the way for new progress. REPLY TO DISCUSSION T.. Francis Ogilvie Untverstty of Mtehtgan Ann Arbor, Michtgan, U.S.A. I just want to say one word besides, obviously, thank you. This same analysis has also been applied to a rather blunt, flat ship by Baba. Iam sorry that I have not been able to read it yet although I have a copy, but it is in Japanese. He also claims some rather remark- able success so perhaps there is something there for blunt ships. 1523 Ogtlvte DISCUSSION Ernest O. Tuck Untverstty of Adelatde Adelatde, Australta I have just one very small comment. The nature of the two- dimensional problem that Professor Ogilvie is solving is, in fact, that of a Cauchy-Poisson problem. The co-ordinate x appears as a para- metric co-ordinate, since it does not enter the governing field equation, so that it plays a role which is identical to the role played by time in a two-dimensional Cauchy-Poisson problem. In fact, with that inter- pretation the problem solved by Professor Ogilvie here actually cor- responds to a growing wedge in a Cauchy-Poisson two-dimensional problem and presumably the Baba problem is the equivalent of a pres- sure distribution on a free surface which is growing laterally as well as increasing with time. REPLY TO DISCUSSION T. Francis Ogilvie Untverstty of Mtchtgan Ann Arbor, Michtgan, U.S.A. I should have recognized that. Of course, Wagner recognized that many years ago when he put these two problems together in one paper. 1524 The Wave Generated by a Fine Shtp Bow DISCUSSION Gedeon Dagan Teehnton and Hydronauttes Ltd. Hatfa, Israél I should like to ask if nevertheless you have tried to compare the experiments with thin ship computation, and I will go farther than that. I think that the slender body theory can be shown to be a kind of particular case of the thin body theory if you let the draft go down to zero, and I wonder if your theory also can be derived as a particular case of the ship approximation. REPLY TO DISCUSSION T« Krancis Ogilvie Untverstty of Michtgan Ann Arbor, Michtgan, U.S.A. In fact I have a comment somewhere in the paper on that. If you think back to the boundary value problem that I solved, I had the same free surface condition which everybody uses in thin ship theory and I had the same body boundary condition which everybody uses in thin ship theory. The only difference was that I dropped one term from the Laplace equation. So to the extent that that term is small then they ought to give identical results. Of course, that term is probably not small and that is part of the reason I am looking at it. So Il am not quite sure what the difference would be. 1525 in We - n enatieeny an eis cqrmias OF ba ee A Sane ith ad oy 7 Alvar Ae") oF “7B 4 y tid ie . wali inca b'y [Jy [hth Ages S& Fico « pete , di wig ar mo Fnoiasoo. two dining e é et) madi i 4 L orc et “4 “ sagem 7 of es ay ; ¥* NS he Lad sity Z ~ons ' “ole2 9eIG OT ‘YI93R evligO eiongs ds +7 eA es } sitet ge re; pr Vv" ip Ts O18 ba SN vt » 4 . 2) aft to toqAe [ abwaseoniee fWoiaaA * syed T 194) 2) my he her t Pavios out gida id ri eaoau Woody Gon cue | oF iv) WAALS). Edad lH Qant pineal Kal Weed “ssehehetogh” tiny wittieioal “70 1 Sy am Se ea ee A » estes syber TE Was Sr okt nince® LA Vite PBA (Ronee /cky Gites: EO Ae eee fe bert esd Or°aeGte Ve ed) 7 68: “who srbisin OF Math AY off? GP eb GI? {ies tt VoSd ede? ty" te O94 ba ict Sy bes “Sta Ps deh uO: vi Pepsi E exon prog Vie ce ibt gg Gin : 1 dnd ewelidote,enlay, “yeonuodc ai) of Aped ini exeu yhodysevs daldw noe ibi09 soubede oom T4v6 O3niw Mac iane Tebnioed yuod sTolea erla & APF TERRA Nes SECS Es ARSE & 1a wittOu! fs i sonts4 of! lo tag wi tad od bloow seshers Vib of? 1efwe nb te pxoti crxad sac Seqqotb | ded) ase soneretib yiso.sfT .yxosd? gig di nodd Tinie er “23 FP41Y cots eae tal” Ae TO widey Fee ML TRANSCRITICAL FLOW PAST SLENDER SHIPS G. -K. .bea Nattonal Setence Foundatton Washtngton, D. C. U.S.A. J. P. Feldman Naval Shtp Research and Development Center Washtngton, D. C. U Gan ABSTRACT The transcritical shallow water flow past slender ships is analyzed using the method of matched asymptotic expansions. A consistent first order approximation was derived which is analogous to the non-linear transonic equation with the Froude and Mach numbers playing similar roles. Solutions are obtained for sinkage and trim in the transcri- tical region and are compared with experimental results, An important result is that both sinkage and trim are functions of Froude number as well as beam to length ratio in the region where Froude number based on undisturbed depth is close to unity. INTRODUCTION In a series of papers Tuck @M @ developed a systematic expansion procedure for the approximate solution to the shallow water flow past slender ships. It was pointed out that a close analogy exists between this problem and the inviscid slender body aerodynamics pro- blem. In fact, Tuck's solution contains the same type of singularity that is encountered in aerodynamic theory and we present here an at- tempt to remove the singularity which occurs in the transcritical re- gion. Thus the shallow water problem that will be examined is concer- ned only with steady translational motion of a slender ship and the associated surface waves so that viscous and compressibility effects are neglected. The Froude number a = U, /V gh, where U,, is the free stream speed, g is the acceleration of gravity, h is the undisturbed 152% Lea and Feldman depth) can be interpreted as the ratio of a characteristic speed to the propagation speed of small disturbances on the water surface in shal- low water theory. On the other hand in aerodynamic theory, the Mach number (M,, = Uy /ao » where U, is the free stream speed and ag is the isentropic speed of sound) is the ratio of the characteristic speed to the propagation speed of acoustic signals in the gas. Thus we see that Froude and Mach numbers play similar roles and this is re- flected in the mathematical formulation of the two problems. For ex- ample, Tuck 2 gave for the first approximation a hyperbolic equa- tion for supercritical flow F, > 1 and an elliptic equation for subcri- tical flow F, < 1. We find the same situation in inviscid compressi- ble flow past slender bodies for supersonic M,,> 1 and subsonic M <= 1 flows. His results for vertical force, trim moment as well as drag contains integrals which relates the source sink distribution to the local hull area and multiplied by the following factor : 2 2 (ie ie ey ie F, for subcritical flow and 2 2— aes pany FE -1 for supercritical flow This factor seem to indicate catastrophic failure at critical flow F, = 1. However, it should be pointed out that aside from the trans- critical region, where | Rie —1 | is small, Tuck's results appears to be more than adequate for most engineering purposes. We shall seek a singular perturbation solution to the problem of shallow water flow past a slender ship with the requirement that the solution must be valid within the transcritical region, This ap- proach is that followed by Tuck'2) and is well documented in books by Cole(?) and Van Dyke @) . The important difference between what follows and the works of Tuck is that two small parameters appear in the formulation, slenderness ratio and Froude number parameter me —1 |, instead of only the slenderness ratio, Appearence of an additional parameter drastically alters the mathematical representa- tion of the problem and the nonlinear effects suggested by Tuck’) are indeed present, I, EXACT STATEMENT OF THE INVISCID PROBLEM Consider a ship immersed ina steady stream of inviscid, incompressible stream with free stream velocity of Un Fy . Atcars tesian coordinate system is afixed to the ship with its origin at the bow and at the undisturbed waterline. The positive x-axis is directed toward the stern of the ship and z-axis is directed vertically upward. 1528 Transerttteal Flow Past Slender Shtps The total velocity in the flow field is given by : Sh Ewer Clee ey + grad — where is the disturbance potential due to the presence of the ship. The dimensional governing equations are the Laplace equation, free surface kinematic and pressure equations, bottom condition and hull tangency condition, These are as follows : igh bw tp ghee ia er a Pak aoe 2t¢/U~, = -(2 0+. +9. +p°) z=$(x,y) (l-c) line qvtzplat pds Tid janie BR PARIS Ap ena LS WEA? ie) where § (x,y) is the unknown free surface and A (x, z) is the given surface of the ship hull. If we are to proceed in a systematic fashion the relative orders of magnitude of the various terms must be establi- shed. One way of accomplishing this is by selecting proper scales for all the dependent as well as independent variables and thereby intro- duce non-dimensional variables of order unity. This does not mean that all quantities will have its maximum of one, but rather that if we choose the correct scale the maximum value could be large as ten units but not one thousand units. We take Uq_ as the velocity scale and the undisturbed depth, H, as the vertical length scale. The selec- tion of a horizontal length scale is a bit more involved as it must re- flect the shallow water approximation include the transcritical nonli- nearities produced by a slender hull, Now, shallow water theory assumes as a first approximation that the vertical pressure variation is purely hydrostatic or that ver- tical accelerations are negligible compare to horizontal accelerations. This can be derived in a systematic manner assuming that the depth to characteristic wave length (H / Ly << 1) is small and utilize Ly, as the length scale in x and y directions and expand — as a power series in H / Ly . We note that shallow water theory is not necessa- rily linearization and the latter results from restrictions that we pla- ce on the type of ''wave maker"! present in the problem. Furthermore, we note that the surface wave system generated by a ship at critical 1529 Lea and Feldman speed is a single wave of translation perpendicular to the free stream. In the absence of viscous dissipation this wave extend to infinity so that the disturbance in the lateral, y, direction is greater than in the axial, x, direction. Thus it seems logical that we should have x=0(1) and y=0 (ge ) at large distances from the ship where € is a small parameter related to the ''shallowness'" of the water. Shallowness implies that depth is small relative wave length (H/ Ly << 1) and slenderness implies that the wave maker, the ship, must be longer than it is either wide or deep. If we define B as the maximum beam, T as the maximum draft, then slenderness means Bi / ds ple de al where Lis the length of the ship. In order to proceed inan order- ly manner some estimates must be placed on the relative orders of magnitude between Ly and L. We note that the dispersion relation- ship for steady progressive free waves in two-dimensions is tanh (27H /e) 7/27 Piya) gH (2) which can be approximated by the following expression after making use of the long wave assumption (H/ L, << 1) Hy ey Sah (2a FF.) Fe =u. /gh (3) The behaviour of this expression in the transcritical region is esti- mated as SDB, Veale REERC TS (4) Furthermore, if we take the depth to ship length ratio H / Las gau- ged by the slenderness of the hull, i. e. H/ L =0(€), then 2 By fis 0 (ei Maser) (5) As the ship approaches the critical flow condition, the characteristic wave length of the surface wave decreases so that in order to retain the transcritical effects and at the same time impose slenderness assumption we take L,/L=0(€ fr ae Fo) = 0(1) We note that in Tuck's analysis(?) it was assumed that as the ship 1530 Transertttcal Flow Past Slender Shtps approaches a line (€—+*0) that} 1 - Fe | remains fixed and of order unity which implies that L , |L = 0( € ) <<1 and is equivalent to the condition Sg! a /V gl = 0 (VE). Thus outside the transcritical region the proper scale length in any horizontal plane is the length of the hull L. However, the situation within the transcritical region is L=0(L, ) so that we can choose either L or Ly as a horizontal characteristic length with the restriction € /VTl - el = Ot 1 jaar = "T+ €& (6) ae where K is some similarity constant which is of order unity. The particular form chosen here is guided by the transonic aerodynamics analysis of the slender airfoil theory since we anticipate a close ana- logy between it and the present shallow water problem. It should be noted that in aerodynamic theory K is not uniquely determined by any analytical approach but depends on the correlation of experimental data. II. FAR FIELD APPROXIMATION Singular perturbation solution is a systematic procedure by which successive estimates to the solutions can be made in the various regions of the domain of solution. If properly applied, the dominate features of each of these regions will be magnified and secondary fea- tures suppressed by scaling of variables. We expect that in the far field details of the ship hull will be lost and that the dominate feature of the problem is that of the surface wave system. As noted in the previous section Lave / Loin = 0( 1) in transcritical region so that for scaling purposes either one would be appropriate and we shall refer to it as simply the characteristic length L. The shallowness parameter € and the slenderness parameter 6 are given by Eso /. 1, = B/ jae yi We shall restrict our attention to that class of problems in which the hull must be more slender than the water is shallow, i. e. the maxi- mum draft be less than the depth, thus lim (O7* a0 A simple relation which satisfies this condition is Sd a nOn- keane el los Lea and Feldman where m is a constant which will be determined by the matching of far field solution to near field solution. It must be remembered that this restriction placed on € and 6& does not imply the existence of a functional relationship between slenderness of hull and depth of free stream. We have chosen the shallowness parameter, € , as a conve- nient gauge function*and use it as a standard for order of magnitude comparisons, The following non-dimensional and scaled variables are in- troduced : Dale =o ee, -* Ge Re x=xL,y=yLe and the non-dimensional variables q=qauU yy it) ate We Lig gycrks = 1 ¢£K.; K =0 (ogg The full inviscid equations become : Potential Equation 2 Zatti Zip + € + =~ (0) - Pe ots! : yy (7 =) Bottom Tangency p(x, Yoni) 3 (7 -b) Free Surface Kinematic rige 2+2p bs PREN(Lb poe tO py fo ones bx, y) (7-c) Free Surface Pressure 2 2 €*¢/ 1 4+eK = ay — [2¢,. +(p_)° +e°P (y)*] en aoe 6 Cx oan) (7-d) We assume the following far field expansions for the disturbance po- tential and the free surface elevation e~ De" eo (x,y,z), t~ 2, emt (x, x) (8) n=] n n Substituting these expansions into the full inviscid equations and equa- ting like powers of € results gives the following : * See Van Dyke (4), pages 23-28 L532 Transerttical Flow Past Slender Shtps Disturbance Potential : prt, (aye +£, Gayle” 4+ le, Gey) - (et /2) 4, Je%... 4 2 +t [£, Gey) - (et / 2) tf) |+0( 7) (ea) Free Surface Kinematic : + rk SM 2 ee! ap tie a! ng rs 5 a 1 ayy out POE pe 0 (9-b) ( bret fon Nokia ck CF Free Surface Pressure : (9-c) The bottom tangency condition is satisfied toO ( € if and the) ''stretch" in the y-coordinate is taken as y = yL /€?= yL /VE or p= 1/2. This is determined by the observation that if p<1/2, then the expansions should proceed as fractional powers of € which cannot be matched to the near field solution, On the other hand, if p > 1/2, then the term fy would not appear to 0 ( 4) anda degenerate case results. Thus the choice of p = 1/2 results ina "distinguished limit process'' as € —» 0? The governing equation for the first approximation to the disturbance potential (», =f, (x,y) is obtained by elimination of second order variables (f, and{,) between the free surface kine- matic and pressure conditions to 0 (£4) and is (K+ 3f),) ave ; Tene 3" i) The mathematical structure of this equation could change locally in the domain of solution depending on the algebraic sign of the term (K + Sfay ). This equation can describe locally subcritical flow (elliptic equation) when ( K + 3f)x ) < 9, supercritical flow (hyperbo- lic equation) when ( K + 3h, ) > Oand the local characteristics have the slope (dy/dx )=+[K + ate) “1/2 (11) The expansion for the disturbance potential ( ) given by equation (9-a) is similar to Tuck's outer expansion; however, it must be noted that our small parameter is based on depth (& =H/L) while Tuck's parameter is the slenderness ratio ( 6 =B / i.) We * See Cole (3), page 46 1533 Lea and Feldman note that the Laplacian operator in the horizontal plane does not occur to 0 ( ¢4) thus in this respect the present expansion for the distur- bance potential is simpler than the linear theory. On the other hand, the free surface kinematic and pressure conditions for —, are deri- ved from higher order approximation which lead directly to the non - linearity in the problem. It would appear that in the transcritical region the nonlinear free surface conditions are dominate and the po- tential nature of the flow is only secondary. In passing we note that equation (10) is mathematically identical to the equation governing transonic flow past two dimensional airfoils. We can now define the relative orders of magnitude between the shallow water parameter ( © ) and the slender body parameter ( 6 ) by examination of the behavior of the far field solution on the body surface. While this can actually be done by the formal matching process, we choose to do it here to simplify the algebra. Substituting the far field variables and expansion into the hull tangency condition, we obtain for the leading terms 3/2 CECE Ghia pee fe (x,0A)= 6A + €Off, (x; A) Ab ol. 2y x - (z+1) f yey 104) a (12) where the ''slenderness"' of the ship hull is exhibited explicitly through 6 A with A = 0(1). Guided by the two-dimensional aerodyna- mic slender air foil theory, we take € 72 -%§ which satisfies our earlier requirement that lim_.o (6/€) =O. Thus it seems to imply that the shallow water problem is analogus to high aspect ratio air- foil problem while the deepwater problem is analogus to the low aspect ratio problem. III NEAR FIELD APPROXIMATION The nonlinear effects are not expected to be important in the near field region where the basic flow pattern is strongly influenced by the hull form. Asa result, one would expect that the near field expansion would yield a series of Neumann problems in the ( y-z ) cross flow plane similar to those derived by Tuck (ey 2 the following non-dimensional and scaled variables are introduced : x=XL, y=ELY, z =€LZ,f =E LS N =E€LN All the remaining variables are as in the case of far field approxima- tion. An additional variable N is introduced such that the unit vector en in the N direction is normal to on and the hull contour (6 A(Y, Z)) 1534 Transertttcal Flow Past Slender Ships at any given cross section, We assume the following expansions for the disturbance potential and the free surface elevation ee eee oeakeitac: ay eee ae ate a raren er se (13) Following the same procedure as for the far field solution, we obtain a consistant approximation in the near field to 0 (e° é ) given by a (e)=e 7 ,~, (E ) SE sae B, eliGapie E Miia = 0 (eas =" a2) (14-a) pi, (KX ¥, -1)=0 ( all n's ) (14-b) ne Ot 0) 0 (Gn, = ee D (14-c) pene (ee Ale —haisias, 1-105 acy = A,/VIFA, (14-d) To the second approximation (n =2 ), the boundary value problems derived are identical to those of Tuck(2) as well as the order of magnitude estimate placed on the disturbance potential? However, we note there is a difference in the estimate placed on the elevation of the free surface and is aaa! Z 2 , 6/2 S Tuck ae alla fie! ppl present } = GRE pee aaht) ti 2 ep 4 , 4/3 ¢ present Pe OE ae ) OD ooze g) Oe Tuck ) Thus we see that the surface disturbance is stronger here than in the linear case. Since the Neumann problems defined by equations (14) have already discussed in detailed by Tuck ') \2 » we shall make use of his results and using the restricted matching technique of Van Dyke 4) to match one term far field to the two terms near field approxi- mation. The important result is the hull tangency condition for the far field equation on a equivalent body and gives a ! fy (x0) =S! (x) /2 (15) *We note here the difference in notation ¢ = = 3/2 Tuck pres. pres. 1535 Lea and Feldman where S(x) is the cross sectional area of the ship hull immersed in the water. IV. RESULTS AND DISCUSSION The resulting nonlinear problem, for the first approximation, can be solved numerically or solved approximately using methods of local linearization developed in transonic aerodynamic literature(5 (©) | We have taken the latter approach due to limitations on compu- ter time and the details of which are given by Feldman\? ?, Here, we shall present some results for the sinkage and trim of a semi-submer- ged spheroidal hull. The cross sectional area of the hull is Pelmde (ker Ee Oe, where B,, 1s the maximum beam. The trim and sinkage are compu- ted at the bow with units of trim measured in terms of ship length and slenderness ratio of 1:10. The results are presented in figure 1 where the Froude number is based on the undisturbed depth. In figure 2, we have presented the same curves but using a different scale so that the linear results computed from Tuck's (2) solution can be viewed si- multaneously for comparison. The apparent discontinuity in slope at F, = 1.0 and F = 1.09 is due to the method of solution and not the model equation. We note that these solutions do indicate the overshoot as well as undershoot of sinkage and trim respectively through the transcritical region which have been measured in experiments such as the works of Graff, Kracht and Weinblum as well as the more recent work of Graff and Binek(9), Sinkage as well as trim data have been computed for more realistic hulls and these will be reported else-where. However, one particular case with experimental results of Graft(8) et al is given here for comparison. The hull chosen is Model A3 of D.W. Taylor's Standard Series and the flow condition is exactly critical (Fe = 1). For computational purposes, the cylindrical hull is approxi- mated by a fourth degree polynomial-arc, we have Experiment (Graff et al) Trim, =a 200; Sinkage/Length, = -,015 Theory Trim. = 2009; Sinkage/Length, |= -. 0123 f=" 5. at ‘df 0 * Paper to appear in the proceedings of the 13th ITTC Conference by Feldman and Lea, 1536 Transcrtttcal Flow Past Slender Shtps The theory appears to be in fair agreement with experiment and indi- cates that this direction or research should be fruitful. V. ACKNOWLEDGMENT We are greatly indebted to Professors Th. Y. Wu and J.N. Newman who encouraged us and nudged us along the path to a possible solution. One of us (G.K. L.) wishes to express his sincere thanks to Mrs. Lea who typed and retyped this manuscript expertly and willingly. REFERENCES 1 TUCK, E.O., "A Systematic Asymptotic Expansion Procedure for Slender Ships"’, Journal Ship Research, 1964. 2: TUCK, E.O., "Shallow-Water Flows Past Slender Bodies"! Journal of Fluid Mechanics, Vol. 26, Part I, 1966. 3 COLE, J.D., ''Perturbation Methods in Applied Mathematics'"' Blaisdell Publishing Company, Waltham, Massachusetts, 1968. 4 VAN DYKE, M.D., ''Perturbation Methods in Fluid Mechanics'"' Academic Press, New-York, 1964. 5 SPREITER, J.R. and ALKSNE, A.Y. ''Thin Airfoil Theory Based on Approximate Solution of the Transonic Flow equation" NACA Technical Report 1359, 1958. 6 HOSKAWA, I. "A Simplified Analysis for Transonic Flows Around Thin Bodies'' Symposium Transsonicum, Aachen, September 3-7, 1962, Springer-Verlag, Berlin, 1964. Ti FELDMAN, J.P. '"Transcritical Shallow Water Flow Past Slender Ships", Ph. D. Dissertation, The George Washington University, Washineton, DiC. Lozi. 8 GRAFF, W., KRACHT, A. and WEINBLUM, G., ''Some Exten- sions of D.W. Taylor's Standard Series", Transactions of the Society of Naval Architects and Marine Engineer, Vol. 72, 1964. 9 GRAFF, W. and BINEK, H., ''Untersuchung des Modelltank- einflusses an einem Flachwasserschiff'' Forschung sberichte des Landes Nordrhin-Westfalen, Nr 1986, Westdeutscherverlag, LOT he 1534 Lea and Feldman $34u930 NI WIdl IIny Teptor9yds pasrourqns-tu19sg YIGWNN JGNOY4 Cl feels avis iT JOVANIS 0°T OILWY Hid3d-Wv4d 10 O1LVa HLONI1-WV48 96 0 26 0 [ eansty 880 80 10 0 MOS LV O1LVY HLONI1-JOVANIS 1538 Transcrtttcal Flow Past Slender Shtps Aytun jo o1je1 yjdop-ureseq pue | ‘9 jo oT}eI YYSUeT-UTeEeq jo TINY [Teptor9yds pasrauiqns-twoes e TOF SOTLOOY} [BOTIIOSUeA] pue TeaUTT 9Yy} UJAMJEq UOSTIedUIODH 7 9AINBIY YIGWAN JGNON4I el Gan a: OT 60 80 L‘0 AYO3HL YW3NIT ib Pes $39¥9I30 NI FIONV WIdL AYO3IHL YVINIT 20 0- ——— Ax03H1 YVINI] 10 0- AYOIHL YVINI] MO8 LV OILVY HLONII-JOVINIS 10 0 L539 Lea and Feldman DISCUSSION irniest OF Guck Untverstty of Adelaide Adelatde, Australia Iam very please to see this done. One thing that bothers me is the condition that delta is order epsilon to the three halves. It seems very surprising that the theory should depend upon such a restriction. In my original linear theory, delta and epsilon were identical, and it seems unreasonable of a theory to demand such a geometrical cons- traint a priori. REPLY TO DISCUSSION George K. Lea Nattonal Setenee Foundatton Washington D.C., U.S.A. The odd power of 3/2 appears is due directly to the critical flow parameter Fyj-1. You must remember that the relation between € and dis an artificial one, Their relative orders of magnitude are es- tablished through matching, with the a priori requirement that a rea- sonable solution can be obtainedas Fy,,~1. This type of proceedure must always be followed when more than one small parameter appears in the problem. The difference between this and your linear result is the difference between non-linear and linear approximations, DISCUSSION Ian W. Dand Nattonal Phystcal Laboratory Feltham, England We have obtained some measurements of sinkage and trim on tanker forms in sub-critical flow, but, using self-propelled and towed models. These showed that the effect of the propeller when the model was self-propelled was such as to modify appreciably the measured trim. I have two questions asa result. 1540 Transertttcal Flow Past Slender Shtps 1. Were the model results, used as a comparison in this paper, obtained with self-propelled or towed models ? 2. As the effect of self-propulsion on trim seems to be quite marked in our experience, is it possible to take account of this theo- retically ? REPLY TO DISCUSSION George K. Lea Nattonal Setenee Foundatton Washtngton D.C., U.S.A. We did not run any model tests ourselves and all comparisons were made with the data of Professor Weinblum and others given in the list of references. Test were towed tests. Self-propelled models may be under additional moments due to the onboard thruster which could cause an initial trim even at zero Froude number. The result is that the water plane area could be dif- ferent and this can be accounted for by changing the area distribution, S(x), in our computations. I would expect the differences between tow- ed and self-propelled models to be the least for trim when you are dealing with full-forms like tankers. ey 7? 7 Waly oe. ain i : \ 26 2 : q Us aq y >. Wat wi MS ; 2 ' ‘ ve “a ‘ wo) ie tues tat lor) eae = ae : _ AZ piety stk ap etveqeras £02 byeo etlveos inborn oth os e es ° elobeom: bowel to heilsioryettea dite Bb: a “wn sjiugy od of errooe oticd ovo ASH LAES GIR io taste adi aA | a . -ood! eidt Io tayoo0s saat ¢2 sidigaog 1 st soneitedes babe, = Ve cuiewt cm 7x vw nga .o) Fe Meeritcr on OF Yeas vey. waves i gaa © tase abhaes er} sf ‘ ‘ re re iad “~e and etgs-ut were bleptical iene ‘hing ‘het tecety sthdiiiak "Maes. a 4u¢c3..o reete = ijt lie ' Ling wr at ee ne Ws + (eG apne wail a Sourmetridal aslee uaerteo® Sas bas covisedd> efee? 6Sarm yne aut Jou oth oW gi cl nsovig é@todio Baa avldanly W x+928810719 to crsh on7 diiw oe .o3003 bawot eqow 4eeT- 2sonS eee hes sl) DISTANT: indoriinbs +s PISi Laat hailaqaiqtia’ otez.te wevs orit? Ishiialt os seoso Divo ASiiv THIS Ad bi soa ~lih ed blocs sets susig ata om Tadi,at Mons: at adamant moltodiia® 2974 920 gaigceen yes oh bu anoDG oe on? aid) bate ~Wol daguied easasi9 Nib ddgavedsy blnow. 4. enoliaiuqmos tee ote vay cally mixt 10) 14551 of} 9d 6f-efebom balieqoiq- ies esodn“? OAL acrrrol} sles aw "i. hee apooare sf w dlreecGy 2 13240 oe a nt. ae Fou mia reas ev that th + oa [ é FiLAtLAR. * f eit ore" . 2 pie: nee — he throigh rnatchen wiadi prLers 2 Lt iopent haga ra j t,t Sok See - « P # . “ee yp" ot ade > " > 6b wnen ns . thn joe Amal, perateter rc \iference between thie ane eour deer te rte - rae -lineaar and linaar apriros nie toene. Lan W@W a A Le PTO wet CF] 5 Pe a ee ‘= Lave ebdtéiiad gaorne mraanremonte 6f stezegs SAd-Ee artes there in out-eeithcal ow, Dot, using exit -axepeligd dat bs " the ahawed (fat the «fect of the propelier whan tha nse eek f=) welled wan ee th 26 tw modlly reppreciably the pee oe . ‘sucettnge ae & trelpgt 1440 COMPUTATION OF SHALLOW WATER SHIP MOTIONS R.F. Beck and E.O. Tuck Untverstty of Adelatde Adelatde, Australta ABSTRACT In previous papers by Tuck (Journal of Ship Research, 1970) and Tuck and Taylor (8th Symposium on Naval Hydrodynamics, 1970), a framework was set up for a complete theory of ship motions in shallow water, in all 6 degrees of freedom. The present paper con- tinues this work by presenting actual computed mo- tions for a full form hull, both restrained andunre- strained, in long waves of various headings. i ANT RODUGTION In this paper we present computed results and/or discussion of motions in all six degrees of freedom of a Series 60, block coef- ficient 0.80 ship, at zero speed of advance in shallow water. These motions are supposed to be induced by incident plane sinusoidal waves of various headings. The shallow water theory of Tuck (1970) (see also Tuck and Taylor, 1970,and Beck and Tuck, 1971) is used to provide the coeffi- cients in the equations of motion. This theory requires that the wave- length be much greater than the depth of the water, which restricts attention to long waves and low frequencies. Such long waves are im- portant for large ships, since they have the greatest potential for mo- tions excitation, even though the low frequency assumption rules out resonance in heave, pitch or roll. In Section 2 we discuss some general analytical features of the equations of motion of a ship in shallow water, and consider the re- lative importance for each mode of motion in turn of various types of inertial, hydrostatic, and hydrodynamic forces. The force balances 1543 Beck and Tuck which dictate the ultimate motions are complicated, but in most cases there are pairs of forces which contribute most to this balance, other forces being formally of a smaller order of magnitude with respect to a small parameter such as beam/length ratio. For instance, in heave and pitch the dominant force balance is between hydrostatic restoring force and the pressure of the inci- dent wave (so-called Froude-Krylov exciting force). Inertia, both natural and hydrodynamic (added mass), damping, and diffraction of the incident wave are all effects of lesser significance in the range of wave periods considered. Indeed, remarkably in shallow water the natural inertia or mass of the ship has the least influence of all these forces. Similar simplifications can be made to the other modes of motion, leading to ''first-order) theories involving only the dominant forces. However, the computations presented in Section 3 for coupled surge, heave and pitch do include all forces, not only those of first order. The first order computations are verified as numerically rea- sonable, and information is obtained about the most significant second order effects. For example, the diffraction exciting force (unfortunat- ely neglected by Beck and Tuck, 1971, in making similar comparisons) appears to be the most significant second-order contribution to heave, whereas pitch is affected more by added hydrodynamic inertia than by diffraction effects. In the case of surge the first order balance is between natural inertia and Froude-Krylov exciting force, and this first order result appears to be remarkably accurate. In particular, there appears little need to worry about coupling with the other modes, Some moor- ing force considerations are discussed in Section 4, the conclusion being that for large ships only surge is likely to be affected, and then perhaps only marginally. The general theory of surging of moored ships has been thoroughly treated in the Civil Engineering literature (see Wilson, 1967, for a bibliography), and perhaps the only new contribution we can make here concerns the correct evaluation of the surge exciting force as a function of hull geometry. This question is given some attention in Section 3 and Appendix III. In Section 5 we continue the theoretical treatment of the very difficult problem of horizontal plane motions, clearing up most but not all of the loose ends left by Tuck (1970) for sway, roll and yaw. The appropriate integral equations which determine the hydrodynamic coefficients in these modes have been set up, but the roll equations have not yet been solved. Computatton of Shallow Water Shtp Mottons Finally in Section 6 we present computed sway and yaw motions, neglecting coupling with roll. This is justifiable, as discussed in Section 2, if the metacentric height is sufficient to remove the roll resonance period from the range of wave periods of interest, a situa- tion which is not unlikely in shallow water. The resulting motions agree well with simple approximate and limiting results, which may be used for estimates in lieu of the very complicated computation procedure needed in the general case. As indicated by Tuck and Taylor (1970), the detailed computations are, however, of importance if swaying is to be in any way resisted, by moorings, fenders, etc. In no case have the present results been experimentally verifi- ed. The apparent lack of systematic (as distinct from ad hoc) expe- rimental measurements of ship motions in shallow water in the publish- ed literature is deplorable in view of the importance of this subject today, and it is to be hoped that this situation will be remedied as soon as possible. II. THE EQUATIONS OF MOTION IN GENERAL The equations of motion for any ship moving sinusoidally with complex amplitude §; at radian frequency o inthe jth mode of motion, the time-dependent displacement being a(t) = ipa dieh (2. 1) are (Salvesen, Tuck & Faltinsen 1970, Tuck 1970) Here Mjj is a generalized mass matrix, i.e. Mo 8 0: Mz 0 M 0 -Mz, 0 Mx, He PGR Ne REMe ahd i ea [M5] OhigaMia AO «Mines WOd validtior all yie= 4, 3:75, and-all) i= 173,585 p Mis nc ero / dxe ee (x) (3. 1) 4 T gia amt Bald. cohhe At Oy (k i wee pe pet py es ap eat Jx- §]) » (3.2) where A(x) = )'S"{ 268) (3. 3) A,( x). 2 Bee) PE A,( x ) = -xB(x) zi yiey (3.5) AL a 4 | -B60 Fi = B- Steal) cos B (3. 6) Here B(x) is the full waterline width and S(x) the section area at station x, while ie is an Hankel function (Abramowitz and Stegun 1964, p. 358). A brief derivation of the above results is given in Appendix I. The physical plausibility of these results, especially the rather com- plicated formula (3.6) for the diffraction exciting force, may be ex- hibited by considering the direct effect on the equations of motion, For instance, in heave, i= 3, the equations of motion (2.2) state 2 -o MS, + C4 8, + Cy. S5 - T35 Sg = 13, 5) + Tey $3 * M35 55 * 157% f 2 _ tae ikEcosB =f J ne if dé[B(E) (f,- 85 5-S5¢ (3. 8) 1552 Computation of Shallow Water Shtp Mottons d if.cosB tz S(é) Serhan oe —e inno, oa (k Ce : ) The terms on the left of (3.8) constitute natural inertia, hydrostatics, and Froude-Krylov exciting forces, all hydrodynamic effects being on the right. The expression eikécosp o (3. 9) is the relative vertical displacement between ship and wave at station £, whereas the term if cos ; Oo P iktcosé jen Shae ant) is the relative horizontal displacement between the (surging) ship and the water particles in the wave. This display of the equation of motion is similar to that given by Newman and Tuck (1964) for infinite depth, except that in infinite depth the horizontal motion terms do not appear. It should be noted that the surge motion ¢, and the horizontal fluid particle motions are large in shallow water, of 0(€-1), which is the reason why the relative horizontal motion is now potentially as important as the re- lative vertical motion in determining hydrodynamic effects. The first step to actual solution for the motions is numerical evaluation of the coefficients Tj; . This is a moderately difficult task, especially as regards the double integrals in (3.2) . This task is carried out indirectly, by Fourier transform techniques as described in Appendix II. An apparently trivial but actually significant point about the numerical computations is the fact that we may wish to avoid nume- rical differentiation of the section area curve S(x) to give S'(x) in (3.3) and (3,6) . In fact a simple integration by parts avoids this difficulty, but raises another question. If the section area S(x) does not vanish at the ends x =+f (e.g. with transom sterns), what do we do about the "integrated part'' after integration by parts ? Thisisa classical end-effect problem in slender body theory, since at least in principle slender body theory is inapplicable to such blunt ships. P55 Beck and Tuck This question is examined further in Appendix III, where it is argued that, at least in so far as the surge exciting force T,, is concerned, the theory remains valid for ''blunt'' ships, provided we discard the terms arising from integration by parts. It seems likely that a similar consideration applies to all expressions involving S'(x). Of course in the absence of transoms etc., i.e. when S(t) = 0, this difficulty of interpretation does not arise, and this is true of the com- putations to be presented here for the Series 60, block 0.80 hull. Figures 3.1 - 3.3 show vertical motions computations in all 3 modes for head seas (f= 180°). The horizontal scale chosen is ship length divided by wavelength, while the vertical scales represent linear displacement amplitudes divided by wave amplitudes. In pitch this is equivalent to vertical bow motion due to pitching alone. The results are given for depths of 1.0 and 2.5 times the draft of the ship (0.062 and 0.15 times the ship length). A depth equal to the draft is of course not safely achievable, but no difficulty arises theo- retically in this case for vertical modes (not so for horizontal modes) and this case may be viewed as a limiting one in practice. The motions shown are those resulting from use of all available information about terms in the equations of motion. In spite of the imbalance in orders of magnitude as indicated in the previous section, no terms have been neglected, and all couplings between all three modes have been included. For comparison purposes however, the first-order results are also shown, these being balances between hydrostatic and Froude- Krylov forces only in heave and pitch, and between natural inertia and Froude-Krylov forces only in surge. In heave and pitch the first-order result is independent of depth at fixed wave-length, whereas the first- order surge varies inversely as the depth. The effects of the second (and third) order terms are quite varied, but some general comments can be made. The main difference between the first order and full heave results in Figure 3.1 is due to the diffraction exciting force. This is particularly true near the mi- nimum of the first order heave (about L/\ = 1.2) , where the heave is substantially increased by diffraction effects. The general trend of the heave results is remarkably similar to those of Newman and Tuck (1964) for infinite depth. The first order heave minimum at about L/A= 1.2 appears in both cases to be shift- ed by second order effects, especially diffraction, to about L/) = 1.4. This is not too surprising numerically in view of the similarity bet- ween (3.8) and the equation of motion in infinite depth. 1554 Computatton of Shallow Water Shtp Mottons Pitch is almost unaffected by diffraction effects, the substan- tial increases shown in Figure 3.2 over the first-order theory at about the pitch maximum (L/) = 0.8) being instead largely due to inertia, especially added inertia. A rough explanation of the nume- rical smallness of the pitch diffraction force is that the first term of (3.6) involving B(x) is a nearly even function for a nearly fore-and- aft symmetric ship ; for reasonably low values of k (thus low L/\) the corresponding value of T is small because A, = -xB is near- ly an odd function of x. The terms of (3.6) involving S(x) corres- pond broadly to surge motion, and lead to small effects when the surging is small, as it is at these wavelengths. All second order effects on surge appear to be small, the first-order balance between natural inertia and Froude-Krylov ex- citing force being (Figure 3.3) remarkably close to the full result. The magnitude of the surge motion is, as expected, quite large for the longer waves, Tyo. < 0.8, say, for which the horizontal particle motion in the wave is large. At fixed L/\ , the first-order surge varies exactly inversely with depth, and the full equations give a similar trend except at high frequencies where surging is in any case quite small. Computations have also been carried out in oblique seas, i.e. for values of B other than 180°. In general, effects of reasonable heading angle on vertical plane motions are mostly accounted for by use of head seas results, but with the effective wavelength 2d sec 6 instead of \ in the horizontal scale. This is exactly true for the first-order results in heave and pitch, and nearly so when second and higher order terms are includ- ed. In surge this effect is combined with a "'cos?8" factor, tending to reduce surging. However, since the effective wavelength is longer than the true wavelength and surge is greatest in longest waves, we should anticipate increased surging, were it not for the cos fac- tor. The net effect at fixed (true) wavelength isa ''cosfB'' reduction factor on surge. Since the computed results agree well with the above qualita- tive discussion, we omit presenting computations for bow seas (B= 135°) . Note however, that the "stretching out'' of the head seas curves due to the sec 8 factor means that heave and pitch are both increased at values of L/A (true) of about 1.2 - 1.6, where the head seas responses were small, In this important range bow seas produce significantly greater net vertical bow motions than do head seas (see Beck and Tuck, 1971). 1555 Beck and Tuck One interesting feature of the heave equation of motion in beam seas (f= 90°) for a fore-and-aft symmetric ship is that all second order terms disappear, so that were it not for the third-order mass terms the heave would exactly equal the wave amplitude. Thus at B= 90° ,@(3.6) gives A, (x) = -B(x) = -A,(x) , and we have T,, = -T,3 - Butalso T,, = C3, hence assuming fore-and-aft sym- 30 metry the heave equation of motion is 2 -o MS, + (Ge. rE), wd bbe ib =" 05, (3, Wp Hence (4 Corre tof [2 - au | 3 0 C33 or 3 (3. 12) = abel doc Of ems) Similarly, if we do neglect all second and third order effects, the first order theory predicts zero surge and pitch, and heave fy = 5 even in the absence of fore-and-aft symmetry. Figures 3.4 - 3.5 show computed heave and pitch motions in beam seas. There is a substantial (60%) increase in heave over the first-order value {| = ‘= as the depth increases, especially at about L/\ = 1.5. The pitch (in bow motion) remains below 25% of the wave amplitude, however, and surge is quite negligible, never more than 2% of the wave amplitude at any frequency. IV. MOORING FORCES As an example of the type of analysis required in order to ac- count for the effect of mooring lines on motions (and perhaps more importantly, vice versa !), we give below a simple discussion of the effect of a single linear bow mooring line on vertical plane motions. More realistic and complicated types of mooring systems can be studied with similar procedures and conclusions, The general con- clusion is that of Wilson and Yarbaccio (1969), who find that ''the spring is quite weak compared to the mass, and the ship can be con- sidered to be floating unrestrained except for restraint against conti- nuous drifting''. If we consider only linear effects of mooring lines, the appro- priate modifications to the equation of motion simply require contri- butions to the restoring force coefficients Cjj in equation (2.2) 1556 Computation of Shallow Water Shtp Mottons Consider for example a linear elastic cable of spring constant k and length R, attached to the bow and initially nearly parallel to the calm water line and nearly lying in the centre plane of the ship. Small angular deviations from this equilibrium configuration have no effect on the restoring coefficients. We suppose there is a mean cable tension Ty at equilibrium due to wind, wave (mean stress) and cur- rent effects. The displacements of the bow as a result of small vertical plane motions are $, longitudinally and f5 - £5 upwards, and from Figure 4.1 we see that the new cable length is a Lene agers Ae oe eee a LO (4. 1) ~ Re LG at adil 2 < the heave restoring force is F. een 2S 1 (4. 4) 2-1 Doi te =£¢.) e557 Beck and Tuck and the pitch moment is a ii se (4.5) Thus the restoring force coefficients due to this mooring are Ci = k (4. 6) and ap mT, bets C SES MENT OS Ca. ae gay Coe = Bete (4. 7) all other C;, being zero, The total restoring force coefficient Cij for use in equation (2.2) is the sum of the hydrostatic contributions given in equations (2.4) and the mooring contributions given in (4. 6), (4.7) above. Equation (4.7) show that there is a small additional contribu- tion to the restoring forces in heave and pitch from the equilibrium tension in the mooring line, independent of its elasticity. Since these modes already possess very large hydrostatic restoring forces, itis very difficult to conceive of equilibrium cable tensions sufficient to produce significant effects on heave and pitch. For example, if we use T = 37 tons and R= 100 feet, the former being computed from Taylor's air resistance formula Z LZ ot= 440, 00Z218..B8 v" (4. 8) where B is beamand V wind speed (assumed 40 knots) , we obtain less than one tenth of a percent change in the computed heave and pitch motions of a 200000 ton ship, For this type of mooring or any combination of such moorings, the equilibrium tension would have to be quite unrealistically large™ for any significant change to occur in the heave and pitch motions. * For a single cable the figure of 37 tons is of course already in this category! 1558 Computatton of Shallow Water Shtp Mottons Surge is rather different, in that the mooring provides the only restoring force (4.6). If we assume, as is clear from the re- sults of the previous section, that all hydrodynamic effects on surge are negligible, we can analyse linear surging as a simple one-degree- of-freedom undamped spring-mass system, with the result that the surging amplitude is Fi Ge 2 Sa (4. 9) 1 Zi k - Mo or (neglecting diffraction) AL: 10 Sel ke ga (4. 10) ] 0 2 k - Mo where aan = Eu is obtained from (AIII. 5) . In fact the surging amplitude in the presence of a mooring is simply equal to the factor k | 0 1 | (4, 11) 2. 2 Mo -k 1-0, /o | times the free surging amplitude, where Cia k/M is the resonant frequency. Figure 4.2 shows this factor as a function of frequency o . Note that unless the wave frequency is less than 70% of the resonant frequency 7p = k/M , the effect of the mooring is to in- crease the motions. For large ships, conceivable values of op cor- respond to periods of minutes or more, so that typical sea or swell gives frequencies well above resonance ; however (Wilson 1959) long period range action in harbors can produce resonance, with disas- trous effects. The condition « <70%o, is in general met only by tides and currents, and indeed the purpose of the moorings must be to overcome these very low frequency excitations. On the other hand if o >>op, itis clear that the mooring is having very little effect on the surge motion of the ship, which moves as if free. The force exerted on the mooring by the ship is then of prime interest, and this may simply be computed by assuming given free ship motions. This also applies of course to motions in other modes (e.g. sway), so long as the wave frequency is again well above 1S59 Beek and Tuck the resonant frequency of the mooring. The actual variable tension in the cable resists only a small fraction of the exciting force under these circumstances, which is just as well, since these exciting forces on large ships are generally enormous, The ratio between the amplitude of variation of the cable tension and the exciting force is ko at k (4. 12) 2 k-Mo : | ni eg ee Ibs WSs which is also shown in Figure 4.2. For example, if oa >5eR (i. e. the wave period is less than one fifth of the mooring resonant period), the mooring bears less than 4% of the exciting force, and the motions are not more than 4% higher than the free motions. V. THEORETICAL CONSIDERATIONS ON HORIZONTAL PLANE MOTIONS The developments of the theory of Tuck (1970) and Tuck and Taylor (1970) on horizontal plane motions were confined in effect to computation of the sway exciting force. The resulting formula for the total exciting force is i i =e a ipghk sin | Sees $08 Eig. (x) (5,1) f where Ag; isa "potential jump" across the ship section, computable from purely near-field considerations, Although (5.1) was only deriv- ed for i=2 (sway) itis also valid for i= 4 (roll) and i=6 (yaw). In the case of yaw, there is no need to obtain Ag, separately, since Ag, = x A¢,. The computation of A b> and Ad, will be discussed later. Tuck (1970) also suggested a connection between the integral (5.1) at B= 0 and the added mass and damping coefficients. For instance we have (552) 1560 Computation of Shallow Water Ship Mottons 3 fe zh Hi aor 4. al where H is the section at station x , and n is outward from the hull (into the fluid). Now the contour integral can be evaluated entire- ly in the near field region as follows. ooo \e BIs< Aa N Q ~ I a | ea ileal Qy gy B|< As Nh ! < Q |S SS) Q rao) + ee ale A=- Q a) | ea aa Bix oe ang Pile N ==, oF os) 1 Tae | QQ x 6 \¥ ees SS where F denotes the free surface, B the bottom, Ry and Le vertical lines at y = +~ and y = -o respectively in the inner (y, z) plane, as shown in Figure 5.1. The first integral above vanishes by Green's theorem and there is no contribution from F or B in the second integral since both ait and — oy vanish on F and B. On Ly, ato = do j whereas on R., 3b == deo . Hence (using also ii 1. 9) } 0 ok ie oy 4 dg i (i oa = is | dz |e, -y sez | to, Su Xi )ise (5. 4) ist ai vier oe But the boundary condition for the inner potential $, is (Tuck 1970) ¢, — yV + 1/2Ag, age y— > +a (5.5) = Hence 1561 Beek and Tuck 362 + %-y 52+ 1/206, as at a (5. 6) i.e Paeeee : and we have s 9 a $4 = he, ( x ) + S(x ) (5.7) H Thus finally : Q A = -po | dx [p65 () + s(x) ms y = -po nf dx Ad, (x) - ¢M (5. 8) a | where y M = 2a dxS( x ) (5. 9) us)! is the mass of the ship. Note that the term involving the mass M was erroneously omitted by Tuck (1970). The new result indicates that the virtual or total inertia, not just the added inertia, is propor- tional to the real part of the exciting force integral (5.1) at B=0. The above analysis may now be repeated for T,;:, for all i,j = 2,4,6 except for the roll self-force term T44 * — example z J 2 v Tg = Teo = pba dxxA¢, (x) -po I, dxxS(x) (5. 10) ee oe ‘| docx" A@, (x) Sig I, at s(x) (5. 11) A i 1562 Computatton of Shallow Water Shtp Mottons 4 5) Boe =: fag = gee :. a dx[S(x)z _ (x) +5B(x)] yy - (5.12) where Z_(x) is the z-co-ordinate of the centroid of the section at x, Unfortunately if i= 4, the element n, = yng - zn, cannot be written as the normal derivative of a harmonic function, so that the two- dimensional Green's theorem cannot be used, as in the above deriva- tion, It would appear that we must leave the formula for T,, in the form 2 T sae ier d At aan (G13) al mag! : 4°4 ; Lt H and evaluate the contour integral explicitly. Computation of all quantities (apart from Tye ) in the horizont- al equations of motion now proceeds via preliminary computation of the potential jumps A o ;(x) . These are related to the inner streaming velocity V;(x) by (Tuck 1970, Equation (54) ) cee? p V.(x) -4|4 + | J seston” (|x - g) ae (5. 14) - dx which comes from the outer expansion, and dah sxe Nsfae vet 1/2 dg. (x) as y—>t o (54215) which is the inner boundary condition. Solving the inner flow problem leads to a connection between V; and Ad; » which in combination with (5.14) gives an integro-differential equation for Ad ;(x) . For example, if we solve the canonical problems indicated by Figure 5.2, i= 2,4,. weshave @, = Voy + (V5-1) ¥, (5. 16) Sos Beek and Tuck and z ; ' re %, V (y +¥,) V4 (5.17) from which follows * Ag, = ZC ve - 1) (5. 18) and Ady KAZE; (Miguel jap-Gs). (5. 19) Thus we have the integro-differential equations AP; (x) Cc, (x) L t Zz alg ie forsee 0 Ole 8) zeta (5:20 which can be converted into integral equations of the form 5 od i (1) l Aelt) wa A [ere (k |x- El) “e| dé 26, (E) sin k(x - ) 0 ce) (5.21) = A. cos kx + B. sin kx Ao dé ——> G c 6) sink(x-&) . where A; , B; are constants to be determined by the end conditions Ad.(+ L) = 0. Although the left side of (5.21) contains the same kernel for i= 4 asfor i=2, the parameter C,(x) which appears on the right has not yet been evaluated numerically, so that in the following section results are given only for sway and yaw. VI. COUPLED SWAY AND YAW As discussed in Section l., there are indications that rollis not a significant mode of motion in shallow water, and that in particu- lar its coupling with sway and yaw is small. Therefore we present here computed free motions of the Series 60, block 0.80, ship in * The quantity C, corresponds to C(x) as in Tuck (1970). 1564 Computatton of Shallow Water Shtp Mottons sway and yaw, with complete neglect of roll coupling. The equations to be solved can be written -T> 'S - thé vy = F, (6. 1) is Ra ae i eee ae one) where ; 0 es = -po h | dx Ad., (x) (6. 3) Lf * * 2 ee ee ea er (6. 4) f ¢ ikpgh sin@ i dx Aq, (x) SPH SOAs, alig G) by I 2 0 y 6 Cyike gh sin of dxx Ao, (x) ois SoBe | (6. 7) hy i Here starred quantities represent natural inertia plus hydrodynamic effects. Note that natural inertia cancels out corresponding terms in the equations (5.8), (5.10) for the unstarred quantities Dam. su lies and To, , assuming the unexcited ship is in equilibrium. However there is a contribution to T,~ if the longitudinal radius of gyration of the displacement of the ship does not equal that of its actual mass distribution, expressed in (6.5) as W(x) per unit length. This extra term in (6.5) is quite small in practice, but has been included in the computed results. The quantity Ag.(x) is obtained numerically by solving the integral equation (5.21), which for i=2 reduces to 1565 Beck and Tuck i (1) he ene Chae +} ddd, ( )Hy (k x- &) =a dé G(E) sin k(x- &) (6. 8) 4 0 ; (6. 8) = + A, cosik= + B, sim icx « k Numerical procedures for obtaining C.,(x) and hence by solving (6. 8), A $, (x x) , are discussed by Taylor ( (1971) and summarized by Tuck and Taylor (1970). Figures 6.1 and 6.2 show the resulting solutions for the sway and yaw amplitudes respectively. At high frequencies, the motions tend to zero rapidly. On the other hand, as the frequency tends to zero (wavelength to infinity) the sway motion tends to infinity, as in the case of surge, because the ship is then following the hori- zontal fluid particle motions. For a fore-and-aft symmetric ship (Tog 5 0) in beam seas ( B= 90°), the sway equation of motion simplifies to p ) . ; rie i dxA®, (x) = §5 . ikpgh I dxA¢,(x) i.e. the integral containing the potential jump A¢2(x) cancels out, leaving simply apes et (6. 9) This remarkable result shows that in this case the sway mo- tion equals the horizontal fluid particle motion at all frequencies, not just as the frequency tends to zero, The small amount of asymmetry in the Series 60 ship does not prevent (6.9) from giving quite close agreement with the curve of Figure 6.1 for B = 90° . Note that (6.9) predicts that sway varies in direct proportion to wavelength (or period), and inversely as the water depth. These qualitative pro- perties are also confirmed by the full computations. Clearly the geometry of the ship, which in general influences C(x) , hence A¢, (x) , has little effect on the free sway amplitude in 1565 Computatton of Shallow Water Shtp Mottons beam seas, since the integrals involving 4¢) (x) tend to cancel out. We may expect a similar conclusion for other headings, and for yaw motions. On the other hand, as indicated by Tuck and Taylor (1970), if the swaying motion is to be restrained, by moorings, fenders, etc. a knowledge of C(x) and henceA¢d > (x) is vital for computation of the required restraining forces. The yaw motion is plotted in Figure 6.2 as horizontal bow motion, analogously to pitch. Note that yaw vanishes identically in both head and beam seas, irrespective of fore-and-aft symmetry, so that maximum yaw occurs at some intermediate heading angle. As the frequency tends to zero, the yaw motion tends toa finite limiting value which may be estimated for a fore-and-aft sym- metric ship as follows. We also assume that we can neglect the second term of (6.5) , which is true if the radii of gyration of displacement and mass are nearly equal. Thenas k— +0, we have Xf { jikegh sna f dxx Ad, (x) [1 +ikx cosB +.. | ay t ve ; roa [exes f af Ok pgh sin B eh dacx“Ag, (x) Again the integrals involving 4¢, (x) cancel out, leaving ¢ sin8 cos8 6 = fe (6. 10) However, this result is of much more limited validity than (6.9), in particular being valid only for low frequency. Note again an inverse dependence on depth, and a maximum at 45° heading. VII. CONCLUSION In the present age of offshore mooring facilities for super- tankers and giant ore carriers, the usefulness of a shallow water ship motion theory is obvious. However, the results presented in this paper are purely theoretical. To the author's knowledge there is very little experimental verification available. Until there are experiments with which to compare the theory, we are forced to rely on the good 1567 Beck and Tuck agreement between experiments and deep water theories to give us confidence in the present shallow water theory. In the case of heave and pitch, the theory presented in this paper (but with neglect of diffraction exciting forces) has been com- pared to Kim's (1968) shallow water strip theory by Beck and Tuck (1971). As expected, the present slender body theory seems to give more acceptable results in the very low frequency range, while the strip theory should be more accurate for high frequencies, In the in- termediate frequency range, both theories give similar predictions of heave and pitch motions, This work was supported by the Australian Research Grants Committee. APPENDIX I DERIVATION OF ee FOR VERTICAL MODES The definition off ‘T;;’is G,j = 0,1, .....- 7) J 2 Bi os 8 Meare | n,¢.dS (A. I ij Tai where ___g ikxcos® coshk(z +h) _iky sin B es Oy hae ‘Cosh kh (An ta2) ik 2 2 Bate CO8P (1 + ay sinB + 0(€) ) (A. 1.3) g and the result of Tuck (1970) is that for j = 1,3,5 Me (a) loca [ee ples (k|x- |) dé (A. 1. 4) where 26x) =) Aa (A. 1.5) H Computation of Shallow Water Shtp Mottons is the flux across the section H at station x in the jth mode. The results (3.1), (3.2) for Tj; g and T;,, i,j = 1,3,5 now follow di- rectly from (A.1.1), while that for j = 7 follows after using sym- metry, Tid = Toi : It remains to evaluate the quantities A;(x) in terms of hull geometry, an easy task for j = 1,3,5 using the elementary results i nad SAP ae ets (A. 1. 6) H and* i nad = -B(x) (A. denim) H with n, = -xn,. For j= 7 we have 2 _ _ 9¢0 SPARE xs ¢O a (A. 1. 8) % 3¢0 340 Oo - stpeas Sy 10 "30" | Carrying out the differentiations of (A. 1.2) and integrating we obtain a 72 Lm . _ikx.cos' 6 i cosB y sin B Zz A(x) =e | Been n, [- =? n, k + al al H which leads to (3.6), using the further elementary results i gee 2 | znad = S(x). (A. 1. 9) H H * The corresponding result in Tuck (1970) has a sign error which occurs twice and therefore does not affect the final answer for T,4 : 1569 Beck and Tuck APPENDIX II COMPUTATION PROCEDURE FOR VERTICAL PLANE INTEGRALS The task of evaluating the integrals (3.2) is simplified by re-casting them in a Fourier transform manner, Thus co ipa di * oT et = = Ate A AI, 1 0 where a Coy = if dxA.( x ) spe : (AII. 2) a bar denotes a complex conjugate, and we adopt the convention that Lory (hi She V heen = iq/ rd? - 1 . The result (AII.1) follows from the integrals 1 2 dA SCZ) 5, ————- _ cos Az : i : Ns, Ge and co Z. Yee? \= 2 a4 Gcosi Zz , 0 T nN = t 1 mithaE hepa re 0 0 0 The Fourier transforms AS (which are incidentally also re- quired for the Froude-Krylov forces Tjg in (3.1) ) are obtained by a modification of Filon's quadrature (Tuck, 1967). Data concerning A;(x) (i.e. beam B(x) and section area S(x)) is supplied at given (not necessarily equally spaced) values of x. Data actually used was read directly from the table for the Series 60, block 0.80 parent form (Todd, 1963) at 25 stations. The Filon quadrature maintains uniform accuracy as the parameter \ increases, The integration with respect to \ in (AII.1) is carried out separately for 0 toy (hoy? waa Ye PRae Se aan daGisia ooh: ginetiaiata meals nt budandaeall Be 1, WPT y nook Ste, Hig panel ah = eis anh ire, 99 gah bans VE NOR MNES 9 | 5 ’ 4 yw 22a Fee ma 34s «af Hi The ¢ ; oe te, al AAted@ Kin bateced he tweed Weary jute retin One we siesder edie al i a 4" We me per Fhe webct oh ‘MAS eu “6 ae sary tog we ve! engih S Se itl / so Apntoped f4e Mea os alin ta she were hen eee vi 4 ‘hea Dear, What de pall i ee plas ME vies ona, Teel ed ats at iy ee ¥ id, Kiznte ie eee tone \enieeectcstam anaamanen a ale yercras eet shee eee sachets & it Re dew WA t perme Yradteduirot Live: ek pega \ «. Z am LI}eF Py Eni O- a) ¥e 4 y an | parbigeene- adi bos. ne fF » CCAR AYR LATS, Met Bara eee OE, HOF wy Salo orth, sla. COLR He’ east, Bil yy diy, uk B23Fh 2 ES fi Cs aid 1 HOSE, ac aatit or the : mine * ; Ten 4G0n fg Said OF ase iy Leos PAG Jy S06 F seal = 3 i¢ ae Arr paei ii : i vo inerae x bid, Die ; tigers DAY bea ra + ty (herd ‘ Vac tt. BAY + widget (te rte OF Be in, (oO mpGles OY S42 in theory, { in ie yay! On, Do tebe Meee ae oo Pet) Via? | Xx i , tee nd W ai Loan eit! Pemdalss 06 ob F gid jetid) at hie bj lise G2 3@ wal or thal Kina" ae hal ae 4% .¥~ itysicre <4 rv) or it 10 O1Si. S14) 0 ' yq i Vale, é ed es {ne wey panes, aor Raw. Dee Leoreloase < | ny *6 sete with Page bleep a SaaS Z ” oe Cuda (ihe sarvectiy Gredicts Taq, WeSroes ehibh Age not ebrte thy walid ia wis vegion, ' & Sits vtne?r Wrelt re tn vhe high frerueney region, Rie Weave added ae OOS tad janes of Compete? hy slender buoy theory go te week . roow Cet hee shoeld paeyarpene to @ Dieizerwebaw TAR SEAKEEPING CONSIDERATIONS IN A TOTAL DESIGN METHODOLOGY Chryssostomos Chryssostomidis Massachusetts Institute of Technology Cambrtdge, Massachusetts, U.S.A. ABSTRACT A procedure leading to the prediction of seakeeping qualities of monohulls in a seaway is briefly re- viewed. The two parameter conformal represen- tation of hull sections is described and compared with the close-fit representations. A proposal for incorporating seakeeping conside- rations into a total ship design methodology with particular emphasis on the identification of the pro- blem areas and design indices associated with sea- keeping is made. The advantages and limitations of the two parameter representation are discussed. The optimization criterion, constraints, and opti- mization scheme used in conjunction with the pro- posed design methodology are discussed and illus- trated by an example. I - INTRODUCTION Attempts to improve the methodology for designing large ocean based systems such as ships have recently appeared in litera- ture, e.g. References 1 and 2. The approach proposed in both these references has retained the iterative nature of the traditional solution method but it has attempted to introduce most of the factors that can influence the overall configuration of the ship as early in the design cycle as possible. In order to do so the proposed approach requires that the largest possible number of alternatives be examined at the outset of 1589 Chryssostomtdts the study and that all considerations that can affect the final decision be introduced at that time. To be able to do so within the time and re- source limitations imposed in all real life problems the proposed ap- proach requires the development of suitable mathematical models that can be used at the different iteration cycles. The mathematical model to be used in the first iteration must be quick (speed is gained by sa- crificing the degree of detail) but of sufficient detail to permit the decision maker to select correctly from among the large number of alternatives that are being investigated. This selection usually invol- ves elimination of all infeasible alternatives. The mathematical model to be used in the second iteration must be sufficiently detailed and re- latively quick to permit the decision maker to select from the alter- natives that were not eliminated in the first iteration. This selection usually involves the elimination of all clearly inferior alternatives. The mathematical model to be used in the final iteration must be fully detailed in order to provide all the information necessary that will permit the decision maker to make the correct final decision. Fig. 1 shows all the steps involved in the proposed design methodology. Seakeeping is a consideration that can affect the final deci- sion because it can affect the system's cost (profit) and feasibility to perform its mission, Therefore according to the method proposed above seakeeping considerations should be incorporated as early as possible in the design cycle. A procedure for incorporating seakeeping considerations in the design cycle is proposed in this study and is described in some detail in the sequel. II - FIRST ITERATION The mathematical model describing the system under inves- tigation during the first iteration of the proposed design methodology must have at least the following two attributes. First, it must be quick to enable its user to investigate the large number of alternatives called for by the proposed methodology and second, it must be accu- rate enough to allow its user to draw the correct conclusions from its results. The method that will permit us to construct the seakeeping part of the mathematical model to be used in the first iteration of the proposed design methodology is given in Reference 3. The highlights of this method are described below. II. 1. Method The notion of using standard series, see for example Ref. 4 1590 Seakeeptng Constderattons in a Total Destgn Methodology OEFINE THE PROBLEM OBJECTIVE TRANSLATE THE PROBLEM OBJECTIVE INTO A DESIGN CRITERION AND INTO PERFORM— ANCE CRITERIA. IOENTIFY THE APPROPRIATE INDICES AND THEIR CONSTRAINTS. DIVIDE THE PROBLEM INTO SUBPROBLEMS I SUBPROBLEM | ===> sabia Tih 2 <=> SUBPROBLEM N t DEFINE THE SUBPROBLEM OBJECTIVES . . dae. ! Bee . SUBPROBLEM SUBPROBLEM + = OBJECTIVE |! SUBPROBLEM|/ OBJECTIVE 2 OBJECTIVE N , . . e . . . ° . . . TRANSLATE THE SUBPROBLEM OBJECTIVE . . INTO A DESIGN CRITERION ANO INTO PER-— Z ¥ © . FORMANCE CRITERIA. IOENTIFY THE . . APPROPRIATE INDICES ANO THEIR CONSTRAINTS. . . ey era de tions Napa T ITERATION bo * 1 GENERATE THE ALTERNATIVES TO BE INVESTIGATED ie i To atetawtialale: Tel a leat cal ELIMINATE ALL INFEASIBLE ALTERNATIVES —~ | ] i | 4 . . 2ND ITERATION | . . 1 : | = | | | ! EVALUATE THE CONSEQUENCES OF SELECTING A 2 PARTICULAR SUBPROBLEM OBJECTIVE. a . = IDENTIFY THE DESIRED SUBPROBLEM OBJECTIVE © . ANO THE MAC*O LEVEL DESCRIPTION OF THE . SYSTEM THAY FULFILLS IT. : | 7 ea Ce Pe ree CY ey error, Cee wer INTEGRATE (INTO A SYSTEM Figure 1 Exploration phase flow diagram r5gt Chryssostomtdts for the determination of the calm water resistance of a given ship is extended to permit the determination of the seakeeping qualities of a ship operating in a seaway. The preliminary results of such an effort are given in Reference 3 where the results for heave amidships, pitch, wave bending moment amidships, added resistance, accelera- tion at stations 0, 5, 10, 15 and 20 and relative motion and velocity at stations 1, 2, 3, 4 and 20 ofa ship operating in long-crested head seas can be found tabulated. The results are given as a function of : Froude No. 0. 105(0..05) 0.30 H 1/3 7 Bp 0.015, 0.020, 0.025, 0.030, 0.040, 0.050, 0.075 and 0.100 LBP/B 5.50 (1::50)68550 B/T 2.00 (1.00) 4.00 and CB - 0.55 (0.05) 0.90 and are applicable for cruiser stern type ships. The 72 hull forms, the six Froude Numbers and the eight non-dimensional sea states (H 13 / LBP) defined above form a grid which allows the user to predict the seakeeping qualities of his ship by interpolation (extrapolation) with all the accuracy called for in the first iteration of the proposed design methodology. A sample table from Reference [3] is included in Appendix I of this study for the reader's convenience. Il,2. Mathematical Model The following describes the method employed in the present study to determine the values of independent variables of the system that will best satisfy an owner's given set of requirements. a) Assign different combinations of values to the independent variables. b) Evaluate the mathematical equations describing the sys- tem under investigation for each combination of values of the independent variables (each evaluation constitutes a sampling cycle of the optimization procedure). 1592 Seakeeping Constderattons in a Total Destgn Methodology c) Eliminate all infeasible designs. d) Evaluate the optimization criterion for all feasible de- signs, and e) Select the alternative that is feasible and satisfies the problem objective. The part of the mathematical model that is directly related to the seakeeping considerations will now be described in some detail. The other elements of the mathematical model can be found described in the literature dealing with the subject of preliminary ship design optimization, see for example Reference [5] , and therefore will not be repeated here. Description of the Environment The environment in which the system under investigation is to operate must be described in order to permit the evaluation of the system's seakeeping qualities. Given the route of operation sucha description can be obtained from the information given in Reference [6] . The complete environment description would require the cons- truction of a frequency histogram as a function of significant wave height, average period, direction,time and geographical location. For the present study the environment description was sim- plified to a frequency histogram which is a function of the significant wave height and geographical location only, and the spectrum des- cribing such seaways (fully developed, long crested) is given by : o* (w ) = aa me exp ( - ae cF where a = 0.0081 23 fe /aet=] and B= 0324 ay feet] In addition only head seas were considered in this study. The simplifications adopted in this study are not considered unrealistic. In the route chosen for investigation (New York-Rotterdam) head seas were predominantly encountered at least 50% of the time i593 Chryssostomtdts spent at sea, see Reference 6 . In addition the seasonal variation is not significant as can be seen from the tables presented in Ref. 6 , and fully developed seas are the predominant seas encountered in the North Atlantic as can be seen from the results of Reference 7 The frequency histogram obtained from Reference 6 for the area of interest is given in Table 1. Table 1 also gives the simplified histogram used in this study. The author did not have access toa computer during the study and had to keep computations to a minimum, resulting in this further simplification. However attention was paid to retain enough detail in the environment description. Therefore valid conclusions can be drawn from the results of the present study. TABLE 1 Significant Wave Height Histogram for the New York-Rotterdam Route Histogram from Reference 6 Histogramused inthis Study Significant Wave Height Percentage Significant Wave Height Percentage ft % ft 7% 4.76 4.16 7. 94 48.25 5.48 6. 81 60,58 26. 14 6. 92 bis oe 13.23 13.47 8.37 19. 96 igs iat 2 7 (. Sf 9. 81 14. 90 212 o 2.20 f1225 11.24 26. 45 1, 89 E2270 42-70 14,14 Be Gl $5258 3. 82 i 03 3.44 18.47 0. 61 19. 91 0.59 21.256 0. 90 22. 80 0.89 24625 0. 3,7 25. 69 0. 44 2 akp 0.34 2:8,:58 0. 22 30,02 On dig 31.46 0.34 35. 79-44. 46 0.01 1594 Seakeeping Constderattons tn a Total Destgn Methodology E.H. P. Calculations The Effective Horse Power (EHP) of a ship operating in the seaway is computed in the following manner. First, the bare hull calm water Effective Horse Power is computed as a function of ship speed from Reference 4 . This es- timate is then augmented by 3% to account for the presence of ap- pendages. Second, the Effective Horse Power necessary to overcome the increased resistance because of fouling is computed. It is assum- ed that the ship is dry docked every year. From the results reported in Reference 8 it is found that increasing the value of ACF by 0.00015 would account, on the average, for the yearly increase of resistance due to fouling. Therefore : 3 0.00015*pxV »* (1. 6889)° * wetted surface P ; = fouling 2.* 550 where V = ship's speed in knots 3 sea water density in slugs/t. and p Third, the wind resistance is computed as a function of the ship speed and sea state (H!3 ). From Reference 9 the Effective Horse Power necessary to overcome the wind resistance is given by 2 0. 00435 *B * Rae BS andi 2 * 325, 66 where B = ship's beam in ft. VR = wind velocity relative to the ship in knots. The wind velocity relative to the ship, VR, is computed by adding the wind speed corresponding to each sea state to the ship's speed V. The wind speed as a function of sea state (H'/3) is determined from Fig. 2. PS Chryssostomtdts o wo P MODEL TOWING TANK ie PIGRSON-MOSKOWITZ SEA SPECTRA (13) LHOIGH 3AVM LNVDIAINOIS Oo om (94S/GV4Y) O AONANDAYS YVINIYID s g oa ae me oe 6 oR Me geAECE E (14) ¥W3d Wld3dS 40 HLON31 3AVM 10 WIND SPEED (KNOTS) ters for fully developed seaways Principal parame igure 2 F 1596 Seakeeptng Constderattons tn a Total Destgn Methodology Fourth, the Effective Horse Power necessary to overcome the mean added resistance in waves as a function of ship speed and sea state (H V3) is computed using the appropriate Seakeeping Tables from Reference 3. Finally, the Effective Horse Power ofa ship operating in a seaway is obtained by adding the bare hull calm water EHP (augmented by the appendage allowance) and the EHP necessary to overcome the increased resistance due to fouling, the wind resis- tance and the mean added resistance in waves. The results of sucha computation can be found in Figures 3 and 4, Speed Calculations without Motion Considerations Given the EHP curves as a function of ship speed and sea state it is possible to compute the average ship speed (assuming no limi- tations due to motion) for a prespecified engine output, SHP,, in the following manner. First, a family of propellers is selected. For the present study the Wageningen B-Screw Series, see Reference 10, is selected. The propeller type used in the present study is the B.4.55 propeller. It is of interest to note that ships similar to the ones investigated in this study operate using propellers with characteristics similar to the B.4.55 propeller. Next the curve of Ky /J* is computed and plotted on the pro- peller diagram to allow the selection of the most efficient propeller for operation in a prespecified sea state. In the present ae it was decided to optimize for the sea state characterized by H 13= 7, 94 ft. because according to information given in Table 1 it is the sea state that occurs most frequently. In the computation of eae EHP * 325. 66 i p* abc ys (1. 6889)>* (ithe) “ee (1-t) the ae and ship speed V was determined from the EHP vs V and H'3 curves developed in the previous section, the propeller dia- meter d was takenasin Reference 4 , i.e. d 20.70 T, and the values for w and t were determined from Reference 4 from the calm water data, These values were assumed to apply to the propel- ler operating in the seaway. Unfortunately the effect of this assump- tion cannot be estimated as very little is published on the subject. However the results obtained from the calculations using this assump- tion appear to be in agreement with published results and it is there- RAZZ Chryssostomidts fore concluded that this assumption is not unrealistic. Once the open water efficiency © is computed the propulsive efficiency 1p, is then calculated IZ O H R iS) where ike = (1-t) / (1-w) 1 = relative rotative efficiency (Reference 9 suggests a value of 1.026 which was adopted for all sea states) uP = shaft transmission efficiency (Reference 9 suggests a value of 0.98 for machinery aft, which was adopted in the pre- sent study) This allows to compute, SHP,, the power required to operate a given ship at a given speed and sea state from the following equation SHP_ = EHP / 1p: The above calculations are repeated by selecting pairs of values for EHP@and? V until SHE 7 — sHP, - This procedure is then repeated using pairs of EHP and V values from the EHP curves corresponding to the other sea states of interest. In these calculations of course the propeller selected for the sea state characterized by H'3 = 7. 94 ft. is always used. Once the speed that can be achieved in all the sea states of interest for a given SHP, , the average speed, Vg, , can be obtained from the following equation 1598 Seakeeping Constderattons tn a Total Destgn Methodology where F, is the frequency of occurrence of the ith sea state (H 1/3, obtained from Table 1. V. is the ship speed in the ith sea state (no motion considera- tions) and N _ is the number of sea states (H 1/3) used to describe the environment (N = 6 for this study). The results of such a computation can be found in Figures 5 and 6. At this stage it is important to point out that in the present study SHP, is an independent variable because the calculations are not done for an assumed average condition, as was the case with Reference 5, but rather for the actual operating environment. Speed Calculations with Motion Considerations : : nS ; The ship speed as a function of sea state (H / ) determined from the previous analysis must now be modified to account for pos- sible further reductions due to motion considerations. The criteria used to determine whether a given ship is motion limited or not are the following a) For the safety of the crew the RMS vertical acceleration at station 15 is not to exceed the value of 0.125 g. This value is determined from Reference 11. b) For the safety of the cargo (i) the average 1/10 highest values of vertical acceleration must not exceed one g anywhere along the ship's length. From Reference 12 V 2 = 1.800 8m__(1- €° 72) 1 = /10 “ where Vm, is the RMS of the response of interest 1599 Chryssostomtdtis and ¢ is the broadness factor assumed to equal 0.60 for the present study. In order to satisfy the requirement that the 1/10 highest values of vertical acceleration does not exceed one g anywhere along the ship's length the inequality Vmo < 0.217 must be satisfied. (ii) the probability ate that the amplitude of relative motion at station 1 will exceed the freeboard at station 1 (f,,) be less than 0.01. From Reference 12 Z yn = -f “i Pei w = See 2 where A = “2(l=e /Z\m rm O c) For the safety of the ship the probability, P station 2 should not exceed 0.01. sg of slamming at 202 2 22 = -(f /A iV A a Te | ( af rm er / a where f, is the draft at station 2 Ms is the threshold critical velocity assum- ed to equal 12 ft/ sec fora 500 ft. ship and scaled according to Froude for other ship lengths 2 and Ay =1 Z(1=% /2)mo, Once the motion indices are calculated the speed determined from resistance considerations is reduced (if necessary) until the motion criteria are satisfied. If the speed is reduced to 3.5 knots (speed assumed to be necessary for the maintainance of a prespecifi- ed course) and the motion criteria are not satisfied, no further speed 1600 Seakeeptng Constderattons in a Total Destgn Methodology reduction is allowed in the model proposed in the present study. When this occurs it is advisable to augment the optimization criterion by outputing the seakeeping qualities of the ship at this reduced speed in order to provide the decision maker with all the information neces- sary for the selection of the ''best" ship. In the present analysis since only head seas are considered it is only meaningful to satisfy the motion limitations by speed reduc- tions. This is not always what happens in actual operation where in heavy seas the operator might elect to change course. It is actually common practice to take heavy seas at 30°-35° off the bow in order to ease the pitching motion. However this limitation in the model is not considered important because in other headings the speed in- creases, see Reference 13 , which partly compensates for the lost time due to extra distance traveled. The average ship speed in a seaway can be computed using the same formula given in the previous section the only difference being that now V,; is the ship speed in the ith sea states including motion considerations. The results of such a computation can be found in Figure 5, Fuel Consumption Steam Turbine is the main propulsion unit adopted in this study. The specific fuel consumption at powers other than 100% power can be obtained from SEC) 904, 1100% 150% Typical values for power, RPM and efficiency 7 are given in Table 2. It is also assumed that for a given power setting if the value of RPM is less than the one shown in Table 2, it will not affect the ef- ficiency of the steam turbine. This is a reasonable assumption since steam turbines are constant power machines, 1601 Chryssostomtdts TABLE 2 RPM - Power - Efficiency Curves for a Typical Steam Turbine RPM Power Efficiency [fraction of design RPM] [fraction of full Power] 0.60 0.216 0. 766 0. 65 0.275 0. 807 0.70 0.343 0. 845 8 7) 0.432 0. 881 0. 80 GC. 51Z 0. 913 0. 85 0.614 0. 941 0.90 On 129 0.965 G.-92 Onno 0. 973 0. 94 0. 831 0.979 0. 96 0. 885 0. 984 0.98 0. 941 0. 988 1,00 1.000 0. 990 TABLE 3 Ship and Propeller Principal Characteristics CASE A CASE B LBP [ft.] 529. 00 666, 50 LWL [ft.] 538.15 678. 03 B [£t.] T5157 95.21 T [£t.] 25. 19 31.74 D at Amidships [ft.] 45. 00 56. 70 D: at Station 1’ [/£t-} 55. 50 69. 94 CB 0. 650 0. 650 CP 0. 661 0. 661 A [long tons] 18700 37400 Wetted Surface [sq. ft.] 48510 77000 dr free.) 17, 50 22.50 Pitch/d 1,00 1, 00 1602 Seakeeptng Constderations in a Total Destgn Metholology II. 3. Example The two ships and propellers whose characteristics are given in Table 3 are the two cases analysed in the present study. Ships A and B are geometrically similar and ship B has twice the dis- placement of ship A. The calculations performed in this section are the calculations that the designer would have to perform in a typical sampling cycle of the solution of a problem where the unknowns are the vessel size and speed of a fleet of ships that will satisfy a pres- pecified transport capability and optimization criterion. Figures 3 and 4 give the EHP vs. speed and sea state curves for ships A and B. The EHP vs. speed curve with the traditional 25% allowance is also shown dashed for comparison purposes. It is of interest to note that in the speed range of practical interest the EHP curve with the 25% allowance is almost identical with the EHP curve for H'3 = 7,94 ft. The SHP, assumed for this study is determined from the value of EHP with 25% margin using a propulsive efficiency equal to 0.75 as suggested in Reference 5. This was done in order to be able to compare the results of the present study with the results that would have been obtained if seakeeping considerations were not in- cluded in the analysis. The values of SHP, used in the present study were 18000 for ship A and 37400 for ship B, both of which cor- respond to a speed of 20 knots. The results of the speed calculations are shown in Figures 5 and 6. From resistance considerations alone, the average speed for ship A is 18.85 and for ship B is 18.81 knots. When motion con- siderations are included the average speed of ship A is reduced to 18.62 knots while the speed of ship B remains unaffected. The speed reduction for ship A was primarily due tothe wet- nesscriterion, Slamming considerations yielded restrictions which were slightly less binding than wetness while vertical acceleration consider- ations were not binding. For ship B the motion considerations were not atall binding andif the shiphad more power available it could goata higher speed. : The ship speed at the low sea states was computed to be lower than 20 knots even though the EHP value computed with the method suggested in the previous section is about the same as the value of EHP computed using the method suggested in Reference 5 This is due to the fact that the propulsive efficiency computed in the 1603 Chryssostomtdts y H*? 26,.45%t. Y, H >=. 27)-)6fe. wn e Hilo omonlette A aeso Ray: Hn) = MO 58Ee, i “ask Olav ee: 25% Margin Vv [knots] . Figure 3 EHP curves for ship A 1604 Seakeeping Constderattons in a Total Destgn Methodology BoHioP-. H Vs oy #21, 168. a OSB lee. BGreg vac Hw’? =10. 58£t. H? = 7.94 ft. Figure 4 26. 45¢€t. 25% Margin V[(knots] EHP curves for ship B 1605 Chryssostomtdts Average Speed. No Motion Average Speed. With Motion V [knots] Speed, Resistance only Py \ \ \ 10 ; Speed, Resistance Ye 9 ‘ and Motion a 8 “* \ Y, \ 7 H [fe.] et

) 40 0PES Hydrodynamic coefficients of a bulbous section 1610 Seakeeptng Constderattons tn a Total Destgn Methodology The computer programs described in References 12 and 15 employ a two parameter conformal representation for the hull sections. This has been found to be very satisfactory when compared to the close fit representation. Figure 7 from Reference 18 shows that the close fit representation approaches the results of the two parameter representation when the number of points employed in the close fit representation is increased (i.e. its accuracy is increased). However as the number of points is increased in the close fit representation the expense also increases and for 40 points it is prohibitive. The limitation of the two parameter fit is that all calculations are perform- ed with the transformed sections and not with the original sections. This however is not considered important because good two parameter description of regular sections and of sections with moderate bulbs is presently available. In addition moderate geometrical changes in the section shape do not affect the seakeeping results. In any case ifa section is to be described accurately in the close fit representation, especially bulb sections, a large number of points is necessary which make it prohibitively expensive. The expense for the use of seakeeping programs can be con- siderable especially if a complete investigation is to be made, In an attempt to overcome this limitation the authors of Reference 3 are extending the notion developed therein and are currently working ona scheme in which the Hydrodynamic properties of a section is stored in a matrix as a function of the two parameters describing the section, A and o , and the non-dimensional frequency 6 . Although the work is still underway, it is expected that a sparsely populated matrix will provide all the accuracy necessary in the second iteration of the proposed design methodology. This will permit the designer to perform the analysis suggested above at almost no cost at all. Before concluding this discussion the author wishes to take this opportunity to suggest research in the area of viscous roll damp- ing under speed and with bilge keels because the state of the art in this area is not satisfactory. fice hind Iteration In the final iteration the author suggests the use of seakeep- ing experiments for the selection of these parameters whose effect cannot be predicted by either models described above such as for example the above water hull shape. In addition these experiments can serve as a confirmation of the prediction made with the seakeeping computer programs especially in the areas where the theory is weak, for example in the prediction of power. The author recognises that seakeeping experiments are time consuming and expensive and there- 1611 Chryssostomtdts fore in designs where previous experience has demonstrated that sea- keeping performance is satisfactory seakeeping experiments should be omitted. However where no previous experience is available, as in the case of a novel design, such experiments are highly recommended. IV - CONCLUSIONS From the results presented in the previous sections the author has concluded that it is advantageous to incorporate seakeeping consi- derations in preliminary ship design optimization programs because of the potential payoff. However seakeeping considerations should only be included where it is meaningful to do so for example when the speed and size of ships are variable. They should not be included when only small changes in the principal characteristics of the ship are contemplated as they will not affect the final outcome. Special attention was drawn to the case of novel designs where seakeeping can be the controlling factor in the feasibility of the system. In this case seakeeping must be considered at the outset of the study. Although the state of the art permits the incorporation of seakeeping considerations in the design of monohulls improvement in the theory in certain areas will be worthwile as it will permit a better analysis. In particular, improvementin the theory to permit better predictions for added resistance, propulsive efficiency and viscous roll damping is considered worthwhile. In addition a better definition of the motion indices is necessary. NOMENCLATURE B beam CB block coefficient EHP effective horse power Fn Froude number g acceleration of gravity Hl/3 significant wave height J advance coefficient K, thrust coefficient 1612 Seakeeptng Considerations tn a Total Destgn Methodology LEE length between perpendiculars LWL lenght on designed waterline RMS root mean square value SHP | available shaft horse power SHP | required shaft horse power + draft Vv ship speed 8 non dimensional frequency ( 6 = Dae / 2,8) ACF correlation allowance ( ACF = 0.0004) € broadness factor IN half bean to draft ratio for each section p specific density of salt water o sectional area coefficient for each section w circular frequency REFERENCES 1 MANDEL, P. and CHRYSSOSTOMIDIS, C., 'A Design Me- thodology for Ship and Other Complex Systems'', London, England, Royal Society, 1972. 2 SNAITH, G.R. and PARKER, M.N., ''Ship Design with Com- puter Aids", Transactions of the North East Coast Institu- tion of Engineers and Shipbuilders, vol. 88, 1972, p. 151-72. 3 LOUKAKIS, T. and CHRYSSOSTOMIDIS, C., ''The Seakeep- ing Performance of an Extended Series 60" , to be published as a Report by the U.S. Maritime Administration. 1613 10 it 12 13 Chryssostomtdts TODD, F.H., ''Series 60 : Methodical Experiments with Models of Single-Screw Merchant Ships'', David Taylor Model Basin, Report No. 1712, Washington, D.C. : U.S. Govern- ment Printing Office, 1963. MANDEL, P. and LEOPOLD, R., ''Optimization Methods Applied to Ship Design"', Transactions of the Society of Naval Architects and Marine Engineers, vol. 74, 1966, p. 477-521. HOGBEN, N. and LUMB, F.E.,''Ocean Wave Statistics", Ministry of Technology National Physical Laboratory, Her Majesty's Stationary Office, London, England, 1967. LOFFT, R.F., "I. T.T.C. Wave Spectrum - Slope Parame- ter'', Proceedings of the 12th International Towing Tank Conference, Rome, 1969, p. 779-80. HADLER,.J.«B.5.,WLLSON, C.J. and BEAL, A. Iu, 0 sue Standardization Trial Performance and Correlation with Model Predictions'', Transactions of the Society of Naval Architects and Marine Engineering, vol. 70, 1962, p. 749- 807. "Principles of Naval Architecture'', 2nd Edition, Edited by J.P. Comstock, Society of Naval Architects and Marine Engineers, New York, New York, 1967. VAN LAMMEREN, W.P.A., VAN MANEN, J.D. and OOSTERVELD, M.W.C., ''The Wageningen B-Screw Series", Transactions of the Society of Naval Architects and Marine Engineers, vol. 77, 1969, p. 269-317. DREWRY, J.T., ''Vertical Acceleration of Ships in Irregular Waves and Associated Motion Sickness", S.M. Thesis Massachusetts Institute of Technology, Cambridge, Mass., 1966. LOUKAKIS, T.A., ''Computer Aided Prediction of Seakeeping Performance in Ship Design'', Cambridge, Mass, : Massachusetts Institute of Thechnology, Department of Naval Architecture and Marine Engineering, Report No. 70-3, 1970. MARKS, W. etal., ''An Automated System for Optimum Ship Routing", Transactions of the Society of Naval Architects and Marine Engineers, Vol. 76, 1968, p. 22-55. 1614 14 15 16 ie 18 Seakeeptng Constderattons tn a Total Destgn Methodology ZUBALY, R.B., ''Causes and Extent of Lost Time at Sea for Dry Cargo Ships", Technical and Research Report No. R-10 of the Society of Naval Architects and Marine En- gineers. CHRYSSOSTOMIDIS, C. and LOUKAKIS, T.A., ''The Sea- keeping Performance of a Ship in a Seaway'', To be published as a report by the U.S. Maritime Administration. SALVESEN, N.; TUCK, EO: and FALTINSEN, ©., "Ship Motions and Seal Loads'"', Transactions of the Society of Naval Architects and Marine Engineers, vol. 78, 1970, Dn ae aa0 be MANSOUR, A., ''Methods of Computing the Probability of Failure Under Extreme Values of Bending Moment", Journal of Ship Research, vol. 16, number 2, 1972, p. 113-23. LOUKAKIS, T. and CHOO, K.Y., ''A Reappraisal of Two Parameter Representation of Ship Sections for Seakeeping Calculations", Report to be published by the Massachusetts Institute of Technology, Department of Ocean Engineering. 1615 Chryssostomtdts APPENDIX I. SAMPLE SEAKEEPING TABLE Definitions of the quantities that appear in the sample Sea- keeping Table. Froude No. Ship Speed/ V g(LWL) Non-dimensional Sea State Significant Wave Height/LBP Heaving Motion RMS Heave Amidships/LBP Pitching Motion RMS Pitch in Degrees Bending Moment (RMS Bending Moment Amidships) 10? Pg (LBP)* Added Resistance (Mean Added Resistance) 10°/pg(LBP)? 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I should like to suggest that the paper does not live up to its title. The problem of seakeeping is both one of tremendous scope and the preeminent hydrodynamic one facing the ship designer, but the reduction of such a problem to the determination of the speed lost in head seas of the Pierson-Moskowitz type, having a height of roughly 8 ft., is, I think, something too brutal. To try to predict the seakeep- ing characteristics of the ship in such a manner is like trying to pre- dict the behaviour of all ladies (and some non-ladies) by that of one's own wife. I should like to think that one might be in for some surprises. To develop the proper perspective note that for over a century the ship designer has equated seakeeping to transverse metrocentric height. It is obvious that one parameter is insufficient to cover all the sins and virtues of a ship, but one cannot overlook the experience of past years, which indicates that of all the aspects that of roll is by far the most important. The greatest motion is that of roll, and the greatest danger comes in roll. But the only time the author mentions the work roll is when he speaks of recommendations on viscous roll damping research, and even that recommendation is unsupported by anything in the text. This I find to be a disturbing omission. Of the three iterations, only one is developed, and that one partially, and the other two are only hinted at. The first iteration is simply a recommendation that the seakeeping behaviour of a ship is assessed in some approximate manner by how a series 60 ship behaves when fitted with a Wageningen B.4.55. propeller. The series 60 was designed to find out how resistance varied with pertinent parameters. But the tests undertaken certainly did not cover adequately the para- meters of form that are important for seakeeping. In fact, Dr. Todd and his collaborators did not even think about seakeeping in those days. But that I mean to say that series 60 good as itis for estimating the resistance of ships of normal form, it is not necessarily relevant to determining the seakeeping characteristics of any ship that a designer might have in mind ; and indeed the relevance of series 60 is nota point to be assumed but a point to be approved, if anything, and I do 1619 Chryssostomtdts not think this can be done, or done very easily. To remain with series 60, the work of Vosser on the motions of this series is totally ignored, In fact, if I study the references I find something strange : a prepon- derance of them are from authors at M.I. T. Now 1am an alumnus of M.I. T. and am very proud of the work that is being done there but I think that the paper is a bit parochial and I have the feeling that it is essentially an advertising brochure for the work done there, DISCUSSION Raymond Vermter Naval Shtp Research and Development Center Bethesda, Maryland, U.S.A. This interesting paper represents a rather ambitious exten- sion of our normal ship design process and I should like to discuss several points made by the author and to advise some caution in the use of the procedure. First, the author states that for the particular example given in the paper, fully developed long crested sea spectra are used and that this simplification is not considered unrealistic, It would appear from some of yesterday's discussion regarding sea spec- tra, however, that this assumption is unrealistic, that the fully risen case is rarely realised and that swell forms an important component of most seaway forcing functions. That the consideration of swell is an important factor for the successful accomplishment of naval mis- sions has been painfully demonstrated to us at the Naval Ship Research and Development Center several times in the recent past and it is never accounted for through the use of standard spectra. It is suggest- ed that multi parameter direction spectral considerations are neces- sary for a valid design study of this type. One might also question the suggested number of iterations concerning the establishment of an EHP value for propeller selection and, even more generally, the real importance of power limitations as regards ship operation ina seaway. It has been our experience that naval ships are never power limited, but that power is voluntarily li- mited from a fear on the part of the captain that he will cause either personnel injury or structural damage if he drives his ship harder. Propeller design considerations for the Naval ship case involve pro- per blade stressing and vibration considerations for maximum power and cavitation free operations to as high a speed as possible, trading 1620 Seakeeptng Constderattons in a Total Destgn Methodology these factors off against the propeller efficiency. So it would seem that improper assumptions have been made in this paper regarding propeller selection, Even in the case of the merchant ship design it is hard to con- ceive of a ship being power limited in a seaway prior to arriving at the voluntary limitation point. Here in propeller design we again con- sider stress and vibration characteristics for full power, but design for maximum efficiency at 80 percent power. While 80 percent is an arbitrary number arrived at through experience, it is a design point of some significance to the ship owner. It represents a real value around which full scale propeller performance can be evaluated and indicates whether the propeller is on or off design. This performance determines payment or financial penalty for the designer. It appears that the proposed procedure may tend to shroud this well defined de- sign point in statistical vagueness. REPLY TO DISCUSSION Chryssostomos Chryssostomidis Massachusetts Instttute of Technology Cambridge Massachusetts, U.S.A. Answering Dr. Saint-Denis first. The thesis of my paper is that in the first iteration of the proposed methodology, once the user decided to adopt a cruiser stern monohull as the solution to his pro- blem, Series 60 ships form as good a basis as any other standard series or for that matter as any other realistic point designs to pro- vide the information needed to make the decisions called for in the first iteration of the proposed methodology. The small variations bet- ween the final design and whatever standard form was used do not in- fluence the outcome of the first iteration and their effect need only be considered as it is suggested in the paper, in the subsequent itera- tions of the proposed design methodology. Roll was not included in the present study, as it was stated in the paper, not because it was considered unimportant but because the state of the art is such that does not permit theoretical prediction of roll with any reliability and therefore a creation of a theoretical de- rived standard series for roll was considered unwise. 1621 Chryssostomtdts Finally answering the comment about the optimization criterion. I do not believe that my considerations were limited to an environment having a wave height of roughly 8 ft. In answer to Dr. Wermter, I recognise that the environment description adopted in the present study was rather limiting. What is of importance is to recognise that the proposed procedure can accom- modate other standard environment descriptions which are consider- ed to be more suitable. I believe, however, that in the first iteration the environment description must be kept as simple as possible, as long as it is realistic, in order to allow the user to investigate the large number of alternatives called for by the proposed methodology and that the appropriate place to use the exact environment descrip- tion is in the second iteration. Answering the question about power. The proposed methodolo- gy treats the available power as an independent variable and permits the user to determine the "'optimum'"' power that one must install ina ship operating in the "'actual'' environment taking into account both voluntary and non-voluntary speed reductions. Answering the question about detailed propeller design. This, in the proposed methodology, is treated in a subsequent iteration. In the first iteration it is necessary to accept a standard but realistic propeller design, in order to be able to examine as early as possible the influence of seakeeping considerations in the decision making pro- cess. DISCUSSION Reuven Leopold U.S. Navy. Naval Shtp chet AIM Center Hyattsville, Maryland, U.S.A When I saw the title of the paper, ''SSeakeeping considerations in a total design methodology", I was looking forward with great inte- rest to the paper itself, which I must admit is rather disappointing. There are several reasons for my disappointment. The first is because I read with great interest and enthusiasm reference 1 ''A design metho- dology for ship and other complex systems" by Professor Chryssosto- midis and Professor Mandel in London earlier this year, which I feel 1622 Seakeeptng Constderattons tn a Total Destgn Methodology laid a very good foundation and framework for a sophisticated ship design methodology, which we badly need, and I thought this might be a second instalment in that direction. The second reason for my disappointment is because I had made repeated attempts to incorporate seakeeping considerations into early design decisions, as this paper repeatedly points out by referr- ing to reference 5 and while I feel that in reference 5 Professor Mandel and myself have described the principles of sucha step, we found that it did not influence those early decisions, The third reason is that because of the latter experience I thought that maybe this paper would show how seakeeping considera - tions can influence early gross design decisions. Unfortunately, the paper does not achieve this objective. In fact, in the conclusions chapter it is even stated: ''The results also suggest that when opti- mization does not involve large changes in the principal characteris- tics of the different alternatives considered, as in the case of ship with constant payload, then seakeeping considerations should not be included in the optimization scheme because they are not expected to influence the final decision". But after all, the normal ship design problem is posed in such a way as: ''Transport or carry a certain payload (say with some future growth and convertibility for a warship) with a certain maximum speed and endurance optimized against some criterion''. Therefore to say what the author has stated in the conclusions part of the paper, which I quoted earlier, seems to me to cut out the majority of design situa- tions. Thus accounting for seakeeping indices, such as acceleration in certains locations along the ship, slamming, wetness and added powering in waves versus not accounting for them in the normal ship design case will not have a significant effect on the gross ship dimen- sions, Iam obviously not referring to a whole host of ship hull cha- racteristics suchas LCB-LCF locations, sheer and freeboard, the detailed design of the hull form itself and the selection of various sta- bilization systems. What Iam referring to, is this : given the mathe- matical model and superimposed optimization technique, reference 5, the existence of a subroutine for seakeeping indices considerations would not result in, say, a 10-20 percent change in gross ships cha- racteristics of a conventional monohull design. There is a big difference between being able to predict the per- formance of a vessel, which is important, versus changing the signi- ficant gross ship characteristics as a result of considering or not con- 1623 Chryssostomtdts sidering seakeeping characteristics. REPLY TO DISCUSSION Chryssostomos Chryssostomidis Massachusetts Instttute of Technology Cambrtdge, Massachusetts, U.S.A. Thank you, Mr. Leopold ; I fully agree with your conclusions in the case of conventional ships with conventional missions, If there are not going to be large changes in the owner's requirements I do not believe it is profitable to include seakeeping considerations in the op- timization scheme of the first iteration of the proposed methodology. For unconventional ships however, I believe seakeeping considerations should be introduced as early as possible in the decision making pro- cess because they might determine feasibility. We have a recent example of this in the form of a small catamaran vessel. I also be - lieve that in unconventional missions even with conventional ships one must introduce seakeeping considerations as early as possible in the decision making process because they might influence the final solu- tion. Recently I was involved in the design of a deep ocean mining ship where seakeeping considerations forced me to accept as "optimum" a much larger ship than I would have accepted if no seakeeping conside- rations were introduced in the investigation, DISCUSSION : Michel K. Ochi Naval Shtp Research and Development Center Bethesda, Maryland, U.S.A. The author has brought up an important subject in the first iteration of the proposed design methodology, namely, the speed re- duction due to ship motion. As part of the criteria to estimate the speed reduction due to ship motion, the author considers the probabi- lity of occurrence of deck wetness at Station 1 and the probability of 1624 Seakeeptng Constderattons tn a Total Destgn Methodology slamming at station 2. It is understood that these two probabilities are considered to be independent in the author's mathematical model. I would like to point out that these two probabilities are both a func- tion of relative motion at stations 1 and 2, respectively. Since these two relative motions are highly correlated, the two probabilities can- not be treated as independent. The correlation coefficient for relative motions at any two forward locations on the ship is usually of the order of 0.7 to 0.8 depending on ship speed. I would like to suggest that the evaluation should be based on the joint probability function of the two relative motions taking into account the correlation between them, REPLY TO DISCUSSION Chryssostomos Chryssostomidis Massachusetts Institute of Technology Cambrtdge, Massachusetts, U.S.A. Thank you, Dr. Ochi, for your recommendation, I will look into this. DISCUSSION Edmund V. -Telfer iitelace Vier Ate Ewell, Surrey, U.K. In my very early professional life this was a subject to which I devoted probably far too much of my time but I now find it rather sad, that although most of that work was probably published before the author was born, it has had practically no influence upon his think- ing. I do not mind this, but I do regret that the author makes no re- ference to the work of Professor Aertssen, and [ think anybody attempt- ing a thesis of this nature who has not carefully studied, and profited, I hope, by the work of Aertssen - and still earlier the work of Kent - will not see the many issues involved as clearly as he ought to inl1972 AD. 1625 Chryssostomtdts It is by no means necessary to go into the esoteric detail that the author does, Very much simpler and reliable information can be obtained by straightforward consideration of the problem as the owner himself sees it, and I would recommend the author, if he has time, to look at some of the earlier work that has been published in the subject. Personnally, I find it quite tragic, looking over the references to many papers in this symposium, to find that very little of the work goes back more than five years. I do stress to my younger colleagues that quite a lot of good work was done before five years ago, and nobody will hold it against them if they refer to and see good in the work done 30, 40, 50 or even a hundred years ago. I still hope, however, despite this criticism, that the author will continue with his subject, gain perspective and, by so doing, add to professional information on the subject. REPLY TO DISCUSSION Chryssostomos Chryssostomidis Massachusetts Institute of Technology Cambridge, Massachusetts, U.S.A. Thank you Dr. Telfer. The list of references included in my paper is not a complete list of all the material consulted. DISCUSSION Edmund Lover Admtralty Experiment Works Haslar, Gosport, Hants, U.K. For some time it has become apparent that a systems analysis including seakeeping considerations is necessary for modern ship de- sign, we should be grateful to the author for demonstrating how these might be taken into account. 1626 Seakeeptng Constderattons tn a Total Destgn Methodology Such an analysis does however require a quantification of per- formance parameters that are very difficult to quantify. The author has bravely stocked this problem but I wonder whether his assump- tions concerning the effects of overall size on hull efficiency elements are not oversimplified. Also for instance SHP,is not always equal to EHP/1, - a paradox that is discussed at length within the I. T. T.C. and elsewhere. The example given is particularly interesting, not only asa demonstration of the method, but also for the result obtained. In this case, the ship with twice the displacement is shown to be better able to maintain speed ina seaway. This result is not surprising. What is surprising however is that the difference appears to be so slight. Here is an example of two vessels, designed for the same speed but with one having twice the displacement, and hence twice the payload of the other. A comparative through costing of performance would therefore need to compare the profitability of one ship versus two and would involve an assessment of profound differences of de- ployment, availability, and manning, as well as first cost. I suggest that these outweigh the seakeeping speed and fuel effects to an extent that one can conclude that these may be ignored when determining the overall size of a large merchant ship design, even when large changes are possible. In other words, the first ite- ration is redundant in such a case and efforts should be concentrated on evolving detailed design improvements - incorporation of adequate freeboard, suitable bow sections to avoid slamming and so on. I am not however suggesting that a systems approach including seakeeping is always unnecessary. Such an approach is appropriate for smaller vessels and is indeed vitally necessary for the design of small warships, where other considerations of crew operation and weapon deployment become dominant, REPLY TO DISCUSSION Chryssostomos Chryssostomidis Massachusetts Instttute of Technology Cambridge, Massachusetts, U.S.A. Thank you Dr. Lover. I agree with all your conclusions. 1627 Ty, -raiy to obihe te mmaspenipaianersaiunvindgesibn tiie niocewlbent Torristerny asthe Rite wes eo okraas ipieraeet 22 iM MmestatwoD tetensioorgy ahead tori vag >is: Ahad wreviba teetiarsuol ‘o> Srbbee, Ap dunes sues eh ai PH sm ketid Ss Ch ediry “ben ede heel DY si oRaD pdt chin lpptre {ns Huse fepeikery ) eeebt foods raedey ie mineore i ie, ey De MVTe toe | a? 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Sargent and Theodore R. Goodman Oceantes, Ine Platnvtew, New York ABSTRACT An important problem associated with establishing a mathematical model that adequately represents the motions of a naval vessel is the question of the pro- per form of the equations, as well as the values of the various parameters entering the equation system. A technique for determining the ability of a particul- ar mathematical form to represent the motions of such a vehicle, together with the determination ofthe numerical values of various parameters(such as sta- bility derivatives, etc.) is carried out by application of the technique known as system identification. The method of system identification is used in this con- text for the means to determine the unknown para- meters in a dynamical system representation from measurements of the time histories of the vehicle trajectories. Different techniques are used for ap- plication to problems that are of transient ncture, following a sudden disturbance or control deflection in a smooth seaway, and for those problems associ- ated with the motions ofa vehicle in a disturbed sea- way where the motionis continuously forced ina ran- dom manner. In addition the influence of noise in its generalized effect asa source of measurement error is also considered in this work. These techniques have been successfully applied to the determination of the stability derivatives (and nonlinear function coefficients) of a conventional surface ship, a hydro- foil craft, and an SES (surface effect ship) craft. 1629 Kaplan, Sargent and Goodman These applications have included vehicle trajectories obtained from computer-generated data, as wellas full scale data. The utility of the techniques is de- monstrated by the results obtained for these appli- cations, together with a discussion of limitations in different towing tanks throughout the world for this purpose, ranging from multi-component balances for staticforceand moment measurementyotating arms, horizontal and vertical planar motion oscillator me- chanisms, etc. (see [1] - [2]). Considering the com- putational and data reduction equipment required as ancillary elements of the measurement devices, as well as the time and expense required to obtain the required parameter values by these means, other methods that may reduce the effort required for de- termination of hydrodynamic coefficients then become attractive(especially tolaboratories or organizations that do not have such involved instrumentation), A particular approach to determine the values of va- rious parameters in a mathematical representation of the dynamics ofan arbitrary system (whether it is a vehicle, a chemical process, control system, etc.) has been developed recently as part of modern con- trol theory. This procedure is known as ''system identification'’, which in the present case is a means of determining the numerical values of the coefficients that enter into a set of mathematical equations that are assumed to represent the dynamic motions of a particular vehicle or system (in addition the proce- dure can also determine the suitability of a particu- lar mathematical model form as well as the sensiti- vity of different modes of motion to particular coef- ficients, as will be demonstrated herein). These va- lues are considered to be the appropriate values re- presenting the system dynamics when they are ob- tained with the same values from a number of differ- ent trajectories of the vehicle motion, thereby insur- ing their uniqueness, The extent of realism inherent in the coefficient values is related to the capability of the mathematical equation model to represent the vehicle motions, since the results of captive model tests in wind tunnels, water tunnels, towing tanks, etc. are aimed at measuring such coefficients ex- perimentally, where the coefficient structure is based 1630 Dynamics of Naval Craft - System Identification upon the form of the equations that will be orare as- sumed for representing the motion of the system. Therefore the basic foundation underliying system identification as a means of representing the vehicle dynamics has (at least) the same degree of validity as any method of dynamic analysis that is presently used as a model of vehicle behavior. What is done in this technique is to obtain responses of a vehicle by measuring "'trajectories'(such as ve- hicle linear and angular displacements, velocities accelerations, etc.) following different types of dis- turbances. With the formulated mathematical model, values for the unknown parameters are then sought so that the solutions to the dynamic equations give a best fit to the data, where this best fit is defined by minimizing the mean square error between the solu- tion of the equations using these coefficients and the actual data record itself. The procedure can be ap- plied to data from both full scale and model scale trajectory observations, thereby increasing its uti- lity for correlation and validation purposes of par- ticular mathematical simulations of naval vehicles. Obtaining stability derivatives from full scale tra- jectories has been standard practice in the aircraft industry from its inception, and this has ordinarily been done by various means of data analysis that are primarily based upon the assumption that the equa- tions arelinear.In addition to considerations of non- linearity which are important for certain naval craft the sensitivity of some modes of motion of particular craft (such as hydrofoils, SES craft, etc.) to surface wave disturbances requires consideration of the in- fluence of random forcing functions, sensor errors and other ''noisy'' disturbances applied to the system and its measured motion responses. These different effects then require particular techniques for their analysis, and the present paper will provide a des- cription of the analysis methods as well as the re- sults obtained when the methods are applied to dif- ferent representative naval craft. The work described in this paper was carried out under the support of different agencies in the course 1631 Kaplan, Sargent and Goodman of various study contracts, viz. the Office of Naval Research, the Naval Ship Research and Development Center, and the Surface Effect Ships Project Office. INTRODUCTION In order to predict the dynamic performance of various naval vessels, different methods of simulation are often employed. The general term ''dynamic performance" used here refers to the deter- mination of dynamic stability, maneuvering and turning properties, response to control input commands, motions in wave systems, etc; the term ''simulation"' includes both free model trajectory tests as well as the use of computer solutions based on a mathematical model that is assumed to represent the craft motions. Considering the basic limits of free model testing, which are associated with limitations in the size of models and/or test facilities (or inherent limits such as in the case of submarine maneuvers), propulsion and control model- ing errors, time constant differences, etc., the major emphasis for motion prediction is the use of computer simulation using mathema- tical models. When considering prediction and simulation studies of ship dynamics with the use of a mathematical model, a vital aspect is es- tablishment of the proper form of the equations as well as the appro- priate numerical values of the various parameters (coefficients, stability derivatives, etc.) entering the equation system. At the present time the main method of determining the various hydrody- namic force and moment coefficients in a desired mathematical model for a particular type of marine craft is by means of captive model tests in a towing tank, together with the associated mathematical analysis of the experimental data in order to provide the required coefficients, Various special purpose apparatus exist. GENERAL DESCRIPTION OF ANALYTICAL TECHNIQUES When considering a vehicle in an undisturbed smooth water environment, transient responses of the craft are excited by means of different initial conditions or excitation inputs (such as a rapid rudder deflection or other impulsive disturbance). The measured outputs (i.e. vehicle motions) are recorded and operated upon by a technique that is essentially a generalization of a Newtonian iteration procedure [3] . The differential equations of motion of the vehicle, whether itis linear or nonlinear, are used together with additional 1632 Dynamtes of Naval Craft - System Identtficatton variables that represent the unknown coefficients in these equations. The coefficients themselves are the actual variables that are sought in this system identification procedure, and different techniques are used within the course of the analysis with the understanding that the variables desired are the coefficients in the equations. Solutions are necessary for all the variables starting with estimated initial condi- tions, where the variables include the state variables of the system as well as the coefficients themselves. Errors between the calculated state variables and the actual measured trajectory data itself are determined, and then modifications of the unknown coefficients are obtained in this procedure. These new values are then inserted again, solutions obtained, modified coefficient values found, and these are inserted again with the method repeated, i.e. an iterative procedure. The main features of this method are the fact that the basic dynamic system itself can be nonlinear (in terms of the state variables) and it is not necessary to measure every response variable in order to obtain the values for the coefficients. Even in the case of a linear system, if each and every response variable, including displacements, velocities, and accelerations of all degrees of freedom are measured, then the only unknowns are the coefficients themselves which can be obtained from solution of a set of linear algebraic equations. However it is often difficult, if not impossible, to measure every variable, derivative, etc., as well as the fact that often such measurements are not very accurate due to instrument limitations. The technique applied here requires selecting just those variables that are easiest to measure and which are available, but nevertheless a certain num- ber of variables must be measured since in a coupled system more than one mode of motion applies; e.g. as an illustration, it is neces- sary to obtain measured data on yaw and roll responses since measur- ing a single mode such as yaw alone would not yield sufficient data to obtain information on roll coefficients, and vice-versa. The original derivations in [3] presented a method for deter- mining unknown parameters in an otherwise known dynamic system using only measurements of the time history of just one state variable. However, practical experience with large systems containing a number of degrees of freedom and many parameters led to a generalization of the procedure involving the use of an increased number of measured trajectory records, (as mentioned above), and this improved proced- ure overcame many difficulties in regard to convergence and unique- ness of the results. A number of applications were made to different vehicles, including aircraft, a surface ship, and a hydrofoil craft, and the results obtained are described in [4] and [5] . A description of the mathematical procedures, and a discussion of results obtained 1633 Kaplan, Sargent and Goodman by use of this method, are given in following sections of this paper. When considering the case of a craft in waves, the effect of continuous random forcing functions (due to the waves) is present. Therefore another technique different from the one used in the pre- vious work described above, which is based on transient responses with no ''noisy'' measurements or random forcing functions, must be used. The method proposed for application to this problem is based on developments in recent literature of modern control theory (maximum principle, two-points boundary value problems, invariant imbedding, and sequential estimation) which are described in [6] and [7] . The basic technique is applied to problems that are gener- ally nonlinear, with the possibility of measurement observation errors and with unknown random inputs. Using continuous time his- tories of the observed output measurements, the task is then to obtain optimal estimates of the state variables and also various parameters in the equations (such as coefficients and other unknown magnitude mathematical forms) by a procedure that is based on minimizing an integral of the sum of weighted squares of residual errors. The errors are the difference between the observed data and the actual desired system outputs (i.e. eliminating the measurement noise), and also the difference between the nominal trajectory of the system and the assumed form of the equation representation (i.e. eliminating the noisy input excitation and achieving a proper repre- sentation of the basic system dynamics). In this case, the unknown parameters are also added as additional variables in the complete dynamic representation. The equations that result for the estimates of the system state and also for the parameters provide an on-line filtering pro- cedure together with a sequential estimation technique, which does not require repeating all calculations after additional observations or measurements are made, as in classical estimation schemes. The resulting equations are of a form that is somewhat similar to that of the Kalman filter [8], but they are applicable to nonlinear systems. In addition the terms entering the equations are not dependent upon a knowledge of the statistical characteristics of the input disturbances or the measurement errors, thereby allowing consideration of vehicles in arbitrary seaway conditions and hence increasing the generality of the approach. The equations developed for this system identification pro- cedure use the continuous measurements of the actual system outputs as observed, and those signals are operated on and processed with the special estimator equations. As time evolves the combined fil- 1634 Dynamtes of Naval Craft - System Identtficatton tering action and identification allows the evaluation of the desired parameters to exhibit themselves as functions of time and arrive at their final steady value. Similarly the estimation of state variables with random disturbances present also evolves as a function of time, and the ability of the estimated state variables to ''track'' the measur- ed trajectories when using the estimated parameter values in the representative system equations is directly exhibited in this proced- ure. Applications of this technique that allows for the presence of "noise'' in the system response have been made for the case ofa surface effect ship (SES craft) as well as for a hydrofoil craft (see [9] and [1 0} ). The mathematical procedures underlying this par- ticular technique, as well as the results obtained in practical appli- cations to different seagoing craft, are described in later sections of this paper. MATHEMATICAL PROCEDURES - ITERATION METHOD The iterative technique used for system identification of dynamic systems for which transient response data is available is described by the following. The dynamical equations representing the system are assumed to be given in the form Me an bts AOE oEaete Ad) where the dot denotes differentiation with respect to time t, a denotes the unknown parameter vector and c denotes the initial value of the solution vector Y and may or may not be totally known. Measure- ments bd, =; bo, > «-.. ofthe state variables “Y, , Y¥>5 <1. St times tm are available, and it is required to find an initial vector c together with a parameter vector a which minimize the sum of the squares of the deviations : 1635 Kaplan, Sargent and Goodman wy py. SAS (EL OP SL A PERM (2) where the weighting factors w are chosen to make each sum non- dimensional and of the same order of magnitude. Thus, the solution of (1) is sought which is in best agreement with the measurements in a least square sense. The parameter vector, a , is suppressed in (1) by considering its components to be additional state variables subject to the equation a= 0 (3) The number n is thereby increased to include the additional state variables and the extended c vector includes the unknown parameter vector in addition to the state variable initial conditions. The parameters of the system are determined in the following way : the initial vector is estimated and (1) is integrated. The es- timated initial vector is denoted by c”* and the resulting solution of (1) by Y*. The deviation can then be calculated and its value denoted by e*. Assuming the initial vector to be changed by an increment 6c, this would cause the solution vector to be changed by an incre- ment 6Y and the deviation by an increment 6e . From (2) itis seen that M M, _ i= wi, by ai ly, (t_) sm bY, (tat... (4) igay = il The equations which the incremental solution vector safisfies are called the equations of differential corrections and are obtained by expanding (1) in a Taylor series and retaining only linear terms : n ad bY, (t) = ) 08; 8, (5) ait tay : j 1636 Dynamtes of Naval Craft - System Identtficatton (the asterisk means that the coefficients are calculated using solu- tion Y™*). Equation (5) is now integrated n times; the jth time the integration is performed the initial conditions are that 6 Y; (0) = 1 and all the other 6Yj (0)'s vanish. This special solution is denoted by 6Yij , and the general solution can then be written, by superposition, ‘as n = ¥ oi Gt bY, > bes 8 i; | ) (6) jel This incremental solution vector is used to express 46Y, , OMgiry . interms of 6c, and upon substituting into (4) and interchanging the order of summation the variation of the deviation becomes where the repeated suffix implies summation on j from 1 to n, The variation of the deviation has thus been expressed directly in terms of the variation of each of the initial conditions. In order for ¢€ to be minimum de must vanish for an arbitrary variation in the initial conditions. This means that if U; is defined to be Waa Saicing, ano y, (t_) “bn éY,, (ti bp Be ns (8) igayresioal thenthe error will be minimized with respect to the ae whenever. Car si Oy ape botaion (9) In general, using the estimated vector c* and the resulting solution vector y*, the values of Uy will not vanish. Denoting the value of Uj , as calculated in this way by u; , the objective is 1637 Kaplan, Sargent and Goodman to make the Uj; vanish by an iteration procedure. Considering the increment in Uj; caused by the incrementin Yj; , from (8) there is obtained 6U. = w >y Y(t) oY, (t_) tees (10) in order for each ys and hence §€ to vanish the condition ou, = sae (11) must be imposed. Upon substituting (6) into (10) and interchanging the order of summations there is finally obtained scat = al Equations (8) with Y = Y*, together with Equations (11) and (12), constitute n simultaneous linear algebraic equations for the n unknowns 6c; . Upon adding the incremental values to the estimated values of c; , improved estimates of the c, are obtained, and the procedure is then repeated until convergence is achieved. A modification of the above algorithm which at times is found to be useful is to introduce some or all of the b's into the right hand sides of Equations (1) and (6) in place of the respective y*'s. A digital computer program for the above procedure was established, and various guidelines evolved for its effective use. One of the problems associated with an iterative procedure is to achieve convergence, and this depends upon the compatibility bet- ween the mathematical model and the actual physical system as well as the ''quality'' of the initial guess of the unknown parameters. Even when these conditions are satisfied there are often cases where con- vergence does not readily follow, and different strategies are used. 1638 Dynamtes of Naval Craft - System Identtficatton Among these are using measurements of additional state variables, applying a gain on the 6c vector (to inhibit any over-correction), and to vary the length of the record in steps (from a short record to longer records, using the converged outputs of each step as the ini- tial guess for the next longer record length until the entire record is processed), Another problem is that of the proper values of the parameters, even if they produce trajectories that match the measured values quite well. Sometimes a parameter that has only a small influence on the particular motion data being analyzed is sought by the system identification technique. In that case very little information related to that parameter is contained in the data, and the value determined by this procedure is spurious (and could also sometimes ''contamin- ate'' other parameter values), Various means of increasing confidence have been developed as a result of experience. Among these are using as many state variable measurements as possible, as well as moni- toring the change in the estimates of the converged c* values as the record length is increased. Since lengthening the record introduces more information, the c* values should begin to settle at some value (i.e. to stabilize), after which no further record lengthening is ne- cessary. Another means of establishing confidence in the results is to vary the weighting factors. When a change in the weighting factors produces no apparent effect on the results, confidence in these results is increased, APPLICATIONS OF THE ITERATION TECHNIQUE The iteration technique described above was applied toa number of dynamic systems for which transient motion response data was available. The naval craft of interest that were treated by this method are a surface ship and a hydrofoil craft. Forthe case of the surface ship, the nonlinear three degree of freedom system of equa- tions describing the steering and maneuvering of that ship are a = f(u) + civ + Cyt m Eau 6.4 rv@ 2 ym C uv + Cpur + Cy en Cou 5 (13) 2 oo Gets Get sat Cun = +C,)u 6 1639 Kaplan, Sargent and Goodman where u = surge velocity, v = sway velocity, r= yaw rate, 6 = rudder deflection, f(u) = known function of u that represents the difference between thrust and resistance, C.= coefficients to be determined. These equations were obtained from (11), and the ''measured" input to the system identification program is provided by a maneuver generated on the computer with known parameters typical of a con- ventional cargo ship, as obtained from [1 1 . The maneuver chosen is a turning circle initiated by commanding a 35 degree rudder angle. Difficulty in converging to the known set of eleven coefficients was encountered with the basic computer algorithm even though the predicted time histories of u, v and r themselves became indis- tinguishable from their respective input records, Efforts to improve the results by sampling at a higher rate or by taking a longer record proved fruitless, and only by retaining an inordinate number of significant digits in the input data did the converged coefficients agree with their known values. However, the accuracy of data measured by real sensors is limited, and, the difficulty in obtaining good coeffi- cients was eliminated by applying the modification to the technique which used the measured state variables instead of the predicted values in the operations of Equations (1) and (6). Using this modi- fication, the effect of varying the sampling period, length of record, number of records, and accuracy of the data was investigated to indicate the measuring requirements for the identification of a real ship (or model) from a maneuver. It was found that a sampling period of from one to two seconds and a record length of from one-half to one minute was adequate for successful identification of a full scale ship. However, it was found necessary to have measured all three variables, namely u, v and r, to identify the eleven coefficients in Equation (13). Using the computer generated data of u, v and r accurate to four significant digits, which was sampled every second for one minute, the identification by this iterative technique was carried out. The resulting C-coefficients are shown in Table | together with their time values, where reasonably close values were found. The identi- fication process converged very rapidly, as only two iterations were required. 1640 Dynamics of Naval Craft - System Identification Table 1 Results of System Identification of Surface Ship True cco 2% 5% Analog Values Data Noise Noise Data Gyiy = 10° eee 03 =e 20? +1071 eeeliai Gs ir Ties : his . 109 -. 589 -1.64 Co 10° =j.gh2 114, suas -. 164 -.091 Chen 10° =optlAs -.149 -. 185 ae lag C, -. 334 oe = a0 -.415 =, 342 feeveilOnk ens 10 PSB 942 796 ae C, x 10° 323 bi PRD 321 .318 e7 10° Beuie2 ee: OT =.O12 2Aets ae Wor 1 397 391 =. 360 JygoM Lalgey ce 24.120 =e -. 097 -. 070 one ae 10° -. 584 -. 583 -. 586 -.591 In order to assess the accuracy requirements, the input was contaminated by adding Gaussian noise of 2 and 5% error magnitudes, where this error is based on the indicated percentage of the maximum value of the variable. The results of identification with these inputs are also shown in Table 1. It is seen that as the noise level is in- creased some coefficients remain close to their true values while others drift away and still others lose all significance. The coeffi- cients C, and C» are seen to deteriorate the most rapidly. These coefficients have been shown in [11] to be of minor importance, and the present results support the relative ordering given in fi] 1641 Kaplan, Sargent and Goodman Ship trajectory data was also generated on an analog computer, using only linear equations in v and r while assuming a constant forward speed. The equations in that case were given arbitrary initial conditions and then allowed to seek equilibrium. The v and r analog signals were sampled every two seconds by A/D convertors to pro- vide the input to the system identification program. The results using the data supplied from the analog experiment are also included in Table 1 and in that case, even though the data can at best be consider- ed to be 1% accurate, better values for the four coefficients consider- ed as unknowns are obtained than that predicted using the noisy di- gitally generated data. This is ascribed to the fact that more accurate values of unknown coefficients can be predicted for a simpler system than for a system with a larger number of unknown coefficients. It is also possible that real ''noise'' from the analog computer output, which is closer to true processed experimental data, may not be as severe as the artifically generated digital noisy data. For the case of a hydrofoil craft, the nonlinear longitudinal equations of motion for a typical hydrofoil craft under autopilot con- trol are : Normal Force Equation L L = F R ° . hi, S_ Sle 17 eases theecs Cc, We C, We (14) Pitching Moment Equation L L Bi . R Gian ey a - ao ; ~ Cc, ee | Fs + C, We (15) The lift on the forward foil is lp Wy Bits Caer whee = ae f Se Ge era * SHLERs Oe m 3 V 5 e + YY aT F 6 6 + 2 F + — Cy v (16) 1642 Dynamites of Naval Craft - System Identtficatton where Wee ae ha Vee. = Ee (17) Lg 2 Pom bap qysediiol., f ..0 fe) Similar expressions are given for the lift Lp on the rearward foil, where the unknown parameters C3 , Ce , or “ Cy are replaced by CC, 4 C, 3 or and Cy Autopilot Transfer Functions 5 bm De a P. # TS 9 * command (ri 6 cho) ‘a2 £ VY13 Irs +1 (18) save (h ) (ry s +1) command Shs)? in terms of the Laplace transform operator s, : PR a: (19) command Love 16 pe yt aoe Aes Sere iy Flap Actuation System Transfer Functions 1 6 = 6 5 le is e aie 7 command’ f | Atta Ww a) —_ -- easy (@) ° : - ies) iy) 3 = a where ha NeaNe, @ “el opiteh atigie, V = craft velocity (assumed constant), je aa Duis = lengths from c.g. to forward foil, real foil and height sensor, = be | " given constants, 1643 Kaplan, Sargent and Goodman Suso = vertical distance from height sensor to undisturbed water surface at design foilborne condition, s = Laplace transform operator, C. = unknown parameters. 1 To demonstrate the feasibility of using system identification to estimate the C's from full-scale tests, computer generated trajec- tories were first used as trial inputs. The sponsoring agency supplied the computer generated trajectories but withheld the values of the C's used in the generation until the results of system identification became known. Three sets of trajectories for each of the three different con- figurations, termed A, B, and C (i.e., for three different sets of C's) were supplied. A run in each case consisted of a step change in the commanded height after the craft had reached equilibrium. Alto- gether, there were 9 sets of trajectories supplied. Even though there was no prior information given as to the proper values for the C's other than the crudest order of magnitude estimates, no difficulty was encountered by the system identification program in converging toa set of C's for each configuration and run. Confidence in the converged results was investigated by varying the record length as described previously, and also by comparing the results from different runs of the same configuration. In all cases high confidence in the results were indicated. The estimates from system identification are given in Table 2, together with the true va- lues. Table © 2 Configuration True (Estimated) A B C C, .0450(. 0439) .0450(.0439) .0950(. 0942) Cc, 14842 (.434) Sheds 129) 161 129) C, .1794(. 179) . 1794(. 179) . 1906(. 190) 1644 Dynamics of Naval Craft - System Identtficatton Cc, Pero. a2 t) . 4446(. 445) . 4446(. 445) C, 1, 912(1. 97) 1. 912(1. 97) 2. 114(2. 18) Cc. . 5097(. 53) . 5097(. 53) . 5579(. 58) C, 2. 668(2. 67) 2. 053(2. 06) 1, 886(1. 87) C. . 4427(. 45) . 4427(. 45) . 5126(. 51) Cy 0 (0(.2)) 10,0. (10, 3) 0 (0(.2)) Ch (0G. 1)) 0 (0(. 2)) 0 (0(. 1)) The parameters Cg and C,, represent coefficients of nonlinear lift. Nonlinear lift appeared to play an insignificant role in almost all the trajectories, making it impossible to obtain firm estimates of these parameters. It was only possible to estimate the order of ma- gnitude of these parameters. This has been indicated in the table by use of an order symbol (as an example, the entry 0(.2) is to be in- terpreted to mean that the value of the parameter is no greater than +,2).In the case of Cg for configuration B, the nonlinear lift term had been artifically increased so that it played a more significant role in the trajectory, and was therefore detectable. The comparison with the true values in Table 2 shows a remarkable agreement between the values estimated by system identification and their respective true values, More detailed information concerning the results obtained for the case of a surface ship and a hydrofoil craft by means of this particular system identification method is givenin [4] and [5] . In addition to consideration of these particular naval craft, results of application of this technique to the case of a V/STOL aircraft using experimental trajectory data from a dynamic free-flight test facility (for vertical plane motion) and also a full scale airplane using flight test data (three lateral modes of motion), are also presented in [5] For those cases the agreement with other techniques of data analysis, or by virtue of matching measured trajectories, also provide verti- fication of the present technique and its ability to successfully iden- tify many unknown parameters in large dynamical systems from measurements of time histories of state variables (a total of eleven stability derivative parameters were determined for the full scale aircraft case). 1645 Kaplan, Sargent and Goodman When considering the utility of system identification tech- niques for analyzing data from full scale maneuvers or from towing tank tests for naval craft, there are various ways in which it could be most efficiently used for such analyses. This is especially true in the case of model testing where the ability to constrain motions enables selected coefficients to be sought independently from the others. The system identification program has the ability to identify simpler constrained maneuvers first and then incorporate the resul- tant or otherwise known coefficients into the more complete model when a complex maneuver is analyzed. This might also relax the indicated measuring requirements as less demand would be placed on the system identification program, In the case of full scale sea trials, where the motions are naturally unconstrained, system identification techniques offera useful method of directly analyzing ship motions when seeking knowledge of a large number of unknown coefficients simultaneously. Another possible use of system identification for surface ship problems is an application to the case of a ship in a restricted waterway, such asa canal, when applied to model testing. Various static force and moment derivatives and related hydrodynamic data can be obtained from captive model tests in a towing tank with spe- cially configured restrictions simulating the canal. However the im- portant dynamic derivatives due to angular velocity, and angular velocity effects combined with lateral velocity and forward velocity, cannot be obtained with ease or without serious questions as to data validity (for oscillator experiments) with ordinary towing tank test techniques. In that case the use of system identification applied to trajectory data from model experiments would allow determination of basic stability derivatives by that method, when normal test proce- dures have basic limitations. Thus it serves as an adjunct to model testing that would allow more complete determination of pertinent parameters, thereby resulting in more reliable prediction of full scale ship performance, The major problem exhibited in the application of this method is demonstrated when noise is artificially added to the observed data, as illustrated in the case of the surface ship. However, the low level noise associated with analog computer output data, for a lower order equation system, did not seem to influence the results. Similarly the full scale data of the aircraft analyzed in [5] also contained some noise in the records, and the influence of this noise was reduced by means of simple smoothing operations applied to each data point (obtaining average values in terms of data points on either side ofa particular data point at each instant of time). While the generated 1646 Dynamtcs of Naval Craft - System Identt fication noise in the surface ship case may have been more severe than noise that would be experienced with actual recorded data on a full scale ship, or in the case of model test trajectories, the indications are that the influence of noise tends to degrade the estimated parameter values given by this technique of system identification. This behavior might be anticipated to some extent in view of the fact that the basic analy- sis method makes no allowance for the presence of noise in the re- corded data, or noise present as a result of an arbitrary (unknown) random excitation. The only requirement is that the resulting diffe- rences between observed and predicted trajectories satisfy the mini- mum mean square error criterion, and that may not be sufficient without other ancillary conditions that would allow for the presence of such noise influences. More extensive investigations of the in- fluence of noise on the prediction capabilities of this method of system identification must be obtained in order to determine its limits when applied to such realistic cases. A discussion of the application of this iteration technique to a full scale case where significant noise dis- turbances were present is given in a later section of this paper, when considering techniques applicable to noisy systems. A description of the mathematical techniques and the results of application to different naval craft where noise has a significant influence is presented in the following sections. MATHEMATICAL PROCEDURES - SEQUENTIAL ESTIMATION TECHNIQUE When considering the use of system identification for cases where the observed data is contaminated by noise or if the system is excited by a random input, the method that is used is based upon a sequential estimation procedure that is derived as illustrated below. The basic problem underlying this system identification technique is that of estimating the state variables and the parameters in a noisy nonlinear dynamical system, and this problem is treated in [7] : which is an extension of the simpler problem where only observation errors ‘occur [6] . Considering the scalar case (i.e. a single state variable), the system is represented by Kode (eilc sit): wcteuatde (sept) a(t) (21) where u(t) is the unknown disturbance input. The measurements or observations of the output are y(t) = h(x,t) + (Measurement errors) (22) 1647 Kaplan, Sargent and Goodman No assumptions regarding the statistics of the unknown input functions or the measurements error is made. With measurements of the out- put y(t), for 0 for these functions seem to ie oak zero, so that the resulting rte of F and F behave in an almost linear variation with ‘Sm R while these F, and F, functions approach an almost constant value as °F R any for the representation used in the case of the compu- ter generated data model. This is shown by the curves in Figures 14 and 15. A means of judging the effect of this difference is to com- pare the values of the products K,F, and KrF. obtained from the full scale tests with the values used in the computer simulation ma- thematical model, since these terms represent the lift effectiveness d 1665 Kaplan, Sargent and Goodman the controls. The product K,F, , which varies slightly throughout the depth range covered in the full scale tests and the simulation data, is found to compare very well using the respective parameter values obtained in each case. This indicates the same lift effec- tiveness being manifested by the forward foil elevator control. The values of the product KF, » representing the rear foil flap effectiveness were quite different, ranging up to about 3-4 times as large in the case of the full scale data value when compared to the mathematical simulation model value. This result indicates a possible difference in the flap areas (or other dimensional changes) between the values assumed for the mathematical model exercise and the actual craft flap characteristics, or it may also imply a defect in the form of mathematical representation used, as given in Equation (44). Still another possibility is that since there were separate port and starboard flap deflections, the use of an average value may not be proper for the present application test of system identification ; there may be some rolling motion developed by the different flap deflections that causes different relative immersions on each side of the craft ; etc. With regard to the question concerning the basic mathem- atical model given in Equation (44), as well as the representation in Equation (42), the quantity F, (or F, ) represents the foil lift coefficient rate Cr for the particular foil. The depth dependence inherent in this type of representation appears to be appropriate, as exhibited in Figure 14, which was the basis for its selection in the mathematical model. However that functional form of F,; and Fy, implies that the same depth dependence is present for the control elements of the foil as is the case for a total foil angle of attack change. This does not appear to be a reasonable assumption, in view of the small sizes of the control element chords relative to the craft foil submergences. It would appear to be more appropriate to assume a realistic depth dependence, with known values similar to the variation exhibited in Figure 14, for the terms associated with the angle of attack variables in Equations (42) - (44). The unknown functional form of F, and F, , which would just be associated with the unknown elevator and flap parameters K, and Ky, would then be sought by means of the present system identification tech- nique to determine the appropriate parameter values and check the resulting trajectory tracking. Thus the results of system identifi- cation provide a means of judging the general validity of particular mathematical model representations that are assumed to represent vehicle dynamics, 1666 Dynamites of Naval Craft - System Identtftcatton The results demonstrated here show that the present sequential estimation technique can provide accurate estimates of parameters as a means of system identification for cases where random-noise interference or excitation is present. For the case where the data was generated on a computer, and the mathematical model was precisely known, quite good agreement between estimat- ed parameter values and the true ones were demonstrated for the hydrofoil case. The analysis of the full scale case was partially successful, in that generally reasonable parameter values were found that yielded good tracking of observed trajectories when using well stabilized values of the parameter estimates. There isa question concerning the proper form of the equations to represent the depth dependence influence on the effectiveness of the controls, as well as a possible influence on the nature of the measured input data on the results, when examining the results of this system identification procedure. These questions aid in the development of more rational mathematical model representations as well as on the nature of data acquisition for use in this type of analysis. There are a number of features of this particular se- quential estimation technique that have been observed in the present work. These features are concerned with methods used to obtain convergence and useful solutions, as well as information on com- putation time requirements, It is easily seen that the computation time generally increases as the number of variables (including unknown parameters) is increased. A general rule is that the time increases as n3? , where n = sum of number of state variables in equations and the number of unknown parameters. For the pre- sent case of the hydrofoil problem, with n = 14, the computation time was 22 times longer that the ''real time'' extent of the observed data, and this is for the case of a sequential estimation technique that would minimize computation time as compared to the classical nonsequential estimation schemes (with the problems being run on a very fast large computer, the CDC 6600). This time could be reduced by about 50% by applying symmetry considerations to the Pij elements, thereby reducing the number of equations to be solv- ed, as mentioned previously. Another possibility is to separate the equations into separate sets of a smaller number of equations, by means of partitioning, which could then reduce the time si- gnificantly depending on the number of partitioned elements. These particular computational modifications were not carried out in this work due to the increased programming effort required, and the fact that the main objective of the work was to develop the capability of system identification per se. These approaches to reduce com- puter time remain as a further step in providing a more ''computa- 1667 Kaplan, Sargent and Goodman tionally'' efficient method for system identification purposes, Various techniques for reducing the time (i.e. real time of the observed data) for convergence of the parameters, which would then also reduce total computer time, were also found. Thus an im- proved convergence time is found when the constant matrix elements on the right hand side of the gain equations (i.e. the P-equations, as shown in Equation (30) ) are made larger, but that improvement does not show significant gains for larger equation systems. The most useful approach is to vary the magnitudes of the elements in the weighting Q-matrix that appears in Equations (29) and (30) in order to reflect the importance of measured state variable data that are well known. Thus larger values of the matrix elements are used for particular variables that are known to be measured accurately and directly, rather than data that is not directly observable or has a lower degree of accuracy due to instrumentation difficulties. Thus these particular strategies are useful means of achieving more rapid convergence for this type of system identification technique. CONCLUSIONS The present paper has demonstrated the feasibility of using different system identification techniques to determine the values of major parameters and coefficients in a mathematical model representing the motion of different naval craft. This was demons- trated by application to a number of diverse vehicles, suchasa surface ship, a hydrofoil craft, and an SES craft, using data that was generated on a computer (with known coefficients and a known mathematical model) as well as from full scale tests. Different techniques are used, in accordance to the extent of the influence of noise on the system (and its measured responses), and their limi- tations as well as capabilities are described in the paper. Certain virtues of the two different methods used are quite important, such as having a means of determining a level of confidence for different converged parameters while carrying out the identification, as well as an indication that a particular mathematical model is not fully appropriate for representing certain features of the craft motions. The two techniques demonstrated here are generally applicable toa number of different stability and control problems of naval craft, for both full scale and model scale data and analyses. Depending upon the degree of accuracy and the procedures used for data acquisition, these methods can be applied to determine stability derivatives, nonlinear coefficients, etc. in a structured mathematical model representation, from data taken in model tanks. Such data would only involve motion trajectory measurements of free models and could 1668 Dynamites of Naval Craft - Identtfication provide a simpler means of determining the required dynamic data without recourse to complicated special purpose test apparatus, More opportunities to apply these methods for this purpose will pro- vide a final demonstration of its utility and practicality. REFERENCES I Symposium on Towing Tank Facilities, Instrumentation and Measuring Technique, Zagreb, Sept. 1959. 2 Proceedings, Twelfth International Towing Tank Conference, Rome, Sept. 1969. 3 GOODMAN, Théodore R., "System Identification and Prediction - An Algorithm Using a Newtonian Iteration Procedure", Quarterly of Applied Mathematics, Vol. XXIV, No. 3, Oct. 1966. 4 SARGENT, Théodore P.,. and KAPLAN, P., "System Identification of Surface Ship Dynamics'', Oceanics, Inc. Rpt. no. 70-72, March 1970. 5 GOODMAN, Théodore R., and SARGENT, Théodore P., '' A Method for Identifying Nonlinear Systems with Applica- tions to Vehicle Dynamics and Chemical Kinetics", presented at the 1971 Joint Automatic Control Conference, Washington Univ., St Louis, Mo., Aug. 1971. 6 BELLMAN, R.E., KAGIWADE, H.H., KALABA, R.E., and STRIDHAR, R., 'Invariant Imbedding and Nonlinear Filtering Theory", Rand Corp., Memorandum RM-4374-PR, Dec. 1964. 7 DETCHMENDY, D.M., and SRIDHAR, R., ' Sequential Estimation of States and Parameters in Noisy Nonlinear Dynamic Systems", J. of Basic Engineering, ASME, June 1966, 8 KALMAN, R.E, and BUCY, R.S., ''New Results in Linear Filtering and Prediction Theory", J. Basic Engineering, Vol. 83, No. 95, 1961. 1669 10 bh) 12 13 14 Kaplan, Sargent and Goodman KAPLAN, P., and SARGENT, Théodore P., "A study of Surface Effect Ship (SES) Craft Loads and Motions, Part IV- System Identification of SES Dynamics and Para- meters'', Oceanics, inc. Rpt. No. 71-84D, Aug. 1971. KAPLAN, P., and SARGENT, Théodore P., ''Determi- nation of Stability Derivatives of Hydrofoil Craft via System Identification''", Oceanics, Inc. Rpt in preparation. STROM-TEJSEN, J., ''A Digital Computer Technique for Prediction of Standard Maneuvers of Surface Ships", David Taylor, Model Basin, Rpt. 2130, Dec. 1965. LEE,E., STANLEY, Quasilinearization and Invariant Imbedding With Applications to Chemical Engineering and Adaptive Control, Academic Press, 1968. HAMMING, R.W., Numerical Methods for Scientists and Engineers, McGraw-Hill Book Company, 1962. Unpublished work at Oceanics, Inc. on system identification for human operator dynamics. 1670 Dynamics of Naval Craft - System Identification observed aS ) = = estimated 4 2 1 / 7 Yr Ft. 0 -2 -4 0 10 20 30 40 tt, Sec. observed 4 — — — —estimated 2 “NN y, ft./sec. -2 -4 ee eee eee Danae, DE SeeE a 0 no) 20 30 40 t, sec. Figure 1. Estimation of states for second order system, transient response, 1671 =e 0 Le) fe Figure 2, Kaplan, Sargent and Goodman true value = 0. 10 20 30 40 t, sec. 10 1 true value = 0. 10 20 30 40 t, sec. Estimation of parameters in second order system, transient response, 1672 A Dynamics of Naval Craft - System Identtftcatton Figure 3. 10 20 30 t, sec. true value = 0.3 Estimation of state and unknown parameter for third order system, transient response. 1673 *‘poinseourzZ *‘Z@g g £ eID SAS IOJ uoTjour yoytd Aoz SoAT}eATIOp AiITIGe}IS JO SoJeWITIS|] PY F1NdTT "Ses ‘3 0S OP O€ 02 OT Ont ee s 3 8 9 8°T i) 3 3 OR <‘ = = SS % e os Ci G < “aes “2 8 0S OF O€ 0 OT 0 BS lo. Sale aaa Saas ee 6°8 0°6 r-G o = 1674 Dynamtes of Naval Craft - System Identtftcatton ‘O= WH = 2 { peanseour g pue ¢g ‘iets SAS OJ uoTOU yo}Id Aoz SoATPeATIEp AjTTIGe}s Jo SoJeUITIS| “G oansryT "oes ‘Q . OF O€ 0z OT 0 —— 6°8 0°6 T'6 Zz, 6 €°6 "Ses ‘Q — Ww oO a oO ~ 2) N oO qo So 5 a PT 2 8°T 0°? Cac >| = IH aa 1675 Kaplan, Sargent and Goodman “70 =‘W= W : peanseour g ‘yer9 SAS IOJ uoTjOW yo}Id TOF SOATLEATIOp AZITIGeIS Jo SeyeuITIS| 9 eain3tT7z "Ses ‘3 OF O€ 02 OT oO 0°6 T°6 7 °6 se © "ses *3 rT aes 8 at 0°? ma] ag = ie 1676 Dynamics of Naval Craft - System Identtf LCatton 9 ‘BIg Ur SUOT}IPUOD 0} SuTpuodsserioOd soaT{eATIop AjtTIqeys Butsn yyeto Sqsg azojz Atojefesz oTsue yoqtd jo uot}ewTIISs | peavuTrsss — — —/—— pearzesqo lL ernst 0 +pea “sot x6 L677 Kaplan, Sargent and Goodman ie (0) C, 0-5 0 5 6 7 8 9 10 qi t;, sec’ 3b (a) Cy ORS 0 5 6 7 8 9 10 Tp t, sec. 4 Ce 2 0 5 6 7 8 9 10 ial t,~"sec. a0) Ce OFS 0 5 6 7 8 9 10 ae: t, sec. Figure 8 Variation of hydrofoil craft parameter values (as a function of time) from computer generated data. 1678 Dynamics of Naval Craft - System Identtficatton 4 Co 2 0 5 6 7 8 9 10 ap) t, sec. 1.0 0.5 Cg 0 5 6 dl 8 9 10 EA t, sec. ae Coy ol 0 5 6 7 8 9 10 aL iG, sec. Figure 8 (continued) 1679 Kaplan, Sargent and Goodman © observed data —— estimated data 2 hye fcc) 0 9 10 12. U2 als) 14 1S t, sec. initial guess t, sec. initial guess 5 initial guess t, sec. Figure 9 Comparison of trajectories (observed and estimated) and parameters of hydrofoil craft, computer generated data, 1680 Dynamics of Naval Craft - System Identtftcatton 15 a9 initial guess 0 9 10 ap 12 13 14 LS t, sec. initial guess 9 10 11 ih 13 14 15 t, sec. initial guess t, sec. LeiO 0.5 initial guess 0 9 10 unl aly. 13 14 15 Creseck Figure 9 (continued) 1681 Kaplan, Sargent and Goodman 0. Sar rad. t, sec. Sey rad. t, sec. Figure 10 Time histories of hydrofoil control deflections, full scale data. 1682 Dynamtcs of Naval Craft - System Identtficatton first pass ---- second pass ip) hw Q N a ws t, sec. Figure 11 Variation of hydrofoil craft parameters from full scale data, Experiment l. 1683 qe N ie) iN) LS) Kaplan, Sargent and Goodman Figure 11 (continued) 1684 first pass —---— second pass Dynamites of Naval Craft - System Identification first pass —-—-—second pass Q r= 9 tw Q = Figure 12 Variation of hydrofoil craft parameters from full scale data, Experiment 4. 1685 x i i Kaplan, Sargent and Goodman first pass —--—second pass Figure 12 (continued) 1686 Dynamics of Naval Craft - System Identtficatton © measured predicted sO Gi, rad/sec , | 6, Gadi. tirsec: Figure 13 Comparison of measured and predicted trajectories using estimated parameters for hydrofoil craft, full scale data, Experiment 4, 1687 Kaplan, Sargent and Goodman x at Figure 14 Depth dependence functions for hydrofoil craft lift forces identified from computer generated data. 1688 Dynamics of Naval Craft - System Identtficatton 6 Spe ft. Figure 15 Depth dependence functions for hydrofoil craft lift forces identified from full scale data. 1689 Kaplan, Sargent and Goodman DISCUSSION D.S. Blacklock Hydrofin West Chtltington, Sussex, U.K. Mr. Christopher Hook, the inventor of the hydrofin had to leave two days ago for England, but he has two friends in this gather- ing, Professor Weinblum and Dr. Saint-Denis and I think another gentleman was to have come here, Dr. Todd. They all helped Mr. Hook in 1951 after he had shown the Red Bug hydrofoil at New York Boat Show. Hook believes in trying to learn through the seat of his pants : ever since 1944 when he converted a scrap cockpit he has been risking his life on hydrofoils with stability devices, and for this he deserves a great deal of credit. It is now thirty years since his 3-ft hydrofin was tested in the sea near Simonstown Naval Base but with the growing importance of Oceanology there is still time for the su- periority of his design to be recognised and for large-scale construct- ion to be undertaken by France, America and ourselves. The paper which we have just been listening to could be entitl- ed ''Out of the test-tank and into the computer'’. I wonder whether we are better off. Perhaps what we tend to do is to be so dazzled that we fail to notice the paramount criterion, parameter or coefficient, viz. "low-g stability''. The other eleven factors discovered by the authors (or nine factors for their hydrofoil) are irrelevant. What we are try- ing to design is a stabilising homeostatic device, something that is stable in spite of speed. If now we invert the criterion of stability we have one of the two components of a sensible unit for costing sea- transport, K/G for ''knots over g-rating'’. What we hope to provide is a service, say ton-miles, where one ton provides space for three passengers, but people pay not only for a service to get across the Channel, la Manche, but also for speed, and speed in rough weather, and comfort in rough weather. Garbage IN, garbage OUT and we can- not afford to build surface-skimmers on bad criteria. Criteria and costing are very important, if you are trying to be commercial. Yet another criterion is the smallness of the craft. This brings to mind the fact that although the Tucumcari gun- boat which has just returned from Vietnam is a wonderful device and the films are absolutely superlative, it was so costly and big that there is no commercial hydrofoil in the United States at the present time 1690 Dynamics of Naval Craft - System Identtftcatton (footnote A). There are a dozen surface-piercing foil craft at Hong Kong, mainly made by the Japanese, but no country has a hydrofin for commerce (footnote B). I realize that my remarks are disjointed and I apologize to the very able Oceanics team. I am myself a statistician and I am enorm- ously impressed by the precision of the authors' predictions. C'est magnifique, but is it what we want ? I am reminded of a paper read to the Royal Statistical Society Journal twenty years ago - and this should should please Professor Telfer - by Barnet Wolf on multi-factor re- gression analysis. He called it the add-a-variant system for he added a factor at a time and tested for significance. Does the added factor improve your predicting of the criterion you are really interested in- namely, stability at speed ? If not, and if you carry on this testing of possible factors, you will end up with a sensible regression equation, having 5 or 6 inputs. Another method is to confine attention to a single factor with a sequence of observed values over time. We cannot afford to have clutter. We just simplify the problem and predict, whether by multi-factor regression or by auto-regression (footnote C). If I may have one more minute I should like to say that after I had used my firm, Plasticol, Ltd., to exhibit the hydrofin in London in January 1969 our Centre of Industrial Innovation at Strathclyde University decided to do a computer study as advocated by the Ocean- ics Team. The report ran to 66 pages and cost £10,000, but it did not produce the low-cost sea-transport. There was no man willing to risk his money in producing hydrofoils for the British Navy or for anyone else. So we turned in desperation to the Daily Express, whose A. The authors do not name their hydrofoil, but Mr Kaplan told me it was the (naval) Plainview of Grumman-Lockheed ; this craft has now joined the costly ladder-type hydrofoils of the Canadian Navy in retirement. 5. My prepared comments had been handed to the Chairman, but they did refer to the re-designing of hydrofin by Cox & Gibbs in 1952, with sonic sensers, for the US Navy. Mr Hook has been asked to de- scribe the system at the January 1972 Symposium of the Aero-Space Corporation, Los Angeles. C. Auto-regression includes the Box Jenkins formula and my own, Orthodiction, with its freedom from swings with runs of errors in prediction sharing the same sign. 1691 Kaplan, Sargent and Goodman reporter is here, and they found a school in Devon with 20 boys and for the sum of £ 500 they have built a 24-foot sailing hydrofin which in my opinion will win the £ 2,000 race next month financed by Player's cigarettes (footnote D). Iam sorry if I have been advertis- ing and I realize now that I have. Would you please edit my remarks before they appear in the Journal. DISCUSSION Peter T. Fink Untverstty of New South Wales Australta It is, of course, marvellous to have a transfer of an advanced control technology concept into this field and obviously a lot of success has been achieved. What we are dealing with seems to be a very so- phisticated kind of curve fitting and I wonder if Mr Kaplan is aware of one thing that was mentioned at the IUTAM meeting on manoeuvrabi- lity and control in London in April. After a discussion of a very im- pressive Ph.D. thesis from MIT by Lt. Hayes of the United States Navy, in which he was adapting similar concepts to the evaluation of a vast number of coefficients describing a deep submergence rescue vehicle, and I think pretty successfully, the discussion after that by Mr Nils Norbin of Sweden revealed something which I found very di- sturbing and which I think must have a bearing on this sort of game as well. He was discussing the question of what sort of law to fit to a yaw rate rudder response curve, say y asa function of x and he took first of all the sum of a linear term, with an x modulus x2term. Then he did it again taking a linear term with a different coefficient, plus an x3 term. He achieved very nice coincidence of the experim- ental points with both of these curves, but in one case the linear coef- ficient was 50 per cent different from that obtained for the other case} So it looks to me as if a very great deal depended on just how you chose these powers, and when you get to much more sophisticated systems like those Mr Kaplan deals with, it must be much harder to D. Iwas wrong but only because the British entrepreneur had to make do with existing moulds for the floats. The British climate is inimicable to the private entrepreneur. 1692 Dynamtes of Naval Craft - System Identtftcatton make the intelligent engineering decisions that he is calling for. It would be interesting to have his comments on that. REPLY TO DISCUSSION Paul Kaplan Oceantes Ine. New York, U.S.A. A number of points are raised by Mr Blacklock. With regard to the particular case of a hydrofoil craft, the design of a control system for such a craft is a function of the basic dynamics of that craft, which differs from that for a surface ship, a submarine or an SES craft. This difference is manifested by finding what are the basic dynamics of the vehicle and then using that information to guide the application of particular sensor outputs to provide commands to the control. This is most often done nowadays for very complex systems by means of computer simulation, and certainly well before actual experiments with the vehicle at sea. The procedure in this paper is not in any way to be consider- ‘ed as statistical. What we are doing here is applying something that is independent of actual statistics but functions within the domain of randomness, which represents the real situation. We are finding values of the coefficients that are supposed to be invariant for a ve- hicle throughout the whole range of maneuvers or motions that it will experience. This is a way of dealing with full scale dynamics, which is really another method of replacing getting data from a towing tank under controlled tests. When you go to sea how is one t> find out what the numerical values of the coefficients are ? All you have is a result- ing trajectory for which there are a lot of possible ways that one can "fit'' some representation to. However there is something that is unique and invariant which must be close to what is being measured in the tank in order to allow an analyst to structure a representation of the original system. That is a feature of this method, since it brings you to that point of comparison with model test data as well as the ef- fective values of the coefficients representing the full scale craft dyn- amics. The next point is in regard to Dr Fink's comments. There are a number of features of the work of Lt Hayes that are somewhat si- 1693 Kaplan, Sargent and Goodman milar to our own results, although the particular cases considered for illustration are different in each paper. There is a problem which has not been mentioned and emphasized in any evaluation of coef- ficients using system identification. This is concerned with the time required for obtaining convergent values of the coefficients, which in the case of the hydrofoil craft required computation time 22 times as long as real time (of the trajectory time history) in order to deter- mine these values. We had 14 state variables, including the coef- ficients, and since computation time increases as the cube of the number of state variables in your system, too many coefficients in a representation will involve a tremendous computational effort. Thus the simplest possible representation of a dynamic system is neces- sary when carrying out system identification, as well as any simulat ion work, as long as the major features of the craft dynamics are adequately represented. With regard to the aspect of fitting the curve that Mr Norrbin presented, I am fully aware of the paper wherein he presented these results, since I have seen it and also looked into the same question. The real issue of importance is notthe value of the linear coefficient that is obtained by using different representations (for the nonlinear portion of the force) in the mathematical form used to express the force. The major point is how well the overall representation fits the total measurement of the force, and most important of all how well whatever mathematical formulation you use for the hydrodynamic forces and moments results in matching the trajectory of the vehicle under varied motions and maneuvers. Thus it is easy to see that there is more than one mathematical form that can represent any set of data, with different coefficients associated with different powers of the variables, and the particular choice may be dictated by consider- ations of formulation,ease of manipulation and subsequent evaluation, etc. I hope that this answers your question. 1694 FRONTIER PROBLEMS Friday, August 25, 1972 Afternoon Session Chairman: Dr, E.G. Maioli Vasca Navale, Roma, Italy Page Non-Linear Ship Wave Theory. 1697 G. Dagan (Technion Israel Institute of Technology, and Hydronautics Israel Ltd. ). On the Uniformly Valid Approximate Solutions of Laplace Equation for an Inviscid Fluid Flow Past Li39 a Three-Dimensional Thin Body. J.S. Darrozés (Ecole Nationale Supérieure de Techniques Avancées, France). Wave Forces ona Restrained Ship in Head-Sea Waves. iN O. Faltinsen (Det Norske Veritas, Norway). Free-Surface Effects in Hull Propeller Interaction. 1845 H. Nowacki (University of Michigan, U.S.A.). S.D. Sharma (Hamburgische Schiffbau- Versuchsanstalt, Fed. Rep. Germany). Shear Stress and Pressure Distribution on a Surface Ship Model : Theory and Experiment. 1963 C. von Kerczek, T. T. Huang (Naval Ship Research and Development Center, U.S.A.). 1695 te, en’ cown Paw wltey eRe Vere 1) eae ap ¥ is . stemiion dco Chet Dee ear. 2 mores ae OY Tn antinnd Oa pel nes (Gea. foe is OF .¢ tlents aniog wy eho nn je Dib eet “Plodg) aber with € ‘euinred for * ee a a weet atlanta, ‘ | en howe thang ARAM w face { fie: & Vea : 2) Vth NAY oto oe > Bite i | ya's hein Vin Ge a mt Le ¢ \ “a } iF ‘’wwahan oe “he ce of un niente eee See STS ee ied PRATER Dtet” conttian bags r wu hati degree tee Oa j apiened HQC? of Vi ¢ ay#iayi ea aed TY 1c als 1a fol: &S Gey 4 i 1 r ¥ ct” yay ty rte isad. ff] seernetrriadl re bemoan .43 abt KG’ ‘ ds ee Pe Hutte their Ve (het © aATh wy: Pwal - paper wicrdin de pPesenteniae ha si W @pctiaore Loo > the some ite =r) Sod te nae oh went ou RS tHe y L crextene ete ache doarad: Fearhaee' r) wauee naib ; ps.) 44 eae aodty aera: is ti, Taondgels ‘ x “hy ine ORE bial’ 9ah: sates be pO Giee, pepe theasmaeph,: Beat MY, cdoer pt dex he iy bed ti ' oh wad a bawid pica twats 2G 2G ibe heersows 5 3 souebe® © tht teppiagere ities eisai’ shangua tT yAge Bi) eA iagageG ako Wie ese tybeacee ae ks wanonet & Opa EES bd » ebet he ae . soe wil & ie yen eo | a cot beh ane i. pres onakk Aly aed otis sod Saeeoiene yeeEed spankea te dere, tC) MeeeHiRE oft 2 *" SieOhLit tobi: sintist yeiisqord Mull mi p?2elta a we : } .meygiioiM lo-vire1esial) tioswe \ 8 wgdttidod atoalarudmel) anvzade a de (wamdieD: qe. bet Jjlajanssdouaze¥ > 2 on noljudictatC etuenet bans Se9716, tae .? Josetiteqxa bas yresdT Ishoh id@ level) gage .T.T ,Asmoze Auge he 9) Ant seine] jae cagale ved. bas doxseer t 2 2054 NON-LINEAR SHIP WAVE THEORY Gedeon Dagan Teehnton Hatfa, and Hydronauties Ltd Rechovoth, Israel ABSTRACT Systematic attempts to extend ship wave theory into the non-linear range are described. The basic deri- vations are carried out for two-dimensional flows and the bearing of the results on corresponding ship wavemaking problems is discussed, The maintopics are: (i) the derivation of the second order wave re- sistance for a body generated byan arbitrary distri- bution of sources, (ii) the wave resistance at low Froude numbers. A uniform solution, valid at low speeds, is forthe first time presented and (iii) a few preliminary experimental results on the bow break- ing wave. I - INTRODUCTION The linearization of the problem of free surface gravity flow past a ship body has (unlike the equivalent 2aerodynamical problem) a two-fold effect : not only the body boundary condition is simpli- fied, but also the free surface boundary condition is linearized. Although these two simplifications are associated with the same first order term ina perturbation expansion in which the uniform flow is the zero order leading term, the mathematical difficulties associated with the nonlinearity of each one are quite different. It is relatively easy with the present large computers to derive a solution which sa- tisfies the boundary condition of zero normal velocity on the hull body. It is extremely difficult, if not impossible, to satisfy the non- linear free surface condition, even by numerical approaches. Itis no wonder, therefore, that effort has been spent in the last years for solving the flow past ship like bodies, while keeping the free surface condition in its linearized version (the so called Neumann - Kelvin problem). The aim of such studies was to determine the range of 1697 Dagan validity of the usual linearized solutions and to improve them, when necessary. This way it was hoped that a better agreement between theory and experiment could be achieved. The present work is dedicated mainly to the influence of the nonlinearity of the free-surface conditions on the wave resistance. Since at this stage we are interested in elucidating problems of prin- ciple and basic concepts, we have carried out the derivations for two-dimensional flows. Two-dimensional solutions are obtained much easier than the three-dimensional ones due to the use of the powerful tool of analytical functions. They permit to find in a simple way quick answers for problems which in three dimensions need a tedious nume- rical treatment. It is realized, however, that the final conclusions about the applicability of the results derived here to flow past ships could be drawn only after their extension to three dimensions. We consider, nevertheless, at each stage of the present study, the impli- cations of the results to associated ship problems. Il - THIN BODY EXPANSION. Il .1 - General. We consider an inviscid two-dimensional flow past a submer- ged body (fig. 5a). Let z' = x'+ iy' be a complex variable, f' = ¢' +iV’ the complex potential and w' = u' - iv' = df'/dz' the complex velocity. We limit our considerations to a symmetrical body parallel to the unperturbed free surface : hj is its submergence depth, 2L' its length and 2T' the maximum thickness. With U' the velocity of uniform flow far upstream, we make variables dimensionless as follows : tee Ay Oy , y= m/e , zaz! f L! ) w=u-iv=w/U’ ynen/L! fo 64+iV= £'/0'L" Sent ae oma spl tay L’, F:U' Agu)’ F2U'/(2 gies (1) n' being the free surface elevation above y'=0. The exact boundary conditions satisfied by w(z) » which is analytical in the flow domain y ¢ n(x), given here for convenience of reference, are as follows 1698 Non-Linear Shtp Wave Theory (y = 7) (2) Im Wdz = 0 (3) u-l (x — - o) (4) Im Wdz = 0 (jxl<1,.yrtet(x)) (5) where t(x) =t' / T' is the dimensionless thickness distribution and w=utiv. For a given body shape w depends on z,e,h and F. We consider now a ''thin body expansion", i.e. an expansion of w for e= o(1) and F, h = 0(1). This is the basic linearization pro- cedure used in ship wave resitance theory. Hence, with 2 Wize €, EF, bh) = 2 + ew, (z, F, h)+ e w, (Z, Fwd). 4 Z; fi(zecojeiech pare + ef, (z, F, h) + « f, (z, BY OF? fee 2 ix, ek, Ooh) =renrte, Be bh) date 15 ; (S50 Ree ta) + 2 (6) the free surface (2), (3), radiation (4) and body boundary condition (5) become at first and second order df Im GF === £.) =p. (x)=0 (7) dz 1 1 (y Sys a ee (8) a (x — — ©) (9) hei tee Ox| 0 for | z - z | see". To satisfy this requirement at second order we have to superimpose a source of strength e%e: uv (x, -h) to the original source of strengthee; and also a vertical doublet of strength - ee) vj (x), - h). Since we consider only second order terms, we disregard the vertical doublets which contribute only at third order. Adding the appropriate sources at Z) and z, and carrying out the computations (for details see Dagan, 1972a) we obtain finally, for the second order streamfunction far downstream, the following expression Zz 2 Me b =ix?/ FE 2 is /E 2it,\/ Fa 5 Oe J € ¥2 jk (x, O)S= Tm € {(eve ees )iB + 2 a ix, /F ise e~! = . J = k ve ei Lle e ee Baie ae Zz i b pertk/F toe lt (x — es) (19) jk where 2 Spgs /F Zz EP Liemeor [ n° Op( bo = oe9] “lag a aa ee moe a jk ek (Fe 2 ie, gE ), jk F 2 i sae 2h / F cost (20) jk jk In deriving (19) and (20) we have assumed that k>j, x, > x and jk= (x, apy aga ee We consider now, the second order free surface correction which results from the linearized pressure P, (x) in (11), acting on the free surface. In the case of a pair of sources P, can be written as follows ig ee €. € P a eee (21) 2,jk Mawes Sip te aye Buk 4m P ix Non-Linear Shtp Wave Theory the different terms resulting from the substitution of the real and imaginary parts of df, jk /dz into the last term of (11). The complex potential of the flow generated by p- , for instance, is (Wehausen Jk and Laitone, 1965) oo = i ~ zZ Z- Ss f. ceca Mi! (s¢%, cee Mth, Ft) w ( ) ds (22) jk nF’ J jk : Fr’ fi and ie , corresponding to pP:: and or of (20), respective- ly, are obtained from fjk by letting x) — x, Carrying out the detailed computations (see Dagan, 1972a) yields for the streamfunction far downstream 2 2 2 - a sae (x, 0) =Ime ix /F {teres [Er 2 git /F )> 438° }4 Vik k where 2 2 s 2 2 gs netic Nea PES es ee et eee 7 2 4h F 2 2 B® = 2e7/F a4 2eh/Fy ieee daly fgets ea Sic a am ere 2 aes 5 Teale Sena ste he 2 Fa jk ico 2 , h be ea 1703 Dagan s 2 yeh i . 2h Eo eS ae epCal a2 6 (eh eae jk oF Lire k, k, Fr’ el 2 + 4h L(+ a) 2 s h/F 2 = 4 1 f Ta (Ff Zeqt*)\reos ik 2 s -h/F ahi es fait =4e (-1+2e °) sin€. 2 Gt _ 2674h/F (24) II .3 - The Wave Resistance of n Sources. In the case of n sources (fig. 1) the streamfunction far downstream is obtained from the solution for two sources as a finite sum. If we write for n sources cos sin. (x — co) (oc 4 "OyR= vi cosx + v sinx v5 (x, 0)= cc cosx + S sinx + Sah (x + oo) (25) where yconst results from the last two terms of (23), then the wave resistance at second order (Salvesen, 1969) is given by 2 3 (26) with 1 cos cos sin sin eo) (27) 1704 Non-Linear Shtp Wave Theory where D =D' / pgLié and D' is the wave drag. By using the results for two sources, D, and D.. Mor ‘n sources are found after some manipulations (Dagan, 1972a) as D, = _-2h/F 3 Se cost. (28) jet het zZ 2 z metas” ettone ALI. 5d [A* sink... +(BPsB “cosh, erhe n ni on 3 ) | ne De ail é, [C5 (sin i + sin cee, - m= j=1 k=j+1 b Ss / b: s + (Ea + Ey) (cos “age cos ban + (To +1 Jeos E “ s ; = ie sint (29) Ds and D, are obtained in (29) by the selection of the coefficients with the appropriate upper index. All the coefficients are given in an analytical closed form in (20) and (24). (29) permit the computation of the nonlinear wave resistance of a body of arbi- trary thickness distribution at any desired accuracy. The function a and £6 (18) may be taken from the ay es Abramowitz and Stegun (1964), taking into account that w (‘¢) = 5 [2 9i - E, (-i¢ I" . of may be easily calculated by using the cease power series of Bal ih). I .4 - Application to Bodies of Different Shapes. We consider first the simplest conceivable case, i.e. the wave resistance of an isolated source. We immediately obtain from (28):and (29)! with ‘He= 1, €y=n2 2 ah Pb Ae 2h/F 1 b Any oe 2 iD, joer Ey 1705 Dagan 2 Zz ae (30) 2 D5 = -16e72/F (1 + 2e The wave resitance (30), of a blunt semi-infinite body in our approximation (fig. 2a), is represented in fig. 2c. We have ooo there the more common drag coefficient Cp = D' Jo GF Te ee) oF" With Cy = Cy, fe (Cpst+ Cys) wehavesy Co, =D, (2 Cpp = D> _/ F2 _. We have also taken in this case L'=h', i.e. h=1, F* =U'*/gh' and €=T'/h'. On the same fig. 2c we have represented the coefficient of wave resistance for a semi-infinite body having a fine leading edge of a wedge shape (fig. 2b), created by distributing ten sources of equal strength at constant spacing. This way we could estimate the influence of the fineness of the bow on the nonlinear wave resistance. In fig. 2c we have represented Cp), as well as the ratios Ge /ay and Cys /Cp4 . The first ratio is a measure of the relative impor- tance of the free surface correction versus the body correction. The second ratio represents the relative magnitude of the second order correction. In these examples there are no interference effects because the bodies are of semi-infinite length. The next case considered was of a closed body generated by a source and a sink of equal strength (fig. 3a and 4a). With n=2 and e€,=-€5=2 in (28) and (29) we obtain in this case 2 Des =o [ty -ye08 (2° / F*)] Ds = - (ecb/FS lise. y Aen = Zc, + K, 5) sin(2/F°) + e (Caml? tepals oe) [1-cos(2 /F°)]} (31) where all the coefficients are given by (20) and (24). Again, we have represented the wave resistance in figs. 3 and 4 in terms of the more conupon egret Cp = Di /2eU' Lis D./2E° Hence with Cp = e Cy, + € Ae: p2. ~we have this time Cy =D, /2E" and Gy'=D, /2F° *yin tie Be cde 1706 Non-Linear Shtp Wave Theory and Gee are represented as functions of the Froude number for a body of length submergence ratio 2L' /h' = 20. In fig. 4b the same curves are represented for the case 2L' /h' = 10. It is em- phasized that the scales of the various quantities are different in figs. 3b and 4b. II .5 - Discussion of Results and Conclusions. Fig. 2 permits to draw a few conclusions on the effect of the bow shape on the nonlinear wave resistance. First, it is seen that the free surface correction Gas is larger than the body correction ens by a factor of three at sufficiently large F = U' / (gh')' When F decreases this ratio begins to increase ina very steep manner. Hence, any conclusion regarding nonlinear effects which is based on the body correction solely is completely misleading, particularly at small Froude numbers. The total nonlinear correc- tion Cy is a small part of Coy at large F . Again, the nonli- near correction becomes unboundedly large as F—0O. In fact, from (30) we have Coo /Cp,~- 16/F* as F-s0o and Bao Cn = ens / Gay ad ih F* as F-0. The influence on the nonlinear wave resistance of making the bow fine is manifest in the medium range of F values, when the bow length and the wave length are of the same order of magnitude. In that range, for a fine bow Cy2/Cp, is almost constant over a large stretch of Froude numbers and is smaller than Cho / Coy of a blunt bow. At small and very large F the behavior is similar to that of an isolated sources. Finally the second order effect is always negative, i.e. it diminishes the wave ee eie ane. Moreover, if ¢€ is not sufficiently small Cy =€ Cp, + e“ Cy> (figs. 3 and 4) may become negative, which is obviously an absurdity. Figs. 3 and 4 display clearly the interference effects. The nonlinear effect is very large for the large length submergence ratio of fig. 3 (2L' / h' = 20) and becomes significantly smaller for 2L' /h'=10 (fig. 4). Obviously, these large ratios have been select- ed in order to emphasize the nonlinear effect. To render it relative- ly small, the body has to be execeedingly thin or not so blunt. Again the body correction Cy, is generally smaller than roe , especial- ly at small F. The nonlinear term Cp> tends to sharpen the peaks of the resistance cuve and to widen its hollows. The nonlinear effect becomes very large in comparison to the first order wave resistance for small F . Again, we may arrive at negative wave resistance near the zeros of Cp, if ¢ is not sufficiently small. One of the stricking results of our computations, which has 1707 Dagan been observed previously by Salvesen (1969) is the singular beha- vior of the waves amplitude and wave resistance at small Froude numbers. If € is kept constant, and no matter how small, the second order wave resistance becomes unboundedly large in compa- rison with the first order wave resistance as F-—0. Hence the linear theory, as well as the second order correction, become inadequate at small Froude number, although both C,, and Cy> tend to zero as F-—0O. This effect is called subsequently ''the second small Froude number paradox". Finally, we believe that our method of computing the wave resistance of n urces by starting with the solution for two sour- ces offers a possible efficient way of attacking three-dimensional problems. III - SMALL FROUDE NUMBERS PARADOXES. III .1 - Introduction. We have seen before that the computation of the wave resis- tance by the thin body expansion, which is the method universally used at present as far as the free surface condition is concerned, becomes doubtful at small Froude numbers. This could be observed only after evaluating the second order terms. Experiments also support the conclusion that the linearized theory fails to predict correctly the wave resistance at low speeds. The aim of Chaps. III and IV is to elucidate this problem The same subject has been considered previously by Ogilvie (1968). Some of his ideas are vali- dated by the present study, but his solution is shown to be incomple- ue: III .2 - Solutions in the Potential Plane. As long as we seek solutions of two-dimensional flows it is more convenient to operate in the potential plane f= ¢@t+iy , as the plane of the independent variable, rather than the physical plane z=x+iy, in order to derive results of principle. The advantage stems from the fact that the free surface is kept at the fixed and known location w=0. Hence, we consider now the solution of w/(f) (fig. 5b) analytical in the half plane y <0 cut along |¢|<1, =-h+0 satisfying the following condition, equivalent to (2), (3) and (4) oh Payee oe Eger =0 ( y = 0) (32) 1708 Non-Linear Shtp Wave Theory ul (¢ —-00) (33) Here, the variables are made dimensionless with respect to U' and Ll’ (fig. 5b) and h is defined as h! /L’ The physical plane is mapped on f with the aid of df Z= hs (34) which leads to an unwieldy integral equation replacing the boundary condition (5). We shall see, however, that in different approximations the body boundary condition becomes quite simple. We consider now two basic types of perturbation expansions of w(f; «, F , h) aimed to linearize (32) : (i) the thin body expansion, considered in Chap. II, 2 mit; <«, © .,h)= 1+ cw. (i: F, wht ew, (f; F, h)+... (35) 1 for *"e=‘o())8 TF°="0(4 ) and (ii) the naive small Froude expansion 0 2 wets .€ yok new. (ipoe sh) Bowe (ie re. ui) os (36) fares = o(l), e=,0(1): Our aim to study the solutions obtained for different limits c= 0, F— 0, III .3 - The Thin Body Expansion. The thin body expansion in the potential plane yields results similar to those obtained in the physical plane. The mapping (34) becomes z=f+t ez, + ae +E &) (37) where z,=-fw,as, ree - fiw, - (w,)° f2 Vs dé 1709 Dagan Using these relationships we obtain the following set of equations for w, and w, , similar to (7)-(14) Im Gr? salty Athg ('y' ='0) (38) re (¢ — — oo) (39) Een My tse tt) (| <1, w= -ht 0) (40) Im (iF* aie we) Be 3 Blu) +) (¥= 0) (41) w,— 0 ( + — ©) (42) Im w, = + (u, +2, ) ({o| , obeying (41)-(43) may be again found like in Chap. II by a discrete source distribution. It will display the same singular behavior for small F as w, (z). Let us consider now a small Froude number limit of w, (44), i.e. an expansion of the type 1710 Non-Linear Ship Wave Theory 0 an Wie Wat Bow, ts: (45) for F* = o(1). To carry out the expansion of (44) we have to find the asymptotic expansion of w for large arguments. It can be shown that the function w(f ) = e's E; (if) has the following asymptotic expansion for large¢é — k! SL ED) Sh Ree eeae rag (-7 = o(1), €, = 0(1) (corresponding to finite thickness, naive small F expansion) with w= w + €5 w' +... we obtain equations similar to (51)-(54) and solutions with no waves. Furthermore, if we let afterwards «,— 0, i.e. w? = €, w + (€, 2 ws +..., we obtain li- mits which are different from those of the preceeding expansion, in the shaded zone of fig. 7. Hence, our model problem leads to the same ''paradoxes'' as the prototype nonlinear problem. Now, let us consider the exact solution for w , satisfying (57) and (58), which can be written at once as Las Dagan where the integrals are carried out below ReA<1l , ImX =h and Rev <.l., Imv =b)m the * }°and .v planes, réespecttver We can use now the series (59) to rewrite (61) as follows 9 ‘tig 0 0 0 1 ame exp(-i ¢/ ea) Aaa taky b 4 ee te er ee exp(id/€,) €>5 “oO ae : 0 0 l l = exp furs shee ne, co Hat...) alan ast valid for finite €; , €5 . We are now ina position to expand (62) for small e, and/or € > . The detailed analysis may be found in Dagan (1972b). Herewith, the main results : (i) the limit €, = 0(1) , €5 = 0(1) of (62) yields the same results at first order, €, w, , as the solution obtained by expanding (57) if, and only if, €, / €, = o(1). This last condition stems from the existence of the ratio e, / €, in the last exponential of (62); (ii) the limit € = 0(1) , €, = 0(1) of (62), wa , does not coincide with that obtained by the naive expansion of (57). The uni- form solution differs from the naive solution in the ''wavy wake" and does comprise ''waves''. Moreover, to obtain a first order complete solution we have to retain in the last exponential of (62) all the writ- ten terms, upto ( eS , in particular €, u,. The "far waves" are obtained by contour integration (fig. 7b) as follows w? = ts exp(-i r/o f [oP + POX) W(A) |. explin/e,) uniform nN cp ks [ [os i wv) | dy } dd (Rela) (63) f Again, it is emphasized that in the last exponential €, uw contributes at O0(1) because of the division with €5 . Going in reverse, the differential equation which yields the uniform first order solution, obtained by the appropriate expansion of (57), is de /ag + (i/e,) (tH + 6m) wo = (i/ €)( oo + Po HY) (64) 1716 Non-Linear Shtp Wave Theory Z where terms up to ( €,) have been retained in 1/(1 +0), the coefficient of (59). (iii) the limit ¢, = o0(1), €2 = o(1) of (62) is not defined un- less we specify the order of ¢,/ ¢,. For ¢ /¢€, =0(1) we obtain again at first order a solution with a ''wavy'' term, the latter having the expression i€ . 0 l Z 0 wW = Ww ans i " uniform ‘i 1, uniform Bias o exp(-if /¢ BI J pf ) bey a 2 . exp(id/e,). exp[ fuer) ar] ar + 0(« fen 2 €5 1 il 2 (Re f= oo ) (65) This solution differs from that obtained by taking the limits «— 0 firstand e,—+0O afterwards in (57). Moreover (65) is obtained from the solution of the differential equation, derived from (57), Or. ds (GO ga er ue (66) IMS) 0 : Pie ge) w = iy? eS) py (iv) the limit = o(1), e,= (1), & / €, = 0(1) yields by the expansion of ie A ie d exp fi) a] =l1+ — f we Hide in (65) Bin i aint the usual linearized approximation 0 ea 0 @.= = exp (-if /e ) fe (A) exp (it /e€_) dA (67) 1 €> 2 4 1 2 satisfying the differential equation 0 0 ; dw, +df +(i/ej)o , = Miidse gh ny (68) a Dagan IV .2 - Application of Results to the Nonlinear Problem (Potential Plane). Due to the similarity between (32) and (57) the results obtain- ed in the model problem can be extended to the hydrodynamical pro- blem at once (for details see Dagan, 1972b). With w=1+W the uniform solution, the key to obtaining first order uniform approxima- tions W° (for F2 =o0(1), ¢=0(1)), €W, (for e=o0(1), F*=0(1)) and e¢ wr €=0(l1), F* =0(1), e/F* = 0(1)) is to expand the coefficient (w)° w of dw/df in (32) ina naive small Froude number expansion and to retain the appropriate number of terms. By doing that we obtain the following uniform asymptotic approximations for W : (i) e= 0o(1), F? =0(1), i.e. ¢€/F* = 0(1) (thin body, finite length Froude number, large thickness Froude number (U'@ /gT'), W=eW, +... . W, coincides with (44), and the usual thin body approximation is, therefore, uniform. (ii) F? = 0(1), €=0(1) (Small Froude number, finite thick- ness), W = w? +... . W® satisfies the free-surface boundary con- dition (similar to 64) 0 z dW im {iF* [(u°)? + F°(u")* (3 tiv’) +... - w ( ¥ = 0) (69) and along the body 1 + wo =w, where w® (51,52) is the naive small F solution. w® is not an uniform solution in the ''wavy wake". The solution of W° subject to (69) is very difficult. (iii) F2 =0(1), €= o0(1); ‘“e/F2 = 0(1)*(small Froude number, thin body, thickness Froude number ule / gl' of order one), W =e Wey Wy; satisfies the free-surface condition (similar to 66) 0, dw, 0 Im Lar’ (nae a Met Wy = 0 (% = 0) (70) where u2 = Re w? is the naive linearized small F solution (56). Also, along the body skeleton Sp (fig. 7) a = w, . By analytical continuation and integration by parts Ww, has been found in a close form as follows 1718 Non-Linear Shtp Wave Theory f 2i w? ety a ewe ae = Oe aire we hai 1 2 1 Ae —co 2 Bie (0 exp (ik /F°). exp[ - =f GS Naval ath (71) Ff For F*>0 WwW, > w, + o(F? ) excepting the ''wavy wake". There, we obtain similarly to (67), by contour integration like in fig. 7b, F 1 0 2 We eMhodiarer 3 > exp [-i(f- in) / F°] f wo?" (s + in) F 5 stih exp (is/F ). exp v) dv] dst+... oo (Ref > o) (72) where w%" and wot are given by (56). w, is obviously not a uniform solution. Only, and if only, «¢/ ee se o(1) (72) degenerates into (47,48). The implications of the different limits are discussed in the following sections. IV .3 - Uniform Solutions in the Physical Plane. It was advantageous to carry out the basic derivations in the potential plane. In applications it is convenient (and in three dimen- sions it is essential) to operate in the physical plane. It is easy to transfer the Gans results to the physical plane. With w(z;e,F, h) = 1+ W(z; e€, ie ,h) we have the following limits : (i) €=0(1), F°“ =0(1) (thin body, finite Froude number), =e Wa | eo WY. , W, » Ws satisfy equations similar to t= (14). Ww, = dt, i, dz is the usual thin body solution, Wa = df, / daz is ihe second order solution (see chap. II). (ii) F* = 0(1), ¢€ = 0(1) (Small Froude number, full body), w-=w? + F? w! +... . The complete first order term we = oe - iV satisfies the free-surface boundary condition 1719 Dagan 0 0 2 2 Zz 4F@ 0 1 ja) +22) a ral ue + 2r* u ve ae y= 0 (y = 0) (73) on the unperturbed free-surface, similar to (64). w? =v -iv® is the rigid wall solution for flow past the actual body and w! is the next term of the naive small Froude number solution. (73) has a simple physical interpretation : it represents the equation satisfied by waves generated on a stream of variable speed w2 +F? w!, beneath y=0. Ogilvie (1968) has retained only the first term in (73), i.e. has replaced (73) by Im fire rh = -w] = 0 (y = 0) (74) He has based the derivation of (74) on the intuitive reasoning that at small F the wave length of the free waves becomes small compared to the body length scale (which governs the rigid wall solution) and, therefore, the waves are travelling on a basic stream of varying velocity. Although the argument valid in principle, (74) is nota uniform asymptotic ap prosime non: as shown in the preceding sec- tion. To determine W~ , satisfying (73), is a difficult task which is not pursued here. (iii) ¢=0(1), F*. =0(1),... €/F* =0(1) (thin body, small length Froude number, finite thickness Froude number), W =eWy + 3, By analogy with (70) we (z) satisfies the boundary condition aw? Im [iF* (1+2ew)) —+ - wi |= 0 Goh eee 1 dz 1 the radiation condition 0 Ww, — 0 (x + ©) (76) and the body condition 0 0 Wie aed (dx)< lg-y.=--h)... (il along the skeleton of the body. wp the linearized rigid wall solu- tion has the expression 1720 Non-Linear Shtp Wave Theory 1 Ohl an B Gpu o,f ert t (s) 1 7 (s) Wan es Maron” ea sae | z-ih-s BS Ridin eat all eae ey = (78) Ww; may be found by analytical continuation across y=0. After some manipulations (Dagan, 1972b) the solution is found to be p : f Zz M23): = a - wi += 2/E ; F : O,u Wag aed ye ae is (A )e e 1 dy (79) -00 which is analogous to (71). The profile of the free waves N = € ND == is derived from (79) as 2 2 0 bs N, =Im Ave Blea Im[ 2p ws athe £20" 9) Be ix/F° oie fy /F dd] e ~ix/F (x + oo) (80) 6 - In (79) and (80), f Sous (v)d» and S§S is the cut |x| ina complex manner. In particular waves are generated by the parallel part of the body (7=0, t= const) and the amplitude changes if the direc- tion of motion of an assymetrical body is reversed. The wave Li22 Non-Linear Shtp Wave Theory resistance is always positive, while in an illegitimate expansion of cs Nt F 5 in a truncated power series it may become negative. Cp isclose to Cpg, (fig. 8) at relatively high F, (i.e. small e2 /F), but shifted towards small FY, - The peaks of the resistan- ce curve of the uniform solution are, excepting the highest peak, much smallerthan those of the usual theory. The disastrous effect of using the second order approxima- tion of the thin body expansion in the region of small Froude numbers is illustrated clearly in fig. 8. It is worthwhile to point out that the wave resistance (81) is integrable only if the leading (or trailing) edge singularity is like w, ~(z - ih+ 1)>“ where a<1. Hence, a parabolical nose ( a=0.5) is acceptable, while a box like shape (a= 1) is not inte- grable. IV .5 - Extension of Results to Three-dimensional Flows. It is easy to proceed along the same lines and to derive the free-surface conditions satisfied by the various uniform approxima- tions as F-—0O in three dimensions. With u, v, w the velocity components and z a vertical coordinate the exact free-surface con- dition, counterpart of (2), may be written as follows (2 z 2 F' [uu,. tuv(v, +u, )+vv,_ + uww,_ + uww, } w =0 x x y y x y (z =n) (85) The naive small Froude number expansion (asy.v% w) = (uo , vo , w? )+ FA (u! Oy hw hockii = Ee n° +... leads to vw , v0 ,w® asa rigid wall solution for flow past the actual body. With e¢« a fineness or slenderness parameter, a further expan- sion of (u, v, w) yields u® = 1 + eu? tan, thode? owl T= e (v9 , wo ) +... . The usual linearized approximations is obtained from (85), by expanding the velocity near the uniform flow MS hl le ceady. bocicrs sy (visuwx' is € (v, »W, > Mm) tees as follows Fou Aaowe f= 20 (z = 0) (86) Let now 1+V® bea uniform small Froude number appro- ximations. By analogy with (73), inthe limit F->0, e« =0(1) © satisfies the following free-surface condition lt2s Dagan F* Aa + aes y 25 ae aoe 7 yg (rhac s + i Vv °7 o) zi 2 a cad Gilg a i Pie ae ie NG i PE 5 Se ME eS yy: xZ y Z (z = 0) (87) @° may be represented along the actual body surface by a distribu- tion of singularities derived 22 2S the rigid wall solution. The rigid wall solution is asymptotic to ® as F390 excepting a ''wavy wake" which this time is generated by rays emanating from the body image across z=0 towards x—oatan angle -§ (arbitrarily small) with the horizontal plane. Todetermine ® , representing waves over a stream of variable velocity, is a very difficult task. The simpler approximation of thin (slender) body € = o(1) small Froude number F = o(1), and finite beam (and draft) Froude number €/F*=0(1), is obtained from (87) like (75) : 0 0 1-2€e wy 0 Bpe_ pol sree ater brass 0 (z= 0) (88) F a may be represented by the source distribution of the rigid wall solution on the body skeleton (central plane, or axis). (88) is the extension of the usual linearized free-surface equation (86) (which is the basis of computation of ship wave resistance via Michell integral) into the range of small Froude numbers, where (86) becomes inva- lid. The solution of a is the object of future studies. IV .6 - Discussion of Results and Conclusions. The two small Froude number paradoxes have been explained with the aid of our model problem. It was shown that the naive small Froude number expansion does not yield a uniform solution, the region of nonuniformity being the ''wavy wake". The elucidation of the second paradox has led to the impor- tant conclusion that the usual thin body first and higher order appro- ximations are valid only. for large thickness Froude numbers. For moderate values a new first order approximation has been derived : it results from taking in the free-surface condition a basic variable 1724 Non-Linear Shtp Wave Theory speed, rather than a uniform flow, as an "unpurturbed" state. This new approximation is the natural extension of the thin body theory into the range of small Froude numbers. The basic equations of flow past a body of finite thickness moving at low Froude number have been also derived. Again, to ob- tain a uniform solution one has to satisfy simultaneously the bounda- ry condition on the full body and a free-surface condition in which a flow of varying velocity is taken as the basic state. This basic flow is derived by solving for the two terms of the naive small Froude number expansion (rigid wall and first order Neumann type problem). To solve for flow past the actual body, while keeping the free-surface condition in its usual linearized form, may lead to erroneous results in the range of small Froude numbers. V - PRELIMINARY EXPERIMENTS ON THE TWO DIMENSIONAL BREAKING WAVE. An important free-surface nonlinear effect, present in the case of blunt bow ships, is related to the breaking wave. At the 8th Naval Hydrodynamics Symposium we have presented (Dagan & Tulin 1972) theoretical models of the breaking wave inception and of the bow jet. Recently, experiments have been conducted at Hydronautics Inc. under the supervision of Mr. M. Altman in order to visualise the two-dimensional breaking wave. The detailed results of these experiments will be reported elsewhere. Herewith a few prelimina- ry observations. A rectangular body has been towed at constant speed in the small Hydronautics towing tank. The water depth was 38 cm and the model has been submerged at (i) 2.5 cm and (ii) 1.25 cm beneath the unperturbed level, such that the effect of the bottom was negligible. The model has been towed at six different speeds in the range 0. 61l- 1.46 m/sec. The model motion has been recorded through the chan- nel glass wall on a 16 mm color film at 64 frames per second. Tak- ing the pictures has started after 3.5 m of run (the tank total length is 24 m). In fig. 9 we have reproduced two pictures for the 2.5 cm model : at 0.61 m/sec and at 1.46 m/sec, the corresponding draft Froude numbers being 1.22 and 2. 93, respectively. The free-surface in front of the body had vertical pulsations which became more violent as the speed increased. This made quite difficult the definition of the average free-surface profile. It seems that the oscillations are related to gravity effects since the periods LY25 Dagan for the two submergence depths were roughly in the same ratio as the square root of the drafts. Satisfactory Froude number similitude has been obtained for the free-surface elevation near the body for the two drafts. The breaking wave inception apparently occurs at a Froude number somewhere between 1.20 and 1.50, which correlates quite well with our theoretical prediction of 1.50. Separation at the cor- ner of the body profile, visible at high speeds, makes difficult the definition of the body shape for an inviscid flow calculation. We did not reach a ''spray regime" in the range of considered speeds. The study of the complex flow pattern of a developed breaking wave is the object of future studies. VI - GENERAL CONCLUSION. We have discussed the pertinent conclusions at the end of each of the preceeding chapters. Here, we will try to discuss their bearing on ship wave resistance. The usual thin (or slender) body first order linearized approximation, leading to the Michell integral, is valid for sufficient- ly large beam (and draft) Froude numbers. In its range of validity this approximation may be improved by taking into account the se- cond order term. It seems that the contribution of the free-surface correction is of the same order of magnitude as that of the body correction in this second order term. As the shape becomes finer, the Froude number limiting from below the range of validity of the linearized approximation, as well as the second order correction, become smaller. For moderate beam (and draft) Froude numbers and, hence- forth, small length Froude numbers, the linearized solution is no more valid and the second order correction worsens the results, rather than improving them. To obtain a first order uniformly valid solution for a thin (or slender) body in this case, one has to take a variable velocity distribution, rather than a uniform, as the basic unperturbed distribution in the free-surface condition. This basic flow, as well as the singularity distribution along the center plane (or axis), may be computed by solving for a rigid wall flow pas the linearized body. Linear free-surface conditions with variable coefficients have been derived also for the case of small length Froude number flow past the actual body (finite beam, or draft, length ratio). To 1726 Non-Linear Shtp Wave Theory obtain a uniform solution, the basic nonuniform flow (on which the variable coefficients of the free-surface condition are based) has to include the rigid wall, as well as the next term, of a naive small Froude solution of flow past the actual body. The singularity distribu- tion on the body surface may be taken from the rigid wall solution solely. This results suggests that solving for the actual body shape, but with a linearized free-surface condition with constant coefficients (the Neumann-Kelvin problem) does not yield a uniformly valid solu- tion at small Froude numbers. The above conclusions are based on the assumption that the results obtained in the two dimensional case may be extrapolated to the ship wave resistance problem, at least in principle. Only solving for actual three-dimensional flows will make the conclusions valid in both qualitative and quantitative terms. Such three-dimensional solutions pose, however, difficult mathematical problems which have not yet been touched. The picture of the nonlinear ship wave resistance theory is not complete unless we refer to two components which are somehow related to viscous effects : the bow breaking wave and the wake. Only the first component has been considered in our studies. ACKNOWLEDGMENT. The present work has been supported by ONR under con- tract No. N00014-71-C-0080 BR 062-266 with hydronautics Inc. Most of the material is based on Hydronautics Rep. 7203-2 and 7203-3 (Dagan 1972a and 1972b in References). I wish to express my gratitude to M. P. Tulin for the stimulating discussions we had on the subject and for his collaboration in the different stages of the work. 1727 Dagan LIST OF SYMBOLS. Dotted variables have dimensions; undotted variables are dimensionless. A amplitude of free waves for downstream. Cy coef. of wave resistance. DD? wave drag. ee complex potential i 2 Froude number based on half length Fs Froude number based on length hi h' submergence of body axis beneath unperturbed level. h. submergence of stagnation streamline far upstream. qi strength of source located at 2. D ik = (x) - x) g/U' dimensionless distance between two sources Iu} body length i length related to the body image in the potential plane N free-surface elevation in a uniform small F expansion t'(x') thickness distribution rE body maximum half thickness u' ,v' velocity components Sie unperturbed velocity w' complex velocity in two dimensions : vertical velocity com- ponent in 3d. W perturbation complex velocity in a uniform small F expan- sion ac! horizontal coordinate positive in the direction of flow y' vertical upwards coord. in 2d, horizontal in 3d z' complex variable in 2d, vertical upwards coord. in 3d. a, B real and imaginary parts of w angle arbitrarily small slenderness parameter 1728 Non-Linear Shtp Wave Theory €) &5 perturbation parameters. p velocity potential in two dimensions } velocity potential of a uniform small F expansion in 3d. y streamfunction in 2d. w(f)=e-! f Pa (i) PONT CO) is the exponential integral n free-surface elevation above the unperturbed level fy g, x, r) v auxiliary variables g,p auxiliary functions ci slope of body contour REFERENCES 1 Abramowitz, M. & Stegun, I. A. Handbook of Mathematical Functions, Dover, 1964 2 Cole, J.D. Perturbation methods in applied mathematics, Blaisdell Publ. Comp. 1968 3 Dagan, G. Nonlinear effects for two-dimensional flow past submerged bodies moving at low Froude numbers, Hydro- nautics Inc. Tech. Rep. 7103-1, 45 p. 1971 4 Dagan, G. & Tulin, M. P. Two-dimensional gravity free- surface flow past blunt bodies, J. F.M., Vol. 51, p. 3; pp. 529-543, 1972 5 Dagan, G. A study of the nonlinear wave resistance ofa two-dimensional source generated body, Hydronautics Inc. Tech. Rep... 7103-2, 197Za. 6 Dagan, G. Small Froude number paradoxes and wave resistance at low speeds, Hydronautics Inc. Tech. Rep. 1103-3, 1972b. 7 Erdely, A. Asymptotic expansions, Dover, N.Y., 1956 8 Ogilvie, T. F. Wave resistance : the low speed limit, Dept. of Naval Arch. & Mar. Eng. Univ. of Michigan, Rep. No. 002, 1968 3 Salvesen, N. On higher order wave theory for submerged two-dimensional bodies, J. F.M. Vol. 38, pt. 2, pp’ 415- 432, 1969 1729 10 11 V2 Dagan Tuck, E.O. The effect of non-linearity at the free-surface on flow past a submerged cylinder, J. F.M., Vol. 23, pt. 2 pp. 401-414, 1965 Van Dyke, M. Perturbation methods in fluid mechanics, Academic, 1964 Wehausen, J. V. & Laitone, E. V. Surface waves, in Ency- clopedia of Physics, Vol. IX, pp. 446-779, Springer, 1960 Re ae = ie 1730 06 0.2 Fig. 2 Non-Linear Shtp Wave Theory Big. a A distribution of sources. Ls0 (a) . Za —U' h—1' — (b) Cp Coe/Cp — Se eee ———S 2CaRICES Cp=D'I9U7T'@EC + €7C,,, € =Tih 2 2.5 3 Ulver" (c) Wave resistance of a body of semi-infinite length : (a) a source generated body, (b) wedge shape leading edge, and (c) wave resistance curves. Brst Dagan Cp*D'lagut w€*Cp,* EC, een (b) Pig. 53 Wave resistance of a source-sink body : (a) the body shape and (b) wave resistance curves for 2L' /W=28 : Ei eS ae / LFS E ancien 10 / x eerie / \ (b) Fig. 4 Wave resistance of a source-sink body : (a) the body shape and (b) wave resistance curves for 2L' /h = 20 E732 Non-Linear Ship Wave Theory y' ge ee ee oe ose er su pre oe k —2ur— (a) (b) Fig. 5 Two-dimensional flow past a submerged body : (a) the physical plane and (b) the potential plane. LLL L Sapa ish, gee omen 2 ee og ee ee -l-ih ih Fig. 6 Regions of uniformity of the small Froude number solutions. LSS Dagan (a) Big il 7 (a) Regions of uniformity of small ¢, solutions of the model problem and (b) integration path for large Ref Cpe” Os -— 2" —— petit Eel hekit FreUiV7g «ext &* Dv2gU7L 04 t= €ll=x2) €=005 h=#0.20 First order linearasd solution (Eq 83 ) ———— Second order lineorusd solution ——— First order small Froude number uniform solution (Eq. 8) ) 03 Cpe? 02 oO. Q3 35 04 Q45 05 Fig.,8 Wave resistance of a biconvex parabolical body by different approximations. 1734 Non-Linear Shtp Wave Theory Haig: 19 Flow in front of a rectangular body . i735 Dagan DISCUSSION Ernest O. Tuck Untverstty of Adelatde Adelaide, Australta In the first instance I want to pass on a comment that Dr. Salvesen asked me to make on his behalf because he could not be here today. Incidentally, I agree with his comment. The comment concerns neglect of the Kutta condition at the trailing edge of these two-dimensional bodies. In fact both Dr. Salvesen and I fell that our own original papers on these two-dimensional problems were defi- cient, in that we should have included the Kutta condition in the first order solutions. We are a little disappointed that in doing this pro- blem again, Dr. Dagan has not seen fit to include the Kutta condition, for reasons which he has stated but with which we do not really agree. My own comment is actually relevant to Dr. Salvesen's work, in that I feel that perhaps insufficient attention was paid to Dr. Salvesen's paper of a couple of years ago. Dr. Salvesen himself was aware of what Dr. Dagan has called the ''second low Froude number paradox''. He analysed this in some detail in this paper, and perhaps some direct reference could have been made to this work. DISCUSSION Edmund V. Telfer i NeAte Ewell, Surrey, U.K. I want to ask a very innocent question. Could the author tell me what he intends to be the significance of the word "'naive''* in the * According to the Oxford Dictionary the meaning of naive is '' Cha- racterised by unsophisticated or unconventional simplicity or artless- ness. 1736 Non-Linear Shtp Wave Theory description of the naive small Froude number? Is it an adjective, or should it correctly be an adverb, meaning ''naively small'' ? But how is anything naively small in the modern interpretation of the word “naave" 2 My second question is, would he care to hazard a guess as to how at very low speeds the wave-making resistance varies? Does it vary as the fourth power of the speed or does it vary as the sixth power of the speed? Could he just give an answer to that, because in a further paper to be discussed this afternoon this point will proba- bly arise? REPLY TO DISCUSSION Gedeon Dagan Technion Hatfa, and Hydronauttics Ltd Rechovoth, Israel Regarding the Kutta-Joukovsky condition, I have stated in the paper that Iam not really interested in solving problems of two- dimensional flow, but only in using the two-dimensional computations as an instrument for understanding and opening the way for the more complex solution of three-dimensional ship problems. Since in the latter case only the thickness effect is generally taken into account, I did not consider the circulation. I agree that if one really wants to solve the problem of the hydrofoil (but I do not see the usefulness for small Froude numbers), the Kutta-Joukovsky condition has to be taken into consideration. The paper of Dr.Salvesen has been amply quotedin the present work. I believe that my main contribution is the solution of the second paradox, and not its discovery. The term 'naive'' as applied to a perturbation expansion has been borrowed from the applied mathematics literature. The word is used in the sense that one expands in an apparently natural way in a power series, without observing the nonuniformity of the expansion. Since the present work is concerned with two-dimensional flow, no attempt has been made to correlate the resistance curve to the power of the Froude number. 1737 inegasq el oi balm vigeste asad ‘ed sOkPVL: Be, 0 lo 19q8q Sh - “OeS8 ad; jo catwise aif! a) nefiediaines afam yro tad} ovoiind Tim i na oe ; ra 9 io ,sviosibs na tw) Veetinen erat k inet’ olen ‘dene wed wi § "[lemne yloviga! geinagre .¢¢evGe ge oe choertreaty.: 4 brow oft le aoltetelasstal esshor ona See ylowien AS Te Re 7 at coeng Pe ewe bb stgs ad Widpy Wt coltesup bagane | sonll Vesivey ascdsiales 4 PA Bees - aye shy cast ghuages wot vio i dinte off ag ¥ie¥ 1) Boob ah Danze FGi Catt we true’ odt oe chosved. tail 01 tewtae Wedeieeh( SE oeD Thangs ods Ye dian, Gie talon wit) copiyrevee epi” buh tn: ath od oF “Sus i. 7 ‘ _ ei 4 a4 : : ; r re] : ‘aut 7 aA e % mks 1 Re os « ~ f ts ions ‘eye ’ £ ay vic SeeNOeN, ViIS38 7 a J s 4 “9 ‘2 t a5 " r¢ at eon ioein, Souble vere % ys HO shav * i ' ; b + ene nl es tak wa ook yet CaP ee RE GS A Acre ee. » » > F A _% ; Tr. sens TP : , : yers.. p» US Re ad t } é ft mi oSjarn sea iF ope 6 Ha TOALK [ Ariu A af Bz bs 295% . ‘ *; Py | eve 5 - au & is af : ; ios a I aS 7 etn vi LASS 4 ia (tee aie Sri ievigci eee Oa rhe mst - as sas Beta t ag Yigu ‘id -RABAL Js ke VIE SAT SOT Ya en? phsgo Rye Bilndstatsacy =ci jae tie a 7 i re ¢ ° Sa? ‘nt spate’ .aresiebiy dida laiwmaseciib-s3 ait w pos Gairmisas wt Aad«<¥ iT sise sy teelle #4 orixs sh sf? tag Se OF Shctw Vilaet’ spot) Je 7o2 gs L at eles cio ail) i othe t sace'ytesy af: wed fac oh | avd) itothyd sii Te nisidoig ae * sd oo enn nottibnoa yaavodon b-atiud edt ,(e7édaind abuoteE Norteateblencos of888 - ns , CUSSOi | (i se - | weet. Oitepalh aif ing ba Bae pO ANB ENS hoksedeiks 2 OF Satine es: "evisen"’ mast eat biow wit o1ptareti) asta’ Ocliggs Al pov boworke bh at yaw leagiern yds taecs Ae ci ohatdee sao teed vanes ont at & noteiadirs 260.46 UddereoL i ngnen <8) gaivtgeda tuodiiw:- estes ‘a i , P’ “ic ae Sue lenorgan, omen iid ton LeveL P He Learn bratve oe 2iteece eds sone” | of ovius aatateians sd? Sigiss1ds oF sbam ceed ead iqare}t § On ; nen ee a Rees LS Pe . cate) oC qeeeewentional eimpllelty &F veri ON THE UNIFORMLY VALID APPROXIMATE SOLUTIONS OF LAPLACE EQUATION FOR AN INVISCID FLUID FLOW PAST A THREE-DIMENSIONAL THIN BODY J. S. Darrozes Ecole Nattonale Supérteure de Techntques Avancées Parts France ABSTRACT The classical solution of the Laplace equation for an inviscid incompressible fluid flow past a three-di- mensional thin body, is shown to be not uniformly valid in the vicinity of the planform edge (A). In order to find a solution which is valid in the vicinity of (A), the technique of ''matched asymptotic expansions" is used, The inner solution in the neighbourhood of a rounded leading edge brings a shift correction tothe classicalouter solution. The inner solution in the vi- cinity ofa sharptrailing edge givesthe starting shape of the vortex sheet. I. INTRODUCTION The study of three-dimensional flows past arbitrary bodies gives rise to many problems of greatinterestin Naval Hydrodynamics, Foraninviscidincompressible fluid, the velocity potential isa solution of Laplace equation, and in the few last years, most of basic works have dealt with the numerical methods used to solve this problem, The greatest difficulty comes from the fact that there is no rigourous ma- thematical theory available for such a problem and numericalattempts may be handled only with addition of physical assumptions, A unique solution could be obtained, only after the difficult analysis ofthe corres - ponding high Reynolds number flow, in the limiting case of an evanes- cent viscosity. As it is not possible to do so, it is necessary to guess [1 ]some results in order to define a problem which has a unique so- lution. For instance, if the physical flow takes place with separation, This work has been supported by the « Office National d’Etudes et de Recherches Aérospatiales » 29, Av. de la Division Leclerc - 92320 CHATILLON 1739 Darrozes we must give an a priori geometric description of the separated flowl?], This is the reason why we will assume later on, that no separation occurs. With this condition, the Kutta-Joukowski theorem leads toa physically correct description, but, even with this requirement, one does not know the starting shape of the vortex sheet in a non-symme- tric flow. So, the solution still depends upon an additive assumptionl9] One can go further, assuming that the body is very thin, be- cause the flow is undisturbed in a first approximation, and the velocity potential @ is formulated in the form of an asymptotic expansion. Unfortunately, new difficulties arise when the classical solution of this simpler problem is evaluated through a numerical analysis. The flow velocity is found to be infinite on the leading edge, the trailing edge and the vortex sheet edges. Up till now, theoretical investigations allow us to know the singular behaviour of the asymptotic expansion of the functions @, in the vicinity of the afore-mentioned lines, but they do not suffice to solve complety the problem] . The classical method is to consider an inner region in the neighbourhood of the singular lines in which we look for the velocity potential in the form of a new asymp- totic expansion called the inner approximation. The technique of match- ed asymptotic expansions, which is the properway of investigation, has been successfully employed by M.J. Lighthilll4]and M. Van Dyke] , to solve the two-dimensional problem of a flow past a thin hydrofoil. In this paper we apply the same technique for the three- dimensional case and it will be seen that the results depend strongly upon the wing geometry. II FORMULATION OF THE CLASSICAL OUTER PROBLEM II. 1 - Basic equations Figure 1 shows the body shape with the following assumptions and notations. The body is very closed to the plane z = 0, and the inci- dent flow is supposed to have a uniform velocity at infinity, parallel to the plane xoy, in the x-direction. The planform (S) is the projection of the body on this plane, in the z-direction and its limiting curve ()) is the projection of a curve ( \) drawn on the body. The curve ()), which parametric equations are x(s), y(s) and € z(s) separates the body surface into two parts. - The upward surface S+ has a given analytic expression : Zo sane£to(acpy ) - The other part S~ is known in the same way : Zameke= (Way yo) 1740 Untformly Valtd Soluttons of Laplace Equatton Figure I The Body shape The straight lines ( L, ) and ( Lz, ),edges of the vortex sheet (2), are tangent to the planform at points i and j, projections of the points I and J belonging to the obstacle. In order to avoid separated flows, the rear part of ( A ) from Ito J is assumed to be a sharp edge. The body is smooth at any other points. The vector nis a unit outward normal. Written under its usual non-dimensional form, the problem to be sol- ved for the velocity potential @ (x,y,z, € ) is the Laplace equation : Ae =0 i'.0 e =o on oe) uo. Cue) (Ty db ~x at infinity It is easily proved in any text-book, that the function @ , may be approximated in term of an asymptotic expansion, the first step being the undisturbed uniform flow $= x }+(x;) yp Zz; ej 4+ eb (x, y, z) tg?) (x,y,z) + ore) (2°) At each step, we have to find an harmonic function, vanishing at infi- nity, and satisfying some boundary conditions, on both sides ( + and - ) of planform S. Ag!') . fe) -- —« © at infinity 1741 Darrozes 5 ge” + 1 ¥ = a AA a (2) + 5 (1) + (1) + 2 ) ee SE SE OS Vix) € (5) We are going to consider now, the validity of the so-called ''outer expansion" (2 ), It could be plausible that some difficulties arise when looking for the behaviour of outer solutions @ '*) in the vicinity of the part (S) _ of the plane z = o, because the liquid cannot go through the body, and the first approximation seems to be a non uniform flow. II, 2 - The region of uniform validity in the plane z =o It is known that, any point in the xoy - plane is regular for the outer expansion (2 ), except the points located on the lines ( A ) ( L,) and ( Lg). This result is easily proved by introducing a new ex- pansion in the following way. Any function A (x,y, z) is written A (x,y, 7z) = A (x,y,z7). The formal expansion of the function A , when z goes to zero, is iden- tical to the expansion of the function A , when 7 goes to zero witha fixed value of z . The 7 -expansion must depend upon the cluster ne \ since A does not depend upon 7 Inserting this formalism into the problem governing the e - term @!'1) of the potential velocity, the behaviour of the solution for vanishing values of z is given by the corresponding 7 - expansion, (ih) a Oxy. 2, 0) <8" my) + 1O["( say, Z) + @ (x,y,7z) =® ae n2A ® hava 922 xy 3p) ae ee ts = a Ze O , (aye ee. Zz The following result is obtained without any difficulty 1 ~ ® Al) 11,363 y ee af 1zf-=1 DW AyyP -=7Z (1) | re eS “11 5 a xyix+55" 2 xy Day B'+0(7 )-1a) 1742 Untformly Valtd Soluttons of Laplace Equatton a(t) @ (x,y) is an arbitrary function and A me stands for the operator a1) aoe oxe gy2 The same procedure is used to know the behaviour of the ; (2) function ® Se 2 mil?) a— +74 = 3 xy ~ (2) ae et) (1) o® ak I ar = 95 ‘eis. el tian ————-= 7 ! ——— + ff! ——— A ce) = 716) Oz - Ox mt Oy i xy : ae) The 7 -expansion in the vicinity of a point (x,y) on the planform (S) is (1) (1) ~(2) (2) i 5 ONaDE- Se" OBI (1) L iden Ae $= (sy) +9 We +f 5 Frifdig Pe “zz AP pole If we consider a point (x,y) on the vortex sheet (£), the Taylor be- haviour of any function @) is: ql ae gu 2 = Po : ne 2, ) + 0( 7 The pressure continuity across the sheet gives additive conditions such as eS |) eal =} z __0¢k _ Ok | Ox Ox Outside SUX, p= . Consequently, rae is a discontinuous function on the curves (A), (L,) and (L,). At these points the flow velocity has no meaning and the outer expansion (2) is not valid. Elsewhere, this expansion is uniformly valid, as it can be shown easily in the follow- ing analysis corresponding to a point (x,y) located inside the plan- form (S). If the approximation (2) is not valid in the neighbourhood of (S), it must be within a region of thickness 0(€), because, in sucha layer, the boundary conditions must be written on the real body, and cannot be approximated by a condition on the plane z=0. Replacing n by €, the preceding results give the behaviour of the inner solution at infinity rewritten with the inner variable Z , as indicated by the matching conditions, 1743 Darrozes re atl) ~(2) b= x te (x, y) + aS E +oaz £! 1F (5) 1~ 0) 380), 9 BB0 ey Se et a + he Sea a + ty Of) ae 2~ 0° 4 oA @’ =- 0 Oz? xy rs + 36 a Sie ~ = OP DS ake ae Sey, f! Od on z= f (x, y) OZ E Ox yy. ay + Matching conditions at infinity. The expansion ( 5 ) is shown to be the exact inner solution. Therefore, the inner region exhibits nothing more than the behaviour of the outer solution, This means that the outer expansion is regular at any point (x,y )€S, and the inner region is not required. Figure 2 Sketch of the flow when € goes to zero. The streamling I generates the streamlines I’ from the leading edge. The streamlines II’gives only the streamline II. Figure 2 shows the flow in the xoy-plane, when € vanishes, There exist physical reasons to explain the singular behaviour of the outer solution on the lines ( A ) and ( L,). Ona round leading edge, the slo- pe is infinite and the normal velocity w = SeAz is o(1) instead of o(e ), At any point of the angular trailing edge, the Kutta- Joukowski condition requires a velocity tangent to the curve ( \ ), which contre- dicts the fact that there is a uniform flow in the first approximation, On the line ( L;) , the streamlines I' closed to the streamline II rolls up, and generates the apex vortex. At infinity downstream, any 1744 Untformly Valtd Soluttons of Laplace Equatton streamline (I') goes to ( L,) and makes the vortex growing. Ill. THE SINGULAR BEHAVIOUR OF THE OUTER SOLUTION ON THE PLANFORM EDGE ( ) ) In this section, we look for the behaviour of the outer solu- tions ol) and »'? , in the vicinity of a round leading edge, ora sharp trailing edge. To this end, a carefull description of the body geome- try is required, III. I The body geometry Let us denote by »(s ), the outward normal to ( } ) in the plane z = o, Figure 3 The leading Figure 4 The local edge geometry (upper curvilinear coordi- surface), nates, We define a reference frame aXYZ, with aZ parallel to the z-axis, and the unit vector » inthe Y-direction. Figure 4 shows the local curvilinear coordinates (s, Y,Z): OM = Oa(s)+Yv(s)+ZK DM =h,rds + vdY +kdZ (6) hy, Sy ear C ( Ss ) C (s) stands for the curvature of the planform edge ( ) ). If this quantity does not vanish for any value of s , the mapping (x,y,z )++(s, Y,Z ) is a one to one transformation in a region of thickness less than s™lco(s)|, surrounding the curve ( ) ).In this region, the body equation may be written under the form : 1745 Darrozes Z=2,(s)+H*(s,¥) (7) with the condition : H-( s, Y )»o when Y_,0 for any s. The behaviour of functions H*when Y goes to zero is obtained using the reverse functions Gt=u7"* The identity : Y= Gt( s, H(s, Y) ) is expanded into Taylor series for small values of H: ee ed a8 +2 YVisOaeHs oct way AHeTew else The following results are found. a) Round leading edge aft = 0, a ~ 0, and conversely for the other point J - When s-.0+t we admit the following expansion ay is: pay vee OCs) a>l1 In that case the behaviour of the body equation in the vicinity of point I, depends upon the way in which s and Y goes to zero. In other words, the expansion (9) is not uniformly valid when s goes to zero. Itis possible to write a uniformly valid approximation of the function H ( a = = - z,(s), for any limiting process (s-.o and Y¥=5 0. )4" rhe corresponding description is given when Y and s go to zero simultaneously: Y=+4(s )¥, withu(s)+.0 when s-—>0, and Y has a fixed value. The function &( s) is determined by using the less degeneracy principle. Writting vas = = == om H (Us, Y )= HCsey ) = Bets: *5 “we khow"that fit SY) or eee s_»o for a fixed value of v » and (ies Ce ween {Bo ( Y = a, H +—*H* + 3H?+., we obtain : one he ak 5 uH(s) Y=3, str (s){ H+...) ESPae (iety| Ba, WA +2 In order to take into account the greatest number of terms in the 1746 Untformly Valtd Soluttons of Laplace Equatton first approximation, we must choose: r(s)-=s°andu(s)=s Then ee ne ro (a, Wee zag F ] +0 ( 5% ( 10) Om i: This expansion is shown to be uniformly valid in a neighbourhood of the point I, after verifying that : the limit Y —-cogives expression ( 8 the limit YW-.o0 gives expression (9 — — III. 2 The vicinity of a round leading edge This paragraph is devoted to the study of the singular beha- viour of the outer solutions "and gi), in the vicinity of a point A belonging to a round leading edge. Let us start our investigation with the function @) solution of the equation (11 ). - fi! ( sheeylewe-n SS (es) which is rewritten using ae curvilinear Sopramnates a) Mc ( —2 94 yc Ae iy ot 8) 8. OEE pis See) ae. G(s oe = 1=YC(s) de® OS. 1-¥C)a5.0 8 = Sia 1 = Yc(s) (zA(s) +H" )- sina( s ) Hy" ae) (12) There are many solutions to this problem, due to the fact that we cannot take into account the condition at infinity, when looking for a beha - viour. Any solution of equation ( 12 ) is obtained from a particular one, after addition of a solution corresponding to the homogeneous problem oe F Z—»0O The homogeneous problem In order to investigate the behaviour of the homogeneous solutions in the vicinity of A , we define the new variables ~ Mie in/ fn Z = (2 -¢2,(#)) / with 7 <1 We look for an 7 - expansion of the function @!") (« n¥, cay 2) +92] = © (ms, *Z8) 1747 Darrozes solution of the homogeneous equation ( 13 ) (1) oO Ays ® ee Oe aes o (*) =6 ( 13°) (1) , as a oa ae Geometry of the outer The complex variable problem Figure 5 eT ae ee ee ; Using the complex variable z=pe = Y+i Z , and its conjugated value z= Y -iZ4,it is easily shown that the n - expansion of gp '1) must be written with the asymptotic sequence 7 V2 CP 2 ee ~ 11) _ (1) Wy vit) =~ (1) x, bi (Ry nh 22, a Os | tO [eee (14) Each term is obtained after resolution of a trivial Hilbert problem on the negative part of the real axis of the plane Z%. The final result is: @=c(s)+7 dj (s)p Sin 2+ "co( s )pcos +n “dl s )Psin 2. O'(7*) Col s), d,(s), co(s) and d,(s ) being arbitrary functions The full problem ~ (1) Olt) We are now looking for_solutions of equation ( 16 ) fe) AMPS SY TOL EY GSO a holes SOR a = gies), gary o) + 7 55 (8) +0(n?) ( 16 ) 3Z pe 1748 Untformly Valtd Soluttons of Laplace Equatton Using the previous result (15), it seems plausible to try an a priori expansion under the form : pit cree 1 0 ep +. (47) It will be seen, later on, that expansion ( KZ) as incomplete, because the asymptotic sequence must contain logarithmic terms, ~(1) Let us consider the problem to be solved in order to find#¢, b)"! ( z,%,) [a a) + alll Z| +i [ By) Bz »| where Au A Y ) and Bit Y ) are real functions on the positive part of the real axis. The holomorphic functions AN z Z )and B!'\ Zz ) are obtai- ned after solving the Hilbert problems deduced from the boundary conditions (1) 361 (at. 2) (et 22K) Pee od 2 SES =O eee 0Z OZ OZ ioe Toe T dee xed aor Be i Vedat Ube Ee at SA Bi tty . sats) cay ee Ar) nel Be ae thy a so §) p V2 A UA imps 4668 dolasthe/ PY (18) pile i Le a Alek § {P eas /p (19) The following notations are used : + 2 OU Sani se REL pile: Ona The solution of equation- (18) is : al’ = Deicke + homogeneous eee of equation (19) is: B" = __bolP)(s)_ z5-log Z + homo- geneous solution. a Jo 2 to the previous variable z = nz =Y+iZ wean see that a OS e)(s) Z—- 5. does not depend upon 7. But the func- beth 1 === 1 - 7 . 1% ie 50!P)(s) ices t i so”) oe ne hon Moe eee afi 1749 Darrozes still depends upon 7 It is easy to see that the logarithmic term is an homogeneous solution, Then, the expansion of the function @ is (P) (p) zs 259, @ BO 260 gt) n/2Log 1 disins + n2p to sgeoss +720(Logesing cos .: O( n 4) The same procedure is used to evaluate the following terms, and after many computations, the final result is found to be : ~(1) $ Cols) + 2% toy n 280 a ein © afPl = ri rotfess (e)~ eas so se io 2\ log p 7 sin Ocos0) +d (s)P % sin. Nlog 1° pcos ie ) +n E sp Sin @- eae 2 (log? p cosO_ end) +c,(s) cos 0 ] 5! ) “The % log n SoH 2054 0(0)5 : O82 nS p% Se ee ay eu +d,(s)sin "38" — Se log P+ eeoi20)(c (s) 50 pee a Oe cern er ner ete tina ceesmenr aie +O ( n% log n ) Ne, ee ey Particular solution of the General solution of the | | | | | | | | | 1 | | | | | | | | | | | homogeneous problem non-homogeneous problem Following the same argumentation, we investigate now the behaviour of the function ®'”) solution of the equation (21 ). cag (2) 2) C(s) ae” 2 376") vcs) d¢ ® y v2" -inycley BY)? ee Be eve os oe 22) : gel) agel gall With the new coordinates Y= Y/n andZ=(Z-é Zp (s)) /n , and using the known expansions ] = + SY cal ail, gh nly | [)s= it + lg 7e 5 + 2, As) sie 1750 Untformly Valtd Solutions of Laplace Equatton nm + + 2 # (2) 5 = ph \\ l il 3 Bil : P + pore wks Sat (alvie o* (ait Pe | = {2} ek 2 og om es 1 = We jE. ce = Logo. oe H, ©, | ad ooh ake Yaar oe Y Y ! 1 n EOME AVE le (tf + Log n |: le FH @, +H, 2 51t 0 + dela [le iP ft Veo he Oe ae ye n + a | Hy 2 ca 5 A eR = oe a 5 es: lyl ae Za Y qo gles, yj anetine vay ant, abe ee ee ay Bao) 0 Viasl 1 3 2 0 2 x (p) z (p) , (p) t (e) is 60 ne 8 ) 6 P er 238 0 i) 1 0 = iat ets P, = - d,+ = +-—- Log IYI, eS = = P u (e) o (e) 1, = - 1 Log | ¥| + 2 - ot = (p) bs (p) ~_ d \"7 (e) pll- =f $9 gir = oP rages ce = - b4 0 Pp eiter ae. 1 + a Log | Y| + 2 ’ 2 = The expanded solution contains some terms o (Log n)?and fe) ( n4 (Log 1)?) because the Hilbert problem : oe See e 6 = - 7 p= ly has the particular solution uw (Z) = =e ( Log z bs After analytical calculations, the result is found to be : 2 2 ~ (2) ~(2) ~(2 2 = ~ $e) (Log n) ®9 + Log 7 rs oP), n%h(Log n) gl?) , nh Log noe ert (2a) Bt pe i ere en. BR) rate ag 202 2 “aed. 4alkty G2 2 > Lot Darrozes 2 Se (Log p T 2 > ~(2) x ia aie (e) 3 = 23% ne E {P) si 6 to Ho | + = ~(2) 8 ™ (put - Hz 3! hy +H by= -—pP4 ba 0 Hy == = - (Log 6 -2)sin—+2 Ocos—}- is + = e + Lo, cos— [3s Be By ass = Ode =| w Zz 2 2 III. 3 The vicinity of a sharp trailing edge This paragraph is devoted to the study of the asymptotic be- haviour of the flow velocity Betis in the neighbourhood of a sharp trailing edge. The function f we (s, Y, Z, 7) must fulfill the following equation : ~ (1) Jolt) ee ie me Pe +. O1( 17) “Slo |) nt a n 6 4 (es) re (s) Y att te By alee ? Y & oO (24) Tv with the addition of the generalized Kutta-Joukowski condition : a6) fas) | ag = = ye | tar | 2 hemes bounded by p=, whenp_.o, *K > 0 (25) When Z_.0* witha positive value of Y , the pressure must be a continuous function : elt) ol) * p= a — + cos o 3 | + 7CY +0 (9? (26) t + The symbols 6;(s) and 65(s) have a new definition in this section (see (9)) 2 dq. = cop a z, (8) - sina /a; (s) 3¢ 2 65 OC - (ap /ay J sina + Ccos a 2, (s) - cosa (aj /a* ) Now, we proceed as in the last section, in the complexe plane =e are 1752 Untformly Valtd Solutions of Laplace Equation Z + { body trailing if edge fl ¥ Figure 6 If theN-expansion of the velocity potential has aterm o (1) gy 2 H = the function ae is a solution of the homogeneous problem A S41" = oz, %,)= Ao (Z) +Ao (Z,) +i (Bo (Z) - Bo (Z)) but A, (Zz) and Bg (Z) are holomorphic functions except on the real axis. ce the negative values of Y , Ao (Z) and Bo (Z) are not defi- ned when Z goes to zero, and the limit functions are denoted At (Y) and B#(¥). In the same way, we denote by AQF) and BO\z) the corresponding limits for oO (31) (ey eet i Gell) bE ves With ida Gres Wns terohech bieaieanis sue oa) ja Z The equation (29) has been solved in the preceding paragraph : Ng laces A(z) © = z Log z+ ¢) (s)% (32) From equation (29) oe Pp) henge By (z) = “ae tie, (s) and using the relation (31), the solution is hes B, (Z) = le | 6 (P) i cote ex(s) (33) From these results, the 7 - expansion of the flow velocity potential is written in the following form : i) (e) = (e) a & = G(s)-e (s)-1Logn 61 Bcos O+ np. wd ( @sin O- Logp cos 8) ui etsy esemacng Bl et S) % 1 sin9- e) ‘(s) cotga cos + o( 772) (34) In this expression, we haxe assumed Bar ae vortex sheet has no thickness, so writing 3 ® Maz | = E32 O , we obtain Cie = Gee = oe. 1754 Uniformly Valtd Soluttons of Laplace Equatton Figure 7 Outer description of the vortex sheet The starting shape of the vortex sheet is given, in this outer solution by the value of 06 “V/aZ when © = 0 or 271.The figure 7 gives a sketch of this departure ye } plane normal to the planform edge ( } ). The slope is found to be bP as in the two-dimensional case of an hy- drofoil without thickness. We conclude the analysis of the outer problem, with the figu- res 8 and 9 which show the singular behaviour of "lin the vicinity of a round leading edge (20) and a sharp trailing edge (34). THE SYMMETRIC PROBLEM LEADING EDGE TRAILING EDGE . ce eel Z Vs BYs wr W+ 1 4 NORMAL VELOCITY Ne Wi + P4 PRESSURE : hog WA \ | 108 p| ¥ DISTRIBUTION P, Figure 8 DSS Darrozes LEADING EDGE TRAILING EDGE GEOMETRY NORMAL VELOCITY PRESSURE DISTRIBUTEUR + Py Figure 9 A discontinuity in the curvature of the leading edge increase the order of the singularity IV. THE INNER SOLUTIONS With the new variables Y=Y/7 and Z = ——*"4 | the ere sae n body equationis : x 2 : at s nh © at 3 (a5) = nae It is easily seen that, in an inner region of order 7 = Bey the bounda- ry conditions must not be written on Z =o, but on the exact body wall : 2Y az Z = + 30am 2.) Therefore, the study of the vicinity of a round leading edge requires the introduction of an inner layer with a thickness 0(€?), The beha- viour of the inner solution at infinity, for this region, is given by the outer solution (20) and (23). 1756 Untformly Valtd Soluttons of Laplace Equatton 22 ao" Ot x(s) + eC(s) he OE SD) sats (P my + Ho + Hpi ne 200%; 84 Ho +H) , 2 Tt 7 2 (p) (. 265 \S) 8 +e€ a4 Ysin.a(s) +P sah cose +9 (Loge sing + cos>)+ d,(s) sin? es ee ~ 2 22) ‘Gall aah -He Se ER Ra ae age clk ade €3 +r = ( Log p) a 5 \ o €?Loge) (35) For the sharp trailing edge, the outer expansion is singular for the symmetric problem, as shown on figures 8 and 9 -g (e) oe p~l1 - € Log p Sapa p=np +o From this result, the study of the vicinity of a sharp trailing edge requires an inner region with thickness of order 7 = 0 ( entfe ). The behaviour of the inner solution at infinity, for this layer, is given by the outer solution (34) : ® = x(s) +€(co(s)t€o(s)) +E ce 4 p-Psina(s)|cosO@ +0 (ee %) (36) qT IV.I The neighbourhood of a round leading edge The full problem, to be solved in the general case of an un- symmetric nose is written below : =e S C = = Ne i (s) \¥ teea ) ) ee RS: = 52.274 0 PG (a,< o) -. Matching condition (35) at infinity Figure IO The local problem for an unsymmetric leading edge Lat Si. Darrozes Taking into account the expansion (35) at infinity, the solution ® is expanded under the form of an asymptotic approximation : ~ ~ ap pV ee — P =%o +€®, + ( € Log €) 2+ é Log 6Pats ys We have obviously Po= x (s) 1= Co(s) > Ae a oe a 2 6\P ut + Ho if?) sina Ho + Ho =e i ra sides Ge Sine OE A nite ee Sy The function ®, is a solution of the following problem Ze (p) 1 (37) nV ©"= 6 -on. ¥ is ; At cd 7,488 P sin— 3 a ain 2 In order to solve this problem, we must change the Z-axis and choo- se the focus of the upper parabola as a new origin Revd. Qiao" Y=¥Y Zee ( 38) We define the conformal mapping Zz = g7 withe@=¥+iZ and § = & +in. In the §-plane, we have to determine a flow, with the uniform complex velocity 4 i 5 {Par , along a step @ (when a> =0) or along a curvilinear wall in the general case @) (when a> #0). The solution, in the first case @ is easily obtained, but has no interest because the infinite velocity at the point O generates separation (except for the ideal angle of attack), The solution in the second case @ has not been obtained, but this solution does not present a great interest, because a slight change in the incidence remove this point of discontinuous curvature, from its present particular position. We restrict our study to the regular nose (i.e. discontinuous curvature). In that case Hd = - Ho and 55'Pl=o . From the expression (35), the behaviour of the inner solution at infinity is : = ~ ) @ =x(s) +€c,(s) + e ; Ysin ato (2.sgcose+d,(s)sins) + a) + ofc’) and is expanded under the form @ = x(s) + €c{'s) + 2 bet o(e?) With the new coordinates defined by (38), the function &, is solution of =~ AyP, =19 1758 Untformly Valtd Soluttons of Laplace Equatton Te 2 ae a 2a 7a is a stream line Figure 1] The local problem for a symmetric leading edge 2 Using the conformal mapping z= $ , the given parabola is transfor- med into the straight line § = 1/ V2\|a2l and the corresponding complex potentialis f£(')=e (5) +iw(e) £ (5 )\y= = siaee(. <2 cy + Cste with bo = = ti 0s fal abies v0 So, the solution we are looking for is: - = = fl y 8. 8 = - gi -2P (| —— —+ 2p in— ®, sin X k Z ci =i coss Bos + K «| and rewritting it, with the coordinates Nej0Z03 4 ® = R{-sindz + 2¢ er ie) (40) 1 0 z lal The identification with the asymptotic behaviour (39) at infinity gives : 6 - hi . id : 2fg ) 2 Tani which is identically satisfie Bo = d,(s) if. which gives the position of the stagnation point P. In the physical plane Y , Z, P is located at the point (eds 4d, / W/ 2g!) 1759 Darrozes The arbitrary function d,(s) may be determined only after the resolu- tion of the whole problem, the location of P depending upon the Kutta- Joukowski condition at the trailing edge. The matching condition is fulfilled up till O(€3). The following term in the behaviour of the velocity potential at infini- ty, given by the matching conditions is : o Stee lee - Pegi a! pcos 9 + Ee” Loge 2 Ho Hie? bse = 2 Jag| 2 3 (e) ~ 2) =A ra) 3 Log Ot - pcos@ + —— 99 cos —— The expansion of $ is then: @ = x(s) +¢¢(s) +e@,te Loge #,+ o(€3) ae being a solution of the problem (41), written with the variables ¥Y and Z Azz %, = 0 = =—2 ¥r= = lo/ Zags jagpmeyo2 is a streamline (41) T This solution is obtained as before : 1 5 4 ~ V2] a5| + iff, 2 2 h hi diti h that f, = d =./7— which The matching conditions show tha B; fe) an V2 tad! laal whic is identically fulfilled. We can conclude by writting the composite expansion uniformly valid in the vicinity of the round leading edge. There is a shifting correc- tion analogous to that found by M. Van Dyke in the 2-dimensional case [8 $= x(s) - Y sin&(s) + nants, Y,Z) + Jel YZ) | 4 (42) +Rfe ‘ies (1 + e Log €) + sual feats i ee . =~Y+iZ 1760 Uniformly Valid Solutions of Laplace Equation IV.2 The vicinity of a sharp trailing edge The new problem that we investigate now is formulated as follows : ~ sash = ~ ~ B = x(s) +e(co(s) * egs)) te“ $(s,¥,Z,€ ) Z Y (=) Figure 12 We must solve : = nA Ass Se ote “e)- =o ao Sta veett nN. =o on = a* (s) with the matching condition at infinity. It is easy to see that the 2 ie L §,(P a © sina : bhatt lee ers =) Viele (44) is a particular solution of the two-dimensional problem, and the matching condition qe SS, requires : Sa T sina U(e)=e | -sinat (c,teh cote«)| (45) We did not prove the expected result : ~ =~ @’ = & + on(-<.4) 1761 Darrozes Nevertheless, the departure of the vortex sheet, given by the expres- sion (44) (see figure I2 ), is not in agreement with the results obtai- ned by Mangler and SmithL6], REFERENCES [2] C. REHBACH, Etude numérique de l'influence de la forme de l'extrémité d'une aile sur l'enroulement de la nappe tourbillon- naire. Rech. Aérosp. n° 1971-6 (Nov-Déc). [2] H. PORTNOY AND J. ROM, The flow near the tip and wake edge of a lifting wing with trailing-edge separation. T.A.E. Report mr 132. [3] R. LEGENDRE, Calcul des ailes subsoniques. Rech. Aérosp. ~ n° 112 (Mai-Juin 1966). [ 4] M. VAN DYKE, a) Subsonic edges in thin-wing and slender-body theory NACA TN 3343, b) Second-order subsonic airfoil-section theory and its practical application TN 3390. [5] M.J. LIGHTHILL, A new approach to thin aerofoil theory. The aero-quaterly,, Vol. lil, -Noys) 51s Part UTI. [6] K.W.MANGLER et J.H.B.SMITH, Behaviour of the vortex sheet at the trailing edge of a lifting wing. The Aeronaut. Journ. of the Royal Acro. Soc. Nov. 70. Bed J.S. DARROZES, The method of matched asymptotic expansions applied to problems involving two singular pertubation parameters Fluid dynamics transactions, Vol. 6, Part II. [8] M.VAN DYKE, Perturbation methods in fluid mechanics. Applied Math. and Mech., Vol. 8, Acad. Press, New-York 1964. [ 9] T.S. LUU,G. COULMY et J.SAGNARD, Calcul non linéaire de 1'écoulement potentiel autour d'une aile d'envergure finie de forme arbitraire. Congrés de 1l'A.T.M.A., Session 1971. 1762 WAVE FORCES ON A RESTRAINED SHIP IN HEAD-SEA WAVES Odd Faltinsen Det Norske Verttas Oslo, Norway ABSTRACT The exact ideal-fluid boundary-value problem is formulated for the diffraction of head-sea regular waves by a restrained ship. The problem is then simplified by applying four restrictions: 1) the bo- dy must be slender ; 2) the wave amplitude is small 3) the wave length of the incoming waves is of the order of magnitude of the transverse dimensions of the ship ; 4) the forward speed is zero or it is the order of magnitude e V2-a ; O is the x-coordinate of the forward perpendicular. Positive x is in the direction of the after perpendi- cular. We do not want to go into detail in this chapter about the ma- thematical expression for ~,) (or wy’ ). For more details see the chapters about the zero-speed and the forward-speed problems. But we should note the following : wg is the solution to the zero-speed problem. So the solution at a certain forward speed U can simply be obtained from the solution of the zero-speed problem by multiplying with Further wy' will not vary with x if the submerged cross-sectional area of the ship is not varying. The second expression of y is then telling us that there is a decaying factor (x + L/2) — as the wave propagates along the ship. The presentation in this paper is divided into the following steps. First we set up a general formulation valid for both the zero- speed and the forward-speed problem. Then we study the zero-speed problem separately. We derive a far-field expansion for a source distribution located on the x-axis between -L/2 and L/2. We obtain an inner expansion of the far-field source solution and study then finally the near-field solution and the matching between the near- field and the far-field solution. Then we have a chapter for the forward-speed problem, which is presented in a similar way as the zero-speed problem. Finally we have a chapter about numerical cal- culations. A computer program has been developed for a ship having circular cross-sections, and comparisons between experiments and calculations have been done. The agreement is shown to be good, II. GENERAL FORMULATION We assume that the ship is moving with constant speed U in the direction of the negative x-axis. The z-axis is upwards, and the y-axis extends to starboard. The origin of coordinates is located in the undisturbed free surface at midship, so that the forward-speed effect appears as an incident, undisturbed flow with velocity U in the 1767 Falttnsen direction of the positive x-axis. =) L/2 Figure 1 Coordinate system The ship is slender and there is no other bodies in the fluid. The fluid has infinite depth. It is incompressible and the flow is irrota- tional, so that there exists a velocity potential ¢ which satisfies the Laplace equation. in the fluid domain. The ship is restrained from performing any os- cillatory motions, and so the boundary condition on the wetted surface of the ship will be = 0 on s4—e-810 ; (29) ae yi) = - Cn ke” on the body. (30) In addition, Y, must match with the far-field solution. A one-term far-field solution is assumed to be the potential associated with a line distribution of sources of density spread along the line y=z=0, -L/2 < x < L/2. That solution has been obtained in a previous chapter, and a one-term inner expansion of a one-term far-field solution can be found from (19). For any fixed x greater than -L/2, it is obvious that a one-term inner expansion is and that the second-order term in the inner expansion is of order V2 compared with the first-order term, (31) should match with a one-term outer expansion of y i(wt- vx . 1 e , as determined from (28), (29) and (30). 1778 Wave Forces on a Restratned Shtp tn Head-Sea Waves Ursell (1968 a)* has given a solution to that problem, but it does not appear to match with (31). However, if we say that the one-term near- field solution is just the negative of the incident wave (this is a special case of Ursell's solution), then (28), (29) and (30) are satisfied, and if we require that dia {2 ) = Ce (32) 27 se [x ¢| k -1/2 kZ -inr/4 € e e then we see that a one-term outer expansion of the one-term near- field solution matches with a one-term inner expansion of a one-term far-field solution. So we have the solution v (X,Y, Z, € ) | | on i] — nae MN IN 7 @ = N i] he 4 Se AN Q Str Fle i] —~ AL wt (Fy eo We solve (32) for 0, (x) formally by letting it be an equality for all x 2>-L/2. We recognize (32) as Abel's integral equation (see Dettman (1965) ), which has the solution os 2 in/4 7 (x)t .2:2ie has Lye) 8° a iG (34) This solution is singular at x = -L/2, which is a violation of the * Ursell's solution will be needed in the second order term and will be discussed then. 1779 Falttnsen assumptions made earlier. However, this is not a serious difficulty if we do not try to use our results very near the bow of the ship. The near-field expansion of which (33) gives the first term, is not uni- formly valid near x = -L/2. In order to examine the solution preci- sely in the neighborhood of x = -L/2 we should construct a separate expansion for a region in which x + L/2 = 0( a ), for some Y > 0. One may expect then that o, (x) is not given in that region by (34) ; rather, 9, (x) will decrease continuously to zero at x = -L/2, as physical considerations require that it must. Using (34) to express 0; (x) produces a higher-order, i.e., negligible, error in the velocity potential, provided that we restrict our attention to a region in which eY = o(x + L/2). We wish next to find ¥, , but first we need to say some more about the far-field. We expect that a two-term far-field expansion is obtained by a line distribution of sources of density i (wt - vx) e (0, (x) + 0, (x)) (35) spread alongthe line y=z2=0, -L/2 < x < L/2. it is assumed that ) (36) A two-term inner expansion of this two-term far-field ex- pansion can be obtained from (19), and itis x i(wt - yx) “1/2 jfk JkZ -in/4 dio (5) ‘= -€ = s] Vx - 2 ~ia/2 1780 Wave Forces on a Restratned Shtp tn Head-Sea Waves It is found that the near-field equations for V5 is 2 z ee ane vPened (38) role 4 0Z ( ae k) y S09 Soon ay = 19 (39) 0Z 2 av, uc 0 on the body . (40) In addition ~Y must match with the far-field solution , (39) and Ursell (1968 a) has derived a solution of (38) (40). It can be written as kZ A, (x) e + fee . [G(kY,kZ;k¢ (s),k n(s) ) Va = C(+) (41) + G(kY,kZ; - k= (s), ky (s) ) J as | where 2 2 G(kY,kZ; kE ,ky ) = Ko [x PA{Y-é) (Z-n) | oe cos coshyutl 1 : : if f ae ere exp [ik(Y -£) sinhy + k(Z+7) cos hu | du (42) 1781 Falttnsen double pole at # = 0, with a corresponding meaning for > ae a modified Besselfunction of the second kind. In (41) C(+) denotes the half of the boundary curve of the submerged cross-section at x for which y > 0; Y=£(s), Z=n%(s) are the parametric equa- tions of the curve C(+). mu(s,k) is determined from satisfaction of the body boundary condition (40). It should be noted that (38), (39), and (40) will be satisfied for an arbitrary A, (x) in (41). A> (x) has to be determined by the matching procedure. The symbol denotes integration along a contour passing wee the In order to match, we need an outer expansion of (41). Ursell (1968 a) has done that. The result is A, (x) fer - yelp Lil i pis, k) og (s) ds | (43) C(+) A three-term outer expansion of the two-term near-field solution, Cen ae (44) can now be written down. It is gil ot = v x) = % -1/2 ae KZ pains bean ‘= me C(+) (45) 1782 Wave Forces on a Restrained Shtp tn Head-Sea Waves The last term is the lowest-order term and the first term is the next lowest-order term. (45) should match with (37) and we see that it does if we set (46) C(+) ( 7 is given by (34) ), and if also /2 2 in /4 % Vik (x + 1/2) ° (47) 1783 Falttnsen which is a condition to be satisfied by ¢, . Equation (47) gives us Abel's integral equation, and it can be solved in principle. It is to be noted that the term in the brackets is a function of x, whichis de- termined in practice by numerical computation. It is the near-field solution that has the primary interest. So let us summarize our result : A two-term near-field solution of the diffraction potential is given by _iV/2 2 im /4 (¥, +¥,) ailwt - x) _ ilt - vx) ree Vric(x+L/2) ° se Hiew Bako C(+) kZ le ~ ieee [G(kY,kZ;k&(s),kn(s)) + G(kY,kZ;-ké(s), c(+) n(s)) as]| (48) where k and C are given by (26) and (27), and G is given by (42). The first term in (48) is just the negative of the incident wave and so (48) tells us that the total (incident-plus diffracted-wave) potential near the body (except near the bow and stern) will have a decay factor OE wile (49) in the x-direction. But note that mu in (48) is alsoa function of x, and so (49) does not give the total x-dependence. However, uw will be the same for similar cross-sections. So, if the cross-sections are not varying much inthe x-direction, the potential will, roughly speaking, drop off with the factor (x + L/2) -1/2 in the lengthwise direction. Note that we have assumed that the wave length is of the order of magnitude of the transverse dimensions of the ship. 1784 Wave Forces on a Restratned Ship tn Head-Sea Waves THE FORWARD-SPEED PROBLEM We write the total potential @¢ as follows is, ¥o2,t) = Ux + Ud, (x,y,z) + or (seaar,t) (50) where U®@, is the perturbation velocity potential in the steady motion problem. It can easily be shown that $7 satisfies = 10 on Sot ple (ony) (52) By combining (3) and (4) and using the assumption about linearity, it can be shown that oT satisfies the free-surface condition Since the time dependence of the incident wave is given by ead: : (see (5)), it is expected that the time dependence for the potential ¢y is also given by e 1@t This implies that we can write equa - tion (52) as Eroo Falttnsen We will write eer as $= @& + ¢ = £2 ca a aoe. (55) T I D = toa ie) where $¢p denotes the diffraction potential. ¢) must satisfy a ra- diation condition. As in the zero-speed problem we are going to use the method of matched asymptotic expansions to find ¢ D: We will assume that the forward speed uw = aie Di. een Ie (56) In the steady forward-motion problem we know that there is a length scale in the x-direction which is connected with the wave length zZ rU?/g. So (56) implies that this length scale is large compared with the transverse dimensions of the ship, and that it can be of the same order of magnitude as the length of the ship. In some way, I expect this length scale will enter our diffraction problem and affect the rate of change of the variables in the x-direction. But it turns out that it will not have any influence on the first two approximations of the diffraction potential. The important length scale in the x- direction will be connected with the wave length of the incoming wave, in the same way as for the zero-speed problem. As we remember from equation (10), this wave length is assumed to be of order e. If, however, we had assumed that ''a'' were zeroin (56), we would have beenin difficulties finding the second approximation to the diffraction potential. The reason must be that there then are two important length scales of order ¢€ in the x-direction, one connected with the wave length of the incoming wave and one connected with the forward speed, and it is difficult to separate out the effect of one of the length scales from the other. Using (7) and (9), we can show that (56) implies that the order of magnitude of the frequency of encounter, w will be qe ie Ore Sse (57) 1786 Wave Forees on a Restratned Shtp tn Head-Sea Waves We then see that the order of magnitude of +t = = is w a =? Dies = i ia ey Sek 2 (58) It is obvious that 7 will be larger than 1/4. This is important, because the solution will be singular when 1 = 1/4 (see Ogilvie and Tuck (1969) ). There are two parts in this chapter: (1) derivation of the far-field source solution due toa line of pulsating, translating sources located on the x-axis between -L/2 and L/2 (see Figure 1) and derivation of a two-term inner expansion of the far-field source solu- tion ; (2) formulation of the near-field problem, and the matching of a two-term near-field solution with the far-field solution. IV.1. Far-field source solution and the inner expansion of the far- field source solution In the far-field description we expect to have waves. Itis difficult to say how differentiation changes order of magnitudes in the far-field. So, to be careful, we would rather keep too many terms in the far-field. But we have to be sure that we have a system of equa- tions that describes a wave motion. Using arguments similar to those in the section ''Far-field source solution and the inner expansion of the far-field source solution" in the chapter on the zero-speed pro- blem, we can find that ¢p must satisfy the Poisson equation, 3 264 pullusd of a ¢ Fe ee te g(a) oe = B(y) 6(z-z) where Zn 0. We write the free-surface condition as follows : 2 (oR) (io + U 2 +4) oT 8 ae =O on ZS 0 where uw is the artificial Rayleigh viscosity, which will approach 1787 Falttnsen zero at a proper later point. This equation system does give waves. The solution to the equation system with z, = 0 can be found in Ogilvie & Tuck (1969). It is i = % 1 (xy, 2, t) = = 5 oe og: ee eae F {o (x) e : Cee (59) H—0 1 eo Ee ara 3 UR ee where -l vx -ikx -ivx F \o (x) e } = dsc. ve a(x) e —-Sso = le oc clade Ua We will rewrite (59) ina way similar to the way we did with (15) in the zero-speed problem. See Appendix C for more details. We can write oy Gry ast) =" Bix, y. 2) te (60) 1788 Wave Forces on a Restrained Ship tn Head-Sea Waves KS 0 i mea. f yy Vl Zz In the first integral, Le ( w + Uk)* jf en -(v-k In o ) ity =e) SB the second and third integrals, Qo = iwv(v-kP - (, + Uk)*/ ge. In the fourth and fifth integrals, ys = - V( Wo + Uk}4 / ge - ( »-kF Further Sit 2 2. k, = oe: (2 U/gt 1), a the parameter introduced by (56), 6B and 6, some very small po- sitive numbers and aa a : -iva x i - vp p ii = | dae o*( v-avp ) ince y al 2 47 é1- 6, a (62) pe a f2 ig 2 =1 yp Z p ley e + Uw Uw (2 a - @ +i E + 2 (1a) |? D2 a i +—a-a)|? g g Falttnsen and 16 ce mor ree l1vax -v J I Spee d o*(vt+va) dl oe a2 Hee a 0 (63) a/ 72 1lvz Veeaal -ivz [Pee AES Oe Tee ES TRE) PTY 4, eee Uw 2 Uw 2. 2 foes f afi ~c +a] Mees -1fr+—2asa) g g& Here, 6, is some very small positive number. (61) is valid for y = 0(1) > 0. We are now going to find a two-term inner expansion of the far-field source solution. We then let y be of order e€ , and we reorder the terms in (61). The procedure is shown in Appendix D. We get (64) Using (60) and the symmetry properties associated with sources, we can now write down a two-term inner expansion of the far-field source solution as 1790 Wave Forces on a Restratned Shtp tn Head-Sea Waves i( wt - vx) Vz abe tik $ y(*, y, 2; t) ~e e = we Nr *-L/2 ed ow, Vw (65) - vly| ¢ (x) - (x) ave As for the zero-speed problem we see that the two-term inner expansion represents waves propagating along the ship in the same direction as the incoming wave. Arguing as for the zero-speed pro- b lem we should therefore expect an integrated effect along the ship as the lowest order term in (65) represents. We note that (65) does not reduce to (19) when U = 0. It should not be expected that (65) reduce to (19) when U =0 since we have assumed T = = > 1/4 and since this assumption has been an important part in our analysis. We should note that the last term in (65) represents a distrubance arising from upstream while the last term in (19) represents a disturbance arising from downstream. IV.2. The near-field problem and the matching We now formulate the near-field problem and perform the matching between the near-field and the far-field solutions. A one- term far-field solution is found to be due toa line of sources with source density. i(wt - vx) 7) (x) e spread along the line y=z=0, -L/2< x < L/2. (See Figure 1). As in the zero-speed problem, a one-term near-field solution is found to be the negative of the incident wave. The matching of the far-field solution and the near-field solution determines o, (x) in a similar r791 Falttnsen way for the zero-speed problem, The two-term near-field solution is given by (92). It should be noted that ''Near-field'' means the region near the body where the distance from the body is 0(€ ). However, we do not expect the near-field approximations to be valid near the bow and stern. We will express the potential of the diffracted wave as follows : i(wt - vx) e.. (ase y (x, y, Z) (66) Using (55), (51), (54), and the fact that the incident wave po- tential satisfies the Laplace equation and the free-surface condition (54), we get that $,, will satisfy the Laplace equation and the free- surface condition (54). Putting (66) into Laplace equation gives in the fluid region. The free-surface condition is ) Z ? fo) (iw + UZ) a ais = O08 on 2 =e (68) Putting (66) into (68) gives (69) 1792 Wave Forces on a Restratned Shtp tn Head-Sea Waves The body boundary condition (52) together with (55) gives ay own 7. KZ Pol ay Ws ae ee ie ae (70) = [ivn - yn gh Ba on. 2 = hte, y) 1 Coane A last condition on wW is that it must match with the far-field solu- tion. As in the zero-speed problem, we stretch coordinates 5 Re Ge ene ee ot = SEIN, on OK Ce (71) to express that ~ varies very slowly in the x-direction compared with the variation of y in the transverse plane. We assume an asymptotic expansion of wy of the form v~>) ¥ (X,Y, Z:€ ) (72) Mhenet gigi =~ oft) as «.— 0) for fixed X,Y, Z. As in the zero-speed problem we introduce ke ==) vie (73) and ex = (74) w fe) 1¥93 Falttnsen The lowest-order equations become 2 2 2 (2. + +, - x") vy, = 0 (75) ay az ee ee ee nad Sn (76) az 1 yes yy. = -Cn ke = on Z= ht, y) Vie aN 1 3 : In addition ¥, must match with the far-field solution. A one-term far-field solution is assumed to be the potential associated with a line distribution of sources of density spread along the line y=z=0, - L/2.< x < L/2. That solution has been obtained in a previous section, and a one-term inner ex- pansion of the far-field solution can be found from (65). For any fixed x greater that - L/2, a one-term inner expansion is Hot nd Fated dia, (&) (78) Vr sah 2 2w oe He =| Va) re) In a similar way as for the zero-speed problem, one see that the only possibility for a solution satisfying (75), (76), (77), and matching (78) is by requiring that 1794 Wave Forces on a Restratned Shtp tn Head-Sea Waves and letting a one-term near-field solution of the diffracted wave be the negative of the incident wave. So Ve eee (80) We solve (79) for o, (x) formally by letting it be an equa- lity for all x > - L/2. We get Abel's integral equation to solve (See Dettman (1965) ). The solution is , [20 +2 Uy im /4 i) (x) = TW, v (x A 1/2) e€ C (81) The discussion that followed the expression of o, (x) for the zero-speed problem (see after equation (34) ) can also be applied for the forward-speed problem. The conclusion was that we had to construct a separate expansion for a region in which x+L/2 = 0( e€”% ), (¥ some positive number) , and that o, (x) is not given in that region by (81). We wish next to find ve , but first we need to say some more about the far-field. We expect that a two-term far-field expansion is obtained by a line distribution of sources of density i (wt - vx) (0, +, whe (82) 1795 Falttnsen spread along the line y=2z=0, - L/2 a (94) and d¢ é. Erp, es when r=a, - w/2 < 6 A On y)} (96) 19 0 The functions S(x,y) and O,,(x, y) will be discussed presently. Ursell considered an infinitely long cylinder and there were no appropriate conditions for x—»> oo that could determine the ar- bitrary constant B, in (96). But we consider a ship, and we have found a condition non x—>+toothat will determine B, . This is si- milar to what we did in finding the solutions (48) and (92). Then we used an integral equation approach to solve (93), (94) and (95). But for this special case with a circular cross-section it is more convenient to write the solution as (96). We will later come back to the determination of B after we have discussed the terms in (96) some more. The source term Se can be written as cos (Coan. L 1 S, (ro (lee ve f f] core in S exp(-vy cos hu + ivx sin hy) dv =c0) = 60 (97) 1800 Wave Forees on a Restratned Shtp tn Head-Sea Waves The paths in the two integrals pass respectively below and above the double pole at w= 0, We are going to rewrite (97) so that we can more easily evaluate it numerically. We introduce me vy sin hu as a new integration variable. We then introduce closed integration paths in the complex a plane properly indented at the branch points of the integrand. By using the residue theorem, we can then write (97) as 0) OO : Zz 2 Ss. ee | i) Oss eee * cos(uy) - » eens [=| : +u 2 2 0 a oe +u (98) The other unexplained terms in (96) are the wave-free potentials O,,. They are given by O'. (9) = Kom of ( vr) cos(2m - 2) 6 + 2K , (vr) cos(2m - 1) +K 6 See mi??? cos2m 2 eles (Ursell denoted these functions by or 1801 Falttnsen K. are modified Bessel functions as defined by Abramowitz and Stegun (1964). The coefficients A, and A,, in (96) are determined by satisfying the body boundary condition (95). This leads to an equation of the form ice 20 - vr cos@ ~ ) A ~~ 6ycos 0 e = = 0 (100) r m = Il * fe) Or We have assumed here that we can differentiate the infinite series in (96) term by term, and we have used the fact that x = peel ; y =’ cos 6 (101) a0 Fes in (100) is obtained from (99) and by using 9.6.26 in Abramowitz and Stegun (1964). So dO m 2m - 2 Fe © covet Ky er gh #2) tomrepe aa aie) OOS + 2 {-K iiepteat (vr) vcos(2m - 1) @ 2m vr 2m -1l m = Be oe K, (vr) } veos 2mé6 We will now describe in more detail how to solve (100) nume- rically and how numerically to evaluate S, in (96) and oe, in (100). Equation (100) is solved by setting up a least-square condi- tion. One assumes that the infinite sums in (96) and (100) converge sufficiently rapidly so that a finite number of terms in the infinite sums gives a satisfactory approximation. One calls this number M. 1802 Wave Forces on a Restratned Shtp tn Head-Sea Waves The least-square condition leads to the linear equation system 80. -(, 8.) IO°(e, 6.) n Be | - m i > Hea Das Or or m= 0 i=l (102) = 80 (r, 8.) - vacos@, n i = p ) GO6,6. 3/2 (M+ 1). It was found that N= 10 gives satisfactory results. 6; have been chosen as us Lanne al! ae one Ghph ee a Gg eae 3 NN (103) (102) can be solved by standard methods. S, in (96) are evaluated in the following way. We introduce (101) in (98) and write S, as 5 - vr cos@ Si c=) (= 2 resin) 6) “é a Z 2 ee ob ae +2 f ee een cos@) e ls ai z a 0 ae a A | 2 3 | du uv -r sin 0 ae 0 i 0 5 (23/2 sin(ur cos@) e (e+ (104) + O(B,) for” 0. =< ee" x72 1803 Falttnsen Pace sin@ v7 + ne where B,= l r sin @ A is chosen so that B, is sufficiently small. It was found that it was satisfactory touse B, = 0.0001. Each integral in (104) was evaluated by first locating the zeroes for the integrand. For the first integrand the zeroes are easily found to be u=0 and u = Spee gore + 1% m= 0, 1g and for the second integrand the zeroes are u = reos gms m =—Osmele 2...Between each zero in an integrand we then used Simpson's for- mula. As is seen above the length of the interval between each zero depends on @ , and so the number of points used in Simpson integra- tion should depend on 6. When @ was close to 7/2, as many as 50 points were needed in the Simpson integration. But when 6 was close to 0, it was only necessary touse 8 points. If A was less than u at the second zero of an integrand, then Simpson's formula was only used between 0 and A. 9So was numerically evaluated in a way similar to that for = So. We have now explained how to obtain numerically the terms in the brackets of (96). We will refer to these terms as "Ursell's solution'' and denote them by @¢ ae So Os B. ¢ (105) (See (96) ). ,, has been plotted in Figure 3 as a function of yr for different values of @ and in Figure 4 asa function of 6 for different values of pr. We now have to find B, in (105). By will of course be de- termined in the same way as we did in the previous chapters where we solved the zero- and the forward-speed problem. We prefer now to use the coordinate system shown in Figure l. If we take an outer expansion of (105), the term which is linear in y will be 1804 Wave Forces on a Restratned Shtp tn Head-Sea Waves y| e (106) (see (96) and (98) ). In accordance with what has been done in the previous chapters, (106) should (for both the zero-speed and forward- speed cases) match with : ly | beg x, (2c) i( wt - vx) (107) where oa 1) can be written for both cases as a (2w +2 Up ) it /4 gh a, (x) = rey (x + L/2) e a (108) By equating (106) and (107) and putting the expression for B, into (105), we can write the potential es acces Zw +2Up ) gh i( wt - yx) 62 wae rw v(x + L/2) upeh,« san Oo Oo Oo (109) Using Bernoulli's equation, it is now easy to find the pressure. To the leading order the pressure will be ap Agh ae a/2 20 FF 2° 7 ¢ ei( wt - vx) (110) pgh 277A) ro v(x + L/2) “u One should note the simple forward-speed dependence in (110). @, and A, will only depend on the wave length. So for a 1805 Faltitnsen given wave length the amplitudes of the pressure, force and moment for a given forward speed can be obtained from the corresponding values at zero-speed by multiplying the zero-speed results by acons- tant factor. V.2. Comparison with Experiments C.M. Lee has measured the pressure-distribution along a restrained, semi-submerged, prolate spheroid which was towed ata constant speed in regular head-sea waves. He used the experimental pressure values to calculate a longitudinal force distribution along the spheroid (C.M. Lee (1964) ). He did not publish the data for the pressure distribution along the spheroid, but he was kind and gave us those data. The surface of the prolate spheroid that C.M. Lee used can be described by the equation : 2 DRETZ x +47 = 1, 2 2 => om oO where { = 19.8" and bg = 3.3". x,y,z are defined by Figure 1. He measured the pressure at cross-sections located at x = -16" (called Cy), x= - 12.5" (called Bp), x=-7" (called Ap), x = 0 (called (QD), x = 7" (calledAy), x= 12.5" (called By), x = 16" (called Cy). He did the experiments for } /L = 0.5, 0.75, 1.0, 1.25, 1.5 and 2.0 where A is the wave length and L is the length of the model. The Froude numbers of the model were F,, = 0..0827,0.,.123, 0. 164,, 0..205,. 0. 246, 07526. In our theory we have assumed that the wave length is of the order of magnitude of the transverse dimensions of the ship. But note that this does not necessarily mean that the theory is bad for larger wave lengths. One can refer to the strip theory which has shown to give good results for a wider range of wave lengths that one rationally has to restrict oneself to. The comparisons with the ex- periments by Lee seems to indicate that our theory is not good for wave lengths \ /L= 0.75 and larger. We will therefore only show the comparisons for A/L = 0.5. There were evidently some irregularities in the experiments 1806 Wave Forces on a Restratned Shtp tn Head-Sea Waves for F,, = 0.328, and so we did not compare experiments and theory for that Froude-number. We decided to present the comparisons bet- ween experiments and theory for Froude-numbers 0.082 and 0.205, but the agreement between theory and experiments was just as good for Froude-numbers 0.123, 0.164 and 0.246. In Figures 5 through 10 are shown the comparisons of the pressure amplitudes for Froude-number 0.085. Figure 5 shows the longitudinal distribution of the pressure amplitude along the keel of the spheroid. It is seen that the experiments confirm the theoretic- ally predicted longitudinal deformation of the wave along the ship. Figure 6 shows the pressure-variation along the cross-section Br. (The index F indicates that the cross-section is on the forward part of the model.) The variable 6 , the abscissa in the figure, is a /2 for a point in the undisturbed free-surface and 0 fora point located on the center plane of the model. It is seen that the agreement bet- ween theory and experiments is reasonably good, Similar compari- sons are made for cross-section A, in Figure 7, cross-section in Figure 8, cross-section A, in Figure 9, cross-section Ba in Figure 10. (The index A indicates that the cross-section is on the after part of the model.) It is seen that the agreement is good, es- pecially for the after cross-sections. In Figures 11 through 16 are shown the comparisons of the pressure amplitudes for Froude-number 0.205. Figure 11 shows the longitudinal distribution of the pressure along the keel of the spheroid. Figures 12-16 show the pressure variation on the cross-sections Br, Ap, 0) , Ag and By , respectively. It is seen that the agreement between experiments and theory is at least as good as in the case of the smaller Froude-number. Since the theory is not as- sumed to be valid near the bow or stern, no comparisons have been made for cross-sections Cp and Cy. In Figure 17 is shown the comparison between theory and ex- periments for the longitudinal distribution of the phase angle of the pressure. The theory predicts that for all Froude-numbers the phase- angle of the pressure is 7/4 before the phase-angle of the Froude- Kriloff pressure. For a given cross-section, the experimental value of the phase angle varied somewhat. So the presented data are ave- rages. The variation is, roughly speaking, not more than + 10°. It is seen that the agreement between experiments and theory is good. 1807 Faltinsen ACKNOW LEDGEMENT This work is based on a doctoral thesis at The University of Michigan (Faltinsen (1971) ). The author was then on leave from his position at the Research Department, Det norske Veritas, Oslo, Norway with a financial support from The Royal Norwegian Council for Scientific and Industrial Research (NTNF). I want to thank the chairman of my doctoral committee, Professor T. Francis Ogilvie. Without his encouragement, helpful suggestions and contributions this work would never have been done. I am also thankful to Professor R. Timman, Technical Uni- versity of Delft. The discussions with him were a turning point in my work, Further, Iam thankful to Dr. Nils Salvesen, Naval Ship Research and Development Center, Washington DC., Dr. Choung Mook Lee, Naval Ship Research and Development Center, Washington DC. and Mr. Arthur M. Reed, University of Michigan for their con- tribution. NOTATION a used in the description of the order of magnitude of the velocity U,U = 0(,'@-# ), Onis ce bf Ze g = Be “o Fn = «Oy V Lg , Froude number. g acceleration of gravity. G(kY, RZsk Eo ky ©) wave source potential (see (42) ). h wave amplitude of the incoming wave. h(x, y) function defining the wetted surface of the ship. k equal to v.€ in the sections about the near-field problem and the matching. Other- wise integration variable in the Fourier transform, 1808 x, Y,Z Wave Forces on a Restratned Shtp tn Head-Sea Waves -£— (20, U/g + 1) U length of ship. coordinate-axis in the direction of the out- ward normal on the wetted surface of the ship. stretched coordinate (see (24) ). (Can also mean a number. ) coordinate-axis normal to and out of a cylin- der with the same cross-section as the ship at a given section. i=1, 2, 3: the x-,y-,z- component of the unit normal vector to the wetted surface of the ship. radial coordinate used in the chapter : "Numerical Calculations" (see Figure 2). time variable draft of the ship midships. forward speed of the ship. Cartesian coordinates (see Figure 1). (The ship moves in the direction of the negative X-axis, zZ is measured upwards, y to starboard), stretched coordinates (see (24) or (71) ). vy -k v | where k is an integration variable. very small positive number. very small positive number. very small positive number. 1809 Falttnsen slenderness parameter. It is a measure of the transverse dimensions of the ship com- pared with the longitudinal dimensions of the ship. C(x, y, 6) free-surface displacement 6 angular coordinate used in the chapter : "Numerical Calculations'(see Figure 2). 6 = 0 isa point on the centerplane. x wave length of the incoming wave. fictitious (Rayleigh) viscosity. (Note that LM K (arg) has another meaning. ) 2 w PA v _ 9 = 8 d p density of water (mass per unit volume). i( wt - vx) , : 5 : : a(x)e source density per unit length in line dis- tribution of sources. foe} -ikx : a *(k) = fp dxe a(x) (the sign * means -00 here inverse Fourier transform). wU a = ea al g ¢ (x, y; z, t) velocity potential in forward-speed problem and in zero-speed problem.” $ (x, We os By velocity potential of the incoming wave. op (* y, Z, t) velocity potential of the diffracted wave. (x, y, Zz) (1/U)* perturbation-velocity potential in steady motion problem. * Note, however, that in Chapter V ''Numerical calculations", it means the time dependent part of the velocity potential. 1810 Wave Forces on a Restratned Shtp tn Head-Sea Waves (x; y> Z, t) time-dependent part of velocity potential. @(x, y, z) see (16) and (17) for zero-speed problem. See (60) and (61) for forward-speed pro- blem. V(x, y, z) see (20) in zero-speed problem. See (66) in forward-speed problem. - N 1= 1 2 a 0) @: 0 6 ‘s> 6 © N, y ~ 52; vi i= 1 wave frequency of the incoming wave. oF vU fe) REFERENCES ABRAMOWITZ, M., and STEGUN, I.A., ''Handbook of Mathe- matical Functions'', National Bureau of Standards Mathematics Series, 55, Washington, DC. (1964). DETTMAN, John W., ''Applied Complex Variables", The Macmillan Company, New York - Collier - Macmillan Ltd, London (1965). ERDELYI, A., ''Asymptotic Expansions'', Dover Publications, Inc., New York (1956). FALTINSEN, O., ''Wave Forces ona Restrained Ship in Head- Sea Waves", Ph. D. Thesis, The University of Michigan, Ann Arbor (1971). JONES, D.S., ''Generalized Functions", McGraw-Hill Book Co., New York (1966). LEE, C.M., ''Heaving Forces and Pitching Moments ona Semi-submerged and Restrained Prolate Spheroid Proceeding in Regular Head Waves'"', Report No. NA-64-2, Institute of Engineering Research, University of California, Berkeley, California, (1964). 1811 10 ry 12 13 14 r5 Falttnsen LIGHTHILL, M.J., ''Fourier Analysis and Generalized Functions", Cambridge University Press, Cambridge (1958). NEWMAN, J.N., ''The Exciting Forces on Fixed Bodies in Waves", Journal of Ship Research, 6:3 (1962) 10-17. OGILVIE, T.F., Unpublished work (1969). OGILVIE, T.F, and TUCK, E.O., “A Rational Strip Taeers of Ship Motions : Part I'', Department of Naval Architecture and Marine Engineering, College of Engineering, The Univer- sity of Michigan, Ann Arbor, Michigan, Report No. 013, March 1969. OGILVIE, T.F., ''Singular Perturbation Problems in Ship Hydrodynamics", Department of Naval Architecture and Marine Engineering, College of Engineering, The University of Michigan, Ann Arbor, Michigan, Report No. 096, October 1970. SALVESEN, N., TUCK, E.O., FALTINSEN, O., "Ship Motions and Sea Loads''", SNAME Transactions, 78, (1970) 250-287. URSELL, F., "On Head Seas Travelling Along a Horizontal Cylinder", J. Inst. Maths. Applics., 4 (1968 a) 414-427. URSELL, F., ''The Expansion of Water-Wave Potentials at Great Distances", Proc. Camb. Phil. Soc., 64, (1968 b), 811. VAN DYKE, M., ''Perturbation Methods in Fluid Mechanics", 1964, Academic Press, New York and London, 1812 Wave Forces on a Restratned Shtp tn Head-Sea Waves APPENDIX A Simplification of the far-field expansion in the zero-speed problem We will show how (15) can be rewritten into (16) and (17). The procedure is based on work by Ogilvie (1969). We first introduce in (15). If we drop the primes we can write (15) as $ p(X ¥izet)| => pbx ynw) a ping (A-1) where ] (x, vi z) = 5 dk eik* o* (k) 4n 5 rm | a Boney ee php O Dey, aN a aedia ace at aes (A -2) We will let y = 0(1) and we are going to. assume that y > 0. We define —y obyeen ONE +(v -k I(k) = lim ae (A -3) ! Falttnsen The poles of the integrand are important in the evaluation of I(k). They are given in the limit u—0 by: Ae (2v -k)k Let us first study the case in which these singularities are imaginary, which means k < 0 or k <2pv_ . Then we study case IL,@in whichy O:< ok =< Zev Gase lo kia 0 ork — Zane. By introducing a closed curve in the complex ( -plane pro- perly indented at the branchpoint i |v -k| of the integrand of (A-3) and using the residue theorem, we will get J Fae v/ 2 ‘ao vz - y Vk(k-2») 0 lvz pe) a re) = MBewioes tnd Seu Loada, fer eee eee mart 2 Z Nice) i Ea -1vz v2 - be og ee Vl 2 p2 -a -i (A -4) v-k : F Here a= It can be shown that the integral term in (A-4) is exponentially small with respect to e Caseul = O0— ke < 2)y)- The poles of the integrand of (A-3) are now real. The Rayleigh viscosity is helping us to determine how to indent the inte- gration path of I(k) around the poles. We get by using the residue theorem in the same way as for Case I that 1814 Wave Forces on a Restratned Shtp tn Head-Sea Waves (A -5) It can be shown that the integral termis 0(€) compared with the first term. By using (A-2), (A-3), (A-4) and (A-5), we can now write ye Ae ikx =y Nk(k = 2» ) o*(ik) @(x,y,z) = - = nn 4 - 00 Vk(k - 27) 0 \ k(2v -k) o* (k) [oe) Pl ba) i - k(k - 2 ; i ve eikx y ( v ) 2v k(k - 2» ) + higher order terms This expression can be rewritten as Eq. (17). 1815 (A -6) Falttnsen APPENDIX B Inner expansion of far-field source solution for the zero-speed pro- blem We will show how a two-term inner expansion of (17) can be written as (18). We let y be of order ¢€ , and we reorder the terms in (17). By expanding the integrands in (17) we obtain oes ikx ¢*(k) @ (x, y, Z) = = Bip = / dk e SSS ey V2 v [ic]. enlLa.8,) vZ ‘3 ve ikx + a y dk e o*(k) __ sist) 0 : v iux g*(ut+2v ) r ve i2vx du e a 2 V2» [ul -(1-6,) 12 l é-(1-t0) Mealy Te / iux o*(ut2y = e due an 2 PU 0 + higher order terms 1816 Wave Forces on a Restrained Shtp in Head-Sea Waves The integration limits Sy “(Ul “6,) role in obtaining (B-1). have played an important Note that the first brackets in (B-1) contain the lowest-order terms, the second brackets the next-lowest-order terms. Because we,want to apply Fourier-transform techniques, we want to set ¢ “(1-9 equal tooo. For the three higher-order terms in (B-1) we could do that ; the effect would be only to introduce higher-order, negligible effects. But we must be careful with the lower order terms. But assuming that o(x) and o'(x) are continuous in the interval -L/2 < x 0 V | F *(k) denote the Fourier transform ofa function F(x). So et n/4 F(x) =—————-_ H (x) (B -3) rie where H(x) is Heaviside step function. We also define F = u< 0 e*q) =] 01 (B -4) eee wd >,0 Vu So re 1/4 G(x) =—————— H(-x) (B-5) We can now write (B-1) as 1817 Faltitnsen vZ 2 1 @ (x,y,z) = 2 | dk ie o*(k) F *(k) + vye a(x) 27 V2, -00 oo vZ i2vx e Lai eee 20 4 2v a) Using the convolution theorem we get -inr/4 [* dia(&) vz @ (x,y,z) = - vee ae / | ———— + vy e a(x) 7/2 V2 wrx LZ yee etx bh dent) ies x By using an asymptotic expansion of the last integral (see Erdélyi (1956)), (B-7) can be written as Eq. (18). Que (a 2 Zu). GG (B -6) APPENDIX.G Simplification of the far-field expansion in the forward-speed problem We will show how (59) can be rewritten into (60) and (61). We first introduce in (59). If we drop the primes we can write (59) as bpleyezt) = (xy,2) eto '™) (C-1) 1818 Wave Forces on a Restrained Shtp tn Head-Sea Waves where co 1 ik Gey, 2) = = / dice o*(k) 2 4n (C-2) oo . f VJ 1)* f2 af fey a (v-k) + . dig a = 2 Zee Z Pa Nee ea -— (w, + Uk = in) We will let y = 0(1) and we are going to assume that y > 0. We define (C-3) The poles of the integrand of (C-3) are important. They are given in the limit by (C-4) p. + I Z 2 = - ot Lal - ( v -k) We have to study the sign of the radicand in order to deter- mine the location of the poles in the complex -plane. Let us define 1819 Falttnsen 1 : (C-5) ~*~ iT] g a eee (c-6) Zz, v2 g It can be shown that when k > k, or k k, : The poles of the integrand of (C-3) are now real. The Rayleigh viscosity mu is helping us to determine how to indent the integration path of I(k) around the poles. By using the residue theorem in the same way as for Case I, we will get a similar result as (C-8). In this case, however, we cannot say that the integral terms are exponentially small for all k. We can write if eVvg psy U fe) oa, es | e e are ars I(k) = + exponentially small terms SS f 1yz S - a for pavee hs? ife! «” E(k) = poten < 2 a 2 wie Uk eat afr 2 1821 Falttnsen + exponentially small terms (Ce Soy. . (C-9) Here 6, and 6, are some very small positive numbers. '"'a'' is defined by (56) andis restricted to 0 Bp Zo tBUr Ls 0 [te a Bey VW vw fe) Oo co v ye ne ikx a: dk e o * (k) 2m Megs : i — ivx mag PE ig Sede o(v + k,) | dve” ¢(v + k,) e . het Teme ew eee ee ene Wey Pe 2a 0 (D. 1) We should note that we have changed the integration limits from + ¢~'1-8-Blto +. This can be justified for the two first integrals, which is the lowest order terms, by using the fact that o *(k)k 3 remains bounded as k>+o. For the three last integrals, which is the highest order terms, it is obvious that we can change Vet a{{=ac the integration limits from f= e mt Big too 1824 Wave Forces on a Restratned Shtp tn Head-Sea Waves By using (B-2), (B-3) and the convolution theorem we can now write (x,y,z) 2 - ———__— te | TL ve Pe) (D-2) ik_x+ vza+ i 7/4 sy : _e : i dia biucnpee tz -L/2 Zw + Z2Up -4 va By using an asymptotic expansion of the last integral (see Erdélyi (1956) ), (D-2) can be written as (64). 1825 Falttnsen noou wi Figure 3 Ursell's solution as a function of Vr for a given value of 8 1826 Wave Forees on a Restratned Shtp tn Head-Sea Waves 0.5 1.0 6 (radians) Figure 4 Ursell's solution as a function of 6 for given value of/r 1827 ptozoyds ay} jo [eey ey} BSuote sxrnssoead ay} Fo uoT{Nqt414sSTp TeuIpNyisuoT g ornsty YS Yo Wy ” a a a NOILWLS Ce St Wl SSG eu Bo = OL 6 8 L 9 S v £ 4 1h tetas Falttnsen AzosuL oO sjuowtzedxg x 1828 Wave Forces on a Restrained Shtp tn Head-Sea Waves 0.5 1.0 6 (radians) Figure 6 Pressure variation along the cross-section Bo. 1829 Falttnsen x Experiments o Theory 0.5 1.0 6 (radians) Figure 7 Pressure variation along the cross-section An 1830 Wave Forces on a Restratned Shtp tn Head-Sea Waves x Experiments o Theory 0.5 Mele) 6 (radians) Figure 8 Pressure variation along the cross-section & 1831 Falttnsen feo A/L Fn 1.4 ~ x Experiments o Theory 12 ——— 1.0 Af Je + 0.5 0.082 0.5 1.0 @ (radians) Figure 9 Pressure variation along the cross-section Ay 1832 Wave Forces on a Restrained Shtp tn Head-Sea Waves egh Wa ee / aleof N ~ \ Sy SS oS ~ 9 1.6 A/L = 0.5 Fn = 0.082 1.4 x Experiments o Theory : pan = ; & 0.6 LZ i z 0.2 0.5 1.0 6 (radians) Figure 10 Pressure variation along the cross-section By 1833 prozoyds ay} jo [2ey ey} BuoTe ernssoad oy} Jo UOTINqIA4SIp [eUIpMTsuoT [][ ens V NOILVLS Sits Si Falttnsen ro) a AzooyL oO squswtzedxqg x v°O 1834 Wave Forces on a Restrained Ship tn Head-Sea Waves x Experiments o Theory é Los a8 ; —— 0.5 1.0 6 (radians) Figure 12 Pressure variation along the cross-section Bo £635 Falttnsen x Experiments o Theory 0.5 2.10 @ (radians) Figure 13 Pressure variation along the cross-section Ap 1836 Wave Forces on a Restrained Shtp tn Head-Sea Waves 6 (radians) Figure 14 Pressure variation along the cross-section 1837 Faltinsen x Experiments o Theory 0.5 1.0 6 (radians) Figure 15 Pressure variation along the cross-section AK 1838 Wave Forces on a Restrained Shtp tn Head-Sea Waves A/L = 0.5 Fn = 0.205 x Experiments o Theory 0.5 1.0 6 (radians) Figure 16 Pressure variation along the cross-section By 1839 Falttnsen degrees A/L = 0.5 Theory For All Froude Numbers —O— Experiment Fn = 0.082 Experiment Fn = 0.123 Experiment Fn = 0.164 Experiment Fn = 0.205 Experiment Fn 0.246 STATION Figure 17 Longitudinal distribution of the phase angle of the pressure 1840 Wave Forces on a Restratned Shtp in Head-Sea Waves DISCUSSION Michel Huther Bureau Verttas Parts, France I have been very interested by the presentation of the author, and I should be very pleased to know his opinion about the limitations and possible extensions of the method. I am interested to know the order of magnitude it is possible to use for the slenderness parameter. Also, have calculations been done with other section shapes as rec- tangular, for example, shapes more similar to the nowaday midship sections than the cylinder ? The author writes that linear superposition can be used. I agree with this point of view for longitudinal motion such as pitching and heaving, but I should be pleased to know the author's opinion in the case of transverse motions such as rolling, where large ampli- tudes are to be considered. DISCUSSION Cheung-Hun C. Kim Stevens Institute of Technology Hoboken, New-Jersey, U.S.A. Dr. Faltinsen proposes a method for improving the present strip method of evaluating the wave-exciting forces. His method is based on the slender body assumption and the comparison was made with the corresponding results of calculations based on the Froude- Krylov hypothesis. I would like to know why the comparison was not made with the corresponding results of calculations based on the strip- wise diffraction theory. 1841 Faltinsen REPLY TO DISCUSSION Odd Faltinsen Det Norske Verttas Oslo, Norway The first limitation that I mentioned in my presentation was the small wave length assumption. I would guess that the upper limit for using the theory would be for a wave length divided by the length of the ship between 0.5 and 0.75. Another limitation is that the theo- ry is not valid near the bow or the stern of the ship. Near the bow there is a singularity in the solution. For practical purposes I think one can use the Froude-Kriloff theory in the bow region. The theory is not restricted to circular cross-sections and it is shown in the main text how to solve the problem for arbitrary cross- sections. But no calculations have been performed for arbitrary ship forms. Further it remains to test how slender the ship ought to be for the theory to be valid. But the slenderness assumption is not expect- ed to be a great problem for conventional ship forms. To get a similar theory for an arbitrary wave length does not seam easy. However, for the long wavelength range one can use the Froude-Kriloff theory to find the pressure. I do not find the question about linear superposition to be ap- propriate in this context. But I agree that the linear superposition principle can be questionable in the case of roll. Further I did not hear good enough Dr. Kim's question about the strip method. DISCUSSION Cheung-Hun C, Kim Stevens Instttute of Technology Hoboken, New-Jersey, U.S.A. Dr Faltinsen evaluated the pressure distribution along the 1842 Wave Forces on a Restrained Ship in Head-Sea Waves keel line according to the method based on the Froude-Krylov hypo- thesis and for relatively short waves. It is well known that the Froude- Krylov assumption is valid only for ery long waves. Would it not be better then to compare the results with calculations based on a strip method which evaluates rigorously the diffraction potential in the strip domain ? REPLY TO DISCUSSION Odd Faltinsen Det Norske Verttas Oslo, Norway I see what you mean. I have not done that comparison. The reason why I showed a comparison with Froude-Kriloff pressure in the presentation of the paper, was to make clear that there is an order of magnitude between Froude-Kriloff pressure and pressure accord- ing to my theory in the low wave length range. I do not agree with Dr Kim's last statement that the usual strip theory evaluates the diffraction potential rigorously in the strip do- main. The usual strip method for zero speed is based on no inter- action between different parts of the ship, and the two-dimensional Laplace equation is used to solve the problem. This can only be true for beam sea, 1843 re -_ Logyd woly72iv shared oc HO bowsd hedjanz edt od pribrossa vhvvot ed) tadt mwool Hew al 1 .covew fray cers rat f 7 RBI OOMP cy PE 8 5! Anmubype ive 8 to melee? antifeli>iasd Pi ivi #3 tagrnns at qitie oni of ee yoy dotisatiilh ah visvoregls asjoulave a> IT Pe A ral Lr Yoel Ud20 OF ¥I98 a ene fr minty gem Re ya tat 1 OR . ee pet \ 4 F ; sed . Beneta BHA) coat ai Glee te tie oe Ape Aa seh of ti, sein. 2Ten the’ . wimtlga, Pom priahee, purpraes if tae & es icity cneor’ tw Tike Sy! CeRPOR, \ ad giteaamos fedy seob Joul 6 ‘re L .neery roy tadw oon Tl as area SSIS HE eo Pos Cine OS abywdrd Ba xe He et Se SBE RATS SAEv'od ane" aA ia 2 yy, J Hor RAY 9 ip ey cd a iy Be ot hae rey fiolr ay nhs: fot PCr et | a by } wn Wortay shane! BAND EE oil? bP oa aes . VE ary & 2 eee UAVS ey Te ee P ‘ ators | Pelatu Blt TRAP Fes Aare 16a Sere Tee seeds joa ob I | hb ar«de i ch vieworoulsy. eiidsior AOI ed Caden eye (leet bea wt bye Bran re) Bare 4 fer Bi th fataennee Tilo esr LD GOF GA) Bh tebe SRM Te Rss he i So tis AA SPIES Ait? BE vide aro état 43 aha sic“ ide’ oo beat al sooeupe oR © ry! 4 04 Pra hy iv 400 ahd Lin (O a r , Msae@eas oS Oe Pipe i ‘ the Cages ith, Purtaer t Geo eaten uv the sietp tiethoG. yam > eonir y ’ uF i Pea (4baet) eva igeted ‘ coc peered Gatti option alodg é FREE - SURFACE EFFECTS IN HULL PROPELLER INTERACTION Horst Nowacki Untverstty of Michtgan Ann Arbor, Mtchtgan, U.S.A. Som D. Sharma Hamburg Shtp Model Bastn Hamburg, Germany ABS TRACT The quantitative role of wavemaking at the free sur- face in the phenomenon of hull propeller interaction is investigated by means ofa general scheme devised to determine the potential, viscous and wave compo - nents of wake and thrust deduction. It requires the concerted application of various analytical tools such as the lifting line theory of propellers, the method of singularities for representing the hull and propel- ler by source distributions and the linearized free- surface theory of wavemaking, as well as model ex- periment techniques suchas the conventional Froude analysis of propulsion factors, nominal wake mea- surements and wave profilemeasurements. The pro- cedure is actually applied to the specific case ofa thin mathematical hull form driven by a four-bladed propeller of simple geometry. It is found that the wave component is dominant inthe wakeand quite si- gnificant in the thrust deduction at Froude numbers around 0.3. Surprisingly, there seems to be an ap- preciable viscous component in the thrust deduction at practically all Froude numbers. 1845 Nowaekt and Sharma I. INTRODUCTION The purpose of this research was to clarify by analysis, com- putation and experiment the quantitative role of wavemaking at the free surface in the phenomenon of hull propeller interaction and con- sequently its contribution to the hydrodynamic propulsive efficiency of the system hull and propeller. Following Froude (1883), hull propeller interaction is con- veniently studied in terms of three propulsion factors : wake, thrust deduction and relative rotative efficiency. The wake is caused by the presence of the hull and the free surface and is a simple measure of the change in propeller inflow as compared to an equivalent open- water condition (free running propeller in an infinite parallel stream). The thrust deduction is really an indirect expression of the fact that the force of resistance acting on the hull is modified (usually augment- ed as compared to the towed condition) as a result of propeller action. With the present state of our knowledge, only wake and thrust deduc- tion are amenable to rational analysis, the relative rotative efficiency being an empirical catch-all for various unclarified effects of relative- ly insignificant magnitude. Since the fundamental work of Dickmann (1938, 39), it has been customary to study hull propeller interaction as a superposition of three basic effects : ''potential'' effects due to an ideal displacement flow about a deeply submerged double body (the zero Froude number approximation), viscous effects due to the boundary layer and viscous wake, and wave effects due to the presence of the free surface. Using standard symbols w and t for wake and thrust deduction fractions respectively, one may write formally = + + 1 w w Ww (1) a ge 2 eee: (2) where the subscripts p,v and w denote potential, viscous and wave respectively. By comprehensive theoretical analysis and careful ex- periments Dickmann demonstrated that the most significant compo- nents were Wp? tb and Wy - Among Dickmann's most impressive achievements were 1) a theoretical relation between potential wake and trust deduction in- volving the thrust loading coefficient, and 2) a reasonable explanation of the effect of the free surface on propulsive efficiency. His main analytical tools were a simple actuator disk model of the propeller 1846 Free Surface Effects tn Hull Propeller Interactton (momentum theory), the method of singularities (Lagally's theorem) for calculating forces on the hull and Havelock's method of images for a linearized treatment of the free surface. In recent years considerable effort has been put into the in- vestigation of potential and viscous effects in hull propeller interaction (see Bibliography). Especially in this country, Beveridge ina series of papers (1962, 63, 66, 68) has refined the technique of calculating the potential thrust deduction to a state of near perfection. At the same time, Hucho (1965, 68) in Germany has made significant contributions to our understanding of viscous effects. The wave effects, however, were persistently ignored for nearly thirty years since Dickmann(1939), until the fundamental treatise of Yamazaki (1967) revived interest in this subject and inspired the recent work of Nakatake (1967, 68) in Japan. Still far from resolving the complex issues at stake, Nakatake's papers are just added evidence of the same conviction that underlies the present study(which, incidentally, was initiated without knowledge of the Japanese effort), namely that the time is now ripe to make a fresh attempt at the further clarification of this admittedly difficult problem. This is due mainly to the following reasons : 1) Major advances in the vortex theory of propellers now allow the use of a far more refined mathematical model of the propel- ler. 2) The recently developed technique of wave profile measure- ment and analysis enables us to verify by (almost) direct measurement the wave effects predicted by analytical theory. 3) The general availability of large electronic computers al- lows the use of more realistic singularity distributions for represent- ing the hull, the propeller and their images in the free surface. Beside the intrinsic interest of a fundamental problem in ship hydrodynamics, a recommendation by the Performance Committee of the International Towing Tank Conference 1966 for specific research in the basic problem area of hull propeller interaction - of which free surface effects are certainly the most intriguing aspect - as wellas the prospect of practical application to modern high speed craft with propellers operating at shallow or even partial submergence were further motivations for undertaking this research, 1847 Nowaeckt and Sharma Il. "GENERAL APPROACH The originality of the present study lies not in the develop- ment of a novel method but in the concerted application of miscella- neous existing analytical, computational and experimental techniques to our specific purpose. Since these numerous tools have to be appli- ed in a rather intricate sequence to get the information desired, it seems necessary in the interest of clarity to precede the account of work done by a brief schematic description of our general approach. The internal details of the individual techniques are only of indirect interest in the present context and will therefore be banished to appro- priate appendices. The basic aim is to determine for a given hull-propeller sys- tem the propulsion factors and their potential, viscous and wave com- ponents by all feasible analytical and experimental means, This dic- tates roughly the following set of operations. First, a considerable amount of basic information can be ga- thered by a number of independent experiments and theoretical calcu- lations which may be executed in any convenient sequence. On the ex- perimental side we may deploy the following more or less routine mo- del tests in the towing tank : E1) Hull resistance test, E2) Propeller open water test (at deep and shallow submer- gence), E 3) Self-propulsion test with hull and propeller, E4) Nominal wake measurements behind the hull in forward and reverse motion, and E5) Wave profile measurements (e.g. longitudinal cuts) for the hull with and without propeller. On the theoretical side only few calculations can be perform- ed without resort to some empirical data ; these are : T1) Wavemaking resistance of the hull, T2) Wave wake induced by the hull in the propeller plane (both in forward and reverse motion), and T3) Potential wake induced by the hull in the propeller plane. 1848 Free Surface Effects tn Hull Propeller Interaction From here on the further analysis is of a semi-empirical nature and must be conducted in an essentially predetermined sequen- ce because at each new step certain information from previous stepsis required . It is helpful to list separately the pure hull analysis , the pure propeller analysis , and the interaction analysis . The purpose of the hull analysis is to verify the mathematical representation of the hull as a source distribution and to establish the degree and range of validity of the linearized wave theory . H 1 ) The total resistance measured in step El can be subjected to a simple form-factor analysis (based on a suitable plane friction formula) so as to yield the viscous and wavemaking components . H 2 ) An alternative estimate of wavemaking resistance can be obtained from a Fourier analysis of the wave profiles measured in step B54) H 3 ) The experimental estimates of wavemaking resistance derived in the two preceding steps may now be compared with the theo- retical calculations of step Tl . H 4 ) For a more exacting test of the theory the experimental and theoretical free-wave spectra can be compared at each speed . H 5 ) An additional test of the theory lies in comparing the sum of the calculated wave wake and potential wake from steps T 2 and T 3 to the measured wake in reverse motion from step E4 since the latter is essentially free of viscous effects . H 6 ) If the mathematical model of the hull flow can be verified in the preceding steps then the calculated wave wake and potential wake may be subtracted from the measured total wake in forward motion to yield the important viscous wake component . The purpose of the propeller analysis is to determine a vortex model of the propeller and to verify the validity of its alternative representation as a source distribution which is to serve as the basis for calculating thrust deduction and wave effects . P 1) A computer program based on lifting line theory in con- junction with the Lerbs ( 1952 ) induction factor method may be used to calculate for any given propeller geometry and assumed foil cha - racteristics the equivalent distribution of bound circulation over the radius and hence by Kutta-Joukowsky's theorem the thrust and torque coefficients as functions of the advance ratio . 1849 Nowaeckt and Sharma P2) The thrust and torque predictions of the previous step are compared with the actual performance as measured in step E2 and the agreement is improved iteratively by adjusting the assumed foil characteristics. Again the crucial link in the algorithm is the circulation distribution, P 3) Using the Hough and Ordway (1965) approximation, the circulation distribution is now translated into an equivalent source distribution over the propeller disk. P4) This source distribution is the basis for calculating the wavemaking due to the propeller by Havelock's (1932) theory. In parti- cular, the axial velocities induced by the operation of the propeller near the free surface, in other words the self-induced free-surface wake of the propeller, can be calculated. P5) This self-induced wake is fed back into the propeller performance program based on lifting line theory to obtain predictions of thrust and torque with the propeller operating at shallow submer- gence. P 6) A comparison of propeller performance predicted in step P5 with actual measurements at the same submergence then pro- vides a check on the correct accounting of free surface effects in the theoretical model. After the mathematical representations of hull and propeller have been verified the actual interaction analysis can be executed as follows. I1) The Froude propulsion factors (mean effective wake, thrust deduction, relative rotative efficiency and propeller efficiency in the equivalent open water condition) are first determined from the results of tests El, E2 and E3 in the usual manner. 12) The radial distribution of nominal wake from step E4 is adjusted to match the mean effective wake from step 11 and fed into the propeller performance program, The output is the circulation dis- tribution of the propeller in the behind ship condition at each Froude number. 13) Again the Hough and Ordway relation is used to translate the circulation distribution into a source representation of the propel- ler in the behind-ship self-propulsion condition. 1850 Free Surface Effects tn Hull Propeller Interactton 14) From the now known source representations of the hull and propeller free-wave spectrum and wavemaking resistance are cal- culated and compared with the corresponding results of the Fourier analysis of the wave profiles measured in step E5. This provides a check on the principle of linear superposition of hull and propeller waves, 15) The mutual flow patterns of hull and propeller can now be calculated and thence by Lagally's theorem the potential and wave thrust deduction. 16) Finally the viscous component of thrust deduction can be estimated indirectly by subtracting the potential and wave components from the total thrust deduction of step I1. III. DISCUSSION OF RESULTS III. 1. Choice of Hull and Propeller Since our work was to consist essentially of a single concrete example of the actual application of the sequence of operations out- lined in the previous section it was rather important to choose as in- structive and useful an example as possible. After considering various alternatives we finally selected the somewhat idealized hull propeller configuration of Figure 1 that has a sufficiently simple geometry for the ease of theoretical calculations and yet quite realistic proportions for the results to be of practical value. The arguments leading to this choice can be summarized as follows. In order to keep the wavemaking calculations manageable it was decided to use a symmetric hull form with parabolic waterlines and frames. The wetted surface is then defined by the equation y apf - (2x/4)™} 41 - (-2/7)" (3) The hull above water is a simple continuation of the underwater form with vertical sidewalls. The integer powers m,n and the form ratios L/B, B/T were chosen to satisfy the following requirements : 1) sufficiently thin hull for linearized theory to be valid, 2) sufficient- ly large angle of run to get measurable interaction with the propeller, and 3) realistic value of block coefficient. This led to the following set of parameters : 1851 Nowaekt and Sharma m Oe he The eA pee Tor By = te Sista ae Syep HORE Cp = 0. 64 ip = ip = arctan 0.4 = 21.8° (4) The absolute size of the model for the towing experiments was dictated by the size of the tank and equipment available : Like va nducO Selina keianaetere Bi itentdg gay ighe ature ey Tee nem Ons OOaun ae, OLOR 4. v = 0.3888 m”> = 13.731 ft? S = 3, AQG2i sane deste Shea (5) The choice of propeller was governed mainly by considera- tions of availability and simplicity. Fortunately, it was possible to borrow a very suitable propeller from the Hamburg Ship Model Basin (HSVA), namely a 200mm diameter model of the Standard Propeller recommended by the ITTC Cavitation Committee in 1960 for compara- tive testing, see Burrill (1960). It has a simple geometry (constant pitch, no rake, no skew) with accurately defined offsets (Figure 2), and performance characteristics were already available from previous tests at the Hamburg and other tanks. Its two-dimensional foil charac- teristics, however, were not known. The center of the propeller was positioned at Xp = t=O ile, = 0 Yp =p =().05.0) an 6 Zp (6) in the coordinate system of Figure 1. This arrangement relative to hull ensured complete submergence (0.75 D at rest) at all speeds and a low axial clearance (0.225 D) with accordingly accentuated interac- tion effects. III.2. Summary of Model Tests In accordance with the scheme outlined in Section 2 the fol- lowing model experiments were conducted : 1852 Free Surface Effects tn Hull Propeller Interactton E 1) Measurement of bare hull resistance over the entire fedsiple speed ranpe of ‘0. ls °F, < 0.45. E 2) Measurement of propeller performance in open water (thrust and torque as functions of speed of advance and rate of revo- lutions) over the range of advance coefficient 0< J <1.2 at four depths of submergence : h/Rp =HS by 2.100, 1. 50° and» T;'00; E 3) Propulsion tests with the propeller operating behind the hull (measurement of thrust, torque and residual towing force as func- tions of model speed and propeller rate of revolutions) at fourteen dis- crete speeds corresponding to ¥, = 3.5 step 0.5 until 8.0 step 1.0 until 11.0, and 12.5. At each speed the propeller revolutions were varied to obtain a sufficient range of loading usually covering both the model and the ship self-propulsion points (for an arbitrarily assumed model scale of 1:80). E4) Measurement of nominal wake in the propeller plane be- hind the hull (x, = -0.51 L) in both forward and reverse motion at three selected speeds corresponding to ¥, = 4.0, 7.0 and 12.5. At each speed the circumferential average of the axial wake velocity was measured by means of calibrated wake wheels at ten different radii R/Rp = 0.2 step 0.1 until 1.1. E5) Measurement of longitudinal wave profiles at a fixed transverse distance (Yo = 0.134 L) from the model center plane in two conditions ; 1) model with propeller running at ship self propul- sion point and 2) model with propeller replaced by a dummy hub, each at two selected speeds corresponding to ¥, = 4.0 and 7.0. Revelant details of the test procedure are given in Appendix A, III. 3. Hull Analysis Figure 3 shows the measured total resistance of the bare hull as a function of speed in the usual nondimensional coefficient form: Cp versus Fy, (or R,). Also shown in the figure are the ITTC 1957 model-ship correlation line 2 Cp = 0.075 ‘a (log) gR,, - 2) (7) and the curve of estimated viscous resistance coefficient The latter is based on the Hughes form factor concept and determined from the measured total resistance at low Froude numbers by the gra- phical method of Prohaska (1966). Assume 1853 Nowaekt and Sharma and further for ah) : 4 Cw = ey Bn : (10) Then Cy/Cy = fil 2 i aly SP (ee /Cy) (11) so the constants (l+k) and c,, may be determined from a linear fit to the plot of Cr/Cr versus F,{/Cr for low Froude numbers. Figure 4 shows that the linear relation implied by Equation (11) applies reasonably well to our model up to Froude numbers upto 0.2. The numerical values of the viscous form factor (l+k) and the coefficient Cy, were found to be (1+) =") OZ5 Cig ee (12) The coefficient of wavemaking resistance thus indirectly derived Cw Sear ( 1+k ) Cr (13) has been plotted in Figure 5 against the appropriate speed-length parameter Y, and compared with the corresponding calculations based on linearized thin ship theory (see Appendix B, especially Eq. (B28)). Although there is a remarkable semblance between theory and ex- periment (e.g. the second, third and fourth humps can be clearly iden- tified in the measured curve), it is disappointing to observe that even for our relatively thin ship (L/B = 10) reasonable quantitative agree- ment between theoretical predictions and experimental reality could be established only over a limited speed range of 2.5 <¥,< 4.5. At higher Y, (i. e. lower Froude numbers) the experimental curve ex- hibits much less pronounced humps and hollows and its general level is only half as high as the theoretical curve. This suggests that the viscous boundary layer and separation probably made the stern quite ineffective in wavemaking. In any case, the two speeds corresponding to Y, = 4(F,, =0. 354) and Y, = 7(F,,=0.267) were singled out from Fig. 5 as the most pro- mising for further investigation. At these speeds the wavemaking re- sistance was evaluated directly from measured wave profiles by the longitudinal cut method described in Appendix B.8. The result, as indicated by the two isolated spots in Fig. 5, showed that the wave- making resistance associated with the wave pattern actually generated by the model was about 30 to 40 percent less than the theoretical pre- diction or the empirical estimate of Eq. (13). Further discussion of the results of wave profile analysis will follow in Section 3, 5. The next step in hull analysis was the evaluation of nominal wake, i.e. the flow perturbation created by the hull in the propeller plane in the absence of the propeller. In order to avoid the compli- 1854 Free Surface Effects tn Hull Propeller Interactton cations invariably caused by viscous effects behind the hull, we first compared the calculated and measured wake in reverse” motion, see Figure 6. The measured values were obtained from calibrated wake wheels directly as circumferential averages at ten discrete radii. The calculated values based on thin ship theory (see Appendix B5, espe- cially Equation (B56)) were available pointwise in the propeller plane and were numerically averaged along the circumference at various radii for the ease of comparison with measurements. It is encourag- ing to observe in Figure 6 the fair agreement between theory and ex- periment, the discrepancy being nowhere larger than 0.03. In par- ticular, both the mean effect of Froude number and the general varia- tion with radius are correctly predicted by theory. However, the mea- sured wake shows some erratic oscillations of unclarified origin at the outer radii. Figure 7 shows an analogous comparison of calculated and measured wake in forward motion, Here we cannot expect direct agreement between experiment and theory since the former contains a substantial viscous component not included in the latter. However, if we subtract the calculated from the measured wake, we notice that the remainder is relatively insensitive to Froude number (see Fig. 7) as we would expect of the true viscous component. This may be inter- preted as indirect evidence that wave effects actually present in the measured total wake are of the same order of magnitude as predicted by thin ship theory. This is quite encouraging, especially in view of the relatively poor agreement between calculated and measured values of wavemaking resistance. For the sake of completeness the conventional ''potential"' or zero Froude number component of wake as calculated by theory (Ap- pendix B.5, Equation (B54)) is also plotted in Figure 7. It is by de- finition independent of Froude number. In view of the foregoing, the trichotomy of nominal wake in potential, wave and viscous compo- nents as displayed in Figure 7 can be regarded as quite meaningful. Evidently, the wave effects are by no means negligible as commonly assumed, * Incidentally, by virtue of the longitudinal symmetry of our hull the "stern'' wake in the propeller plane x =x, in reverse motion is equi- valent to the ''bow'' wake in the reflected propeller plane x = -Xp in forward motion, 1855 Nowaekt and Sharma IlI.4. Propeller Analysis Measured propeller performance characteristics for three depths of submergence are plotted in Fig.8 in the usual nondimensional coeffi- cient form. The largest depth (h/R, = 3.47) was the maximum attainable with the propeller boat available for open water tests, andthe smallest (h/Rp= = 1.50) corresponds exactly to the immersion selected for self- propulsion tests (Zp = -0.5T) describedlater. Apart from verifying the measurements eaiactea previously ataneven lager depth (h/Rp = =\4'.0) in the Hamburg Ship Model Basin (HSVA), the principal conclusion from these tests was that free-surface effects are negligibly small for depths h/Rp 21. 50. At the shallowest depth investigated , however , with the pro - peller disk just touching the static water level (h/Rp = 1.0), pronounced free-surface effects were measured , see Fig.9 . The observed loss of thrust and torque as compared to the deeply submerged condition , the steady accentuation of the effect with increasing loading (i.e. de- creasing advance coefficient) , and a slight drop in efficiency are to be naturally expected from the combined effects of ventilation and wave - making at the free surface . It is not intuitively obvious , however , why the thrust and torque should suddenly break down at some "'critical" advance coefficient , here Jx0.41 . Similar discontinuities have been measured by others, notably by Shiba (1953) . Flow observations reveal that the discontinuity is accompanied by a sudden transition from partly ventilated to fully ventilated condition . A satisfactory theoretical ex- planation of this phenomenon would certainly require an intricate ana- lysis of the stability of partly ventilated flow . It is also intriguing to note that the drop in thrust and torque is nearly proportionate so that the discontinuity is hardly perceptible in the curve of efficiency, This lends some credibility to Dickmann's (1939) simplified treatment of propeller ventilation as a mere reduction in the density of the medium due to a mixture of air with water! For the sake of completeness it should be reported that ventila- tionalsooccured to some extent at two ofthe deeper immersions,namely h/Rp =1.5 and 2.0, especially in the bollard condition and at the low- est advance coefficients. It was distinctly audible and often visible as a vortex from the free surface to the propeller tip, but its effect on thrust and torque was obviously too small to be measurable (see Fig.8). The measured thrust and torque characteristics (in the deeply submerged condition) were transformed into an equivalent vortex model of the propeller by means of a computer program based on lifiting line theory and using assumed (or adjusted) two-dimensional foil characte- 1856 Free Surface Effects tn Hull Propeller Interactton ristics as the connecting link between propeller geometry and forces after taking account of the velocity perturbation induced by the trailing vortices . Without going into details , which are given in Appendix C , Fig. 10 is presented as evidence for the close fit finally achieved bet- ween calculations and measurement . Note that the results of two dif- ferent calculations are displayed . The four sets of crosses mark the calculated performance of a series of hypothetical propellers indivi - dually designed at each respective advance coefficient so as to produce the known measured thrust with a minimum loss of energy (i.e. optimum distribution) . The exact agreement with the measured Ky values is therefore trivial , while the good agreement with the measured Ky values proves that hydrodynamic losses were reasonably estimated in the calculation and that the actual performance of the propeller is near- ly optimum over the range 0.6 Crp) /4™ own * For a truly self-propelled system the towing force F)=0, and then Equation (19) agrees with Equation (15). 1862 Free Surface Effects tn Hull Propeller Interaction and the thrust deduction fraction = + Ss t om ats R,) ip Lik fa J 2 =)1) =, 4S /OD>) (ees Cep)/ Kay (20) Now read from the open water characteristics (Fig.8) the advance coefficients J_, at thrust identity (K_ = K,. ) and J_ at torque identity (K. = K = . Calculate ry from Hie aero (07)? £~and read the eae eilien oan water efficiencies nov , 70Q » 70M from Bagads ahs Do. 55 Sh sil respectively . Calculate effective wake se hes £5 Q M fractions ell Gace “Tr Jo/ Sey ae is Io/ Tey UM cece ies (21) hull efficiencies ee ak ee 9 MQ = (1+t) / Uw) tum = O-B/ C-wyy) (22) and relative rotative efficiencies "ep = Dor tar Vota I/9Q "HO Tam = "p/"om "HM (23) This completes the analysis. The result of one such evaluation , out of fourteen actually carried out , is reproduced in Fig. 24 . Since this is generally typical of all others , the following remarks are relevant . First , the thrust deduction fraction and relative rotative efficiency are relatively insen- sitive to changes in loading. Second, the equivalent open water efficiency decreases with increasing loading (decreasing J_,) as expected . Third the effective wake fraction , and consequently the hull efficiency de - crease with increasing loading . This is in contradiction to the 1863 Nowaekt and Sharma theoretical behavior in potential flow (see Appendix D) . However, in a real flow the decrease in effective wake with increasing loading can be explained qualitatively by a supposed contraction of the viscous wake due to propeller suction as first pointed out by Dickmann (1939) , see also next section . Fourth , all propulsion factors vary slowly and al- most monotonically with changesinloading, so that the arbitrary choice of one particular loading (e.g.that corresponding to the self-propulsion point of a ship of \ = 80) for further investigation is not liable to hide any important phenomena . Fig. 25 shows the various propulsion factors as functions of Froude number over the range 3.5 <¥% <12.5, all evaluated at the self-propulsion point of a smooth ship of Y = 80 . ( This choice of scale ratio is arbitrary , but not crucial as just pointed out ). The following features deserve special mention . First , all factors depict- ed exhibit a significant and oscillatory dependence on Froude number . Second , the self-propulsion point advance coefficient J,, , and con- sequently the equivalent open water efficiencies ng , depend mainly on hull resistance , and hence reveal humps and hollows in inverse phase to the coefficient of wave resistance (compare Fig. 5) as expect- ed . Third , contrary to common belief , the thrust deduction and effective wake fractions vary significantly with Froude number , the most remarkable feature being the sudden drop around Yo= 5 . The hull efficiency zz merely shows their combined effect . Fourth, the relative rotative efficiency np is exceptionally low but approaches normal values at higher Froude numbers . Fifth , there is an unusually large discrepancy between thrust and torque identity points , but it tends to decrease with increasing Froude numbers . The last two effects are presumably due to strong nonuniformities in the viscous wake of the hull , which would also explain why they are relatively weaker at higher Froude numbers . Iit."G. 2: ~“Wailee The next step in interaction analysis was an attempt to corre- late by theory the measured wake and thrust deduction . This required first the generation of a mathematical model of the propeller in the behind hull condition . Again the computer program described in Appendix C was used . The inputs to the program were the advance coefficient J), at the ship self-propulsion point , the corresponding thrust coefficient Ky, the radial distribution of measured nominal wake w (R), and the two-dimensional foil characteristics already established on the basis of open water characteristics ( see Propeller Analysis ) . In order to account for the difference between nominal and effective wake the program was allowed to determine by trial and error a wake corrector k,, , with which the nominal wake w(R) was 1864 Free Surface Effects tn Hull Propeller Interactton multiplied, such that the calculated thrust coefficient equalled the measured K-ypy;,. The primary output of the program was the distri- bution of bound circulation along the radius. In addition, it also fur- nished a calculated torque coefficient K and a mean effective wake Wo (based on thrust average rather than volume average) from which followed the equivalent open water advance coefficient Jp. This elaborate analysis was done only for three selected Froude numbers corresponding to y_=4.0, 7.0 and 12.5. The results are shown in Fig. 26 and 27. The effect of wake on circulation distribution is quite evident in Fig. 26 where the circulation maxima have been shift- ed toward smaller radii as compared to the open water condition of Fig. 11. Turning now to Fig. 27, the good agreement between cal- culated and measured advance coefficient J, is a confirmation of the realistic simulation of thrust generation in ine theoretical model, while the lack of agreement between calculated and measured torque coefficient Kp ;; points up the shortcomings of the theoretical model, specially the total neglect of all circumferential nonuniformities and the associated lack of any simulation of the relative rotative efficiency. However, we would not expect these defects to have any serious effect on the intended calculation of thrust deduction. Before passing on to the evaluation of thrust deduction we pause to consider briefly the issue of nominal wake versus effective wake. Conceptually, the distinction is clear : Nominal wake is the flow perturbation created by the hull in the propeller plane with the propeller removed, while effective wake is the flow perturbation due to the hull in the propeller plane with the propeller in place and oper- ating. In practice, however, the relative magnitudes of these two wakes have been a topic of considerable controversy and confusion in the literature on hull propeller interaction. It is generally agreed that there are two fundamentally different reasons why these two wakes need not be identical. First, there is a genuine physical effect of the propeller on the flow perturbation caused by the hull. This has three partially counteracting components. a) The potential component, which may be understood as the additional flow induced by the image of the propeller in the hull, tends to increase the effective wake com- pared to the nominal wake, since this image consists predominantly of sinks in the afterbody. b) The viscous component, which results from a contraction of the viscous wake, is specially pronounced if the line of boundary layer separation is shifted rearward by propeller suction and generally tends to decrease the effective wake compared to the nominal wake by bringing more undisturbed flow into the pro- peller disk. c) The wave component, referred to as a pseudo-non- linear effect of the propeller on the wavemaking properties of the hull in Section 3.5, can act in either direction depending upon Froude number. Second, there is a spurious computational effect due to dif- 1865 Nowackt and Sharma ferent methods of averaging. The measured nominal wake is conven- tionally averaged over the disk on a volume flux basis, while the mean effective wake is measured by the propeller as a calibrated thrust (or torque) generating device which tends to put maximum weight near the radii where the circulation is a maximum, The follow- ing table, a by-product of our calculations, is likely to shed some light on the relative importance of these two effects. Effective Speed-length Measured Corrected parameter nominal nominal wake wake wake 2 Simu-| Meas- lated | ured ~ Ww w w etl 4.0 [0.354 | 0.208] 0.230 | 0.704|/ 0.146| 0.162 |0.153| 0.145 7.0|0.267 | 0.291| 0.322 |0.892 | 0.259) 0.287 |0.291 | 0.285 12.5 |0.200 | 0.304| 0.370 | 0.933 | 0.284] 0.346 | 0.377 | 0.360 First, note that the wake corrector k is a measure of the true physical difference between nominal and effective wake since, as explained earlier, it was determined by trial and error as the requir- ed multiplier of the measured nominal wake in the computer program to ensure that the simulated and measured thrusts were equal. This difference is here seen to vary from -7% at the lowest Froude number to -30% at the highest. That it is strongly negative, suggests that the viscous effect mentioned above was probably dominant in this case. Second, the residual difference (up to +33%) between the cor- rected volume average wake kw and the thrust average wake wer must be attributed to the difference in the methods of averaging. Note that this spurious effect is greatest at the lowest Froude number where the concentration of bound circulation over the inner radii was also the most pronounced. Third, the good agreement between the computer simulated and the experimentally measured mean effective wake is rather encouraging. Fourth, note that the effective wake is much better approximated by the corrected nominal wake at 0.7 radius, kw(. 7Rp): than by its disk average, kw. This observa- tion has direct relevance to the design of wake-adapted propeilers. Finally,asa word ofcaution, it should be noted that the relative ma- gnitudes of the nominal and effective wakes as well as the quantitative rankings of the different effects found here may be peculiar to this model and therefore should not necessarily be generalized. To complete the discussion of wake, Fig. 28 shows the meas- ured versus calculated wake as.a function of Froude number. The fol- lowing quantities are plotted: 1) The disk average of the measured 1866 Free Surface Effects tn Hull Propeller Interaction nominal wake w. This was available at three speeds only (compare Fig. 7). 2) The disk average of the potential wake w_,, calculated by thin ship theory, see Appendix B.5, especially Equation (B 54). This is a zero Froude number approximation. 3) The disk average of the sum of potential and wave wakes (w, + w,,) also calculated by thin ship theory, see Appendix B.5, especially Equation (B 53), 4) The quantity (wp -wp-wWy) as an approximate estimate of the viscous component w,, , see Equation (1). The striking correlation between the measured effective wakes and the calculated wave wake certainly suggests that the observed oscillations of wake with Froude number are indeed free-surface effects and that the thin ship wavemaking theory despite all its weaknesses does give a reasonable estimate of this phenomenon, Even the quantity (wp-wp-w,,), which as the dif- ference of a measured effective wake and calculated nominal wake components must be regarded with due caution, gives a credible im- pression of the magnitude of viscous wake w,. However, one cannot put much faith in its observed oscillations, III. 6.3. Thrust Deduction We now turn to our final goal of calculating the thrust deduc- tion fraction and its components. This was done to two different de- grees of approximation, At the three selected Froude numbers, where the calculated circulation distribution was available (see Fig. 26), the Hough and Ordway relation, Equation (B 16) in conjunction with the simulated effective wake k,,w(R) , was applied to generate the equi- valent sink disks. At all other Froude numbers we had to be content with Dickmann's approximate relation between thrust coefficient and source strength, Equation (B 15) in conjunction with the measured ef- fective wake wp. The numerical difference between these two ap- proximations is illustrated in Fig. 29. Evidently, the Hough and Ordway approximation yields slightly higher mean values and, in ac- cordance with the distribution of bound circulation, effects a concen- tration of sink strength toward the inner radii. It is believed to be more accurate than Dickmann's uniform sink disk since the vortex model yields a more realistic flow pattern than the simple momentum theory. In either event, the sink disk was used to calculate first the wavemaking resistance of the propeller alone and of the system hull and propeller as explained in Appendix B. The wavemaking resistance (and free-wave spectrum) of the propeller in the behind hull condition calculated in this way were found to be in reasonable agreement with the corresponding results of measured wave profile analysis at two Froude numbers as already discussed in Section 3.5. Given the wave- making resistances of the hull Rwy» propeller Rywp., and total system Rwy, only one additional quantity 5;4Rwp =, see Eq. (B 64), 1867 Nowaekt and Sharma was needed for calculating the combined potential and wave thrust deduction force 6,Ry,,, see Equation (B 65), from which followed the thrust deduction fraction (th + t,,) by Equation (B 66). The po- tential component t, alone was obtained from a simple degenerate case (zero Froude number wake) of this calculation, see remark fol- lowing Equation (B 67). The final results of this calculation are shown in Fig. 30 in comparison to the measured total thrust deduction t- replotted from ‘Fig. 25: Let us try to interpret the salient features of Fig. 30. First, the wave component of thrust deduction t,, is small, but not negligi- ble compared to the potential component t,. Second, the oscillations in calculated thrust deduction are not due to t,, , but are already present in t, . This can be understood by reference to Equation (B 63) which defines thrust deduction as the Lagally force on the hull sources due to the axial flow induced by the propeller sources. Since our hull sources were assumed independent of Froude number and since the flow induced by a source upstream of itself is almost monotonic with Froude number, the observed oscillations of calculated thrust deduc- tion can only be due to variations of propeller source strength with Froude number. This is indeed the case, for by Equation (B 15) the source strength depends on loading and wake, which were both found to oscillate with Froude number, Asa result the calculated thrust deduction t, (as wellas t,,) correlates strongly with advance coef- ficient J,;, and effective wake wp (compare Fig. 25). Third, the oscillations in the measured thrust deduction t are much stronger than in the calculated (t, + t,). This means that either the residual viscous component of thrust deduction t, , see Equation (2) , oser= lates appreciably with Froude number or that our assumption of the hull sources being independent of Froude number was invalid. This point cannot be decided at the moment. But in any case it points toa significant interaction of viscous and wave effects at the stern, pre- sumably intensified by propeller suction. For instance, if the line of boundary layer separation is pulled rearward by the propeller, the result would be a negative viscous thrust deduction as well as a rela- tive increase in the effective sink strength of the afterbody. Fourth, specifically the steep variation of measured thrust deduction around Yo = 5 cannot presently be explained, except as a possible viscous effect, i.e. a reduction in the extent of boundary layer separation under the combined influence of a negative wave wake (Fig. 28) anda high propeller loading (Fig. 25). Fifth, the thrust deductions calculat- ed from the Hough and Ordway sink disk are significantly higher than . those calculated from the Dickmann sink disk and are in better agree- ment with measurements. This is a direct consequence of the signifi- cant difference between the two sink disks, both in average intensity and in its relative distribution over propeller radius, see Fig. 29. 1868 Free Surface Effects in Hull Propeller Interactton IV. CONCLUDING REMARKS It has been demonstrated by practical application to a speci- fic example that our conceptual scheme for determining the potential, viscous and wave components of wake and thrust deduction is indeed workable. It has required the concerted application of miscellaneous analytical, computational and experimental techniques, The varying degrees of success achieved with the individual techniques have been discussed in detail in the appropriate sections and need not be repeat- ed here. Several results were obtained by more than one method, for instance by independent calculation and measurement, and in most cases there was fair agreement, at least there were no striking con- tradictions except perhaps in the calculated and measured wavemaking resistance at low Froude numbers, which came as no surprise. It would be rash to try to derive general conclusions concern- ing the quantitative role of wavemaking at the free surface in the phe- nomenon of hull propeller interaction on the basis of one single exam- ple. However, two salient results do seem to have a broader signifi- cance. First, it was found that contrary to common belief the wave component can be dominant in the wake and quite significant in the: thrust deduction at Froude numbers around F, = 0.3. Second, there seemed to be an appreciable viscous component in the thrust deduc- tion at practically all Froude numbers. Moreover, the undulating va- riation of this component with Froude number points to a complicated interaction of viscous boundary layer, hull wave pattern and propeller suction near the stern, These two effects are of direct relevance to the hydrodyna- mic design of fast ships and also to the methods of extrapolating pro- pulsive performance from model to full-scale. It is recommended that further studies of this nature be under- taken to resolve the remaining issues and to collect systematic design data on the effect of wavemaking on the propulsive performance of ships. V. ACKNOWLEDGMENTS We want to sincerely thank the following individuals and orga- nisations for their valuable contributions to this project : Mr. J.L. Beveridge and Dr. W.B. Morgan of the Naval Ship Research and Development Center for crucial consultations in the planning stage of this work, 1869 Nowaekt and Sharma Mr. W.S. Vorus, Ph. D. candidate at the University of Michigan, for a major share of the computer programming and test evaluation, NB bainal Or Ostergaard, visiting scientist at the University of Michigan, for the computer program for propeller design originated at the Technical University of Berlin and extended at the University of Michigan to cover the off design performance, Mr. A.M. Reed, graduate student at the University of Mich- igan, for the computer programs for wave profile analysis and the- oretical wavemaking resistance, Mr. B. Hutchison and Miss S. Pian, students at the Univer- sity of Michigan, for their assistance in writing the computer pro- gram for hull wave flow calculations by thin ship theory, Mr. E. Snyder of the Ship Hydrodynamics Laboratory, Uni- versity of Michigan, for his expert execution of the model test pro- gram, Messrs. T. Little, A. Toro, and B.L. Young, students at the University of Michigan, for their enthusiastic help in conducting the model tests, The staff of the Ship Hydrodynamics Laboratory, University of Michigan, for their careful and patient handling of all jobs pertain- ing to the model tests, The Hamburg Ship Model Basin (HSVA), Hamburg, Germany, for lending a model propeller for our experiments, and The Naval Ship Research and Development Center, Carderock, Maryland, for providing copies of computer programs for propeller design (Lerbs induction factor program) and potential flow calcula- tions (Hess and Smith program), 1870 Free Surface Effects tn Hull Propeller Interactton LIST OF SYMBOLS Note : - The standard symbols recommended by the ITTC Presentation Committee have been used wherever possible . See also Section B.1 for the special notation used in Appendix B . B Beam of hull Ca Block coefficient of hull form Cp Drag coefficient of propeller blade section Cyop Value of Cp at design point Jp CF Coefficient of friction , Equation (7) Crp Coefficient of residual towing force = 2Fp/p sv CFrM Value of Cp at model Reynolds number Crs Value of Cpr at ship Reynolds number CL Lift coefficient of propeller blade section Crp Value of Cy, at design point Jy Cm Midship section area coefficient Cp Longitudinal prismatic coefficient CT Coefficient of total resistance = 2R-/p sve CTh Thrust loading coefficient , Equation (B15) Cy Coefficient of viscous resistance = 2Ry/pSv* Cym Value of Cy at model Reynolds number Cvs Value of Cy at ship Reynolds number Cw Coefficient of wave resistance = 2Ry /pSv- Cwp Waterplane area coefficient Gus Fourier cosine , sine transforms, Equation (B50) e og Modified Fourier cosine , sine transforms , Eqn. (B70) D Diameter of propeller E (u) Free-wave amplitude spectrum Exq(u) E (u) of hull alone E Ju) E (u) of propeller alone Levi Nowaekt and Sharma E (u) of total system hull and propeller Special function , Equation (B39) Sine component of free-wave spectrum Subscripts H, P, T apply asto E (u) Residual towing force in self-propulsion test Froude number = V / Noe Submergence Froude number = V/ gh N sy ),a2.0303 Special functions , Equation (B24) Free-wave spectrum of propeller in a coordinate system with origin in the propeller plane , Eqn. (B73) Non dimensional bound circulation = [/ 7 DV Green's function of point source , Equation (B33), (B57) Partial derivative of G , Equation (B58) Cosine component of free-wave spectrum Subscripts H,. P, T apply as to E (u) N=1, 2, 3 : Special functions , Equation (B47) Imaginary part of Modified Bessel function of zero order Advance coefficient of propeller =V /nD for free-running propeller = Va/nD for propeller operating behind hull Value of J at the design (optimum) point Virtual advance coefficient of propeller operating near the free surface , Equation (14) Advance coefficient of propeller based on hull speed = V/nD Mean of Jg and JqT Value of J at torque identity Ko Kou KTH Torque coefficient of free-running propeller =Q/pn Value of J at thrust identity Ky 2p5 Torque coefficient of propeller behind hull = Qy;/en2D° 1872 Free Surface Effects tn Hull Propeller Interactton Thrust coefficient of free-running propeller = T/pn2D* Thrust coefficient of propeller behind hull = Tyy/pn2D4 Length of hull Coordinate system, see Figure l Pitch of propeller Propeller torque in open water Propeller torque behind hull Real part of Propeller hub radius Reynolds number of hull = VL/» Propeller tip radius Total resistance of hull Viscous resistance of hull Wavemaking resistance R.,, of hull alone W Rw of propeller alone Rw of total system hull and propeller Polar coordinates in propeller plane , Equation (B9) Coordinates of source point in propeller plane Wetted surface area Draft of hull Propeller thrust in open water Propeller thrust behind hull Speed of advance of hull Speed of advance of free-running propeller Speed of advance of propeller relative to wake in the behind hull condition Speed of model Number of blades of propeller 1873 Nowaekt and Sharma aj, ao Empirical constants defining propeller foil characteristics , Equation (C13) b Half beam of hull = B/2 c Chord length of propeller blade section Cy Empirical constant , Equation (10) dD Drag generated by blade element, Equation (C10) dL Lift generated by blade element, Equation (C8) £ (se; <2) Function defining hull surface , Equation (B4) g Acceleration due to gravity Submergence measured to propeller axis i Imaginary number = J-1 in Induction factor for axial velocity , Equation (C5) ip Angle of entrance of hull ip. ~ sAngle,of run ofthull iT Induction factor for tangential velocity , Eqn. (C6) k Empirical factor defining propeller foil characteristics , Equation (C12) k Circular wave number (Appendix B) ky Empirical wake corrector , Appendix C.3 k+1 Viscous form factor , Equation (8) L Half length of hull = L/2 m Hull form parameter , Equation (3) n Hull form parameter , Equation (3) n Rate of revolutions of propeller ry Distance between field point and source point Equation (B33) To Distance between field point and mirror image of source point , Equation (B33) s Function of u, 3. = (1+v) /2 t Thrust deduction fraction 1874 Free Surface Effects tn Hull Propeller Interactton Potential component of t Viscous component of t Wave component of t Transverse wave number Axial velocity induced at the lifting line by the vortex trail of the propeller Tangential velocity induced at the lifting line by the vortex trail of the propeller ld oba(oerora, (ne Wl 5 We = VJ 1+4u2 Longitudinal wave number (only in Appendix B) Wake fraction (Unless otherwise specified , the disk average of the nominal , axial wake is implied ¥) Self-induced free-surface wake of propeller Potential component of wake w Viscous component of wake w Wave component of wake w Circumferentially averaged value of w (R,0) Subscripts,.£.¢ . Doi s,s) apply as to w Local nominal wake fraction at point (R ,0 ) Subscripts f, p, v, w apply as tow Effective wake fraction Mean of wo and wr Effective wake fraction from torque identity Effective wake fraction from thrust identity Simulated effective wake fraction , Equation (C18) Weights in iteration formula , Equation (C11) Longitudinal coordinate , positive forward Longitudinal coordinate of center of propeller Coordinates of field point Coordinates of hull source point Transverse coordinate , positive to port L675 Nowaekt and Sharma Transverse position of longitudinal wave profile Transverse coordinate of center of propeller Vertical coordinate , positive upward Vertical coordinate of center of propeller Bound circulation along propeller blade Geometric pitch angle , Equation (C3) Step size in time t Step size in wave number u Step size in distance x Angle of attack of blade section , Equation (C2) Value of a at design point Jp | Hydrodynamic pitch angle , Equation (C4) Hydrodynamic pitch angle at design point Jp Nondimensional speed-length parameter = gL /2Vv% Increase in propeller wave resistance due to presence of hull Increase in hull wave resistance due to presence of propeller = force of thrust deduction Hull form parameter in Appendix B, Equation (B5) Drag / lift ratio in Appendix C , Equation (C18) Free-surface elevation at point (x, y) Propulsive efficiency , Equation (15) Hull efficiency , Equation (16) Additional subscripts M , Q , T defined in Egn. (22) Openwater propeller efficiency = KJ i 2rKo Additional subscripts M,Q,T, applyasto J Relative rotative efficiency , Equation (15) Additional subscripts M,@Q, T defined in Eqn. (23) Direction of wave propagation in Appendix B Polar coordinates in propeller plane , Equation (B9) 1876 Free Surface Effects in Hull Propeller Interactton Bos Polar coordinates of propeller source point » Scale ratio v Kinematic viscosity of water v Summation index , Equation (B50) bn’ Relative field point coordinates , Equation (B36) pil; Gd Relative source point coordinates , Equation (B36) p Density of water o Source strength = Source output / 47 o (x, z) Density of hull source distribution a(R, @) Density of propeller source distribution a(R) Circumferentially averaged value of o(R, 6) T Draft/half-length ratio , Equation (B36) ge Velocity potential of perturbation flow 9x 7Fy, 58 Partial derivatives of ¢ eH Longitudinal flow induced by hull eP Axial flow induced by propeller e(N) N = 1, 2, 3, 4; Components of ¢,, Equation (B34) w Angular velocity of propeller = 27n Avi Displacement volume of hull REFERENCES _Note : - The following is a fairly complete list of recent publi- cations , not necessarily cited in our text , directly dealing with the subject of hull propeller interaction . References to older literature can be found in Dickmann's (1939) monograph . We hope that this list will serve as a useful handy reference to future workers in this field . AMTSBERG , H.: Investigations with axisymmetric bodies into the interaction between hull and propeller (in German), Jahrbucn STG 54, 117 - 152 (1960) AMTSBERG , H., and ARLT ,W.: Thrust deduction studies of bodies with bluff ends (in German) . Schiff and Hafen 17 , 786-790 (1965). 1877 Nowaekt and Sharma BASSIN ,A.M. : Theory of interaction between the propeller and hull of a vessel in an infinite fluid (in Russian) . Bulletin of USSR Academy of Science , Division of Technical Sciences 12 ,1723 - (1946). BEVERIDGE , J.L. : Thrust deduction due to a propeller behind a hydrofoil . DTMB Report N° 1603 (1962). BEVERIDGE, J.L. : Effect of axial position of propeller on the pro- pulsion characteristics of a submerged body of revolution. DTMB Report N° 1456 (1963). BEVERIDGE , J.L. : Pressure distribution on towed and propelled streamline bodies of revolution . DTMB Report N° 1655 (1966) . BEVERIDGE , J.L. : Analytical prediction of thrust deduction for submersibles and surface ships . Journal of Ship Research 13 , 258 - 271 (1969 ) = NSRDC Report N° 2713 (1968 ) CHERTOCK , G. : Forces on a submarine hull induced by the propeller. Journal of Ship Research 9 , 122 - 130 (1965). DICKMANN , H.E. : Thrust deduction , wave resistance of a propeller, and interaction with ship waves (in German) . Ingenieur Archiv 955452 = 2BGribgssH)o, DICKMANN , H.E. : Propeller and surface waves (in German) : Schiffbau 40 , 434 457 ( 1938 ) DICKMANN , H.E. : Wave resistance of a propeller and its interaction with ship waves . Proc. International Congress for Applied Mechanics 5 ,-( 1938 ) DICKMANN , H.E.: Interaction between hull and propeller with spe- cial consideration to the influence of waves (in German) . Jahrbuch STG 40, 234 - 291 (1939 ) DREGER , W. : A procedure for computing the potential thrust deduct- ion (in German) . Schiffstechnik 6 , 175 - 187 (1959). ” FROUDE , R.E. : A description of a method of investigation of screw propeller efficiency . Transactions INA 24 , 231 -255 (1883). HORN , F. : Study on the subject of interaction between hull and pro- peller (in German) . Schiff md Hafen 7 , 601 - 604 (1955). 1878 Free Surface Effects tn Hull Propeller Interaction HORN , F. : Relation between thrust deduction and wake in pure displacement flow (in German) . Schiffund Hafen 8 , 472-475 (1956). HUCHO , W.H. : On the influence of a stern propeller upon the pres- sure distribution and boundary layer of a body of revolution (in German) . Institute of Fluid Mechanics , Braunschweig , Germany , Report N° 64 / 65 (1965 ) HUCHO , W.H. : On the relation between pressure thrust deduction , frictional thrust deduction and wake in flow about bodies of revolution (in German) . Schiff und Hafen 20 , 689-693 ( 1968 ). HUNZIKER , R.R. : Hydrodynamic influence of the propeller ona deeply submerged submarine . International Shipbuilding Progress 5 , 166-177 (1958 ) ISAY , W.H. : Propeller Theory - Hydrodynamic Problems (in German) Springer Verlag , Berlin/Heidelberg/New York (1964). ISAY , W.H.: Modern Problems of Propeller Theory (in German) Springer Verlag , Berlin/Heidelberg/New York (1970) . KORVIN - KROUKOVSKY , B.V. : Stern propeller interaction with streamline body of revolution . International Shipbuilding Progress’ 3° 3’ 24 (1956') LEFOL, J. : The interaction between ship and propeller (in French). Bulletin ATMA 46 , 221 - 252 (1947 ) MARTINEK , J. , and YEH ,G.C.K. : On potential wake and thrust deduction. International Shipbuilding Progress 1 , 79-82 (1954). NAKATAKE , K. : On the interaction between the ship hull and the screw propeller (in Japanese) . Journal of Seibu Zosen Kai 34 , 25-36 (1967 ) NAKATAKE , K. :Ontheinteraction between the ship hulland the screw propeller (in Japanese) , Journal of Seibu Zosen Kai 36 , 23 -48 (1968 ) NIEMANN , U. : Survey of investigations on the interaction between hull and propeller for partially submerged propellers ( in German ). Forschungszentrum des Deutschen Schiffbaus , Hamburg , Germany , Report N° 5 (1968 ) 1879 Nowaekt and Sharma NOWACKI , H. : Potential wake and thrust deduction calculations for shiplike bodies (in German) . Jahrbuch STG 57 , 330-373 (1963) . POHL , K.H.: On the interaction between hull and propeller (in German) Jahrbuéh SFG9S5 5, 2552305) o( 196i POHL, K.H.: Investigation of nominal and effective wake in the pro- peller plane of single-screw ships (in German), Schiffstechnik 10; 23 = 28 (1963). St. DENIS , M. and CRAVEN , J.P. : Recent contributions under the Bureau of Ships Fundamental Hydromechanics Research Pro- gram . Journal of Ship Research 2 , 1 - 36 (1958) TSAKONAS , S. : Analytical expressions for the thrust deduction and wave fraction for potential flows . Journal of Ship Research 2 , 50 - 59 (1958 ) TSAKONAS , S. and JACOBS , W.R. : Potential and viscous parts of the thrust deduction and wake fraction for an ellipsoid of revolution . Journal of Ship Research 4, 1 - 16 (1960) TSAKONAS, S., and JACOBS , W. : Analytical study of the thrust deduction of a single-screw thin ship . International Shipbuilding Progress 9 , 65 - 80 ( 1962 ) WALD , Q. : Performance of a propeller in a wake and the interaction of propeller and hull . Journal of Ship Research 9 , 1-8 (1965 ) WEINBLUM , G.P..: The thrust deduction . Journal ASNE 63 , 363 - 380 (1951 ) YAMAZAKI , R. : Introduction to propulsion of ships in calm seas (in Japanese) . Journal of Seibu Zosen Kai 33 , 177 - 196 ( 1967 ) 1880 Free Surface Effects tn Hull Propeller Interactton ADDITIONAL REFERENCES Note : - References not belonging by virtue of their subject matter into the foregoing Bibliography but cited in our text for some specific reason are listed below BURRILL , L.C. : Propeller Cavitation Committee Report . Proceed- inp ae PETES, ,o353e2355001960! )t: BGGERS;. K.W.H.; SHARMA’, "S.D. , and WARD , L.W. : An assessment of some experimental methods for determining the wavemaking characteristics of a ship form . Transactions SNAME 75 ,112 - 144 (1967 ) GRADSHTEYN', 15S.; and RYZHIK , I.M. : Table‘of Integrals, Series and Products . Academic Press , New York and London (1965). HASKINS , E.W. : Calculation of design data for moderately loaded propellers by means of induction factors . NSRDC Report N° 2380 ( 1967 ) HAVELOCK , T.H. : The theory of wave resistance. Proceedings of Royal Society (A) 138, 339-348 (1932 ) HESS , J.L:°, and SMITH’; A.M.O. : Calculation of non-lifting potential flow about arbitrary three - dimensional bodies . Douglas Aircraft Co. Report E.S. 40622 (1962 ) HOUGH'S GtR.’ , and ORDWAY, D.E:7 "The generalized actuator disk . Developments in Theoretical and Applied Mechanics 2 , 317-336 (1965 ) KUCHEMANN » D. and WEBER , J. : Aerodynamics of Propulsion . Mc Graw-Hill Book Company , New York/Toronto/London (1953). LERBS , H. : Moderately loaded propellers with a finite number of blades and an arbitrary distribution of circulation . Transact- ions SNAME 60 , 73-123 (1952 ) LUFT , H. : Wave probes for model tanks . Hamburg Ship Model Basin Report N° F 46/67 , translated from German by W.H.Roth and S.D.Sharma , University of Michigan , Depart- ment of Naval Architecture and Marine Engineering ( 1968 ) . 1881 Nowaeckt and Sharma LUNDE , J.K. : On the linearized theory of wave resistance for displacement ships in steady and accelerated motion . Tran- sactions SNAME 59, 25-85 (1951 ) MEYNE , K. : Experimental and theoretical considerations on scale effect in propeller model tests (in German) . Hamburg Ship Model Basin Report N° 1361 (1967 ) MICHELL , J.H.: The wave resistance of a ship . Philosophical Magazine 45 , 106-123 (1898 ) OSTERGAARD , C. : On computer-aided propeller design . The Uni- versity of Michigan , Department of Naval Architecture and Marine Engineering Report N° 88 (1970 ) OSTERGAARD , C. , KRUPPA, C., and LESSENICH , J. : Contri- bution to problems of propeller design (in German) . Schiff und Hafen 23 ,, 531-538 ( 1971 ) PROHASKA , C.W.: A simple method for the evaluation of the form factor and the low speed wave resistance . Proceedings ITTC 11, 65-66 (1960 ) SHARMA , S.D.: An attempted application of wave analysis techniques to achieve bow-wave reduction . Proceedings of Symposium on Naval Hydrodynamics 6 , 731-773 (1966) SHARMA , S.D. : Some results concerning the wavemaking of a thin ship . Journal of Ship Research 13, 72-81 (1969) SHIBA , H. : Air-drawing of marine propellers . Transportation Technical Research Institute of Japan , ReportN° 9 (1953). English translation published by Unyu-Gijutsu Kenyujo , Tokyo , Japan . WEHAUSEN , J.V. ,. and LAITONE , E.V. ©: Surface Waves. Encyclopedia of Physics 9 , 446-814 (1960 ) Springer Verlag, Berlin/Géttingen/Heidelberg 1882 Free Surface Effects tn Hull Propeller Interactton APPENDIX A EXPERIMENTAL PROCEDURES All hull and propeller model experiments were conducted in the towing tank of the Ship Hydrodynamics Laboratory of the Universi- ty of Michigan following essentially standard test procedures, Some of the more interesting, but less obvious details are documented here for the sake of record. A.1. Hull Resistance Test The University of Michigan ship model No. UM 1201, built out of wood to the shape and size determined by Equations (3) through (5), without appendages was used for the hull resistance test. An un- usually high freeboard (equal to full draft) was provided to enable test- ing at high Froude numbers up to F,, = 0.5. Circular cylindrical studs of 1/8 inch diameter and 1/8 inch height were fitted at 5/8 inch spac- ing center-to-center along the entire girth of station No. 1 (that is 0.05 L abaft of the vertical stem) to stimulate turbulence. Departing from standard practice, the model was almost ri- gidly attached to the towing carriage by means ofa three-point system of vertical supporting rods in addition to the usual grasshopper type anti-yaw guides at the two ends. This constraint was necessitated by the marginal transverse stability of the model and by the desire to preclude dynamic trim and sinkage for the ease of comparison with theory. The model was correctly weighted before making the connect- ions, and static draft, heel and trim were verified before and after each test. The resistance was measured by means of tare weights and a horizontal load cell built into a floating beam arrangement between the model and the carriage. Carriage speed was measured from wheel contacts and displayed on a calibrated digital counter. The speed range was extended up to V,, = 9.75 ft/sec (about Ra 0.45), which was the highest attainable within the limitations imposed by tank length, model freeboard, and instrumentation, A.2. Propeller Performance Test The Hamburg Ship Model Basin model propeller No. HSVA 1222 with a standard nose fairing piece as shown in Fig. 2 was used for the propeller performance tests in open water. The propeller ma- terial is bronze, 1883 Nowaekt and Sharma The test procedure was to keep a constant rate of revolution and measure thrust and torque at various speeds of advance so as to cover the entire range of advance coefficient from the bollard condi- tion (J = 0) up to the zero thrust condition (J~P/D). A standardKempf & Remmers propeller dynamometer was used, The measured torque was corrected for bearing friction determined under identical test con- ditions with the propeller replaced by a dummy hub. No "dummy" hub correction'' was applied to the measured thrust. The Reynolds number for open water propeller test is conven- tionally defined as 2 nD Cc 2 z (E) =—— (~) Jot Oia) n 0.7Rp v D0 .( hs D with the design advance coefficient Jp usually approximated by O.75 P/D. Given the propeller geometry Dt=t0.12 ms, (c/D) oR =,0;, 328 2} “B/D =1 and our test conditions Tia BS ce t= 69°F = 0, 9904. = Tigers. = it is seen that the Reynolds number was about 3.4 - 10°. This might appear io be barely sufficient to avoid scale effects due to laminar flow. However, we obtained satisfactory agreement with previous tests run at the Hamburg Ship Model Basin at a Reynolds number of 3.6° 10-. By contrast, a test series run at the Institut fOr Schiffbau in Hamburg with the same propeller at a Reynolds number of 6.0: 10 showed systematic scale effect at advance coefficients J<0.6, cf. report by Meyne (1967). A. 3. Self-Propulsion Test Special care was taken in the self-propulsion tests to ensure that test conditions were identical to those of hull resistance and pro- peller performance tests. The model was constrained in the same fashion as in the resistance test and the towing force was measured by the same instrumentation used for resistance measurements. The mo- del propeller was driven by an electric motor at predetermined rate of revolutions and thrust and torque were measured by the same dyna- mometer used for the open water tests. A streamlined tail fairing * This was the highest rate of revolutions possible without over- loading the propeller dynamometer in the bollard condition. 1884 Free Surface Effects tn Hull Propeller Interaetton piece (see Fig. 2) was fitted to the propeller hub. The measured torque was corrected for bearing friction determined by replacing the propeller temporarily by a dummy hub. No ''dummy hub correction"! was applied to the measured thrust. The self-propulsion points were determined by the so-called British method, i.e. for each test run the towing speed and propeller rate of revolutions were preset while thrust, torque and residual tow- ing force were the quantities to be measured when the steady state condition had been reached, For each Froude number investigated, five to eight test runs at the same towing speed but varying rates of revolution were conducted to cover a wide range of propeller loading aroundthe ship self-propulsion point (and usually extending up to and beyond the model self-propulsion point). There was some indication (a characteristic knocking sound familiar from the previous open water tests) of mild ventilation at the highest propeller loadings encountered in the self-propulsion tests. However, there was no visible effect on the measured thrust and torque values. A.4. Wake measurement A set of standard Kempf & Remmers four-bladed wake wheels was used to measure the nominal wake in the propeller plane behind the hull in both forward and reverse motion, The diameters of the wheels available ranged from 40 to 220 mm in steps of 20 mm, The wheels were designed to yield directly the circumferential average of the axial flow velocity at the wheel radius. There was provision for turning the wheels around by 180 deg on their axis to ensure that the direction of flow relative to the blades was the same for both forward and reverse motions of the model (thus requiring only one set of cali- brations). The wheels were first calibrated in open water at a submer- gence of 150 mm (identical to that used for the model wake measure- ments) by means of a special towing device also supplied by the manu- facturer. In principle, the calibration curves (i.e. wheel rate of revolution as a function of towing speed) should have been linear. In practice, a few wheels showed pronounced nonlinearities and even mild discontinuities at some speeds, presumably due to flow instabi- lities. However, all calibrations were highly repeatable. For the actual wake measurements, the wheel towing device was mounted rigidly to the inside of the model with only its axis pro- jecting out of the stern tube on to which each respective wake wheel 1885 Nowackt and Sharma was mounted at the appropriate propeller clearance (45 mm from wheel center to the vertical stem profile). Every measurement was repeated at least once. It has been noted elsewhere that the measured wake in both forward and reverse motion showed somewhat erratic undulations at the outer radii (see Fig. 6 and 7). This could possibly be blamed on the method of measurement. We had no way of establishing just how accurately the uniform flow calibrations could be relied upon for de- termining the circumferential averages of a varying axial velocity in a complex nonuniform flow involving significant circumferential and radial components. A.5. Wave Measurement A stationary wave probe was mounted at a point about midway along the length of the towing tank and at a fixed transverse distance Yo = 605 mm from the center plane of the model (which coincided nearly with the center plane of the tank itself), Hence, a time record of local wave height at the probe, while the model passed by, was ob- viously equivalent to a longitudinal cut z = {(x,y,) through the steady wave pattern of the model ina coordinate system Oxyz moving with the model. A thin light beam was set up across the tank at a known fixed distance (x; = 336 mm) upstream of the probe. During the run a shutter affixed to the model at a known fixed distance (x, = 2933 mm) forward of the midship section interrupted the light beam and generated an event signal marking the point x = x,-x, on the wave record, thus defining the coordinate origin. The wave probe itself was of the conductance wire type adapt- ed from the HSVA design of Luft (1968) to match the available Sanborn carrier frequency preamplifiers. The circuit output was fed into one channel of a Sanborn strip chart recorder. The overall sensitivity was set at 6 to 9 mm deflection per inch of wave height so as to produce full scale deflection at the measured wave peaks. Wave height records were manually read off at about 350 points at equal time intervals At = 0.03 sec, thatis ata step size Ax = -VAt, and key-punched on IBM cards. All further analysis was done by computer programs, It should be noted that at the highest Froude number investi- gated the length of useful record (taken before running into tank wall reflection) was not really adequate to establish with confidence the asymptotic character of the wave profile behind the model which is needed for the application of a truncation correction (see Fig. 16). However, this was due to a purely geometrical constraint resulting 1886 Free Surface Effects tn Hull Propeller Interaction from the given ratio of model length to tank width, so there was little we could do about it. APPENDIX B WAVEMAKING CALCULATIONS All calculations concerning the wavemaking of the hull and the propeller were based on the strictly linearized theory and there- fore involved the usual assumptions of irrotational flow, infinitesimal wave heights etc., see e.g. Lunde (1951) or Wehausen and Laitone (1960). The following is essentially a compilation of the important formulas used in the present study without attempting to give complete proofs o derivations, B.1. Nondimensional Notation Throughout this Appendix a special nomenclature particularly adapted to the analysis of steady-state gravity-wave problems will be used. This differs from the nomenclature in the rest of the report only in that all * dimensional variables have been consistently rendered dimensionless by reference to a set of three fundamental quantities, namely the acceleration due to gravity g , water density p , and ship speed V. Thus if Q is any dimensional quantity involving only the units of mass, length and time, its nondimensional counterpart Q is defined simply as Q= Q/gaphv" (B1) where the choice of a , B and Y is obviously unique. For instance, =i eee =i. 2 leet 0 ig ee On =210) Ve V =. -2 6 (B2) Rw = Rw/g pV where x is the longitudinal coordinate, L the half-length of hull (now identical to the dimensionless speed-length parameter Y, used else- where in the report), o the density of a surface distribution of sources, Ry the wavemaking resistance etc, With this notation the quantities p, V and g can be formally eliminated from the analysis, thus lead- ing to a considerable simplification of many formulas without any es- sential loss of generality. * Where dimensional variables are nevertheless required, e.g. for purposes of definition, they are identified by underlining to avoid any possible ambiguity. 1887 Nowaekt and Sharma B.2. Source Representations All wavemaking calculations were based on the Havelock ( 1932 ) theory of sources moving under a free surface . It was therefore neces- sary to first define mathematical representations of the hull and pro- peller by means of source distributions . The standard first order (linearized) approximation in thin ship theory is to represent the hull by a center-plane source distri - bution of density SCOT Sx YP 2x (B3) fl 3 boop) (B4) ol; 72) where ; y| defines the hull surface (see Fig. 1) . The results obtained by Have- lock's theory are then identical to those of Michell (1898) The family of hull forms considered in the present study is defi- ned by (BS) y= ab{i = (x/f)2m\{y — «(2/7 where b, £ and T are half-beam , half- length and draft respectively, while ¢€ is a flat-bottom parameter that can vary from €= 0 (wall sided hull with completely flat bottom) to € = 1 (sharp keeled hull with completely curved bottom) . By virtue of Equations (B3, B4) this form is represented by the polynomial source distribution (x2) = (m/m) (b/L) (x/L) 2" fy. 6-2/7} (B6) over the rectangular plane nilsson ds 5 N tg esd tout rena (B7) Following Dickmann (1938) the propeller can be represented by a continuous distribution of sources of (negative) density o (R, @) over the propeller disk x = Xp > Ry p> -7r< O6< fF (B8) where R, 9 are polar coordinates = 5 = +R i y R4cos_6 Z Zp sin 0 (B9) the point (x), 0, zp) is the geometrical center of the propeller and 1888 Free Surface Effects tn Hull Propeller Interaction Ry , Rp_ are the hub and tip radius of the propeller respectively . There is no simple way of relating the source strength directly to propeller geometry , speed and rate of revolutions . However , using momentum theory , Dickmann derived two useful approximations con- necting propeller source strength to thrust loading , one of which yields a uniform sink disk of source density a(R, 0) = -(¥l+Crp -1)/4-7 a Se Soy (B10) over and the other a discrete point sink of source strength -(4)1+C yy - 1) (Rp* -R,,7)/4 at (xp,0,2p) (B11) where Z er fen) a oes (BRE (B12) is the thrust loading coefficient based on disk area (excluding the hub ). In addition to the above we have also used the following alter- native relation due to Hough and Ordway (1965): meer 5.6) == ZG / 4.5 (B13) where GR), sa all shu Ze elec ties (B14) is a non dimensional function representing the radial distribution of bound circulation [’ along each blade of a Z bladed propeller at advance coefficient J = V / 2 a Rp . Here the source density is a function of radius R , but still independent of angle © . Since the circulation is obtained numerically from a computer program at dis - crete radii, 6 will generally be defined merely as a tabulated function. Unless analytical interpolation is used for further processing , it is tantamount to a radially stepped sink disk 1889 Nowackt and Sharma A certain ambiguity arises in interpreting the speed V in the above relations when the propeller is operating in a wake behind the hull . We believe that the physical sink strength of the propeller should be determined by the local speed of advance VA while its wa- ve pattern must be characterized by the speed V_ relative to the fluid at infinity . Hence the corresponding relations in the presence of an effective axial wake Ww, become git (Bon by = -(¥l+Cry - 1)(1l- wp )/4n = 2 2 Cr, = 2 D/ » Va* (Ry - Ry) (B15) in the Dickmann approximation , and @ (Bon @). =n, Zia (RY, ME JS eR) in the Hough and Ordway approximation . In either case , the left hand side is the appropriate dimensionless source density ¢9 to be used in the subsequent calculations of wavemaking and thrust deduction It may be noted that source disk representations of the propel- ler are only useful for calculating the induced flow field (outside the slipstream) . For calculating propeller performance (thrust and torque) resort must be taken to the correct vortex model . In princi- ple , it is possible to calculate also flow field and wavemaking directly from the vortex model , cf. e.g. Nakatake (1968) . However , the increased computational effort is hardly justified in view of the other approximations in the analysis. B,.3.. Free-Wave Spectrum A useful description of the wavemaking characteristics ofa ship is provided by its free-wave spectrum as defined for example in Eggers , Sharma and Ward (1967) . Given an arbitrary source distribution o (x, y, z) over adomain D , its complex-valued free-wave spectrum (as a function of transverse wave number u ) becomes G (u)i+ iF (u) = Pb ue aw ies y, Zz) exp {8° z+i(sxtuy) } ie D v = y l+4u2, ss and~—s 8 = yt /2 (B18) The significance of the free-wave spectrum lies in its ability to yield a simple description of the asymptotic wave pattern behind the ship . where 1890 Free Surface Effects tn Hull Propeller Interaction For x—4 -0 ; 1 oo CO6 3) =ae [iro sin (sxtuy) + G/(u) cos (sxtuy) } du (B19) Here s and u can be interpreted as the circular wave numbers in- duced by a free plane wave (moving with the ship) in the x and y direction respectively . Hence F(u) and G(u) are called the sine and cosine components of the spectrum and its amplitude is given by E (u) = F4 (u) + G2 (u) (B20) The phase of the free-wave spectrum depends on the choice of the coor- dinate origin but its amplitude does not . The associated wavemaking resistance Rw suse as (u) du (B21) is determined solely by the amplitude spectrum . By virtue of formulas (B6) and (B17) the free-wave spectrum of the hull becomes bo Ma a0 on wry of Ly fo PHA HA Ga/a- exp (s*ztisx) (B22) or after some simplification Gy (u) = 0 1 F,, (u) 2 pele Fr! ) (m, sf) Fl?) (9, € Seg ) Vv (B23) 1891 Nowaekt and Sharma with F(3)(n,q) = q if fe dkp Ge sal ae (B24) 0 The integrals pl) and (3) can be solved in closed form [ see formulas 2.6334 and 2.3212 in Gradshteyn and Ryzhik (1965) ] or evaluated by recurrence formulas : F() (0,p) = 0, F() (m,p) = 2m { (2m-1) [ sin(p)-F((m -1,p)] /p-cos (p)}/p F(3)(0,q) = 1-exp(-q) , FG)(n,q) = -exp(-q) + nF@G)(n-1,q) /q (B25) Similarly , the free-wave spectrum of the propeller can be written as l+v Gp(u)+iF p(u) = Eyl dR frao fo (R,8) exp {s°(zp#R sin 9) + i(sx Xp + uR cos0)) | (B26) If the propeller source distribution o is a function of radius only , then this simplifies by virtue of transverse symmetry to l+v Gy (u)+iF (u) = oe v exp( 5? Zz p'is*,) fe )dR yi exp(s¢R sin@)x cos (uR cos @)d@ l+v Ne = 1674 { = exp ( s2zptisxp) rf Ro(R)I,5(sR)dR Ry (B27) 1892 Free Surface Effects tn Hull Propeller Interactton where I, is the modified Bessel function of zero order [see Gradsh- teyn and Ryzhik (1965) , formula 3.9372 ] . This last integral can be easily evaluated for any numerically defined function o (R) B.4. Wavemaking Resistance The individual wavemaking resistances of the bare hull and the free running propeller Rwp are found directly by substituting the appropriate free-wave spectra (B23) and (B27) into the general formula (B21) ; 1 (16d 2ov RwWH ="8 f {— Fm, sh) F'2)(n, <, Ts(?)) | du Rp 1 l+v 2 Vv = ag ifs [ 16” { = | exe sz) f Ro(R)Io (sR) aR Tay de 0 Ry (B29) To calculate the wavemaking resistance Rwyr of the total system hull with propeller one can use the principle of linear super- position of free-wave spectra , that is Gr (u) Gy (u) + Gp (u) Fr (u) + Fp (u) (B30) Fey (u) provided both spectra are expressed in the same coordinate system . The general formula (B21) can then be applied to the total spectrum 2 2 Eq(a) = yor (u) + Fp “(u)} Evidently , in general Rwr # RwH + Ryp (B31) 1893 Nowaekt and Sharma and the difference Rwr - Rw - Rywp = = if {GH (u) Gp (a) + Fy (a) Fp (u)| — du 0 (B32) is a measure of the interference between the wave patterns of the hull and the propeller . The interaction term can be positive or negative B.5. Wave Flow due to Hull The perturbation flow induced at the propeller plane by the motion of the hull under the free surface can also be calculated by thin ship theory . We start with the Green's function of the problem as defined by Equation (63) of Eggers , Sharma and Ward (1967 ): 1 1 Gi (sis) aie Wie = IS r] +> Ep 2! ee a Ai 2 eee hs [ex [-w |x-x"| tiu(y-y')+iY¥w -u"(z+z')] ay = 90 |u| ne -u2 -iw2 J 4s +im— + -sgn(2x-x')b {=z exp {is(-x") + iu(y-y') +s*(z+z') \ du with (B33) ry ae Pe oe (y-y')" + (ee + (yay!) + (zt2t)* This is the velocity potential due to a point source of unit strength at (x', y', z') . Integration over the hull source distribution (B6) yields the velocity potential of the hull ¢ (x,y,z) , and subsequent differen- tiation with respect to x,y,z yields the components of perturbation velocity yx , Cyn Og It is convenient to break up each expres- sion into four parts corresponding to the four terms of the Green's function (B33) . For instance ¢, (x,y,z) = e{1) + e) - e (3) .? (4) (B34) 1894 Free Surface Effects tn Hull Propeller Interactton with Deg te) 2m-1 a Sas aif ad] ae ! -e(-2'/7)™h x cy a es Me (B34) vy! eee dz' 3/2 {(xc-x1 )2 + (y-ys2+( 2-2")? } (B35) etc . It turns out that in the resulting integrals the x', z' integrations can be carried out in closed form , while the u,w integrations must be performed by numerical quadrature . It takes some algebra to reduce the expressions to a form suitable for computer programming . We will show this for one exam- ple . Substitute in (B35) Bis/l, t= x'/l, nay/h, b22/l, 0% =2° fk, r= 7T/h (B36) Then Wy ee Alehee) [ ¢ ] bar ( tie be Wola (2k) athe Px ar xf dé! VA Digi, Lbs eter se) =e ee) Now put (B37) (een ae oa fee ie st (B38) Then ; oy fai f (eCepeie elt aint; Vie Hm fat ee = ae ak MiG, Pe eee : es ftr ees ce tale Now factor out the constants and apply the binomial theorem to get 2m -1 : at} SB (-) i ( 2m-1 ) ee, i i=0 1895 Nowaekt and Sharma n i+j Gm e) 2-0) ay oe agit = ‘ Tr Pe Py ) i ay = 4 i, i=0 j= where ai ins i 3 Ej (bab + ) = dé as é ‘i 3/2 ig kl ge ee E+] on ti ee } (B39) This double integral E;,; has a closed form solution (for constant limits of integration) amenable to numerical evaluation by a recurrence formula . (The authors are indebted to Drs K. Eggers and H. Kajitani for this suggestion ) . Consider the indefinite integral (B40) By repeated use of formula 2.2631 from Gradshteyn and Ryzhik ( 1965 ) it can be shown that 1 . : 4 fh—S> whol : jtl Fa Bij Lit) ee Ey, )-()( F201 +y Fs, ;)f 1 eet i+] 2 (B41) Hence, starting from the four fundamental solutions EQ 0 = ve arctan (yr/xz) Eg = - In (x+r) Ei o = - ln (ztr) Pe oe (B42) 1896 Free Surface Effects tn Hull Propeller Interaction any element E; ; can be constructed by recurrence , for example os vg ¥ Pp 3 3 3 E> 2 = 3 + In (z+r) acy In(xt+r ) +4 arctan (xz/yr) an ae? E33 re os y* (2 +2% -2y*) + 2 (xt+2%) - 222} (B43) For a computer algorithm , however , it is not necessary to develop the analytical expressions explicitly since the recurrence formula (B41) can be applied numerically to each of the four summands (cor- ner values) of the definite integral (B39) . A similar analysis can be applied to the second term of (B34) , but but the final result is obtained more easily by considerations of sym- metry < o ex, y,z)= = e Moxy, -z) (B44) Restricting our attention to field points behind the hull (x <-1), the third and fourth terms of (B34) can be simplified as follows . gle) 4 Peeping te Jeg pee none e(-2'/T) 4h. ice Mgt ce a w exp [w(x-x') tiuy +i a au“ (z+z' i} ne w -u- -liw bre =" af Parx2L,min, ¢ 7) cos(uy) du (B45) 0 where Pn Os [u(x+ZL) 4 Nap agg i exp(iz one an, < ei ee “hice a ao dan alia a 8 exp(-t) dt (B46) 1897 Nowaekt and Sharma with ¢ DP a I Ce ee and c" exp(-iq¢) af (B47) The integrals g(t) and G(3) can be solved in closed form or evaluated by recurrence formulas : cg) CO. py = O_, G'om,p) = m {(2m-1) [(1-e7P)/r+2c!(m-1,p)/p)/p-(1+e*P) /} elt = (1-6 V4 Gna) = fag) (n-t,q) - 21/3 (B48) The integral I has an exponentially decaying factor in the integrand and is therefore suited to Gauss-Laguerre numerical quadrature . The real part in (B46) need not be evaluated analytically , if complex arithmetic may be used in the program . Similarly , for x <-J2 £9 oe ). tm f au fax! fae as a {1-« (ia T nt Pr ‘of = exp { is(x-x") + iuy + s“(2+2') | a ) sin ( sx ) F"!) Gn, sl) F? es €, °T)| cos(uy) du (B49) 1898 Free Surface Effects tn Hull Propeller Interaction where pl) and p) are the functions already defined in (B24) The integrals (B45) and (B49) can be truncated at a sufficiently large value of u and approximated by the known recurrence for- mulas for Fourier series . Suppose co Cc. + is =m I(u) {cos(uy) + i sin(uy) } du : N ~ Au = (u, ) { cos(u,y) + isin(u vy) be, ie with 1 i = vou, €, sco ae e, = "Itor 9S! 116 where Au isa suitable step size and N is sufficiently large . Then cat C= 2. (Uj-U and S = U, sin(y Au): Au (B51) where the U, are defined by the sequence Pitas UN NM Ge Se yO U, = I(u,) + 2cos(y a ak PE rach ete | (B52) This completes the wanted algorithm for all four terms of (B34) . By our definition , g¢ evaluatedin the propeller plane is iden- tical to the total potential wake fraction , 1.e. the sum of the so-called potential wake (zero Froude number effect) and the wave wake (finite Froude number effect) . Thus (B53) wp(R, 6) + wy (R,0) = x(x, y, 2) if | | x = Xp» y = R cos®, 24= 2, +R sin © 1899 Nowaekt and Sharma By evaluating ¢, ata sufficiently large number of field points the circumferential and disk averages of the theoretical wake can be es- timated. Moreover , it can be shown that the zero Froude number wake, the infinite Froude number wake and the bow wake (x > L) can also be derived from the four components of expression (B34) (1) (2) Biot Ob. Myer). apy see - ~ n w p x x (B54) ree ae (2) Be, oo: Wp Wee = ae he (B55) x >: %, (x,y,z) = ea y,z)+ ¢,,() (-x, y, z) + Pes (-x, y, Z) (B56) This last quantity evaluated at x = -Xp yields by virtue of longitu- dinal symmetry the desired theoretical wake in the propeller plane "behind" the hull in reverse motion’. B.6. Wave Flow due to Propeller In order to calculate the perturbation flow induced by the mo- tion of the propeller under the free surface , we start with an alter - native expression for the Green's function (B33) , see formula (56) in Eggers , Sharma and Ward (1967) a2. ] 1 2 Go= (-s2 5 ae debice sec “6 15h eee ‘+iai) Jak l 2 BS 35/2 k-sec29 n/2 2 fa + Im 2 f sec’ @ exp [sec”9 (z+z'+ia) ]de 1/2 (B57) with @ = (x-x') cos 6+ (y-y')sin® and r),r> as befo- re . Differentiating with respect to x, and taking advantage of the symmetry in 0 , we obtain x-x! x-x'! 4 m [2 G,(x-x',y-y',z,z') = wr Rawgulsoy f se<8 de. ea tT? f = k dk fi exp[k(ztz') ] sin [k(x-x') cos 09]cos[ k(y-y') sin ©) aaa k-sec20 0 (contd. . ) 1900 Free Surface Effects tn Hull Propeller Interactton 1/2 raf sec?Q exp [ (z+z') sec*O | cos[ (x-x') sec 9] P cos [(y-y') tan® secQ|d0 (B58) Since we are interested only in the flow induced by the propeller in its own plane (the so-called self-induced wake) , we confine further ana- lysis to the case x = x' = x,. The first three terms of (B58) then vanish and in the last we substitute u = secO@tan@, v = yin’ tats = (l+v)/2, du/d@= sv to get a (28) Gay yty’ 3, 2’) pan exp [(z+z')s“ ]cos[(y-y') u] tty du 0 (B60) Now integrating over the propeller source distribution o (R) and taking advantage of its transverse symmetry , we get for the self - induced wake the following expression R P 7/2 we (R, 0) ules dR bos G,(0,y-y',z,z') o(R') R'do ) : oa =4_7 fi “7 }exp [ (zt+zp)s°] of R'o(R') I(sR')aR't 0 Sie ieae (uy) du (Bé1) where the integral formula quoted after (B27) has been applied again. Since the function o(R') is in general not analytic , the R' integral must be evaluated by numerical quadrature (e.g. Simpson's rule) for suitable values s(u . such that the u integral can be approximated by the recurrence formulas for Fourier series , see Equations (B50-B52) . By proper choice of the fields points y = Rcos®, z= ZptRsin 0 the self-induced free-surface wake wf(R,0) can be calculated at sui- table points (Rj , 9x) on the disk , from which the circumferential ave- rage wy (R) and the disk average wy can be obtained by numerical integration . 1901 Nowaekt and Sharma A useful check on the numerical accuracy of the calculated values of w,(R,@) is obtained by using them to determine the wave- making resistance of the propeller by virtue of Lagally's theorem [see Equation (11) of Eggers , Sharma and Ward (1967) }: Rp 7/2 forltawt arf R a(R) f ws (R,@)d@ dR R -7/2 (B62) Analytically , of course , this is identical to the more direct formula (B29) based on the free-wave spectrum . Numerically , we found that the differences were negigibly small for a reasonable step size AO< 7/6. B.7. Thrust Deduction We wish to calculate the force of thrust deduction , i.e. the augmentation of hull resistance due to propeller action . Let us call it S>RwH . Conceptually , the most direct approach would be to use Lagally's theorem., i.e; 6 R = 92) au >P (x, 0,2) Pp WH = a2 mf dz {0 (2) Viz » 0, \ aaa, This would seem to necessitate the explicit calculation of the longitu- dinal perturbation flow eP induced by the propeller on the center-plane of the hull y = 0O , which is not quite easy due to the singular double integral in formula (B58) . However , we will circumvent this diffi- culty by an indirect approach . Let us denote by 6yRwp the augmen- tation of propeller resistance due to hull action . Then again by Lagally's theorem w/2 Rp - buRwp = Zan aR f R dO (o(R,0)ex (x, R, 8) Ry - 1/2 (B64) where yl now is the axial perturbation flow induced by the hull in the propeller plane , i.e. the wake as already defined by (B53) On the other hand , by virtue of previous definitions we have Rwt = RwH + Rwp + 6pRwH + 6yHRwp Hence, (B65) 1902 Free Surface Effects tn Hull Propeller Interactton where the sum of the first three termsas defined by (B32) is relatively easy to calculate . Thus we see that the calculation of thrust deduc- tion requires no further effort beyond that already expended for the calculation of hull wake and the wavemaking resistance of hull alone , propeller alone and the total system hull with propeller . In particular, it is needless to calculate the flow induced by the propeller on the hull. Note that 6pRwy, is the total potential thrust deduction , i.e. the sum of the so-called ''potential'' component (zero Froude number effect) and the ''wave'' component (finite Froude number effect) . The thrust deduction fraction becomes to Hplyes dSpRwy/Ly = (8,2, /K_.,) (V2/gD)*(v/nb)? 2 4 2 (6 PRwy/ Kpy) (L/D )2(F,) 4( yy) ae At zero Froude number , of course , Rwt = Rwy = Rwp = 9 and therefore = R 'pRwy Su “wP (B67) Thus if we use the zero Froude number wake (B54) in (B64) to calculate 5yRwp , we can determine the '"potential'' component of thrust deduction t, from the simple principle of reprocity (B67) already exploited by Dickmann (1939) It must be emphasized that the force 6;;Rwp apparently exer- ted on the propeller by the hull does not necessarily have a physical meaning since the source disk is an inappropriate model for calculat- ing propeller forces . However , it is a perfectly valid mathematical artifice for a simple , although indirect , determination of the quantity dpRwr which is a real force exerted on the hull due to propeller action, viz. the force of thrust deduction. 1903 Nowaekt and Sharma Incidentally, ifthe source strength is uniform over the disk , as in Equation (B15) , the integral (B64) simplifies to byRwp = 44°(Rp* - Ry”) o (R) (w, + wy) (B68) where wp and w,, are the disk averages of the potential and wave wake respectively . Moreover , if only the potential component of thrust deduction is wanted , substituting from (B15) in (B68) and taking advantage of (B67) one obtains eee fa thea & Nibin Ne al get (B69) This is slightly different from the classical result of Dickmann (1939), cf. his equation (15) . However it agrees with Tsakonas' (1958) equation (12) , except that he does not distinguish between the poten- Pp and the total wake wp B.8. Wave Profile Analysis tial component w The purpose of wave profile analysis was to establish the true or experimental free-wave spectrum (and associated wavemaking resistance) of the hull and propeller as opposed to the theoretical spectrum based on linearized source representations discussed in the previous sections . The longitudinal cut method of Sharma (1966) as described in Eggers, Sharma and Ward (1967) was used. The essential steps of the analysis are given below. Let z =$ (x, yg) be a longitudinal cut through the wave pattern of the model as measured at a fixed transverse location y = y, in the coordinate system of Fig. 1 . Define modified * Fourier transforms C* (avg) iS te tepid) =f V2 -1 § (x, yo) exp(isx) dx (B70) * The asymptotic nature of the wave pattern behind a ship is such x/|y (—-» seo : (x,y) ~ exp(ix)/ «/c-x that the modified Fourier transform remains finite for any s , while the ordinary Fourier transform becomes infinite at s=1. 1904 Free Surface Effects tn Hull Propeller Interactton Then the free-wave spectrum of the model is given by G(u) =——— z {C"(s,y,)eos(uy,) - S*(s,y,)sin(uy,) | ; * F(u) =——— {c*(s,y )sin(uy_)+S (s,y_)cos(uy )} (B71) re) fe) fe) ) 2s -l 2 i , g y where u = sl s -l in accordance with (B18). By applying this pro- cedure separately to the model hull with and without propeller one can obtain the spectrum G-r(u), F.p(u) of the total system hull and propel- ler and the spectrum Gy,(u), Fy ,(u) of the bare hull respectively. The spectrum of the propeller alone Gp(u) , F p(u) then follows from the principle of linear superposition. lO Us cer er II I hy OD a — eos Oy 1 1 yO ae Ss (B72) For the ease of comparison with theory the propeller spectrum may be transformed to a new coordinate system Oxyz which has its ori- gin in the propeller plane. If x = X-Xp ; yoy, Bee then a) = 9G u)cos(sx,,) + F,(u)sin(sx p! p) (B73) F(a) = F,(u)cos(sx,) - G u) sin(sx p!| p) The associated wavemaking resistances of bare hull (Ry ), bare propeller (Ry p) and total system hull-propeller ee are obtain- ed from the respective spectra by use of the general formula : = \ Mae Ry == if fei PG tape dia (B74) 8 r > 0 1+ Vl+4u It is assumed here that the wave pattern has transverse symmetry. 1905 Nowaekt and Sharma APPENDIX C LIFTING LINE CALCULATIONS C.1. Problem Formulation The principal method for calculating thrust deduction and free- surface effects due to a propeller as described in Appendix B pre - supposes that a lifting line representation of the propeller , i.e. the distribution of bound circulation along the radius , is known . In order to be able to apply this method to a given propeller we need a scheme for determining the circulation distribution for any given operating condition of a propeller of predetermined geometry . This is essential- ly the classical "performance" problem in propeller theory ( as opposed to the design problem , in which a certain performance criterion is prescribed and the optimum propeller geometry is sought for ) . Physically , the problem can be formulated as a set of rela - tions which must be satisfied at every propeller radius between the hub and the tip . Using standard symbols , these relations are ct. = Cc, {@) CGT) =" = 6. (G2) tan b= P/2rR (C3) tan B, = (1-w)V+u,t/ jek - apt (C 4) ee aria dR! eget ee ae reo Re (C 5) Dae E: » 7) §at(R')|_ aR! Shi ah i -(R/Rp» 8B,» 2) ) aR (ROR : (c 6) r= Cyc {(1-w )V tugl /(2sinB (C 7) 1906 Free Surface Effects tn Hull Propeller Interaction Here , Equation (C 1) represents the predetermined two - dimensional foil characteristics , i.e. the lift coefficient C, asa function of angle of attack a . Equations (C2) to (C4) establish the local angle of attack o@ as the difference between the predetermined geometric pitch angle @ and the unknown hydrodynamic pitch angle f; . The velocities ug and uy induced by the free vortex trail of the propeller may be obtained from Biot-Savart's Law and are expressed in Equations (C5) and (C6) as integrals involving the slope of the bound circulation [(R) and two special functions in and i- (of three variables ) , the so-called induction factors , see Lerbs (1952) . Equation (C7), finally , is the relation between lift and circulation in accordance with Kutta-Joukowsky's theorem . Mathematically , the problem is an integral equation for the unknown function 8; (R/Rp) ; which can be solved by iteration if efficient algorithms are available for computing the induced velocities ug and u-+ The solution of the above problem yields the distribution of circulation , and hence lift , over the radius dL = p {(1-w )V tug} aR/(sin 6; ) (C 8) Now the drag can also be estimated from the known foil charac- teristics Cy = on (a ) (C 9) dD Sa R {(1-w) v tug }?c ar /(singi)” (C10) Hence , by resolving lift and drag along the axial and circum- ferential directions and integrating over the radius , one can calcula- te the thrust and torque produced by the propeller . C.2. Method of solution Our method of solution was dictated by the computational tools and the information on propeller characteristics available tous . The principal computational tools at our disposal were two computer pro- grams for propeller design , both based on the lifting line theory and incorporating efficient algorithms for numerical evaluation of induced velocities . One was obtained from the Naval Ship Research and Deve- lopment Center and the other from the Technical University of Berlin by courtesy of Dr. O stergaard . They are well documented in the lit- terature , cf. Haskins (1967) and Ostergaard (1970) , and therefore 1907 Nowaeckt and Sharma need not be described here in detail . After several test runs had re- vealed that the two programs yielded practically identical solution of the design problem , we chose the Berlin program for further use and adapted it to a solution of the performance problem . The principal modification necessary was the following . In the design problem the hydrodynamic pitch angle 8; (R/Rp) is generally prescribed to fulfil the optimum ( minimum energy loss ) condition thus eliminating the need to solve an integral equation . In the performance problem , how- ever , the integral equation must be solved . This was done by the method of successive approximations to the unknown function B;(R/Rp). Starting with an initial guess ($j ),5, say corresponding to the opti- mum condition , a better approximation was found by cycling through Equations (Cl) to (C7) . In order to prevent the iteration from diverg - ing it was found necessary to weight the successive approximations as follows (B°x),42 os w, (Bin i wo (Bin +1 (C11) With w,; = 0.9 and wz = 0.1 the final error in B {(R/Rp) after ten iterations was found to be generally less than 1 % A major handicap in this algorithm was that the two-dimensional foil characteristics of our propeller (see Fig.2) were not explicitly known to us . We therefore back-calculated the foil characteristics from the known measured performance (thrust and torque) of the pro- peller (in the deeply submerged open water condition) . This was done by treating any given operating condition as a small perturbation from an assumed design (optimum) condition , i.e. | eee (C12) Cy. (fo) Rees & + 2m (a-ay ) LD a ie ® 2 Cy (a) = a, |Cz,(a) -C, 5) pote (C13) The design angles of attack a and the corresponding lift coefficients Crip were specified indirectly by the choice of a design advance coef- ficient Jp. Let Bip be the optimum hydrodynamic pitch angle at any radius at the design point Jp. Then for calculating the performance at any other advance coefficient J (and assumed pitch angle Be ee i: is only necessary to evaluate the differences (see next page) 1908 Free Surface Effects in Hull Propeller Interaction (a-ap) = Bi, -Bi (C14) Cy (@)-Crp = 2nr(a-apn)k (C15) Boia me ay |Cy, (a) - Cyp|“? (C16) It is thus seen that in all five arbitrary constants (Jp, Cpp>k:, ay» a>) were usedto match the calculated propeller performance to the actual measured performance . By trial and error the following best fit values were determined for our propeller : Fiavesc lal ads.aGinmnaaan ee k = 0.67 Ba ee alts” gb oe =" 1G (G17) Note that the only critical parameter here is the factor k , which may be interpretedasan empirical catch-all to account for viscous losses and miscellaneous three-dimensional effects . The degree of agreement finally achieved between the measu- red and calculated (more precisely , simulated) performance of the propeller is obvious from the following table and from Fig. 10. Further details of this method of calculating off-design perfor- mance and results obtained with other propellers are reported ina recent paper by Ostergaard , Kruppa and Lessenich (1971) Measured Simulated 1909 Nowaekt and Sharma C.3. Applications It should be obvious from the foregoing that as far as the dee- ply submerged open water condition is concerned , our computer pro- gram as described above did not really predict propeller performance analytically but rather simulated the known measured performance by means of a lifting line model . This was perfectly acceptable because our primary aim here was not to predict propeller performance , but to determine the equivalent lifting line for use in calculating thrust deduction and free-surface effects . However , in two subsequent ap- plications this program was indeed used to obtain certain genuine pre- dictions of propeller performance. First , for estimating the effect of free-surface on propeller performance at shallow submergence the program was run with the calculated self-induced wake wf (R) of the propeller as an additional input (see Section B.6) without any attempt to manipulate the foil characteristics fixed once for all on the basis of the deeply submerged condition , see Equation (C17) . Hence , the thrust and torque calcu- lated for shallow submergence as plotted in Fig. 13 and reproduced in the following table are , in a certain sense , true predictions of the free-surface effect h/Ry, = 3.47 h/Rp= 1.00 h/Rp= 1.00 Measured Measured Calculated It may be noted that the wake w;(R) input to the program was here calculated for the lifting line corresponding to the deeply submerged propeller . In principle, it would be possible to run a second iteration with the wake w,(R) recomputed for the new lifting line determined 1910 Free Surface Effects tn Hull Propeller Interaction by the program for the shallow submergence. However, this refine- ment is considered unnecessary. Second, for evaluating the propeller performance in the behind hull condition again the same procedure was used, with the measured nominal wake w(R) substituted in Equations (C4) and (C7). However, the calculated thrust and torque were found to deviate substantially from the measured values. Since the primary purpose of this calcula- tion was to obtain a realistic simulation of actual propeller perfor- mance by a mathematical lifting line, it was decided to enforce a thrust identity. However, this was accomplished not by further mani- pulating the assumed foil characteristics but by multiplying the input wake w(R) with a constant wake corrector k,, whose final value was determined by iteration. Thus the program was here used not only to determine the equivalent circulation distribution but also to simulate the physical difference between the nominal and effective wake through the factor k,. Moreover, the program also calculated a mean effec- tive wake wT which was based on a thrust average rather thana volume flux average. This was defined as R ne alee) {Rag} {re ¢ tan 6, } r(R)dR are w. = R (C18) Uh {oR : ugh {t- « tan 6, | T(R)AR Ru where not only the nominal wake w and the circulation [ but also the quantities up ,8; , and e€= Cp/Cy, vary with radius, even though this has not been explicitly indicated in the formula. The numerical values obtained for k,, and wr and their practical significance have been discussed in Section 3, 6.2. 1911 Nowaekt and Sharma APPENDIX D DOUBLE BODY CALCULATIONS D.1. Motivation All calculations described in Appendix B are based on the first-order thin ship theory in which the hull is represented by a li- nearized (with respect to the beam) source distribution on the center plane, see Equation (B6). This has the great advantage that "potential" (i.e. zero Froude number) effects and wave (i. e. finite Froude number) effects can be calculated consistently using the same source distri- bution. However, the accuracy of the results depends in an uncontroll- ed manner on the ''thinness'' of the ship. In order to obtain a quanti- tative estimate of the error involved in the application of thin ship theory to our hull, a few wake calculations were also performed by the method of Hess and Smith (1962), which does not impose any res- trictions on hull geometry. As is well known, the Hess and Smith al- gorithm provides a general solution of the Neumann problem of non- lifting potential flow about arbitrary bodies by means ofa surface dis- tribution of sources, Due to the enormous amount of numerical com- putation involved, however, the application of this method to the cal- culation of flow about ships is still limited to the so-called zero Froude number approximation, in which the ship (including the pro- peller) is conceptually replaced by a deeply submerged double body generated by reflecting the under water form about the static water plane. In our terminology, therefore, only the pure potential effects (as distinguished from the viscous and wave effects) can be evaluated by this method, An improved version of the original Hess and Smith computer program was made available to us by the Naval Ship Re- search and Development Center. D.2. Results Without going into the intricate details of the Hess and Smith method we report here only a few relevant results obtained by this program, First, a series of nominal wake calculations was perform- ed with the propeller disk assumed in its proper transverse and ver- tical position (¥p= 0 , z4= -0.5 T) but at five different longitudinal positions as shown in the following table. Since the accuracy and com- puting effort in this method depend critically on the number and size of the body surface elements, we tried four different arrangements involving N = 100, 125, 145 and 150 elements. As our double body had three planes of symmetry, the elements are understood to cover only one eighth of the total body surface. To ensure finer detail near the stern the element size was not uniform over the entire length of the hull but made increasingly smaller toward the ends. 1912 Free Surface Effects in Hull Propeller Interaction The results showed that the arrangement with 125 elements yielded adequate accuracy for our purposes. Moreover, the average wake was practically identical to that calculated by thin ship theory. (This gave us, of course, more confidence in applying the thin ship theory also to the finite Froude number case where no such accuracy control was possible. ) Potential wake Wp averaged over the propeller disk Thin Ship Theory Hess and Smith Program N =100 N =125 2x,,/L 0.1785 0.1769 OTrSazu 0.1502 0. L307 DE 1Z9IS 0.1136 OTL 27 0.0998 0.0990 Next, a series of effective wake calculations was conducted with the propeller located in its proper position (xp = -0.51 L) and assumed operating at the advance coefficients J,,; = 0.733,0.889 and 1,131 corresponding to the ship self-propulsion points at Y, = 4.0, 7.0 and 12.5 respectively, see Section 3.6. This involved two extra complications compared to the previous nominal wake calculations. One, the presence of the propeller destroyed the longitudinal symme- try of the flow so the number of significant hull surface elements had to be doubled from 125 to 250. Two, the flow induced by the propeller (and its mirror image about the plane z= 0) onthe hull surface had to be given as an additional input to the Hess and Smith program. For this the Hough and Ordway source disk representation of the propeller (see Fig. 29) was used. The algorithm for computing the flow induced by source rings at arbitrary field points was taken from Ktiichemann and Weber (1953), pp. 310-316. The results are summarized in the following table. 1913 Nowaekt and Sharma Calculated effective wakes WEp versus nominal wakes Wp \ Hess and Smith Program 0...2 0.3 0.4 05 0.6 On7 0.8 0:59 20 It is seen that the calculated effective wake wp is 2 to 4% higher than the calculated nominal wake Wp > depending upon propel- ler loading of which the source strength o is ameasure . This is exactly what one would expect from theoretical considerations , since the propeller sinks induce extra sink strength in the afterbody which in turn induces an additional positive wake in the propeller plane . However , the difference is too small to have a significant effect on the calculation of thrust deduction . 1914 Free Surface Effects in Hull Propeller Interactton Figure ] Hull propeller configuration and coordinate system FOES Nowaeckt and Sharma QJQ0NVH Aajoutoe3s sz9{Jedorg 7 o1n317 1HOIY NO1193410 ONINBNL $33930 o- g IVY 2 'S30V78 40 W38WNN *v/*v ‘Olive vi¥v J0v7E d * (ANVISNOD) HOLIG 0 ‘¥313WVI0 ZZ) -49113d0Ud VA SH ViVO ¥3173d0¥d ISSi YsivmM N3d0O 4OJ4 3SON ‘Oe i OUr SYSLAWITIIW NI 3aVv SNOISN3WIO Vv isa NOIS1NdOSd YOS TIVL CO-nmMnOR 1916 Free Surface Effects tn Hull Propeller Interactton «107? =—=——_ BU ages en ee cee | Ch 0 2 4 6 8 LOR “12 x10 Figure 3 Measured total resistance 1917 Nowaekt and Sharma Figure 4 Measured total resistance at low Froude numbers. Deter- mination of viscous form factor 1918 Free Surface Effects tn Hull Propeller Interactton © Derived from wave cuts Calculated from linear theory Measured Cp-(1+k)C, Figure 5 Calculated and measured wave resistance KOLO Calculated 0, 20:1 "0 <= pi & Pe 7.0 4.0 = 0.15 Yo SEO ig 4s eG O20 = 7F8 Been see 4.0 0.05 0 0 0.2 Nowaekt and Sharma w. (R)+w, (R) =. Figure 6 Calculated and measured wake in reverse motion 1920 Free Surface Effects in Hull Propeller Interactton 0.6 Difference w(R) = w(R)-w,(R)—-w,(R) 0.4 RK ~~ NS > ee O82 ; ee ~ Calculated w_(R) << 7s p ~~ - (@) Yo \ 1 easured | \, 2 is 6 \ 7.0----- _- = 0. w(R) \ Q 4,.0—— — NN XN \g a. = N\ ‘N 0.4 NE 28 ™ ra SS ae NN \ a we br eck — --— ~ 0.2 aS ke ce —-—-~ Calculated he en wy (R)+w,(R) Sees es 10) (@) 0.2 0.4 0.6 0.8 1.0 Figure 7 Calculated and measured wake components 1921 Nowaeckt and Sharma Symbol h/Rp + Sia Oo 2.00 ros 20 0 0.2 a a 0.8 a) M1. 2 Figure 8 Propeller characteristics at deep submergence 1922 Free Surface Effects tn Hull Propeller Interactton h/R, = 3.47, 2.0, and 1.5 —K— h/R, = 1.0 Figure 9 Effect of low submergence on measured free-running pro- peller characteristics 1923 Nowaekt and Sharma 0.8 Symbol Oo Calculated for given propeller x Calculated for optimum propeller _—— Measured Figure 10 Calculated and measured propeller characteristics at deep submergence 1924 Free Surface Effects tn Hull Propeller Interactton Figure 11 Calculated distribution of bound circulation for free- running propeller at deep submergence 1925 Nowaekt and Sharma Figure 12 Calculated self-induced free-surface wake of free-running propeller at submergence h = Rp 1926 Free Surface Effects tn Hull Propeller Interaction x Calculated by lifting line theory O Estimated from Je for sink disk (Hough and Ordway) — Estimated from Jp for sink disk (Dickmann) ----- Measured es —-— Measured at deep submergence Figure 13 Calculated and measured propeller characteristics at shallow submergence 1927 Nowaekt and Sharma Ogre Rwp Uniform sink disk (Dickmann) oe oe 0.10 i Generalized sink disk 0.08 VY tne and Ordway) ‘N cox pe ‘ ! \\ 0.0 ! \ \ Discrete point sink | . (Dickmann) ! \ | \ ° I \ \ / \ \ 0.04 | \ } Ae, ! \ ] | ! \ 0.0 / } ~ / : 7 TA 7 0 (0) OF2 0.4 J 0.6 0.8 qa 9) ae ips pee aa arn, = en a eee a ae oe a (@) 0.4 0.8 P Ne 1.6 2.0 2.4 nh Figure 14 Calculated wave resistance of free-running propeller at shallow submergence h = Rp 1928 Free Surface Effects tn Hull Propeller Interactton Hull with propeller Je = 0.389 Hull with dummy hub tg/v*? Figure 15 Measured wave profiles at F,, = 0.267 1929 Nowaekt and Sharma Hull with propeller Jy, = 0.733 Hull with dummy hub tg/v? >_ Figure 16 Measured wave profiles at F, = 0. 354 1930 Free Surface Effects tn Hull Propeller Interactton lg 1o) 0 Measured Toy Me SE Calculated = ; Se oe Se 0 2 4 6 8 10 Figure 17 Free-wave spectrum of bare hull at F,, = 0.267 1931 Nowackt and Sharma Measured Calculated Figure 18 Free-wave spectrum of hull with propeller at F, = 0.267 1932 Free Surface Effects tn Hull Propeller Interactton Measured Calculated Figure 19 Free-wave spectrum of propeller at F, = 0.267 1933 Nowackt and Sharma 1 < cu) 0 Measured Di ee a ) ra ee ee ee Calculated -1 Figure 20 Free-wave spectrum of bare hull at F,, = 0. 354 1934 Free Surface Effects tn Hull Propeller Interactton Measured Sis a ier. Calculated Figure 21 Free-wave spectrum of hull with propeller at Fy, = 0.354 1935 Nowaekt and Sharma Measured Le ey OO ee eee Mee ai Ses Calculated Figure 22 Free-wave spectrum of propeller at F,, = 0.354 1936 Free Surface Effects tn Hull Propeller Interactton Ship self-propulsion 10*CEp= Or 1785 0.60 0.65 0.70 Jy 0.75 0.80 6.85 Figure 23 Typical result of a propulsion test and the determination of self-propulsion points, Min sot 2 P9s7 Nowaekt and Sharma 102G FD 0.4 WwW 0.2 we Wr i oes 0 — 0.72 0.76 Jy 0.80 0.84 Figure 24 Variation of propulsion factors with loading for Y, = 4 1938 Free Surface Effects in Hull Propeller Interactton hoe 0.8 0.4 (57 0 0 Figure 25 Variation of propulsive factors with Froude number at the ship self-propulsion point 1939 Nowaeckt and Sharma Figure 26 Calculated distribution of bound circulation for propeller behind hull at self-propulsion point 1940 Free Surface Effects in Hull Propeller Interactton 2 Measured in propulsion test: Kay Kou? Jue Jn Calculated by Vortex Theory:© Figure 27 Calculated and measured characteristics of propeller operating behind hull at self-propulsion point 1941 Nowackt and Sharma OD Symbol (a) w (Measured ) | 2 SA ae PA Le ie Tin a nies e q 0.5 0.4 0.35 O<3 P 0.25 0.22 0.2 Figure 28 Calculated and measured wake fractions as functions of Froude number 1942 Free Surface Effects tn Hull Propeller Interaction -0.08 me) Hough and Ordway SSSSs= Dickmann Jaa -0.06 7 -0.04 =O) 02 ) 0 Figure 29 Calculated distribution of source strength for propeller behind hull at self-propulsion point 1943 Nowackt and Sharma 0.5 Symbol = t Measured SSS =e ty Calculated from uniform sink disk —- _ t_¢t_ Calculated from uniform sink disk 0.4 PP ih! — ty Calculated from generalized sink disk So == ttt Calculated from generalized sink disk Figure 30 Calculated and measured thrust deduction fractions as functions of Froude number 1944 Free Surface Effects tn Hull Propeller Interactton DISCUSSION Edmund V. Telfer ets Nets Ewell, Surrey, U.K. Anybody who has attempted to read this paper, as I have during the past two or three days and this morning at 2 o'clock after coming back from our delightful banquet, will appreciate that it con- tains a lot of matter for real thought. The first point I would like to deal with is the authors' attack on the subject of relative rotative ef- ficiency and their quite innocent reference to relative rotative effi- ciency being an empirical ''catch-all' for various unclarified effects of relatively insignificant magnitude. I am not sure whether that is fair. Undoubtedly during the thirties, a lot of attention was given to the subject. In 1951 I published a North East Coast Institution paper on various aspects of the propeller/hull interaction problem and I made a suggestion that the real meaning of relative rotative efficiency could be very simply understood by plotting Kp toa Kop base so getting the well-known propeller polar. Suppose we have such a plot of all propeller polars over a range of pitch ratio, we will then have the higher pitch ratios on the right and the lower ones to the left. What then happens in self-propeller tests is that the measured Kp value does not locate on the correct pitch polar with the correspond- ing Ko value. In most cases it will be found that the Kp lies to the low pitch ratio side of the actual open polar and it is this that produ- ces the phenomenon of relative rotative efficiency. Values of relative rotative efficiency exceeding unity are then obtained, which to the lo- gical mind appears to be impossible, but if it is realised that what has really happened is that the actual correlation of the behind thrust and torque has left the open line and come to a smaller Kp value or in other words the centroid of the thrust of the behind propeller has been moved radially inboard, as a consequence of the normal wake distribution in a single-screw ship, having the heavier wake towards the shaft centre. Thus when we see this taking place we realise that all that positive relative rotative efficiency is showing is that there has been a change in the wake distribution compared with the uniform distribution of the open condition, Therefore, really, relative rotative efficiency is merely a wake distribution factor and if one thinks of it in that way one can get a much clearer understanding of the problem. 1945 Nowaekt and Sharma R.E. Froude, when he first introduced the idea of rotative efficiency was then principally concerned with twin screw naval ves- sels, and there the erratic variation of rotative efficiency with oppo- site screws certainly puzzled him. Part of the trouble was due to the fact that the starboard and port propellers,being of opposite hands, were never made with identical accuracy ; and one frequently found different rotative efficiencies for the two screws, so offering some justification for the authors'own statement that rotative efficiency was earlier known as a ''catch-all'', Model manufacturing inaccuracy should no longer be allowed to occur and so cloud a fundamental issue. The effect of a positive rotative thus indicates a change in the wake distribution and therefore a bigger wake towards the root. If one makes the thrust wake integration as the authors have done ra- dially, it will be found to yield a correct rotative efficiency (in the real meaning of the word) which is actually less than unity. Thus if one does not presume the propeller to do the integration but if, as the authors did, one takes each radius separately, from such integration a lower rotative efficiency and a higher wake will be obtained. It is rather significant however that in the authors' tests the thrust wake is less than the torque wake. This is most unusual. Most single screw ships show the reverse, certainly with ordinary testing methods. When this is more deeply considered, as the authors have done, it becomes clear that the wake has actually increased, and one has then to recredit what was previously known as rotative efficiency to an in- crease in the wake due to the difference in the wake distribution. I am extremely glad that the authors have brought out this particular point and I hope my remarks are understandable in the light of their own work. One is tempted to believe that the original description of the words ''rotative efficiency'' was a justifiable one, but we now see that there is no justification for thinking that the pro- peller normally can work with a higher efficiency behind the hull than in the open, Yet certainly in the thirties, Teddington was reporting rotative efficiencies of about 1.2 and even higher, and Dr. Baker him- self expressed the view that these results were evidently ''phoney" and there was some other explanation to be found for the meaning of rota- tive efficiency. I suggest that the wake distribution effect is the real interpretation. The authors state that the wake is caused by the presence of the hull and the free surface. This is not quite correct. It is not caus- ed by the presence of the free surface, but provided there are waves on the free surface then admittedly they may have some effect on the 1946 Free Surface Effects tn Hull Propeller Interaction wake. One could have, for example, a submarine generating waves on the surface when the submarine itself was totally submerged. There could then be a wake change depending upon the orbital motion of the waves passing the propeller. If there were no surface waves, however, there could not be any additional wake, at least not in my opinion, Finally, I should like to refer to the authors' determination of the form effect of their model. They have used the ITTC line with its 75 numerical coefficient to determine the form effect deduced by the Prohaska method. If one allows for the fact that the low value they obtained of about 21/2 per cent, by using the 75 coefficient should be changed to refer to the Hughes' two-dimensional basis using a 66 coefficient, one will find a form effect of 161/2 per cent. On the other hand, when one investigates what the equivalent plank would be for the model - that is a plank having the same length as their model and the same measured surface as their model - one finds the mini- mum value to be some 22 1/2 per cent. This throws doubt on the Prohaska method which is insecurely based, in my opinion, on the unjustifiable assumption that the specific resistance initially varies as the fourth power of the Froude number. Older work, and certain- ly the very early work of Hovgaard, Taylor and others, appeared to produce a certainunanimity inthe finding that the specific resistance initially varies as the square of the Froude number. One has just the choice between the fourth and the second power and if the Prohaska method is used with either of these assumptions one finds that using the square relation gives a much smaller form effect than using the fourth power of the Froude number. It is suggested therefore that the method is not sufficiently acceptable for determining form effect ; and when this is also associated with the basic defect of the ITTC line in its inability correctly to extrapolate to the ship, I think we have to fall back on the only alternative way of correctly determining a form effect, which is to have a geosim series for each particular ship. Against this it is always held that a geosim series is much too expen- sive, and this is certainly true. But I suggest that the cost of truth has to be faced and in the general euphoria is soon forgotten, I would say, in conclusion, that I have greatly enjoyed this paper and strongly recommend it to all who are interested in the sub- ject. 1947 Nowackt and Sharma DISCUSSION Georg P. Weinblum Institut fiir Sehiffbau der Untversttdt Hamburg Hamburg, Federal Republie Germany There exists a small number of investigations on the inter- action problem of the complete system hull-propeller-rudder. Herbert Voigt showed (JSTG 1932) that the thrust deduction t ofa single screw ship with an old fashioned square rudder post and a plate rud- der could be appreciably reduced by fitting a stream lined rudder. Results of a systematic investigation of the ''complete'' problem have been published by Ivchenko (publications of the Krylov Institute, Ship hydrodynamics II ; unfortunately, I was unable to obtain the re- port by the same author which show the pertinent proofs, Krylov Ins- titute, No 146). Ivchenko's findings are supported by extended expe- rimental work. Most advanced ideas on the subject are due to Yamazaki ; the first of a general and therefore rather programmatic character, the second one presenting the application of theory to a thin ship, showing an unexpected increase of t due to a streamlined rudder. Because of the complex character of our problem the authors justly restrict themselves to the classical hull-propeller system, emphasizing free surface effects while a recent paper by Dyne treats pertinent scale effects. It may be permitted to state, that the authors lived up to the high standard, set by Dickmann in his classical work on the subject. Especially they used all modern methods partially de- veloped by them and were not anxious to perform tedious calculations to obtain quantitative results. The choice of a simplified mathematic - al model is justified by the fundamental character of the investigation leading to the conclusion (Fig. 30), following which the wave thrust deduction t,. contrary to Dickmann's conjecture becomes heavily dependent pod the Froude number for F,=0.30. Indications of this dependence ty(Fn) have been communicated by Pien (SNAME Jubilee meeting, 1968). Because of the extremely bad wave making properties of the model this speed limit may be higher and the effect much less pronounced with usual practical forms, I disagree with the authors statement in concluding remarks that ''contrary to common belief the wave component can be dominant in the wake.... at F,#0.3". 1948 Free Surface Effects tn Hull Propeller Interactton The dependency of the wake uponthe Froude number was an important finding by F.Horn, several times published by him and his school, also in English, and recommended as design principle. The authors being thoroughly familiar with the German school, I would prefer a statement that their findings nicely support these well known results. It is a matter of opinion and of considerable importance with regard to scale effects that the difference between measured and cal- culated t-values is due to an appreciable viscous component, Ivchenko corrected the disagreement by a factor caused by non unifor- mity and unsteadiness. As pointed out by the authors here much re- mains to be done. DISCUSSION Kiaus W. Eggers Institut fiir Schtffbau der Untversttdt Hamburg Hamburg, Federal Republte Germany I have a short comment regarding the analytical method for determining the wave-thrust deduction, which was performed here following Eq. (B65) in order to avoid explicit calculation of the flow generated by the propeller singularity at the hull surface. I want to suggest a more elegant and direct method. As the wave flow generat- ed by the hull within the propeller area is available through closed form integration over the hull, the wave induced drag on the hull due to the propeller field can easily be calculated as a Lagally-force on the propeller singularities induced by the inverse flow due to the hull. i.e. the flow for the case that the ship travels astern with the same speed. I wonder if such a calculation would confirm the numerical re- sults obtained so far only indirectly through (B65). 1949 Nowaekt and Sharma DISCUSSION Carl-Anders Johnsson Statens Skeppsprovntngsanstalt Goteborg, Sweden I should like to emphasise Professor Telfer's plea for taking the relative rotative efficiency seriously and draw the authors' attent- ion to an earlier work by Yamazaki in 1966 which contains some in- teresting calculations of the relative rotative efficiency. I should also like to ask the authors why they use wheels for measuring the wake and not pitot tubes. Some years ago we used wheels at the Swedish tank but we had difficulties in getting consistent results. DISCUSSION Gilbert Dyne Statens Skeppsprovntngsanstalt GOteborg, Sweden The investigation presented in this paper is a valuable con- tribution to the understanding of the important hull propeller interact- ion problem. The paper illustrates clearly how fruitful a combination of theory and experiment can be when treating complicated flow phe- nomena, The strength, o , of the sinks representing the propeller is directly related to the propeller induced mean axial velocities at the disk. Following the definitions used by the authors, o , is given by ie bead u 1 Al] We o(R,8) = - ENC ee I: | 1. Then the behind-hull Kypy, vs. Koy polar will lie to the left of the open-water Kr vs. polar and hence a virtual propeller of lower pitch will have to be chosen, This would indicate a lower value of J , which in turn due to the convex shape of the ng vs. J curve would yield a higher value of 7 o/(1-w) for the real propeller, But since 7, and t are independent of the choice of wake, so is the quantity Me No/(1-w) = 2)/(1-t) Free Surface Effects tn Hull Propeller Interaction Consequently, the increased value of 19/( -w) in Telfer's analysis would automatically lead to a lower value of mR: If we had started out with the conventional analysis yielding Ip - aaPon fF wa ce ot Cearn i i ihe ’ errs 7” . Pern Bee § ime i341 e TSS aTpory ds ven sre credt ine; Nia Se et Vee ew ee ate yr oe@ ~ EL RA tem ots: epee OT8 on ik n aeiOu ak Loe RA A Li woot sent eer? a e ety ta (Sipe RAT aSS Va baw Monn tamonyeecrs id mitrigat On ony apple? “tde ta lear Biase esbinnteddaaAl: oclod Uae , ‘ ies a gov St) a vi valpi hey. 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ABSTRACT This paper presents the results of an experimental and theoretical investigation of the distribution of shear stress and pressure on BRIAN BORU, a 20- foot model of a Series 60, Block 60, surface ship. Boundary layer calculations were carried out using the Cumpsty-Head-Smith momentum integral me- thod under the small crossflow assumption; the po- tential flow was obtained from slender body theory for zero Froude number. Surface shear stressand pressure distribution were measured at sixty points on the hull ; the magnitude and direction of shear stress were determined from high-aspect ratio hot films and Preston tubes, Additional data on total resistance, sinkage, trim, and wave profiles on the hull are also presented. The experimental and theoretical results are com- pared for a range of Froude numbers(0.2 made calculations similar to Gadd's. They used Cooke's 16 method for three-dimensional boundary layers and Guilloton's!”? method for the potential flow. Their calculations 1965 Huang and von Kerezek show mainly the effect of the wave-induced pressure gradients since Guilloton's potential flow method cannot be used to trace, accurately, the streamlines on the lower half of the hull surface. Thus, their results are restricted to boundary layer characteristics along streamlines on the hull near the free surface. Finally, von Kerczek'8 has applied the Cumpsty-Head-Smith”® method and the slender body potential flow program of Tuck and von Kerczek* to calculate the boundary layer on a double hull of the LUCY ASHTON corresponding to the experiments of Joubert and Matheson !? , von Kerczek'® found that the effect of streamline con- vergence and divergence is of overriding importance. Computed skin friction coefficients are in good agreement with the measured values. The present study uses von Kerczek's!8 computer program to compute the boundary layer for BRIAN BORU, the series 60 block 60 model at zero Froude number. The computed results are compared with experimental data at Froude numbers equal to 0. 22, 0.28, and 0.32. These comparisons are a first step towards deve- loping a boundary layer computation method for a ship at arbitrary Froude number, Shear stress distributions on the ship hull at arbitrary Froude number has been measured by Steele29 (Tanker Model), Steele and Pearce®! (High Speed Linear), and Tzou2* (Series 60 block 60 model). All of these shear stress measurements were along waterlines only and considerable oscillation of the shear stress along the waterlines was noted 292! | In none of these experiments was shear stress direction measured. In the present study we use hot-film and Preston tubes located at sixty points on the hull to measure the shear stress dis- tribution on BRIAN BORU at various Froude numbers. The probes were located along four zero Froude number streamlines and along one waterline (14% draft). The direction of shear stress was deter- mined by the hot-film shear probe. In addition, pressure distribution, total resistance, trim, sinkage, and wave profiles were also mea- sured for a range of Froude numbers. A completely-detailed set of experimental ship resistance data is collected and presented, Avai- lable theories are compared with the corresponding experimental data. Only the boundary layer computation method will be described briefly in the next section. The other theories will be used without derivation, CALCULATION OF THE TURBULENT BOUNDARY LAYER We calculate the turbulent boundary layer on the ship hull 1966 Shear Stress and Pressure Distrtbutton on a Shtp Model by the method of Cumpsty and Head? and Smith® with the addition- al assumption of small crossflow. A complete description of the method is given in References (7) and (8) anda description of its application to ship hulls in Reference (18). We will only reproduce the final formulas here. In the following we have non-dimensionalized with the follow- ing units : half the sae length, L/2, for length, the ship speed, V, for velocity, and PU ies for stress, where U isthe (dimensional) inviscid velocity at the edge of the boundary layer. Let @ 4 and 6 i> be the momentum and displacement thickness of the streamline com- ponent of the boundary layer flow, where N is the coordinate normal to the body, U, is the inviscid velocity at the edge of the boundary layer, u is the boundary-layer time- averaged velocity component in the streamline direction and 6 is the nominal boundary layer thickness, The relationship between 6,,, 67 , and Cy, = 75, /0U“/2) where 7, is the wall shear stress in the streamline direction is determined by assuming small crossflow and integrating the approximate momentum-integral equation d 6 gt 16 HU 6 Bee G = se eaareliahi ld alana iA legidantliioe Vga de oe Ss) de and the auxilliary rate-of-entrainment equation d( 6 G) 1 dU 6 a = it + wha s ee F (G) (3) da S de along the streamlines, where a is the arc length parameter along the streamlines, In equations (2) and (3) K, is the geodesic curvature of the equipotential line, H = 6 11/67 is the shape factor and G is the parameter ( 6 - 6*)/@,and F is the empirical rate- of- entrainment function, The empirical correlation of F to G to H is given by Standen 23as 1967 Huang and von Kerezek -0. 653 a o ii] 0. 0306 (G - 30) 2st DS 4 Q i" 1535. (ft -‘0.°7) ce The skin friction coefficient C,, is given by the two-dimensional formula niet tig H. +1 Q fe) 0. 0146 ( om ) Ce S i log, 4 (2R 6, ) SS log) (2R 0, ) + 0. 4343] Hap» = PLAsaY dog R + 0. 9698 fe) 10 B44 (4) Q = 0.9058 - 1.818 log,,H Q. a Q(H,) where Rg, = R,U, 6), and Ry, = VL/2» (recall U, and 6,, are non-dimensional). This is Granville's (24) formula and is an extension of the Von Karmann-Schoenherr flat-plate skin-friction formula to flows with a pressure gradient. This skin friction formula was chosen over others, for instance the Ludweig-Tillman formula, because it is more accurate at high Reynolds numbers. With the Mager 25 profile assumptions for the crossflow, the crossflow momentum integral equation is transformed into an equation for the angle 6 that the shear stress makes with the streamline direction. Then 6 is de- fined by tan B = where Cy = Ty2/PU%/2) and Twz2 is the wall shear stress in the crossflow (e.g. normal to the surface streamlines) direction. The shear stress magnitude is then given by C =" G a: . CIN SRR Se 1968 Shear Stress and Pressure Distribution on a Shtp Model B remained less than about 8.5° on all streamlines except in some local regions near the stern on the keel and waterline, Here the transverse section curvatures are very large and the boundary layer approximation is not valid anyway. Note that equations (2) and (3) correspond to the axisymme- tric boundary-layer momentum-integral equations for a body of revolution with local radius r if r is defined by ] dr r el ek The dominant three-dimensional effect on slender ships is simply the streamline convergence and divergence represented ty the equi- potential line geodesic curvature Fe Any potential flow that supplies the streamlines and invis- cid velocity distribution thereon can be used with equations (3) and (4), but we have used the very simple zero Froude number, slender- body potential flow theory given by Tuck and von Kerczek” in con- junction with the surface equation for the hull described in von Kerczek and Tuck . This combination seemed to give fairly good results for the LUCY ASHTON when compared to the double-model experiments of Joubert and Matheson, We used exactly the same procedure for the Series 60 block 60 calculations and will compare the results to model experiments in a towing:tank, The extension of the Tuck-von Kerczek slender-body potential flow program to include free-surface effects is underway. We have used the zero Froude number potential flow because of its availability and the ex- pectation that wave effects on the boundary-layer flow will be rela- tively minor on the lower portions of the hull, This calculation will serve mainly to illustrate the suitability of the Cumpsty-Head- Smith ‘’~ boundary layer method for moderate block-coefficient hulls, THE MODEL, EXPERIMENTAL SETUP, AND PROCEDURES The ship model used was a 20-foot Series 60, block 60, wood model, The name ''BRIAN BORU" was given to the model as a counterpart to the British research ship model ''LUCY ASHTON", A photograph of the model is shown in Figure 1, The body plan is shown in Figure 2, A slight hull modification aft of station 18 (which preserved sectional areas) was made to accommodate a propeller shaft used earlier for propulsion and vibration experiments, This 1969 Huang and von Kerezek model has been tested by many of the towing tanks in the worldasa standard vibration model. The streamlines on the ''double model" were computed by Douglas -Neuman? and slender -body ‘ potential flow methods. In Figure 2 good agreement between the streamlines computed by each method is noted. The slight difference in stream- lines plotted is due mainly to the fact that the starting points in the two computations are not exactly the same. A coordinate system 0'x'y' with its origin in the undis- turbed free surface and another coordinate system ox yz fixed in the ship are used, Both are right handed coordinate systems moving with the steady velocity of the ship. The plane o'x'z' is on the un- disturbed free surface, o'x' is in the direction of the ship motion and o'z' is upward. The plane oyz contains the midship section, plane oxz the center plane section and the plane oxz the design water plane. The locations of the shear probes and pressure taps in the oxyz coordinate system are tabulated in Table 1. The reference length used to non-dimensionalize all lengths is L/2 where L is the length between perpendiculars. When the model is tested in the free- to-trim condition, the two coordinate systems are no longer coinci- dent. We denoted the vertical distance from the axes o'x to ox by h(x) (positive above the undisturbed free surface). The sinkage is defined as -[h() + h(1)]/2, trim by bow by -[(h@) - h(1)} and trim angle by tan ~1 (ona Pn yas Provisions were made for sixty interchangeable shear probes and pressure taps spaced evenly along a total of four zero Froude number streamlines, designated by A, B, C, and D, on the double model, and along a waterline E (14% draft). These probes have to be mounted flush to the hull surface. At each location a one-inch dia- meter teflon mounting plug was .sunk into the hull with its axis paral- lel to the normal of the ship surface, and its face flush with the ship surface, the surface of the plug was carefully polished to follow the original contour of the hull. The hot-film shear probe penetrated the plug and was fastened by four screws, The depth and angle of the probe with respect to the hull were adjustable. A photograph of this arrangement is shown in Figure 3. The depth of the probe with res- pect to the surface was carefully set to protude less than 0.002 inches out of the hull surface by using a flat face pressure transducer as a probe-protuberance feeler. Preston tubes and static pressure taps were placed on the hull through the same mounting plugs. Dynasco pressure transducers were used to measure the pressure from the Preston tubes or the pressure taps. Seven-channel DISA (Franklin Lake, New Jersey) constant temperature anemometers were used for the hot-film shear probes which were manufactured by Lintronics 97-0 Shear Stress and Pressure Dtstrtbutton on a Shtp Model Laboratory (Silver Spring, Maryland), The use and calibration of the shear probes is given in the Appendix. The electrical output of the calibrated transducers was digitized, averaged for 100 seconds, and analyzed in various dimensionless forms using an Interdata computer. The results were printed out immediately after each experiment. At each Froude number the model was first run free to trim. This trim condition was then fixed for subsequent runs at the same F, . The total resistance was routinely measured by a floating girder and a block gage. The trim and sinkage were measured using two potentiometers located at FP and AP, and wave profiles were traced along the hull using a colored pencil and then measured. These provided a complete set of experimental data for BRIAN BORU. RESULTS AND DISCUSSION The experimentally measured wave profiles along the hull, sinkage and trim, total resistance, and pressure and shear stress distributions at various Froude numbers will be presented and com- pared with relevant theories and numerical results. (1) Wave profiles along the hull. Photographs and a dimen- sionless plot of wave profiles along the hull at six different Froude numbers are shown in Figures 4 and 5, respectively. Figure 6 shows the measured profiles at Fn = 0.22 and 0.28 compared with the profiles predicted by Guilloton's method. Although the forward quar- ter of the predicted wave profiles on the model compare favorably with the measured profile, the agreement becomes poorer downstream, The prediction not only overestimates the magnitude of the last trough, but also misses the location (phase). (2) Sinkage and trim. The measured sinkage and trim,com- pared with the first-order thin ship theory computed by Yeung”4 re shown in Figure 7. The measured values of sinkage are all smaller than that predicted. However, the measured sinkage and trim coeffi- cients agree rather well with the calculated values using the zero Froude number, slender body(4 or Douglas-Neuman 3)theoretical pres- sure distribution. It should be noted that the measured trim does not vary with Froude number as much as that predicted. Since the sin- kage, and trim and the wave profiles predicted by the thin ship theory are not in good agreement with the measured values, the thin ship theory may not be suitable for this model which has a flat bottom and a moderate block coefficient. No further comparison of experimental data with thin ship theory is attempted. 1971 Huang and von Kerezek (3) Total and residual resistance. The measured total resis- tance and the residual resistance, Cp = Cr - Cr , are shown in Fi- gure 8, where the 1957 ITTC friction line was used to detente Cy for all of the data. The other data of the geosims by Todd ?®, and by Tsai and Landweber ”” (parent hull without stern modification are also shown. The 6-, 10-, and 14- foot models were tested in the Iowa Tank. The Cp ofthe 10- and 14- foot models is higher than that of 6- and 20- foot models. The 20 model was tested in NSRDC basin I and II. However, reasonable agreement for the 6- and 20- foot models is noted. The Cy of the present 20- foot model was measured in NSRDC basin I and II by a floating girder and a block gage. Turbulence stimulation, a row of studs, 1/8 -inch in diameter, 0. l-inches in height, and 1-inch in spacing was used in one of the tests. No significant difference in C. with the turbulence stimulator was found. The discrepancy in Crp's among the four present tests is less than 2%. (4) Pressure distribution. If the flow is assumed irrota- tional, then the Bernoulli equation in the o'x'y'z' coordinate system is P2 p Pp : 1 2 a 2 oe = NM Peer = 18872 are Px + ey + e2'| = where V is ship speed, Pg is the atmospheric pressure and ¢ the perturbation potential. We define the pressure coefficient by Hed ALLO AK AORN AA G0 th —— : y : p v’/2 xia (6) which yields Cp =1 at the stagnation point where %, = V and Yy = ¥z = 0. If the linearized free-surface boundary condition is used, at any waterline below the undisturbed water surface ¥ reduces CO Pie Phy 92) exp (gz'/v*)¢ (x', y', z' = 0), From Equation (6) we have gud tuileli aah: cxp| ete (7) 1942 Shear Stress and Pressure Distribution on a Shtp Model where {¢ is the wave height inthe oxyz coordinate system and h is the distance between o'x' and ox, The measured CC. along streamlines A,B,C, and D, and along waterline E and the corresponding approximate C, computed from Equation (7) and for the double models are shown in Figure 9. As shown in Figure 9-e the wave approximation (Equation 7) is in good agreement with measured values near the free surface (14% draft) for the three Froude numbers tested. However, this approxi- mation is not valid near or on the ship bottom. On the after half of the flat ship bettom the measured C_ is in general close to the C predicted by the double model (Figure 9-a through 9-c), and the effect of the surface wave (Froude number) there is small. However, some effect of waves (Froude number) on C_ onthe forward half of the ship bottom is noted. It should also be noted that C.. near the keel (streamline A) ata Fn=0.22 is very close to the C. predict- ed by the double model. Figure 9-b shows the C, of the double model model computed by the Douglas-Neuman theory * and slender body theory *'8 Close agreement between the two computations in the middle of the ship is noted. (5) Shear Stress Distribution. The local shear stress coef- ficient is defined as C; = tw/ (PV*/2), a vector tangent to the hull surface. The shear stress magnitude and the angle of the probe relative to the waterline for points on the ship side and to the buttock lines for points on the ship's bottom were measured by rotating the probe to three angular positions (0,+ 0 ). This can be used to compute the magnitude and angular position of shear stress vector on the hull. This information along with the direction cosines of the waterline or buttock line tangents and the surface normal calculated from the sur- face equation © , were sufficient to decompose the shear stress vector into three components (C, 3 Cry eS ) relative to the body axes (x, y, z,). The measured direction cosines of Cy rela- tive to the (x, y, z) axes are shown in Figure 10 at the high and low Froude numbers of the experiments. Note that these direction cosines do not vary much with Froude number except near the station of maximum wave slope (i.e., between x = -0.7 and -0.5). In Figure 11 we present the measured and calculated distri- butions of Crs, along various streamlines and on waterline E. Note that the agreement between experiment and calculation is better at low Froude numbers and is fairly good on streamline A for the entire range of Froude numbers of the experiment. In these cases, wave effects were at a minimum and this indicates that the Cumpsty-Head- Smith boundary layer calculation is adequate for moderate block- 1973 Huang and von Kerezek block-coefficient hulls at small Froude numbers. Crossflow effects are very small throughout. The main discrepancy is near the stern and in the region of maximum wave slope. The discrepancy near the stern is due to a combination of inadequate boundary layer theory, poor pressure distribution prediction, and poor body-geometry fitting by the surface equations there. Improvement of prediction techniques for this region requires special attention. In the region of maximum wave slope, we find that the calculated shear stress magnitude and the measured shear-stress magnitude agree fairly well, so that the differences between C;, measured and Cy, calculated shown in Figure 11 is mainly due to differences in streamline direction at zero and finite Froude number. In the last graph of Figure 11 we have included the calcu- lation of Ce, along waterline E of Webster and Huang!> . Here the pressure gradient effects on the boundary layer due to body geo- metry and the waves are very small, and thus there is little diffe- rence from flat plate values except very near the stern and near the maximum wave slope. An interesting observation is that the Webster- Huang!> calculation seems to have not predicted the effect of the wave satisfactorily. This effect is mainly due to change in the shear stress direction in accordance with the streamline direction. Webster and Huang!> used Guilloton's'’ potential flow method to calculate the inviscid velocity on the ship surface. Due to the rather crude approximation of the body and the potential flow, accurate stream- lines and consequently accurate values for streamline convergence and divergence are not obtained. The present calculation is also not expected to predict the shear stress near the free surface since the effect of the free surface is neglected in the potential flow computa- tion. From these considerations it seems that it is very important to obtain an accurate discription of the potential flow streamline and pressure distribution in order to adequately calculate the proper magnitudes of the shear stress components (Cy. 4 Cry . Cr, }. It also should be noted that both the present computation and the computation of Webster and Huang!5 overpredict C,, near the stern, It is not possible to predict thick boundary layer charac- teristics near the stern by these methods, CONCLUSION Comparison of the measured pressure and shear stress distributions, trim and sinkage, and wave profiles along the hull of BRIAN BORU at various Froude numbers with various theories and boundary layer calculations allows the following conclusions to be 1974 Shear Stress Pressure Dtstrtbutton on a Shtp Model drawn : 1, The sinkage and trim, and the wave profiles along the hull predicted by thin ship theory are not in good agreement with the measured values. Thin ship theory is not satisfactory for ships having a flat bottom and a moderate block coefficient. 2. The measured pressure distributions on the after half of the ship bottom are rather close to those computed on a double- hull model and show little effects from waves (Froude number). Thin ship theory does not give a good approximation of the flow in this region. Near the free surface the pressure distributions behave like a linearized wave, which agrees with the thin ship approximation, The flows near and on the forward half of the ship bottom are affect- ed by the combination of the free surface waves and the details of the ship geometry. A new physical model is needed in order to pre- dict the flow over the after half of the hull. 3. The measured shear stress vectors at selected points on the model show that the shear stress vectors are oriented in nearly the same direction as the local streamlines indicating, as has been found previously, that boundary layer crossflow is small on moderate block coefficient hull forms. Although the local shear stress values depart little from equivalent flat plate values, the trend of the departure is fairly well predicted by the Cumpsty-Head- Smith boundary-layer calculation method with the small crossflow assumption, especially along streamlines where wave effects are negligible. This indicates that boundary layer calculations carried out along the streamlines, taking into account pressure gradients and streamline convergence or divergence, using momentum integral methods can be quite useful. It is important, however, to develop an accurate potential flow calculation methods and methods for calcu- lating thick boundary-layer approaching separation, ACKNOW LEDGMENT The authors are indebted to J.H. McCarthy of the Naval Ship Research and Development Center for his stimulation and in- terest during the course of this work. The authors would also like to thank Messrs. N. Santelli, G.S. Belt, and L.B. Crook for their assistance during the experiment. Mr. C.W. Dawson is also thank- ed for performing the exact double hull potential flow computation. This work was authorized and funded by the Naval Ship Systems Command under its General Hydromechanics Research Program, Task SR0090103. 1975 Huang and von Kerezek APPENDIX EXPERIMENTAL TECHNIQUES FOR MEASURING MAGNITUDE AND DIRECTION OF SHEAR STRESS In order to determine the shear stress vector distribution on the hull, it is necessary to measure the distribution of the magni- tude of the shear stress vector 7, and its angle 2 with respect toa convenient direction on the ship hull. Two useful measuring devices are considered in this Appendix : the flush-mounted hot-film shear probe, and the Preston tube and the directional Preston probe. (1) Hot-Film Shear Probes The principle of the hot-film shear probe is that skin fric- tion is a function of electrical current required to maintain a metal film at the constant temperature placed on the hull surface 3132. The output of the hot-film anemometer is a nonlinear power function of shear stress. The ideal response of the hot-film is that the output of the instrument is directly proportional to the shear stress measured. This ideal response can be accomplished by processing the nonlinear output from the anemometer through a linearizer which is commer- cially available (e.g. DISA type 55D15 linearizer). The functional relationship between the output of the linearizer and the shear stress is obtained through calibration, and slight nonlinear response is tolerable. Most commercial anemometers and linearizers can be adjusted to achieve almost perfect linearization. Hot-film shear probes designed and built by Ling 31,32 Were used in this study. A strip of platinum film about 0.1mm wide and 0.8 mm long is fused under high temperature to the polished end of a pyrex rod 0, 1l-inch in diameter and l-inch long. Figure Al shows the outputs of a hot-film anemometer and linearizer before and after the test versus the shear stresses measured by a Preston tube. A special wall-jet calibration facility, in which the wall shear stress on a flat wall two feet from a 1/2 inch jet can be varied from 0 to 0.5 psf, was built for this study. This facility using towing basin water (not to vary chemical properties and temperature), is essent- ial for the proper calibration of the hot-film shear probes. The dir- ectional response of the hot-film shear probe is calibrated by rotat- ing the hot-film element with respect to the flow direction. Typical results are shown in Figure A2, The directional response is propor- tional to cosine @ upto @ = 65 degrees. The difference between a 1976 Shear Stress and Pressure Ditstrtbutton on a Shtp Model misaligned probe is also shown. Since the angular response is a cosine function, the angle between the maximum shear stress anda reference line, 2 can be obtained by rotating the probe +6 with respect to this line, i.e., cei ens Os 2i 250 we! (Al) where (e)+, is the output when the probe is rotated at an angle equal to t g respectively. The values of @ used in the study were 45 deg. and 30 deg. depending upon the angle of the shear stress vector with respect to the reference line. The film on the probe element was aligned parallel to the waterlines for points on the ship side and parallel to the buttock lines for points on the ship bottom. The angle between the maximum stress and the reference line is obtained through Equation (Al). The magnitude of the shear stress along the reference line and the angle 2 , along with the direction cosines of the waterline and buttock line tangents and the surface normals cal- culated from the surface equation 2© were sufficient to decompose the shear stress coefficient vector Cy into the components (Cy, , Cry, C4,), relative to the body axes (x, y, 2). One of the difficulties in using the hot-film shear probe is mounting the probe perfectly flush to the surface. As shown in Figure A3, the response is very sensitive to the probe protuberance. In order to keep the accuracy within 5%, the probe protuberance should be kept within 0.002 inches. This was accomplished by using a flat face pressure transducer as a probe feeler. (2) Preston Tube and Directional Preston Probe The Preston method of measuring skin friction in the turb- ulent boundary layer makes use of a circular pitot tube resting on the wall. The Preston tube pressure, together with the static pres- sure at the same point, permits the computation of the skin friction at that point. The use of the Preston tube is based on the assumption that the tube lies within the law-of-the-wall region of the boundary layer. In this study we limit the diameter of the Preston tube to less than 15% of the boundary layer thickness in order to satisfy this assumption, The calibration of a Preston tube reported by Landweber and Siao 33 , by Patel?* , and many others is shown in Figure A4. The Preston tube used was also calibrated ina 1l-inch pipe flow. PSFe Huang and von Kercezek The present calibration is in good agreement with references (33 and 34), Patel ?* also found that a Preston tube can be used with acceptable accuracy (maximum error of 3 percent) if the pressure gradient parameter is limited to the range -0.005 <»/(pu2 ) dp/dx <0.01 where dp/dx is the pressure gradient along the flow direction and U, = Vtw/p is the shear velocity. The validity of using a Preston tube in boundary layers with large crossflows is not known. However, it is believed that the crossflow on the present ship model is rather small (the crossflow angles calculated are all less than 15 deg. ). The directional response of the Preston tube has been cal- ibrated and is shown in Figure A5, It is not practical to rotate the Preston tube flush on the three-dimensional ship hull and therefore the Preston tube is not used for measuring the angular position of the shear stress vector. The three-tube directional Preston probe, as shown in Figure A6, has very good directional response. One drawback of the Preston tube and the directional Preston probe is that several of them can not be used close together because they will not only cause an increase in the ship model resistance but will also have an interference effect on the downstream probes (see Figure A7). LOTS [2] [3] [4] [5] [6] [7] [s] [9] Shear Stress and Pressure Distrtbutton on a Ship Model REFERENCES LUNDE, J.K., ''On the Linearized Theory of Wave Resist- ance for Displacement Ships in Steady and Accelerated Motion", Transactions of Society of Naval Architects and Marine Engineers, Vol. 59, p. 25 (1951). TUCK, E.O., "A Systematic Asymptotic Expansion Procedure of Slender Ships", Journal of Ship Research, Vol. 8, No, 1, 1964, HESS, J.L. and SMITH, A.M.O., ''Calculation of Potential Flow about Arbitrary Bodies", Progress in Aeronautical Sciences, Vol. 8, Pergamon Press, New York, 1966. See also, DAWSON, C.W. and DEAN, J.S., ''The XYZ Potential Flow Program", Naval Ship Research and Development Center Report 3892, 1972. TUCK, E.O., and von KERCZEK, C., "Streamlines and Pressure Distribution on Arbitrary Ship Hulls at Zero Froude Number", Journal of Ship Research, Vol. 12, No. 3, Sep 1968. BRADSHAW, P., ''Calculation of Three-Dimensional Turbul- ent Boundary Layers", Journal of Fluid Mechanics, Vol. 46, Patts3,<1.97, U. NASH, J.F., ''The Calculation of Three-Dimensional Turbul- ent Boundary Layers in Incompressible flow'', Journal of Fluid Mechanics, Vol. 37, Part 4, p. 625, 1969. CUMPSTY, N.A. and HEAD, M.R., ''The Calculation of Three-Dimensional Turbulent Boundary Layers, Par 1: Flow Over the Rear of an Infinite Swept Wing"', The Aeronaut- ical Quaterly, Vol. 18, 1967. SMITH, P.D., ''Calculation Methods for Three-Dimensional Turbulent Boundary Layers", Aeronautical Research Council, Report and Memorandum No. 3523, 1966. COOKE, J.C. and HALL, M.G., "Boundary Layers in Three Dimensions'', Progress in Aeronautical Sciences, Vol. 2, Pergamon Press, New York, 1962. 1979 [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] Huang and von Kerezek LANDWEBER, L., "Characteristics of Ship Boundary Layers", 8th Symposium on Navai Hydrodynamics, Office of Naval Research, Pasadena, California, 1968. NEWMAN, J.N., ''Some Hydrodynamic Aspects of Ship Maneuverability", 6th Symposium on Naval Hydrodynamics, Washington, D.C., Office of Naval Research, 1966. GADD, G.E., ''The Approximate Calculation of Turbulent Boundary Layer Development on Ship Hulls", Transactions of the Royal Institution of Naval Architects, Vol. 113, No. 1, LS a WU, T.Y., 'Interacticn Between Ship Waves and Boundary Layers'', International Symposium on Theoretical Wave Resistance, University of Michigan, 1963. UBUROI, S.B.S., ''Viscous Resistance of Ships and Ship Models", Hydro-og Aerodynamisk Laboratorium Report No. Hy-13, Lyngby, Denmark, sep 1968. WEBSTER, W.C., and HUANG, T.T., "Study of the Bound- ary Layer on Ship Forms", Journal of Ship Research, Vol. 14, No. 3; sep 1970; COOKE, J.C., "A Calculation Method for Three-Dimensional Turbulent Boundary Layers", Aeronautical Research Council, Report and Memorandum, No. 3199, 1961. GUILLOTON, R., ''Potential Theory of Wave Resistance of Ships with Tables for its Calculation'', Transactions of Soc- iety of Naval Architects and Marine Engineers, Vol. 59,1951; also KORVIN-KROUKOVSKY, B.V. and WINNIFRED, R.J., "Calculation of the Wave Profile and Wave Making Resistance of Ships of Normal Commercial Form by Guilloton's Method and Comparison with Experimental Data'', SNAME Technical and Research Bulletin No. 1-16, dec 1954. von KERCZEK, C., ''Calculation of the Turbulent Boundary Layer on a Ship Hull at Zero Froude Number", Journal of Ship Research. Volk. 7, INos2, June L973: JOUBERT, P.N., and MATHESON, M., ''Wind Tunnel Tests of Two Lucy Ashton Reflex Geosims", Journal of Ship Research, Vol. 14, No. 4, dec 1970. 1980 —" — [20] [21] [22] [23] [24] ee [26] [27] [28] [29] a Shear Stress and Pressure Dtstrtbutton on a Shtp Model STEELE, B.N., ''Measurements of Components of Resist- ance on a Tanker Model", National Physical Laboratory Ship Division Report No. 106, 1967. STEELE, B.N., and PEARCE, G.B., "Experimental Deter- mination of the Distribution of Skin Friction on a Model ofa High Speed Linear'', Transactions of Royal Institution of Naval Architects vol. 110, p. 79, 1968. TZOU, K.T.S., 'An Experimental Study of Shear Stress Variation on Series-60 Ship Model", Iowa Institute of Hyd- raulic Research Report, No. 108, 1968. STANDEN, N.M., ''A Concept of Mass Entrainment Applied to Compressible Turbulent Boundary Layers in Adverse Pressure Gradients'', American Institute for Aeronautics and Astronautics No. 64-584, 1964. GRANVILLE, P.S., "Integral Methods for Turbulent Bound- ary Layers in Pressure Gradients", Naval Ship Research and Development Center Report 3308, apr 1970. MAGER, A., ''Generalization of Boundary Layer Momentum- Integral Equations ot Three-Dimensional Flows Including Those of a Rotating System", National Advisory Committee for Aeronautics Report 1067, 1952. von KERCZEK, C., and TUCK, E.O., ''The Representation of Ship Hulls by Conformal Mapping Function", Journal of Ship Research, Vol. 13, No. 4, dec 1969. YEUNG, R.W., ''Sinkage and Trim in First-Order Thin-Ship Theory", Journal of Ship Research, Vol. 16, No. 1, 1972, TODD, F.H., "Series 60 - Methodical Experiments with Models of Single-Screw Merchant Ships'', David Taylor Model Basin Research and Development Report 1712, 1963. TSAI, C.E., and LANDWEBER, L., "Total and Viscous Resistance of Four Series-60 Models", 13th International Towing Tank Conference, Berlin, Hamburg (Sep 1972). WEHAUSEN, J.V., and E.V. LAITONE, "Surface Waves", Encyclopedia of Physics, edited by S. Flugge, Vol. IX, 1981 [34] [a By [4] [4 [34] Huang and von Kerezek Fluid Dynamics III, Springer-Verlag, 1960. LING, S.C., "Heat Transfer Characteristics of Hot-Film Sensing Elements Used in Flow Measurement", Transac- tions of American Society of Mechanical Engineers, Journal of Basic Engineering, Vol. 82, p. 629, 1960. LING, S.C., etal., "Application of Heated-Film Velocity and Shear Probes to Hemodynamic Studies", Circulation Research, Vol. XXIII, No. 789, dec 1968. LANDWEBER, L. and SIAO, T.T., ''Comparison of Two Analyses of Boundary-Layer Data on a Flat Plate", Journal of Ship Research, Vol. 1, No. 4, 1958. PATEL, V.C., ''Calibration of the Preston Tube and Limitat- ion on Its Use in Pressure Gradients", Journal of Fluid Mechanics, Vol. 23, pp. 185-208, 1965. RAJARATNAM, N., and MURALIDHAR, D., ''Yaw Probe Used as Preston Tube'', Royal Aeronautical Society Journal, Vol. 72, No. 1060, dec 1968. SIGALLA, A., ''Experiments with Pitot Tubes Used for Skin Friction Measurement", British Iron and Steel Research Association Report, mar 1958. 1982 _— eT ——EeEeEEeEEOeeEeEeEeeeEeEeEe—e———eeeT a Shear Stress and Pressure Distribution on a Shtp Model TABLE 1 - PROBE LOCATIONS ON THE MODEL HULL pas ae a a re | -900}0.0120/-0. : 0.0287] -0.0246 . 700] 0.0462 ‘ 4 . 0696] -0.0409 ADDITIONAL POINTS -0.600] 0. 1093} -0.0400 -0. 100] 0. 1338] -0.0400 0. 300] 0. 1308] -0.040 0.950) 0.0090) -0.0752 L = 20 feet 0.950 0.0042] -0.0250 1983 Huang and von Kerezek T2 POW (09 ‘0 = d D) 09 SPtteg JoOT-07 : NNO NvIUd . T oanst 7 1984 Shear Stress and Pressure Dtstrtbutton on a Ship Model suoT}e00'T Oqoig pue ‘souTfuIeeryg ‘ueTq Apog ‘7 eansTy 10 02 “TM,0O 21 / y " / We / $U01}2907] aqoug aunssaug pue sPaYs 54 (9) p48Z4r uo, pue y49nNL) Apog sapuals “TM,O 02 (43 Lus pue ssaq) 39exq L2pow aLqnog uo seuLjwess3s 1985 Figure 3. Huang and von Kerczek Photograph of Hot-Film Shear Plug 1986 Probe and Mounting ————————aEooooo Shear Stress and Pressure Distrtbutton on a Shtp Model Figure 4, Wave Profiles at Various Froude Numbers 1987 Huang and von Kerezek SIoquInN epnoi gy snore, ie TINY ey} SuoTe peinsesy, so[tjorg sAcM SseTUuOTsUSIIq ‘“G oan3r7 O'T 8°0 9°0 7°0 G0 0 tao 9°0- 9°0- 8°0- OMS u (ad yore Ww TIny ey suoTe poanseou e19m sjutod ¢Z) 7ZZ'o0 O hg v Sjuoutiedxy eee 1988 Shear Stress and Pressure Dtstrtbutton on a Shtp Model SOTIJOId VACM P2}DTIpeTg pue poazanseosyf jo uostaedutoy ‘9 oansty peanseay 1989 -(h(-1) + h(1))/2_, Sinkage v [parkas ; Trim by Bow = (hitb) sah) v I 2B Huang and von Kerezek 4 (double hull) slender body theory -0.08 -0.04 ‘First-order Thin-ship Theory computed by Yeung 0 0.04 0.08 0.15 0.20 On25 0. 30 0.35 = u n —— Figure 7. Dimensionless Sinkage and Trim 1990 Shear Stress and Pressure Distribution on a Shtp Model 0.007 0.006 eS raven 0.005 OTD AKM YX yc. Sn ae W w & is 4 = Q , “a J i} aa 4, x At — ose cy 0.004 | PRESENT sTuDY, [pp = 20 ft. CARRIGE | MEASURED BY 2 ae TSAI AND LANDWEBER 2! I BLOCK GAGES P a 14 ft. FLOATING GIRDER O 10) II FLOATING GIRDER Vv 6 ft. I FLOATING GIRDER bioae TURBULENT STIMULATOR ) STUD Upp —Q— 20 ft. PRESENT STUDY we TSAI AND pq oO —A—- 14 ft. — LANDWEBER ! 0.002 Bes —-f>}— 10 ft. Ne vee-GP-+ 6 ft. 2 --—@-— 20 ft. ropp28 c 1957 I.T.T.C. f CORRELATION LINE 0.001 10) 0.15 0.20 0.25 0.30 0.35 ee ia: Figure 8. Total Resistance and Residual Resistance of the Various Length Models 1991 Huang and von Kerezek dv Oe TINH 94} uo UOTINQTIISIG ®InsseIg pee[NoTeD pue peans eayy 2/1/* g*0 9°0 1°0 z°0 0 2*0" 1° O- *6 orInsty GIATA ALINIANI NI ‘TINH WIanod 4a SINAN TY Ad X4 Y+t)F e/rAd [(u+€)&-“d1-d 1992 Shear Stress and Pressure Distribution on a Ship Model (penutjuoD) TINH uy uo UOTINIAJSTIG ®AINssoig poye[NoTeD pue poranseospwl “6 9anst zy av z/1/X ZieNe ((4y+#)4@-d)-d ¢ rus 3 sso PINTA @ITUTFUL UL TINH aTqnog J squewpiedxg "es @ITUTJUT UT TINH aTqnog —————— a. ae Z£°0 fra ey . 4 a0) eA al (age) 42 Coe i iene 1993 Huang and von Kerezek (penutju0D) TINH 4} uo UOTINAIAISTG ®9INssxetg poe}eTNoTeD pue peansespw, “6 9INIsT ZT 2/1/* da GINTH GALINIGNI NI TINH WZiENnod Z£°0 INGNTYadXa 1994 Shear Stress and Pressure Distrtbutton on a Ship Model sjueuoduroy ssoaiajg- Iesyg pernsesyy Z/1/X 0 c°0- auTTWe=eIAS §6(q) OT ernst gy oTqe¥ L995 Huang and von Kerezek (penutju0D) sjuouodui0oy sseajg-a1eeyg pernsesy ‘QT eansty c/1/X 014°Y 027° U 1996 Shear Stress and Pressure Distribution on a Shtp Model oO Ea co oO Le] \o (0) 5 ue) So ee! iS) q fe) : o — wn ~ q 7) ¢ : 4 23] = g - ° g 0 W) o Q n 9 ome ) 3 = H = a 1 ~ u es 3 at aT op) ae) 0 u : : S 2 ! o © ro) Oo ee | o) u 3 tot oa — fy oOo 1 og on 1997 Huang and von Kerezek TI°H 4} wo SUOTINGIIJSIG SS91}JG TeeYSG poe,eTNoOTeD pue poransespyy “TT 29aNndT ZZ 7/1/X €00°0 VY eupqTuees3s (aisyueoyos) a2PTd IeTA WTT4-10H eqn uojsead xj squeuttedxq €00°0 1998 Model . tp Sh ton on a hear Stress and Pressure Dtstrtbutt oi re) suotjnqriajstq ssoaqg 1 2) (419yue0yIS) aqeld ITs (penutzu0D) TIn}H ey} uo BOYS peye[NoTeD pue peanseow “TI ernst y cL Vv x da C@ autTTweea4s auTTmeetqsg (3) thd-J a uojse uotqoTpelg Lone’ wT Td-J0H qn uoqse1g sjUusuTiedxy ic 10= 7°O- 9505 8°0- OL 100° c00° €00° 700° TON" c00° £00° 700° 1999 Huang and von Kerezek (PepntsuoD) TINH ey} uo SuOTINATIISTC SSotjg Te9Yg pejye[Nds[eD pue pornseowl “TT ean3ta7 (VT°0 = H/Z) a @UTT 197 eM (2) T/X Z/ a da 8°0 9°0 v} (0) 7°0 0 Ga0= yp (0}= 97.0> Sz0= (OV 0 q[nsel queseirg Sa OliT AS (0) V cp en R 193sqomM ——--——f£y'T 82°0 | T00°0O 3[Nsel Qusesolg CoA) pee: oO @ UOT IIT petg OT > ana aq WTty-70y4 eqn} uojsead <3 ff) squsutioedxy z00°0 €00°0 700°0 e 2000 Shear Stress and Pressure Distrtbutton on a Shtp Model Probe 00 06 CTA ch.6 Cold Resistance = 5.43 ohms Operating Resistance = 5.70 ohms Over Heat Ratio = 1.05 ®@ Ambient temperature = 71.5°F Se Oo ae ; ogy a © eo vA Output, Voltage : y| Linearized Anemometer Before the ie O 0 Experiment After the Experimen .e Figure Al, Typical Calibration Curve for a Hot-Film Shear Probe 2001 Huang and von Kerczek UOTJEJUSTIO AeTNSsuy fo uotTJOUN ese oqoig Ie9ys WIT7-10H e jo ssuodsoy TeotdkA], “zy oanst 7 ae18ep ut 9 adOud GUNOTIVCIN VY HOd AANND ASNOdSHY AFYNSVaN I Gd0Ud GUNOTTVY ATLOAAYAd Y HYOd HANNO ASNOdSHY AHYNSVEN Id Adoudd 2002 Shear Stress and Pressure Dtstributton on a Shtp Model 0.2082 0.1570 0.1119 -0.004" -0.002" 0) 0.002" 0.004" 0.006" Figure A3, Effect of Probe Protuberance on Hot-Film Response 2003 eee eee eee eee LL. ee i Ae Huang and von Kerezek 33 ' Landweber & Siao * - 1.388 for 5.¢x £8. at Q" 6.0576, d,/d = 0.722 present study, d= Pipe I.D. 5 Calibration Curve for Preston Tube in a Pipe 2004 Figure A4., Shear Stress and Pressure Distrtbutton on a Shtp Model GE YVHOIIVaNW GNV WYNLVUViVvad 40 NOIS34 Viva eqn], uoJserg aeTsutg jo ssodsey jeuoyoe1Iq $35N9I0 NI 6 "GV oanst gy ds 2005 Huang and von Kerezek (seeaseq Gp = © o[suy ®SON) eqn], uoJserg [euoT}IeAIq Jo ssodsey ‘gy canst yg $3349] NI 6 2006 Shear Stress and Pressure Distrtbutton on a Shtp model LdS pue ddq Jo }90}}q SOUSTezTOJUT “Ly CANST GT LIdS 4O ddd AHL 40 WVHULSdN AdONd ANWNC AO AONVLISIG WVAALSdA ddd AWNNG ON WL 400) & SO OILVa 2007 Huang and von Kerezek DISCUSSION L. Landweber Untverstty of Iowa Instttute of Hydraulte Research Iowa Ctty, U.S.A. Iam very pleased to see such a basic study of viscous cha- racteristics of flow about a ship at nonzero Froude number. There have been too few such studies. As the authors realise, the method that they used to calculate the three-dimensional boundary layer on the ship is suitable only for fine formes, free of the cross flows, bilge vortices, secondary flows and separation. I hope that, in the continuation of their work, they will use methods which are suitable for fuller forms, possibly with separation, extending even into the thick boundary layer zone near the stern. I would like to call attention to a book on three-dimensional boundary layers, recently published by one of my colleagues at the University of Iowa, V.C. Patel. It is available only directly from him ; he is his own publisher. Concerning the potential flow, we at the University of lowa have tried the technique that is used in this paper, that of Tuck- Kerczek, and some of our work on this method was reported at the Seventh Symposium in Rome. The method is a very attractive one. It appears to enable one to obtain the parametric equations of a ship form in a very compact way. Our exprerience has been, however, that the method cannot give a sufficiently accurate representation of the hull at the bow and stern, so that it could not be useful for obtaining source distributions or for calculating wave resistance, especially since, for wave resistance, the result is very sensitive to small deviations from the original form. It appears, however, that the me- thod is suitable for determining the streamlines at low Froude number and can serve as the basis for determining a streamline co-ordinate system for boundary-layer calculation, as the authors have shown. One results in the paper is not clear to me, and may be incorrect. On page 10 the authors give a separable form for the po- tential function. Their assumption is that the free-surface condition can be applied below the undisturbed water surface. I think that this result 2008 oe lee lee ees eee eee ee ee aa aaa HO Shear Stress and Pressure Ditstrtbutton on a Shtp Model is valid for two dimensions, not for three. The authors have shown some of our data for a family of series 60 geosims ranging in length from 6 to 14 feet. The results for the 14 feet model should be corrected for blockage, as we plan to show at the forthcoming ITTC in Berlin. The authors are to be congratulated on their very interes- ting and valuable paper. DISCUSSION John V. Wehausen Untverstty of Caltfornta Berkeley, Caltfornta, U.S.A. Bruce Adee has recently made a computation which seems very similar to yours, except that instead of using the rigid free- surface boundary condition, he actually used the linearized free- surface boundary condition. I wonder if you have had a chance to com- pare your results with his, or if even he sent you a copy of his thesis ? REPLY TO DISCUSSION Thomas T. Huang Naval Shtp Research and Development Center Bethesda, Maryland, U.S.A. We have not yet received a copy of Dr. Adee's thesis for comparison, I thank Professor Landweber for his kind suggestion. This is a complicated problem involving the free surface boundary condi- tion and turbulent boundary layer. This topic would open many inte- resting investigations in the future. We shall take Dr. Landweber's suggestion very seriously. 2009 Huang and von Kerezek Dr. Landweber's comment on page 10 is right, but the measurements show that the wave along the mid-ship is almosta two-dimensional wave. The blockage correction is a very interes- ting problem which has to be considered. The 14-ft and 10-ft models appeared to need some blockage correction in the lowa tank. It is interesting that the 6-ft model and the 20-ft model agree very well. DISCUSSION Jean-Francois Roy Basstn d'Essats des Carénes Parts, France ( Translated from French) All the results presented in this paper are very interesting. And especially, I think it is exceptional to know both distributions of pressure and shear stress ona hull. In sucha case, I would have been tempted to check, by integration, the values of the total resis- tance and its two components that have been measured by other means, May I ask the authors whether they did make such a verification. Thank you. REPLY TO DISCUSSION Thomas T. Huang Naval Shtp Research and Development Center Bethesda, Maryland, U.S.A. Because of time, the actual integration of the pressure dis- tribution and shear stress distribution has not been done. We want to improve our computation methods for the potential flow as well as boundary layer first and the actual integration can be done if we have time, and probably will be carried out in the near future. It is not a major point of our paper. 2010 AUTHOR INDEX 1543,,1 552.1563, 158i. BE ETOWS; Ge, 563. BENEN,.b/, 419, BESCH Pk. 5+ 543, 3:954396,397. 398, 399. BINDEL, Sio4 1182. BLACKLOCK, D.S., 1690. BRARD, R.o.257, 746,1187, 1280, BECK, R.F.,; 1282, 1284. BRESLIN, J.P., 397,398,664, 1000, 1283. CHANG, M..,. 1331, 1368, CHENG,. FM.,.411. CHRYSSOSTOMIDIS, C., 1589, WGZ1, l624, 1625, 1626,.1627. CUMMINS, W.E., 1365. DAGAN, G., 1525, 1697, 1737. DAND, 1.W.; 1540. DARROZES, J.S., 1739. DELMONTE, R.C.,; 761. BERN, J.C., 1003, 1106. DIMANT, Y., 1305. DOCTORS, 1.3., 35, 96, 255. DUPORT, J. 3; 29. DYNE; G., 1950, EDSTRAND, H., EGGERS, K.W., FALTINSEN, O., EE EDMAN, J.P.,;. 1527, PINK, lL. 1265, 1304. PMNEG 2. 1278, 1692. GERASSIMOV, A. V., 1079. GCOOMAN, T.R.; 1629. MOOK, G., 339;. 579, FUSING, Li t., 1324, 1963,.2009, 2010. HUSE, E.,, 666. HUPHER, M., INE, Ts, 687. JOHNSSON, C.A., 581, 656, 659, SS. 745, 1948. 1366, 1394, 1842. 660, 661, 663, 665, 668, 1179, 1950. 2011 1763, 1843, 1844, MiA LASORATORY Se eee LIBRARY WOODS HOLE, MASS. W. H. 0. I. JONES, EBwA.. 293. KAJITANL, Hy, 68. KAPLAN, ©., 256; 258, 681, 1629, 1693. KIM; GC .H., 793, 2583, 1587, 1842; 1843, KOSTILAINEN, ¥.. ,..158Z,, LAC KENBY., H. . 661 , LANDWEBER, L.,. 756, 2008. LANG, TGs, 549; 573 .510n5005 580. LAG. Keown LOL7 Se, boo o. LEE, C.M., 463, 543, 545, 792, 996. LEOPOLD, RR... 340,341, 395,396, 399, 417, 460, 461, 1622. LEWISON, G., 685. LIBBER, P.; 1304; Lig, ¥.N.3 343. LOVER, EP. , 577; e26; MAZARREDO, L., 1184. MERCIER, J.A.;, 1793. MERRIE) way Sa 30% MICHELSEN, F.C., 662. MORGAN, W.B..,. 29; 415,655; P1238. MORI, K.., 687, 746, 749, 757, MUR THY, E. KS. 99,255, 256; 251,250, 259, 544, NAUDASCHER, E., 1285. NEWMAN, J.N., 541. NOWACKI, H., 415, 1845. OCH, MK... 955, 1624. OGIE Vii, LI. .5 14383; 1523; P5245 F525. OOSTERVELD, M., 658. PERSHITZ, R.Y., 1079. PETERSON,: F.B., 1131, 1184. PIEN,;. P.C., 463. PLATE, E.d.5, 999, 1371, 1395. POREH, M., 1305, 1326, PROKHOROYV, S.D.; 26h; RAKHMANIN, “N.N., 1079,1107. TUCK, E.O., 684, 1127, 1524, ROGDESTVENSKY, V.V., 1129. 1538, 1543, 1736. ROPER, J.K., 419. VAN OORTMERSSEN, G., 957, ROUSETSKY, A.A., 401. 998, 999, 1000. ROY, dF... 200. VISCONTI RE. 3.) a0) SAINT-DENIS, M., 459, 679, 1367, VOITKOUNSKY, J.J., 1325. 1619. VOLEOYV,. 6D, ers SALVESEN,N., 574. VON KERCZEK, C., 1963. SARGENT, T.P., 629. WEHAUSEN, J.V., 2009. SAVITSKY, D., 419, 460, 461. WEINBLUM, G.P., 1948. SCHMITKE, R.T., 293,339,341. WERMTER, R., 1620. SHARMA, S.D., 1845,1953. WHITNEY, A. 11917 Yi277 eee SILBERMAN, E., 1181. 1130. SONTVEDT, T.,. 58! WIEGEL, R.1L.; 761, 792. STROM-TEJSEN, J., 95, 413. WU; Der l ass. TELFER, ESV... 660, 1325, 1625, WU, To Y!, Ili, vies 1736, 1945. YACHNIS, M.N., 1432. TIMMAN, R., 1523. YANG, I.M., 671, 680, 683, 685, TOURNAN, R.,° 1394. 686. TRESHOCHEVSKY, 14.) 201,269, YIM, €.S.,, 139%, basen 412, 416, 417, 1281. yx U.S. GOVERNMENT PRINTING OFFICE : 1975 O—565-527 2012 a 2+ ay pam TC ww = —- hae MAK e - ty 6 ae = ay Se ee = SS SSS = reese: izi= = SSS SS SS SSS SSS SS Se SS eS ——S { { i Sa Se = SSS SSS SSS SE ee === == —— H : i ih tH te i i | i 1 ‘ ( # ! 4 ti f i hy | f Me A “Ath : Hi it Hit it ut itt La Ht it | i. if Hats ae Pepsi nit i Mi fH iif i : i t! H i = a =. =- == se == =t ree See ee § 1} i ! | ! i ot i i Hh ith i ul } =ts> Sasa 5S = Sta SS= = eS Sess Sb ese. = es eee Se t=set- <2 s= <= — 3352S e > ———— =r 23S SS * =i 323 Ss SS eae Siti isis iriisiisti Sits His BS 2 SS SS SS SS SE ; :