[ith Syeopesham MAMAL H¥DRODYNAMICS Se MOTIONS DRAG REDUCTION PROFESSOR |. K LUNDE ~ STANLEY W. DOROFF i EDITORS SUULELEARNET 1 apr PRR EERLSRNS NEY RL RES es RG PRR ER PLALEV ELEC LE S. a GREPRPLPLPLA LONER VEN VORP RAAPAPR ELF PLR TR ERT EEN ERA BP Te Tar AP RT PT RES POF PN PTA ALS FATA AED EREAE VOUS FES ASES RAE HED SA TRAE RTT WERE REY eT Citice of Navel Research He Department of the Novy | | He HRS 1s i Ms soy Cae n i sc ebtehou qoeo U wh ON jOHM/ 198 Fifth Symposium on = HH NAVAL HYDRODYNAMICS SHIP MOTIONS AND DRAG REDUCTION Sponsored by the OFFICE OF NAVAL RESEARCH and the SKIPSMODELLTANKEN MARINE BIOLCGICAL LABORATORY LIBRARY WOODS HOLE, MASS. W. H. O. I. September 10-12, 1964 Bergen, Norway ACR-112 OFFICE OF NAVAL RESEARCH— DEPARTMENT OF THE NAVY Washington, D.C. PREVIOUS REPORTS IN THE NAVAL HYDRODYNAMICS SERIES _ . “First Symposium on Naval Hydrodynamics,” National Academy of Sciences - National Research Council, Publication 515, 1957, Washington, D.C.; $5.00. 2. “Second Symposium on Naval Hydrodynamics,” Office of Naval Research, Department of the Navy, ACR-38, 1958; Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., Catalog No. D210.15:ACR- 38; $4.00. 3. “Third Symposium on Naval Hydrodynamics,” Office of Naval Research, Department of the Navy, ACR-65, 1960; Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., Catalog No. D210,.15:ACR- 65; $3.50. “Fourth Symposium on Naval Hydrodynamics,” Office of Naval Research, Department of the Navy, ACR-92, 1962; Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., Catalog No. D210,15:ACR- 92; $6.75. ol . “The Collected Papers of Sir Thomas Havelock on Hydrodynamics,” Office of Naval. Research, Department of the Navy, ONR/ACR-103, 1963; Superin- tendent of Documents, U.S. Government Printing Office, Washington, D.C., Catalog No. D210.15:ACR-103; $4.50. 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 Depart- ment or of the naval service at large. For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C., 20402 - Price $7.25 li PREFACE This Symposium is the fifth in a series each of which has been concerned with various aspects of Naval Hydrodynamics. The first (held in September 1956) presented critical surveys of Hydrodynamics that are of significance in naval science. Subsequent meetings were to be devoted to one or more topics selected on the basis of importance and need for research stimulation, or of particular current interest. In keeping with this objective, the second symposium (August 1958) had for its subject the areas of hydrodynamic noise and cavity flow; the third (Sep- tember 1960) was concerned with the area of high performance ships; and the fourth (August 1962) emphasized the topics of propulsion and hydroelasticity. Still continuing with the original plan, the present symposium selected for its dual theme the areas of ship motions and drag reduction, thus emphasizing, among other things, the interest in the current problems and latest accom- plishments associated with: theoretical and experimental determination of the coefficients of the equations governing the motions of ships in a seaway; the characteristics and designof motion stabilizers; the reduction of frictional resistance by the introduction of additives; and the design of bulbous bows to reduce wave drag. The international flavor of these meetings continues to be an outstanding feature, and in this case, has been enhanced by virtue of the setting, the par- ticipation, and most particularly by the joint sponsorship by the Skipsmodell- tanken of Trondheim, Norway and the U.S. Office of Naval Research. The address of welcome by Dr. Weyl! and the speech opening this sym- posium by H.R.H. Crown Prince Harald more than adequately describe the background and objectives of this meeting, thus leaving little more to be said other than to express our gratitude to all those who contributed so much to the success of this symposium. However, taking the liberty of speaking both for the Office of Naval Research as well as the international scientific community of hydrodynamicists, I should like once again to express our deepest appre- ciation to Professor J. K. Lunde, to his associates Dr. H. Aa. Walderhaug and Mr. O. Skjetne, and to the Norwegian Ship Model Experiment Tank, Trond- heim for their outstanding efficiency and care in managing the many varied aspects of this symposium. RALPH D, COOPER, Head Fluid Dynamics Branch ili CONTENTS Preface Address of Welcome F.J. Weyl, Deputy and Chief Scientist, Office of Naval Research, Washington, D.C. Opening Address H.R.H. Crown Prince Harald of Norway Introductory Remarks G. P. Weinblum, Institut flr Schiffbau der Universitat, Hamburg, Germany RECENT PROGRESS TOWARD THE UNDERSTANDING AND PREDICTION OF SHIP MOTION T. F. Ogilvie, David Taylor Model Basin, Washington, D.C. PROBLEM AREAS IN SHIP MOTION RESEARCH W.J. Pierson, Jr., New York University, New York, N.Y. SOME REMARKS ON THE STATISTICAL ESTIMATION OF RESPONSE FUNCTIONS OF A SHIP Y. Yamanouchi, Ship Research Institute, Tokyo, Japan RESPONSE TO COMMENTS BY WILLARD J. PIERSON, JR. T. F. Ogilvie, David Taylor Model Basin, Washington, D.C. CURRENT PROGRESS IN THE SLENDER BODY THEORY FOR SHIP MOTIONS J. N. Newman and E. O. Tuck, David Taylor Model Basin, Washington, D.C. DISCUSSION H. Maruo, National University of Yokohama, Yokohama, Japan COMMENTS ON SLENDER BODY THEORY E. V. Laitone, University of California, Berkeley, Calif. REPLY TO DISCUSSION : J.N. Newman and E. O. Tuck, David Taylor Model Basin, Washington, D.C. SLENDER BODY THEORY FOR AN OSCILLATING SHIP AT FORWARD SPEED W.P.A.Joosen, Netherlands Ship Model Basin, Wageningen, Netherlands APPLYING RESULTS OF SEAKEEPING RESEARCH E. V. Lewis, Webb Institute of Naval Architecture, Glen Cove, Long Island, New York DISCUSSION G. Aertssen, University of Gent, Gent, Belgium DISCUSSION G. J. Goodrich, National Physical Laboratory, Teddington, England Page iii x1i xiv xvi 80 Oi 127 129 162 165 166 167 187 210 211 DISCUSSION H. Lackenby, British Ship Research Association, London, England DISCUSSION W.A.Swaan, Netherlands Ship Model Basin, Wageningen, Netherlands DISCUSSION L. Vassilopoulos, Massachusetts Institute of Technology, Cambridge, Massachusetts REPLY TO THE DISCUSSION E. V. Lewis, Webb Institute of Naval Architecture, Glen Cove, Long Island, New York THE DISTRIBUTION OF THE HYDRODYNAMIC FORCES ON A HEAVING AND PITCHING SHIPMODEL IN STILL WATER J. Gerritsma and W. Beukelman, Technological University Delft, Netherlands DISCUSSION E. V. Lewis, Webb Institute of Naval Architecture, Glen Cove, Long Island, New York DISCUSSION J.N. Newman, David Taylor Model Basin, Washington, D.C. DISCUSSION OF THE PAPERS BY GERRITSMA AND BEUKELMAN AND BY VASSILOPOULOS AND MANDEL T.R. Dyer, Technological University, Delft, Netherlands REPLY TO THE DISCUSSION BY E. V. LEWIS J. Gerritsma and W. Beukelman, Technological University, Delft, Netherlands REPLY TO THE DISCUSSION BY J. N. NEWMAN J. Gerritsma and W. Beukelman, Technological University, Delft, Netherlands A NEW APPRAISAL OF STRIP THEORY L. Vassilopoulos and P. Mandel, Massachusetts Institute of Technology, Cambridge, Massachusetts DISCUSSION Winnifred R. Jacobs, Stevens Institute of Technology, Hoboken, New Jersey DISCUSSION M. A. Abkowitz, Massachusetts Institute of Technology, Cambridge, Massachusetts DISCUSSION O. Grim, University of Hamburg, Hamburg, Germany DISCUSSION W.R. Porter, Massachusetts Institute of Technology, Cambridge, Massachusetts DISCUSSION A—THE INFLUENCE OF THE ADDED MASS FORMULATION UPON THE COMPUTER MOTION PREDICTIONS P. A. Gale, Bureau of Ships, Washington, D.C. DISCUSSION B—THE PITCH AND HEAVE OF TEN SHIPS OF DESTROY ER-LIKE FORM IN REGULAR HEAD WAVES— COMPUTER PREDICTIONS COMPARED WITH MODEL TEST RESULTS P. A. Gale, Bureau of Ships, Washington, D.C. vi 212 al3 214 aia 219 247 248 249 250 251 253 356 360 364 364 366 368 REPLY TO THE DISCUSSION L. Vassilopoulos and P. Mandel, Massachusetts Institute of Technology, Cambridge, Massachusetts SOME TOPICS IN THE THEORY OF COUPLED SHIP MOTIONS J. Kotik and J. Lurye, TRG Incorporated, Melville, New York KNOWN AND UNKNOWN PROPERTIES OF THE TWO-DIMENSIONAL WAVE SPECTRUM AND ATTEMPTS TO FORECAST THE TWO- DIMENSIONAL WAVE SPECTRUM FOR THE NORTH ATLANTIC OCEAN W.J. Pierson, Jr., New York University, New York, New York DISCUSSION A. Silverleaf, National Physical Laboratory, Teddington, England REPLY TO THE DISCUSSION W.J. Pierson, Jr., New York University, New York, N.Y. FORCE PULSE TESTING OF SHIP MODELS W.E.Smith and W. E. Cummins, David Taylor Model Basin, Washington, D.C. DISCUSSION OF FOUR PAPERS L.J. Tick, New York University, University Heights, Bronx, New York DETERMINISTIC EVALUATION OF MOTIONS OF MARINE CRAFT IN IRREGULAR SEAS J. P. Breslin, D. Savitsky, and S. Tsakonas, Stevens Institute of Technology, Hoboken, New Jersey DIS CUSSION J.F. Dalzell, Southwest Research Institute, San Antonio, Texas DISCUSSION M. Fancey, Institut Za Brodska Hidrodinamika, Zagreb, Yugoslavia DISCUSSION G. J. Goodrich, National Physical Laboratory, Teddington, England DISCUSSION S. M. Y. Lum, Bureau of Ships, Washington, D.C. REPLY TO THE DISCUSSION J. P. Breslin, D. Savitsky, and S. Tsakonas, Stevens Institute of Technology, Hoboken, New Jersey TESTING SHIP MODELS IN TRANSIENT WAVES M. C. Davis and E. E. Zarnick, David Taylor Model Basin, Washington, D.C. DISCUSSION E. V. Laitone, University of California, Berkeley, Calif. REPLY TO THE DISCUSSION M. C. Davis and E. E. Zarnick, David Taylor Model Basin, Washington, D.C. PREDICTION OF OCCURRENCE AND SEVERITY OF SHIP SLAMMING AT SEA M. K. Ochi, David Taylor Model Basin, Washington, D.C. DISCUSSION G. Aertssen, University of Gent, Gent, Belgium vii 405 407 425 434 435 439 457 461 498 299 500 500 504 507 541 542 545 589 DISCUSSION E. V. Lewis, Webb Institute of Naval Architecture, Glen Cove, Long Island, New York DIS CUSSION W.A.Swaan, Netherlands Ship Model Basin, Wageningen, Netherlands DIS CUSSION L. Vassilopoulos, Massachusetts Institute of Technology, Cambridge, Massachusetts REPLY TO THE DISCUSSION M. K. Ochi, David Taylor Model Basin, Washington, D.C. THE INFLUENCE OF FREEBOARD ON WETNESS G. J. Goodrich, National Physical Laboratory, Teddington, England DISCUSSION E. V. Lewis, Webb Institute of Naval Architecture, Glen Cove, Long Island, New York DISCUSSION R. F. Lofft, Admiralty Experimental Works, Gasport, England HYDROFOIL MOTIONS IN A RANDOM SEAWAY B. V. Davis and G. L. Oates, De Havilland Aircraft of Canada, Limited, Downsview, Ontario, Canada DISCUSSION H. D. Ranzenhofer, Grumman Aircraft Engineering Corporation, Bethpage, Long Island, New York DISCUSSION A. Silverleaf, National Physical Laboratory, Teddington, England THE BEHAVIOUR OF A GROUND EFFECT MACHINE OVER SMOOTH WATER AND OVER WAVES W.A.Swaan and R. Wahab, Netherlands Ship Model Basin, Wageningen, Netherlands DISCUSSION W.A. Crago, Saunders-Roe Division of Westland Aircraft Limited, Wight, England DISCUSSION R. F. Lofft, Admiralty Experimental Works, Gasport, England REPLY TO THE DISCUSSION W.A.Swaan and R. Wahab, Netherlands Ship Model Basin, Wageningen, Netherlands BEHAVIOR OF UNUSUAL SHIP FORMS E.M. Uram and E. Numata, Stevens Institute of Technology, Hoboken, New Jersey DIS CUSSION E.V. Lewis, Webb Institute of Naval Architecture, Glen Cove, Long Island, New York A SURVEY OF SHIP MOTION STABILIZATION A.J. Giddings, Bureau of Ships, Washington, D.C. and R. Wermter, David Taylor Model Basin, Washington, D.C. DISCUSSION P. DuCane, Vosper Limited, Portsmouth, England Vili 591 591 593 595 Sb A) 606 607 611 688 689 691 712 714 715 717 745 747 801 DISCUSSION J. F. Dalzell, Southwest Research Institute, San Antonio, Texas DISCUSSION S. Motora, University of Tokyo, Tokyo, Japan DISCUSSION E. Numata, Stevens Institute of Technology, Hoboken, N.J. DISCUSSION K. C. Ripley, J. J. McMullen Associates, Inc., Washington, DAGr DIS CUSSION A. Silverleaf, National Physical Laboratory, Teddington, England REPLY TO THE DISCUSSION A.J. Giddings, Bureau of Ships, Washington, D.C. and R. Wermter, David Taylor Model Basin, Washington, D.C. A VORTEX THEORY FOR THE MANEUVERING SHIP R. Brard, Bassin d’Essais des Carenes de la Marine, Paris, France DIS CUSSION N.H. Norrbin, Swedish State Shipbuilding Experimental Tank, Goteborg, Sweden DISCUSSION A. J. Vosper, Admiralty Experimental Works, Gasport, England REPLY TO THE DISCUSSION BY NORRBIN R. Brard, Bassin d’Essais des Carenes de la Marine, Paris, France REPLY TO THE DISCUSSION BY VOSPER R. Brard, Bassin d’Essais des Carenes de la Marine, Paris, France THE REDUCTION OF SKIN FRICTION DRAG J. L. Lumley, The Pennsylvania State University, University Park, Pennsylvania DISCUSSION S. K. F. Karlsson, Brown University, Providence, R.I. DISCUSSION—A BASIC THEORY THAT COULD EXPLAIN DRAG REDUCTION IN A FLOW CARRYING ADDITIVES A. C. Eringen, Purdue University, Lafayette, Indiana DISCUSSION A. Kistler, Yale University, New Haven, Connecticut REPLY TO THE DISCUSSION J. L. Lumley, The Pennsylvania State University, University Park, Pennsylvania THE EFFECT OF ADDITIVES ON FLUID FRICTION J. W. Hoyt and A. G. Fabula, U.S. Naval Ordnance Test Station, Pasadena, California DISCUSSION H. Schwanecke, Hamburg Model Basin, Hamburg, Germany DIS CUSSION M. P. Tulin, Hydronautics, Incorporated, Laurel, Maryland 802 803 809 810 811 812 815 908 908 910 Sit al 915 9B9 944 945 946 947 959 960 REPLY TO THE DISCUSSION J. W. Hoyt and A. G. Fabula, U.S. Naval Ordnance Test Station, Pasadena, California AN EXPERIMENTAL INVESTIGATION OF THE EFFECT OF ADDITIVES INJECTED INTO THE BOUNDARY LAYER OF AN UNDERWATER BODY W.M. Vogel and A. M. Patterson, Pacific Naval Laboratory, Victoria, British Columbia, Canada DISCUSSION T. G. Lang, Naval Ordnance Test Station, Pasadena, Calif. REPLY TO THE DISCUSSION W.M. Vogel and A. M. Patterson, Pacific Naval Laboratory, Victoria, British Columbia, Canada AN EXPERIMENTAL STUDY OF DRAG REDUCTION BY SUCTION THROUGH CIRCUMFERENTIAL SLOTS ON A BUOYANTLY- PROPELLED, AXI-SYMMETRIC BODY B. W. McCormick, Jr., The Pennsylvania State University, University Park, Pennsylvania DISCUSSION T. G. Lang, Naval Ordnance Test Station, Pasadena, Calif. PROBLEMS RELATING TO THE SHIP FORM OF MINIMUM WAVE RESISTANCE H. Maruo, National University of Yokohama, Yokohama, Japan EXPERIMENT DATA FOR TWO SHIPS OF “MINIMUM” RESISTANCE W.C. Lin, J. R. Paulling, and J. V. Wehausen, University of California, Berkeley, California DISCUSSION P. C. Pien, David Taylor Model Basin, Washington, D.C. DISCUSSION L. W. Ward, Webb Institute of Naval Architecture, Glen Cove, Long Island, New York DISCUSSION G. P. Weinblum, Institut fur Schiffbau der Universitat, Hamburg, Germany REPLY TO THE DISCUSSION W.C. Lin, J. R. Paulling, and J. V. Wehausen, University of California, Berkeley, California SOME RECENT DEVELOPMENTS IN THEORY OF BULBOUS SHIPS B. Yim, Hydronautics, Incorporated, Laurel, Maryland THE SHIP, BULB A. Nutku, Technical University, Istanbul, Turkey THE APPLICATION OF WAVEMAKING RESISTANCE THEORY TO THE DESIGN OF SHIP HULLS WITH LOW TOTAL RESISTANCE P. C. Pien, David Taylor Model Basin, Washington, D.C. DIS CUSSION G. P. Weinblum, Institut fur Schiffbau, University of Hamburg, Hamburg, Germany DISCUSSION K. Eggers, Institut fur Schiffbau, University of Hamburg, Hamburg, Germany DISCUSSION J.N. Newman, David Taylor Model Basin, Washington, D.C. 961 975 a 999 1001 1015 1019 1047 1060 1061 1061 1062 1065 1098 1109 1137 1139 1140 DISCUSSION L. W. Ward, Webb Institute of Naval Architecture, Glen Cove, Long Island, New York DIS CUSSION A. Silverleaf, National Physical Laboratory, Teddington, England DIS CUSSION S. W. W. Shor, Bureau of Ships, Washington, D.C. REPLY TO THE DISCUSSION P. C. Pien, David Taylor Model Basin, Washington, D.C. AUTHOR INDEX xi 1141 1144 1145 1148 1154 ADDRESS OF WELCOME F. J. Weyl Deputy and Chief Scientist Office of Naval Research Washington, D.C. Your Royal Highness, Professor Lunde, fellow wayfarers on the road to Bergen: It is my pleasure in the name of the Norwegian Ship Model Experiment Tank and the United States Office of Naval Research to bid you welcome at this our goal. We are most appreciative for this opportunity of descending, maybe a bit disorganized but full of friendliness, upon your country; and we look for- ward to discovering more of its social fabric, its forests and fjords. Perhaps we are, inadvertently, redressing in these latter days a long term balance by the confusion we may cause in your hostelries and shops in return for that caused by visits which were made from these parts to very nearly the whole of the here- represented world during the centuries of Norway's birth. Most deeply grateful we are to the staff of Skipsmodelltanken, its distinguished director, Professor Lunde, and his associates, for having taken on the task of being host for the Symposium, and thus to look after our well-being, both temporal and spiritual, during our days in Bergen. We shall express our thanks in work reported and new endeavors initiated, in legends told and traditions started. The ocean is a strange and wondrous thing, not only to the historian who traces the role which it has played in the fates of men and people, but no less to the scientist. Let us first give a look at its geometry. Its characteristic hori- zontal dimension exceeds its depth by three orders of magnitude. Bounded by the atmosphere above, it presents a mightily agitated surface. Massive currents and huge eddies characterize the motion of the basins, driven by gravity and the rotation of the earth. Unexplored heat transfer phenomena across its bottom vitally influence the energy balance. In short, it is all boundary and yet presents itself so unbounded. To a technology peculiarly matched to life in the atmosphere where the sig- nal speed is that of light and even the fastest form of locomotion constitutes but one thousandth of one percent of this speed, the ocean again presents a radically different concert of parameters. Opaque to electromagnetic radiation, the char- acteristic mode of signal transmission is acoustic. Complicated reflection and refraction phenomena are caused by the layer structure, multipath phenomena obscure reception, and scattering is a fact of life rather than being encountered at the very fringe of usefulness of the information carrier. Moreover, measured in terms of signal speed, locomotion is now two orders of magnitude faster than in our wonted atmospheric medium. Lastly, let me point out the tremendous range of time scales encountered in the dynamic behavior of the oceans. The waves and wave patterns onthe surface xii cover a range from minutes to days. The large currents and the eddies which they generate show an erratic behavior whose large scale changes are meas- ured in weeks and months. Seasonal variations characterize the major features of stratification; and finally, in the deepest layer of the oceans, memory appears to be measured in centuries. All of this presents us with engineering challenges and tasks of tremendous scope, of which those so ingeniously solved with ever increasing competence by the designers and developers of ships are but a small yet highly characteristic part. Reaching out towards an ultimate goal, where we can freely traffic and go about such business as we may choose throughout the volume of the oceans, an exciting spectrum of new problems and opportunities opens up to scientist and engineer alike. The integrity of the hull requires that new ground be broken in the physics of materials, in the ingenious invention of geometries, and in advanc- ing processes of fabrication. Propulsion and maneuver control place demands on the marine engineer which force him to look to the very boundaries of science and in many instances beyond before he will be confidently able to meet them. And, finally, there are the pioneering adventures in experimentation and instru- mentation which alone can secure for us the scientific knowledge and the opera- tional experiences that are prerequisite for ultimate mastery of the medium. Viewed in this light, the preoccupations, past and future, which will constitute the substance of our discussions here during the next few days, appear as an advanced salient of an onward sweeping front of competence and knowledge which we shall surely see broadened with great vigor during our lifetimes. In short, to quote with slight adaptation the modern American lyric poet, E. E. Cummings: ''There's a hell of a nice universe out there, let's go!" His Royal Highness, the Crown Prince of Norway, has graciously accepted our invitation to come here and to open this the 5th Symposium on Naval Hydro- dynamics. We are most particularly appreciative of the fact that, even with the duties and responsibilities of Head of State on his shoulders now during the King's absence, he has consented to be with us this morning, giving added sig- nificance to the occasion. I have, therefore, the honor at this time to call on His Royal Highness to open the proceedings. OPENING ADDRESS H.R.H. Crown Prince Harald of Norway Mr. Chairman, Ladies and Gentlemen, May I firstly thank Dr. Weyl for his kind words of welcome. Perhaps it is typical of the universality and internationalism of our times — and indeed a promising feature — that an American scientist shouldaddress us here in Bergen, Norway, in his capacity as Host. My father, His Majesty The King, who addressed a similar symposium — the 7th International Conference on Ship Hydrodynamics — ten years ago in Oslo, has asked me to bring you all his greetings and best wishes for a successful and enjoyable stay — both beneficial to science and conducive to pleasure. As a user — one of those who benefit, or suffer,as a result of your findings — Iam particularly happy to behere today. I have no doubts that the great majority of ideas tested are found not suitable —perhaps even dangerous — and thereby you spare us anxiety and economic losses. On the other hand, we live in a com- petitive world—in politics, in economics and in sports. When a new idea is thought of and found fruitful, we, the users, would like to keep it to ourselves. You have the scientific attitude; you like to share your findings, for the better- ment of mankind, to develop your findings, and indeed to further the science you represent. The world has come a long way from pieces of wood drifting in rivers and on the sea, thereby giving man the idea to try to float himself on the first raft or boat. Sturdiness and stability, particularly in the serious and often fatal question of top-weight, were the first problems to be solved. Then followed a long epoch of the shipbuilder as an artist, and now science has more and more taken over. The modern shipbuilder is no more a fifth generation artist in his field with a saw and axe, but a serious, studious mathematician with drawingboard and slide rule. Perhaps we have come too far; perhaps we shall have to take one or several steps back, searching for something important overlooked in the rapid develop- ment. That in itself may be one of the findings, here or elsewhere. I wish you all every success in your endeavours to improve ships and boats for the benefit of all. May your discussions be fruitful and not too long. XiV When you leave may you feel that you have benefited technically, and also that you have made good contacts and established friendships. With every good wish to all of you for a useful, successful and pleasant congress and stay, I declare the Fifth Symposium on Naval Hydrodynamics to be opened. KV INTRODUCTORY REMARKS G. P. Weinblum Institut fur Schiffbau der Universitat Hamburg, Germany Ten years ago the International Towing Tank Conference held a meeting in Oslo. At that time, I had the privilege of lecturing on the subject, ship motions, before His Royal Highness Crown Prince Olaf now His Majesty the King. Itisa highlight of my professional career that today in your Royal Highness' presence a team of gifted younger scientists will report on the impressive progress reached in our field during the recent years. They will prove the well estab- lished fact that we, in engineering sciences, usually overestimate what can be accomplished within one year but fortunately underrate what can be done within ten years. Now that your Royal Highness has graciously opened the session we shall start immediately with our work. Our kind hosts have carefully included short curricula of the lecturers in the abstracts. Thus the need for introducing the speakers to the auditorium is eliminated. There is another reason why it is perhaps not so important to follow this well established habit of introduction: although our young speakers have already reached a high scientific reputation, their future is still more important to our profession than their past. Calling now Dr. Ogilvie, the head of the Free Surface Phenomena Branch of the David Taylor Model Basin in Washington, D.C., to deliver his lecture. It is my pleasant duty to emphasize that during his stay as liaison scientist of the Of- fice of Naval Research in London hehas earned universal esteem and friendship. Xvi Thursday, September 10, 1964 Morning Session SHIP MOTIONS Chairman: G. P. Weinblum Institut fur Shiffbau der Universitat Hamburg, Germany Page Recent Progress Toward the Understanding and Prediction of Ship Motions 3 T. Francis Ogilvie, David Taylor Model Basin, Washington, D.C. Current Progress in the Slender Body Theory for Ship Motions 129 J. N. Newman and E. O. Tuck, David Taylor Model Basin, Washington, D.C. Slender Body Theory for an Oscillating Ship at Forward Speed 167 W. P. A. Joosen, Netherlands Ship Model Basin, Wageningen, Netherlands 221-249 O - 66 - 2 1 a . ei “F et a “f s > p Nl ae t t o AF ap P ae a j Perel “ eee Ah \" ale - , is ' ' i i w ie | ‘MTRODUCTORY REMARKS) im ¢ - . ne Siri tie Ses B hy: . vi 4 7 a 2" t ee + ™ af q : ; aii 3 t (J AA 2) a 1a a i : f ant lnwpliges 'F (crate iets na it J ree | J loa! i; Jd 7 YVR Yo UGG : Ls . RY PR hs eid lt ‘Gn Ute foazads Ciwes i 7°)! 2 CA OD aioe , v ‘ as j ie, a0 : 74 y howd “sbire 7) Gu Bas Q0%4 ‘tne Wek Mitsuya het iss? -Onoend bedee 7c potwab Me ea oo 3) lentil roe WG hae ; “itt? nlifoet Hk 720i wetowy! vist See Bi isboM ote® ebasltedisA .neeodh A AS ahnigligorniay Wreeteti | 7 RECENT PROGRESS TOWARD THE UNDERSTANDING AND PREDICTION OF SHIP MOTIONS T. Francis Ogilvie David Taylor Model Basin Washington, D.C. ABSTRACT Since the Symposium on the Behavior of Ships in a Seaway (Wageningen, 1957), many papers have been published on the theory of ship motions. The present paper is a survey, collation, and evaluation of those con- tributions which have led toward a rational theory for predicting ship motions. During this period, evidence has accumulated which demonstrates the validity of the superposition principle for ship motions in a seaway. This concept was stated as hypothesis eleven years ago by St. Denis and Pierson (and also sixty years ago by R. E. Froude); its validity may now be considered as proven, beyond the fondest hopes of earlier investigators. With this principle established, attention once again returns to the pre- diction of motions in small-amplitude regular waves. The best prac- tical approach to making ship motions predictions is probably still through use of strip theory. However, the two-dimensions assumptions of strip theory are so pervasive that the validity of the resulting anal- ysis is always questionable except in the most routine problems. In the past decade, the concept of the thin ship has been extensively applied to ship motions problems. Many elements in the complete pic- ture have been developed on this basis, and in addition, thin-ship theory has been highly systematized. This latter effort, involving the estab- lishment of a rigorous development of the theory on a set of carefully stated assumptions, has pointed up some basic shortcomings in applying thin ship ideas to motions problems. Very recently, much attention has been devoted to developing a slender ship theory for predicting motions. The motivation and basic ideas are discussed; more thorough consideration will be found in other papers at this symposium. Ogilvie INTRODUCTION Background The hydrodynamic theory of ships was born in the last half-decade of the nineteenth century, and it was a spectacular beginning, for within three years there appeared three papers by Krylov and the famous paper by Michell. Unfor- tunately, the response to these papers was not what they deserved, and many years passed before naval architects again considered their problems as sci- entific problems. We look back and see a few hardy souls struggling to progress against the apathy of their own profession. Not until almost 1950 was there a general renaissance of interest in the possibility of finding scientific solutions to the naval architect’s hydrodynamics problems. Then, in 1953-4, there was another spectacle comparable to the one over fifty years earlier. In these two years there appeared the papers by St. Denis and Pierson (1953) and Peters and Stoker (1954). The former suggested the procedure for relating to reality the highly idealized hydrodynamic theory of ship motions (as it then existed). The latter provided a logical foundation for this idealized theory and, in particular, it set forth clearly the hypotheses involved. Neither of these two papers presented the final words on the subject; on the contrary, each raised more questions than it answered. But these authors were more fortunate than Krylov and Michell, for their papers were followed by an explosion of activity. By 1957, it was possible for the Netherlands Ship Model Basin to sponsor a symposium on seakeeping at which there were presented nearly fifty papers, some on the most basic scientific aspects of seakeeping problems. Now, seven years later, we have again come together to (1) assess our progress, (2) discuss our latest findings, and (3) orient ourselves toward further discoveries on "'the way of a ship in the midst of the sea."" My own purpose is concerned primarily with the first of these three, viz., to look back over the last few years and attempt to evaluate our progress. I shall be discussing al- ready published work almost exclusively. Of course, I cannot ignore work that is in progress, but a few words on such will suffice, for other speakers here are ready and willing to present their latest findings. Neither can I ignore the future, and in fact my whole presentation will be somewhat biased towards what I consider the most auspicious recent trends in research in our field. Scope In naval architecture, as in all branches of engineering, the designer is faced with immediate demands. During the past decade it has become evident that it would be not only desirable but perhaps even feasible to calculate the motions of a ship, given only a geometrical description of the ship and adequate information about its sea environment. Of course, shipbuilders and shipowners want this information now, and so it has been incumbent on the naval architec- ture profession to produce techniques as good as the state of the art allows. 4 Understanding and Prediction of Ship Motions Some important results have been obtained in this effort. However, my presen- tation is not very closely related to these efforts, for I shall discuss progress toward what I call a ''scientific solution" of the problems of ship motions. Perhaps I should be more specific in defining a "scientific solution.” By this I mean that one starts with a mathematical model of the fluid. It may be — and in fact must be —a highly idealized model, but the implications of the ideali- zation are probably well-understood in a general sense. To this mathematical model, one must add a set of boundary conditions and also possibly initial con- ditions, all of which should be stated as precisely and accurately as possible. Even though the fluid is represented by an idealized model, the resulting prob- lem is always intractable. Therefore one must put forth a set of additional as- sumptions which reduces the problem to manageable proportions. When this analytical problem has been solved, one makes calculations and compares them with experimental data. There will be discrepancies, and so one goes all the way back to the beginning and tries to relax one of the restrictive assumptions, find a more general solution, etc., etc. Two parts of this process qualify it as a ''scientific solution" by my defini- tion, viz., all of the assumptions are stated at the beginning, and improvements are made by modifying the assumptions rather than by trying empirically to patch up faulty results. In practice, the engineer may not have the time to do all of this, or it may be simply impossible. Still, he must make predictions. So, if he is a good engi- neer, he improves his first poor predictions in any way he sees fit. This prog- ress requires great ingenuity and skill, and its accomplishment is an essential element in the working of our technocracy. However, I shall not discuss such attempts, important though they may be. Other speakers here are much better qualified for this, and I leave it to them. Summary of Contents Generally speaking, we wish ultimately to supply certain statistical infor- mation to the ship designer. We may justify such an approach either by reason- ing that he cannot really use more precise information or by accepting the fact that we cannot hope to provide anything better. In either case, we begin with a statistical description of the sea, assuming that the water motion can be de- scribed as the sum of many simple sinusoidal waves, each of which is described separately by the classical Airy formulas of linearized water wave theory. It was the great contribution of St. Denis and Pierson (1953) to suggest (a) that the statistical nature of the sea could be expressed by allowing the phases of these components to take on random values and (b) that the response of a ship to the sea was the sum of its responses to the various components. They only sug- gested these hypotheses, and it may be claimed that both had been made earlier, but these authors were the first to state them in precise, quantitative terms. Their suggestion (a) relates more to the oceanographer's problem, and so I shall not consider it here. However, (b) will be discussed in some detail, for it has received much attention in recent years and it is at the heart of our problem. Ogilvie Today we may consider that it has been confirmed, for most practical purposes; some of the evidence will be presented. St. Denis and Pierson used an extremely primitive set of equations of mo- tion, and we must now conclude that those equations are quite unacceptable. They were the best available ten years ago, but we can now do much better. The use of second order ordinary differential equations to describe the rigid body motions of a ship is quite artificial. Under appropriate conditions and with proper interpretation, they provide a valid representation, but such equations certainly cannot have constant coefficients in the usual sense. The form of the equations of motion can now be stated with considerable confidence, and this will be done. Actually, the discussion of rigid body equations of motion is somewhat of a digression. Basically, having accepted the linear superposition principle, we need only to find a means of determining the transfer function (or frequency re- sponse function) of the ship. This may be done experimentally, in which case the whole subject of equations of motion need not be introduced, or it may be done by the use of hydrodynamic theory, in which case the information provided by the equations of motion comes out automatically. Nevertheless there are important reasons for studying the equations of motion per se. On the one hand, the direct experimental procedure treats only input (the exciting waves) and output (the motions). It provides no insight into the particular ship characteristics which cause different ships to respond dif- ferently in a given seaway. On the other hand, the hydrodynamic theory of ship motions is not yet highly enough developed to tell us comprehensively which ship characteristics are most important in seakeeping and why they are so important. Perhaps the largest portion of the literature on ship motions during the past decade has been concerned with the calculation of individual elements in the equa- tions of motion. Some of the methods used have been quite sound scientifically, and some of the results have shown quantitative agreement with experiments. For example, the damping (due to wave radiation) in heave or pitch can be ana- lyzed straightforwardly in certain situations, and recently it has been demon- strated how to calculate the added mass or added moment of inertia through knowledge of the damping. Although some of these analyses have led to remarkable results, there is also a basic difficulty of principle in using them, and this problem was already clearly pointed out by Peters and Stoker (1954). Since the free surface prob- lems involved must all be linearized before any progress can be made, these authors set out to perform the linearization in a clearly stated, rational way and to investigate the logical consequences of the simplification. They obtained the linear mathematical model from a systematic perturbation analysis, ship beam being the small parameter. The results were disappointing, for they obviously do not correspond to reality: In the lowest order motion solution, there appear undamped resonances in heave and pitch. The physical interpretation of this result is that the wave damping is of higher order (in powers of the small pa- rameter) than the exciting force, restoring force, and inertial reaction force. Understanding and Prediction of Ship Motions Attempts were subsequently made to correct this situation by reformulating the perturbation problem. In particular, the Peters-Stoker assumption that the ship beam can be used as the sole characteristic small parameter is open to question; the amplitude of the incident waves is a small quantity which is quite independent of beam. A multiple-parameter perturbation scheme takes care of this problem theoretically, but it does not lead to practicable results. The the- ory for motions of a thin ship still stands in this unsatisfactory condition. There seem to be at least two logical ways out of this predicament. We must have at least one small parameter associated with the hull geometry, in order that the ship travelling at finite speed may cause only a very small dis- turbance. (This is necessary for any linearization to be valid.) We could try to select this small parameter so that the damping due to vertical motions is in- creased in size by an order of magnitude. Such a result is realized, for exam- ple, in a flat-ship theory. But there are at least two objections to this; first, the practical solution of the flat-ship approximation is very difficult, involving a two-dimensional integral equation, and, second, the original difficulty would pop up again in consideration of horizontal modes of motion, namely, in surge, yaw, and sway. The second logical escape is to use a small parameter which leads to no resonance at all in the lowest order non-trivial solution. This is accomplished by assuming that the ship is both shallow and narrow, i.e., slender. Then it can be shown that the inertia becomes an order of magnitude smaller than in the thin-ship theory, whereas the damping order of magnitude is unchanged. But slender body theory for ships also has its problems. In particular, a theory for ship motions should be part of a general theory which includes steady transla- tion as a special case. We now know that slender body theory in fact gives poor results for the wave resistance of a ship in steady motion. Nevertheless, slender body theory appears promising for predictions of ship motions. I shall only outline the ideas involved, for, if I presented the de- tailed modern theory as it stands in the published literature, I would be out-of- date before this morning session is over. The following speakers will present some of the evidence which suggests the promise of the approach. It is obvious that much remains to be done in the theory of ship motions. There is still a problem of developing a logical approach which gives answers agreeing with experiments. Furthermore, most of my discussion relates only to motions in the longitudinal plane of the ship; we have barely begun to attack the corresponding problems involving yaw, sway, and roll. SHIP MOTIONS IN CONFUSED SEAS It has long been recognized that the sea is a complicated thing, but it was only with the war-time and post-war development of random noise theory that the means became available for providing a realistic description of it. The kind of statistical description to be employed in describing the sea de- pends on the specific aspect of the ship motions problem which happens to be of Ogilvie immediate interest. The engineer who must evaluate the likelihood of fatigue failures is obviously concerned with different data and different theoretical for- mulations from the engineer who must design equipment for helping aircraft to land on a carrier. One might say that the ship captain will not be satisfied with statistical descriptions at all; he sets an absolute standard: the safety of the ship. So we must state carefully what problem concerns us before we choose a statistical model. Long term phenomena, such as the fatigue problem, must still be treated on a strictly phenomenological basis. At present, we cannot hope to specify ship motions or any other ship-related variables for the whole variety of conditions which a ship encounters in its lifetime. Even if we could suddenly obtain perfect oceanographic prediction data, such an enterprise would be out of sight in the future — and probably not even desirable. Also beyond the scope of this paper is the problem at the other extreme, that is, the prediction of the specific short-time motions of a ship, given its immediate, detailed history. We shall here be concerned with a problem somewhere between these, namely, to predict the probability of occurrence of various phenomena when a ship is travelling in certain well-defined environments. Since we are limited by the available tools of probability theory, we restrict ourselves to the case ofa stationary random sea. Such an environment is probably highly non-typical, but its study does give valuable information and it is in any case the best we can do at present. Following St. Denis and Pierson (1953) and others, we first describe the seaway by the energy spectrum of the wave height. This function specifies the fraction of the total energy which is associated with any given band of wave fre- quencies. The assumption of an energy spectrum description implies nothing about the possibility of linearly superposing wave trains on each other. It sim- ply means that one measures the wave height at a point for an (in principle) in- finitely long time and then calculates the spectrum by a standard technique which is found in many textbooks. Next, one generalizes the spectral description at the point so as to obtain a description valid over an area of the sea. It is here that the assumption is in- troduced that the sea can be represented as the linear sum of elementary waves, each travelling in the manner described by the classical Airy formulas of line- arized water wave theory. If one starts with a wave height record at only a sin- gle point, many possibilities are available for making the generalization. Of all these possibilities, two have special meaning for us, because they correspond to situations of physical interest: 1. We may assume that all of the wave components travel in the same di- rection. Such a thing does not happen in nature, of course, but it is the situa- tion which many towing tank operators have attempted to produce. 2. We may assume that the energy in any bandwidth is distributed among wave components travelling in a continuous distribution of directions. Insofar Understanding and Prediction of Ship Motions as the sea can sometimes be described as a stationary random process, such an assumption can lead to a description of a real sea if the angular distribution is properly chosen. Without question, such a description can represent the short- crestedness of the sea. The particular distribution of energy as a function of angle will vary greatly with sea conditions, and it is not clear at present if there is a standard distribution which will lead to generally useful results in connec- tion with ship motions predictions. Our knowledge of the hydrodynamics of ship motions is such that we are well-advised to limit our attention to the first of the two choices above, although it is unrealistic. Stated bluntly, the fact is that we have far to go on the simpler problem, and we cannot hope to understand ship motions in multi-directional seas until we first understand what happens in artificially-produced uni- directional seas. This statement need not apply if we are content to obtain fre- quency response functions strictly by experiment. But the principle purpose of this paper is to consider the prospects for entirely analytical predictions of ship motions. With such a goal in mind, we must accept that we cannot solve all of our problems at once. Therefore I shall restrict myself generally to long- crested seas, recognizing that a broader outlook is desirable and will ultimately be necessary. In calculating the energy spectrum from a given wave height record, one effectively discards the information which relates to relative phases of the var- ious component waves. The energy spectrum gives us information only about about the amplitude of the components. From the point of view of probability theory, all wave height records which yield the same energy spectrum are equivalent.* Then, if one wants a general representation of the surface eleva- tion corresponding to a particular energy spectrum, one must allow complete ambiguity in the relative phases of the frequency components. For the long- crested sea, St. Denis and Pierson (1953) proposed the representation: CG) = | cos [Kx- at - e(w)] V[f(a)]2 dao , (1) 0 (x,t) = surface elevation at position x, time t, [%(w)]* = energy spectrum of (x,t), a function of frequency, «, K = w?/g, g = acceleration of gravity, and €(#) = a random variable, with equal probability of realizing any value between 0 and 27.7 *That is, they are all members of an ensemble which is characterized by a sin- gle energy spectrum. We assume not only that the processes are stationary but also that an ergodic hypothesis is valid. TA more precise definition is that Pla, <«(w) 0 as t > ©, so that all transforms existed in the conventional sense. The character of a ship is such that this assumption may well not be warranted, and for generality a modification of the results is necessary. In Ap- pendix B, it is shown that if the ship system is stable, that is, if all a,(t) re- main bounded for all time, then Eq. (15) is still valid even if the transform of 37 Ogilvie a,(t) does not exist, provided only that we replace 5{a,} by d{a,}/io. There will also be a singularity at « = 0, and so the modified (15) is not valid for w-0. It is also shown in Appendix B that o,(~) will generally be zero unless some of the c;,'s are zero. This is, of course, quite reasonable, since the c;,'s are restoring force coefficients (even though they are not hydrostatic restoring force coefficients). We have now shown that the two types of equations of motion are equivalent for both sinusoidal and transient motions, provided only that the system is sta- ble. In the third situation of interest to us, viz., a ship moving in a stationary random sea, the usual arguments of generalized harmonic analysis can be used to show that the two descriptions are again equivalent. Actually, in order to carry out the conventional spectral analysis, we need only to be certain that the assumption of linearity is valid, and evidence was presented in Chapter II to show that it is indeed valid in at least certain modes of motion. The value here of having equations of motion is in the capability which they provide for predict- ing the effects of various parameters on the spectral properties of the ship. They also enable us to develop test procedures, which have been called "pulse techniques,"' which are an order of magnitude more efficient than regular wave tests in determining the frequency domain characteristics of a ship hull. See Cummins and Smith (1964). From (14) or (15), it is natural to define the following quantities: @ ar, 1 Hj, () = added mass coefficient = Ce a = K5,,(t) sin ot dt ; 0 (16) @ bj, (@) = damping coefficient = bi, + J K;,(t) cos ot dt. 0 Of course, following Cummins, we previously defined 1;, as an added mass. This situation merely shows how arbitrary the definition of this quantity is. The added mass defined in (16) depends on frequency and speed, and so in one sense it is not so natural as the previous definition. However, for the special case of sinusoidal oscillations, it is as reasonable a definition as the other. In the time-domain equations, it was not possible to identify any one quan- tity as a "damping coefficient"; some or all of the damping was included in the forces represented by the convolution integral of (13'). In the case of sinusoidal motions, we can readily pick out certain quantities which we identify as "damp- ing coefficients,'' but we must be careful in interpreting the label thus applied. If there is sinusoidal motion in just one mode, say the k-th mode, then the aver- age rate at which the ship performs work on the water depends only on the com- ponent of force which is in phase with the velocity in that mode; in other words, the dissipation of energy depends only* on b;,, which is properly a damping coefficient. The force components in phase with acceleration or displacement *This is true only in the reference frame in which the water is streaming past the ship. Otherwise ship resistance is involved with the work done. 38 Understanding and Prediction of Ship Motions may be called "reactive" forces; they are associated with the local disturbance of the water, but they are not related to the average rate of transfer of energy. If there are two or more modes of motion occurring simultaneously, then, as we have seen, there will be coupling between modes, and we may expect that there will be, say, j-component forces in phase with a 5 ( t) which result from the accelerations and displacements in the k-th mode. This will be demon- strated explicitly in the next chapter. This means that the damping in the case of coupled motions will involve the coefficients Hit and c;,. It is still conven- ient to refer to the coefficients 5,, b;,, and c;,, for k + j, respectively, as added mass, damping, and restoring force coefficients, but it must be remem- bered that all are involved in the damping. In the two expressions appearing in (16), the frequency dependence enters only through the integral terms, and it is important to note that these integrals are, respectively, the sine and cosine transforms of the same function, K;,(t). This fact will lead to the establishment in the next chapter of a formula relating added mass and damping coefficients. In concluding this chapter, I would comment much as I did at the end of the previous chapter. We can now continue to use the old second order differential equations, as we did years ago. But now we know that we can, when desirable, turn to the true equations of motion, for it is these which give broader meaning physically to the equations which are valid only for sinusoidal motions. We also know that we must allow the "constants" in the differential equations to be func- tions of frequency. And finally we have obtained from this study of the equations of motion some powerful new tools: pulse methods of testing, which are an order of magnitude more efficient than the older methods, and an extremely valuable relation between added mass and damping (to be proved presently). PROPERTIES OF TERMS IN THE EQUATIONS OF MOTION This chapter will be devoted to some special relationships for the various terms and coefficients in the equations of motion. Specifically, the following facts will be proven: 1. The added mass matrix can be determined from the matrix of damping coefficients, and vice versa. 2. The exciting forces at zero speed can be deduced from knowledge of the far-field potential for the problem of the ship oscillating in calm water, i.e., the diffraction problem can be avoided. 3. The diagonal elements of the damping coefficient matrix can be calcu- lated from the same far-field potentials used in (2) above. If the ship has zero speed, all elements of this matrix can in principle be found in this way. In other words, if we can find velocity potentials for the six problems cor- responding to the sinusoidal oscillations of a ship in calm water, we can evaluate 39 Ogilvie these potentials far away from the ship (effectively at infinity) and from the re- sulting simplified functions determine some of the damping coefficients. From the same asymptotic forms of the potentials we can also find the forces on a ship due to sinusoidal incident waves from any direction, without having to solve the problem of determining the diffracted waves around the ship. In both prob- lems we avoid the necessity of integrating the pressure over the ship hull. It is only necessary to integrate over a simplified mathematical surface far away from the ship. Finally, in any case for which we know the damping coefficients, we can find the corresponding added mass coefficients. These relationships all depend on our use of a linear model to describe the ship and fluid motions, but they do not depend on a specific mathematical repre- sentation of the ship. In general, we shall be talking about the frequency-domain equations of motion; the concept of ''damping coefficient" has no meaning in the time-domain equations which were developed in the last chapter. In order to use any of the relations proved in this chapter, we must be able to find the velocity potentials for the oscillating ship problems, or at least the far-field asymptotic forms of these potentials. Finding these functions requires the assumption of a particular mathematical model for the ship, for the velocity potentials cannot be found until we have formulated appropriate boundary condi- tions for the whole problem, and this obviously requires some statements about the flow near the ship. Two general methods of finding the velocity potentials will be discussed in the following chapters. It may perhaps be argued that all of these relations are academic, for there are several important gaps. To fill these gaps requires the integration of the pressure over the hull, and thus the complete potential, including the compli- cated local flow, must be considered. It then follows that perhaps one may as well solve the whole problem by evaluating the local potential and integrating pressure over the hull to find the forces. This may turn out to be true, but the simplicity of using the far-field potentials is so attractive that I have considered it desirable to present these partial results, hoping that someone may be able to fill the gaps in an equally simple manner. Relation Between Added Mass and Damping Coefficients It was pointed out previously that the frequency-dependent parts of the added mass and damping coefficients, as defined in (16), are proportional to the sine and cosine transforms of a single function, K5,(t)- From the theory of Fourier transforms, it is well-known that, if K;,(t) is well-enough behaved, either of these transforms uniquely determines the inverse transform. Therefore, if either transform is known, the function K;,(t) can be found, and from this the other transform can be determined. In the language of the ship motions problem, this means that if we know 55 (®) for any single frequency and the damping coefficients for all frequencies, we can obtain the added mass for any frequency. This result is sufficiently im- portant that it deserves to be stated explicitly in formulas. 40 Understanding and Prediction of Ship Motions If (oo) I K5,,(7) dr 0 is absolutely convergent, then the Riemann-Lebesgue lemma* says that foo} lim i K;,,(7) sin w7 dr = lim i K5,.(7) COs @r cr = O. > oo 0 @ @> © Then, from (16), it is evident that Ban Ce) ; so that we know the constant term b,, if we know bj, () for all w (as assumed). The inverse of the cosine transform is given by: foe) 2 Kane) = =| [Pi - b3x| cos at da. 0 Then the added mass is: (oo) {ee} 2 Hi 4(®) =) ee 4) sin at | [ei Co") = dix | cos w t dw dt. 0 0 The last formula would be rather awkward for purposes of computations, and indeed a much simpler formula is possible. For the moment, let the upper limit of the outer integral be a large positive number, M, and interchange the order of integrations. Then we can use the Riemann-Lebesgue lemma again in letting M > o, finding: 2 fo} M ne) Rak as im { das’ bj, (#") “la] |) sin wt cos wt dt 0 0 {oo} foo) MEST : < : cos (w' + w)M 1 ; 7 7@ jin | f bre )- bj. | | @' +o — ot iw da foo) ‘¢ } cos (@' -w)M 1 ; E [ Exe ) iy. | ee = Beal du} l ale oO » (pea = Gee KH = ee Sy a i} lop = eS) — a, & i = & Ss 4 | & | Qu. 3 | Se) sa) on — & me Te & 4 if o a eS SI lfon & nN *See, for example, Whittaker and Watson (1927), p. 172. 41 Ogilvie @ ? indicates that a Cauchy principal value is to be used. Similarly, we find: 0 @ Pe 4 2 * ; aa! = dene BE ital NBG, th ae Bae )- Hx mee (17b) Equation (17a) is useful if we know 1;, = 4j,(~), but it is easily shown that knowledge of ,.;,(«) for any single frequency is sufficient. The same comment applies to (17b) with respect to bj,,(«). The values at w = » are most likely to be amenable to calculation, although in some cases it may be easier to calculate the values at w = 0. In either of these two limiting cases, the free surface prob- lem degenerates into a much simpler problem, and in fact numerical solutions for quite complicated geometries are possible by methods such as that of Hess and Smith (1962). Equations (17a) and (17b) have been proven by Kotik and Mangulis (1962) for the special case of heave disturbances at zero forward speed, and these authors surmised that a similar result would be valid for all modes, with or without for- ward speed. Their argument was based on the observation in other fields of sci- ence* that such formulas are obtainable whenever the system response obeys a linear law and there is a clear causality relation between input and output. In the ship motions problem, linearity has been demonstrated for certain types of motion, as described in Chapter II, and Cummins' analysis was based on a line- arity hypothesis. Therefore it is not surprising that the formulas can be derived from Cummins' results and the experiments indicate that we should expect the formulas to be valid. Also, there can hardly be any question about the validity of the causality assumption. t An alternative derivation of these relations is presented in Appendix C, wherein we avoid the double transform operations which were used to derive (17a) and (17b). It is seen in the Appendix that the formulas are really just corollaries of Cauchy's Integral Formula. Two points should be made with respect to use of these formulas: 1. Contrary to statements by Kotik and Mangulis, it does not follow that an approximate formula tor, say, a damping coefficient can be used in (17a) to ob- tain an approximate formula for the corresponding added mass coefficient. The reason for this is that an approximate formula for damping coefficient may give good results in the range of interest for damping coefficients (especially near resonance) but the asymptotically wrong at extreme values of the frequency. *Such relations are known as the ''Kramers-Kronig relations" in statistical me- chanics. They may be interpreted as Hilbert transforms. Davis and Zarnick (1964) have questioned this, because in their experiments they observed a response before t= 0 when an impulse occurred at t=0. How- ever, I consider their paradox a result of their choice of time coordinates and their definition of an impulse. Certainly there can be no ship disturbance until the ship encounters a free surface disturbance, and soa causality hypothesis is valid. 42 Understanding and Prediction of Ship Motions Since (17a) depends on the value of the damping coefficient over the whole spec- trum, one may expect that the added mass will be incorrectly predicted unless b;,(#) is approximately correct over the entire frequency spectrum. The "'Hi- Fi'' approximation espoused by Kotik and Mangulis has proper asymptotic limits, and so this effect does not vitiate their calculations. However, an example to the contrary may be found in slender body theory, where added mass and damp- ing coefficient predictions both break down at high frequency. Equations (17a) and (17b) cannot be used with predictions based on slender body theory. 2. Throughout this survey, I assume that the amplitude of all disturbances is bounded for all time, and this assumption is necessary for taking Fourier transforms of (13'). If there is an instability such as static divergence (which can occur in yaw with an inadequately controlled ship) or if there is negative damping at any frequency (which can in principle occur at high speeds), then these formulas are invalid for all modes unless the modes of difficulty are con- strained to have zero amplitude. Exciting Forces One of the most difficult parts of using any equations of motion for analyti- cal predictions of ship motions is calculating the forcing functions, that is, find- ing the force and moment exerted by the incident waves on a restrained ship. Quite often in the past, the practice has been advocated of using the pressure in the undisturbed wave and integrating it over the actual surface of the ship. In other words, it is assumed that the presence of the ship does not affect the pres- sure in the water. This assumption, often referred to as the ''Froude-Krylov"' assumption, is obviously not generally correct, although under certain circum- stances it may not be grossly in error. Properly, one must formulate a bound- ary value problem in which there are included both the incident waves and the diffracted waves. The two systems of waves must be such that the total fluid velocity on the ship surface satisfies the correct boundary condition there. In addition, there will be waves generated by the motions of the ship. From a hydrodynamic point of view, this presents an easier problem than the incident- diffracted wave problem, because the normal velocity component on the hull is a fairly simple, known function, depending only on the shape of the hull and on the six rigid body modes of motion. Therefore Haskind (1957) made a considerable contribution to our problem when he showed that the forces due to incident waves could be calculated from solutions of the forced oscillation problem. Specifi- cally, he showed that if we can solve the hydrodynamic problems involved in the oscillation of a ship in an otherwise calm sea, then we can also compute the force and moment on a ship restrained in incident waves. Haskind proved his result only for the case of a ship at zero speed. His solution is rederived in a paper by Newman (1962). Recently, Newman has shown that an analogous result can be obtained for the case of a ship with for- ward speed. However, there is a logical difficulty in such case, for it is not certain that the diffraction-wave potential should satisfy the ordinary linear free surface condition. This problem is discussed in the next chapter in con- nection with thin ship theory. We shall limit our discussion here to the pub- lished case of a ship at zero speed. 43 Ogilvie Suppose that the ship oscillates sinusoidally in the j-th mode. Let the po- tential be the real part of iwt V 59) (%1)%>%3) e where v; is a real amplitude. Then 9; satisfies the usual free surface boundary condition, a radiation condition, and a condition on the hull, dQ : ear = f 5(X1)%5)%3) OnE Sr (18a) where f; depends only on the geometry of the hull and on the mode being con- sidered. We may look on the quantities f; as modal weighting functions. For example, f, = cos(n,i,). If the ship surges, the fluid disturbance due to an element of the hull surface is proportional to this direction cosine. Haskind's formula arises because this same quantity, f,, plays a role in the inverse prob- lem: If there is an external disturbance to the fluid, the surge-force contribu- tion of the pressure on this element will again be proportional to f,. (See Chertock (1962).) To see how this works out, we must consider the potential function for the diffraction problem. Let wt iwt i Po(X1+X_.X3) & Pq(X1>X_,X3) & be the functions of which the real parts are the potentials, respectively, for the incident wave and the diffracted wave. (The ship is fixed in this problem.) The functions 9, and 9, satisfy the same free surface condition as 9;, and 9, satis- fies the radiation condition as well. », is known everywhere, but it clearly does not satisfy the radiation condition, since it represents a wave which is in- cident on the ship. These two potentials yield normal velocity components on the hull surface which are equal and opposite, since the hull is not moving; that is, ee on S. (18b) The force in the j-th mode on an element of the surface of the hull is just proportional to f j 2» 80 that we may write for the generalized force X; = = J pf,ds, S where p is the hydrodynamic pressure: 4 re) iat = ; iowt p = -p = Re {(9,+ 0g) e”'} = -Re {iap(o,+ og) ©}. We substitute this expression into the previous equation, and we also make use of (18a): : iowt 99 j X; = Rejiaoe (%,+ 4) 3, ds é S 44 Understanding and Prediction of Ship Motions Since », and 9, Satisfy the same radiation condition and the same free surface condition, Green's theorem yields the fact that 80; 004 4 >, 38 = Oe eee Ss S) The second equality follows from (18b). With this formula, we can eliminate 9, from the expression for X;: 00 ; 20, X; = ze {ee eave | le. ce hd: 22 ssh. (19) Green's Theorem can be used again to show that the integral need not be evaluated over the actual hull surface, S, but may be evaluated over any control surface enclosing the ship. In particular, we may choose a surface arbitrarily far away, Say a cylindrical surface extending from the free surface far down into the water, closed on the bottom by a horizontal surface. (The latter, as its depth becomes infinite, will contribute nothing to the value of the integral.) This is a particularly valuable result, because we avoid considering all of the local disturbance effects in 9,. In fact, we need only asymptotic expressions for 9,, valid far away from the ship, and such expressions will represent simply the radiated waves in the forced oscillation problem. These asymptotic forms of the potential will be the same functions which are needed to predict the damping coefficients, as will be seen in the next section. Calculation of Damping Coefficients In an oscillating ship problem, the existence of damping implies that the ship is performing work on the water, that is, energy is being put into the water. Since we consider always a nonviscous fluid, this energy cannot be converted into heat but must be radiated in outgoing surface waves. Thus we expect to find a relationship between the damping coefficients and the outgoing waves far away from the oscillating ship, and this relationship will be based on the law that there can be no non-zero average rate of accumulation of energy in any region of the fluid. Inthe derivation which follows, due to Newman (1959), it will be shown that there is a simple formula giving the diagonal elements of the damping coefficient matrix in terms of the velocity potential at infinity. Also, it will be possible to obtain a formula which relates the sum of symmetric pairs of the same matrix to the potential at infinity, but it has not yet been found pos- sible to determine these off-diagonal elements completely separately except in the special case of zero forward speed. We establish formulas for three energy flow rates. First, we assume that the ship is being forced to oscillate sinusoidally in some mode or combination 45 Ogilvie of modes, by means of an artificial external system of forces. From knowledge of the force and the ship velocity in each mode, we can calculate the average rate at which the external force system performs work on the ship. Since the ship cannot absorb energy steadily over a long period of time, this energy is then transmitted to the surrounding water, and we calculate the average rate at which work is performed on the water. Finally, we visualize a large fixed mathematical surface far away from the ship which completely encloses the ship. There can be no average rate of accumulation of energy in the fluid region between the two surfaces, and so the rate of flow of energy out of this control surface must equal the two previous rates of energy flow. First, suppose that the forces F,; cos(at + 5;) are applied to the ship by some external means (there are no incident waves), and let the motions be designated by a,(t) = a; cos (at+€;)- (We suppose further that there is a superimposed steady flow past the ship at speed y. Of course, there will be a net drag force, but there will be no work done by the drag force, since the ship has no forward speed in the coordinate system chosen.) Let the equations of motion be: 6 a {mix tC] a. (t) + b5,,() a,(t) Tea. a(t} = F; cos (at + 55) : k=1 The rate at which work is done on the ship by the external forces will then be 6 W = ye a s(t) F; cos (at + 5;) 1 6 6 Sy SG) sie > a; a, sin (ot + €;) j=1 k=1 x {[2%m), + Hix) + Cx | e0s (Ot te) i OD jk sin (at + “o} : The average value, W, over a whole cycle will be 6 6 TTL eee 1 2 * . * W = 7 > : a; ay, { [+ (mj, + Hj) = Ci sin Cen ea) + ob 5 i cos ca 6)I A fel Sk=1 We note that m;, = m,;, and so the generalized mass terms cancel each other, due to the presence of the antisymmetric factor sin («, - €;)- Thus We 7 @ DE va As Oy { [othe ee] sin Caer) + wb 5 cos ce 6) : The rate of increase of energy of the fluid within a closed surface can be written: ae J [e%(®,-Vq) ~ PV, | dS. Ss 46 Understanding and Prediction of Ship Motions (See p. 14 of Stoker (1957).) Here, n is an outward unit normal vector, and v_ the velocity component of the surface normal to itself. The surface may be a. physical surface, always containing the same fluid particles, in which case ®, = V,, it may be a mathematical surface following an arbitrary prescribed law, or it may be a combination of real and mathematical boundaries. The same formula may be interpreted as the energy flux rate across a non-closed surface. However, one must be careful to note that a positive flux rate is to be taken in the direction opposite to the standard normal vector. On the ship hull, which is a physical surface, we have ®, = V_, and so the average rate at which the ship does work on the water is: W = - { pv, as. S) As a control surface far away, we take a vertical right circular cylinder extending from the free surface far down into the water, capped on the bottom by a flat horizontal surface. This is a fixed (mathematical) surface on which Vg 0. Furthermore, we assume that all disturbances vanish sufficiently rapidly with increasing depth so that there is no contribution at all from the deep horizontal surface. Then the average rate at which energy passes outward through the con- trol surface is: Wass J po,®, ds, where > denotes the cylindrical control surface. (Physically it is clear that no energy can pass through the free surface. Mathematically this statement follows from the fact that the free surface is both a physical surface, so that ®, = V and a zero-pressure surface.) n? We now have three expressions for w. Actually, we do not need the second one, for the desired result comes from equating the first and third: 6 6 ae 1 : sO Me 25% | [He ein] Salma @er €;) + wb}, cos (, - «| = -p| ®, @ ds. jel ken y (20) We have here one equation relating all of the hydrodynamic coefficients in the equations of motion with an integral of the velocity potential far away (at "infinity,'' for talking purposes). The problem still remains of separating as far as possible the various coefficients in (20). At the beginning of this deriva- tion, it was assumed that the ship was forced to oscillate in an arbitrary mode or combination of modes by an external force system. Because of this arbi- trariness we can separate a number of special cases of (20), by selecting the amplitudes a F and the phases €; in appropriate ways. First, let us assume that only one particular a; is non-zero. If ©; is the velocity potential for such motion, then 47 Thus the diagonal damping coefficients can be calculated from the velocity po- tential at infinity for this mode of oscillation. Next, let just two a,'s, Say a, and a,, be non-zero. We can choose the relative phases so that <,- «, = 7/2 or 0. In these two cases, then, from the respective potentials at infinity we can calculate respectively Nig i ace [oun - Mba) . (Cab ba) | en W = 5 ad. (Be, BL (22) From the latter, we obtain the sum of any symmetric pair of coupling damping coefficients, but it is not generally possible to find them separately. If the ship has zero forward speed, then c,, is just the hydrostatic coupling coefficient, which is easily calculated from the ship lines. In this case we can use such a calculation, together with (21), to find u2, - uf,, and from the equa- tions (previously proved) relating added mass and damping coefficients we can in principle calculate the difference between the two damping coefficients: w! 2 dw’ * * 2 * ’ * ‘ Dab ¥ Dba Far 2f ae 3 5 My al © )| 2°53 0 ! w'2-w (It may be recalled that b,,, = b3,() = 0 for zero forward speed, and also Lab = Hab(®) = Mpa-) In this special case, we now have both the sum and the difference of b*,(«) and by,(), from which they can be individually calculated. These formulas are useful generally only when we have found the appropri- ate velocity potentials for the oscillatory ship motions. The problem of finding these potentials will be taken up in the next two chapters. In the meantime, it may again be pointed out that all of the results of the present chapter require the knowledge of the potentials only at a great distance from the ship. Further- more, there are no problems here in deciding whether the pressure must be evaluated on the actual hull position or the mean hull position. This is a great simplification in carrying out computations, but, as has been seen, there are several gaps in the results. In particular, the coupling damping coefficients cannot be found in the case of non-zero forward speed, and therefore the added mass coupling coefficients cannot be determined either. These gaps would not exist if we could calculate the c.,'s, but doing this is a rather formidable undertaking; it must be recalled that these coefficients are not just the hydro- static coupling coefficients, but, rather, they depend strongly on hydrodynamics and they involve the complicated local flow around the ship. 48 Understanding and Prediction of Ship Motions THIN SHIP THEORY Historically, the thin ship idealization was introduced by Michell in his famous study of ship wave resistance. In order to formulate a consistent linear- ized free surface problem for a ship moving at finite speed, it is necessary to assume that there is some identifiable property of the ship which makes the ship produce a very small disturbance, in spite of its moving at an arbitrary finite speed. Michell chose to consider "thin ships," that is, ships with such a small beam/length ratio that they may be pictured as knife-like. If we were concerned only with ship motions at zero speed of advance, such problems would not concern us. However, we certainly do not want to restrict ourselves in such a way. Furthermore, a rational theory of ship motions which includes forward speed effects should include the special case of steady forward motion (without time-dependent perturbations). Therefore we are forced to give consideration to the linearization problems which have so disturbed mathemati- cians working on wave resistance theory. Michell assumed that, in addition to linearizing the free surface condition, he could replace the boundary condition on the hull by a requirement that there be a certain anti-symmetrical component of velocity normal to the ship center plane. In recent years it has been demonstrated that the latter simplification follows logically from the assumption of small beam, if only we assume that the potential flow can be continued analytically into the hull up to the center plane; it is not a separate linearization.* See, for example, Wehausen (1957) or Stoker (1957). There has been much discussion of this point in recent years, naval architects arguing that there ought to be an improvement in predictions if the hull boundary condition is satisfied exactly — even though the free surface con- dition remains linearized. At the risk of offending both naval architects and mathematicians, I must insist that this remains an open question. Certainly, from the point of view of thin ship theory, such a patching-up of procedures is at least inconsistent and could give misleading results, but the grounds for accepting the thin ship ideali- zation are not very secure either. I would hope that some day numerical results may be presented which are based on such a hybrid approach. ' Then it may be possible to compare these results with the predictions of the strict thin ship the- ory and with experiments, to find out whether the present apparent shortcomings of the theory can be laid to the simplification of the hull boundary condition.‘ If such appears to be the case, then we shall have to presume that the premises of thin ship theory are at fault. *Newman has claimed that even the assumption of the possibility of analytic con- tinuation is not needed. See p. 39, Newman (1961). +We had had hopes of obtaining just such results from the work at the Douglas Aircraft Co. See Smith, Giesing, and Hess (1963). Apparently the problem is still too complicated for present day methods, even with computers such as the IBM 7090. +Before this is possible, there will have to be a tremendous improvement in our understanding of the experiments as well. 221-249 O - 66-5 49 Ogilvie Some of this controversy has been stimulated by the observations, largely in Japan and Germany, that if a centerplane source distribution for a thin body in an infinite fluid is determined by the recipe of thin ship theory, and if the streamlines which result from this source distribution are actually traced, it is found that the body which is generated is quite different from that which was originally prescribed. This observation certainly suggests that one should be more careful than heretofore about satisfying the body boundary condition. However, I can only assume that the investigators who discovered this fact have never tried to trace streamlines in a linearized free surface problem. Figure 6 shows the results of tracing streamlines in a very simple linearized free surface problem. A dipole is located at (0,0) in the figure, and there isa steady superimposed flow from left to right. Of course, a dipole exactly gener- ates a circle in the flow of an infinite fluid (in two dimensions). For the flow depicted, the dipole potential has been modified to satisfy the linearized free surface condition on y = 2. We would expect that under these conditions, the "free surface dipole'' might generate a somewhat distorted circle. However, we note first of all that it does not even generate a closed body; the forward and after stagnation points lie on different streamlines! The streamline containing the after stagnation point passes right out of the lower half-space, as if it were part of a vertical jet flow. This is immediately followed by a downward jet, as the same streamline re-enters the lower half-space. The double-jet pattern repeats every cycle of the wave behind the "body.'' On the other hand, the or- dinary linearized-theory free surface condition gives the broken line as the free surface shape —a not unreasonable looking wave, although its amplitude is rather extraordinary. This figure was prepared by Dr. E. O. Tuck, to whom I am indebted for al- lowing its use here. He will be publishing a paper soon which will include a dis- cussion of the problems pointed up by this calculation. Here it must suffice to say that, although the case depicted is so severe that one would be suspicious of linearized theory, one would not expect the streamlines to do such ridiculous Ne NU a7 Orso kes ee Cae ee ee Sere > = RADIUS OF CIRCLE (IN INFINITE FLUID) Fig. 6 - Streamlines around a dipole under a free surface (linearized problem) (by courtesy of Dr. E. O. Tuck) 50 Understanding and Prediction of Ship Motions things. The non-existence of a closed body can be rationalized easily. However, the manner in which the streamlines cross the linearized free surface curve ap- pears to be most unreasonable. At the least, it suggests that much more study must be devoted to these streamlines before we jump to far-reaching conclu- sions about how best to improve satisfaction of the body boundary conditions. Perhaps it is more important to satisfy the free surface condition exactly. Tuck has made force calculations which suggest that this may be the case. It must be emphasized that the strange behavior of the streamlines depicted in Fig. 6 has nothing to do with nonlinearities, except inasmuch as we are neg- lecting them. The streamlines shown are those which result from solution of the first order (linearized) problem. We usually accept the idea that solutions of linearized free surface problems may be physically invalid just because the problems are linearized, that is, because we have omitted and/or simplified some terms in the boundary conditions. This example demonstrates that the linearized solution can be meaningless because it is internally physically con- tradictory. Regardless of these problems, we can formulate a self-consistent math- ematical theory for the motions of a thin ship, and the theory will include the Michell-Havelock wave resistance theory as a special case. This was first done in a general way by Peters and Stoker (1954). Their work is quite well known in our field, especially since most of it was reproduced by Stoker (1957) in his monograph on wave problems. Only a very brief discussion of it will be presented here. Peters and Stoker first formulate the exact nonlinear problem of a ship performing arbitrary motions in a nonviscous, incompressible fluid with a free surface not able to sustain surface tension. They then assume that all variables can be expanded in perturbation series in powers of 8, a small parameter which may be considered as the beam/length ratio. Some quantities must be allowed to have a zero-order term; in particular, ship speed is assumed to have the expansion: SCE) = SC) + BsaCe) + [Sey arc os However, most of the variables are assumed to represent small disturbances, and so their expansions start with terms linear in 6. For example, the dis- turbance potential is written: ®(x,y, z,t) = Bo,( x,y, Z, t) + BuO Ca Zo) aF a . The free surface elevation, the motion variables, the thrust, etc., all have similar expansions. These expansions are all substituted into the various conditions and equa- tions, and the terms are all arranged according to powers of 6. Before solving any boundary value problems, Peters and Stoker make a number of observa- tions. For example, ds,/dt = 0, which means that s(t) represents a steady forward speed with perturbations superposed on it, the perturbations being of order £. The usual conditions for hydrostatic equilibrium are also obtained. ol Ogilvie The equations of motion in the longitudinal plane are all found to contain only second and higher order terms. They then restrict their attention to the case of a ship in sinusoidal head waves. These incident waves have an amplitude with order of magnitude 6. The resulting second order problem is then straightforwardly separated into a time- independent problem and a time-dependent problem. The solution of the former leads to the Michell-Havelock solution of the resistance problem, unaffected by the incident waves or the motions. The solution of the latter, the time-dependent problem of lowest order, is carried out without the necessity of treating any more boundary value problems. The resulting ordinary differential equations of motion represent simply a two-degree-of-freedom coupled spring-mass system, without damping. Even the coupling is removed if we assume that the centroid of the waterplane is in the same cross Section as the center of gravity of the ship. In this case, the solution predicts an undamped resonance in heave and in pitch. In the heave mode, the spring constant is the hydrostatic restoring force per unit deflection, and, in the pitch mode, the spring constant is the hydrostatic restoring moment per unit pitch angle (plus a non-hydrodynamic contribution which results from the condition that the center of gravity is generally below the origin, i.e., below the pitch axis). The resonance frequencies are then ob- tained as the square roots of spring constants divided by mass and moment of inertia, respectively. The disturbing force is just a ''Froude-Krylov" force. That is, the heave or pitch excitation is obtained simply by integrating the pres- sure in the incident wave over the hull, with direction cosines and lever arms as weighting functions, as appropriate. The presence and motion of the hull do not affect the values to be used for the pressure. Obviously, some of these results must be rejected on physical arguments, especially the prediction of undamped resonances and thus of infinite amplitudes of motion. Unfortunately, heave and pitch resonance frequencies quite often oc- cur within the important range of wave excitation frequencies, and, if this hap- pens, it is evident that the narrow spectral band around resonance covers the frequencies of most interest. Even if this theory is valid for other frequencies, it is not of much help in predicting real phenomena. In spite of these difficulties, the results come directly out of the hyptheses. There can be no arguing with the logic used by Peters and Stoker in deriving conclusions from their formulation of the problem, and so the difficulty must be sought in the formulation. This situation will be resolved presently, but for the moment let us note that the anomalous behavior at resonance can be explained non-mathematically. There are three types of quantities which are assumed to be of order £, and we can start a catalog of orders of magnitude by listing these: ship beam, waterplane area, volume, mass 6 amplitude of incident waves B amplitude of oscillations of the ship B Speaking in terms of orders of magnitude, we can say that: (a) exciting force = (amplitude of incident waves) X (waterplane area); (b) restoring force = (ampli- tude of ship motions) x (waterplane area); (c) ship inertial reactions = (ampli- tude of ship motions) x (ship mass); (d) amplitude of motion-generated waves = 52 Understanding and Prediction of Ship Motions (amplitude of ship motions) x (waterplane area); (e) motion-generated fluid force = (amplitude of motion-generated waves) X (waterplane area). We can now add to our catalog of orders of magnitude: excitation by incident waves B? hydrostatic restoring force Be ship inertial reactions Be added mass and damping force /° At resonance, the restoring force and inertial reaction add up to zero, and there are no second order forces to counter the excitation force. Therefore the re- sponse amplitudes are unbounded to this order of approximation. This violates the assumption that motions are of order (, but it would not be proper in the perturbation analysis to try to modify the resonance prediction through use of higher forces, simply because they are of higher order and therefore small by comparison. Peters and Stoker criticized Haskind for assuming a priori the orders of magnitude of the various kinds of forces, but it can be seen a fortiori that Peters and Stoker have done essentially the same thing, for they also assumed the orders of magnitude of certain quantities (not the forces) and arrived at un- tenable conclusions. These authors recognized and noted this anomaly, and they suggested sev- eral escapes from this predicament. For example, they discussed the ''flat ship" linearization. However, such an approach simply shifts the same diffi- culty to the lateral motion modes. They also considered a ''yacht-type" ship, which would avoid the trouble in all modes except surge. Such a mathematical model is quite artificial, but it might produce successful results if it could be worked out. However, the basic difficulty with thin-ship theory may be looked at in an- other way which suggests a totally different method. It was assumed that ship beam, ship motions, and incident waves were all small, of the same order of magnitude. However, it was found that motions near resonance could be very large —to an extent that invalidated the assumptions. Newman (1960) proposed that there should be more than one small parameter in the statement of the problem, and he worked out a development in terms of three parameters. He retained 6, the beam/length ratio, and he added a parameter y which indicates the order of magnitude of the unsteady motions and another parameter 5 which indicates the order of magnitude of the incident waves. Such a triple expansion allows for consideration of two important points: 1) There is no reason at all to assume that ship beam is related in size to the amplitude of the incident waves; 2) it is not necessary to make any a priori assumptions about the magnitude of ship motions relative to the magnitude of ship beam or incident waves. With regard to point 1), we note that ship beam and incident wave amplitude remain as independent parameters throughout the problem, whereas, with regard to point 2), we expect that the solution of the problem will provide us with infor- mation about the actual amplitudes of motion. 03 Ogilvie Newman expands each of the dependent variables in multiple series expres- sions. For example, the potential is made to depend on all three small param- eters: 9(x,y,z,t) = yr Bi yi sk Viju(®YyzZ,t), Leyak whereas the motions are represented by double series, e.g., for heave, z(t) = ab Biyi ig a i,j (The unsteady motions depend on y¥, by the definition of this parameters, but we also expect steady displacements, which will depend on 8. This is the reason for including both parameters in the expansion here.) Since Newman develops his analysis on the assumption that it will be neces- sary to include higher than first order effects, he carefully introduces other needed expansions, which will not be written out here. For example, he trans- forms from a body-fixed coordinate system to a steadily translating system, both for calculation of the potential functions and for calculation of the pressure and forces; this transformation involves the parameters 8 and y. Finally he obtains a sequence of problems, each homogeneous in each of the small param- eters, and he solves explicitly for the following potential functions: 9,,,, the potential for the steady translation problem; »,,,, the potential for incident waves; 9,,,, the potential for small motions of the ship in an otherwise undis- turbed ocean; 9 ,,,, the diffraction potential. The first is just the Michell- Havelock potential, and the second is the classical potential for sinusoidal waves on an infinitely deep ocean. The third and fourth potential functions are somewhat more interesting and deserve some further comment. If a ship model is forced to perform small oscillations of order of magni- tude y, perhaps by being driven by a mechanical oscillator, then the appropriate potential function is 9,,,- The one complication of interest here lies in the specification of the body boundary condition. In the initial formulation of this problem, it is necessary to state the boundary condition on the actual, instan- taneous position of the body surface. Then, by a systematic procedure, this condition can be translated into a (different) condition on the mean position of the body. This problem has already been mentioned; see the discussion accom- panying Eq. (10b). There is an interaction between the ship oscillations and the steady flow past the ship which produces effects in the lowest order unsteady solution. This interaction is lost if we assume immediately that the body bound- ary condition can be satisfied on the mean position of the hull. Such an error has occurred frequently in work in this field; the first correct treatment is ap- parently due to Hanaoka (1957). Newman (1961) discusses the problem quite ex- plicitly and shows that the difference in the potential functions, corresponding to the two methods of satisfying the hull condition, is equivalent to the potential of a line distribution of oscillating sources located on the mean keel line. It must be emphasized that this is not a higher order effect, and the problem is not related, for example, to the arguments about how to satisfy the body boundary condition in the steady motion (resistance) problem. The elegant formulation of 54 Understanding and Prediction of Ship Motions the boundary condition by Timman and Newman (1962) provides the most con- venient procedure for handling this difficulty. Again, see Eq. (10b). The potential 9,,,, aS found by Newman (1961), points up an interesting difficulty which is still not well understood or appreciated. This potential rep- resents the diffracted flow around the translating restrained ship. It satisfies a straightforward boundary condition on the hull, providing a normal component of velocity which just offsets the corresponding velocity component of the inci- dent wave system. However, its boundary condition on the free surface is unique among the potential problems formulated by Newman. When the series expansions are substituted into the free surface condition and the resulting con- ditions are modified so as to apply on the undisturbed free surface, the poten- tialS 9150 0012 and 9,,, all satisfy a homogeneous condition: However, 0, 9,, must satisfy a nonhomogeneous condition; there is a nonzero right-hand side in the corresponding equation, and this right side contains terms which are essentially products of 9,,, and 9,,,. This situation is some- what analogous to the problem of satisfying the body boundary condition. There is an interaction between the incident wave system and the steady Kelvin wave system such that an apparent pressure distribution is applied to the free sur- face, and this apparent pressure gives rise to an unexpected addition to the dif- fracted wave. This complication with the diffracted wave is an excellent example of the value of systematic perturbation analyses. We could have set up the diffraction problem much more easily. It would have seemed quite reasonable to assume that the usual linearized free surface condition would apply, and so we would have found a potential function which satisfied that condition and which also off- set the normal component of the incident wave system on the ship hull. New- man's systematic approach shows that this is not proper. We need another con- tribution to the potential which satisfies a homogeneous condition on the hull and a non-homogeneous condition on the mean free surface. This extra part will be of the same order of magnitude as the potential which we would obtain by the more naive approach. We should note specifically that, since this effect is due to interaction of the incident waves with the steady wave system of the translating ship, this is a problem only when the ship has non-zero forward speed. The effects of this difficulty may be quite pervasive. In particular, the Has- kind relations for predicting wave-induced forces (discussed in Chapter IV) were derived only for a ship at zero forward speed, and the extension of these relations to ships with non-zero forward speed will depend on a further satis- factory resolution of the problem discussed here. For the thin ship moving through sinusoidal waves, Newman's solution is complete to first order in £, and he obtained a set of formulas for the coeffi- cients in the force expansions, complete to second order in 6. The expressions are quite unwieldy, and one can not be very optimistic about being able to use them for practical calculations. However, it is of some interest to point out 55 Ogilvie how they resolve the problem left over from the work of Peters and Stoker: How do we explain away the predicted infinite amplitudes of motion at resonance? In Newman's formulation, the lowest order forces have the following orders of magnitude: excitation by incident wave, B3; hydrostatic restoring force, By; ship inertial reactions, BY; added mass and damping forces, 67,. Away from resonance, the excitation must be equal to the sum of hydrostatic plus ship inertia forces. Under these circumstances, it is clear that we can set: y = 6 + smaller terms. At resonance, the forces of order fy total zero, and so the excitation force must equal the added mass and damping forces, which are the lowest order non- vanishing forces. Therefore, at resonance 8 (y= may smaller terms. Since y must still be a small parameter, we must require that 5 << £, if the perturbation analysis is to remain valid. Such a requirement appears reason- able. As a practical approach, if we were to try to use Newman's formulas for the forces, we could now follow the procedure which is usual in perturbation analyses, viz., absorb the small parameters into the force and motion variables. We would then calculate the forces, including the higher order added mass and damping forces, and from these calculate the motions. Away from resonance, the higher order contributions should be negligible (if the conditions of the the- ory are really satisfied), and the results should reduce to those of Peters and Stoker. At and near resonance, the higher order forces should dominate the lower order forces and control the predicted responses. This approach is logical, at least insofar as a ship may really be consid- ered as thin, but the results are not very useful because of their complexity. The damping coefficients are the only elements of the problem which fall out in a fairly simple fashion, and it has already been seen that at least some of these can be calculated in a much simpler manner, from radiation considerations. In order to evaluate the potential usefulness of the thin ship idealization in predict- ing ship motions, some calculations of damping coefficients have been made for Series 60 models and compared with experiments by Gerritsma, Kerwin, and Newman (1962). Figure 7 shows some typical results from their paper. The heave damping coefficient (b},) is plotted against frequency for a sequence of values of Froude number. The ship concerned is the C, = 0.60 form of the Series 60. The agreement is at least qualitatively good. 36 Understanding and Prediction of Ship Motions KEY TO EXPERIMENTAL POINTS FOR VARIOUS FROUDE NUMBERS O=0.15 4=0.2 GO =0.25 4 =0.3 NONDIMENSIONAL HEAVE DAMPING COEFFICIENT Fig. 7 - Heave damping coefficient for series 60, C, = 0.60, Model at various Froude numbers, experiments and cal- culations (from Gerritsma, Kerwin, and Newman (1962)) SLENDER SHIP THEORY A slender body theory, whether for aircraft or for ships, is formulated on the assumption that all dimensions in a cross Section of the body are small compared to the length of the body. Also, the rate of change of transverse di- mensions (with respect to the lengthwise coordinate) must be small in a simi- lar sense. Such an approximation appears attractive for ship problems, since ships generally fit such a qualitative description. Nevertheless, the application of slender body theory to ship problems has been long in coming, in spite of the fact that the aerodynamic version of the theory is forty years old. o7 Ogilvie There are probably several reasons for this long delay, and a quick inspec- tion of these reasons will suggest something about the nature of slender body theory. One important problem arises immediately which distinguishes the slender body approach from thin ship analysis. If we introduce the slenderness parameter into the formulation of the boundary value problem, the effects of the free surface are generally lost and we are left with an infinite fluid problem. This is clearly quite unsatisfactory, because we seek primarily a description of just those phenomena which result from the presence of the free surface. Fur- thermore, such a formulation turns out to be equivalent to a set of two-dimen- sional problems, and the effects of interactions between various cross sections are apparently quite ambiguous. These problems can be resolved by a refor- mulation which is altogether different from the thin ship approach, but then the final formulas depend on the geometry of the ship in an elementary way — which is offensive to the naval architect, for it implies that the complicated geometry of a hull is of little importance for ship motions. These difficulties all demand that our attempts to apply slender body theory in ship problems be done with great care, by a systematic procedure. Using a perturbation analysis, we can answer to all of these problems, even the last, for, by being systematic, we can (in principle) proceed to higher approximations which involve more and more details of the hull geometry. Before proceeding to the logical development of slender body theory, we should note that Grim (1957, 1960) anticipated much that would later come out of the theory. He pointed out that there were apparently two general approaches to representing the ship in studies of ship motions: (1) The ship can be represented bya set of three-dimensional singularities which clearly predict three-dimen- sional effects but which are loosely connected to ship geometry. (2) The exact shape of the ship in each cross section can be generated as if that cross section were part of an infinitely long body of uniform shape, with no account taken of three-dimensional effects (either interactions or forward speed effects). In order to combine the advantages of both, he proposed to solve the potential prob- lem corresponding to the second approach, representing the potential as a two- dimensional multipole expansion about a line in the centerplane, and then at each section to replace the two-dimensional singularities by three-dimensional sin- eularities of the same strength. The resulting potential would then be used to calculate pressure and force. In other words, he proposed to use strip theory only to find singularities for representing the ship and then to use truly three- dimensional potential functions to represent the flow. Grim (1960) published the details of the analysis and some calculations, all for the case of zero forward speed. He also stated that the theory had been worked out for forward speed cases as well, but apparently he has not yet published that. There is considerable similarity between Grim's procedure and the rules for calculation which follow from slender body theory. However, the two are not identical, and it is obscure as to what meaning should be attributed to the differences. It will appear that slender body theory actually gives simpler re- sults than Grim's, and one may say that the slender body results fall into the category which Grim criticized for not providing a realistic representation of the ship geometry. But the systematic approach is logical if the assumptions 08 Understanding and Prediction of Ship Motions are correct, and, if they are correct, then there is no need to use Grim's more complicated formulas. The questions raised here can not yet be answered. The first comprehensive attack on ship motions problems by slender body theory was by Vossers (1962a), and I shall follow his approach in essence. (However, other methods are possible. See, for example, Ursell (1962), Tuck (1964).) After formulating the problem exactly, we must introduce the slender- ness approximation and this proves to be a difficult task. Vossers's analysis and results are very complicated and moreover they are somewhat suspect. Newman (1964) has had more success in obtaining approximations, at least for the case of no forward speed, and his first numerical results are very encour- aging. However, the calculations are still at a very rudimentary stage, and it is too early to predict the quality of the outcome. Whether all of this effort will lead to valid and useful formulas is not yet known. I leave the two following authors (and their discussers) the opportunity to speculate on the future of slender body theory for predicting ship motions. In formulating the slender body problem for ships, Vossers assumes that the ratio (ship beam)/(ship length) is a very small quantity, which we call ec. The purpose of his investigation is to find solutions which become more and more accurate as « becomes smaller and smaller. Vossers expands various quantities as perturbation series in powers* of «, substitutes these into the various mathematical conditions of the problem, and non-dimensionalizes all quantities and equations. In the last process, a number of special non-dimen- sional ratios arise, and the nature of problem and solution depends on the rela- tive sizes of these quantities. The important non-dimensional quantities are, besidesie, — aL/2V, a reduced frequency, 2V?/gL, a forward speed parameter, proportional to (wavelength of waves travelling at speed v)/(ship length), #°L/2g, proportional to (ship length)/(wavelength of waves with frequency «), #*B/2g, proportional to (ship beam)/(wavelength of waves with frequency ~), oV/g, a parameter for describing the pattern of radiated waves (usually called "7" in the American literature), where L = ship length, B = ship beam, » = circular frequency of exciting or motion-generated waves, and Vv = forward speed. *This is really not correct, and it shows the danger of loose assumptions. One should, as it turns out, use double series containing factors e"(loge)". Itis also possible to avoid this trap altogether by assuming only that the potential can be expanded: ¢ = i¢,, with én4+1 = 9(¢,)- By such an approach, one must determine in turn the actual order of magnitude of each term. 59 Ogilvie The nature of the free surface problem depends primarily on the length of waves (of frequency ~) compared with the ship dimensions. If these waves have length comparable with ship length, that is, w*B/2g = 0(c«), then Vossers shows that the free surface condition reduces to the rigid wall (low frequency) condi- tion. If the waves are short compared with ship beam, the high frequency de- generate boundary condition applies. Only if the waves are comparable in length with ship beam does one obtain an interesting problem, for then the free surface condition becomes (in dimensional form): =—— + g =—— = — on x, = 0), (23) pane 0) Ox ha Ox i Ox s to (24) 34 320 xe Oxne In words, the problem reduces to a set of two-dimensional problems; for each cross section, we must find, in two dimensions only, a solution of Laplace's Equation satisfying (23), the usual free surface condition, and (as Vossers shows) the usual boundary condition on the body. This problem should be quite familiar, for it corresponds exactly to ''strip theory.'’' There is no effect of forward speed and no interaction between cross sections. Vossers' formulation shows clearly then that strip theory is a natural consequence of assuming that the disturbance waves and ship beam are com- parable in size, and accordingly it should be valid in problems of ship rolling in short beam waves, for example, but not for problems of pitching and heaving in waves comparable with ship length. It should be noted specifically that solutions which satisfy (23) and (24) are functions of x,, since the body boundary condition depends on x,. Moreover we can add to such solutions any other function of x, which we desire, without violating (23), (24), or the body condition. This arbitrary additive function may be interpreted as expressing the interaction between sections. But it is unknown. In this formulation of the problem, we can do only as in strip theory, namely, assume that there is no interaction. If we are interested in pitch and heave problems (and most of this paper is concerned with just these problems), then we must consider wavelengths com- parable with ship length, and it is apparent that we do not obtain a satisfactory formulation by the above procedure. Therefore Vossers proposed a different tack, viz., that we write down the solution in a general way by using Green's theorem and then use the slenderness approximation to simplify the resulting integral equation. In other words, we effectively establish an integral equation for the solution of the problem involving a general body (not a slender body); we 60 Understanding and Prediction of Ship Motions cannot solve this equation, but we simplify it for the special case of the slender body, and the resulting equation can be solved. It turns out that the integral equation which must be solved relates to a set of two-dimensional problems again, but this time we obtain an additional part of the solution which explicitly represents interaction effects between sections. It seems to be desirable at this point to restrict ourselves to the case of zero forward speed, since it has been worked out in detail and we can be rea- sonably confident of the results. In detail, the approach which follows is that of Newman (1964); the general concept is still Vossers’. In order to simplify matters, let us assume immediately that all disturb- ances are Sinusoidal in time. For the potential we write Re {x) eit}, and we have similar expressions for all other variables. This is not necessary and perhaps not desirable, but it is certainly convenient. We also stipulate that ¢(x) represents the potential for motion-induced diffraction waves, but not for inci- dent waves. By Green's theorem, we can write an expression for the potential at any point in the interior of the fluid: 1 é oG(x, £) a) = Te I {ocx p - $(€) Se | do. (25) G(x,€), a Green's function, is any function which satisfies Laplace's Equation (in three dimensions) except at x = €, where it has the behavior: Cas) = COW = Gl) pay ae The domain of integration, =, must be a closed surface with x in its interior, and € is the dummy variable which ranges over =. Under these conditions, (25) is a very general equation, and its usefulness for us depends on our selecting G(x,é) in a meaningful way. We choose Gx, é) as the potential function of a pulsating source located at é, the potential satisfying the linearized free surface condition on oo Oraldyan appropriate radiation condition at infinity. Also, we define the closed surface 2 aS S,+S,+S,, where s, is uae wetted surface of the ship, S, is the mean free surface, that is, the plane x 3 = 0 outside the ship, and S| is a closing surface far away from the ship, at "infinity."' If now we assume that $(x) satisfies the linearized free surface condition and a radiation condition, the integrals over S, and S_ vanish, leaving only the integral over S, in (25). We have left a gap in our logic by assuming that we can use the linearized free surface conditions. However, we get away with it in the case of zero for- ward speed, for we actually have two means of supporting the linearization: If there are incident waves, we may assume them small, so that the ship motions will also be small (even if it is a 'fat ship''), or we can concentrate on the as- sumption of slenderness of the ship, in which case even finite amplitude mo- tions will produce small amplitude disturbances. It is evident that the linearized 61 Ogilvie condition will be appropriate and we proceed to use it without further justifica- tion. However, the forward speed problem would require much more care. With = now replaced by S,, Eq. (25) is much simpler, but we still can do nothing with it in its present form. We assume that 3¢/on is a known quantity on S,, but ¢(x) is not known, and so this is an integral equation for ¢(x) on S, * To reduce it to a simpler integral equation, Newman now introduces the slender- ness parameter. This is essentially a rather tedious exercise in estimating the relative sizes of various quantities, and I shall be satisfied here to state his re- sult. He finds that ¢(x) can be written as the sum of two terms plus an error: P(x) = [byn(x) + FO] [1 + O(e Log €)] (26) where ¢,, is the solution of the two-dimensional problem in which the body cross section performs its motions in the presence of a rigid wall at x, = 0. The function f(x,) is obtained explicitly: f(x,) = aI RG, y (ats Gully = igh = eal et 2iJ,(Klx,- €,1)} dé, 1 L/2 ‘ Di eS | i F + OT og iL, sen Cx) Si) eer = (Sa ) ey, ’ -L/2 where eke F(x,) = aa ot 5 C is the contour around the hull in the cross section at x,, KS Og, J, = Bessel function of the first kind, Y, = Bessel function of the second kind, and H, = Struve function. It is easily seen that F(x,) is just the fluid flux through the hull surface at the cross section at x,; it is zero for yaw and sway motions but not for heave, pitch, and roll motions. (We avoid mention of surge motions. They can be ana- lyzed by slender body theory just as well as the other kinds of motion, but the results can be expected to be rather special with respect to orders of magnitude. To some extent this is true for roll also.) *In order to obtain the integral equation on S,, we must let x approach S,, and then the factor 1/47 changes to 1/27. 62 Understanding and Prediction of Ship Motions In the formula for f(x,) we have two integrals over the length of the hull, and we should distinguish between their meanings. The first integral, involving the Struve and Bessel functions, represents a free surface effect. It can also be looked upon as expressing an interaction between sections — an interaction caused by the presence of the free surface. The second integral also represents an in- teraction, but it would exist even in the absence of the free surface. We could combine the latter with ¢,, and look on the sum as the slender body approxima- tion for the three-dimensional body in the presence of a rigid wall, with the first integral supplying a correction to account for the free surface effect on the three-dimensional body. We note that the interaction, due to either term of f(x,), involves the ship geometry in a very simple way. In fact, F(x,) depends only on the ship beam at section x,, and thus f(x,) depends only on the waterplane shape. However, ¢,, depends on the detailed shape of the cross section. (Fortunately, the finding of ¢op is not too difficult, since it is not really the solution of a free-surface prob- lem.) Thus, the solution does depend on the hull geometry in a detailed manner, but this dependence is shunted off to the mathematically easier field of problems with fixed, rigid boundaries. We should now refer back to the earlier statements which resulted from various assumptions about orders of magnitude. We have assumed here that wave length is comparable with ship length. Under these conditions, Vossers showed that the three-dimensional boundary value problem would reduce to a set of two-dimensional problems in the cross sections, with the free surface condition replaced by a rigid wall condition. This is exactly what Newman ob- tains. However, we now have an explicit formula for the interaction term, f(x,). Perhaps it should be emphasized that (26) is valid only very near to the ship hull, at distances which are of order of magnitude have the values which we associate with the undis- turbed ship, whereas the factors (n - i;) vary with time. Therefore we find it somewhat easier to calculate the components xX; directly. To the expression for X; we add and subtract a quantity, as follows: 70 Understanding and Prediction of Ship Motions Boks) X; {f (n-i;)p dS of (n-ijyat | ‘vos 1 oh + | (n-ijat | Bday i? ’ where L is the line of intersection of the ship hull at any instant with the undis- turbed free surface, and ((x,,x,,t) is the free surface elevation. I have as- sumed that the ship is wall-sided near the free surface. It can now be recog- nized that the quantity in braces is just an integral over that part of the hull surface which is wetted when there are no waves and no ship motions. It has the same shape and size as S_, but it is displaced from the equilibrium position. The direction cosines, (n-i;), in fact have the values appropriate to S, itself, but the pressure must be evaluated on the actual position of S. However, we can use the Taylor expansion of the pressure to convert it into a function to be eval- uated on S,. The quantity in braces is then just: nde io} J (n i;){p CS D738) OTB) oi S The correction term, the integral over L, is expressed partly in terms of each of the two coordinate systems. It is perhaps easier to retain the quantity (n- i,) as it stands, and so we must express the inner integral in terms of the primed coordinates. From the geometry, it is found that this correction term can be written: C-(E+Oxx) +i, i (n-i, dt | acy PP) (Go) VB at L 0 ie} where L, is the intersection of the undistrubed free surface with the hull sur- face, the latter being in its undisturbed position. The upper limit of x 3 1S in error by a small quantity which does not affect the result. If we now work out this integral, systematically keeping only lowest order small quantities, we obtain 2 Wo 1 or V or wv J atm ip sf. ie Sta emd MST ot Sle Mee L 1 1 The complete result for x; is: : C) X; = ef aS(n-i;) | -wy~ 8685+ #94192) = ae te NY aE - Vo.- s ° 30, +V[1+ (€+ Oxx)-V] = bev df(n-i.) = 1 =-—-- set (E+ 94-12,)}- L 1 Ogilvie We can interpret the various terms here readily. The first term of the first integrand is just the hydrostatic pressure at equilibrium, and the second term yields the hydrostatic disturbance force. The following three terms are un- steady force contributions, and the last term in the first integral yields the re- sistance and the unsteady force which results from interaction between the steady flow and the displaced position of the hull. The line integral represents an interaction force arising from the superposition of steady and unsteady mo- tions; the factor °®9,/ox, can be recognized as being proportional to the wave height in the steady motion problem, and is proportional to the unsteady wave height (omitting a term containing both 9, and 9,). It may be noted that the line integral is zero for j= 3. This follows from the assumption that the ship is wall-sided near the waterline, which means that n lies in the plane of i, and i, for points on L,. Furthermore, if we consider only motions in the longitudinal plane, then the line integral is also zero for j = 2. Finally, if the steady motion problem is linearized in any of the usual ways (thin, flat, or slender ship approximations), then n-i, is small, of the same order as 9, and the whole integral is of second order in terms of the perturbation parameter for the steady motion problem. The moments acting on the hull can be calculated in a similar manner, with only slight complications appearing. Let us again choose to work directly with the moments about the unsteady (body) axes: Kings = J ey he) Gln Then the moments about the steady axes will be given by: M=M'+@xM'+éxX where M = (X,,X,,X,) and X = (X,,X,,X,), and similarly for the primed quantities. We proceed to calculate X;,, in the same manner as we did X; and so the details will not be repeated. The factor (n -i; xx‘) must initially be kept within the inner integral in the correction term, since x‘ depends on x,, but if we set x, = 0 in this factor, we cause only higher order errors, as is easily verified. The result is: ‘ 39 ce | dS (n-i ;xx) {-exs~eC8s+ 291-18) Fae fe ase VO,°VO4 Ss ° +V[1+ (€+0xx)-V] 29, lo 1 1 y 2 ow | df{(n-i: po) > o{2 Sry Set (Est 21 175) fe io} 72 Understanding and Prediction of Ship Motions Since we set x, = 0 in the line integral, the vector x now lies in the x, - x, plane, as does n also. Therefore the correction term is non-zero only for j = 3. That is, it contributes only to the yawing moment. For motions limited to the longitudinal plane, the correction term is clearly zero. To conclude this appendix, we write down the complete expressions for force and moment components in the steady coordinate system, using the poten- tial function in essentially the form proposed by Cummins, Eq. (11): 6 6 6 Ki(t) = Kio Do mix H(t) - Dy ju Ht) - Do ej HCE) kia k=1 k=1 6 t 6 t - J a,(T) nan ache = ay | a.(7) Miy(t-7)dr, (A1) k=1 - © k=1 - © where fy 015 (%) 80, se =D : ae 7 Sage Nae ds , (A2) OW, (x) PE en =e | a Wi, (x) dS, (A3) S OW (*) OW () Bee ty ea) We acannon 2h oir easement ol ae ou) |g (A4) ' B50) fay) | Shik = 2 on aay Ox, FY atu Wo) + gl, -hy (x) Ss 20, Wr, ;(x) [ 20 WiC) WV epeel Ch yo sn a ee -alnx)) eda, (A5) iy 0 0 0 Cie (SC 0 0 0 -X, 0 +X, 0 0 On Pee | Axe 0 Cik C ik + . i ’ (A6) Oe sKs mK OHM Ae -X, Ou ge eX uae Ke 0 aX, +Xy -X, 0 +X. aN 0 73 Ogilvie iho ted Kets oo hy. (x) = Ts (A7) seus 2S 9 k = 4,5, 6, op 5) ) 0 Lint) = 0 | B= (se ae t Mo 8) Hale td Ss to} eub, ((s)) Oa (2) IV 1j\= Or= 3 wee +S on mos (= my se) ence dt, (A8) iL, 1 ° ul OW, 5X) C) C) Fe erapallcree tay fa ect aie s °o Ob, (x) 99,(x) pV 1j\= ot = re) ) duces | on Ox, (=- Y se) XC 8 dt. (A9) te} It may be noted that the quantities (c;,-cj;,) arise from the transformation of force and moment components from the unsteady to the steady reference frames. There is some ambiguity in choosing the best representation of the convolu- tion integral terms. There would be no basic difficulty in carrying along both sums, but it would lead to much extra writing later on, and so we choose to com- bine the two sums of convolution integrals into a single sum. We can partially integrate the sum containing the L,,'s: t t t : ic) i a.(T) LGta aan = a(T) Lc GS a?) | + ( a4( 7) AE Lj,(t-7)d7, -o - © -@ and then this sum has the same appearance as the other one, since the integrated part vanishes. However, for reasons of convenience later, we choose now to handle these integrals in the opposite way: We assume that a partial integration can be performed on the sums containing the M.,'s and that the integrated terms will vanish, so that we can write the last two sums in Eq. (A1) as follows: t 6 6 : ; dX mone ae a) Be Cerra ie 2 J ay (7) Ky, (t- 7dr. (A10) A further discussion of this point appears in Appendix B. 74 Understanding and Prediction of Ship Motions APPENDIX B Systems Lacking Some Restoring Forces We want to treat here the case that, after a transient disturbance, some of the a,(t) may not return to zero. As explained in the main text, we specifically exclude the case that any a,(t) may continue to grow indefinitely, for then the linear analysis must certainly break down and there is no meaning to applying Fourier transforms to the equations of motion — even though they may be valid in the initial stages of the motion. If a,(t) + a,(w) + 0, as t > w, the Fourier transform of 2, does not exist in the classical sense. However, we can still treat it by using the concepts of generalized function theory. (See Lighthill (1958).) Let a(t) = [a,,(t) — a,( 00) H(t)] + a,(o) H(t) , where 0 fOr (¢ <@, je((ie)) = The Fourier transform of the quantity in brackets exists in the usual sense and is easily shown to be: B {a,(t) - (a) H(t)} = 2 [4a - a,¢@] , au where, for brevity, I have introduced the notation Co Se. (Lhe We also note that a,(0) = i SCs) Gle = ee (D2 In the language of generalized functions (see page 43 of Lighthill), SCE = Ce) = eCay A ig) Therefore, d Guy, (2) SAG = Baye 5 4,(0) Oe iw Since we assume that a,(t) remains bounded, there is no difficulty about taking transforms of the terms containing 4, or 4,, but the convolution integrals 75 Ogilvie in the equations of motion, (13), require care. Rather than attempt to transform these terms as they appear in Eqs. (13), I find it desirable to retrace steps somewhat. Equation (Al) of Appendix A states that the j-th component of the motion induced force is: 6 6 6 Gs gato Je eMC ee ope CNG) — a eam dcn (i) k=1 k=1 k=1 6 t 6 t ps ye | ay.(T) ans = wicker - me J a,(T) M;,(t-7)dr. (B1) k=1 ~-o@ k=1 ~-o The last two sums are compressed into the single sum of convolution integrals in (13). However, it is now more perspicuous to consider the two separate kinds of integrals. In order to study the behavior of the kernels in these integrals, let us as- sume that there is a velocity impulse in the k-th mode, so that (CE) = @h- BCE) 2 Then, for t > 0, t Xx; = ce Gly es + Lj, (t) +{ wore. 0 The term L ;x(t) represents the actual unsteady force due to the velocity impulse itself. Physically, this must approach zero for large t, since the impulse mo- tion generates only a finite-energy wave system, and these waves rapidly radiate away. The terms iG le: + { ron 0 represent the force which results from the steady deflection of the translating ship after t = 0. The integral term of this expression must approach zero eventually, because c;,, by definition, is the constant of proportionality be- tween steady perturbation force and displacement. Obviously, if the integral of M jx approaches zero as t becomes infinite, then M ik itself approaches zero. We can now find the Fourier transforms of the convolutions. For the first, involving L;, and the disturbance velocity, we have from the convolution the- orem: : : ) 1 a(7) Lit = nar| = &, (©) L jy (©) ; -@ 76 Understanding and Prediction of Ship Motions For the other convolution, we must perform a little manipulation: t t off a. (T) ut] = off [o,.(7) — A, (0) H(7)] M;,,(t = Dar} 20984 oo -o 0 t u cor} M M;,.() = Mi Co D{a,(t) - a,(0) H(t)} + a,(«) x ee w Guy, () = oe) oe M5 1.(@) a aes iw iw au w) i@ = Mj, (2) The two kinds of convolution integrals can now be combined, as in Eq. (A10), and their sum will have as its Fourier transform Gin((@)). (Te st us (2) jp G0) Ca] i 1W The above derivation amounts to a demonstration that the integrals in (13), t | ay(7) Kjy(t- 7dr, do indeed exist and have conventional transforms. The transform of (xX ; 7 Xj.) can now be written out explicitly: 6 3{X; eae = 5 (a) my Ci (©) k=1 2 is ze ay, (@) = oe {-2? Hit + iob;, + cj, + io Lj, () + M,C) } — . (B2) k=1 This can be substituted into the transform of (13), and we recover (15) — with three changes: 1. 4,(«)/iw replaces 4,(«). 2. L;,(@) + M,,(#)/iw replaces Kj, (®). 3. There is an extra sum on the left-hand side: 6 5) ase ae k=1 dole / COL/ Ogilvie Item (1) is of no consequence if 2,() exists, for it then equals Gy, (w)/ie. Even if a,() does not exist, our assumptions imply that a () does exist, and so the applicability of (15) has been extended. Item (2) is also of no consequence, for we have just shown that the two ex- pressions are equivalent. Also, M;,(«)/iw exists even when w = 0, because of the fact that @ \ M547 )d7 = 07 0 Item (3) is not quite so easily disposed of. We note that, if the external forces represented in (15) are well-behaved transients, then 0] F (a) + Gi(«)| > 0 as w>0. Also, 8(#) = 0, forall . Therefore, if we multiply Eq. (15) (as now modified) by w and let w = 0, we are left with: 7s C ik ea(ONn = ye Cr UC) aOe: k=1 k=1 This result is hardly unexpected, for it says simply that c;, must be zero if ay, (@) + 0, unless two or more a .(®) are non-zero in such ‘a way that this sum vanishes without the individual terms all vanishing. It is now evident that the above sum can be omitted from (B2) and thus from the modified (15). However, the equation will not apply when w = 0. With this exception, Eq. (15) remains valid even when 2,(«) does not exist, provided only that we replace 4,(w) by 4,(w)/io. APPENDIX C Alternative Derivation of (17a) and (17b) Equations (17a) and (17b), relating the added mass and damping coefficients, can be derived in a way which avoids inverting one of the transforms and using the inversion to find the other transform. Thus it also avoids the double inte- gration and the interchange of limiting operations (which were not proved valid) in the derivation of (17a). The proof which follows is not entirely rigorous either, but it shows some of the physical bases on which the final formulas stand. The kernels K.,(t) in the convolution integrals all have the property that K(t) = 0 for t <0. (I shall omit the subscripts hereafter.) Furthermore, they approach zero as t > o. Therefore the Fourier transform of K(t) can be written: 78 Understanding and Prediction of Ship Motions K() = i KGeye, dt la) Kia) ~ 1 K_(2).. 0 Also, foe) K(-w) = i K(t) e?@tdt = K.(@) + iK (w) = K(w).* 0 These relations depend quite explicitly on the condition that K(t) = 0 for t < 0; this has frequently been described as an effect of ''causality,"’ i.e., the system does not respond before t = 0 if the disturbance comes at t = 0. If now we consider w as a complex variable, it is clear from the definition of the Fourier integral K(w) that the convergence of the integral can only be quickened when Im w < 0. Then K() cannot have any singularities in the lower half of the w-plane. But for an analytic function we can use Cauchy's integral formula: pats { K(w')dw' Pipl w@!'- Ww K(w) = - Cc See the figure. If K(w) vanishes far from the origin in the lower half-plane (no matter how slowly), the contribution from the semi-circle vanishes as the radius grows to infinity, and so we can replace the integral over the closed contour C by a con- tour integral along the real axis from -o to +o. Now we let approach the real axis (from below), and we indent the contour above the real axis, so that for real w os or os 74 ees el K(@' do! pee ee ie 1 K(@')da' aa anf Si Scag. ag oes embapeeT al lloeee HRS *The long bar denotes the complex conjugate quantity. 79 Furthermore, i pe Ogilvie K(w' )do’ Gi)! a5 @) f = Kaye - © Finally, we express the real and imaginary parts of these equations separately Vv = = © 'K 1 d 1 K.(@) = 4 Ko) + K(o) | = -4 ees ie: gin Oi a 27 #&'K,(a")do' y a Tv ils =a P Kacy LOPS ilk) = K(a) | ue 2} eons (a2) = (69) From (16), we substitute the definitions: Ko) = and this leads immediately to (17). * i (a) = 1s 2 | K(@) = -o[2°(@) - 4]; PROBLEM AREAS IN SHIP MOTION RESEARCH Willard J. Pierson, Jr. New New INTRODUCTION York University York, New York There are a number of subjects that were not covered by the various papers on ship motions given at this symposium. Three of the subjects are: (1) Ships in short crested waves, (2) coherency and resolvability of spectral and cross spectral shapes, and (3) the solutions of specific problems that are nonlinear. It is the purpose of these comments to discuss these subjects and their rela- tionship to the papers that were pr esented so as to complete the record of this 80 Understanding and Prediction of Ship Motions symposium. These comments apply in particular to the papers by Dr. Ogilvie, Dr. Ochi, and Drs. Breslin, Savitsky, and Tsakonas. The papers by Fuchs and MacCamy (1953) and by St. Denis and Pierson (1953) have been cited as initiating the study of the motions of ships in real waves, the first from the time domain viewpoint and the second from a spectral viewpoint. The first is often thought of as deterministic and as having little to do with the Gaussian properties of real waves. The second is thought of as highly dependent on the assumption of Gaussian behavior for the waves and on the principle of linear superposition. The first is nevertheless highly dependent on many of the same assump- tions of St. Denis and Pierson. Since waves are very nearly Gaussian, it is useful to get the spectra and cross spectra that describe the response of a ship to long crested waves in order to obtain an accurate time domain operator for the application of the procedures of Fuchs and MacCamy. Their model is just as linear and just as dependent on a linear hypothesis as that of St. Denis and Pierson. In actuality the work of Fuchs and MacCamy is much more restrictive than the work of St. Denis and Pierson, and much of the work described at this sym- posium is too restrictive for direct application to real ships in real waves. The work of Fuchs and MacCamy is strictly applicable only to ships in long crested waves. Long crested waves are an abstraction not met in nature. The work of St. Denis and Pierson, and its completion so as to include co- and quadrature spectra, by Pierson (1957) is applicable to actual ships in actual waves and pro- vides valuable guidance in the study of ships and other floating objects in real waves. Studies such as those of Canham, Cartwright, Goodrich, and Hogben (1962) and O'Brien and Muga (1965) show the value of spectral and cross-spectral analysis. More can be done in a full utilization of these results, however. The concept of linearity invoked by St. Denis and Pierson is not as essential to their theory as it seemed at the time although even for such extreme condi- tions as slamming, the theory yields useful results as in the work of Tick (1958) and in the paper of this symposium by Dr. Ochi. Recent work has extended the linear model of the seaway to a number of nonlinear models, and for specific problems nonlinear models concerning waves and the effect of waves on ships and some other objects have been developed. SHIPS IN A SHORT CRESTED SEAWAY If 7(x,y,t) is the sea surface, and if S(#,¢) is the variance spectrum of the waves, one can write that TCS5 Me 2) = i | cos ee cos 0 + ysin@) - at + <(~,0)] V 2S(@, 0) dwd@. (1) 0 TTI Consider a point moving in the negative x direction with the velocity, v. The coordinates of this point are given by Eq. (2) as x,,y,. _ 221-249 O- 66-7 81 Ogilvie x = x + vt (2) Vows Ova s Such a coordinate system moving with this velocity would record or see a sea surface given by Eq. (3) as a function of position and time with reference to the moving origin. iki iad 2 2 WC Xa View =) -f{ cos [Cx cs ae ie sin 8) - (2 -<-veos a) t~e| V 2S(@, 8) dwdé . Oa (3) For region I of St. Denis and Pierson, the spectrum, S(w,@) becomes the spectrum of encounter as given by 1 -/1-4a.v cos 60/ eeiasleee 2s cel ee ae 2w. v cos @./g (Ae rece ea ea Ae Se (4) Vl a A@v cos (G: /g and the seaway of encounter is given by Te CRANE)» = it ‘i cos {(o.t + €) - | (2(@9)) ‘/e| (Sg CosiOs Gaye sia a.) Region I x Dna upon yidande. & (5) The above steps can be repeated with appropriate modifications for regions II and III with the result that Nel Xe Vert) F Net(%erVet) 3 Net1(%erVet) a Netti( %e: Ve: t): (6) If the center of gravity of a ship is located at the point x, = y, = 0 Kq. (5) can be written as ence) = 2 Acosm(@ tea cy 2onqan, oa) Noes (7) where a partial sum is indicated as an approximation to (5). The important point to note at this stage of the derivation is that the same frequency of encoun- ter, »,, can result from many different waves coming from many directions, @ and can be associated with many different wavelengths. e? In practice for a given 6,, and w,,, given the wavelength, the region can be determined. One single term in the double partial sum of (7) is thus sensed as in Eq. (8) by a wave recorder located at the moving origin. Ne(t) = a cos (@,,tt«).- (8) 82 Understanding and Prediction of Ship Motions In Eq. (8) the other important parameter for the waves, that is, the direction of encounter, @., is no longer in evidence. If w, and @, and the region of definition for that portion of the spectrum are known then a particular wave, 7.,(t) can be associated with the motion of a ship as caused by this one sinusoidal wave. The concept of a transfer function as evaluated at a particular frequency for a particular direction is then valid and the response of the ship can be considered as a part in phase with the forcing waves or 180 degrees out of phase and a part in quadrature with the forcing waves having a phase of either 90° or 270° with the forcing waves. The func- tions, c,(@,,9.), 4,(@.:9,), and so on, can be determined either by means of experiments or by theories. The response to a single forcing wave is therefore given by Eqs. (9), (10), and (11) for heave, pitch, and roll. ZaCt) w= ac, (@.472 21) COs (Ot Fe) aq,(@.4) 9.4) sin (w,,t + €) (9) w(t) = ac y(.4)Fe1) Gos. (Gee ert aqy(@.1> 9.4) sin (w,,t + €) (10) P(t) = acy(@.,,0,,) cos (@,,t + €) + Aad 4(@.119—1) Sin (w,,tt €)- (11) The difference between short crested waves and long crested head seas is most striking here. In general, if ¢, is changed, c,, q,, cy, and q, may or may not change but they can change even for the same frequency of encounter. For example, one wave at +30° into the course of the vessel and another at -30° to the course of the vessel will look the same at the center of gravity of the vessel, but they will produce rolling motions that are 180° out of phase with each other. From Eqs. (9), (10), and (11) by means of the definition of the seaway of encounter given by Eq. (5), it is possible now to write down the vector Gaussian process that describes the time histories that would be recorded for the forcing seaway (at the center of gravity of the vessel if it could be observed there) and the heaving motion, the pitching motion, and the rolling motion. NAGE) = 22-cos (@,t + €) ¥28.(@,,9,) Ao,, NO. ZG \ =) => [cos (CBSE sr a) GO) ae Sin (CORE 2) GOs) 8.) V 28S..(@,,9,) Aw, AP, (12) Wie) = SD [cos (aki €) Cy, Gap san (@,t + €) Dy (Ge, é.)| V28..(@,,9,) Ae,, ANON $(t) = 22 [cos (wt + €) c4(o,,0,) + sin (w,t + €) dg(e> Fe)] V25.( 9.) Aw,, AO, - These double summation partial sums are to be evaluated over the same net in », and 6, for the same random phases. If there is any "phase relation- ship'' between the various motions and the forcing seaway these phase relation- ships will be preserved. However, by virtue of the remarks made above in con- nection with Eqs. (9), (10), and (11), it is not necessary for a particular phase relationship to manifest itself. Indeed, in general, there are cases where no 83 Ogilvie phase relationship exists and there are other cases where a phase relationship appears to exist. An ensemble of such vector processes as defined by Eqs. (12) can be gen- erated by choosing different sets of the ¢'s at random in a large number of partial sums. One can then compute ten different expected values of various time lagged products of different combinations of these motions. E [ye(e)a aCe + | ENlzce), z(t +7) E (tye | eC eee) Dineen etry BD fme) beer] 2 (mca), eCet a) EB (z(t), Y(t») E (z(t), d(t + 7) and E [y(t), d(t + 7) One of these expectations as evaluated is Eq. (13): E [ne z(t + T)| = IS@@ee) [Ex@nes) GOS Gur + GiL(@.5¢).)) Sim eT | dé... (13) The cospectrum between the forcing waves and heaving motion is thus given by Eq. (14) as this is the even part of the Fourier transform of Eq. (13). C.(a@) = [Ney 8.) 66a. 985) Ges (14) NZ The quadrature spectrum is given by Eq. (15). av = TNO pO) Gene ces: (15) It is to be noted that the cospectrum and the quadrature spectrum still in- volve an integration over ¢,. The way in which c,(#,,6,) and q,(#,,9,) vary as a function of 6, for a fixed », can evidently have a marked effect on the cross spectra. A complete analysis yields four spectra, six co-spectra, and six quadrature spectra. The spectrum for the heave, for example, is given by Eq. (16). S(2.) = [8(%.6.) | (e2(%e8e))? + (4:(20%)) | de. (16) The co-spectrum between heave and the waves and the quadrature spectrum be- tween heave and the waves are given by Eqs. (17) and (18). Clay) = ISa.,2) [e(@.9.)] dé, (17) @, (@,) = J8(e,.6,) [as(os.2)) de. (18) The cross spectra between heave and pitch are given by Eqs. (19) and (20). 84 Understanding and Prediction of Ship Motions Coy(.) = SS(@,5 96) [c,(w,, 94) e Conon) 4 8an (Gn) dy (er Fe)] dé. (19) Qiyl@e) = L8(%19e) [ex He1 Fe) CylHerFe) ~ Al erFe) Ay(erF_)] dA, - (20) The response amplitude operator defined by St. Denis and Pierson is seen to be given by Eq. (21) in terms of the spectrum for the heaving motion and the spectrum of the seaway of encounter. This is simply the square of the re- sponse of the vessel to a unit sine wave at a particular frequency of encounter and direction of encounter. TOo10,-5) = [ce (@e, 8) + [a,(,, 9.) : (21) The coherency for short crested waves takes on a new and essentially dif- ferent feature, however. Consider, for example, the coherency between the forcing waves and the heave. It is given by Eq. (22). = 2 = 2 : Viczeo here) (22) ne) = S.(@,) $,(@,) This can be rewritten in full as Eq. (23), and the top expression can be shown to satisfy the relationship given by Eq. (24). [S.(., .) C2(,,6.) dé, |? - [JS .(2,; 0.) F(,,6.) deals Se (23) JS.(0,,6,) dO, *fS.( 0,8.) (coer e)) 2+ (4,(@,,6,)) ‘| dé. ce ak ®.) Gster poe 2,0.) des) sisi) » THEORETICAL VALUES COMPUTED FROM THE SMOOTH SPECTRUM OF FIGURE | © THEORETICAL VALUES CONVOLVED WITH TRIANGULAR FUNCTION + 300 + 200 QUADRATURE SPECTRA h ny Fig. 2 - Cross spectra obtained by varous procedures 90 Understanding and Prediction of Ship Motions 1.4 @ COMPUTED FROM OBSERVED VECTOR PROCESS © COMPUTED FROM OPEN CIRCLES OF FIGURE 2 x COMPUTED FROM CONVOLVED SPECTRUM OF FIGURE | (CROSS SPECTRUM NOT SHOWN IN 2) SAUL SSU Er aero MONN AIM M2 nISh Nia) 1S eG milli n Sian IOms20 h, he Fig. 3 - Coherencies obtained by various procedures the cross spectra given in Eqs. (28) and (29) when substituted in the equation for the computation of coherency should yield one. The problem then is why are the computed coherencies so much lower than the theoretical coherencies ? To find out, the spectrum shown in Fig. 1 was smoothed and points were read from it at five times the spacing of the h values indicated on the horizontal scale of the figure. The smoothed higher resolution curve is shown by the dash- dot curve. Total variance was preserved with reference to the area under the smoothed curve. It was then possible to take Eqs. (28) and (29) and from them compute what the co-spectrum and the quadrature spectrum should have been. These values are shown by the small black dots in Fig. 2. By definition they would yield the coherency of one. Now the window through which the co-spectra are viewed is roughly trian- gular in shape, and if, for example, it were peaked at the value h=9 on one of these figures, it would fall approximately linearly to zero at the values h=7 and h=11. A linear combination of seventeen of the values given by the black dots thus represents the value given by a diamond. This operation is called a convolution and the results of a seventeen-point centered convolution with a linear growth to the middle value and a linear descent from the middle value is 91 Ogilvie shown by the open circles in Fig. 2. More points are available than were avail- able in the spectral estimates. The black circles are those points that corre- spond to the diamonds as far as the horizontal axis of the figure is concerned. This approximation to the spectral window yields values for the black dots that compare favorably with the values of the black diamonds that were obtained di- rectly from the spectral computations. The results show that both the shapes of the cross spectra and the location of the zeros in the cross spectra are lost due to poor resolution. The rapid variation in the values indicated by the black dots when convolved with a broad triangular weighting function results in a decrease of the peaks of the estimates for the cross spectra and shifts the zeros to erro- neous values. In Fig. 3 the coherencies as computed from the open circles are shown in order to compare them with the black diamonds. The loss of coherency caused by the shape of the window is confirmed. The above computations were based on the assumption that the spectrum as smoothed in Fig. 1 was the true spectrum. It is in fact, an estimate of the true spectrum that also has been convolved with a window of roughly the same shape. The original spectrum cannot be recovered and hence the computations do not completely describe the full effect. An attempt was made to determine what the effect of the convolution is by convolving the true spectral estimate once more and then computing the coherencies that would be obtained by using this new spectrum and the spectra and cross spectra obtained from the open circles and the filled circles of Fig. 2. The result of the computation is shown by the crosses in Fig. 3. Some of the rapid variations in the circles have been re- moved but the same general trend is evident. There are two ways to avoid the low coherencies that were obtained in this example. The first requires that a considerably longer record be obtained and that the resolution of the analysis be anywhere from five to ten times greater than that used in this example. The rule is, of course, that the convolution op- erator, which spreads over four frequency intervals, must operate on a portion of a curve that is slowly varying. The work of Dr. Yamanouchi is important in this connection. The computations illustrated by Pierson and Dalzell (1960) show that when the resolution is increased in such spectral and cross spectral estimates the peaks become much higher and the coherencies improve. A sec- ond procedure is to require that, for the same resolution, the estimates of the cross spectra be less rapidly varying. This can be achieved by a ''false"” time shift of one record with reference to the other. For example, the cross variance function obtained by computing all lag products of 7(x,t) and 7(x+L, t+7) for L fixed yields a function that has a maximum at a certain value of 7, Say, 7,. If the covariance function is then considered to be time shifted so that the value of +t, iS zero, one achieves a very nearly even function about this new time origin. The co-spectrum and quadrature spectrum computed from this new time origin in general have the property that the co-spectrum is quite large and the quadrature spectrum is near zero. The coherency between these cross spectra and the original spectrum is then quite large. Pierson and Dalzell have illustrated these ideas by studying both higher resolution spectra analyzed in the same way and by studying time-shifted covariance estimates. Coherencies that fell from 0.9 to 0.5 over a range of six or seven ordinate values were raised by these techniques to from 0.8 to 0.9 by higher resolution and to values greater than 0.9 by a time shift. 92 Understanding and Prediction of Ship Motions The high coherencies that should actually occur in an adequately resolved properly designed analysis of a vector Gaussian process associated with a long crested random seaway are reflected finally in the papers of the symposium in which the motion of the ship was predicted from the forcing waves. Accurate transfer functions are needed to construct the time domain operators for such predictions. These so-called "predictions" are not strictly predictions in that they use data from the future to a certain extent in order to compute the mo- tions of the ship. A true prediction would be one in which the forcing waves were known only up to a certain time, t = t, and the motions of the vessel in all degrees of freedom were known up to this same time. The problem would then be to predict the observed value of one of the motions at a time, say, 10 sec- onds into the future, based on just the amount of data available at t =t,. Itis evident that this true prediction problem can be solved more accurately for models in long crested waves. From a discussion in this section and from the results on short crested waves it should prove to be more difficult to predict the motions of an actual ship in actual waves 10 seconds into the future. Never- theless, the ability to do this is still needed, and an adequate investigation of this problem needs to be made. The past history of the motions of the vessel and the past history of the waves as observed at some point near the vessel can be combined to provide a prediction of the motion of the vessel at some future time. THE SOLUTION OF SPECIFIC PROBLEMS THAT ARE NOT LINEAR From different assumptions, a large number of linear models to describe ships in waves have been developed. The more advanced models may even be nonlinear in the beam parameter and still linear in the forcing wave systems. Nevertheless, roll and certain extreme motions will eventually have to be treated by nonlinear equations. A number of realistic problems have been formulated in wave theory and in ship motion theory that are not linear. These problems have been solved. The assumption of linearity by St. Denis and Pierson and by Pierson (1957) is no longer a restriction due to the lack of techniques for solving problems that are not linear. An interesting example of a procedure that does not get too deeply involved in the intransigent aspects of the subject is the analysis of the forces due to waves on a vertical piling as given by Pierson (1963) and by Pierson and Holmes (1965). Consider a pile in water with long crested waves passing it. The velocity field due to the waves in the water will exert a force on a small segment of the pile given by Eq. (31). f(t) = k,Uct)|Uct)| + k{UCt) . (31) Given the depth of the water, the spectrum of the waves, and the depth of the pile segment at which the force is measured, the spectra of Wt) and Wt) can 93 Ogilvie be found. Also the variance of U(t) and U(t) can be found and designated as y, and y,. Then the joint probability density function U and U can be given by 1 -U?/2y,-U?/2~, PARE : (32) P(U,U) = Given this equation, the probability density function for f can be derived. It is given by a: es th x a P( f)dt a e 8 e 2 ve be +e SZauyd ae ae cif e (33) an Veo 0 aa The time history of f is not given. The time history could be obtained by generating U and U as functions of time given the free surface 7(t) in a manner quite similar to that of some of the other papers in this symposium. The non- linear operation corresponding to U(t) |U(t)| could then be carried out in the time domain and the time history of the force on the pile could be constructed. This has in fact been done by Reid (1958). In this analysis, however, the only thing desired is knowledge about the probability density structure of f(t). This density structure can be obtained simply by reading off equally spaced values of this force and plotting the histogram of the values that are read. This probability density function as given by Eq. (33) has been compared with values obtained directly from measurements of the forces on an actual pile. Though there appear to be a number of parameters involved in Eq. (33) there are really only two, the second moment and the fourth moment, since P(f) is an even function. When these two parameters are determined from the data consisting of a twenty-minute long record of the fluctuations in this force, and used to con- struct P(f), the resulting probability density function agrees remarkably well with the observations. The probability density is not Gaussian; in fact, it pre- dicts probabilities about ten times those of the normal distribution, three stand- ard deviations from the origin. It is also interesting to comment that the computation of the bi-spectrum of f(t) would not have been particularly revealing because the bi-spectrum resolves the third moment of a distribution into frequency pairs. The third moment of this distribution is essentially zero. Work described by Tick (1963) has shown that it is possible to take a linear Gaussian model for the long crested seaway and construct the model that would satisfy the equations of motion in the Eulerian frame of reference to second order. One result is that there is a correction to the frequency spectrum. A second result is that the profile of the waves changes as a function of time at a fixed point. The crests become higher and sharper and the troughs become 94 Understanding and Prediction of Ship Motions shallower. The density function for the waves observed as a function of time at a fixed point will then have a certain amount of skewness that could be investi- gated in terms of bi-spectra. Wind waves have been carefully measured by Kinsman (1960) and found to be non-Gaussian. His data have been used by Longuet-Higgins (1963) to verify a theory for the probability structure of the waves and this probability structure has been adequately represented by a modified Gram-Charlier series. The mathematical techniques of Longuet-Higgins would be applicable to the study of the nonlinear aspects of certain ship motions. Another example of great interest to this symposium is the example pro- vided in the comments of Dr. Yamanouchi. He has solved the very complicated problem of the nonlinear damping of the rolling motion of a ship in irregular waves in terms of a random process and second order nonlinear correction to the motions. Just as the work of Dalzell was cited by Dr. Ogilvie as establishing the principle of linear superposition assumed by St. Denis and Pierson, some investigator now needs to study the rolling motion of a ship in long crested beam seas in order to see if it is possible to verify the nonlinear probabilistic theory of roll damping propounded by Dr. Yamanouchi. It is quite likely that this non- linear theory will verify quite well and that the spectra of the rolling motion as predicted by this theory will agree with the observations. Nonlinear roll in short crested waves requires very careful control of resolution, sampling variability, and coherency calculations in the analysis of the time series that would be re- corded. Still missing is the probability structure of the rolling motion. Per- haps the techniques of Longuet-Higgins (1963) could be applied here to obtain it. CONCLUDING REMARKS The essential feature of the work of St. Denis and Pierson now appears to be that of expressing the short crested waves and the resulting ship motions in terms of a probabilistic description instead of in terms of a deterministic one. The assumption of linearity so convenient in order to obtain results on the probability structure of the resulting ship motions is no longer absolutely es- sential toward the further understanding of the motions of ships at sea. Insofar as the actual waves that force the ship satisfy nonlinear differential equations, it should be possible to model these waves with these essential non-linear fea- tures as accurately as desired by means of continued efforts along the path out- lined by Tick in the work cited above. At the same time whenever it should turn out that the differential equations that describe a particular phenomenon are nonlinear, a perturbation technique such as the one described by Dr. Yamanouchi should make it possible to obtain the spectra of these motions and certain of the statistical properties of these motions. Even at times the probability density functions that provide considerable information about the phenomenon can be ob- tained by analogy to the results on the forces of waves on a pile. The strength of the techniques that have been developed quite obviously then lies in the as- sumption of randomness and the use of the very powerful tool of probability for the derivation of results and the very powerful tool of statistics in the analysis and interpretation of observations. 95 Ogilvie ACKNOWLEDGMENTS The work by the writer described in these comments was supported at var- ious times by the Office of Naval Research, Geophysics Branch, by the David Taylor Model Basin, and by the U.S. Naval Civil Engineering Laboratory. REFERENCES Canham, H. J. S., D. E. Cartwright, G. J. Goodrich, and N. Hogben (1962): Sea- keeping trials on OWS ''Weather Reporter.'' Trans. Roy. Institution of Naval Architects, London, v. 104, pp. 447-492. Cartwright, D. E., and N. D. Smith (1964): Buoy Techniques for Obtaining Di- rectional Wave Spectra. Trans. Buoy Technology Symposium, Marine Technology Soc., Washington, D.C. Fuchs, P. A., and R. C. MacCamy (1953): A linear theory of ship motion in ir- regular waves. Tech. Rept., Series 61, Issue 2, Inst. of Engineering, Univ. of California, July 1963. Kinsman, B. (1960): Surface waves at short fetches and low wind speeds - a field study. Tech. Rept. XIX, vols. 1, 2, and 3, ref. 60-1, Chesapeake Bay Institute, The Johns Hopkins Univ. Longuet-Higgins, M. S. (1963): The effect of non-linearities on statistical dis- tributions in the theory of sea waves. J. Fluid Mech., v. 17, Part 3. Longuet-Higgins, M. S., D. E. Cartwright, and N. D. Smith (1963): Observations of the directional spectrum of sea waves using the motions of a floating buoy. Ocean Wave Spectra, Prentice-Hall, Inc. O'Brien, J. T., and B. Muga (1965): Sea test of a spread moored landing craft. (To appear in the Proc. of the 9th Conference on Coastal Engineering, approx. Feb. 1965.) Pierson, W. J. (1957): On the phases of the motions of ships in confused seas. Tech. Rept. No. 9, contr. Nonr 285(17), New York University, College of Engineering, Research Div. Pierson, W. J. (1963): Methods for the time series analysis of water wave ef- fects on piles. U.S. Naval Civil Engineering Lab., Pt. Hueneme, Cal. Pierson, W. J., and J. F. Dalzell (1960): The apparent loss of coherency in vector Gaussian processes due to computational procedures with applica- tion to ship motions and random seas. Tech. Rept. contr. Nonr 285(17), New York University, College of Engineering, Research Division. 96 Understanding and Prediction of Ship Motions Pierson, W. J., and P. Holmes (1965): The force on a pile due to irregular waves. (Submitted for publication to ASCE, Journal Waterways and Harbor Division.) Reid, R. O. (1958): Correlation of water level variations with wave forces on a vertical pile for non-periodic waves. Proc. Sixth Conf. on Coastal Engi- neering, Gainesville, Florida (1957). St. Denis, M., and W. J. Pierson (1953): On the motions of ships in confused seas. Trans. Soc. of Naval Architects and Marine Engineers, v. 61, pp. 280-357. Tick, L. T. (1958): Certain probabilities associated with bow submergence and ship slamming in irregular seas. JSR SNAME, v. 2, no. 1, June (p. 30). Tick, L. J. (1959): A nonlinear random model of gravity waves. J. Math. and Mech., VIII, 643-652. Tick, L. J. (1963): Nonlinear probability models of ocean waves. Ocean Wave Spectra, Prentice-Hall, Inc. SOME REMARKS ON THE STATISTICAL ESTIMATION OF RESPONSE FUNCTIONS OF A SHIP Yasufumi Yamanouchi Ship Research Institute Tokyo, Japan INTRODUCTION In connection with the papers by Dr. Ogilvie, Prof. Lewis, Mr. Smith and Dr. Cummins, Drs. Breslin, Savitsky and Tsakonas, I would like to make several comments on three problems concerning the statistical estimation of the re- sponse functions of ship. The first problem concerns the statistical considera- tions that occur in estimating the frequency response function, which no paper has referred to in this symposium and which still has several difficulties to be solved. The second problem is about the impulse response of ship motion which has been one of the main topics in this symposium. The third problem is about the effect of nonlinearity on the response of motion, on the computation of spec- trum, and on the estimation of the frequency response itself. 221-249 O - 66 -8 97 Ogilvie 1. A PROCESSING SCHEME FOR THE ESTIMATION OF FREQUENCY RESPONSE FUNCTIONS Beginning with the work of Blackman and Tukey [1], many contributions have been made to the problem of the estimation of the statistical characteris- tics of time series. Most of them, however, treat mainly the estimation of the spectrum, and very few have been concerned with the frequency response itself. For the sake of establishing a standard procedure for obtaining the frequency response, our group did some work [2] that followed the results of Dr. Akaike and myself [3]. Skipping over the items that are already commonly clear in text books, several items closely related to the selection of parameters in the estimation of the frequency response function will be mentioned and examples for the case of ship oscillations will be given. Choice of m If m is the maximum number of lags in the correlation function used for the compu- tation of the estimates, then m should be chosen so as to Satisfy mAt > 27/B, where B is the bandwidth of the peak of |H(o)| 1H(w) | of main concern as defined by Fig. 1. An excessively large m implies an in- crease of sample variance, and an exces- sively small m implies that bias occurs and usually gives an underestimate of |H(«)|. Also, the amount of shift, mentioned later, should be taken into account in choosing m. Figure 1 For the case of ship rolling where $+ abt wo2d = wo2¢,, under the condition where «x << 1, the bandwidth of the peak of |H(#)| is calcu- lated as B = 2xw,. The Effect of Windows Before starting to discuss the choice of the amount of shift, the effect of various windows is examined approximately. 98 Understanding and Prediction of Ship Motions Assume that the output y(t) and input x(t), as well as the noise nt) that contaminates the output y(t), have the Fourier transform. Then 217 QT 27 QT a) . a(S 4) eo +N (Su) - Accordingly, the estimate of H() at w = (27/2T)y evaluated from the cross spectrum is =Wwy -X A (ZF 2) br Aire eee QT 2 WSK oe : ii = : = : 2W,H {= (u- »)} KT Sere : =W, Hy okra =W, Xo, Te WX ee xe where W, are the weight factors that describe various windows. The noise and the input can be considered as mutually uncorrelated. Therefore 277 = e [n(3a )| = £ EW, HS (u- yf Mo ee Die psec SWEX x Vv [b-v p-v o) Petia encom laga al aF Sex (3 #) EW Hop CH} Sy (rH) fe as Here the variation of S,,(#) around » = (27/2T)y is assumed to be smaller than that of H(w). Accordingly eee al) sen Boneh. Now let us consider the case where H(w) in the form |H(w)|e}7‘*) has the local approximation 2m w {co 9} =H Bes) (tea (Bee) + 0 GE)". where k, = - do(#)/dw as in Fig. 2. We have assumed that |H(~)| varies locally asa 2nd. order curve, and that the phase o(w) increases linearly with frequency. Then, 99 Ogilvie ow) ieee Figure 2 w 27 at 2a mie aT = 27 2 PA 27 —vk j—vk 2 j—vk ve Zi Sn (Ze OTH (Ft) me te (F») aT EW HY Sr (uv) = u(t ) {=M, « Ee, ar) e HB aM, ar) & 7 = H (221) {w(k,) + ay WK) i joes ick ,)} where w(k ,) = = w(7)| wk.) | = A 4 wy T=k be This shows, through the averaging effects of the weights, W,, that the estimated frequency response is much affected by the variation of phase angle do(w)/dw around that particular frequency. This fact was entirely disregarded in the paper by Goodman [4], which treated the confidence limit of the frequency re- sponse. For example, if the k ts has values that are shown in Fig. 3, even if on= Oh and 76) ==0 bt (Ea)] ~ eum can v} = Ba) wen and w(k,) is much less than 1. As the result, the estimated frequency response becomes much less than the true value. This has been the reason for the low coherencies obtained in some experiments as already pointed out by Mr. Dalzell and me [5]. As an extreme case, when ele 100 _ Understanding and Prediction of Ship Motions Figure 3 =W.H 27 (u- »)} 29. This window also causes bias in phase estimation. Generally, however, £ is much larger than a, when |H(w)| has a peak, and then do(w)/d» usually has a large value. Accordingly the effect of (a/27)wk,) is rather small compared to that of w( k,) and [@/(27) *)w(k ,) . This shows that the effect of a window on the phase shift, o(w), is rather small compared to the effect on the amplitude gain, |H(@) |. If we shift the data window by Ke and bring the origin of the window to k,, then w(r) = 0 and w(t) = 0. Moreover if we can choose the shape of the window so as to have w(r) = 0, then, as the result, the bias due to phase shift can be eliminated completely. As the natural results of the above-mentioned considerations, the following results are obtained. In these results sAt is the amount of shift of the data window for the computation of the sample cross spectra. Choice of sAt To compensate for the bias due to the phase shift, sAt should be chosen so that we have 0.15 mAt W, sAt - {- so} | < < 0.30 mAt + for W, dw 0.40 mAt W, namely if 0.15 mAt W, [3 See ONS fe ora dw 0.40 mAt Wee 101 Ogilvie we do not need to shift the data window or the output. W,, W, and W, are the windows which are explained later. Otherwise, |H(~)| will be underestimated by more than 5%. sAt should also be chosen so as to satisfy lsAt | < 0205 NAG . Otherwise, adopt y(t+sAt) as y(t). This is the clear solution for the elimina- tion of the effect of windows on frequency response function. For example for roll, expressed by the equation of motion, cc : 2 je p+ Wkwg+ woh = og Ka) = b(w) ane where y(w) is the effective wave slope coefficient and 2 7 er C(w) e’? is the wave slope expressed by the wave height ¢(w) measured at the C.G. of the ship. The amount of time shift is computed as follows: ad (ya?/g)? (w? - aw?) + (2kw,w) ? —2KwW_W aa) = Gam™ ( 2 )+4 @ (eo) |H(w)|? = do() d ‘ ~2KWw i" ~2Kw,( a? + w*) = da tan Z PRiGGar 1h Lan ne da (a2 - w*)? + ( 2K ,w) ? therefore 1 K@ W=W fo) ° do( w) dw iH Namely if we want a good estimate (with small bias) of H(«), the amount of shift sAt is 1 K@, ae = MEASURING POINT This shows that the output of roll y(t + sAt) should be taken as y(t). When the wave height C(w) is measured at the distance, D, as is shown in Fig. 4, the phase difference between 102 Understanding and Prediction of Ship Motions the measured wave height and the wave slope at the C.G. of ship is 7/2 -(w?/g)D, and this gives - 2Kw, w a 2 e(ie)) = tan~! {—————_] + aasaeo ost and therefore, m do(w) SAt = ae W=W Oo B o When the character of the frequency response, H(), is known before we start the computation, the above-mentioned value can be estimated. This is, however, not the case usually. At that time, as is shown in Fig. 5, T,, can be taken as a good estimate of sAt. This value can be decided after computing the cross covariance function. Namely, in the case where the input is considered to be fairly white in the range of our concern, we can obtain a fairly good overall estimate of H(w) by shifting the center of the lag window or the origin of the time axis of the cross covariance function to that time point T_ where the maxi- mum of the absolute value of the sample cross covariance function occurs. Pierson and Dalzell [6] showed very interesting analyses for two cases, one for wave measurements where the apparent low coherency between two wave meas- urements was improved and another in which the coherency between the waves and the ship response was improved. I have also described [7] the intuitive and practical method to find the amount of shift. The above-mentioned theory shows more generally the way to find the proper amount of shift. A large bias because of the large phase shift which usually occurs at the peak of the amplitude gain, induced by the use of windows, is prevented in this way. Ryx(T) 103 Ogilvie w, W,: Windows Several papers have been written on the design of windows so as to provide spectral estimates with various properties. Hamming and the hanning windows are the most popular ones commonly used. Just a few comments will be made here on this subject. Putting X as the Fourier transform of x(t) (-T a fe) Figure 6 Many windows have been designed [8], and have been checked [3] for the case of simple oscillations like linear rolling in waves. The following windows, W,, W,, W,, which have been designed, are the optimum as the 1st, 2nd and the 4th order type and are free from biases up to the Ist, 2nd and the 4th order respectively. 104 Understanding and Prediction of Ship Motions Raed sok. 1, [esate As an example, these windows can be applied to the estimation of the frequency response of a simple oscillation such as linear rolling. 2 () MPS TI CA SENS (M5 —~@°) + QjKa,o W,, W,, W, have been checked to be adequate for use if is kept less than 1. (Usually this is satisfied if mAt > 27/B, where Ba elke, eo = == « The window Q, which is a modification of W,, is generally recommended for the estimation of the power spectra, cross spectra and of the frequency response function of a linear time-invariant system. If the very careful estimation is necessary, the following is recommended. Apply the windows W,, W, and W, successively, and if there is a significant dif- ference, say, of order greater than 10% between the results, it is advisable to repeat the whole computing process using 2m in place of m, and, at the same time, use a correspondingly increased N, if necessary. W, tends to produce the deepest troughs and highest peaks, W, the next deepest and highest, and W, the shallowest. R(w): RELATIVE ERROR OF H(e#); CONFIDENCE BAND The relative error of the estimation of H(«), obtained by the procedure mentioned above was evaluated relative to the estimated value of coherency. The results are as follows: Assuming that ¥(@) = X(@) H(w) + N(w) Syy(@) = S,,/H(#)|7 + S..(@), 105 Ogilvie |H(@) | : 3xx(@) Snn() Coherency y2(w) = ——<———— ? 1 - = Syy() Syy() Then 1 n- 1 y?(w) - 1) F[5, 2, 2(n- 1)] where n is the integer nearest to also F[5, 2, 2(n- 1)] is defined as Prob. dEso44) < Pls, 2200-1) ]p =0 8) At » = 0, and 27/2At,F[8, 2, 2(n- 1)] should be replaced by F[5, 1, (n-1)]. Ac- cordingly, the confidence band is drawn as Prob. {|H(w) - H(#)| < R(w) |Ai(o)| and | Arg [H(«)] - o(o) | ~ sie! R(«)} > 30 R(w) should be put equal to 1.00 to indicate the relative error greater than 100%, when the value inside the square root of the definition of R(w) is greater than 1 or less than 0. The value 5 = 0.95 is used usually in our group. When R(~) had the value, 1.00, the estimate of o(w) often showed sudden change of magnitude +7, Which shows unreliability of the results. The following approximation formula for F-values can be conveniently used for computer application: 10.00 F(2, n, 0.95) = 3.00 + SoA is 10.006 F(1, n, 0.95) = SR cern Try . Figures 7 through 14 are the examples of the results obtained by this pro- cedure. The cross correlograms of wave-roll and wave-heave in Figs. 7 and 8 are the ones where the origin was shifted by 9At and 7At respectively. Corre- lations were computed to very large lag number +200, however, m, the largest lag number was taken as 90 in the analysis. All windows W,, W, and W, were applied to the calculation of the spectra; however, the results came out so close that it is difficult to show on one sheet. Accordingly, in Figs. 7 through 13, only 106 Understanding and Prediction of Ship Motions MAAR AAR AN Ra AAA ARR AAS iil qe 1104 O'1 WV¥9013Y4NOd O- i Ht 08! O91 Ob Oz! O01 08 09 oF oz. Os Whe p> WS elo om (rio (sie (fi RR. pooh gh, on dh ry A ct A a “aN oh Do Prckr Po £3 oo FAS RRA S Po t% i | 1g R r ! qh 9 f A Ph s oh Po tf dy JY Ph, \ ‘Be! TO fi PPP ra eh MENT PRA ANE areca (6 L4IHS) TI0N-3AWM O11 O1- {= ‘ os! O91 Ob Oz! oo' og o9 I i 402- Ob- 09- = TR eroonvannnndiintana An aaRRnd 107 Ogilvie CORRELOGRAM 10 WAVE-HEAVE mm? +sec N =698 160, Power of wave m = 90: | ; sat ig 0 Be a, =0.2434 tip 100;- 80}- 60}- 40+ 20/- nT 2.0 3.0 40 5.0 60 7.0 8.0 90 100 =a) ae CD sec Figure 9 108 Understanding and Prediction of Ship Motions Power of roll N =698 m = 90 deg”. sec W, 30 (a Ziad a, =0.2434 25 20 15 10 0-10 20 3.0 40 5.0 6.0 7.0 80 9.0 10.0 —- W sec Figure 10 Power of heave N =698 m= 90 mm*-sec W, 160 [ee mae] @, =0.2434 140}- 120 100 80 60 40 20 — Piel hese Reo pee, Ae 0 1:0 2.0 3.0 40 50 60 7.0 8.0 9.0 10.0 =e) sec | Figure ll 109 Ogilvie Coherency N =698 m = 90 WwW, a, =0.5132 a, =0.2434 % 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 100 ———) (f)) sec Figure 12 Coherency Figure 13 110 Understanding and Prediction of Ship Motions Relative error 0.8) \ Coherency Phase shift —2x oN =698 m= 90 W, a, = 0.6398 a, =0.2401 a, = — 0.0600 Amplitude gain “ Fourier Inversion of Impulse Response 4 5 6 Response of roll to wave height ao ee, Figure 14 the results obtained by using W, are shown here as examples. The amplitude gain in roll was calculated as the ratio of the roll angle to the wave slope in Fig. 12, from the original results of gain of roll angle to wave height as is shown in Fig. 14, where the result obtained by using W, is shown as an example. The relative error is shown in Fig. 14. 2. ON THE EVALUATION OF THE IMPULSE RESPONSE FUNCTION The impulse response function is one of the forms by which the character of the response of a linear system is expressed completely. This has long been used very conveniently in many engineering fields. However, it has been rather unfamiliar to naval architects. Fuchs and MacCammy [9] wrote a paper in 1953 and made it clear that the time history of the heave and pitch can be synthesized by the convolution of impulse response and the time history of the waves. They computed the impulse response function theoretically for a cylinder, and as the Fourier inversion of the frequency response function obtained from tank experi- ments for a ship form. The present author [10] has already called attention 111 Ogilvie to this function and has also made clear the way to get the impulse function from a free damping test. Mr. Smith and Dr. Cummins insisted in their paper that the step compulsory input is not applicable to get the impulse response function, because the step function includes components of very wide range of frequency, theoretically from zero to infinitive, and this incurs the noise which comes from the reso- nance of model itself, guide, restraining frame or others to high frequency component of input. However, if we are careful on a few points, this author be- lieves, we can obtain the impulse response even from the inclining test, espe- cially if the response has very low natural frequency as that of rolling. This author obtained successfully the very complicated frequency response function of a ship with Flume type anti-rolling tank as the Fourier inversion of the im- pulse response obtained by free damping test. The results show that this method is a useful way to guess the frequency response just by a simple free oscillation test. The impulse response function is also useful in this example, and here some topics related to the statistical estimation of the impulse response will be described. When the system is linear and time invariant, {oo} y(t) = i h(7) x(t=7) dy = \ h(t-7) x(7T) dt = h(t) * x(t) where * represents the convolution operation. The impulse response function h(r) and the frequency response function H(~) are related to each other by Fourier transformation as H(w) = | h(T) eI °Tdr , hén) = =) H(«) ef ?7dw. The Fourier transforms of output and input are connected by the frequency re- sponse H(~) as Y(w) = H(@) - X() . Through manipulation, we will get ok) = I | h(v) h(v) R,,(7- w+ v) dvdu = h&r) * A(T) erik (4P)) O° This corresponds to the relation in frequency domain Syy() = H(a) He) S,,(0) = |H(a)|* S,.(@) - In the same way 112 Understanding and Prediction of Ship Motions JS ACT) | h(#) R,(7- #) du = i h(7- #) Ry,(H) du =) h(7 Ree Ch)y- This corresponds to Syx() = Hie) + S,,(@) . Namely R,,(7) is connected with R, (7) by the impulse response function h(7) just as y(t) with x(t). Asis very clear, the relation between R, (7) andR,,(7) is much stabler from the statistical point of view than that between y(t) and x(t). When the computation is carried out digitally from the sample of data taken at interval At, Rm) = DaliG)\ REA Cae)e Nee be Putting At as 1 for the purpose of simplification Ri) = Sw) Ras oS ph The impulse response can be obtained discretely as the form of a weight func- tion, for example as h_,, h_.,,,---h,,---h,_,, h,, by solving the simulta- neous equations e() R,,(0) R,,(1) s cilRean(m)jy wees! Re(Ba) ia (ou al) Ree Gh) R, (0) Soule yoas Reams) lines R,,(0) R,,(n) Ro Gus sh) io (0) »» -Ry(n) In\e Reed) RE ANS TD) LS (OND WSR (opel Cy) bane, Rea) R,,(2n) Rea(Zne ID) secltue(@)y\ = se ctteed() hy Figure 15 shows the impulse response function obtained by solving a 59th order simultaneous equations from R,, at 7 = -29~ +429 and R,, at 7 =0 ~ 58 of roll and wave height correlation, shown in Fig. 7. In the same figure, the impulse response function, calculated as the Fourier inversion of H(w) that was obtained through the spectrum analysis as in Fig. 14, is drawn. The latter was calculated aS h_4, ~ hg,, and the figure shows the main part of it. The two weight functions are not necessarily the same. To our surprise, however, inversely, the frequency 221-249 O - 66-9 113 Ogilvie Impulse Response 401 From Ry:(t) =A(r) «R,,(7) Fourier Trans of H(w) Figure 15 response function obtained as the Fourier transform of this impulse response function calculated through the simultaneous equation shows just about the same values as the frequency response (amplitude gain) obtained from the results of spectrum analysis as is demonstrated in Fig. 14. The example of synthesis of y(t) from the history of x(t) using this im- pulse response functions is shown in Fig. 16 which shows pretty good agreement with the actual observation of y(t), whichever impulse response function is used. Attention should be paid on the fact, however, that an analysis in the time domain by means of RD) and R,,(7) is inferior to the analysis in the frequency domain in the following reasons: 1. The choices of At, m, and N are difficult from the statistical point of view. This makes it difficult to decide on the really important part of the cor- relogram to be used for analysis as that part must include enough information. 2. The evaluation of error is not easy as in the frequency response func- tion. In the frequency domain, the coherency plays a big role, and makes the estimation of relative error practicable. We have to be very careful because of the above-mentioned defects. After these points have been made clear once, however, we can utilize the method to obtain the impulse response function from the correlations directly and use that to predict the future response. For this purpose, some special type computer 114 Understanding and Prediction of Ship Motions lone Loa DG a IP aL i a Wave Elevation observed Time - Roll ek: 8°P RAP AL Heo VAN Ti - ca, Coen ae ~T0see fn A | ihe \ |\ Jaan Synthesized oe . 8 °p Pe —20™n x by Fourier trans ; H (@), oby h;from R,,=h; *R,, viv AMA Aha i POO Eye ee ete 2 a ee ee eer eee reer ed gee el) Figure 16 such as an analogue computer can be used in addition to the general purpose computer. 3. ON THE EFFECTS OF NON-LINEAR DAMPING ON CALCULATION OF THE SPECTRUM [11] When the irregular input, for example the sea waves for the ship, can be considered as a Gaussian process, the output of a linear system, such as the linear oscillation of ship, is also a Gaussian process and can be expressed using the spectral expression. | lice dZ(w) WAC Ey i E[dZ(@) dZ(w')] = dS ,y() S(w- w') S,(2) = [H(o)|" S,,(2) . When a non-linear component is included in the response character, the output is no longer Gaussian, and cannot be expressed by the spectral form which premises the superposition theory. Even for that kind of case, the autocorrelation function as 115 Ogilvie T Ryy(7) = Ely(t+7) ye) 2 +| y(t +7) ye) dt = 1 can be calculated. Accordingly if we adopt the definition of a spectrum in a wide sense as the Fourier transform of the correlation function, we can com- pute the spectrum. Here a trial has been made to show how the non-linear element — here non-linear damping as described by velocity square damping has been considered — affects the computed spectrum, using an approximation method. For example, for rolling, the equation of motion with velocity square damping is Id + No + Ni GId] + Kid = M(a) eit, (3.1) Now for the purpose of simplicity, all coefficients on the left-hand side of this equation are considered to be constant and do not vary with frequency. Then for input of a general irregular moment, the equation is b+ 2ab + Bbld| + w2h = g(t). (3.2) As the zero order approximation ¢,, the solution of the linear equation without velocity square damping is taken as db, + 2ap, + OI@. | = fy(te) 6 (3.3) Then QO. = | Ce 7) ae) clr = i a ®) Gi( ie = 9) Clr (3.4) h,(7), being the unit impulse response function of this linear system to the compulsory moment g. h,(r) is decided from Eq. (3.3) and, of course, is con- nected to the frequency response function H(w) by a Fourier ‘transformation and its inverse. Here Kq. (3.2) is modified and the compulsory force g(t) is assumed to change to {g(t) - 6¢|¢|}. On substitution of ¢, into this ¢, the 1st order ap- proximation is taken as the solution of the equation $, + 20d, + w2¢, = w(t) - Bbol ey). (3.5) The left-hand side of this equation is just the same as that of Eq. (3.3). There- fore using the same response In 4) g Cb) = | ne = Ww) eG) = Bele) 2 leaueyii cw = | h(t - 2) e(u) dy | h(t-~) {¢,(4) Al} du = $(t) - o4(t), (3.6) 116 Understanding and Prediction of Ship Motions ¢,(t) being the modification term and is ede) = | ho(t- 2) {6,(4)-lécu) |} du. (3.7) Of course the convergence of this approximation should be certified strictly at first; however, here the convergence is assumed as far as the linear damping a/w, is rather large and the non-linear damping £ is pretty small. The autocorrelation function of this 1st approximation ¢,(t) is then, Ryo, = Eld(t+7)-4,(t)] = E[A(tt7) - o(t+7))] [d,(t)-d1(t)] SEBO rence Elesces 7) exces) = Blot +7) o,(e)) + Ele(t+ 7) o(t)] = Bit) OC) = AWE leew) Bie] = Wiens a) exCey|. (ee) Accordingly the spectrum Ss ed («) of the 1st approximation, ¢,(t), is 1 1 ‘ - jor 7 Pee = Sag Co) al e 7 QRE[P(t+7) H(t] dt to { ef" Edict +7) - dict] dr. (3.9) - @ The substitution of Eq. (3.7) into this equation yields = Sy 5 Oe Bak I, i. ha(t- a) e 507 Blo (t+ 7) -b,(4)16,(4) |] dude +a | | | ine tear = Bb) Joye = 2) x e dT Eld (uw) 16,(4)1-°$,(7) 1650) |] dudy dr foe} K, mek SNE = 2a. ge) = i h,(t — 4) eet e) au | - @ (3.10) (Cont.) NL’ Ogilvie x BP IOC HERB) E[¢,(t + 7) $,(4)°16,(2) ] dr (oo) Fe | ise te 7 = (8) eden er) ar | DAC — 2) eIeCsre) av | xe SeCH-¥) BL (u)-1d,(0) 1-407) 14,07) |] du = Sp9,(@) ~ 2K [SH,(4) °S, (o)| oreo el + BLO Spo Sie Cae (3.10) $5°|¢0|+45°| 40! Now g(t) is the compulsory moment that comes from the waves, and, so, if the waves are Gaussian, g(t) is also a Gaussian process. From Eq. (3.4), $,(t) is also Gaussian, and $,(t) = | h(t -'s) 2(s) ds . (3.11) This shows ¢,(t) is also a Gaussian process. Here in order to calculate (3.10), we have to evaluate the. expected value of [,(a) -¢,(b) > 1é,(b) |] and [¢,(~) - 16,(4)| -4,() - |é,(~)|] concerning two Gaussian process ¢,(t) and ¢,(t). #,(t) and ¢,(t) are correlated to each other, of course by the correlation coefficient pe. Here, two Gaussian variables ¢ and 7, connected by the correlation coef- ficient o are assumed. The two-variable Gaussian probability distribution function is ote = 20-0 En) + a? ne | (3.12) ano ,0, J1- p? eee ACil= pe") It is necessary to evaluate E[é, 7 -|7|] and E[é-|é| -7-|7|] by means of this distribution function. The absolute values are what make the problem somewhat intricate. These evaluations could not be found in any text books and papers, and so were calculated by this author. The results are E[é, 7. |7]| = [2 co, 22 = ie pee (3.13) 118 Understanding and Prediction of Ship Motions Ia 20 ? - Cag 2 2 -1 E(é-|é|-n-Inl] = f= )p2) x18 + 2(1 4.27) tan joe E-|El-n-|n Sen TS where = 1 we 1 = =o, ee Noe) + 3 a0, Rz,(7) + ist os Rz,,(7) + p= Re, oo, From these results and the relations he) = (-j@) Sea @) Se @) ts (®) , PoPo CaP a the result is - O W) = Mees er HN 2 f Sng amees iN oy tale n Saag es Accordingly the 2nd term of the Eq. (3.10) is where RHE) Ste tea Co) = 2 /20, eee (0 (oO RE jo H()} $5°%5|%o| /2 2 =9 7g BSe 6 (% xa 4( ”) 3 go) = Q{H(#)} = - 119 (3.14) (15) (3.16) (3.17) (3.18) (3519) Ogilvie Also 1 5 Ret Sqr ol We F re ee) (3.20) The 3rd term of the Eq. (3.10) is the Fourier transform of the functions that is the product of Eq. (3.20) and some other function. The following relation is used, in this calculation: ~ R(7) Bigs ein es S(o) , (Pp) = i S() et qa (3.21) - @ Ble By cher. ot llaley er a eee a ie R°(T)e alr = eal e Rr | S(@ je dw dr = al haaugs ye R(T) | ee ise at | S(" yaa dr -| sco") | S(o") eal era a a Rm Ee dos” Z ic S(w') i S(w") S(w-w'-w") dw'da" . (3.22) Similarly val R5(7) @ dr = iF S(@,) (e S(,) i S(@5) ir S(@,) x S(@- w,- @,- 0, - @,) do, dw, dw, da, . (3.23) However, because of the small value of the coefficient of R°(7), Eq. (3.23) was omitted in further analysis. Then, the 3rd term of Eq. (3.10) is 120 Understanding and Prediction of Ship Motions B°|H,(~)|° 4 foe (@) + el tee sts oe OMS; 4 (a8; ; (o-o,- ay do,de,] Eye H()|7|--22 28, , (a + 5) e g as bob 302 , (3.24) where S(e) = ie ig Serie . ee a ae @,- w,)dw, dw, . (3.25) By the substitution of Eqs. (3.19) and (3.24) into Eq. (3.10) and from (a) 1 IO( @) H = = =" PH ; ge ®) G(w) ee w? + Qjaw M(@) 0 @) 2aw 1{@) 2 = =< = ORL). (a2 - 0)” + 4070? and also from 2 2 So.9(2) = HCl? SpCoy = {Hoyt at rah Sp2(2) we = {H,(2) So? We} Spp(a) 2 Wo) w2 os a alk x iL [Ar(@)] 4 0 Se) KO (3.26) {(o? Baye 42a} 4 @ G = 2| S. . (w) dw $ A ease fo} the spectrum of the 1st order approximation ¢,(t) can be calculated. As an example, the spectrum of the non-linear rolling of a model with a, = 3.85 and a/w, = x = 0.06705, 6 = 0.08 has been calculated. For the waves, the Neumann shape spectrum which has its peak around «, = 3.85 was used. In these kinds of computation which include the convolution of a spectrum, the spec- trum should be defined in the range of -~ ~ + of w, and accordingly the definition 121 Ogilvie which is usually used by oceanographers that define the spectrum only on 0 ~+m range of w is inconvenient. The spectrum of the waves, |H,(«)|?, S4.¢.2 Sig, and also the convolution S are shown in Figs. 17, 18 and 19. From Fig. 19, we can see the character of single and double convolutions. In the single convolution, a large power appears at w = 0 and at the frequency twice of that of the peak. In the double convolution, the large power appears again at the frequency of the peak of the original spectrum, and at the triple of that frequency. Because of the large decay of |H()|? at these high frequencies, the latter peak did not modify the spectrum. From these results, S3,4, was ob- tained as in Fig. 20. This is a pretty reasonable example of the effect of non- linear damping, which reduced the height of the peak that appeared in the spec- trum of zero order approximation. There is the possibility that when the linear emé@ (SEC) SP blu) RAD@ (SEC) SHb(w) RAD2 (SEC) Fig. 17 - Spectra of wave, roll, and roll angular velocity 122 Understanding and Prediction of Ship Motions 180 000000000000 C00 150 120 90 (DEGREE) 60 30 0.014 Sao sec) Fig. 18 - Linear response of roll to the compulsory moment 123 Ogilvie RAD9/sec5 CONVOLUTION OF SPECTRUM, S(w) Figure 19 e LINEAR S®#(w) © NON-LINEAR (+8¢7) S49 (w) (ste) Fig. 20 - Computed spectrum of non-linear rolling 124 Understanding and Prediction of Ship Motions damping a/w = x is very large and H(~) does not decay so much at 3w,, that a small peak will appear around 3a,, when the velocity square damping exists. NOMENCLATURE (22) circular frequency = 27f, f = frequency, time, time interval between adjacent data values (sampling time interval), input to the system, assumed to be a weakly stationary stochastic process, output of the system, under the input x(t) and usually contaminated with noise n(t), x(nAt), y(nAt), frequency response function of the system, when the system is linear and time-invariant; otherwise, that of the corresponding linearized system, amplitude gain, Arg {H(~)}, phase shift defined by H(#) = |H()| exp {jo(w)} (j?=-1), impulse response function of the system when the system is linear and time-invariant, maximum number of lags of correlation computed, T,, = mAt, number of data used, NAt = total length of observation, covariance function, power spectrum function. REFERENCES 1. Blackmann, R. B. and Tukey, John W., "The measurement of power spectra from the point of view of communications engineering,'' Part I and II, Bell System Tech. Jour., Vol. XXXVII, No. 1, Jan. 1958 and No. 2, March 1958. 2. Akaike, Hirotugu; Yamanouchi, Yasufumi; Kawashima, Rihei and others: "Studies on the statistical estimation of frequency response function,"' An- nals of the Institute of Statistical Mathematics Supplement III, 1964. 125 10. alae Ogilvie . Akaike, Hirotugu and Yamanouchi, Yasufumi, ''On the statistical estimation of frequency response function, ''Annals of the Institute of Statistical Math- ematics, Vol. XIV, No. 1. Goodman, N. R., ''On the joint estimation of the spectra, cospectrum and quadrature spectrum of a two dimensional stationary Gaussian processes," Scientific Paper No. 10, Engineering Statistics Laboratory, New York Univ., March 1957. . Dalzell, John F. and Yamanouchi, Yasufumi, ''The analysis of model test results in irregular head seas to determine the amplitude and phase rela- tions of motions to the waves,'' ETT Report 708, Stevens Institute of Tech- nology, 1958. Pierson, Willard J., Jr. and Dalzell, John F., ''The apparent loss of coher- ency in vector Gaussian process due to computational procedures with ap- plication to ship motions and random seas,'' Tech. Report, College of Engi- neering, Research Division, New York Univ., Sept. 1960. Yamanouchi, Yasufumi, ''On the analysis of the ship oscillations among waves, Part II,'' Journ. of the Society of Naval Architects of Japan, Vol. 110, Dec. 1961. Akaike, Hirotugu, ''On the design of lag window for the estimation of spec- tra,'' Annals of the Institute of Statistical Mathematics, Vol. XIV, No. 1. Fuchs, R. A. and MacCammy, "'A linear theory of ship motion in irregular waves,"' Tech. Report Series 61, Issue 2, Institute of Engineering Research, Wave Research Laboratory, Univ. of California, July 1953. Yamanouchi, Yasufumi, ''On the analysis of ship oscillations among waves, Part I,"" Journ. of the Society of Naval Architects of Japan, Vol. 109, June 1961. Yamanouchi, Yasufumi, "On the effect of non-linear damping on the calcu- lation of response spectrum," 24th general meeting of the Transportation Technical Research Institute, 1962. * * * 126 Understanding and Prediction of Ship Motions RESPONSE TO COMMENTS BY WILLARD J. PIERSON, JR. T. Francis Ogilvie David Taylor Model Basin Washington, D.C. I appreciate Professor Pierson's comments very much. Since he was co- author of one of the most important papers ever written on the subject of ship motions, any worker in our field should listen carefully when he enters the discussion. It is rather difficult to reply to his formal discussion, since his comments generally refer to what I did not say. My paper was too long as it was, and so a large amount of oceanographic data and statistical theory were omitted. In fact, Figs. 2-4 of my report, which I took from Dalzell's work, were modified to the extent that I cut out Dalzell's reported results on coherencies, since I wished to avoid detailed arguments about such matters. Perhaps this was wrong. Nolo contendere. Furthermore, I have been very close to this whole subject for several years, and I have come to accept certain statements as being so obvious that one need no longer state them. For example, I would have been quite surprised if anyone were to suggest that the coherency between roll and wave height in head seas were not extremely small. However, if Professor Pierson considers that such facts should still be restated in 1964, I may have again committed a sin of omission. The comments in the section, 'Coherency and Resolvability of Spectral and Cross Spectral Shapes," do not seem to be relevant to my paper and so I shall not offer any response to them. Professor Pierson's comments on nonlinear problems are relevant and I welcome them. It is very encouraging and stimulating to observe recent prog- ress in the probabilistic treatment of nonlinear physical problems. It appears that the oceanographers and statisticians have in fact stepped far out ahead of the hydrodynamicists. Unfortunately, there is much more to the prediction of ship motions than the establishment of statistical laws. Eventually, we would hope to be able to start with geometric and dynamic descriptions of the ship, add to this an ade- quate description of the seaway, and then predict any desired motions-related quantity. Professor Pierson's comments almost imply that all of this can now be done, because the oceanographers have supplied the tools. Actually, we can- not make very good predictions of heave and pitch in long-crested head seas, where the simplest concepts are most nearly valid. We are still lacking in basic methods for treating the hydrodynamics of such problems, and under such circumstances the statisticians' impressive accomplishments are of limited 127 Ogilvie usefulness. Certainly their value must rest very largely on the use of empirical substitutes for hydrodynamic theory. It was for considerations such as this that my paper was totally lacking in discussion of short-crested-seas problems. How can we hope to solve such problems when we have not been able to solve the long- crested-seas problems? Nevertheless, it is pleasant to anticipate the prospect that, if and when the hydrodynamicists make breakthroughs in the future, the statistical apparatus will all be waiting for them — ready to cover not only linear problems of short- crested seas but nonlinear ones as well. 128 CURRENT PROGRESS IN THE SLENDER BODY THEORY FOR SHIP MOTIONS J. N. Newman and E. O. Tuck David Taylor Model Basin Washington, D.C. ABSTRACT This paper describes current work towards a complete systematic the- ory for the motions of a slender ship in a seaway. Part I contains an introduction and a general discussion of the results which are obtained, and presents calculations of pitch and heave response at zero speed. Part Il contains a complete derivation of the zero speed theory for har- monic oscillations in the presence of oblique incident waves. Part III contains a derivation of a more general theory with forward speed, for arbitrary forced oscillations in calm water. A significant feature of this paper is the splitting of the velocity potential and forces into parts which are dependent on free surface effects, plus parts which corre- spond to the motion of the double body in infinite fluid or specifically to the case of a rigid free surface. I. INTRODUCTION A fundamental motivation of the theoretical physicist is his desire to bring a sense of order to the physical world, by means of mathematical models which are derived from the basic physical principles governing the problem at hand. Practical problems in ship hydrodynamics have resisted this ordering process, however, not because the basic physical principles were unknown, but because their mathematical representation has been comparatively intractable, at least by comparison with most other problems in classical mechanics. The predic- tion of ship motions in waves is typical of this situation, and in spite of con- certed efforts we are still short of our desired goal of giving engineering pre- dictions of ship motions from a rational theory. It is generally accepted that, for most purposes at least, the desired theory can be attained by considering that the water around the ship to be an ideal (in- compressible, inviscid) fluid, and by linearizing the unsteady motions (wave heights and ship motions). Within this framework there have been several dif- ferent approaches, which can be distinguished according to the assumptions made concerning the hull shape and forward velocity (Table 1). At zero speed it is possible to proceed without any assumptions as to hull geometry, and we 221-249 O - 66 - 10 129 Newman and Tuck Table 1 Rational Linear Theories for Oscillatory Surface Ships Pen [oe [oe [a [ee re Fat Ship Thin Ship 0 0 or 0(1) Slender Ship 0 or 0(1) Flat Ship 0 or 0(1) Strip Nomenclature: B = beam, T = draft, \ = wavelength, w = radian frequency. indicate this situation by the designation 'fat ship theory"; this approach has the advantage that no assumptions are made concerning the hull geometry, but closed form solutions are not obtainable, and moreover the theory is restricted to zero speed by the requirement that the disturbance of the free surface be small. The thin ship model, which is most familiar in wave resistance theory, has been applied to ship motions in longitudinal (head or following) waves and both first- and second-order theories have been developed; criticisms are first that conventional ships are not thin (B/T is usually greater than one), secondly that the first order theory contains an unbounded resonance in pitch and heave while the second order theory is extremely complex, and thirdly that the use of this model for oblique wave motions results in a lifting-surface type of integral equation. The flat ship was proposed in order to overcome the objections of the thin ship, but its analysis is still incomplete, and one may note that it suffers from drawbacks similar to the thin ship, but with the vertical and transverse modes reversed. The strip theory and slender ship theory are based upon identical geomet- rical assumptions, namely that the beam and draft are both small compared to the ship's length; intuitively this assumption seems reasonable for conventional ships. They differ however, in regard to the additional characteristic length of the problem, namely the length of the incident waves. The strip theory, which assumes two-dimensional flow in transverse planes at each section of the ship, is rational only if the wavelength is small compared to the ship length. If this is the case, interference between the bow and stern will be negligible, since they are many wavelengths apart, and the three-dimensional hydrodynamic 130 Current Progress in the Slender Body Theory problem can be reduced to a sequence of two-dimensional problems.* An addi- tional drawback of the strip theory is that, by hypothesis, it cannot be rationally applied with forward speed. Slender body theory, on the other hand, attempts to account for longitudinal changes in the flow, either from interference effects or from the effects of forward motion, but at the expense of transverse interference phenomena since the beam is assumed small compared to a wavelength. Thus it would seem natural to apply the techniques of slender body theory, which have been well established in aerodynamics, to the prediction of ship motions in waves. However, this seemingly obvious union was not consummated until recently. Now progress is being made by several workers and we can optimistically hope that a rational and successful theory for predicting ship motions in waves will be forthcoming in the near future. This paper contains an outline of some recent developments toward the above goal. Our results are still incomplete, and to some extent disjoint, but they are sufficient to suggest the practical utility of a truly rational approach to ship motion predictions. To support this statement we will show numerical computations for practical ships which, at least in parts of the domain of inter- est, are as accurate as available experimental data. Our paper will be divided into three parts and these will be presented in the inverse order from that which is customary, so that the most important results are exhibited before we become engrossed in the details. The Essential Features of Slender Ship Motions Our theory assumes the ship to be long compared to its beam and draft, to be floating on the surface of an ideal incompressible fluid, and to be excited in unsteady motion either by external forces or by an incident plane progressive wave system. We assume moreover that the unsteady motions are of small amplitude compared to all of the other characteristic lengths (i.e., the ship dimensions and wavelength) so that linearization is possible. Finally we as- sume that the wavelength of the incident wave system or the waves radiated from the body is of the same order as or greater than the ship's length. The last assumption ensures that the transverse dimensions of the body are small compared to a wavelength and greatly simplifies the interference effects be- tween points on the ship's surface. It is convenient to introduce a small parameter ¢«, which may be defined as the beam-length ratio of the ship. The slender body solution of our problem is then developed by formulating a boundary value problem for the appropriate velocity potential, whose gradient represents the unsteady fluid velocity vector, and then finding an approximate solution of this boundary value problem which is asymptotically valid for small values of «. The first order slender body *However, if incident waves are present from any angle other than abeam, the resulting two-dimensional problem is governed by the wave equation rather than Laplace's equation. This complication is frequently overlooked in analysing the exciting forces from strip theory. 131 Newman and Tuck theory results from retaining only those contributions to the velocity potential and forces acting on the body which are of leading order in «, and higher order approximations follow by systematically including the next-higher-order con- tributions, etc. However, our problem is complicated by the fact that a slender ship, oscillating in six degrees of freedom, will produce hydrodynamic disturb- ances in the various modes and encounter hydrodynamic, hydrostatic, and iner- tial forces in each mode, which are of different orders in «. It is clear, for example, that the surging or rolling oscillations of a slender ship will not gen- erate disturbances of the same order as pitching or heaving modes. Moreover, even within one given mode, say heave, certain types of forces will dominate others; for example the hydrostatic restoring force will be of the same order as the waterplane area, or 0(«), while the inertial force will be of the same order as the ship's displaced volume, or 0(«*). Asa result many of the accepted components to the total forces and moments acting on the ship are higher order, and do not appear in the first order theory for each mode. This situation is illustrated in Table 2, which shows the order of magnitude, for each mode of oscillation, of various physical quantities. These include the normal fluid ve- locity 3¢,/0n on the ship's surface induced by its oscillations and by the incident wave system, the corresponding body velocity potential ¢, representing the dis- turbance of the fluid by the ship, the hydrodynamic body force F, due to this disturbance, the hydrodynamic force F,, due to the pressure field of the undis- turbed incident wave system (the ''Froude-Krylov" force), the hydrostatic re- storing force F,,, and the inertial force F, due to the body's own mass or moment of inertia. For each mode the forces of leading order are underlined, and the first order equation of motion is written symbolically in the last column. Conceptually this table can be derived most easily for uncoupled motions, but in fact the inclusion of coupling between modes does not affect the order of magni- tude in each case (assuming that the origin is taken at the center of gravity). Table 2 Relative Orders of Magnitude for each Degree of Freedom First Order Equation To illustrate how the entries of Table 2 are obtained, consider the case of surge. The normal velocity on the ship's surface is proportional to the direction cosine in the longitudinal direction, which is 0(«) for a slender body. The 132 Current Progress in the Slender Body Theory magnitude of the potential can only be established rigorously by solving the problem, but it can be estimated heuristically by considering the corresponding two-dimensional problem in the transverse or ''cross-flow" plane, where the ship's submerged area is pulsating at a rate proportional to the longitudinal rate of change of sectional area, and it is easily verified that a pulsating cir- cular cylinder of radius R will have a potential, on its surface, of magnitude proportional to R logR times the normal velocity. The hydrodynamic forces follow from Bernoulli's equation and the fact that the longitudinal projected area of the ship is 0(«?). (The potential of the incident wave is of course 0(1) since it doesn't depend on «.) There is no hydrostatic restoring force in surge and the inertial force is proportional to the displaced volume, or 0(«?). The leading order forces are the Froude-Krylov exciting force and the inertial restoring force. Thus the leading order equation of motion for surge oscillations does not depend on the hydrodynamic disturbance generated by the body. We note that a similar conclusion holds for heave, roll, and pitch. Thus, at least in these four modes, the familiar damping and added mass forces are secondary and the Froude-Krylov hypothesis has a rational basis. Certain fundamental conclusions follow from Table 1: 1. In every mode the leading-order equations of motion are homogeneous in «. Thus the response in each mode, to incident waves, will be 0(1) in terms of «, and of the same order as the wave height. 2. For surge the dominant forces are inertial and Froude-Krylov, with the effects of the ship's own disturbance small by the factor «? log e. 3. For sway and yaw the ship's hydrodynamic disturbance must be ac- counted for even in the first-order equations of motion. 4. For roll the Froude-Krylov exciting moment and hydrostatic restoring moment are dominant, with other effects small by the factor e«. 5. For pitch and heave the dominant forces are Froude-Krylov and hydro- static, with effects from the ship's hydrodynamic disturbance small by a factor e loge. It follows that the first-order equations of motion for pitch and heave will not contain resonance effects, but there will be a bounded resonance in the second-order equations. At first glance the above conclusions may seem trivial, at variance with physical observation, and a step backward in our scientific development. One noted critic has even stated that ''at least thin-ship theory predicts resonance." The best counter-argument is to show some results from the application of the first-order theory for pitch and heave (Figs. 1 and 2). These show the pitch and heave response of an aircraft carrier at zero speed. The solid curves are the results of solving the coupled first-order equations of motion, equating the Froude-Krylov exciting force and moment* to the hydrostatic restoring force *Actually we employ the slender body limits of the Froude-Krylov force and moment, in which the surface integrals over the hull are replaced by simpler line integrals. 133 inch in degrees per Pitch Response Hea ve Response Newman and Tuck LEGEND A REGULAR WAVES O TRANSIENT TEST (e) .2 4 6 8 1.0 Frequency in cycles per second Fig. 1 - Pitch response calculated from first order theory and compared with zero speed experimental data a l LEGEND A REGULAR WAVES © TRANSIENT TEST (0) .2 4 6 8 1.0 1.2 Frequency in cycles per second Fig. 2 - Heave response calculated from first order theory and compared with zero speed experimental data 134 Current Progress in the Slender Body Theory and moment. The experimental points were obtained both from regular wave tests and from transient or "pulse" type tests, as described in the paper by Davis and Zarnick at the present Symposium; these experiments were made in the Maneuvering and Seakeeping Facility, so that wall effects are minimized. It is clear that under these conditions the seemingly crude first order slender body theory gives a very good prediction of pitch and heave, in fact much better than is usual in this field. The above results are less surprising if we recall that they are for zero forward speed, and in this condition the resonant frequencies of conventional ships in pitch and heave correspond to very short wavelengths, on the order of 50-75% of the ship length, or much shorter than the range of practical signifi- cance. In other words, at zero speed with conventional ship forms the practical frequency range for heave and pitch is substantially below resonance. Clearly, however, the situation will change when forward speed is involved, at least in ahead waves, since the frequency of encounter will be increased. This is illus- trated in Figs. 3 and 4, showing the same theoretical curves compared to ex- perimental data with forward speed (at a Froude number 0.14). There is nowa resonant peak within the domain of interest, although the data are essentially unchanged away from resonance. This suggests that a second order slender body theory, including the mass of the ship and all other effects of equal order, might be sufficient to give predictions with forward speed of the same accuracy as those shown for zero speed. It is for this reason that we have been examining the second order slender body theory for ship motions in waves, which includes LEGEND & REGULAR WAVES o TRANSIENT TEST =—— THEORY PITCH RESPONSE IN DEGREES PER INCH O 2 4 6 8 1.0 1.2 FREQUENCY IN CYCLES PER SECOND Fig. 3 - Pitch response calcu- lated from first order theory and compared with experimental data at 0.14 Froude number 135 Newman and Tuck LEGEND 4 REGULAR WAVES © TRANSIENT TEST Heave Response O .2 4 6 8 1.0 Frequency in cycles per second Fig. 4 - Heave response calculated from first order theory and compared with experimental data at 0.14 Froude number all the familiar complications of damping, added mass, and the diffraction of the incident wave system by the presence of the ship. The complete second order theory at zero speed is presented in Part II, this work being an extension of the results of Newman (1964). Figures 5 and 6 show the resulting pitch and heave response for the same conditions as Figs. 1 and 2. The first order theory and experimental data are repeated for compari- son. It is apparent that there are only minor differences between the first- and second-order results. At finite speed neither a complete theory nor calculated responses are as yet available; the theory is presented in Part III for the case of forced oscilla- tions only, leaving the exciting forces still to be determined. The.theoretical results of Part III (e.g., Eqs. 3.36) are presented in the form of double integrals involving the cross-sectional area curve S(x) and/or the waterline beam curve B(x) multiplied by a complicated kernel function K(x,,U) where is radian frequency and U forward speed. If U= 0, K reduces as in Part II to a combina- tion of Bessel and Struve functions which are tabulated; on the other hand for non-zero U, K remains an untabulated function defined at the moment only in the form of a Fourier integral. More work is needed on investigation and tabu- lation of this function before computation of responses at finite speed can be carried out. 136 Current Progress in the Slender Body Theory ——-— FIRST ORDER THEORY —— SECOND ORDER THEORY OQ EXPERIMENT PITCH WAVE SLOPE Fig. 5 - Pitch response from second order theory compared with first order theory and experiments In deriving the second order theory, the fundamental result we use is that any velocity potential ¢ representing a regular disturbance of the fluid by the ship can, near the ship, be written in the form cows) = COMING G2) + HSE) (1.1) where ¢‘*4!") is the potential for the identical problem but with the free sur- face replaced by a rigid surface or ''wall'' (which is, by reflection, the problem for a double body consisting of the ship hull plus its image above the free sur- face in an infinite fluid). The function f‘*S (x) contains all the free surface effects (and in particular is dependent on the acceleration of gravity whereas pi*4tT) is not), and is defined by an integral transform of the form 137 Newman and Tuck ——— FIRST ORDER THEORY SECOND ORDER THEORY O EXPERIMENT HEAVE WAVE HEIGHT 5.0 3.0 2.0 1.5 1.0 0.8 0.5 0.3 0.2 X Fig. 6 - Heave response from second order theory compared with first theory and experiments Pox) = | KEx- 4,0) WO aE (1.2) L where ae = | sea (1.3) Cc is the flux through the cross section C of the ship at the station x. Since 9o¢/dn is given from the hull boundary condition, Q is calculable as a function of hull geometry and the motion amplitudes. On the other hand the function K isa 138 Current Progress in the Slender Body Theory kernel function independent of hull geometry and of the motions, the calculation of which may be carried out once and for all, this being one of the chief objec- tives of the present theory. For sinusoidal oscillations kK is a function of the radian frequency ~; for general motions, however, K may be interpreted in the usual sense of control theory as a transfer function. Using this splitting up of the potential we may now calculate the hydrody- namic forces on the ship, which will be split up in a similar manner. In partic- ular for sinusoidal oscillations we can define in the conventional manner a matrix of frequency dependent damping, added mass, and exciting force coef- ficients, each of which can be decomposed into ''wall'' and "free surface" por- tions. In this paper we shall focus attention on the latter half of the problem, although in Part II the classical slender body theory is used to find the "wall" forces for oscillations at zero speed. Il. THE ZERO-SPEED THEORY Motions in Oblique Waves We shall outline the general analysis for zero forward speed, constructing the velocity potential from Green's theorem in the manner suggested by Vossers (1962). Further details of the present analysis can be found in the recent paper by Newman (1964). A slender rigid ship is floating with zero mean velocity in the presence of plane progressive incident waves, of amplitude A and angle of incidence 6 rela- tive to the longitudinal x-axis. The resulting fluid velocity vector can be rep- resented by the gradient of a velocity potential ¢(x,y,z) e°i®*, including both the known incident wave potential iCuynzZ)n = = exp [K(z+ ix cos£ + iy sin #)] (250) and the unknown disturbance potential ¢, due to the presence of the body. Here «w denotes the circular frequency, g the gravitational acceleration, K = w*/g is the wave number, and the z-axis is positive upwards with z = 0 the plane of the undisturbed free surface. It follows from Green's theorem and the boundary conditions of the problem that the disturbance potential satisfies op 3G Pp(X,y,Z) Soles, Af floceveene = = Ppl S, 7, 19) 26 Jas, (2.2) Ss where the integral is over the submerged surface S of the ship, the direction of the normal n is out of the ship, and the Green's function is defined (cf. Wehausen and Laitone, 1961) by the expression Go Great iG a (2.3) 139 Newman and Tuck SE [@} I Maes eran CO ie ICR er HGR le ere) 4 G4) (=) i (oo) dk i yf 2) 1 2 | k-K on 7, (kfcx- 2? rm) ; (2.5) c= The contour of integration in the integral for G, is indented below the singu- larity k = K, in order to satisfy the radiation condition of outgoing waves at infinity. Physically the Green's function G represents the potential of an oscil- latory source, located beneath the free surface at the point x = €, y = 7, z = %; the function G, is the elementary source function 1/R plus its image above the free surface, and the function G, represents the necessary correction to account for free-surface effects. The above statement of the problem is exact, and Eq. (2.2) can be regarded as an integral equation for ¢,. If the body is slender, however, major reductions can be affected. It can be shown that the term ¢,(0G,/0n) is small compared to the remainder of the integrand, by a factor 1 + 0(« log «), and the surface inte- gral of the term G,(0¢,/0n) can, to the same degree of accuracy, be reduced to a line integral over the length. The resulting integral equation, for points (x, y, z) in the near field (i.e., a distance 0(«) from the ship), is then Beh 3G b= - Ef (=~ 4 Se) ass £0 , (2.6) where og ne a all C(Gz,0,0,€,0,,0)) (J a) dé L Cc sinia¥ ail G,(x,0,0,€,0,0) Q(¢) dé. (2.7) L Since G, is the Green's function for the rigid free surface problem, it can be shown that (2.6) is equivalent to (WALL) (FS) eee hf in oe C2) Thus, as stated in Eq. (1.1), we can express the velocity potential explicitly in terms of the solution of the corresponding wall problem plus a function f‘**(x) containing the free surface effects. (The f£‘?*)(x) of (2.8) is in the form (1.2) with - (1/47)G, for the kernel kK.) It is now a straightforward matter to find the hydrodynamic forces due to the disturbance of the fluid by the body. From Bernoulli's equation the linear- ized hydrodynamic pressure is 140 Current Progress in the Slender Body Theory . B= ogee "5 (2.9) and thus the six forces and moments are he iope "| [¢ cos (n,x,) dS, (2.10) where cos (n,x;) denotes the direction cosine for i=1,2,3 and the generalized direction cosine X29 COS (N)X;_3) > *,2, COS (a,x, _.) for i= 4,5,6. Substituting (2.7) and (2.8) in (2.10) it follows that (WALL) (FK) Te -iot Eoc= B: + F. + qe eRe i dS cos (n,x;) 1 1 1 : if QE) G,(x,0,054,0,0) dé, (2.11) where F‘**? denotes the ''Froude-Krylov" exciting force from the undisturbed incident wave potential ¢,. The last term in (2.11) contains all of the free sur- face effects due to the presence of the body. This can be reduced further by noting that G,(x,0,0;€,0,0) = -7K {H,(K|x- |) + Y(Klx-€|) - 2iJ,(Klx-é])} , (2.12) where H,, Y,, and J, are the Struve function, Bessel function of the second kind, and Bessel function of the first kind, respectively. One important consequence of (2.11) is that for transverse oscillations (sway, roll, and yaw), ae poe i ro 1 1 1 Ci DAB (2503) Of course higher order terms including free surface effects could be re- tained. In particular the damping coefficients for sway, roll, and yaw can be found fairly easily from the energy flux at infinity (Newman, 1963), in the form 2 Bo, ee S(x) + m,,(x)/p 1 ; iKx cos 6 1 Bee aq CK? f dé sin’ om S(x) Z9(x) + 7p B°(x) - m,,(x)/e pdx] , Bes 0 L x S(x) + m,,(x)/p (2.14) where m,,(x) and m,,(x) are the two dimensional added mass coefficients of the section for the sway force due to sway and the sway force due to roll, 141 Newman and Tuck respectively, and for the rigid free surface condition, and where Sx) is the sectional area, z,(x) is the vertical coordinate of the center of buoyancy at each section, and B(x) is the beam of the section at the waterline. We note that B,, and B,, are 0(«*) while B,, = 0(«®), and as indicated in Table 2, all three are of higher order compared with other terms in the equations of motion. Pitch and Heave in Head Waves We shall illustrate the above theory by considering in more detail the im- portant case of pitch and heave motions in head waves. If ¢, and ¢. denote the (complex) amplitudes of heave andpitch, the boundary condition on the ship hull is 22 = =. ($,+¢g) = —i@l, cos(n,z) + iol, [x cos(n,z)- z cos (n,x)], (2.15) or, for the disturbance potential, C) .s 2 og we -iot -l:. B iL 2|x- | o pe Oaiae i; ( (x) 3 og L sen (x- ¢) x & [Bcé) (one ideik*) | dé dx ~qvexte tet [ (T\acxy [Bey (0,-xt,- idet*) {Hy(Klx- él) ib, L .s + ¥,(KIx- 1) - 2iJ,(Klx-€l)} dé dx se iopAie sto i) [Bcx) - KS(x)] etkXax + 0(€3) , (2.21) L >.< where the last term represents the slender body approximation of the Froude- Krylov exciting force and moment, to second order in «. The total force and moment will include the hydrodynamic components, represented by (2.21), plus the conventional hydrostatic restoring force and moment (of first order in <) and the inertial force and moment of the ship's own mass (of second order in <). Setting the sum of these equal to zero yields a consistent set of equations of motion, accurate to second order in «. We note that the first order contributions include only the hydrostatic terms plus the first order Froude-Krylov contributions. The solution of this first order sys- tem was illustrated in Figs. 1 and 2. There are various second order contribu- tions in Eq. (2.21), each of which is interesting by itself. The first integral gives the "strip theory wall forces" involving the stripwise zero frequency added mass times the relative acceleration, including the incident wave height. The first double integral gives the corresponding "wall" three-dimensional cor- rection to the added mass and exciting force. The second double integral con- tains the free surface effects, including an added mass contribution from the real part of the kernel H, + Y,, and a damping contribution from the imaginary term -2iJ,. Note that in all cases the relative displacement ¢, - x%, - iAe*** *This is nota unique definition by itself. We may say that m,(x) isthe coefficient of the force associated with the pressure iwp¢‘?P) eit, and (7?) must be of the form ¢(2)) we tog(¥?t+74)/L2 as y2 + 2230. 143 Newman and Tuck between the body and the undisturbed incident wave height is of principal im- portance; in this way the theory accounts for the diffraction effects, or the cor- rection to the Froude-Krylov exciting force due to the presence of the ship. It is important to note that both double integrals contain truly three-dimensional effects, with the disturbance at one station of the ship (&) affecting the force at another (x). Til. FORCED OSCILLATIONS AT FINITE SPEED OF ADVANCE Introduction In this portion of the paper we suppose that the ship is being forced to make small oscillations about an equilibrium fixed position, while an otherwise uni- form stream U flows past in the positive x direction. These forced oscillations, which need not in general be sinusoidal or even periodic, will be described by given functions of time (,(t), ¢,(t), ¢,(t) for the linear displacements in surge, Sway and heave, and (,(t), ¢,(t), ¢,(t) for the roll, pitch, and yaw an- gles. Similarly we denote the resulting hydrodynamic force component in the ith mode by F,(t). Under the usual control theory assumptions of linearity and causality there will be a linear relationship between F, and all Cj, j =1,...6, which we may write symbolically as lig = »» Ci | Cj (3.1) ya for some set of linear operators C;,. Alternatively (3.1) may be interpreted literally as a linear algebraic relationship between the Fourier transforms of the variables F;, ¢;, with coefficients C;; = C; ;(~) called "transfer functions." Here the Fourier transform of C,(t) is defined as C5 (a) =| cieteisaan@c)) (3.2) (the use of the same symbol for a function of time and for its Fourier transform is common and convenient, and will not cause confusion). Clearly C; ;(@) is the Fourier transform of the force C,.(t) in the ith mode due to a unit impulse 8 t) of displacement in the jth mode, the actual relationship between ¢ j(t) and F(t) being thus a convolution integral with Cc, j(t) as kernel. In the case of sinusoidal motion at a real radian frequency w, the real and imaginary parts of the functions C; ;(~) define the frequency response of the forces to sinusoidal displacements of unit amplitude. Historically these quan- tities as used in ship problems have been calculated in the form of ''added masses" 144 Current Progress in the Slender Body Theory C; .(@) - C;;(0) M;;(#) = Re oe) (3.3) ( = iW) 2 in phase with the accelerations (-iw)” Che and ''damping coefficients" (Ca a (Cea) = 1J B;;(#) = Re (eae) (3.4) in phase with the velocities (-iw) Cj. Thus there are three interpretations of the linearity Eq. (3.1). When (3.1) is to be viewed as an operator equation we write C;; = C;;(~) but reserve the combination ''-iw'' to mean the operator ''d/ot.'' The second interpretation views C; ;(#) as a transfer function with » asa (complex) Fourier transform variable, while the third views C; ;(~) as the frequency response for real »o. The following analysis may be given any of the three interpretations although it is mainly expressed in the language of the second of them; that is, we seek a set of 36 complex valued transfer functions C; ,;(~) of the complex variable ~. How- ever, in practice one need calculate the c;; only for real ~, so that added masses and damping coefficients would be obtained directly, as in the third in- terpretation. Evaluation of the Velocity Potential Near the Ship Firstly let us linearize with respect to the amplitudes ¢ ;(t) of motion, which are assumed to be small of order a, for some small parameter a which measures the general size of the motions. Thus we expand the velocity poten- tial in the form Pp = Poo (YZ) + P1)(* YZ, t) + Pe 2y)(*% ¥2,t) Heche hens (3.5) where $:9) is the steady flow due to a uniform stream U past the ship fixed in its equilibrium position, while ¢,,) = 0(c) is the first approximation to the un- steady potential for small oscillations of order a about this position. Further terms ¢,2) ... describe non-linear effects due to not-so-small oscillations and will not be investigated in this paper. Now if the ship is slender, each of the potentials ¢.,), ¢,,,, --- may be further expanded in terms of the slenderness parameter «, in a manner typified by the expansion of the steady term 2.0) which has been obtained previously (Tuck, 1964). Thus near the ship we can write Po) = Ux + Exerc + £9 | + 0(€3 log? e), (3.6) where the term ''Ux" represents the free stream and is of zero order in «, while the contents of the square brackets are of order «? log « and represent the steady disturbance to the stream due to the presence of a fixed ship. This 221-249 O - 66 - 11 145 Newman and Tuck disturbance potential is of the class described in Part I, and the terms oy "y and Wey ' have the significance discussed after Eq. (1.1). The "wall" potential can itself be further decomposed into (WALL) (2D) (WALL) Hay Cone) = Oy Coe) Bay CDs (3.7) where, for constant x, ¢{ >|” satisfies the two dimensional Laplace equation with respect to y and z. Both terms fon and a represent interactions between sections of the ship and were determined explicitly in Tuck, 1964, in the form ae z | i" Ky enter) us’(é), (3.8) aes | dé KS (x- 6) USE), (3.9) with on ae Ss = - [sgn x log 2\x\]. (3.10) (FS) il. el Kio) (x) = -& & [Ho(Ge) + (2+ sen ve (| F/I (3.11) The above results are in the form of Eq. (1.2), since "'US‘(é)" is the flux through the cross section at € due to the steady motion, S(¢) being the immersed area of the cross section. Now a similar analysis holds for the linear unsteady potential $1) which can be written as WALL Po1y(% ¥, zt) = $45 se, Ba + £5 8) + 0(€? loge) , (3.12) with the wall potential further decomposed into (WALL) AL Ck SCS) 5/2519) Peay (HY. zt) + ae Peseta (3.13) : : : : ( WALL) (FS) : if desired. The interaction terms f,,, and f,;, are not now simply related to the area curve as in (3.8) and (3.9) but, since the unsteady flow is produced by linear oscillations of the ship, will be linear functions of the magnitude of these motions. Thus we can write 146 Current Progress in the Slender Body Theory (FS) ° See ea eae (3.14) yen with f; = f;(x,) as the transfer function between motion in the jth mode and the free surface interaction potential at station x. A similar relationship would hold for the wall interaction f‘""'"?, which is not of interest to us in the pres- ent work. tee The free surface interaction transfer functions f; are obtained in Ap- pendices I and II by the use of the hull boundary condition, the result being again in the form of Eq. (1.2), i.e., f - | dé K1)(*- 4) Q;(S) , (3.15) where Q(x) = -|- ses eI 4) S'(x) A dx OF Es) == 9 Q,(x) = =| iw + wel B(x) (3.16) Q,(x) = 0 Q.(x) = + (-ies Us| x B(x) aes) Sd are flux transfer functions, and K_,, is an absolute kernel independent of hull geometry and defined by 1 -ikx Mey) = =) de B(k) coth A(k) , (3.17) where = 5 2 coshes¢k) = (ie e|k| (for the case w real, see Eq. (A3.6)). The kernel K_,,(x) depends on the frequency w and speed U as parameters as well as the variable x, and will sometimes be written K,,)(x,#,U) to emphasize 147 Newman and Tuck this. In particular for U=0, K_,, reduces to the zero speed kernel of Part II, Fiqee (2-12), es, ( )s al = 2iJo( ) (3.18) On the other hand, for zero frequency, K,,, reduces (as it must) to the steady kernel Eee of Eq. (3.11), i.e., 2x wx wx 1d Kenge) 2.5 ge H, (E) * + sen vo (|S) 3 (3.19) For non-zero values of w and U, K_,, has not yet been tabulated, and further work is needed to investigate the properties of this function. We may note that by a suitable non-dimensionalization K,,)(x,#,U) may be expressed as a func- tion of two dimensionless variables only, for instance _ & : w*x a K(1)(%@,U) = oar function of Ss aa) It is easy to see that K,,, has the familiar singularity when U/g = 1/4, which may complicate the task of numerical evaluation of the kernel. Pressure Calculation From Bernoulli's equation the hydrodynamic portion of the pressure field is = 6 1 2 1 2 p = =6 ries oy el = wl (3.20) (here "'-iw' may best be interpreted simply as the operator ''3/ot''), which gives on expanding with respect to a that 1 2 1 . Ds S52 E IVb. 9 (% ¥> 2) = = Ur a MOP sy WP eg Bo 2) ‘ VG.1)(% > Zz, t) 6 0¢a%)| , that is, p = Po )(% YZ) + P(1)(*y,Z, t) tS ees where p_,) is the steady pressure field Proy = 7p E lb, 1? wr | ? (3.21) while p,,) is the term of first order in a, namely 148 Current Progress in the Slender Body Theory ys -p|-iad, rake y V1] ; (3.22) Equations (3.21) and (3.22) give the steady and linear unsteady pressure fields for an arbitrary body. Now if the body is slender, both pressures may be con- sistently approximated in the form (WALL) (FS) Po) a Pio) (GCkg% 2) ar Pio) (x) (3.23) (WALL ) (FS) Pens) re 2 ily) (2%) Wo Bae) iF Poi) G8) ? (3.24) as was done for the potentials. We shall not write down the "wall" pressures, which are not required for the present analysis; the free surface contributions are (FS) (FS)! Qa ee (3.25) iS) De ee (iow = Fig Cee) + (3.26) These formulas give the free surface dependent part of the pressure everywhere in the field of flow. In particular from (3.14) we can express the unsteady pres- sure field as a sum of contributions from each mode of motion, with appropriate transfer functions. In order to find the forces on the ship we require the pressure on the in- stantaneous hull surface. This is obtained by evaluating the pressure on the equilibrium hull surface and adding a correction term to account for the dis- placement of the ship in the non-uniform steady flow field. Thus if p,,, now denotes the unsteady pressure evaluated on the equilibrium hull surface, then the unsteady pressure on the actual hull is ; 2 Poiy + & * VPp9y + 0(2°) . where c is the vector displacement of the hull at any point. In particular the correction to the FS part of the pressure is (FS Ss a -VPr05 (*) 5 CS Py (*) = SG Use aoa (3.27) where {,(t) is the surge displacement (note from (A2.1) that the x component of a also contains terms involving the pitch and yaw angles @.(t) and ¢ ,(t), but these contributions are negligibly small in « compared with other retained terms). Thus the only contribution from this correction is in surge excited mo- tion, and we can write for the pressure on the actual hull 149 Newman and Tuck (FS) - j=l where the pressure transfer functions p ; = P(x.) are given by = =O|=i@ + cl f= Wee P, = -pl-ie+ = 1 p (0) (SX) 5 (3.29) eee C =e By = p(-ioru 2) f, sy bP =2 1S tees (OM and where the f,; are those of Eq. (3.15). Forces and Moments The forces and moments now follow directly by integrations over the hull from the formulas Nay Se IP ae EI - |{ pads (3.30) Hen an iebiees abba -|[prxnas. The splitting up of the pressure p into a wall and a free surface part leads to a similar splitting of the forces, viz., (WALL) (FS) Fae ie lie Aa Es (3.31) 1 1 i for all i =1,...6. But since Por is a function of x (and time) only, the re- sults of Appendix I, Eqs. (A1.3) and (A1.4), may be used to show that (FS) : (FS) F, = faxstoo Po) (GX 18) FS Ey ep edo (FS) (FS) Pa = ae Por) Ga) (3.32) (FS) 4 = 0 (FS)- (FS) Pi = - [ ax x Bex) Poi) (x,t) FS FC i 0 150 Current Progress in the Slender Body Theory Now we view the force in each mode as the sum of contributions from displace- ments in all modes, with transfer functions C,, as in Eq. (3.1), putting for the > j free surface portions: 6 (FS) (FS) Eee ake (3.33) jeu Thus (FS) : ee eke [ axs'¢s) P ;(x, @) Ss ek dei (FS) C3; = J axa P; (x, ®) (3.34) Cres 0 (FS) Co; = - [ ax x Bex) P ;(x, ®) Ss ons ) = 40" In terms of the f; of Eq. (3.15), the non-zero eae are s a“ ce ) = =p | ax S'(x) (io + U =) i = ou | ax S'(x) ae 6S Ss ee ye -p | ax S'(x) (-ie + U 35 fe CS) es , 5 Cus ie! 7 p{ ax S'(x) (-io + U =) fe (FS) 5 eee Be oe ERC (-ie ea a f,- pUfdx B(x) f(g, (x) (3.35) (Cont.) (FS) cr = -p | ax B(x) | -1@ + us) i (FS)" fst ou | ox x B(x) fio) (x) 151 Newman and Tuck Gasp ts 3 ere = p| dx xB(x) |-iw + U = i (3.35) ACES i 3 5 = p| dx xB(x) |-1 + Wa lips (FS) Finally, using the convolution integral representations (3.9) and (3.15) for f (0) and f, respectively, we have Gilet o| dx dé S'(x) S'(é) [K(x-¢,0,U) - K(x-€,0,U)] Cis = P| [axdé 8'(x) BCE) K(x-E,0,0) cE? = -9 | faxde 8'(x) EB(2) K(x-€,0,0) Cue o { faxae B(x) $/(E) [K(x-4,a,U) - K(x-€,0,U)] cM =o [faxdg Bex) BCE) K(x-E,0,0) (3.36) Che = -p [fax dé Boo EBLE) KCx-Z, 0,0) Cann ~p | faxae xB(x) S'(é) [ K(x-€,0,U) - K(x-€,0,0)] CRN ap | faxae xB(x) BCE) K(x ~&, «,U) Coe = P| fade B(x) BCE) K(x-E,0,0) where : a\2 K(x,@,U) = (-ix + U ral K( 1)(*, @,U) é G3) This K is the kernel for all heave and pitch motions, but for surge induced mo- tions the kernel is is - lim a aO>0 the additional correction being only of importance for non-zero forward speed and arising from the correction (3.27) to the pressure field due to displacement of the ship in the steady flow field. But we can easily see that without this cor- rection the results for (say) et ae would be nonsensical, for as +0, C{**) must represent the restoring force in surge (i.e., change in wave resistance) due to a 152 Current Progress in the Slender Body Theory unit lengthwise displacement of the ship, and this is clearly zero. On the other hand, if j +1 then c‘*S) does not in general vanish at zero frequency and yields for w-0 the trim forces and moments on the ship in steady motion. Since in evaluating added masses by (3.3) we must in any case subtract off these trim forces C; ;(0), the kernel "K a lim K" wo70 (FS) ‘ : may be used for all C;,; ' whenever trim forces are not required. At non-zero frequencies w of purely sinusoidal motion we can split the 9 Eqs. (3.36) into their real and imaginary parts yielding 18 added masses and damping coefficients from Eqs. (3.3) and (3.4). In order to compute these 18 quantities we require just two functions s(x) and B(x) describing the geometry of the ship and one universal kernel function K(x, #,U) which can be computed once and for all. Of course the complete added masses and damping coefficients are the sum of the "wall" values plus the values obtained from Eqs. (3.36), but the determination of the former is, as described in Part I, a much less difficult task. For the lateral modes of sway, roll, and yaw, where i or j takes the values 2, 4 or 6, the ca” vanish, so that the remaining 54 added masses and damping coefficients are dominated by the "wall'' values. Since the latter are frequency independent, this conclusion is equivalent to the conclusion that all lateral and damping coefficients are independent of frequency, to leading order in slender- ness. Any frequency dependence must come from higher approximations in «. ACKNOWLEDGMENTS The authors wish to thank Mr. Werner Frank and Miss Evelyn Woolley, who developed the computer programs used for the zero speed calculations. REFERENCES Newman, J. N., 1963, "Lectures on the Theory of Slender Ships,'' Unpublished notes, Department of Naval Architecture, Berkeley, California. Newman, J. N., 1964, "A Slender Body Theory for Ship Oscillations in Waves,"' Journal of Fluid Mechanics, Vol. 8, Part 4, pp. 602-18. Timman, R., and Newman, J. N., 1962, ''The Coupled Damping Coefficients of a Symmetric Ship,'' Journal of Ship Research," Vol. 5, No. 4, Reprinted as David Taylor Model Basin Report 1672. Tuck, E. O., 1964, "On Line Distributions of Kelvin Sources,"' Journal of Ship Research, Vol. 8, No. 2. 153 Newman and Tuck Vossers, G., 1962, "Some Applications of the Slender Body Theory in Ship Hy- drodynamics,"' Unpublished Thesis, Delft. Wehausen, J. V., and Laitone, E. V., 1961, ''Handbuch der Physik," Vol. 9, Sur- face Waves, Berlin:Springer-Verlag. APPENDIX I SOME GEOMETRICAL IDENTITIES Gauss's theorem applied to a closed surface consisting of the immersed hull surface S, together with the waterplane, indicates that J Jonas = thal V p dx dy dz - ih pk dx dy (A1.1) interior water of hull plane and J for-nas =| f [rxve axay az - { fecyi- xi) dx dy , (A1.2) for any sufficiently regular scalar p(x,y,z), n being the outward unit normal and r the position vector xi+yj+zk. The following identities are obtained from the above for some simple special choices of the function p(x, y,z): [ [eco nas 2 fax p(x) [i (-S"(x)) +k (-B(x))] (A1.3) J feo rxnds = fax p(x) i («Boo = Eee ayas)| (A1.4) f feco ZndS) = fax p(x) E - & | {zayae) + scx) (A1.5) J foco Zane Si [ ax p(x) E (-x so) - = [fet ayaz)] (A1.6) [ feco ial = [ax P(x) [i S(x)] (A1.7) | Jecoysxnas 2 fax p(x) E (- | [zayaz 2 = B3(x) +k xS(x) + & fv? avez) | 3 (A1.8) For instance, to prove (A1.3) we note that 154 Current Progress in the Slender Body Theory (aig Vp(x) dxdydz = i \ dx p'(x) ° ff dy dz interior length cross of of section hull ship te - i fax p'(x) S(x) = i fax p(x) S'(x) , where S(x) is the area of the cross section at x and is for the last step assumed to vanish at both ends of the ship. On the other hand ROHL Sy SO | MEY RP ay water- length width of plane of waterplane ship at x -k [ax P(x) B(x) , where B(x) is the waterplane beam at station x. The remaining identities (A1.4) to (A1.8) may be proved similarly. The above identities are exact for a hull of arbitrary shape (providing it is symmetrical with respect to y) and for an arbitrary function p(x). Their prin- cipal use, of course, is for evaluating the forces and moments on a slender ship, in which case p(x), zp(x), yp(x), Will be identified as terms in a Taylor series for the pressure on the hull. In addition, if the ship is slender, some of the terms in (A1.3) to (A1.8) may be dropped to a consistent order of approximation in «. For instance, in (A1.4) the term = a {| z dy dz is of order <* whereas xB(x) is of order ¢; the former will be neglected when (A1.4) is used in obtaining (2.21) and (3.32). Equations (A1.7) and (A1.8) are used to obtain the sway, roll and yaw damping coefficients (2.14). It was not necessary to use explicit representations of the components of the unit normal n in the above, but for later reference we now derive these using a particular equation Zi OZ (CXERY) describing the hull. Clearly then PO : : aly 2 Dh GIA ta Ze be) faze; 22) : 155 Newman and Tuck Also since in terms of this hull equation the magnitude of the element of surface area is 1/2 dS = dxdy (1+ Z} + Bee L we have for the outward vector element of surface area mash 6UZ eZ) kk )adxidys: This may be written in a manner not dependent on the choice z = Z(x,y) of hull equation, viz. adS = de E ~ (zdy) + j dz - eal (A1.9) where dy and dz = Z,dy denote components of arc length along the cross sec- tion curve. The area of a vertical strip from the free surface to the cross sec- tion curve is -zdy so that the x component of ndS represents the decrease in the area of this strip in passing from station x to station x+dx. The integral iden- tities (A1.3) to (A1.8) may also be proved directly using the expression (A1.9) for ndS, but appear to be more easily derivable from Gauss's theorem, as indi- cated above. APPENDIX II EVALUATION OF FLUX TRANSFER FUNCTIONS Q; AT FINITE SPEED The hull boundary condition for ¢,,,, on an arbitrary body with unit normal n at a point where the hull displacement is eS (A ECoG CeCe) sei (Cay EGC) GCE ee, E211) can be written = n+ [-iwa + V* (Ex Von) (A2.2) (Timman and Newman, 1962). The second term inside the square brackets gives the induced normal velocity due to non-uniformity of the steady flow ¢,) in which the body oscillates. On separating out the contributions due to each mode ¢,(t), j =1,... 6, we can write (A2.2) as CNG) - Spat DL, Ba@) Ga (A2.3) j=1 156 Current Progress in the Slender Body Theory where the hull velocity transfer functions ¢g,;(~) are given by ; 3 WEG Neg te ae — Lene cme Mero. (A2.4) 1e, 7 18, Kee = eX (2 eye Ries) Oy) - No slenderness assumptions have been made up to this point and the boundary condition (A2.3) is still valid for an arbitrary rigid body making arbitrary small motions in an arbitrary steady field ¢,,,. Now if the ship is slender n lies nearly in cross-sectional planes (or, more accurately, from (A1.9) we see that n, = 0(«)n, = 0(e)n,) so that the g,; may be consistently approximated in the form : rn) e) ( 2D) Arie De rec 2 a (n, Sv BB 2) Poy, : O) 0) (2D) Ean = = I@in, = (n, ay * n3 2) Poy, : 0) C) (2D) Bo) =. = eras WS (n, iy n3 =) Poy, (A2 5) & C) C) (2D) Ba S95 en | oe =| P0) Bey = Sra) Uuale Be S S885 a Wins all with error a factor 1 +0 («log «) which is mainly due to the replacement of ¢(0) by its slender body approximation Ux + ae Re f(y, from (3.6) and (3.7); of these terms only ¢{;}’ contributes to the g;. Notice that in the expression for g, we have used the slender body approximation to the boundary condition for ¢,,),, namely 3 a \ CaS x (n, ag” Mg 2) Pom ee UTE (A2.6) Notice also that the surge and roll velocity transfer functions g, and g, are smaller by a factor 0(«) than the other g,, Since a slender body is an inefficient exciter of motion in these modes. *Recent investigation has shown that Eq. (A2.6) cannot legitimately be used to simplify g,- The resulting values for the fluxes Q;, Q4 in these modes are un- changed, although the given derivation for Q, is no longer valid. 157 Newman and Tuck Now as a consequence of slenderness, 0¢,,)/0n aS given by (A2.3) with the g, of (A2.5), is the normal velocity across the cross-sectional curve in planes normal to the x axis. Hence, the net flux across this curve may be calculated in the form Ee 2% o1) = ° Oiges = Ailias = SSI (fe; 48) = ae yy vidas (A2.7) where Q; dx = Jejas is the flux transfer function for the jth mode, the integration proceeding around the equilibrium hull cross section curve beneath the plane z =0, with dS obtained from (A1.9). Thus at. HONG: Q, = Ea feay where S(x) = - {/zdy is the area of cross section below the equilibrium free surface z =0. ; (2D) (2D) Q, = =10) dz ~ | (a Poo) = ohy P0) ). yy yz ’ satisfies the 2D Laplace equation J (2D But since ¢,) = =ialz)s) + [40], (A2.8) where [ ],_, indicates that the difference between the values of the enclosed quantities at the two points of intersection of the cross section with the free surface is to be taken. But ¢'>}) is by definition the 2D double body potential, i.e., Bie = 0 on z =0, So that vA 158 Current Progress in the Slender Body Theory and thus, from (A2.8), Q, = 0. Similarly But now if the waterline of the ship is described by 1 y = +> B(x), then [y],_, = B(x). Also it is clear that the boundary condition (A2.6) for es reduces to (2D) nites, Toy masa ae ce at the free surface z = 0; i.e., (2D) , #0) | = WE) o Yjz= Thus QF =) 1OB Ce) = UB Gx) S45 (-io + ie BED Now : : (2D) (2D) iv [yay + iw Jeaz = Jaz Poy, = Jey Pro), 5 iw [y?+ z?| z=0 Ber i O b i] = (0). provided the ship (and hence the steady flow $e a) has transverse symmetry, which is the case of interest. Finally, by similar reasoning to that for Q,, Q,, we have that 159 Newman and Tuck Q, = (-ias U Z| xB). Q, = 0. These Q;, j = 1,... 6, are reproduced in Eq. (3.16) of the text. Note that the dominant flux transfer functions are those for heave and pitch, for which Q, and Q, are of the order of B(x), i.e., 0(<). The flux in surge is of order S’‘(x), i.e., 0(e«7), while the flux in transverse modes of sway, roll and yaw vanishes. APPENDIX III EVALUATION OF THE KERNEL FUNCTION KD AT FINITE SPEED Now at any finite distance from the ship (i.e., such that y?+ z? is large compared with the small lateral dimensions of the slender ship) the effect of the motions of the ship for vanishing slenderness is that of a line distribution of sources of strength Q,.,, per unit length. These sources are ''wave sources," i.e., ordinary sources modified to satisfy the linearized free surface condition g 224 (-ie + U 2) ¢ = 0 On 4=(); (A3.1) This potential may be obtained from well-known results on such wave sources (e.g., Wehausen and Laitone, 1961). One way of writing the source potential is in the form of a Fourier transform with respect to x, putting @ P(1)(% Yr 2z,t) = | de go Oe Bn Ip Zo ©) OeCSe) iH {oo} | dives" 4 O77 (it), * etc., where we have for the Fourier transformed potential Pore Tubes / Fay = Way -* (1k Jy?+ 2?) x =idy +2Vk7+X2 + F Cie - itu? | dh ee sj WRONG ea E Re 2 (sie = wR W)Z (A3.2) 160 Current Progress in the Slender Body Theory Here K, is a modified Bessel function of the third kind and gives the source be- havior of ¢,,), while the integral with respect to \ is the correction required to satisfy the free surface condition (A3.1). Now the interaction term in the potential near the ship is found by investi- gating the source potential near the line of sources, i.e., for y*+ z* small, in which case 1 1 Ora = = Or. os Jy? + 2? + log = C|k| 4 4 (ilar in)? | _Aiecan c= MELTS ate ira ee 8 NS) -o Vk24+r2 (g Jk? +A? ¥ (-iw- ikU)?) (log C = y =0.577...). The first term of (A3.3) corresponds to ¢/ 7}? in (3.13), the second to f({}""? and the third to f‘"*?. The last is the quantity of interest here, and the \ integral involved can be integrated explicitly to give fail’ hie =O A(k) coth A(k) , (A3.4) where £(k) is defined by ANNE: BS QR (A3.5) g|k| with |Im6| < 7. The last condition appears to fail for w real, since then cosh 6 is real and negative. The correct interpretation is, however, obtained by taking -iw to have a small positive real part (corresponding to decaying transients) which we may then let tend to zero, giving 2 i(m- a(k)) , iis GOS G{(k)\ye= Cod By SpilpeOes as) i=. e|k| 2 sp GK) Seii(@)sP RUD) 5 SE Gosia Gk) = a Zeit, SOV Sragk) (Str g The inverse Fourier transform of (A3.4) may be taken by use of the convolution theorem, giving foray = | dé K.1)(*- §) Q1)($) » where 221-249 O - 66 - 12 161 Newman and Tuck 1 -ikx Ko = a dk e7?** &k) coth B(k) , this being the kernel function of Eq. (3.17). x * * DISCUSSION H. Maruo National University of Yokohama Yokohama, Japan Dr. Newman and Dr. Tuck have achieved a rigorous and systematic devel- opment of the slender body theory in the problem of the motion of ships among waves. It must be one of the most important achievements in the theory of ship motion, because it enables a consistent formulation for the damping force and the added mass which cannot be realized by the thin ship theory. According to the results, the effect of the free surface by which the frequency dependence ap- pears, is not important unless the finite speed of advance exists. Therefore the results with finite forward velocity seem to be more important. The ultimate aim of the formulation is to enable the prediction of the hydrodynamic forces by means of the theory. In this respect, the present analysis is not yet conclusive. The reason is that the formulas for the forces and moments given by the Eqs. (3.36) and (3.37) with the kernel function (3.17) are not convenient for the nu- merical computation. An important thing is that the final result should be given by a convergent form. However, the formulas given here involve divergent integrals. In order to obtain a formula which is suitable to the computation, another expression is needed. For this purpose, an expression for Green's function which was obtained by Hanaoka some ten years ago is recommended. It takes the following form: GiGe yn Zee yz ke ae g| See vin? +n? - im(x-x')] va ue -@ 0 lm? + n2 © 9 x {cos (nz + €) cos (nz’'+<€) - cos nz cos nz’} dn my | exp [(z+ z')(m- w,)?/x - |y-y'| K, - im(x-x')] (m= oy)? & E 1 2 |b 2 Hh) t : ; : ; dm = O59 (Ge Al NG eh) 8 = Nyy [Sp tin (ee= =") (Bae) ™4 (Cont.) 162 Current Progress in the Slender Body Theory cif? E | . ‘ dm _# exp [(z+2z')(m- a)?/x - ily-y |K, - im(x-x | Os a 0 . 2) 3 1 . / dm +2 exp [(z+ 2')(m- w)?/x tily-y | K, - im(x-x')] fae Se mid m4 foo) ; ; , dm —— + exp [(z+z')(m+a,)?/x -ily-y Kee erm (sen) (m+ wy) XK.’ 2 0 m ties i Ghat yl Oa es 1 GaSe 2 PiGiay Doe cae Tg. x = Aue @, = o/U Ky Syne Gite Gh eee K, fan (mon 77 xn m, ; = ang (x AD, yx? + 4x dy ) mo m T(x - 205 + Vx? = 4x04) > w —— II On applying the slender body approximation, the asymptotic expression for Green's function along the line y = z = 0 becomes PLE NEE Se iD Cm GCk ys Za oS) Be iseela & | sae) = en d pale m3 : ; (m+ w,)? SO 4 anniek) LAE oe ee dx == ET -m, m, m./m*x (m Wy ) (Cont.) 163 Newman and Tuck m mjm?x? - (m+a,)* where (m+ @,)? (m+ @, )? @(m) = cos) when x|m| > (m+ @,)? ae 4 x|m x*m (m + @p ) | | (m+ @ ye (m+ @, ) = z cosh’! o when x|m| < (m+a@)?. (m+ a, )4 - x? m? x|m| Making use of the above, the hydrodynamic forces can be expressed by con- vergent forms. The component of the force in heave for instance, is given by the following form: + 7 htc salt Pal es =m, m 4 m2Jm?2 x2 - (m+ w,)4 (m + a, )* dm melee teers (ial Wee ane” where © dBCx) x ore | ea eo dhe -L/2 This formula resembles Michell's integral for the wave resistance in uniform motion. Hanaoka has given a similar formula for the hydrodynamic forces and moments of an oscillating thin ship. The discussor wishes to propose that the above formula will be called Hanaoka's integral. There is another type of the expression, which is given by repeated integrals of a kernel function and has some resemblance to Vosser's formula for the wave resistance of a slender ship. 164 Current Progress in the Slender Body Theory (FS) C Bae ee pu? faruayarcuaeay BHC L722) | B"(x) K(x-L/2)dx L/ 2 = B’(-L72) i B"(x) K(x +L/2)dx =-L/2 L/2 L/2 | | B"(x) B"(x) K(x- x’) oan jb L/2 where 2 m+ @ 2 = (2) = + ®(m) ( a ) SOS Te. - dm m2 al f ys (m+ a@,)* (cos mx- 1) ai m*./m = (m+ w,)4 7 eat Ba ig (m+ @,)* (cos mx - 1) += -| a: >| Se 7 -@ -m, m3 m*+ Gea \e SmI Since the above expression involves divergent integrals, the finite part of the integral should be taken. COMMENTS ON SLENDER BODY THEORY E. V. Laitone Professor and Chairman University of California Berkeley, California It should be noted that the singularities noted in the integrals for the source distribution can be always evaluated by using Hadamard's concept of the ''Finite Part" of the integral. This is a generalization of Cauchy's ''Principal Value," and can usually (but not always) be most simply determined by "Integration by Parts." 165 Newman and Tuck Also the question arises as to how deep must the thin ship be in order to avoid the three-dimensional effects that correspond to the differences (k, -k,) in the virtual mass coefficients for a body of revolution (see Lamb: "Hydro- dynamics," p. 155), or to avoid the fineness ratio effects corresponding to the complete elliptic integral (E) determination of the virtual mass of a thin plate (see Lamb, Eq. (16), p. 154). REPLY TO DISCUSSION J. N. Newman and E. O. Tuck David Taylor Model Basin Washington, D.C. As Professor Maruo correctly points out, the kernel function K(x) of Eq. (3.37) (which is proportional to the 4th derivative of his function K(x)) has a high order singularity at x = 0, so that if Eqs. (3.36) were to be used as they stand to calculate the c{{*?, some juggling (such as integration by parts, as suggested by Professor ‘Laitone) would be needed in order to get a finite answer. However, in presenting the results in the form (3.36), we did not imply a rec- onumendaiion that this particular form of the integrals was suitable for direct computation. Just as Michell's integral can be manipulated into many different forms, so also can the integrals for the transfer functions c{**), and Professor Maruo has shown an alternative form due to Hanoaka which is clearly better for numerical computation than that given in (3.36), and which avoids the difficulty with the singularity. In fact our initial attempts at numerical computation have used precisely this form, which can be derived directly from Appendix III by use of the Fourier transform convolution theorem. The form in which we gave the results in the paper was chosen for pedagogical reasons, since it illustrates most clearly the simplicity of the formulas in their dependence on B(x) and S(x). The three-dimensional effects mentioned by Professor Laitone are of smaller order of magnitude according to slender body theory than the contribu- tions we calculate. As far as possible we have indicated by order of magnitude statements the size of the error in each equation, but there is probably no way other than comparison with experiment to test whether or not the ship is suffi- ciently slender for all the neglected terms (not only those mentioned by Profes- sor Laitone) to be small. 166 SLENDER BODY THEORY FOR AN OSCILLATING SHIP AT FORWARD SPEED W. P. A. Joosen Netherlands Ship Model Basin Wageningen, Netherlands ABSTRACT A linearized theory is developed for an oscillating slender body which is moving along a straight line on the free surface of an ideal fluid. Green's function is used to formulate the velocity potential. Some as- sumptions are made about the order of magnitude of the Froude number and the frequency with respect to the slenderness parameter. The firstorder term of the potential is derived byasymptotic expansion. INTRODUCTION During the past few years several papers have been published on the subject of slender body theory for surface ships. In the slender body theory the beam- length ratio « is supposed to be small with respect to unity and of the same order as the draft-length ratio. This is in contrast with the thin ship theory, where only the beam-length ratio is assumed to be small. The principal task of the theory is to provide the expansion of the velocity potential in terms of the Slenderness parameter «. Ursell [1] has solved the problem of an oscillating slender body of revolu- tion at zero forward speed for the case of small and moderate frequency param- eter as well as for the case of large frequency. He derived two terms in the series expansion. Newman [2] followed another approach, suggested by Vossers [3] starting from Green's theorem. He treated the problem of an oscillating slender body of arbitrary shape at small or moderate frequency in the presence of incoming waves. He derived the first order terms of the velocity potential and of the forces and moments. A difficulty arises in the equation of motion for pitch and heave, because it appears that the force due to hydrostatic pressure and the Froude-Krylov force is of lower order than the hydrodynamic forces (added mass and damping). A similar result was obtained already by Peters and Stoker [4] in the thin ship theory. 167 Joosen Simultaneously Joosen [5] derived the solution of the same problem without waves, also using Green's function for two conditions. More precisely for the case where the frequency parameter is of order unity and for the case where the frequency parameter is of order <«'!. In the first case the final formula for the velocity potential consists of two terms, one term corresponding to the prob- lem of a pulsating double body in an infinite fluid and another term representing the longitudinal interference effects. In the second case the result leads to the conclusion that the flow in each cross section is independent of the flow at other sections. It is therefore a rigorous justification for the use of the two-dimensional strip theory such as is applied by Grim [6] and Tasai [7], who calculated the added mass and damping coefficient for a family of cross section curves. The agreement between theoretical values and experimental data is very good, of course, especially for the higher frequencies. The problem of a slender body moving at a steady speed on the water sur- face has also drawn attention. Vossers was the first who attacked the problem starting from the three-dimensional formulation with Green's theorem. Using the method of inner and outer expansions, Tuck [8] solved the problem for a body of revolution. Starting from the formulation with Green's function Joosen [5] obtained the solution for a body of arbitrary shape under the condition of straight vertical lines at bow and stern. It appears that the influence of the end point terms is dominant and that the series expansion is not uniformly conver- gent for arbitrary shape of the bow and stern line. From the numerical value of the wave resistance it can be concluded that the results are not in closer agree- ment with the experiments than the Michell theory. The reason for this seems to be the behaviour near the end points and the fact that the Froude number is in most practical examples of the same order of magnitude as the slenderness parameter. The more satisfactory formulae for this case will be obtained as a by-product of the present work. The result contains only integrals along the bow and stern line. In the following sections the full problem of an oscillating slender body at forward speed will be considered. In the usual strip theory forward speed ef- fects and three-dimensional effects are not present. In the past several authors have considered the forward speed effect in damping and cross-coupling co- efficients; see Grim [6], Korvin-Kroukovsky [9]. Although this work seems to be in good agreement with experimental data (Vassilopoulos [10]), a consistent theory, based on a rigorous asymptotic expansion of the three-dimensional for- mulae is still lacking. Recent experimental work of Gerritsma [11] has shown the relatively small effect of forward speed on the total value of damping and added mass coefficient for heave and pitch, but an important influence on the distribution of the damp- ing over the ship length. In order to verify these results an asymptotic theory is set up in this paper with the assumptions that the Froude number is of order e1/2 and the frequency parameter of order «'!. It is expected that the result consists of that of the two-dimensional strip theory extended with some terms representing the three-dimensional and forward speed effects. 168 Slender Body Theory for an Oscillating Ship The case that both parameters are of order unity is also treated here. Al- though the same difficulties in the equations of motion can be expected as in the corresponding problem with Froude number equal to zero it is nevertheless worthwhile to carry out the calculations in order to get some insight in the range of validity of the theory. FORMULATION OF THE PROBLEM In the coordinate system used in the following the x,, y, plane coincides with the free surface. The origin moves with the ship speed Vv in the same di- rection as the ship and the z, axis is taken positive in upward direction. The hull surface in equilibrium position is assumed to be of the form Va = LCase Zap eSem Yique (221) As an additional condition the bow and the stern have the shape of sharp wedges. Between B! and S! the bottom of the ship is flat. The length of the ship is L, the beam B and the draft T. A cross section contour is denoted by C(x,). The bow contour and the stern contour are denoted respectively by |}, and !’,. Only heaving and pitching motions of the ship are considered, which are harmonic in time with angular speed ». The same pro- cedure can be followed for swaying and yawing motions. In the inviscid fluid a velocity potential exists defined by ®(X1,¥,,2,>t) = SxS + O(X,,Y15Z,,t) ’ (2:2) $(X,,¥,,Z,,t) must satisfy the Laplace equation AD = Dis (2.3) the linearized free surface condition for z, =0 On Ores = NC a Bez. 0 (2.4) and the boundary condition on the hull 169 Joosen BS ey Ge ay a eco (2.5) where H(x,y,z,t) = 0 is the part of the ship under the water surface at the time tie The following dimensionless quantities are introduced: aly ¥ ib e IL PUB x, = S1> PSG a UhIe, Za Ga Sie (2.6) E Ve wh wB oV er Oa sep) Sow = ary NCS Gye obey = ab! Sie = el” €, = Oe ”’ Y 7 Tails The displacements and rotations of the ship are supposed to be small and consequently the problem can be linearized. Rec ipere eNOS | ot aller cae (2.7) H can be written in dimensionless form as H(E,7, 6,t) = e{n- £(€,D} - o(€)- Ws) fre *** + (eo). (2.8) Because of the linearity of the problem it is possible to split up 4(x,,y,,2,,t) in a time-dependent term and a term independent of t: 2 Lie Le -ia P(X15¥412,>¢) = € O.(S,27Ma0 Ga) + E2 = O5((Ean Mins Sa) e€ ° (2.9) ae 2a After substitution of (2.8) and (2.9) into (2.3)-(2.5) and omitting higher order terms the conditions for 9, and 9, are obtained: Nin = Or Ap, = 0 (2.10) fOr wan Ob: Py Pie 2, + Pre, = 0, =E5, Gi, = BG, Pree, + Di EYD ae + Por, : (2.11) for 4 = (Sas Ge eee Or ae iL. ae t noe is Die gee f, ae ee = hp ayo’ ae ous Pan, a ane ies am a = S16 So Wo S1) “ee He ot ee 5 ee ey (2.12) 170 Slender Body Theory for an Oscillating Ship Of primary interest is the leading term in the series expansion with respect to e of 6, and 5,. Before starting this derivation it is necessary to introduce some statements as to the order of magnitude of o, £, and ¢, with respect to e. In this paper two cases will be considered. Te Gh Se Bes eB Fe = (2:13) with 8, = 0(1) and &, = 0(1); Lie or Pema OCIe nes 01) (2.14) The first case is related to the problem of low Froude number and high fre- quencies and one may expect that it corresponds with a ship moving in head waves with small wave length. The second case deals with the problem of high Froude number and low or moderate frequencies and it seems to be a good approximation for a ship moving in following waves with moderate wave length. As far as the magnitude of the frequency parameter is concerned it is of course evident that the ratio wave length to ship length is of much importance. The potential 9; can be written in the form 1 Q; = | dé | BiCs; C) Ga Gans Baap (S) dé 1 c(€) where G; is Green's function for the free surface condition and F(é, 2) is the source distribution to be determined from the boundary condition (2.12). A further notation is introduced: oO. = 9) + 9; (2.15) with 1 QM = | dé | BCG .6) -1 e(€) 1 z ne ———_—_————eeEe jar (2.16) gia sere Cig fot (a Ow Ca eyo meer, 8) er( le Oe and 1 We = | az | FG OnG. (SE 29 Sess ©) 6 - (2:17) a e(€) tet Joosen The formula for G, can be found, e.g., in [4]: GAG Waker Soo) = C,§o)ati(s,-s 6 2 ie oo q pe 1+) ati 1 )q cos eae hata, = ENG 6} dq = ae | eS 0 IG.Ce COSO 2 Dye cose + & =G i = 6 9 ia ao { a ReC Sa sche CS i €)qcos ae ee eae 6} da a, M, B)d° cos? + Qyq cos + & = q (£,+¢6)q-1(€,-€)q cos 9 Tia eee) $* 1 1 ee eG Gj f\acine ad is M E.or GOS" = Dye cose + Ee =a 2 with ee! cos O, Sat (E765 6m) = [Gl mo ono], (2.19) Ls If the roots in the denominators are denoted by q,, q,, q;, q,, the contours M, and M, are defined by 1- 2y cos 6+ iV4y cos 6- 1 Ne. Cos 1 Sse cos @ ay —- . l= By cos @ = 1 Way cos GS i Gg - dg = Shi ag 28, cos?é 1+ 2y cos 6-1 + 4y cos @ q — ‘ 28, cos*é 1-- 2y cos 6-1 = 4y cos 0 q = : 2 BRECOS ae cos 6 < —. Ye 1 - 2y cos 6 +1 - 4y cos @ =e cos 24 1+ 2y cos 6+ V1 + 4y cos @ Dercosac GW, - 172 Slender Body Theory for an Oscillating Ship The first order term in 9, is well known, see [5]: Q) = -2 { {an VOg=* Ey= 0? an Jona EO? | FE yO) al. (2.20) c(€,) In the next sections the corresponding term in 9, and 9, will be derived for the two cases (2.13) and (2.14). THE CASE OF LOW FROUDE NUMBER AND HIGH FREQUENCY First the potential 9, for this case will be considered. From that result the final form for 9, can easily be obtained. The Greens’ function G, is trans- formed into a slightly different form by separating the poles and introducing a new variable. Greets 6. o) = \ 1 2 @ tC +t) +18 = la | d ( = 5 9%) T our with il (Ean = Pe 1 Daya + V¥ (Dez = Dy jie ae 403, | and 1 EER) Se re Dy V@ee- Dye + 4D: ps (3.7) The first order term in (3.5) originates from the neighbourhood of the sta- tionary point and therefore if (€,,6,) is an interior point, this term becomes: i See Nas qB(q, 6, )e* mide finn =a eco ye SE =OGee ene 5 = | Féy, Oat | Sameer at ¢ 0) alc ( 0?) 46 ECS) L, = {ey = 00 = re | Fé Dat | eu sva Ooi coe RUS Ee gigs ore (3.8) ae 2 (q-1) yPQ Inserting for D(é,@) the expression (3.6) the result becomes cos @5) = OE), Dez = O(€), Deg = sna Oe). (3.9) Ea = Ep Key Dog = XC) % D(Sie oe =. 2 (ORS BEC = 8! With (3.7), (3.8) and (3.9), for (3.5) the result SSS F(é,, Ode BG Gay ae (3.10) és ee : ih 2 Gqrall is obtained. From (3.1) and (3.10) the first order term of 9, follows easily: 175 Joosen dq €3(6,+¢)q oR tes cos {fa(71.~- fa} Grq - (3.11) IEG) = a | FE, Ode { ey) The By changing the integration contour (3.11) is transformed into: 2 (oye wea ee ileal = 4 | F(E,, Ode aD A? © ies ree SE.) 0 +| aes In (Gi) > Cl GE a. (3.12) Following the same procedure as discussed before it appears that the first three integrals of (3.1) with €, = 0 produce no first order contribution to the value of o, except for the case that bow and stern region are of order «. The last two terms of (3.1) become with ¢, = 0 G, = 2 ee ee ee eee ee ere WERS@)— eG D> ee et Gy GO) After expansion with respect to « and with the condition of sharpness at the end- points for 9, is obtained: Tie = ate Ber Inv) Canoe (G4) 1 +4] sen(é,-€) In 2lé,-elae [Fee at. (8.13) = il c(€) The function 9, +9, can be considered as the potential associated with the translatory motion of a body in an unbounded medium. The function F(é,¢) can be determined by the boundary condition (2.12). A discussion of the results of this section with a view to experimental re- sults obtained elsewhere, will be postponed till section 6. THE CASE OF HIGH FROUDE NUMBER AND MODERATE FREQUENCY The velocity potential 9, for this case is already known (see e.g., [5], [8]). 176 Slender Body Theory for an Oscillating Ship = 1=-2] F(é,, 6) {in Von, fie ee CG aan ee int Crk £7 a (+5? | at c(€,) raf ag | H(E) Odo sence ae c(é) lé,-4| o 2 s(n x ( rc + N}3 ° Pa Oy, | fo) | Oy Sa ! The Greens function (2.18) associated with the potential », is written in the form: 2 = Ci ee, G5 Gay, (4.2) GREG eR Os ECan O)- where C,t+f)qtiq(é,-& g eens pe CRS CD nea 3 {e(n,- f)a sin 6} dq pany -2 6 A (q, 0) G.= Saag dé 5 (ELGl? Coste =: Ohi] GOS @ + Gi = pes C,+¢ i - 2) -9 fe. ao { Aa; 9) es a any Seah pata cos {e(7,-f)q sin 0} dq p i [Exe COSTE 4 2G GOS OE = Ue p €(C,+¢f)q-iq(é,-€) cos 9 ; % =| /2 ao BCehOe— 1 cos {e(n,-f)q sin 6} dq (4.3) Bade cos?@ - 2yq cos @ + g.= Aves ydancose te) yqicosio). | (By 8 = )G.q7 cose 2 yqncas] (4.4) 4 The term containing G, in 9, is integrated with respect to €. The first order term of 9, can easily be obtained by putting « =0 in the formulae with G,, because all the integrals remain convergent. It is assumed that { F(1,0)dt = f Rtas = 00 e(1) c(-1) The result becomes: 221-249 O - 66 - 13 ie Joosen OA Erasmas Sa) =a (G55) In WCqa= fie 1 (Sy + OV dc e(£,) 1 +4 | sencé,-4) In 2lé,- €lae [ Fe.oae ail c(&) 1 1 ’ d = d F ’ G d ’ 4.5 + ut dé oe t)G, dt IF As 1 OO) alt (4.5) where -iq(é-€,) cos og G -2 { De 40 [ A,(q; 9) e 7 A 6 Bal" cos*6@ + 2yq cos 6 + Gal =4 4 Q 5 ft ene ae aa x ral a | ea CE ee ee TU Ss in (SAS COSTE! 4 Ohy7cl COS Ob E= Gi y 1 i > 6 s 1/2 B (eye €,) cos ig a Ge eee (4.6) i 0 M, By a? cos*@ - 2yq cos 6 + ey -q with A, = B, = ¢, A, = =i1(/SnGl cos 6 + 2y) , B, = i(/8iy I cos 6 -2y). (4.7) By changing the integration contour and introducing some new variables (4.6) can be transformed into a formula that is more convenient for computation: 1 2 3 GP eGo tenet iG ee: n n n v where (1) ty ypecs 1/2 Gon = /2 | RCP) {( vag emg? + m,) 0 1/2 -p|é4-¢ d - i sen m, (Vn? + nz - m,) hesels | ae ae m2 + m” BOGS aa 0F 4 -i st (E-€,)7 B) 1 d Cease sin | tye 1° — 0 /Cha + 2y)* =4he 72 178 Slender Body Theory for an Oscillating Ship d © i = (€-€,)7 = sid | Diy ee” beta eat oy 1 Var - 2y~)* = 4d? 7? with 2 mee, 2,pe) —)(4y77o0) pt . om al er yb Sens oan, Ra = ly = (Cy See Se See aif Ree oP sen CS = Sis 27/2), ieee an a) ¥ L, = -i(tr+ 2), c, = i(S7- 2). Dye ee Gee y= tray, For y < 1/4: and Bor y > 1/4: and =496 cs ae Eb 0 1 = flee Phy a» Vile oye C= 2) ey 1 2) ay Se n= Le Dey Stes + Be {/8(S7+ I(1-T) +i8r sen(é-€,)} Ni(7)e dt : (87 + 2)(87+ 1)(1-7)T 179 Joosen Na= 4. N, = /S(omiy@ia) sen(e—4, e- 5 (87 2y- D> — 47 le For €, = 0 these formulae become: Sas lea=6l G, = 0, oe afte (5 )= {2 = sen(é,- 0) a z ) 0 0 which is in agreement with (4.1). For £, = 0 the result is: Ce OF, Ch = ae inary acl ACGulenstGl er 2 linen Ga= Sly - This has been obtained already in the papers [1], [2], [5]. THE ADDED MASS AND DAMPING COEFFICIENT The varying part of the pressure exerted by the water on the body equals: 2 -1io P(x, y, Zit) = = gLpi 94(¢, 1), C) 3 1 we Poe(S; 1), GD) e€ : * (5.1) D) a Considering only the heaving motion the vertical force acting on the ship be- comes: A re Ve ’ = Bos = gL? pie a dé i f(€, 6) { PC b) 3 = BacCEn Of dé e(€) a =m Zin + mize ; (5.2) where 8m, et : SF eg Ae = Me = Ee = = 1m | dé | fy be - 2) dl (523) 4c&, by “1 c(E) : Nea Al RAI -ne fs ae i ap \ die ae peer sere. 2 c(é) fu are the coefficients for added mass and damping. For the case I these coefficients are already calculated for a family of cross section curves, see [6], [7]. 180 Slender Body Theory for an Oscillating Ship In order to obtain the three dimensional and forward speed effects the terms originating from (3.2) must be added to (3.12). A comparison of these results with Gerritsma's experimental data show a qualitative agreement. The influence of forward speed, as expressed in (3.2), involves F(é,¢). This func- tion assumes positive values at the bow, negative values at the stern. The deviations from the midship behaviour in Gerritsma's results show the same character. For the case II the coefficients can be obtained by computation of the re- sults of section 4. If the forward speed is zero M, and N, become: 35 A Mea fe, ft ee tet, [ED Imveay- 7 HCGHD? 26, So -1 alae) 1 63 i =e | End, | bxD sen(é,-5 In 2le,- Elae = il -1 e3€. 1 1 oat i b(é,)dé, I; b(é) {H,(é,1é,- 41) + ¥,(& 1€,-€1)} dé (5.5) ere 1 1 Nein 5 b(S1)dey { BCE) T(E, 18, - £146 (5.6) -1 = il Here b(é) is the beam at the point é. The damping coefficient and the part of the added mass that depends on the frequency is calculated and represented in the graph below. For b(x) is taken b(x) = 2 cos 7 x and ¢€ = 0.2. 0.016 O 0.012 -0.004 No va 0.008 -0.008 0.004 -0.012 fo) 1.0 2.0 3.0 aOum ——_ gL Joosen Up till € ~ 2.5 the curves have a character that can be expected for three- dimensional bodies. It can be compared, e.g., with the curves for a sphere cal- culated by Havelock [13]. Experimental data are only available for frequency parameters higher than 2.5, but there obviously the theory is not valid anymore. CONCLUSIONS It appears to be very useful, in dealing with the problem of a slender ship performing oscillatory motions at different forward speeds, to express the Froude number and the frequency parameter in terms of the slenderness pa- rameter ¢«. For practical purposes the range of low Froude number and high frequency parameter is most interesting. In this paper the first order term of the velocity potential is derived for the case where the Froude number is of order «//? and the frequency parameter is otvorder wear! : The theory presented here can easily be extended such as to determine the motion of a slender body in waves. Then a consistent pair of equations of mo- tion for heave and pitch will follow. The analysis of section 4 is resulting in the two dimensional strip theory if the slope of the bow- and stern-line is of order unity or smaller and the only problem then is to solve an integral equation for each cross section separately. If the slope is larger three dimensional and forward speed effects are present as well. The resulting integral equation can be solved by an iteration process, but an alternative method is to start the analysis from Greens’ theorem instead of a source distribution on the hull. Apart from the problem of the oscillatory motion of the ship an interesting result is obtained for the steady advancing slender ship. For the case where the Froude number is of order «!’? the only first order contribution to the velocity potential and the wave resistance originates from the source distribution on bow- and stern-line. From this fact it becomes clear that it must be possible to affect the wave resistance by adding another singu- larity in the bow and stern region. The strength of the singularity might be determined from a condition of minimum wave resistance. By adding a dipole at the bow the concept of a bulbous bow could be treated in the frame work of slender body theory.* *See comments by Laitone on paper by Newman and Tuck. 182 10. ie 12. 13. Slender Body Theory for an Oscillating Ship REFERENCES . Ursell, F., "Slender Oscillating Ships at Zero Forward Speed," Journal of Fluid Mechanics, 1962. . Newman, J. N., ''A Slender Body Theory for Ship Oscillations in Waves, Journal of Fluid Mechanics, 1964. . Vossers, G., 'Some Applications of the Slender-Body Theory in Ship Hydro- dynamics," Thesis, Delft, 1962. . Peters, A. S., and Stoker, J. J., ''The Motion of a Ship,"’ communication on pure and applied mathematics, Vol. X, 1957. . Joosen, W. P. A., "The Velocity Potential and Wave Resistance Arising from the Motion of a Slender Ship,'' Seminar on theoretical wave resistance, Ann Arbor, 1963. Grim, O., ''A Method for a More Precise Computation of Heaving and Pitch- ing Motions Both in Smooth Water and in Waves," Third Symposium on Naval Hydrodynamics. . Tasai, F., 'On the Damping Force and Added Mass of Ships Heaving and Pitching,"' Journal of Zosen Kiokai, July 1959. Tuck, E. O., "The Steady Motion of a Slender Ship,'' Thesis, Cambridge, 1963. Korvin-Kroukovsky, B. V., ''Pitching and Heaving Motions of a Ship in Reg- ular Waves,'' SNAME, 1957. Vassilopoulos, L., "The Analytical Prediction of Ship Performance in Ran- dom Seas,'' Publ. Massachusetts Institute of Technology, 1964. Gerritsma, J., and Beukelman, W., "Distribution of Damping and Added Mass Along the Length of a Ship Model,"’ Publ. Shipbuilding Laboratory, Techn. University, Delft, 1963. Jones, D. S., and Kline, M., "Asymptotic Expansion of Multiple Integrals and the Method of Stationary Phase," Journ. of Math. and Phys., 37, 1958. Havelock, T., ''Waves Due to a Floating Sphere Making Periodic Heaving Oscillations," Proc. Roy. Soc., London, A231, 1955. 183 pricey | pes ‘gts (ome pee noteawens ; - Yen 7 ; 5 “are Pec ee i , f t i d bi ee. gies ie Uk Cee ery Tea at ar Sih. “ix Shih i 5 eT yy Wee: 2s rh Fs ) Weshieay ; VitS Lah is — . ost - itty] an et af a eee A > ’ aes a “em Uk a, Pir. tas SRG RL ETAL: Oe cand * DOr tes aie Who TSO PRS erway tec hi prietei Liawk 2 i aheint ‘ashi = ROCF 34 AohAER He ais ‘ ; 4 ‘ . re iN 4 : ~~ Nave a! ANG elIO aie % eMart YECR sabsestce # ys aft ' i = a care) ’ GP Aen e 4 ART a . mo " \ t ice Ay, tf Pil i ' ~ HK shade Y ‘ ok hg toy ae tetys ¢ ‘ iy Ray petore i G re pele ‘ “7 Wh ya sasa x 1 i 8) bea Maps oe ee Ea re ae, j ’ ~ 45 0 wn 2 29 ; 7 ; { aL eae | i , , ta heh y y ¥ sis « en) j Ry de hee! , ht f 1 hens Af i revs i j a tee ’ ve + oad Fy re} Lae ft Tee vy ye Xf oe Piet Corre Be } ia heen erie od he bh} Roz : i i) ‘ » pees i ASA ite 6k i L . , r ~ ri hie ; om { F k Thursday, September 10, 1964 Afternoon Session SHIP MOTIONS Chairman: R. Brard Bassin d'Essais des Carenes de la Marine Paris, France Applying Results of Seakeeping Research Edward V. Lewis, Webb Institute of Naval Architecture, Glen Cove, Long Island, New York The Distribution of the Hydrodynamic Forces on a Heaving and Pitching Shipmodel in Still Water J. Gerritsma and W. Beukelman, Technological University, Delft, Netherlands A New Appraisal of Strip Theory Lyssimachos Vassilopoulos and Philip Mandel, Massachusetts Institute of Technology, Cambridge, Massachusetts Some Topics in the Theory of Coupled Ship Motions J. Kotik and J. Lurye, TRG Incorporated, Melville, New York Known and Unknown Properties of the Two-Dimensional Wave Spectrum and Attempts to Forecast the Two-Dimensional Wave Spectrum for the North Atlantic Ocean Willard J. Pierson, Jr., New York University, New York, New York 185 Page 187 219 253 407 425 ayn Sees ie ou aha saith fevis deur ih, u et 7 2MONOM SiH oes Mg . rs, a te Ptiwat a ne Ep gh soamehe Sal Biase “Oo niege CHaTy BIT es 4 / . \ 7 e . f . ey 1 ye ' - it rt treet ac if | } : act. ace Siunaut ¢ ae ¥ winiel ae ae “ oe q 7 ive hai 197 innybotbyit ort 30 ‘ete telat Live i shone : % ti iti ar HAs ‘ve ret J eek (ite & [ haben ty ~ . = as yxo9 J? lo ioe tip ate Rah Aor Winn’ Bits AI af} Hae oY AAA wineudonendM oghiiceia) goles, wg / ‘Oe eu iiot Prodcjare “onal atti ate , MTitiOM ate rode: ‘a a PEt oe! fe, si ; es ian vai) fxroiation Mi-awl oii lo eorMtane tT One f encdénamiin-dwt 416 14298109 OF. Saye AT Ex) a way) SiN Sia 3 4Ov1 ei OF crt + G aravintl #yo AGF. at er ; 7 % \ uyoy ver - i i r i .. ; 3 “ws a ul ho y di t ! P4, as: 5 yr Pe ul Ly i . i Sets hay y a 104) oie! \ ae UES Lie utr APPLYING RESULTS OF SEAKEEPING RESEARCH Edward V. Lewis Webb Institute of Naval Architecture Glen Cove, Long Island, New York ABSTRACT Although developments in the theory of seakeeping are still continuing rapidly, this paper points out that presently available research results can be effectively applied to practical problems of ship design. The most useful tool is the method of superposition whereby almost any ship response to irregular short-crested seas may be predicted--pro- vided the responses to regular waves are known. Pending the develop- ment of completely satisfactory methods of calculating these responses theoretically at all headings to waves, results of systematic model tests can be used. A calculation procedure to be followed in making such predictions of ship behavior in irregular waves is outlined, and typical results of cal- culations are presented. These include trends of wave bending moments with ship size, speed, and heading. In the same manner, trends of rel- ative bow motion are presented under the influence of similar factors. Some general conclusions are drawn regarding the effects of ship size, proportions, speed, and heading on seagoing performance of ships. Needs for further oceanographic data, systematic model tests in waves, and advances in seakeeping theory are outlined. Future possibilities in the use of such research in developing improved naval ships are ex- plored, with particular emphasis on the optimization of ship designs in relation to seagoing performance. INTRODUCTION Professor B. V. Korvin-Kroukovsky in the introduction to his classic paper on the theory of ship motions in regular waves [1] called attention to the need at times to apply "vigor" as well as "rigor.'' The emphasis of this symposium has been rightly placed on rigor — on refining and improving our theoretical tools for calculating the motions of ships in waves. This paper, along with certain other presentations, meanwhile, attempts to demonstrate that the application of vigor — even with our presently available tools — can yield valuable conclusions for the guidance of ship designers. 187 Lewis The basic theoretical tool available to us is the principle of superposition first applied by St. Denis and Pierson [2] to the study of ship responses to ir- regular seas. The essential empirical data that make it workable are system- atic model tests, such as those of Vossers [3,4], and observational data on ocean wave spectra, such as those of Pierson and Moskowitz [5]. However, for prac- tical application of even the best theory it is necessary to have a suitable calcu- lation procedure. This may or may not be programed for electronic computer computation. Furthermore, for practical people to accept the results of such calculations it is necessary that they be able to visualize the factors involved and understand the trends obtained. It is the purpose of this paper to describe a convenient procedure whereby the performance of a ship in realistic irregular seas can be predicted and then to show the sort of trends and conclusions that can be obtained by the method. The work discussed here has been carried out largely in connection with research sponsored by the American Bureau of Shipping and Society of Naval Architects and Marine Engineers. The paper itself has been prepared under ONR Grant Nonr(G)00063-64. NON-DIMENSIONAL REPRESENTATION It has been previously pointed out [6] that the dimensional characteristics of the conventional form of presenting sea spectra and ship response curves make it difficult to understand and interpret the results of the calculations, par- ticularly when comparing geometrically similar ships of different size. Accord- ingly, a quasi-non-dimensional method of presentation was developed at Webb Institute based on a sea spectrum showing component wave slopes as a function of the logarithm of wave length [6]. Since the original proposal was made, it has been found that a suggestion of Dr. Y. Yamanouchi to use log. instead of log, \ results in a truly non-dimensional representation which appears more suitable for general adoption. Here » is circular frequency, 27/T, T is wave period, and \ is wave length. In this log-slope scheme not only is the sea spectrum independent of the units used, but geometrically similar ships will have similar response operators. Hence, it will be shown that the effect of ship size and form, sea spectrum shape, etc., can be clearly visualized. It is unnec- essary to convert to frequency of encounter as originally proposed [2]. In order to explain the new form of presentation, reference is made first to Fig. 1 showing the transformation of a typical wave amplitude spectrum (a), [r(w)]? vs w, to log-slope form (c). The first step is the transformation from w to log,«w base. This is accomplished by finding the increment on log, scale, 5(log, ~) that corresponds to $a, thus: d (log, ~) dw 6 (log, ~) oo) (a) Hence, for an incremental area to be the same in both systems, 188 Applying Results of Seakeeping Research [r(log.«)}* = [r(w)]? . It so happens that the range of log. of general interest to us is negative in sign. (2) 62- (nor SPECTRUM (PIERSON) (2) [ries Oo 0.2 O¢ 26 28 ZO [rilos. «)]” 20) [rw] 2 |rltegee)]. WAT ane 4 2 ge [rllasee]- 5 Srey 0 23 -4 -6 -8 -10 =j2 -|4 Loge. wW Fig. 1 - Transformations of sea spectrum 189 Lewis Finally, for the present purpose the spectrum must be transformed from amplitude (b) to slope (c) form. In general, maximum wave slope is 27¢,/). Since A = 27g/w?, maximum slope can be expressed in terms of , ae © g a where ¢, iS wave amplitude. The square of the amplitude of a wave component is given by:* [r( log, «)] 25 log, @ where [r(log,«)] 2 represents the wave spectral ordinates on the log,» base. Therefore, the square of the slope is given by: a [r(log, w)]? 8 log, ®. g? Hence, if the spectral ordinate plotted in (c) represents = [r(log, #)] F g an incremental area will represent the square of a component wave slope. Fur- thermore, the area under the spectrum (with finite limits) can be interpreted as a mean wave slope. The most obvious difference between the log-slope form and the conventional form of spectrum is the suppression of the spectrum peak which is so prominent in the conventional form of presentation. This calls attention to the fact that the wave components at the peak of a conventional spectrum are usually less steep than at the higher frequencies. It has been found that for many ship motions wave amplitude in relation to length, i.e., wave slope, is more important than wave amplitude directly, or energy. For such motions, the log-slope form is preferable for the study of ship behavior. For example, pitch angle is directly related to maximum slope. In fact, as wave lengths become very long and the frequency of encounter is far from reso- nance with the ship's natural frequency, pitch amplitude will approach wave slope asymptotically. The manner in which the new form of log-slope sea spectrum may be used in predicting ship responses is shown in Fig. 2 for the case of pitching motion. The figure shows the simple case of a ship heading directly into a long-crested *The original concept of [2] is used here, in which the spectrum represents am- plitude squared. In some systems a factor of 1/2 is introduced in order to represent wave energy. 190 Applying Results of Seakeeping Research a = 2 © QQ 3 ~~ (a) SEA L SPECTRUM LO (6) RESPONSE OPERATORS NX 2 15 P.4 al “> R 1.0 cs (c) FES PONSE 2 SPECTRA o, 2 ) =2 -4 -.6 =5 -/.0 =12 -|.4 Loge & Fig. 2 - Non-dimensional representation of pitching response to irregular sea 191 Lewis irregular sea (a), but it will be shown later that a short-crested sea and differ- ent headings can also be taken into account. In determining the form of the re- sponse amplitude operators, it must be recognized that the parameters describ- ing ship performance should be non-dimensional. Pitch angle is a satisfactory measure of angular motion, for it will be the same for ships of different size in comparable situations as well as being related to wave slope. Fig. 2(b) shows the pitching response amplitude operator in the form Cm = oe Mi x pee Sa with 6, in radians. It is clear that if the response operator curves are ex- pressed non-dimensionally — here pitch angle/wave slope — they will be identical in shape for geometrically similar ships at the same Froude Number. However, they are separated horizontally by an amount equal to log.,/w,. Furthermore, points at corresponding values of )/L will have the same ordinates, where L is ship length. If, as in this case, one ship is twice the length of the other, we have L, =2L,, and at equal values of \/L, \, = 2\,. Hence, o, = /2 w,, and the separation of corresponding points is log, w, - log, w, = log, #,/w, = log, V2 = 1/2 log,2 = 0.3468. Finally, we may multiply the wave slope spectrum (Fig. 2a) by the pitch re- sponse operators (Fig. 2b) to give the non-dimensional response spectra (Fig. 2c). These non-dimensional response spectra are of direct quantitative signifi- cance, since they represent (pitch amplitude)? and the mean pitch amplitudes will be a function of the areas under the curves. Similarly heaving acceleration — or vertical acceleration at any point along the length of the ship —is properly referred to wave slope. For in long waves, if we neglect forward speed, the vertical motion of the ship will approach that of the surface wave particles, whose vertical acceleration is, when expressed non- dimensionally, («*/g) ¢,. Maximum wave slope at any particular frequency is the same, for PS 2 2 r Hence, if vertical acceleration is referred to wave slope, this is equivalent to relating it to the wave particle accelerations at the particular wave frequency. The response amplitude operator for heaving acceleration (or vertical acceler- ation at any point) can therefore be expressed as ZVe |: Za Pe fol) ee | Heaving motion is somewhat different. If one is concerned with the absolute value of heaving, then the conventional wave amplitude spectrum is appropriate, with a response amplitude operator in the form: 192 Applying Results of Seakeeping Research 2 Ee eal r 2a Wave amplitude I Gn j But when a non-dimensional relationship is appropriate, one may divide by a ship dimension such as length, giving a ratio, Z,/L. This means that we con- sider two ships to have equivalent heaving behavior in comparable conditions if the ratios of heave amplitude to ship length are the same. (This is in contrast to the conventional procedure to comparing heave amplitudes directly.) The pa- rameter Z,/L will be the same for geometrically similar ships in similar waves. The response amplitude operator may be obtained by dividing by the wave slope, which is also non-dimensional, thus: ZTE al Zacealle OA a Ge |) g a This operator goes to infinity as wave length becomes very long and Z, ap- proaches infinity. Similarly, vertical velocity, Ze = w,Z,, can be non-dimensionalized by mul- tiplying Z,/L by VL/e, giving Z,//gl which is a sort of Froude number. The re- sponse amplitude operator is then, Eee) Z,/Ve |’ anl,/d oo a : This operator also goes to infinity as wave lengths become very long, but not so rapidly as the above. Multiplying the non-dimensional velocity Z,/Vel again by w, VL/g gives the non-dimensional acceleration previously discussed, Z,/g. Similarly, any other response that is non-dimensional may be related to wave Slope. For example, relative bow motion, S,, in relation to length, L, is more significant than the absolute value, S,, and therefore S,/L is an appropri- ate non-dimensional parameter. Although similar in appearance to the heave parameter, it tends toward zero in very long waves. The response amplitude operator for relative vertical velocity between bow and wave, which is of significance in relation to slamming, can be obtained by multiplying s,/L by w, vl/g, giving S_@ S Vener ae which is a non-dimensional relative velocity. The response amplitude operator then is, . 221-249 O - 66 - 14 193 Lewis Eo _ | Sa/vet |’ ant,/ > a Also wave bending moment, if expressed in non-dimensional form, may be di- vided by wave slope, giving [11]: fib I" ed where h, is effective wave height and h,/L is a non-dimensional bending mo- ment coefficient, M Ww 3 cogL BC, where M,, is wave bending moment in irregular sea (such as average or highest expected value in 10,000 cycles), C isa static bending moment coefficient = static wave bending moment in L/20 wave/pgL” B (L/20)C,,, e is mass density of water, g is acceleration of gravity, L is ship length, B is ship breadth, C,, is waterplane coefficient. An important step in the application of the superposition principle to ship behavior was taken by Gerritsma [7] when he showed that the added resistance, power, torque, or propeller revolutions in waves could also be handled in this way. However, the work of Maruo [8] had indicated that these quantities are roughly proportional to the square of wave amplitude and therefore should not be squared as are motion amplitude operators. The non-dimensional coefficient of power increase, AP, used by Gerritsma was AP pe62VB2/L oa = which is also the response amplitude operator. Swaan has applied the superpo- sition procedure to predicting trends of power and speed in waves [9], using this coefficient and a conventional amplitude or energy spectrum. 194 Applying Results of Seakeeping Research For use with a slope spectrum it is convenient to adopt the modified coeffi- cient, pec2VB Study of Gerritsma's model results [7] shows that the trend with \/L indicated by this coefficient is roughly correct for values of \/L greater than about 1.0, but it reverses for \/L < 1.0. Nevertheless, the coefficient appears to be en- tirely suitable for use with a wave slope spectrum. So far mention has been made only of the simple case of ship response to long-crested irregular head seas. The method of presenting data can easily be extended to the case of short-crested seas and any ship heading — provided, of course, that model test results in oblique seas are available. The short-crested sea is represented by a family of curves showing the magnitude of wave compo- nents coming from different directions. Response amplitude operator curves are also prepared for different wave directions, and each of the directional spectrum curves must be multiplied by the appropriate response amplitude op- erator curve. The resulting family of response spectral components can be in- tegrated to obtain a single response spectrum on a base of log,w. This proce- dure will be illustrated in the section on results. The computations required to obtain the curves that have been discussed can be conveniently carried out by slide rule or desk computer with the use of a suitable computation form. The form and procedure developed at Webb Institute of Naval Architecture, mainly in connection with work for the American Bureau of Shipping, is described in [10]. It has also been programed for solution on an IBM 1620 computer. RESULTS — WAVE BENDING MOMENTS The application of the procedures discussed above can be illustrated first by considering trends of wave-induced bending moments for a series of ships for which model results in regular waves were available [3]. This work was carried out under the sponsorship of the American Bureau of Shipping. Figure 3 has been prepared to show graphically the calculation for the case of the 0.80 block ship heading into short-crested irregular seas. The upper portion of the figure shows a spectrum based on the average of the 13 worst records reported by Pierson [5], with directional components obtained by apply- ing a "spreading function" of 2/7 cos? 1, to approximate the effect of short- crestedness. The second part of the figure shows the family of curves repre- senting the response amplitude operators derived from the model test results, each curve for the 600-foot ship length defining the response of the model to the waves coming from a particular angle. The curves are labeled with the angles indicating the responses to the same angular wave components as those shown in the sea spectrum. Each of these component response curves was derived from the model tests at a particular angle to the waves by picking off the results at the appropriate angles. 195 Lewis ANGULAR ComMPONENTS DIRECTIONAL SEA SPECTRUM AVERAGE OF IS Severe NortH ATLANntic Storms (5) =1.0 -1.5 Log, w 1000 2000 5000 A, FEET 300' 600’ 900 1200’ ~— SHIP LENGTH Suip RESPONSE OPERATORS _ YE /80° eae een INTEGRATED [CESPONSE : ‘10,000 S 600! PECTRA h (L . 046 per w 4 Se )| he O36 Note: Highest B Mt. Coerr EXPECTED IN 10000 CycLes oe » 18 PROPORTIONAL 1200' O THE SQUARE Feoot OF THE .027 AREA UNDER THE RESPONSE Curve. O -.5 -].0 -1.5 Loge W Fig. 3 - Bending moment prediction for C, = 0.80 ships 196 Applying Results of Seakeeping Research Also shown in this plot are the head sea response operators expanded to ship lengths of 900 and 1,200 feet, and reduced to 300 feet. The other angular components for these lengths have been omitted from the figure for clarity. A comparison of the operator curves for different ship lengths demonstrates the advantage of the form of presentation used in these calculations — the response operators for any series of geometrically similar ships plot as a set of identi- cally shaped curves, shifted on the log, axis according to the absolute sizes of the ships. Portions of the curves shown by broken lines are extrapolated be- yond the measured data. The product of a sea spectrum component for a certain angle y»,, and the response amplitude operator component associated with that wave direction gives a response spectrum component curve. The family of curves obtained in this way (one curve for each wave component) is then integrated over direction (angle) to obtain a single response curve. Four such integrated response curves for the four ship lengths are shown in the lower plot of Fig. 3. The angular components of the response spectra have not been plotted. Finally, the integration of a response spectrum curve over wave frequency gives the cumulative energy density, R, for the bending moment coefficient. From values of R for each ship size statistical parameters, such as the average value of the highest expected wave bending moment coefficient out of a total of N oscillations, may be calculated from the expression, h,/L = CR where the mul- tiplier Cc takes different values depending on the number of oscillations consid- ered. For example, assuming a Rayleigh distribution, Average h,/L = 0.866 /R Average of 1/10 highest h/L = 1.800 VR Highest expected h,/L in 100 oscillations = 2.280 /R Highest expected h,/L in 1,000 oscillations = 2.730 /R Highest expected h,/L in 10,000 oscillations = 3.145 V/R.: The variation of wave bending moment with ship speed is shown in Fig. 4 for a ship heading directly into a severe 62-knot spectrum [12]. It is evident that increasing the speed of a ship does not in general increase the wave bending moments. Decreasing speed can, in fact, increase the wave bending moments slightly. No consideration is given here to two other effects of speed, namely the increase in the bending moment caused by ship-produced waves as speed in- creases and the effect of speed on slamming which may increase midship hull stresses. The former causes a shift of the mean value; the magnitude of the effect of slamming requires further detailed study. The vertical wave bending moment is also influenced by the direction of the ship's travel relative to the waves. In a short-crested sea the wave components come from various directions simultaneously, so that regardless of its heading the ship reacts to waves coming from many angles. The heading of a ship is defined here as the angle between the direction of ship's motion and that of the 197 Lewis O 5 10 1S 20 Piss Supe Speep - Knovs Fig. 4 - Bending moment for series 60 ships in Pierson 62-knot spectrum as a function of speed (highest expected value of h./L in 10,000 cycles) dominant waves, i.e., of the wind. The calculated bending moments are the re- sult of superimposing the ship's response to all wave components present for each heading. The effect on wave bending moment of ship heading is shown in Fig. 5 for ships of 600-foot length in both short and long-crested seas, corresponding to the 62-knot spectrum [11]. This figure indicates that maximum bending mo- ments are reached in head seas, as expected, and are then less in realistic short-crested than in hypothetical long-crested seas. It also shows the reduc- tion in bending moments in beam seas is comparatively small when the waves are short-crested, especially for fine ships. 198 Applying Results of Seakeeping Research FOLLOW/AIG BEANT HEAD SEAS SEAS -OG SHIP LENGTH = GOO’ — — — Lowe CeesTed Sea~ G=.80 = 6.50 ] O 45 FO 135° /8O HEADING , DEGREES Fig. 5 - Variation of effective wave height and bending moment coefficient with heading (vertical bending only) The comparatively high values of bending moment calculated in beam seas seems reasonable on the basis of the principle of superposition. However, it should be noted that the application of this principle to ship behavior in short- crested seas has not yet been confirmed through model tests. It is to be hoped that facilities for generating realistic short-crested seas in a model tank will be developed by some laboratory in order to check and confirm the superposition principle. 199 Lewis The results of the calculations for tanker type vessels with C, = 0.80 in the average severe spectrum (Fig. 3) are shown in Fig. 6, which gives effective wave height as a function of length [11]. A low ship speed of Froude number = 0.10 (8.25 knots for a 600-foot ship) was considered to be a reasonable maximum Speed in an extremely rough sea. The curve crosses the L/20 line at L = 500 feet, and coincides with the 0.6L°:° wave that has been proposed from about 500 feet to 650 feet. The matching of the calculated trend with these other criteria thus provides a sound basis for the comparison of the larger ships with those of 500 to 650 feet, even if the absolute significance of the statistical parameter is doubtful. The calculated trend indicates that at lengths greater than 600 feet the increase in effective wave height with length is less rapid than is shown by the other criteria. The results for the finer ships are also shown in Fig. 6. A somewhat higher speed (Froude number = 0.15; 12.4 knots for a 600 foot ship) was used since the finer ships could be expected to make better speed in rough seas. Possible in- creased stresses caused by slamming were not included. The trend with length is similar to that for the fuller ships, and from 15 to 20% lower. Thus the bend- ing moment coefficient is not quite proportional to block coefficient, since in that case the reduction would have been 25%. However, it should be noted that fullness is already taken into account in the bending moment coefficient h,/L which includes the waterplane coefficient. RESULTS — BOW MOTIONS The trends of ship motions in irregular seas have also been investigated, with particular reference to relative bow motion. This work has been carried out under the sponsorship of the Society of Naval Architects and Marine Engi- neers, Panel H-7 of the Hydrodynamics Committee. Calculations are based on Vosser's Series 60 model tests in regular waves [4], showing the effect of both speed and proportion. Figure 7 shows the results for a ship of C, = 0.70 and L/H = 17.5 at vari- ous speeds in short-crested head seas, using one of the severe sea spectra used in the bending moment study [12]. It may be seen that the response amplitude operator peaks increase steadily with speed. They also move to the right with increasing speed, which has a favorable effect — because of the downward slope of the wave spectrum. However, the overall effect of speed is unfavorable, as shown by the response spectra at the bottom of the figure. Figure 8 shows ina similar way the effect of varying the L/H ratio when heading into the same sea at constant speed. It may be seen that the reduction in height of the response amplitude operator peaks with increasing L/H results in a corresponding reduction in response spectra. The trend with ship speed is shown more clearly in the upper part of Fig. 9. Also shown in the figure are two points from Fig. 8 for ships of different length/ draft ratio at the same speed. 200 SUOTJETNUIIOFZ JYSTOY 9APM T9YIO UTM uostiedurods ut ‘seas peay IP[NPIaIIAIT p9jSeI1d-JLOYS B9IDAVS UT JUSTOTJZION YOoTq Qg'O pue 09°0 Jo sdtys roy payndutos yysuez drys yzIM “Y yYstey eAeM BATIOEZZO JO puery, - 9 ‘81g “4d “HL5NAT Applying Results of Seakeeping Research 201 285 - LHDIBH JAY SAILOFZ445— °y Lewis DIRECTIONAL SEA SPECTRUM TOA Pierson 62 «tr (/2) =O% -04 -O.65-08 0 -"'2 loge W FESPONSE SYVIPLITUDE OPERA TORS -02 -04 -06 -08 -|.0 1.2 Lage u) a. we INTEGRATED LESPONSE SPECTRA Gas i) b} /o= 180° (HEAD SEAS) aL ‘y= 1750 eh 05: Gs = 0.70 L= 500’ TFS (OKO) SOAS” -0.2 -04 -O06 -08 ~\.0 -12 ’ Log. w Fig. 7 - Relative bow motion, series 60 ship in Pierson 62-knot spectrum at various speeds 202 Applying Results of Seakeeping Research 30 %S Directional SEA SPECTRUM : Cae re Pierson 62 Kt. (12) 3 es 1.0 . RESPONSE AMPLITUDE ia « OPERATORS Ns - 180° Me Caen es 0.0 -0.2 -0.4 - 0.6 -0.8 -1.0 Loge wd ks} x INTEGRATED F7ESPONSE SPECTRA = 1,0 ‘ 4.2 | GO (HEAD SEAS) . C= .70 Series 60 SuHiPs 0.0 -0.2 -0.4 -0.6 -0.8 -10 = Lag, Fig. 8 - Relative bow motion of series 60 ships in Pierson 62-knot spectrum showing effect of L/H 203 Lewis S = RELATIVE BOW MOTION ra = i) Wrenite (% = AVERAGE OF 4g HIGHEST H es Me VALUES OF Sy ee 10 Cn - Laas ee i ae fe) x ese = ‘hy = 24.0 = 2 10) Sy. Ly io 05 H |@ 1.28 KT 11.0 ]428.3'[38.9'] 52.50° 17.5 |500.0 | 28.6 | 47.50 24.0| 555.5 | 23.1 0) 0 5 10 15 20 25 SHIP SPEED (KNOTS) 100 Cp = 0.70 Ly = 7.00 A = Bac TONS 75 FBD @ BOW | —=E = 0.09 2) lJ o Lyy = 24.0 Ny = 24. & 50 ae ye 5 FOREFOOT uae) EMERGENCE Re eee xs Se 25 ary Lyy= 1.0 FOREDECK IMMERSION Lyy = 24.0 beans —_Ly, = 17.5 + H fe) Sr as fo) 5 10 15 20 25 SHIP SPEED (KNOTS) Fig. 9 - Relative bow motion for series 60, = 0.70 ship in Pierson 62-knot spectrum, trends with speed Of more direct interest in evaluating a ship's seagoing performance than relative bow motion are two derived quantities: (a) Foredeck immersion, as an index of shipping water. (b) Forefoot emergence, as a rough indicator of possibility of slamming. For a particular forward freeboard or draft these quantities can easily be worked out statistically from the response spectra. A convenient form of pres- entation is in terms of percentages of cycles of motion in which the foredeck is immersed or the forefoot emerges, as shown in the lower part of Fig. 9. It is interesting to see from this figure that increasing speed is even more unfavorable to wet decks than was suggested in the upper part of Fig. 9. For increasing the speed from 7-1/2 to 20 knots almost doubles the frequency of 204 Applying Results of Seakeeping Research foredeck immersion. It also shows a big advantage of L/H = 17.5 over L/H = 11.0. In all cases bow freeboard is 9% of length. Considering the question of forefoot emergence, Fig. 9 shows again a dis- advantage in speed. Conversely, it shows that slowing down will always amelio- rate the situation. However, it also shows a distinct disadvantage for a slender ship with high L/H value. This is because, although the shorter ships have more relative bow motion, their greater draft serves to reduce the frequency of bow emergence. Definite conclusions regarding slamming cannot be drawn, however, because the form of the more slender ships involves less flat of bot- tom and therefore less tendency to slam when the bow does emerge. Further investigation is clearly needed, but the calculation procedure described does in- dicate the trends of forefoot emergence. Finally, the effect of ship heading can be considered. Figure 10 shows the trend of relative bow motion with ship heading for the case of one particular ship at one speed. The improvement shown in behavior as the bow falls away from the sea is to be expected, but it is perhaps surprising to see such small changes for all headings between a beam and a following sea. FOLLOWING BEAM HEAD ga SEAS SEAS 05 .04 RMS SL Ol 0 45 90 135 180 HEADING (DEGREES) Fig. 10 - Relative bow motion effect of heading in Pierson 62-knot spectrum series 60, C, = 0.70 ships 205 Lewis CONCLUSIONS A method of computing the response of a ship to irregular waves is avail- able which is non-dimensional and convenient for graphical presentation. Use of this log-slope form of plotting shows that for most ship responses the wave components at the peak of the wave spectrum may be less significant than the shorter wave components. It also shows that the cut-off point, or maximum wave length present, is of considerable importance. Samples of the application of the procedure lead to certain general conclu- sions: Non-dimensional wave bending moment coefficients in very severe seas show a distinct downward trend as ship size increases. Wave-induced bending moments in severe storm seas are affected relatively little by increase of speed, but relative motion between bow and wave is appre- ciably affected. High values of length/draft ratio show a distinct advantage in this respect which leads to less shipping of water forward with a given freeboard/length ratio. But possible danger of slamming from greater forefoot emergence should be considered. Change of heading has a significant effect on wave bending moments, but much more so in long-crested than in short-crested seas. Relative bow motion is greatly reduced by a change from head to beam seas. FUTURE POSSIBILITIES It is of interest to consider some of the further possibilities in the applica- tion of results of seakeeping research. One of the obvious steps being under- taken at M.I.T. [13] and elsewhere is to make use of calculated response ampli- tude operators instead of model test values. This requires perhaps some further refinement in ship theory along the line of work by Grim [14] and Gerritsma [15]. It also requires that the theory be extended to oblique seas in order that short- crestedness can be properly taken into account. Preliminary investigation of this important problem indicates that it may not be too difficult [16]. Lalangas [17] has shown that pitching and heaving motions in oblique seas can be predicted quite well simply by allowing for the effect of heading on effective wave length, frequency of encounter, and ship-wave interaction effects such as "Smith effect." In due course it will be possible to evaluate the seagoing performance of any number of alternative designs entirely by electronic computer. Meanwhile, systematic model tests at all headings to regular waves can provide the needed inputs (response amplitude operators) into our calculations. For ships of very unusual characteristics, such as semi-submerged types for supercritical operation [18], model tests are the only reliable basis for the cal- culations. The excellent work of Vossers [3] should be extended to cover a wider range of ship characteristics and speeds. From the viewpoint of naval ship design, the need for systematic model tests is particularly great, for very little complete information is now available. For example, many reports on 206 Applying Results of Seakeeping Research naval ship motions give pitch and heave amplitudes in regular head seas, but no information of phase angles that would permit relative motion between bow and wave to be computed, or vertical acceleration at any point along the length of the ship. Furthermore, motions in oblique seas are unknown. It is to be hoped that vigor will be applied here in systematic experimental work. A related development of great importance is the application of electronic computers to the preliminary "feasibility study" stage of ship design. Pioneer- ing work in the field of merchant ship design [19] is now being applied to the naval design problem. The outstanding result of this work to date is the clear demonstration that, insofar as the ship design problem for ideal, calm water conditions is concerned, there are many possible technical solutions. Assuming certain required characteristics, such as payload, range, and speed, the princi- pal technical requirements to be met, which can be expressed in equation form, are: displacement, volume, stability, and freeboard. But the number of ship variables to choose from — as dimensions, fullness, power, etc. — is much greater. In short, there are more unknowns than there are equations, a situa- tion which is disturbing to a mathematician but intriguing to the naval architect who discovers he has a wider freedom of choice than he had previously realized. Following the traditional trial and error approach, the designer was apt to feel, when a satisfactory compromise of all the factors was reached, that this was the only possible design solution — or at least that he could not depart far from it. But results show [19] that very wide variations in overall dimensions are possible with only slight changes in the cost criterion used (capital charges plus fuel). The significance of all this to the seakeeping problem is that the availability of a method of realistically evaluating the seagoing performance of widely dif- ‘ferent alternative ship designs opens the door to definite improvements in the economic efficiency of merchant ships and the military effectiveness of naval vessels. The procedure is visualized as follows for the case of a destroyer- type ship whose primary mission is patrol duty in the North Atlantic, for exam- ple. A wide range of possible ships is determined, each of which has the re- quired speed, payload, and range. The potential performance of each design is then predicted on the basis of some criterion such as percentage of time that a stated speed or speeds can be attained at sea without shipping water. Cost fac- tors and operations research techniques must finally be brought into the picture to ascertain which design is best from the viewpoint of military effectiveness. It is my firm belief that the optimum ship designed in this way will not be the same as that designed for minimum displacement, minimum power on trial, or other purely technical criteria. In short, vigorous application of techniques now at hand should lead to better ships for the Navy. These future developments will be greatly enhanced in value if much more complete information on ocean wave spectra encountered on various trade routes becomes available. The excellent work of Pierson [5] is only a beginning. Fur- thermore, there is a real need for additional short-crested sea spectra, such as those obtained by the National Institute of Oceanography in Britain [20]. Here again vigor in obtaining and analyzing ocean wave data is the most urgent need. 207 Lewis Finally, progress in applying results of seakeeping research requires much more complete information on criteria of seagoing performance for different types of ships. What values of acceleration are acceptable? How much water can be shipped over the bow before speed must be reduced? How severe can slamming be in terms of hull stress or local pressures before remedial action must be taken? For being able to predict ship performance at sea is not enough. We must be able to determine at what speeds and in what seas any particular design is satisfactory or unsatisfactory. In conclusion, it is felt that valuable tools are now available to determine significant trends of ship behavior in realistic sea conditions. It is urged that in planning research in the field of seakeeping vigorous efforts be applied to the systematic accumulation of basic data on the sea, model series results in waves, and criteria of seagoing performance. Then our future improved theories and computation techniques can be verified and applied rather than set to gather dust on library shelves. ACKNOWLEDGMENTS The assistance of various members of the Webb staff is gratefully acknowl- edged, particularly Professor Robert B. Zubaly and Mr. Roger H. Compton. Mr. Larry Liddle, Student Assistant, prepared the figures. The courtesy of Ameri- can Bureau of Shipping and Panel H-7, SNAME, in permitting reproduction of research results is appreciated. REFERENCES 1. Korvin-Kroukovsky, B. V., "Investigation of Ship Motions in Regular Waves, TRANSACTIONS OF SNAME, Vol. 63, 1955. 2. St. Denis, M. and Pierson, W. J., ''On the Motions of Ships in Confused Seas,"’ TRANSACTIONS OF SNAME, Vol. 61, 1953. 3. Vossers, G., Swaan, W. A., and Rijken, J., ''Experiments with Series 60 Models in Waves,"’ TRANSACTIONS OF SNAME, Vol. 68, 1960. 4. Vossers, G., Swaan, W. A., and Rijken, J., "Vertical and Lateral Bending Moment Measurements on Series 60 Models,'' INTERNATIONAL SHIP- BUILDING PROGRESS, Vol. 8, No. 83, July, 1961. 5. Moskowitz, L., Pierson, W. J., and Mehr, E., "Wave Spectra Estimated from Wave Records Obtained by the 0. W.S. WEATHER EXPLORER and the O. W.S. WEATHER REPORTER," New York University Research Division Report, Part I, November, 1962, and Part II, March, 1963. 6. Lewis, E. V. and Bennet, Rutger, 'Lecture Notes on Ship Motions in Irreg- ular Seas,'' Webb Report, October, 1963. 208 " 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. Applying Results of Seakeeping Research . Gerritsma, J., Van Den Bosch, J. J., and Beukelman, W., ''Propulsion in Regular and Irregular Waves," INTERNATIONAL SHIPBUILDING PROG- RESS, Vol. 8, No. 82, June, 1961. Maruo, H., ''The Excess Resistance of a Ship in Rough Seas,'' INTERNA- TIONAL SHIPBUILDING PROGRESS, Vol. 4, No. 35, July, 1957. Swaan, W. A. and Rijken, H., ''Speed Loss at Sea as a Function of Longitudi- nal Weight Distribution,"", TRANSACTIONS OF SNAME, Vol. 79, January, 1963. Zubaly, R. B. and Compton, R. H., ''A Computation Procedure for Determi- nation of Ship Responses to Irregular Seas,'' Webb Spring Seminar on Ship Behavior at Sea, May, 1964. Zubaly, R. B. and Lewis, E. V., "Ship Bending Moments in Irregular Seas Predicted from Model Tests,'' Webb Report, December, 1963. Pierson, W. J., "A Study of Wave Forecasting Methods and of the Height of a Fully Developed Sea on the Basis of Some Wave Records Obtained by the O. W.S. WEATHER EXPLORER During a Storm at Sea,'' Deutsche Hydro- graphische Zeitschrift, Band 12, Heft 6, 1959. Vassilopoulos, Lyssimachos, ''The Analytical Prediction of Ship Perform- ance in Random Seas,'' M.I.T. Department of Naval Architecture and Marine Engineering Report, February, 1964. Grim, O., "Die Deformation Eines Regelmassigen, In Langsrichtung Laufen- den Seeganges Durch Ein Fahrendes Schiff,'' Symposium Uber Schiffstheorie Am Institut Fur Schiffbau Der Universitat Hamburg, 25-27 January, 1962. Gerritsma, J. and Beukelman, W., ''Distribution of Damping and Added Mass Along the Length of a Shipmodel,"" TNO Report No. 49S, March, 1963. Lewis, E. V. and Mumata, Edward, "Ship Motions in Oblique Seas,"’ TRANS- ACTIONS OF SNAME, Vol. 68, 1960. Lalangas, Petros, ''Theoretical Determination of the Pitching and Heaving Motions of a Ship at Oblique Headings,"' Thesis, Stevens Institute of Tech- nology, 1960. Lewis, E. V. and Breslin, J. P., "Semi-Submerged Ships for High Speed Op- eration in Rough Seas,"’ Third Symposium on Naval Hydrodynamics, Sche- veningen, Netherlands, September 19-22, 1960. Murphy, Sabat, and Taylor, ''Least Cost Ship Characteristics by Computer Techniques,'' Chesapeake Section, SNAME, October 23, 1963. Canham, H. J. S., Cartwright, D. E., Goodrich, G. J., and Hogben, N., "'Sea- keeping Trials on O.W.S. WEATHER REPORTER," TRANSACTIONS OF RINA, Vol. 104, 1962. 221-249 O - 66 - 15. 209 Lewis DISCUSSION G. Aertssen University of Gent Gent, Belgium This paper is an excellent approach to the trend of the wave-induced mo- ments in extreme Seas and the investigation comes at the right moment. It is known that in extreme seas some waves are exceptionally high (heights of 80 ft have been recorded in the North Atlantic). But on the other hand strain gages applied to the stringer plating of the main deck of usual cargo ships of 500 ft showed in these extreme seas bending moments which were not greater than the calculated bending moment, the ship being poised on a trochoidal wave of a length L,, and a height L,./12, i.e., a height of about 25 ft. It has been argued that the reason for this was the ability of the ship to adapt herself to the actual shape of the sea, especially when it is considered that in this extreme sea the ship of 500 ft is hove to at a speed of about 5 knots. Prof. Lewis comes to a better explanation when applying to the bending mo- ments the superposition principle and accepting a spreading function for the en- ergy of the assumed short-crested sea. The surprising result is that for the 300 ft cargo ship having C, = 0.8 the wave induced bending moments are then quite the same as the conventional static bending moments. A second important result of this work is the deviation from the L/20 law for long ships. It was known that for these ships a smaller wave height must be taken and a wave height 0.6L°-° was proposed. This again, as Prof. Lewis shows in Fig. 6, is a very good approximation for all bulk carriers and tankers now under construction and ranging from 500 to 800 ft. There is an old rule limiting the bending stresses calculated on a basis L/20 to 5L + 500 Kg. per sq mm, L is ship length in m. This rule holds good for L = 150m where the allowable stress is 1250 Kg. per sq cm and for the largest bulk carriers up to L = 200m where the allowable stress is 1500 Kg. per sq cm, which means 20 percent more for the longer ship. This allowance of 20 percent is exactly — as Prof. Lewis shows in Fig. 6 — the error in excess when applying for large bulk carriers the static method on a base L/20. This is a support for the static calculation based on L/20, even for ships up to 200m, provided the allowable stress is given by the formula 5L + 500 Kg. per sq cm. There are other remarkable results emerging from Prof. Lewis' paper. Stresses and bow motions in the realistic short-crested sea are reduced in beam and in following seas as compared with head seas but less than would be expected and this is especially true for the stresses. The relative bow motions are roughly the same in beam and in following seas. The writer recently, in a paper 210 Applying Results of Seakeeping Research to the North East Coast Institution of Engineers and Shipbuilders,* gave the re- sults of observations on two trawlers in rough seas. It is evident from Fig. 8 of this paper that in rough seas pitching is the same in beam seas and in following seas. Rolling of these trawlers in extreme seas is roughly the same in bow seas and in beam Seas, as is evident from Fig. 11 of the same paper. Altogether the short-crestedness of extreme seas has in a certain way a smoothening effect on stresses and on motions. These irregularities — and others — of random seas al- low in certain circumstances to turn a cargo ship of 12,000 tons even in a sea H1/10 = 35 ft of which some waves are as high as 50 ft. Finally a question. A speed of 12 knots of the fine 600 ft ship ahead in an extreme sea is somewhat surprising. Has Prof. Lewis some information as to what extent this 600 ft ship was able to maintain this speed in such a severe sea? DISCUSSION G. J. Goodrich National Physical Laboratory Teddington, England Prof. Lewis has, as usual, produced an extremely practical paper. It is obvious to us all that research in seakeeping, to be of worth, must ultimately produce design data for the improvement of ship performance. I would question the practical use of pitch and heave response operators in regular waves, for such waves never exist. Uni-directional long-crested seas are rare and even one node two dimensional spectra are few and far between. The sea in general consists of multi-nodal spectra and predictions should ulti- mately be made for such conditions. However, full scale sea data on such spec- tra are almost non-existent and it is probably sufficient for the present time to consider the one node two dimensional spectrum. There is no doubt that ship operators look to those of us working in the field of seakeeping, for help in assessing new designs and we must look extremely closely at what we consider to be the important features of a design which should influence the choice of such a design. Prof. Lewis has suggested in the closing paragraph of his paper some of the important criteria which should be consid- ered. *Aertssen, Ferdinande and De Lembre: Service Performance and Sea Keeping Trials on Two Conventional Trawlers, Trans. North East Coast Institution of Engineers and Shipbuilders, November 1964. 211 Lewis I would suggest that for the merchant ship the loss in speed to be expected is of prime importance; other factors such as accelerations, bow wetness and slamming, are others that will influence the captain in reducing power and hence producing a further reduction in ships speed. It is not enough however to consider such criteria for single sea states. Predictions must be made on a long term basis, for after all, the more severe sea states occur at low probability and the consequences of high seas may be negligible in relation to the all year round operation of the ship. In conclusion I think the method of analysis and presentation in terms of log » used by the author is useful for visualizing what is happening to the re- sponses of the ship as various parameters are varied. * * * DISCUSSION H. Lackenby British Ship Research Association London, England The subject of Professor Lewis' paper is of particular interest to me, namely the application of results of seakeeping research. A considerable amount of work has been carried out on this subject over the past few years, but it has not always been very clear as to the design applications in many instances. A contributory factor in this has doubtless been the apparent complexity of the subject. Against this background Professor Lewis' paper is very timely, and I would just like to raise a point of principle which was touched on this morning. As I understand it, the essence of the theory and analysis is based on the principle of superposition and the principle of linearity, that is, relatively small angular displacements, wall-sidedness of the model or ship within the range of the motions and so on. From the practical point of view, however, I think it is the larger angles and the question of whether or not water breaks over the decks which are the more important. This state of affairs appears to be well outside the linear range, but from the discussion this morning it seems that the princi- ple of linearity applies beyond the range that one would expect. The instances quoted however have referred particularly to model tests on a destroyer form and I should be glad if Professor Lewis would care to comment on this aspect, more particularly as far as the fuller merchant ship is concerned. In other words, to what extent can we use — or perhaps one should say abuse — the linear principle and get away with it for practical design purposes ? 212 Applying Results of Seakeeping Research DISCUSSION W. A. Swaan Netherlands Ship Model Basin Wageningen, Netherlands The paper gives a clear review of the possibilities of applying the results of presently available and future seakeeping research. I do agree with the au- thor in his conclusion about the need for more data on the sea, model Series in waves and criteria for seagoing performance. Especially the lack of sea data is a great obstacle in providing useful behaviour predictions for new ship designs. It appears doubtful to use the expression ''response amplitude operator" for the power coefficients because it is something essentially different from the "response amplitude operator" for ship motions. In the case of the power co- efficient the result of the described procedure is a mean value; the zero fre- quency component of the power in irregular seas. In the case of ship motions and bending moments the results have an oscillatory character with a zero mean. Therefore it might be better to indicate the power coefficient as "re- sponse operator" and leave the expression "response amplitude operator" for oscillating phenomena. The comparison of the wave bending moment in Fig. 5 for short-crested and long-crested seas is very illuminating. It can serve as a warning against using long-crested irregular seas in an overconfident way. Figure 6 shows the highest expected vertical bending moment in 10,000 cycles. Because the ships in this diagram have lengths between 300 ft and 1300 ft and speeds from 6 knots to 18 knots one would expect the time interval cov- ered by these 10,000 cycles to be a function of ship length. This involves a dif- ferent risk when small and large ships are compared. Because the author does not convert the spectrum to the frequency of encounter it is not quite obvious in which way one can reach some definite conclusion about the time covered by 10,000 cycles. There is another difficulty involved in using the highest expected value in 10,000 cycles. The relation given in the paper between the spectrum area and the value of the average highest amplitude is only valid for a narrow band spec- trum in which no negative maxima and positive minima occur when the zero level is taken at the mean position. It seems therefore much easier to discard the use of bending moment amplitudes altogether and return to the Gaussian distribution of the bending moment values when they are determined at constant time intervals. Because the variance of this Gaussian distribution is equal to the area of the spectrum it is not difficult to determine the percentage of time in which a certain bending moment is exceeded. From this it follows that for a narrow spectrum the average highest amplitude in 10,000 oscillations is equiva- lent to the deviation (absolute value) which will be exceeded during 0.0009% of the time or 3/4 seconds per day. The number of times it occurs is left undefined 213 Lewis in this way so there is indeed no reason to use a frequency of encounter spec- trum. For a broad spectrum, containing negative maxima and positive minima, only the number of times may be different but not the percentage of time. This makes it superfluous to make any assumption on the shape of the spectrum. Therefore the interpretation of the results in this manner is more rigorous without being less vigorous. DISCUSSION L. Vassilopoulos Massachusetts Institute of Technology Cambridge, Massachusetts I cannot help but basically disagree with the philosophy behind this paper as well as the alleged usefulness of the procedures which Professor Lewis pro- poses. The points of the paper, which bear directly to the profession's real needs at present, are unfortunately obscured and are only very briefly treated while the author mainly reiterates his recently proposed technique for interpret- ing results of seakeeping research rather than applying them. Despite the fact that we are almost ready to commence an evaluation of the importance of seaworthiness considerations in preliminary ship design, there still exists a definite need for: (a) a scrutiny of the validity and applicability of the basic procedures with which the results of ten years active research have been obtained, and (b) the establishment of a generalized philosophy for applying our knowledge to the actual design process of all ship types. With respect to the first item, one notes that members of the profession on occasion fail to adhere to the fundamental notions and implications behind the St. Denis - Pierson approach. It is the writer's opinion that the present paper introduces unnecessary confusion and complication. The author seems to believe that our present procedures rest on such sure principles that we are in a posi- tion to modify and transform these principles. It is with this belief that I dis- agree. Professor Lewis has actually recast, without any formalism, the basic Wiener-Kintchine relation of the theory of random processes to suit what he terms the needs of the ship designer. He forcedly transforms the components of the equation 2 ®0(%) = |H(o,)]” ©, 5(o,) (1) where ®, ,(~,) = input function amplitude density spectrum, 214 Applying Results of Seakeeping Research system complex frequency response (system transfer function), and H(@,) ®,,(%.) = Output function amplitude density spectrum into a so-called non-dimensional form, but in so doing forgets the precise notions behind each quantity and assumptions and reasoning behind the derivation of Eq. (1). Let us examine the problem more carefully by fixing attention on the inde- pendent variable involved in Eq. (1). The frequency domain analysis of linear systems in other engineering fields is precise in the sense that the analysis in- volves a single, unambiguous "frequency.'' Unfortunately, in ship work this is not the case, for we have two "frequencies" to play with; the absolute wave fre- quency and the encounter frequency. This adverse fact causes much trouble and the ensuing complications, especially in astern seas are, of course, due to the fact that sea wave celerity is a function of wavelength. The question which arises is which is our fundamental variable and why? Professor Lewis arbi- trarily employs the logarithm of the absolute wave frequency and states that it is "unnecessary to convert to frequency of encounter as originally proposed."' The writer disagrees with this choice and suggests that the frequency of en- counter is the basic variable because of the following reasons: (a) The frequency of encounter is the frequency which the ship feels and to which it responds. (b) The ship-system is "non-stationary" and furthermore "directional." Hence, ship speed and wave direction are not simply labels to families of graphs but must be embedded in the encounter frequency. (c) The mathematical model of the ship system involves the frequency of encounter and not the absolute wave frequency. (d) Equation (1) is strictly applicable only to system functions derived from the mathematical model and relates them via the actual input density spectrum to the actual response spectrum. There is, furthermore, a delicate point in the statistical process which merits some attention. First of all there is an ambiguity as to what constitutes the actual input function to the ship system. Is it the wave or is it the load (force or moment) caused by the wave? The answer depends on the definition of the system. The physical system (the ship model), presents no difficulty and what we measure in say a unit amplitude wave system is definitely related via Eq. (1) to the wave. The mathematical system needs special care however; if the Korvin-Kroukovsky type differential equations are used, then strictly speak- ing, the calculated response must be related to the load, whereas if the Cummins- type differential equations are used the calculated response must be related to the wave. Whichever the case, however, the important point is that as regards "inputs,'' wave amplitude or wave-induced load amplitudes have a definite physi- cal meaning whereas ''wave slopes" do not. Incidentally, the area under the 215 Lewis Lewis log-type spectrum is equal to the mean squared wave slope and not the mean slope which presumably must be zero just like the mean wave amplitude is considered to be zero. There is next, a definite and precise meaning attached to the complex fre- quency response and I suggest that arbitrary interpretations had better be avoided. What the meaning of the now accepted word-response amplitude oper- ator should be, is simply the square of the response amplitude measured in its own units due to a unit amplitude of the excitation, be it wave or wave-induced load. In advocating his non-dimensional procedure, Professor Lewis is forced, on account of the large number of ship responses, to examine and adopt differ- ent parameters which will non-dimensionalize each individual response. Hence, the cause of such confusing statements like "wave slope is more important for pitch motion than wave amplitude.'"' What Professor Lewis means is that if one wants to non-dimensionalize an angular displacement he had better divide by a (dimensionless) angle such as maximum wave slope. Clearly then, because we have many and different "responses" in the ship-system case, non-dimensional- ization is of no real use and only adds to undue complication. I also fail to see the legitimacy of multiplying two arbitrarily derived func- tions in order to get a response spectrum, unless these functions indeed repre- sent quantities which specifically relate themselves to the fundamental notions behind the theory of linear systems. The advantage that the author claims is that the effects of ship size can be readily shown. But by size, Professor Lewis limits himself to length only. What about variations in say breadth or draft or water-plane coefficient when the "useful shift'' of the curves doesn't take place? Do we have to start all over again with new non-dimensionalizing ? An important final point is that in the end of our analysis, we should not be satisfied with simple families of curves. The trends, once established, are only a palliative; the really useful information to the designer is rather numbers like the ones Dr. Ochi has discussed in his paper. The author presumably makes a plea at the end of his paper that we should avoid masses of dusty information. Personally, I can think of no better way to fill drawers than by attempting to col- lect curves for all possible variables. The last section of the paper is the most interesting and it is a pity that the author did not amplify the basic problem. As I see it, there are now three things to be done before we can really say that we are incorporating our knowl- edge in ship design. The first thing is related to the oceanographers and here we must wait for their answer to the basic question. In a given year (or even better in a period of years) and over a specified ship route what are the sea spectra encountered by a ship and what is their individual time occurrence ? The second thing is to determine in numerical terms exactly what ship op- erators mean by unacceptable wetness, untolerable number of slams or unbear- able acceleration. 216 Applying Results of Seakeeping Research Third and final, we must attempt to devise an approach which will discrimi- nate between a family of ships all meeting the owner's requirements, and will choose the one that exhibits the best capacity for sustaining a preassigned speed in rough water. REPLY TO THE DISCUSSION E. V. Lewis Webb Institute of Naval Architecture Glen Cove, Long Island, New York Mr. Lackenby's comments are appreciated. In reply to his question, I would expect to find linearity apply to merchant ships as well as to destroyers. He is quite correct in pointing out that the larger angles are most important, when water is shipped over the bow or slamming occurs. However, since we are in- terested mainly in identifying when these non-linear events occur, rather than to determine how deep the bow is immersed or how far out of the water it emerges, we do not need to push the assumption of linearity too far. Mr. Goodrich suggests that uni-directional sea spectra are adequate for the present. However, we have found in our calculations at Webb that short- crestedness has a significant effect, and therefore even an approximate allow- ance for it is better than none. Mr. Goodrich is quite right in pointing out that to draw significant conclusions one must take into effect the combined effect of different sea states based on their probabilities, and he has illustrated this fur- ther step very clearly in his own paper before this Symposium. Professor Aertssen has called attention to particular features of the paper and indicated their possible implications for ship design. His comments based on his own wide experience in making measurements on ships at Sea is greatly appreciated. As for the speed of 12 knots for the 600 ft ship, this was simply the lowest speed for which model test data were available, and I doubt very much that it would be maintained in an extremely rough sea. Mr. Swaan has made a number of good points, and I concur with all of them. It is certainly true that different numbers of cycles should be used for large and small ships when comparing them over the same long period of time. However, the difference in predicted stress will not be great. Although most ship motion spectra seem to be narrow enough to make predictions of the highest expected value from a Rayleigh distribution reasonable, working directly with the Gauss- ian distribution of points in the record is a useful and simple approach. I feel that Mr. Vassilopoulos has given a little too much emphasis to follow- ing strictly the mathematics of linear systems theory without recognizing the peculiarities of the ship-wave problem. In particular, we must recognize that 217 Lewis frequencies of encounter can vary either with ship speed or wave length, and this leads to a great deal of confusion. The procedure outlined in the paper, if properly used, gives the same numerical results as the conventional procedure and therefore cannot be incorrect. Moreover, the graphs that can be prepared in the course of the work are much more meaningful than the numbers alone, as one finds with practice. Mathematics should be a tool, not a straitjacket. It is correct that the area under the log-slope wave spectrum is equal to the mean squared wave slope rather than the mean slope. I agree entirely with Mr. Vassilopoulos' closing paragraphs and have been working in the directions he suggests for a long time. Dr. Yamanouchi has made a valuable contribution* which can stand on its own as an important paper. Therefore, I shall simply thank him for presenting it to the Symposium. I wish to thank all of the discussers for their interest in my paper and for their very valuable comments. *See remarks by Yamanouchi on paper by Ogilvie. 218 THE DISTRIBUTION OF THE HYDRODYNAMIC FORCES ON A HEAVING AND PITCHING SHIPMODEL IN STILL WATER J. Gerritsma and W. Beukelman Technological University Delft, Netherlands ABSTRACT Forced oscillation tests are carried out with a segmented shipmodel to investigate the distribution of the hydrodynamic forces along the hull for heaving and pitching motions. The vertical forces on each of the seven sections of the shipmodel are measured as a function of forward speed and frequency. By using the in-phase and quadrature components of these forces, an analysis is made of their distribution along the length of the shipmodel. The experimental results are compared with the results of a simple strip theory, taking into account the effect of forward speed. The comparison shows a satisfactory agreement between theory and experiment. INTRODUCTION The calculation of shipmotions in regular head waves by using a strip theory, has been discussed in a number of papers. Recent contributions were given by Korvin-Kroukovsky and Jacobs [1], Fay [2], Watanabe [3] and Fukuda [4]. In these papers the influence of forward speed on the hydrodynamic forces is considered and dynamic cross-coupling terms are included in the equations of motion, which are assumed to describe the heaving and pitching motions. In earlier work [5] it was shown that a relatively small influence of speed exists on the damping coefficients, the added mass and the exciting forces, at least for the case of head waves and for speeds which are of practical interest. On the other hand, forward speed has an important effect on some of the dynamic cross-coupling coefficients. Although, at a first glance these terms could be regarded as second order quantities, it was pointed out by Korvin-Kroukovsky [1] and also by Fay [2] that they can be very important for the amplitudes and phases of the motions. This has been confirmed in [5] where the coupling terms 219 Gerritsma and Beukelman are neglected in a calculation of the heaving and pitching motions. In this calcu- lation we used coefficients of the motion equations, which were determined by forced oscillation tests. In comparison with the calculation where the cross- coupling terms are included and also in comparison with the measured motions, an important influence is observed, as shown in Fig. 1, which is taken from Ref. [5]. Further analysis showed that the discrepancies between the coupled and uncoupled motions were mainly due to the damping cross-coupling terms. The influence of forward speed has been discussed to some extent in Voss- ers' thesis [6]. From a first order slender body theory it was found that the distribution of the hydrodynamic forces along an oscillating slender body is not influenced by forward speed. Vossers concluded that the inclusion of speed dependent damping cross-coupling terms is not in agreement with the use of a HEAVE AFTER PITCH Fn =.20 — = DEGREES PHASE ANGLE 1.0 0.5 HEAVE AMPL WAVE AMPL. —_—_ 1.5 w ilo a =|n < 1.0 Ww x|> oO} < F\|= = —O— EXPERIMENT CIRCULAR FREQUENCY rad/sec 1 IN PHASE COMPONENT + 100 +200 +300 Fig. 4 - Components of force on section 2, pitching motion PRESENTATION OF THE RESULTS Whole Model It is assumed that the force F and the moment M acting on a forced heaving or pitching shipmodel can be described by the following equations: Heave GO er ba wet) Gz) i) = Aisin (ate) (1) Dz, + Eze GZ 25 = Mersin (@t+ 75) Pitch 2 oo U A (A+ ky yev) 0+ BO+ CO = Mz, sin (at +) (2) can Gas gO = -F, sin (at + 8) For a given heaving motion z, = z, sin wt, it follows that: Bea Sina. SM ssa a Z 40 Ls Z,@ (3) GzZa0 = he cos .a: pz. + M cos 6 a = = /eY/ = EEE Z,0* z,07 221-249 O - 66 - 16 225 Gerritsma and Beukelman Similar expressions are valid for the pitching motion. The determination of the damping coefficients b and B and the damping cross-coupling coefficients e and E is straightforward: for a given frequency these coefficients are propor- tional to the quadrature components of the forces or moments for unit amplitude of motion. For the determination of the added mass, the added mass moment of inertia, a and A, and the added mass cross-coupling coefficients d and D it is necessary to know the restoring force and moment coefficients c and C, and the statical cross-coupling coefficients g and G. The statical coefficients can be determined by experiments as a function of speed at zero frequency. For heave the experimental values show very little variation with speed; they were used in the analysis of the test results. In the case of pitching there is a considerable speed effect on the restoring moment coefficient C. C decreases approximately 12% when the speed increases from Fn = 0.15 to 0.30. This reduction is due to a hydrodynamic lift on the hull when the shipmodel is towed with a constant pitch angle. Obviously this lift ef- fect also depends on the frequency of the motion. Consequently, the coefficient of the restoring moment, as determined by an experiment at zero frequency, may differ from the value at a given frequency. As it is not possible to measure the restoring moment and the statical cross-coupling as a function of frequency, it was decided to use the calculated values at zero speed. This is an arbitrary choice, which affects the coefficients of the acceleration terms: for harmonic motions a decrease of C by AC results in an increase of A by AC/w? when C is used in the calculation. The results for the whole model are given in the Figs. 5 and 6. The results for the heaving motion were already published in [13]; they are presented here for completeness. Results for the Sections The components of the forces on each of the seven sections were determined in the same way as for the whole model. As only the forces and no moments on the sections were measured two equations remain for each section: Heave (A OP) 2. 2 ls 2. ez = PS sin (re a) 5 Pitch (4) (d* + pPY'X;) 6 +e0 + gO = =19'5, sin (wt + 5*), where pv'x, is the mass-moment of the section i with respect to the pitching axis. The star (*) indicates the coefficients of the sections. The section co- efficients divided by the length of the sections give the mean cross-section co- efficients, thus: 226 Distribution of Hydrodynamic Forces on a Shipmodel HEAVING MOTION oO oO Oo w ~ N w /oas By IN3I3144509 SNIdWvG ~ q 10 0 Oo ao wo wt N oO w/pes 64 SSWW aadav ~—» 928 64 1N319144509 SNI1dNOI*_ 3 Sh G0 7 GU (Sy Ge GT) 2295 64 IN3IDI44509 ONIIdNOI~—— a mono SAO Ody hou cece Fig. 5 - Experimental results for whole model 227 A —_—. ADDED MASS MOMENT OF INERTIA kgm sec* d__. COUPLING COEFFICIENT kg sec? Gerritsma and Beukelman PITCHING MOTION I) .—) = S) a oi a (=) 9° ul B—__. DAMPING COEFFICIENT kgm sec 1) 5 10 15 5 10 15 WwW —___,_ rad / sec Gj sees rad/sec e—. COUPLING COEFFICIENT kg sec 0 5 10 als) (W) es rad/sec Fn =.15 eee eee Fn =.20 Be eee Nisa 25 oe Fin e730 Fig. 6 - Experimental results for whole model 228 Distribution of Hydrodynamic Forces on a Shipmodel * a Lpp/ and so on. Assuming that the distributions of the cross-sectional values of the coefficients a‘, b’', etc., are continuous curves, these distributions can be de- termined from the seven mean cross-section values. In the Figs. 7, 8, 9 and 10 the distributions of the added mass a, the damping coefficient b and the cross- coupling coefficients d and e are given as a function of speed and frequency. Numerical values of the section results, a*, b*, etc., are summarized in the Tables 2, 3, 4 and 5. Table 2 Added Mass for the Sections and the Whole Model kg sec ?/m Fn = 0.15 Gerritsma and Beukelman Fn = 15 Fn = .20 CES rad/sec Z| b _ S Z| = W=8rad/sec Nee ea a : ‘W=|Orad/sec iB Bea : CA Y > —=S Fig. 7 - Distribution of a over the length of the shipmodel 230 Distribution of Hydrodynamic Forces on a Shipmodel Fn =. 15 ‘4 <— »! kgsec/m* ——& Fig. 8 - Distribution of b over the length of the shipmodel 231 Gerritsma and Beukelman In Fig. 8 it is shown that the distribution of the damping coefficient b de- pends on forward speed and frequency of oscillation. The damping coefficient of the forward part of the shipmodel increases when the speed is increasing. At the same time a decrease of the damping coefficient of the afterbody is noticed. For high frequencies negative values for the cross-sectional damping coeffi- cients are found. Table 3 Damping Coefficients for the Sections and the Whole Model kg sec/m Fn = 0.15 rad/ Sum of Whole Sections | Model 232 Distribution of Hydrodynamic Forces on a Shipmodel The added mass distribution, as shown in Fig. 7, changes very little with forward speed but there is a shift forward of the distribution curve for increas- ing frequencies. Negative values for the cross-sectional added mass are found for the bow sections at low frequencies. For higher frequencies the influence of frequency becomes very. small. Table 4 Added Mass Cross-Coupling Coefficients for the Sections and the Whole Model kg sec? Fn = 0.15 rad/ ae of | Whole Sections | Model 233 Gerritsma and Beukelman ee oe ee a Sa ese ees eeceeen em ON ee ec a Ty, ae re a a ae 1.0 0.5 1.0 OF w=4rad/sec. 0.5 W = citeloe =0.5 O -0.5 1.0 0.5 Ve rae 1.0 a Of w= “oan 0.5 ei sls Woon -0.5 O 2 Cap ah AY Ne E 0.5 Ke) eo \ & 05 = OF 2 0 % | LO® en e 2 EI YORS 1.0 (0) ™ 05 -0.5 (0) -0.5 1.0 0.5 1.0 0 0.5 -0.5 {e) -0.5 Fn = .25 Ss es es oe ee ee MD ip | | a | |S | | 1.0 / “0 OS Cae 05 W =4 rad /sec ii 0.5 a3 BN 5 -0.5 1.0 \ wivat : abeals ae! eal. O.Gen4s £ es es — EOS a 1.0 ~ 0 W = » 0.5 g-0.5 fo Sercayce a iG SUpaP=EiS 1.0 = [ON se ee ie sles alpen] AE 0.5 Se aro ms A ww abe -0.5 1.0 0.5 ON ER hes Fig. 9 - Distribution of d’ over the length of the shipmodel 234 kg sec /m ———& { Saou kg sec/m ————@> e! Distribution of Hydrodynamic Forces on a Shipmodel Fn = .I5 Fn = .20 W =6rad/sec PETE se ' Woua nu °o kg sec /m ———& BSA sf bedewel TS Sh ibe 4 ° e! kg sec/m \ 4 oun ououn i] ou Oa Fig. 10 - Distribution of e over the length of the shipmodel 235 Gerritsma and Beukelman Table 5 Damping Cross-Coupling Coefficients for the Sections and the Whole Model kg sec Fn = 0.15 ieee ee d aah 7 | Sum of | Whole Sections | Model The distribution of the damping cross-coupling coefficient e varies with speed and frequency as shown in Fig. 10. From Fig. 9 it can be seen that the added mass cross-coupling coefficient depends very little on speed. For higher frequencies the influence of frequency is small. As a check on the accuracy of the measurements the sum of the results for the sections were compared with the results for the whole model. The following relations were analysed: 236 Distribution of Hydrodynamic Forces on a Shipmodel Sa’ = a [ d'xde= L Sbaseb J e'xdx=B L =d* =.d J atxdx =p L Se =e [ bi’ xax = E. L The results are shown in Fig. 11 fora Froudenumber Fn = 0.20. For the other Speeds a similar result was found. A numerical comparison is given in the Ta- bles 2, 3, 4.and 5. It may be concluded that the section results are in agreement with the values for the whole model. No influence of the gaps between the sec- tions could be found. N oO & o € 10 a 4 is Ss 2 ogere 3 t 2 8 t | t aq so 6 18 a 0 5 10 15 ° w —®rad/sec 4 (0) 5 10 By w —® rad/sec cy 2 z N g t fe) = i (@) 5 10 5 4 WwW —® rad/sec es 0 5 10 5 WwW —® rad/sec (9) 5 10 15 | I 3 1 77) 40 : a I x 4 \ Ww (0) 5 10 15 WwW —® rad/sec b ——® kg sec/m 1) o) e@ SUM OF SECTIONS o WHOLE MODEL 10 @ -@ kg sec te) 5 10 15 (0) 5 10 15 wW—® rad/sec WwW —® rad/sec Fig. 11 - Comparison of the sums of section results and the whole model results for Froude number Fn = 0.20 237 Gerritsma and Beukelman ANALYSIS OF THE RESULTS The experimental values for the hydrodynamic forces and moments on the oscillating shipmodel will now be analysed by using the strip theory, taking into account the effect of forward speed. For a detailed description of the strip the- ory the reader is referred to [1], [2] and [3]. For convenience a short descrip- tion of the strip theory is given here. The theoretical estimation of the hydro- dynamic forces on a cross-section of unit length is of particular interest with regard to the measured distributions of the various coefficients along the length of the shipmodel. Strip Theory A right hand coordinate system x,y,z, is fixed in space. The z,-axis is vertically upwards, the x,-axis is in the direction of the forward speed of the vessel and the origin lies in the undisturbed water surface. A second right hand system of axis xyz is fixed to the ship. The origin is in the centre of gravity. In the mean position of the ship the body axis have the same directions as the fixed axis. Consider first a ship performing a pure harmonic heaving motion of small amplitude in still water. The ship is piercing a thin sheet of water, normal to the forward speed of the ship, at a fixed distance x, from the origin. At the time t a strip of the ship at a distance x from the centre of gravity is situated in the sheet of water. From x, = Vt + x it follows that x = -V, where v is the speed of the ship. The vertical velocity of the strip with regard to the water is z,, the heav- ing velocity. The oscillatory part of the hydromechanical force on the strip of unit length will be U ates me Fy = ae 4) SIN oe CBE 9 where m’ is the added mass and N’ is the damping coefficient for a strip of unit length and y is the half width of the strip at the waterline. Because Gin = Gl vs dt dx : it follows that Fe ee UN eee a eee Ree ome (5) H fo) dx (0) PBy @) > For the whole ship we find, because Distribution of Hydrodynamic Forces on a Shipmodel FS c (J dx) Za (J N‘dx) Di pie MZ (6) L L where A,, is the waterplane area. The moment produced by the force on the strip is given by My = -xFy = Gam!) 8, + (Nix - vx aus) i ae POLI (7) Because My = (J n'dx) Za (J N'xdx + vn) Zee TRIS EZ s (8) L where S,, is the statical moment of the waterplane area. For a pitching ship the vertical speed of the strip at x with regard to the water will be -x@+Vé, and the acceleration is -x@+ 2v6. The vertical force on the strip will be : ey Gene VO) NU (CxC-A VG)! Soe yO) 2 = Be eps ett or Wi es boos Nase? me dm‘ Z 5) Gite oo 1 p= n'xé + (N’x 2Vm' -xV in) B+ (2egyx+v “aR N'v)@. (9) The total hydromechanical force on the pitching ship will be (J n'x dx) 4 (J Nixdx Va) + (rss,-v J n'ax) a. (10) ib L L The moment produced by the force on the strip is given by 1 ' poe ent ; dm’ \ : dm’ ; M, = ~xF, = -m'x?6 - (x x? -2Vm'x - x?V ee (2ceyx? + Vx ae N vx) 8. (11) The total moment on the pitching ship will be Me meen (J mix? dx) 6 - (J Nix? ae) 8 = (ve I,-Vv?m-v[ Nixdx) 2, (12) L L L 239 Gerritsma and Beukelman because vy Se ae Soy { maddy. dx L L A summary of the expressions for the various Coefficients for the whole ship according to the notation in Eqs. (1) and (2) is given in Table 6. Table 6 Coefficients for the Whole Ship According to the Strip Theory (13) For the cross-sectional values of the coefficients similar expressions can be derived from the Eqs. (5) to (12). For the comparison with the experimental results two of these expressions are given here, namely: a ae dm‘ bo) = N V age (14) e’ = N'x - 2Vm’ - xV dm! dx Also it follows that and (15) 240 Distribution of Hydrodynamic Forces on a Shipmodel Comparison of Theory and Experiment For a number of cases the experimental results are compared with theory. First of all the damping cross-coupling coefficients are considered. From Eqs. (13) it follows that: 3 = [ nixax+ Vm ib (16) e = [ Nixax - vm. ib The first term in both expressions is the cross-coupling coefficient for zero forward speed. For a fore and aft symmetrical ship this term is equal to zero. For such a ship the resulting expressions are equal in magnitude but have oppo- site sign, which is in agreement with the result found by Timman and Newman [7]. The experiments confirm this fact as shown in Fig. 13 where e and E are plotted on a base of forward speed as a function of the frequency of oscillation. The magnitude of the speed dependent parts of the coefficients is equal within very close limits. Extrapolation to zero speed shows that the e and E lines in- tersect in one point which should represent the zero speed cross-coupling co- efficient. Using Grim's two-dimensional solution for damping and added mass at zero speed [9] the coefficients e and E were also calculated according to the Eqs. (16). The distribution of added mass and damping coefficient for zero speed is given in Fig. 12 and the calculated damping cross-coupling coefficients are shown in Fig. 13. pie rikg sec/m* Fig. 12 - Calculated distribution of a and b for zero speed 221-249 O- 66-17. 241 Gerritsma and Beukelman W =6 rad/sec CALCULATION ws W =8 rad /sec W=10 rad/sec W =12 rad/sec i) —<——— Fig. 13 - Comparison of calculated and measured values for e and E 242 Distribution of Hydrodynamic Forces on a Shipmodel The calculated values are in line with the experimental results. The natu- ral frequencies for pitch and heave are respectively » = 7.0/6.9 rad/sec and in this important region the calculation of the damping cross-coupling coefficients is quite satisfactory. The zero speed case will be studied in the near future by oscillating experiments in a wide basin to avoid wall influence. Another comparison of theory and experiment concerns the distribution along the length of the shipmodel of the damping coefficient and of the damping cross-coupling coefficient e. From Eq. (14): 1 el ee dm’ bey = ON SV. Eee e = Nex —2Vme x, a dx Again using Grim's two-dimensional values for N’ and m’, these distributions could be calculated. An example is given in Fig. 14. Also in this case the agreement between the calculation and the experiment is good. For high speeds negative values of the cross-sectional damping in the afterbody can be explained on the basis of the expression for b’, because in that region dm'/dx is a posi- tive quantity. Finally the values for the coefficients A, B, a and b for the whole model, as given by the Eqs. (13) were calculated and compared with the experimental results. Figure 15 shows that the damping in pitch is over-estimated for low frequencies. The other coefficients agree quite well with the experimental re- sults. RM? la Saale Gal EXPERI rad/sec Ss kgsec/m* See ke Fig. 14 - Comparison of the calculated distribution of e and b with experimental values for Froude number 0.20 243 Gerritsma and Beukelman CALCULATION CALCULATION EXPERIMENT EXPERIMENT g__. kg sec?/m A__.. kgm sec* 15 : W—.rad \/sec = 2 r E a EXPERIMENT iS CALCULATION an x 1 ! a EXPERIMEN 0 5 10 15 W—_~ rad /sec w—_— rad/sec Fn =.15 Sse aes=-— Fn =.20 ee BE nera2S Fn =.30 Fig. 15 - Comparison of calculated and measured values for a, b, A and B (whole model) LIST OF SYMBOLS aT oe : } coefficients of the motion equations (hydromechanical part), ee eal the same for a section of the ship, ADS Ge F vee } the same for a cross-section of the ship, (a. AG" Cz Block coefficient, Fn Froude number F, amplitude of vertical force on a heaving or pitching ship, 244 Distribution of Hydrodynamic Forces on a Shipmodel Fy F oscillatory part of the hydromechanical force on a heaving or pitching ship, Pp g acceleration of gravity, k longitudinal radius of inertia of the ship, IW length between perpendiculars, M,,M, amplitude of moment on a heaving or pitching ship, MyM, oscillatory part of the hydromechanical moment on a heaving or pitching ship, m’ added mass of a cross-section (zero speed), N' damping coefficient of a cross-section (zero speed), t time, V forward speed of ship, xy Z right hand coordinate system, fixed to the ship, X5,Yo.Z, Yright hand coordinate system, fixed in space, z vertical displacement of ship, x, distance of centre of gravity of a section to the pitching axis, a,8,y,6 phase angles, @ pitch angle, P density of water, ® Circular frequency, Y volume of displacement of ship, and V* volume of displacement of section. REFERENCES . Korvin-Kroukovsky, B. V. and Jacobs, W. R., ''Pitching and heaving motions of a ship in regular waves,'' S.N.A.M.E., 1957. Fay, J. A., ''The motions and internal reactions of a vessel in regular waves," Journal of Ship Research, 1958. 245 10. 11. 12. 13. Gerritsma and Beukelman Watanabe, Y., ''On the theory of pitch and heave of a ship,'' Technology Re- ports of the Kyushu University, Vol. 31, No. 1, 1958, English translation by Y. Sonoda, 1963. Fukuda, J., "Coupled motions and midship bending moments of a ship in regular waves," Journal of the Society of Naval Architects of Japan, No. 112, 1962. . Gerritsma, J., ''Shipmotions in longitudinal waves," International Shipbuild- ing Progress, 1960. . Vossers, G., ''Some applications of the slender body theory in ship hydro- dynamics," Thesis, Delft, 1962. Timman, R. and Newman, J. N., "The coupled damping coefficient of a sym- metric ship,"' Journal of Ship Research, 1962. Golovato, P., "The forces and moments on a heaving surface ship," Journal of Ship Research, 1957. Grim, O., ''A method for a more precise computation of heaving and pitch- ing motions both in smooth water and in waves,"' Third Symposium of Naval Hydrodynamics, Scheveningen, 1960. Tasai, F., a. "On the damping force and added mass of ships heaving and pitching," b. 'Measurements of the waveheight produced by the forced heaving of the cylinders," c. 'On the free heaving of a cylinder floating on the surface of a fluid," Reports of Research Institute for Applied Mechanics, Kyushu University, Japan, Vol. VIII, 1960. Goodman, A., "Experimental techniques and methods of analysis used in submerged body research,'' Third Symposium on Naval Hydrodynamics, Scheveningen, 1960. Zunderdorp, H. J. and Buitenhek, M., "Oscillator techniques at the Ship- building Laboratory,"' Report No. 111, Shipbuilding Laboratory, Technolog- ical University, Delft, 1963. Gerritsma, J. and Beukelman, W., "Distribution of damping and added mass along the length of a shipmodel,"' International Shipbuilding Progress, 1963. * * * 246 Distribution of Hydrodynamic Forces on a Shipmodel DISCUSSION E. V. Lewis Webb Institute of Naval Architecture Glen Cove, Long Island, New York This is a noteworthy paper in an important series by Professor Gerritsma and his colleagues that is of vital importance to ship motion theory. This con- tinuing work has been characterized by unerring choice of the right research subjects and by extraordinary experimental skill. The results have served to clarify the so-called "strip theory" of ship motion calculations and to provide step by step confirmation of the different elements of the theory. Thus the tre- mendous power of this comparatively simple approach to the problems of ship motions is being reinforced and the value of the pioneering insight of Korvin- Kroukovsky and others confirmed. It may not be generally realized that this type of experiment, in which forces on seven different sections are measured, is of unusual difficulty, not only because of the many simultaneous readings to be taken, but in the need for accurate determination of in-phase and out-of-phase force components in spite of extraneous noise. The authors have mastered this difficult problem. The particular value of the resulting research is in showing that when the ship velocity terms are included, excellent predictions of the longitudinal dis- tribution of damping forces are obtained. Furthermore, the nature of the cross- coupling coefficients, E and e, has been clarified by the demonstration that they should be equal at zero speed and differ only by the term +Vm at forward speeds. (Incidentally, m is not defined, but is apparently equal to - a.) Incidental features of the paper are simplifications in the coefficients, which are not immediately obvious. It is mentioned that [es Gi ake -av | m'x dx, dx which makes the B coefficient, Eq. (13), much simpler than given in (1). Also [x@ dx = [xan! = Jn‘ ox = -m (= a), dx and therefore the e coefficient is also simplified [Eq. (13)]. Hence, the simple relationship between e and E emerges in Eq. (16) and Fig. 13. It is hoped that this important work strengthening the strip theory approach will be continued, including oscillation tests at zero speed and restrained tests in waves. My congratulations to the authors for a beautiful piece of research. 247 Gerritsma and Beukelman DISCUSSION J. N. Newman David Taylor Model Basin Washington, D.C. First of all let me congratulate the authors on yet another in the series of excellent papers which we have come to expect from Delft. Certainly one of the most valuable results obtained recently is the very simple forward speed correction to the strip theory, as outlined in the strip theory paragraph, and the correlation of this theory with experiments. It would seem that all important speed effects are taken into account simply by replacing the time derivative in a fixed coordinate system by that for a moving coordinate system, or As a result, the added mass coefficient contributes both to the acceleration and velocity terms of the equations of motion, since However this process seems rather arbitrary; why not repeat it for the second time derivative, so that Ce d? 1 d ’ eit ea or easier ae el) 1 dm’ : 2 d?m' dN’ ; n'a, -(N w SB) 4, - (2e8yev ea =| Zoe It is clear from the experimental results that too much cross-coupling would result, and thus that the last equation is ridiculous both in appearance and in practical utility, but Iam left wondering why the equation used in the paper is so much better. Is it possible to give any rational explanation for this? Finally, since Professor Vossers is not here to defend himself, let me point out that, in general, forward speed will have an effect on the distribution of hydrodynamic forces along an oscillating slender body. Vossers reached the opposite conclusion only for the special case of high frequencies of encounter and very slow speeds. 248 Distribution of Hydrodynamic Forces on a Shipmodel DISCUSSION OF THE PAPERS BY GERRITSMA AND BEUKELMAN AND BY VASSILOPOULOS AND MANDEL T. R. Dyer Technological University Delft, Netherlands The paper by Vassilopoulos and Mandel rigorously examined seakeeping theory, with valuable emphasis on practical ship design. The paper by Gerritsma and Beukelman contains significant experimental results and a clear concise strip theory, thus relating theory and physical phenomena. However, the paper by Vassilopoulos and Mandel agrees only partially with Gerritsma and Beukel- man, and with Korvin-Kroukovsky. The papers were examined by this discusser with the following results: 1. Complete agreement exists as to (a) which motion derivatives appear in each coefficient, and (b) the appearance of velocity dependent terms arising purely from the mechanics of a fixed axis system. 2. Disagreement exists as to the importance of the effect of forward speed on strip theory, but this is the only point of disagreement. This disagreement led to different evaluations of some motion derivatives. Direct comparison of the coefficients in the two papers does not reveal all dis- agreement, because of the cancellation of terms due to strip theory by terms due to the mechanics of a fixed axis system. The disagreement in the strip the- ory specifically arose in two ways: (1) Gerritsma and Beukelman consider sec- tional added mass to be a function of time, as suggested by Korvin-Kroukovsky. This is a "three-dimensional correction" and is justified experimentally by a velocity dependence in the b‘ term for the three-dimensional end sections of Gerritsma and Beukelman's model. (2) Gerritsma and Beukelman consider the distance x, between the body axis origin and the hypothetical sheet of water, to be a function of time. This is independent of dimensionality. The second differ- ence is confusing; for Vassilopoulos and Mandel do implicitly take x as function of time when converting from movable to fixed axes, but do not when applying the strip theory. The strip theory of Gerritsma and Beukelman was re-derived, eliminating these disagreements. The results agreed completely with those of Vassilopoulos and Mandel. Application of integrals quoted by Gerritsma and Beukelman showed agreement between that paper and Korvin-Kroukovsky. This therefore showed no errors in Korvin-Kroukovsky's work, only disagreement with Vassilopoulos and Mandel as to the role of forward speed on the strip theory. Conversion of Gerritsma and Beukelman results to a movable axis system revealed no diffi- culties, but clearly showed which speed terms result from mechanics and which from strip theory. 249 Gerritsma and Beukelman The differences, therefore, are seen to be completely a result of a different assumption of the importance of forward speed on strip theory, independent of what axis system is used. The assumption of Gerritsma and Beukelman seems to be justified by experiment. The derivation of the equations of motion by Vassilopoulos and Mandel, due to Abkowitz, seems the most rigorous and satis- fying. However, the evaluation of the motion derivatives by Gerritsma and Beukelman, due in part to Korvin-Kroukovsky, seems to yield better results. This discusser therefore feels it most practical to use the former work to study the mathematics of motion and the latter to evaluate the motion derivatives. * * * REPLY TO THE DISCUSSION BY E. V. LEWIS J. Gerritsma and W. Beukelman Technological University Delfi, Netherlands The authors are grateful to have Professor Lewis' comments on their paper. The definition of m, which is omitted in the paper, is given by [ max =m=a. It should be noted that L L and not J xan’ = i m‘dx, ii L as suggested by Professor Lewis. The work reported in this paper was recently extended for the zero forward Speed case. These tests were carried out in a wide basin to avoid wall influence, due to reflected waves. The results support the conclusions of the present paper. Within the very near future the restrained tests in waves with the segmented model will be carried out in our Laboratory. The results will be compared with calculated values. 250 Distribution of Hydrodynamic Forces on a Shipmodel REPLY TO THE DISCUSSION BY J. N. NEWMAN J. Gerritsma and W. Beukelman Technological University Delft, Netherlands For a fully submerged slender body of revolution in unsteady motion, the total hydrodynamic force on a transverse section is equal to the negative time rate of change of fluid momentum. By taking the time derivative in the moving body axis system the expression is found. For the surface ship, it is assumed that the flow over the submerged portion of the ship is similar to the flow over the lower half of a fully submerged body with circular cross sections. Corrections are then necessary for the shape of the sections and for free surface effects. It is assumed that these corrections are introduced by using Grim's values for the sectional damping and added mass coefficients of cylin- ders having ship-like cross sections oscillating at a free surface. It is admitted that this assumption is more or less intuitive and it was clearly necessary that the assumptions being made had to be verified by experiments, as shown in the paper. The authors cannot give a similar physical interpretation of the procedure put forward in Dr. Newman's discussion; they have therefore no rational expla- nation why such an approach is not successful. In addition, the result would certainly not agree with the experiments. Vossers' results are discussed too shortly in our paper, and the authors are grateful to Dr. Newman for his additional comments. However, for the actual ship form, as tested in our case, the forward speed effect cannot be neglected, even at quite low speeds, say Fn = 0.15. For pitch, the method, as given in our paper, is valid for such combinations of forward speed and frequency that the motion of the ship in the stationary sheet of water does not depart too much from a harmonic motion (see Ref. [2]). * * * 251 : ; = es ae Ae piel : ; | jarontegd? Wea? iain od i iL ORIARS Ye a Fe RAMEN TOOT E OP tee iii LP 1 a a, . 7 5 i ia Phebe) Ih t . ey oh é ae 8 d > gf : $ rat. hm | 4 it + +} JF Tin: wigan m Orly 4 ; = i ; ws) ’ i ri] - j ‘Oi VieAY ¢ ‘ y . 1 | ‘ - i ' 7" \ +. < P , . ral i. ri ' i ‘ ‘ 2 iG IASG ‘ ts Mil & A ae pt \ | € i — AD - Ls e ee > wots . oxTOray : sa ~ i ay ' ‘ i ‘ a) 2 4 4" yee ‘ 7) ? ; ’ y ese j : 4 ‘ ‘ "x iom be | : ‘ 4 7a 1 " P 4 2 ee i 7 ' eo y , ‘4 r , ae . t i wD te ze P| = j ary ) 7 P i ) a A NEW APPRAISAL OF STRIP THEORY Lyssimachos Vassilopoulos and Philip Mandel Massachusetts Institute of Technology Cambridge, Massachusetts ABSTRACT After a brief historical review, this paper presents the results of the broad comparison between experimentally measured and theoretically computed ship motions and phase angles first reported in Ref. [6]. Tank data for a wide range of Series 60 models in regular waves were extracted from N.S.M.B. publications and correlated with model re- sponses calculated by a digital computer program which is based on the Korvin-Kroukovsky linear theory of ship motions in conjunction with Grim's latest results on added mass and damping. Seas from both di- rectly ahead and astern are considered and emphasis is paid to the ef- fects of variations in hull form shape and weight distribution. Methods which will improve the applicability of strip theory and ad- vantages to be gained by modifying its analytical description are next presented in anticipation of further development of the theory. New theoretical data on added mass and damping are also discussed. Al- though no definite statements are as yet made with regard to some apparent inconsistencies in the Korvin-Kroukovsky analysis, there is reason to believe that certain modifications and corrections can be made which will generally improve the procedure and render it more useful. INTRODUCTION About seventy years ago, Captain Kriloff laid the foundation of what today is known to be the strip theory for computing pitching and heaving motions of a ship in regular waves [1,2]. Yet, it was only in 1950 when Weinblum and St. Denis launched a new era in seakeeping research [3] that Kriloff's seldom read paper received the recognition it deserved. During the last decade, Korvin- Kroukovsky addressed himself to the problem of improving and refining the analytical procedure and in this work he was assisted by numerous complemen- tary studies made by other researchers. The culmination of all this activity led to the publication in 1960 of a guide by Jacobs et al. [5] making possible the ready application of strip theory as it was understood at that time. In a recent report [6], one of the present authors utilized strip theory es- sentially as it was set forth by Jacobs et al., with a view towards evaluating seaworthiness performance in random seas along analytical lines. The present 253 Vassilopoulos and Mandel paper rests heavily on the results reported in Part I of Ref. [6] which includes an extensive correlation of results of strip theory calculations with model re- sults for .a wide range of hull forms. The present paper also reports on the further analysis made and experience gained since the publication of Ref. [6]. In contrast to the more rigorous thin-ship, raft and slender body theories, strip theory is undoubtedly the crudest and relies on the most limiting assump- tions. However, in advocating a less rigorous approach, the proponents of strip theory were presenting to the profession a procedure for immediate practical application, something which more rigorous approaches to the ship motion prob- lem have still failed to fulfill adequately. In the words of the quotation selected by Korvin-Kroukovsky [9], the advocates of strip theory have had to truly sacri- fice rigor in favor of vigor. Strip theory has reached its present state through a series of distinct stages during which significant contributions and corrections were advanced from time to time. In fact, a perusal of its evolution indicates that the method was built on a series of estimations, adjustments, and tedious accounting. Too many approx- imations which were neglected for "obvious" reasons in the beginning had to be incorporated at a later stage and many "'essential" truths had to be finally neg- lected. The fact that the strip theory procedure was not initially developed on a rigorous physical and analytical basis caused, and still causes, much doubt as to its validity in practical applications. For example, Cummins has referred to it [7] as a "shoe that doesn't really fit." On the other hand, at this stage of progress in seakeeping research, we cannot yet afford to reject a useful device which simulates nature fairly effectively, albeit, by force. In support of the previous statement, one of the objectives of the work re- ported in Ref. [6] was to assess the degree of correlation between strip theory and experiment. It is mandatory to note that since both experimental and theo- retical approaches contain sources of systematic errors, this comparison at- tempt was characterized by the absence of a distinct norm. Thus, it is only to be interpreted as being an attempt to match the products of strip theory and ex- periment in the hope that further light will be shed. After a brief historical review, this paper presents the results of the broad comparison between experimentally measured and theoretically computed ship motions and phase angles first reported in Ref. [6]. Tank data for a wide range of Series 60 models in regular waves were extracted from N.S.M.B. publications and correlated with model responses calculated by a digital computer program which is based on the Korvin-Kroukovsky linear theory of ship motions in con- junction with Grim's latest results on added mass and damping. Seas from both directly ahead and astern are considered and emphasis is paid to the effects of variations in hull form shape and weight distribution. Methods which will improve the applicability of strip theory and advantages to be gained by modifying its analytical description are next presented in antici- pation of further development of the theory. New theoretical data on added mass and damping are also discussed. Although no definite statements are as yet 254 A New Appraisal of Strip Theory made with regard to some apparent inconsistencies in the Korvin-Kroukovsky analysis, there is reason to believe that certain modifications and corrections can be made which will generally improve the procedure and render it more useful. It is important to note that the prime objective of the authors' research ef- fort is to ascertain the importance of seaworthiness considerations in prelimi- nary design. Since, however, strip theory of all suggested theoretical approaches had been brought closest to practical application, the decision was made that it was the most appropriate building block upon which to erect further structure. This report constitutes the authors’ thoughts as to the accuracy of the strip theory as currently understood and suggestions for improvements. HISTORICAL NOTES The earliest and least refined version of strip theory was presented in Ref. [8], where the authors essentially amplified the studies of Kriloff, Weinblum, St. Denis and other pioneers in the field of ship oscillations. The major advance- ment in Ref. [8] was the inclusion of some of the cross-coupling coefficients in the equations of motion. The first complete presentation of the procedure fol- lowed in 1955 [9] and was subsequently corrected and improved two years later [10]. In this effort, various discussers of Ref. [10] and in particular Kaplan [11] and Abkowitz [12] were instrumental in pointing out certain mistakes of the 1955 exposition, while Fay's analysis [13] motivated a more accurate definition of the velocity dependent terms in the equations of motion. Finally, Jacobs [14] at the suggestion of several discussers of Ref. [9] presented a more precise expres- sion for the exciting force (and moment) and hence extended the procedure to the analytical calculation of ship bending moments, as a result of which a unified computational approach was outlined in [5]. The most recent discussion on the coefficients of the equations of motion and excitation terms appears in Ref. [15], whereas for a complete summary of the whole problem as it was understood by Korvin-Kroukovsky the interested reader is referred to Ref. [4]. Since the appearance of Ref. [6], the inclusion of hull-shape nonlinearities was achieved by Parissis [16]. This latter work represents a further refinement of strip theory and provides, with the aid of Kerwin's polynomial hull represen- tation [17], some interesting answers with regard to the validity of linearity and effect of hull-shape non-linearities on ship responses. Although this quasi- nonlinear work is valuable in its own right, it is of no direct use in statistical analysis which is solidly tied to linear systems. STRIP THEORY VERSUS EXPERIMENT The first objective of the investigation reported in Ref. [6] was to attempt to assess the accuracy of the strip theory in the form it existed at the time of writing, Ref. [5]. This was accomplished by correlating theoretical computa- tions to experimental data published by the Netherlands Ship Model Basin in Refs. [18] and [19]. The results reported in the latter publications were chosen as the main source for the comparison attempt because they contained the most 255 Vassilopoulos and Mandel systematic experimental data in seakeeping obtained to date and also because they dealt with the effects of extensive variations of hull form shape and hull weight distributions on model motions. The experimental data contained in the NSMB reports covered variations in: 1. block coefficient (C,), 2. length to beam ratio (L/B), 3. length to draft ratio (L/H), 4. longitudinal radius of gyration (k,) for motion in regular waves of height (double amplitude) equal to 1/50 the model length at four different speeds (Froude number of 0.10, 0.15, 0.20 and 0.25) and several heading angles. To accomplish the extensive calculations involved in strip theory, a com- puter program was written, debugged and used on the IBM 7094 digital computer of the Computation Center at M.I.T. For a detailed description of the program and its use, the interested reader is referred to Ref. [6]. The basic steps in- volved in the digital computation are essentially similar to the ones proposed in Ref. [5]. Thus, the whole computation is broken down into suitable packages which can be easily modified or extended if this is deemed necessary. The ship hull, however, and certain of the coefficients of the equations of motion are more accurately defined in the M.I.T. computer program than in Ref. [5]. Also, a subroutine based on Grim's theory for calculating damping and added mass co- eae was incorporated, in preference to the graphical data presented in Ref. [5]. The results of the theoretical computations were compared with only a part of the results reported in Refs. [18] and [19]. In particular, consideration was given to non-dimensional pitching and heaving amplitudes together with their associated phase angles which correspond approximately to directly ahead and astern seas. The word "approximately" is used, since the experimental results of Refs. [18] and [19] referred to actual heading angles of 10° and 170°, and therefore some corrections had to be made for direct comparison at y = 0° and x = 180°. These corrections were based on a suggestion of the authors of Ref. [3] in which the model is considered to move at a modified speed in a fictitious train of waves of the same amplitude but different wavelength. This suggestion was recently justified by Lewis and Numata [20] for the case of small heading angles. The correlation between theory and experiment was considered in two dis- tinct phases. The first phase dealt solely with the effects of variation of hull shape geometry and was accomplished for the range of hull parameters shown schematically in Fig. 1. Table 1 indicates the main particulars of the family of models chosen for the correlation. Further information required for the com- putations, such as sectional area coefficients and load waterline shape were ob- eae from Tables 4, 6, and 8 of the original paper on the Series 60 models PAL ||. 256 A New Appraisal of Strip Theory Fig. 1 - Three-dimensional config- uration of model hull parameters under examination Table 1 Series 60 Model Characteristics Model aed (tect) Gas Le (from Fo) 10.00 | 1.818 *As calculated for fresh water based on above particulars. All models have a radius of gyration kp, = 0.24L = 2.4 feet. 221-249 O - 66 - 18 257 Vassilopoulos and Mandel The results of the first phase of the correlation are reported herein in the form of graphs of non-dimensional motion amplitudes versus wave length to ship length ratio for constant Froude number (Figs. 2-57). Heave is divided by the wave amplitude h, and is considered positive upwards; pitch in radian measure is divided by the maximum wave slope (27/\)h, and is defined positive when the bow is up. Amplitudes of motions are considered positive for both ahead and astern wavelengths. Phase angles are superimposed on the same graphs and are defined as lags when referred to the maximum wave elevation amidships; their range is restricted to +0-180° only. The second phase of the correlation was concerned with the effect of longi- tudinal weight distribution on ship motions. The experimental data required in this case were obtained from Figs. 16 and 17 of Ref. [19]. In the latter work, Model C of Fig. 1 was ballasted in four additional ways so as to yield non- dimensional radii of gyration of k, = 0.21, 0.225, 0.255 and 0.270. The previ- ous discussion with regard to presentation of data applies also in this phase of the investigation with the following exceptions due to insufficient model data: a. Only amplitudes of pitch and heave were compared. b. Only directly ahead seas (x = 180°) were considered. c. The results are given for only three Froude numbers of 0.15, 0.20, and 0.25. Since Figs. 2-57 all pertain to the case of k, = 0.24, Figs. 58-65 deal with the remaining four values of k, only. Wherever the wavelengths for resonance came within the range of values shown on Figs. 2-65, arrows are drawn to indicate their critical values. KEYS TO FIGURES 2-65 Motion Motion Non-Dimensional Phase Angle Amplitude (Lag) Theoretical Experimental Theoretical Experimental 258 A New Appraisal of Strip Theory 5* (DEGREES) oO 8* (DEGREES) | | (6) (e) (eo) (o) Fig. 2 - Model A in directly ahead seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 259 Vassilopoulos and Mandel 5* (DEGREES) tees pate So a oe) (e) 5* (DEGREES) Fig. 3 - Model A in directly ahead seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 260 A New Appraisal of Strip Theory (DEGREES) e* WL +150 +100 I + a o) a e* (DEGREES) -!00 —I50 0.6 09 12 LS 1.8 ML Fig. 4 - Model A in directly ahead seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 261 Vassilopoulos and Mandel (DEGREES) ETE | oO (e) on je) fe) fe) ak +150 +100 e* (DEGREES) ier + Oo Oo wa 6) (oe) GQ © © (oe) Fig. 5 - Model A in directly ahead seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 262 A New Appraisal of Strip Theory (DEGREES) | oO je) 5* -100 -150 AL +150 +100 5* (DEGREES) re + on Oo on on (o} es © © © 0.6 0.9 1.2 1S 1.8 ML Fig. 6 - Model A in directly astern seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 263 Vassilopoulos and Mandel (DEGREES) §* +150 +100 S5* (DEGREES) et + oO (eo) oO (6)) So © © © © ML Fig. 7 - Model A in directly astern seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 264 A New Appraisal of Strip Theory 0.6 0.9 1.2 15 1.8 WL Fig. 8 - Model A in directly astern seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 265 (DEGREES) e* e * (DEGREES) Vassilopoulos and Mandel (DEGREES) 1 | 1 oa Oo a (e) (oe) (oe) * AL uJ e* (DEGRE 0.6 09 1.2 U 15 1.8 Fig. 9 - Model A in directly astern seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 266 A New Appraisal of Strip Theory §* (DEGREES) +150 +100 ! Do SOm 5* (DEGREES) te a oO S) ©) /L Fig. 10 - Model B in directly ahead seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 267 Vassilopoulos and Mandel (DEGREES) 8* AL 1 ‘4 © 8* (DEGREES) ile us a oO oO Oo 0.6 0.9 12 Re) 1.8 ML Fig. 11 - Model B in directly ahead seas, heaving non-dimensional amplitude and phase angle vs wavelength te shiplength ratio 268 A New Appraisal of Strip Theory AL 0.6 09 12 15 18 WL Fig. 12 - Model B in directly ahead seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 269 (DEGREES) e* €* (DEGREES) Vassilopoulos and Mandel (DEGREES) AL (DEGREES) 0.6 0.9 1.2 15 1.8 ML Fig. 13 - Model B in directly ahead seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 270 A New Appraisal of Strip Theory Fig. 14 - Model B in directly astern seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 271 Vassilopoulos and Mandel + oOo (e) fe) (DEGREES) +150 +100 S* (DEGREES) ae a + (6)) (oe) (6) Oo oO 3 So Oo Fig. 15 - Model B in directly astern seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 272 A New Appraisal of Strip Theory +150 +100 +50 0 lu xc 08 S = 50) << -100 -150 AL oO WwW ld c oO lJ = 0.6 0.9 L2 ie) 1.8 ML Fig. 16 - Model B in directly astern seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 221-249 O - 66 - 19 273 Vassilopoulos and Mandel (DEGREES) e* i gS o e* (DEGREES) | | a o Oo oO Fig. 17 - Model B in directly astern seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 274 A New Appraisal of Strip Theory §* (DEGREES) WL +150 +100 | + 3 BO. so 5* (DEGREES) —100 50 d/L Fig. 18 - Model C in directly ahead seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 275 Vassilopoulos and Mandel (DEGREES) 5* 1 S o 5* (DEGREES) | | Oo {e) (eo) [e) 0.6 0.9 1.2 i) 1.8 XL Fig. 19 - Model C in directly ahead seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 276 A New Appraisal of Strip Theory AJL 0.6 0.9 1.2 ie) 1.8 ML Fig. 20 - Model C in directly ahead seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 277 (DEGREES) ex (DEGREES) e* Vassilopoulos and Mandel AL Fig. 21 - Model C in directly ahead seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 278 (DEGREES) e* (DEGREES) e* A New Appraisal of Strip Theory =IS{0) Fig. 22 - Model C in directly astern seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 279 5* (DEGREES) 5* (DEGREES) Vassilopoulos and Mandel Fig. 23 - Model C in directly astern seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 280 S5* (DEGREES) (DEGREES) * A New Appraisal of Strip Theory 1.6 1.4 +150 1.2 +100 1.0 + 50 — 0.8 O 0.6 as - 50 0.4 -100 0.2 -150 0.0 D : 0.6 09 1.2 1.5 1.8 ML Fig. 24 - Model C in directly astern seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 281 e* (DEGREES) e* (DEGREES) Vassilopoulos and Mandel A/L Fig. 25 - Model C in directly astern seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 282 e* (DEGREES) e* (DEGREES) A New Appraisal of Strip Theory (DEGREES) * +150 +100 I ar oy ©) 5* (DEGREES) | | oO je) (oe) (eo) Fig. 26 - Model D in directly ahead seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 283 Vassilopoulos and Mandel +150 +100 + 50 | oS © 8* (DEGREES) =|00 -150 A7L 8* (DEGREES) Fig. 27 - Model D in directly ahead seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 284 A New Appraisal of Strip Theory e* (DEGREES) AL oO e* (DEGREES) — 50 —100 —150 0.6 0.9 1.2 LS 1.8 ML Fig. 28 - Model D in directly ahead seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 285 Vassilopoulos and Mandel ML 0.6 0.9 1.2 5 1.8 AL Fig. 29 - Model D in directly ahead seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 286 e* (DEGREES) e* (DEGREES) A New Appraisal of Strip Theory Fig. 30 - Model D in directly astern seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 287 5* (DEGREES) 5* (DEGREES) Vassilopoulos and Mandel ZL Fig. 31 - Model D in directly astern seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 288 S* (DEGREES) 5* (DEGREES) A New Appraisal of Strip Theory e* (DEGREES) | S o e* (DEGREES) Be fo) fo) —150 Fig. 32 - Model D in directly astern seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 221-249 O - 66 - 20 289 Vassilopoulos and Mandel Fig. 33 - Model D in directly astern seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 290 (DEGREES) e* (DEGREES) A New Appraisal of Strip Theory 1.2 ML Fig. 34 - Model E in directly ahead seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 291 8* (DEGREES) 5* (DEGREES) Vassilopoulos and Mandel 0.6 0.9 1.2 ie) 1.8 X/L Fig. 35 - Model E in directly ahead seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 292 (DEGREES) 8* 8* (DEGREES) A New Appraisal of Strip Theory i oO oe © e* (DEGREES) ie oO [e) (e} (oe) AL +150 Tee el + + (@) Seer ot Lom onte «* (DEGREES) Fig. 36 - Model E in directly ahead seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 293 Vassilopoulos and Mandel AL 0.6 0.9 12 ie) 1.8 X/L Fig. 37 - Model E in directly ahead seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 294 e* (DEGREES) (DEGREES) e* A New Appraisal of Strip Theory (DEGREES) | a (eo) * -100 —150 WL . (DEGREES) Fig. 38 - Model E in directly astern seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 295 Vassilopoulos and Mandel (DEGREES) 5* +150 +100 S* (DEGREES) Hic al + ()] (eo) (6)) eS oo Jd 3 © Fig. 39 - Model E in directly astern seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 296 A New Appraisal of Strip Theory (DEGREES) * +150 +100 (DEGREES) Ea + oO (e) on oO to) So © © © € *K Fig. 40 - Model E in directly astern seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 297 Vassilopoulos and Mandel e* (DEGREES) (e) e* (DEGREES) | | oO (eo) {o) (e) Fig. 41 - Model E in directly astern seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 298 A New Appraisal of Strip Theory +150 +100 + 50 (DEGREES) | (9) ] fo) (eo) 5* -100 -150 +150 +100 8* (DEGREES) ieee Dero ii oO [o) (61) Oo Oo Jj So © Fig. 42 - Model F in directly ahead seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 299 Vassilopoulos and Mandel (DEGREES) — 50 5* a 19) fe) i} S o 8* (DEGREES) ipeerle a oO ©) © Fig. 43 - Model F in directly ahead seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 300 A New Appraisal of Strip Theory e* (DEGREES) AL 1.6 1.4 +150 1.2 +100 1.0 = O-+ 50 4 re 9 0,6 05 kno Q 0.6 = 50) ~~ * 0.4 -100 ~ 0.2 -150 0.0 S 0.6 09 1.2 15 1.8 ML Fig. 44 - Model F in directly ahead seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 301 Vassilopoulos and Mandel U 3 © «* (DEGREES) lial a oOo Q © AL e * (DEGREES) 0.6 09 1.2 15 1.8 VAE / Fig. 45 - Model F in directly ahead seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 302 A New Appraisal of Strip Theory 5* (DEGREES) A/L +150 +100 +50 W c O° 2 - 50 * =;00) —|150 0.6 09 12 15 1.8 ML Fig. 46 - Model F in directly astern seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 303 Vassilopoulos and Mandel 5* (DEGREES) (e) 5 * (DEGREES) La a oO oS © Fig. 47 - Model F in directly astern seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 304 A New Appraisal of Strip Theory e* (DEGREES) WL e* (DEGREES) [Pe alk all, oS ASS Fig. 48 - Model F in directly astern seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 221-249 O - 66 - 21 305 Vassilopoulos and Mandel AL 0.6 0.9 12 LS 1.8 X/L Fig. 49 - Model F in directly astern seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 306 (DEGREES) e* e* (DEGREES) A New Appraisal of Strip Theory 1.6 1.4 +150 1.2 +100 1.0 + 50 tt [oa tog ari (a) 0.6 - 50 a (2.0) 0.4 -100 0.2 -150 D OOS Toe 09 12 15 1.8 WL +150 +100 (oe) §* (DEGREES) - 50 -—100 =I I5}0) 0.6 09 1.2 LS 1.8 M/L Fig. 50 - Model G in directly ahead seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 307 Vassilopoulos and Mandel +150 +100 + On (o) 1 Soo 5* (DEGREES) -!00 4 |Fr =0.20 + -|509 Fig. 51 - Model G in directly ahead seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 308 8* (DEGREES) A New Appraisal of Strip Theory ! on [e) slice a oO o Oo AL +150 +100 | + Oe e * (DEGREES) —!100 0.6 09 1.2 i) 1.8 ML Fig. 52 - Model G in directly ahead seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 309 e* (DEGREES) Vassilopoulos and Mandel (DEGREES) (je) «* (DEGREES) i. Me a oOo oO © Fig. 53 - Model G in directly ahead seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 310 A New Appraisal of Strip Theory -150 +150 +100 | + Bi so) 26 5* (DEGREES) —100 al) a fo) 0.6 0.9 1.2 LS 1.8 M/L Fig. 54 - Model G in directly astern seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 311 S* (DEGREES) Vassilopoulos and Mandel (DEGREES) S* +150 +100 5* (DEGREES) ae + a (e) (S)) fe) (oe) 3 fe) (e) Fig. 55 - Model G in directly astern seas, heaving non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 312 A New Appraisal of Strip Theory a LJ WW [eal oO iva) 2 * w AL +150 +100 +509 W c OB [@) -— 50 = * w —!100 -150 0.6 0.9 12 L5 1.8 ML Fig. 56 - Model G in directly astern seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 313 Vassilopoulos and Mandel e* (DEGREES) AL Fig. 57 - Model G in directly astern seas, pitching non-dimensional amplitude and phase angle vs wavelength to shiplength ratio 314 A New Appraisal of Strip Theory 0.6 09 2 1.5 1.8 A7L Fig. 58 - Model C (ky, = 0.210) in directly ahead seas, heaving non-dimensional am- plitude vs wavelength to shiplength ratio 315 Vassilopoulos and Mandel 0.6 09 1.2 L5 1.8 X7L Fig. 59 - Model C (k, = 0.225) in directly ahead seas, heaving non-dimensional am- plitude vs wavelength to shiplength ratio 316 A New Appraisal of Strip Theory 0.6 0.9 1.2 1.5 1.8 A7L Fig. 60 - Model C (k, = 0.255) in directly ahead seas, heaving non-dimensional am- plitude vs wavelength to shiplength ratio 317 Vassilopoulos and Mandel 0.6 0.9 1.2 1.5 1.8 A7L Fig. 61 - Model C (k, = 0.270) in directly ahead seas, heaving non-dimensional am- plitude vs wavelength to shiplength ratio 318 A New Appraisal of Strip Theory 0.6 0.93 12 (LS) 18 A7L Fig. 62 - Model C (k, = 0.210) in directly ahead seas, pitching non-dimensional am- plitude vs wavelength to shiplength ratio 319 Vassilopoulos and Mandel 0.6 09 12 15 18 NL Y Fig. 63 - Model C (k, = 0.225) in directly ahead seas, pitching non-dimensional am- plitude vs wavelength to shiplength ratio 320 A New Appraisal of Strip Theory O Fr =015 0.6 0.9 1.2 1.5 1.8 A7L Fig. 64 - Model C (k, = 0.255) in directly ahead seas, pitching non-dimensional am- plitude vs wavelength to shiplength ratio 221-249 O - 66 - 22 321 Vassilopoulos and Mandel 0.6 0.9 1.2 5 1.8 A7L Fig. 65 - Model C (k, = 0.270) in directly ahead seas, pitching non-dimensional am- plitude vs wavelength to shiplength ratio 322 A New Appraisal of Strip Theory DISCUSSION OF RESULTS Previous correlations between strip theory and experiment shown in Refs. [10,14,22,23] were for the case of ahead seas only and the agreement was de- scribed as very satisfactory. No direct comparison can be made however be- tween the results reported in the above references and the ones described herein, since the earlier correlations were based on more approximate hand computations and in some cases different formulations and/or experimental data for the coefficients and excitation terms were used. The current comparison for models of different hull shapes, shown in Figs. 2-57 indicates that, in directly ahead seas, reasonably good correlation is achieved for heave amplitude. An exception is found in the case of Model E, C, = 0.70, B/H = 1.57 (Figs. 34 and 35), where theory fails to reveal the cor- rect trends and grossly exceeds measured values. At the time Ref. [6] was pub- lished, it was suspected that this was due to numerical errors which probably arose for low B/H ratios in the subroutine of the computer program which cal- culates damping and added mass according to Grim's theory. As discussed ina later section, this suspicion was confirmed by subsequent analysis. Apart from this discrepancy and for all wavelengths, except those corresponding to reso- nance, agreement can be termed satisfactory. For wavelengths corresponding to resonance, theory overestimates experimental heaving amplitudes by 15-20%. The above deviations are due to underestimation of heave damping by the ana- lytical approach which is in accord with previous findings [1 0,14,23]. Agreement in directly ahead seas is much better for pitch than for heave although the trends are not the same for different models. For models A, B, C, and D (Figs. 2-33) at all speeds, theoretical results are below the experimental data and the effect is more pronounced as the wavelength is increased. In the case of Model E (Figs. 36-37) large discrepancies are not observable as with heaving motions. The best agreement in this case is found in the case of Model F (Figs. 44-45). With respect to phase angles in directly ahead seas, it will be seen that the theoretical predictions are usually higher than experiment and this is true for both pitch and heave. Since phase angles are more susceptible to both compu- tational and experimental errors, agreement should perhaps be interpreted as satisfactory whenever deviations are less than about 15-20%. Discrepancies usually occur at wavelengths equal to model length. Apparent disagreement is also observable at very short wavelengths, but this is mainly due to the manner in which experimental data have been presented in Ref. [18]. Pitching phase an- gles as computed by theory are much closer to the experimental data than heav- ing phase angles and this is particularly obvious in the case of Models F and G (Figs. 44-45 and 52-53). In the case of directly astern seas, heaving motion is underestimated by theory and the deviations increase with wavelength. For most models, agree- ment of pitching motion amplitudes in directly astern seas is excellent, although in the case of Models A and B (Figs. 8-9 and 16-17) theory is considerably lower than experiment. No comparisons have been made of phase angles in astern seas, but general indications are that theory reveals the expected trends. 323 Vassilopoulos and Mandel The second part of the current correlation which is concerned with the ef- fect of weight distribution is illustrated in Figs. 58-65. For small radii of gy- ration, theoretical heaving motion amplitudes are slightly less than the experi- mental ones, but the situation is reversed and slightly worsened as k, is increased. Agreement in pitching motion appears to be similar and the tendency here is for the experimental data to be 15-20% higher, particularly at high wave- lengths. All figures for ahead seas indicate that for wavelengths less than about half the length of the model, both pitching and heaving motions are negligible while for wavelengths higher than about twice the model length, heaving becomes equal to the wave amplitude and pitching corresponds to the maximum wave slope. The range 0.9L < A <1.5L excites the models the most. On the contrary, astern seas do not induce large responses in heave or pitch and amplitudes tend to in- crease in a linear fashion with wavelength. These findings are in accord with those of earlier investigations [24] which showed that for all wavelengths and speeds, conventional ships suffer only small responses in astern seas in con- trast to the more severe resonant responses that do occur in ahead seas. It was noted in the introduction that both experiment and strip theory, as they were utilized for the purposes of this paper, are replete with errors. The inadequacies of the strip theory as it was employed so far in this paper will be discussed in the next section. Some of the shortcomings of the experimental approach, particularly as they relate to a comparison with theory (not to a com- parison with a full scale ship responding to hypothetical regular waves) are as follows: 1. The models used by Vossers et al. [18], were free to move in all six degrees of freedom. Theory treats motion in the longitudinal plane of symmetry only and furthermore presumes the absence of surge. 2. All models tested at NSMB were equipped with bilge keels and were furthermore self-propelled. Bilge keel damping and oral lsr thrust fluctua- tions are ignored by theory. 3. Errors in measurement of wave heights. 4. Wall effects especially at low speeds and small wavelengths. THE EQUATIONS OF MOTION FOR A SURFACE SHIP MOVING IN WAVES General Remarks A constructive appraisal of a given theory is best accomplished by examin- ing the issue from different points of view. In this case, many such different points of view exist. We will therefore examine the validity of the linearized theory of ship motions as developed by Korvin-Kroukovsky by using a different approach which is backed by physical reasoning. The approach will involve two cardinal steps: 324 A New Appraisal of Strip Theory 1. Develop rigorously the linearized equations of motion. 2. Use the strip theory technique to compute the values of the coefficients of equation of motion as well as the excitation terms from elementary arguments based on the results of two-dimensional flow theory. In fulfilling the latter step, the reason why strip techniques are employed, the assumptions implicit in strip techniques, as well as the upper limit of accu- racy that can be expected from strip techniques will be considered. We will start from the basic concepts of the mechanics of rigid bodies fol- lowing Abkowitz [12,25,26]. His approach, similar to those used by aerodynami- cists, provides a concise statement of the kinematical and kinetic problem and readily identifies all of the physical mechanisms involved. After the equations of motion have been developed in an accurate manner, all that remains to be done is to determine the values of the coefficients of the equations as well as the forcing functions. It is here that use will be made of the cross-flow hypoth- esis and two-dimensional hydrodynamic theory. The reader may well at this juncture question the consistency of the paper; first an extensive investigation using an existing theory is presented and then the very foundation upon which the theory rests is questioned. This is true. However, it is only after using a certain procedure that one can really appreci- ate and question it. Furthermore, it is suspected that the inconsistencies which seem to exist in the Korvin-Kroukovsky approach will not radically affect the final result. This is probably due to the fortunate cancellation of errors, but this remains to be verified. A basic difference between the approach formulated by Korvin-Kroukovsky and the approach proposed in this paper is that hydrodynamics will be employed after dynamics have been utilized. Furthermore no attempt will be made to force fit the mathematical model to conform to experimental results, but rather a ra- tional approach will be developed with the hope that eventually, refined experi- ment will agree with refined theory. Derivation of Equations of Motion In this section the mathematical model describing the six-degree of freedom motion of a surface ship in regular waves will be developed first and the results will then be specialized for the case of pitching and heaving motions. The fol- lowing assumptions will be made in developing the equations: 1. The ship will be considered as a rigid body. The high-frequency vibra- tion modes of the hull excited by the low-frequency wave encounters will not be considered here. 2. The size, geometry, mass and mass distribution of the ship are assumed known and invariant in time. 325 Vassilopoulos and Mandel 3. Rudders and other control surfaces and mechanisms are assumed "locked" in zero position. 4. In deriving the equations of motion for pitch and heave the coupling be- tween the latter two and the other degrees of freedom is totally neglected. For seas from directly ahead or astern, this is a reasonably valid assumption. In particular, surge effects are ignored which in turn implies that propeller thrust fluctuations are negligible. 5. As a consequence of the last statement in 4., the ship speed is assumed to be constant. 6. Forces and moments due to wind action, tow-lines, etc., are not consid- ered. The external excitation is to be that due to waves only. 7. Since a linear theory will be developed, the translatory and angular de- partures of the ship from and about an inertial reference are assumed to be very small (first order). 8. The ship is assumed to be originally on an even keel. 9. Motion of the ship is assumed to take place in a given, idealized fluid which is unbounded in all directions. 10. The wave excitation is that due to uniform, infinitely long-crested sinus- oidal waves of small amplitude which come from directly ahead or astern, i.e., the direction of ship motion is taken to be normal to the wave crests. Two orthogonal, right-handed systems of coordinates will be employed in the development of the coupled pitching and heaving equations of motion. Con- sistent use of right-handed systems is advantageous because it allows a conven- ient check in the analysis by simply permuting the terms of various expressions. The first system of axes will be fixed in space with its origin located at an arbi- trary point on the still water level. This will be regarded as a Newtonian frame of reference with respect to which the wave configuration and body orientation in space will be referred. The second system of axes, usually referred to as "body" axes, will be fixed in the ship with a convenient point as origin. In rigid body dynamics, the origin of the "body" axes is usually chosen to be the center of gravity of the body. However, in the ship problem case it is usually advanta- geous to fix the origin of the "body" axes at the intersection of the midship sec- tion, the longitudinal vertical plane of symmetry and the waterplane through G. This not only simplifies the computation of the hydrostatic and hydrodynamic forces but is also convenient because the midsection plane is fixed in a ship, whereas the position of the center of gravity is variable. It is pertinent to note at this point that, from 1954 [8] onwards, Korvin- Kroukovsky assumed for simplicity in all his analyses that the vertical plane of the center of gravity and midship section coincided. Whereas, it is true that the LCG is usually a small fraction of the ship length, this simplification is some- times responsible for wrong interpretations of phase angles, leading to errors up to 10° for certain ships in short wavelengths. 326 A New Appraisal of Strip Theory For reasons of consistency and systemization in future analyses, we herein suggest the use of and shall adhere to the nomenclature of Bulletin No. 1-5 of SNAME [25]. Following the above definitions it is shown in Refs. [12,26,27] that in order to obtain separate vector force and moment equations, the principles of linear and angular momentum must be used for the center of gravity of the ship, but must be measured relative to the "body" axes fixed about the point defined previously. If we therefore denote by F the total external force and by G the total moment of the external forces about the center of gravity, then, the princi- ples of linear and angular momentum give, d F = ae (m Ua) (1) and d where Foz exe ve ele), (3) G = ik+ jM+ KN. (4) Following the principles of dynamics and carrying out the operations indicated above, it may be shown [26] that the complete six-degree of freedom motion of the ship is characterized by: X = m[u+ qw-1v-xg(q?+1r?) + yg(pq- ft) + zg(prt+4)] (5) Y = m[v +ru-pw-yg(r?+p?) + zg(ar -p) + xg(aptr)| (6) Z = m[w+pv -qu-zg(p?+q?) + xg(rp-q) + yg(ratp)] (7) K = I,p + (1,-I,) ar + m[yg(w+pv- qu) - zg(v +ru- pw)| (8) M = I,q+ (1,-I,) rp + m[zg(a+qw-rv) - xg(wtpv-q)| (9) N= Uw Ces )ypa4 m[xg(v+ru-pw) - yg(utqw-rv)| (10) where the various symbols are defined in Ref. [25]. From this general approach, it may be seen that if G, the center of gravity, is identified with the origin of the "body" axes, then the above equations reduce to the well known Euler equa- tions: X = m(u+ qw- rv) (11) 327 Vassilopoulos and Mandel Y = m(v+ru-pw) (12) Z = m(w+pv -qu) (13) Keele pen Clee) ar (14) hee, st CG Sl) ig) (15) NS Te (k= pag (16) If all other degrees of freedom except pitch and heave are now ignored from Eqs. (5)-(10) and if the center of gravity is assumed to be located on the longitu- dinal body axis at a distance x, from the origin of the "body" axes, then, the problem reduces to the examination of the coupled pitch and heave equations as given by: m(w-qu- xqgq) = Z (17) . eg +mxXgw = M. (18) By the same token, the equivalent set corresponding to Eqs. (11)-(16) becomes, m(w- qu) = Z (19) Gl M, (20) where the mqu term in Eq. (19) represents the main distinction between the or- dinary Newtonian equation with axes fixed in space and the equation of motion with axes fixed in the ship. Turning now to the examination of the loads, we note that in the most gen- eral case, the total external force F and moment G about the center of gravity must depend on: 1. The characteristics of the body 2. The properties of the fluid 3. The parameters which describe the relative motion between the body and the fluid. These may be listed as follows: 1. Characteristics of Body Characteristic length (size) Geometry Mass and its distribution 328 A New Appraisal of Strip Theory 2. Properties of Fluid Density Viscosity Surface tension Elasticity Vapor pressure Pressure Also thermal, electric, magnetic properties, etc. 3. Relative Motion Parameters Orientation parameters: x,,y,,2,,9,9,¥,h. o> Puerco wat Le. tie) is. For a surface ship of fixed geometry, mass and mass distribution moving at constant speed in a sufficiently idealized fluid, the total force and moment depend only on the parameters describing the relative motion between the body and the fluid. For pitch and heave motions of a surface ship, these are: a. Body and fluid orientation parameters: z,,0,h. oe b. Body and fluid dynamic parameters: w, q,h,w,q,h. The most convenient way of defining an unknown function is in terms of its multivariable Taylor expansion about some convenient equilibrium point. Since we will be content with developing a linear theory, only the linear terms in the expansion need be retained. A convenient equilibrium point about which to ex- pand the total force and moment and hence their components Z and M is that characterized by: (a) constant ahead speed, u = u,, and (b) zero orientation and dynamic parameters. The Taylor expansions for the heaving and pitching forces then result in: 4, = Z(h, h,h, t) + =) (2-2, + = (PE) ak (22 (w-w,) OZ O) a ine Q) beets a = CQL S CE as ae QUSWe) & e) (qi Gee (21) M = M.(h,h,h,t) + = (2-2, )+ eq (0 -6,) + (= (w- we) O26) | 329 Vassilopoulos and Mandel where the zero subscript denotes the dynamic equilibrium condition and, for reasons to be subsequently discussed, the wave action forces and moments have been lumped conveniently in Z.(h,h,h,t) and M,(h,h,h,t). Such a linearization indicates that the forces and moments acting on a pitching and heaving ship may be conveniently considered to be of two sorts: a. Wave-induced forces and moments acting on a restrained ship, and b. Forces and moments brought about by the motion of the ship in calm water. Noting that z, = 6, = w, = w, = 4, = 4, = 0 and using the notation OZ a (3), | etc., Eqs. (21) and (22) become, Z Z(b,hyh,t) + Z, zo + 20+ Zw + Zea + Zw + Z. 4, (23) = I M.(h,h,h,t) + M, Zev Meret M. w + M.a- (24) Since it is desirable to express the differential equations in terms of the orien- tation parameters z, and @ and their first and second time derivatives, an ex- pression must be found for w and w in terms of z, and 6. From the following sketch, it follows that, w = Z, cos 9 + u, sin 6 (25) or since within linearity, cos 6 = 1 and sin 6 = 6, we get 330 A New Appraisal of Strip Theory Wi SZ ee Hao (26) w= Zo+ulé. (27) Substituting Eqs. (26) and (27) in Eqs. (23) and (24), calling q = 6 and q = @ and rearranging terms in Eqs. (17) and (18), we finally obtain the coupled pitching and heaving equations of motion in the form: (m-2,) 25+ Zy2g > Zz % ~ (Z,+%g) 6 - (Z,+u,Z.) 6 oO Zo - (Zg+u,Z,)@ = Z(h,h,h,t), (28) (I, - M6 5 iE ets M.)@ = (uisee Whe ube) = (M. + mxg)Z, Me Ze SW vee = M.(h,h,h,t). (29) If the above set were developed on the basis of Eqs. (19) and (20), i.e., for the origin of the 'body" axes at the center of gravity, then the resulting equations would differ in form from Eqs. (28) and (29) only in that the mx, term would be missing in the coefficients of pitch and heave accelerations in Eqs. (28) and (29). CALCULATION OF THE COEFFICIENTS The solutions to Eqs. (28) and (29) are in principle easily obtained provided that the twelve coefficients and the two forcing functions are known. Although, rational treatment of the problem does provide a precise identification of each term, the present state of art permits only rough approximations on the basis of theory and it is to this end that resource will be made to strip techniques. De- ferring the discussion of the exciting terms until the next section, attention is here focused on the forces and moments brought about by the ship's oscillatory motion in calm water, and which are identified as the terms on the left hand side of Eqs. (28) and (29). In order to exhibit the relationship between the new approach and that of Korvin-Kroukovsky, we shall, without loss of generality, substitute for each co- efficient of Eqs. (28) and (29) a letter and thus obtain the set: a(w,) Z, + b(,) an CZ ce d(w,)6 + e(w,)6 + = Z.(h, h, h, t) (30) A(@,)6 + B(w,)8 + CO + D(w,) Zz, + E(w,) 2, + Fz, = M,(h,h,h, t) (31) ie} where «, is the frequency of encounter with sinusoidal exciting waves and hence the frequency of the forced responses also. In the interest of brevity and since the assumptions and steps to be followed in the computations by strip theory will be similar for all coefficients, we shall next examine in detail the manner in which one of the coefficients of the equation 331 Vassilopoulos and Mandel of motion can be computed on the basis of a strip technique. The extension of this approach to the other coefficients will be fairly apparent so that the values of the other coefficients will be given without derivation. Equation (28) indicates that, after linearization, the coefficient of the heave acceleration term a(,) consists in fact of two additive terms; the mass of the ship, m, which is known and is constant with time and the partial derivative of the total vertical hydrodynamic force with respect to heave acceleration, Z-. This is the force that is exerted on the body when oscillating in smooth water and its derivative is computed at the equilibrium condition characterized by a constant ship speed u = u,, and by z, = 96=w=q=w=q=-0. The statement of the problem has been given but the exact solution for the complete three-dimensional body is available only for special mathematically defined forms. Theoretical results are however available from two-dimensional theory; hence, it will be assumed that an arbitrary three-dimensional body can be replaced by the sum of a large number of two dimensional segments or strips. This is the essence of strip theory. It involves the following simplifications: a. The underwater hull geometry is defined by an arbitrary number of typi- cal sections. b. These sections are arbitrarily assumed to be equally spaced. c. To date, these sections are defined in terms of two geometrical parame- ters; the sectional area coefficient, o(x) and the beam/draft, B(x)/H(x), ratio of section or its reciprocal. d. Each of the strips is assumed to belong to a specific infinite cylinder oscillating at zero forward speed and its behavior is assumed independent and isolated from the neighboring strip. e. Longitudinal perturbation velocities which exist in the three dimensional problem are totally neglected. f. Since the available theoretical data to be used are based on an ideal fluid, viscosity is ignored. Following strip theory, +L/2 Z.. = - | Z.(@,;%) dx ; (32) -L/2 where the integrand is the partial derivative of the force on the strip, which on the basis of extensive theoretical data is defined as bBo) = Role, © INGS)- « (33) The integrand is more commonly known as the added mass of the section or strip where the constant c = 7/8. 332 A New Appraisal of Strip Theory By similar reasoning, Z,, the coefficient of the heave velocity term is ap- proximated by +L/ 2 . Vhs -{ N(x) dx (34) -L/2 where the integrand is the damping coefficient of the section and is calculated by the Havelock-Holstein [4] formula N(x) = (A)? ee? (35) a3 where A = ratio of the amplitude of the wave created by the oscillation of the body to the amplitude of the oscillation of the body. With the exception of the restoring coefficients c, C, f, F which can be evaluated on the basis of elementary hydrostatics, the remaining hydrodynamic derivatives of the equations of motion can be computed based on the knowledge of added mass and damping coefficients of cylinders of various shapes. The proposed expressions are summarized in Table 2 and compared to those devel- oped by Korvin-Kroukovsky and his associates. Since the equations of Korvin- Kroukovsky were developed with the origin of the body axes fixed at the center of gravity of the ship, the new coefficients refer to the modified set of equations in which x, is set equal to zero. Proper consideration was also given to the different definition of the total vertical force existing between the two approaches. Thus the expressions of Korvin-Kroukovsky have been corrected to allow for the fact that the total force is to be taken positive downwards. Table 2 shows that the expressions for four of the newly proposed coeffi- cients do not agree with those derived by Korvin-Kroukovsky. The differences in the Korvin-Kroukovsky coefficients e(o,), B(w,), C and E(w.) appear to be mainly due to an erroneous time differentiation of a fixed body coordinate with the result that: (a) a factor of 2 appears in the velocity dependent terms of e(@,) and B(w,), and (b) a pseudo-three-dimensional term is introduced in co- efficients e(~,), B(#,), C and E(#,). It would also appear that the introduction of terms dependent on the rate of change of added mass over the ship length is inconsistent with the use of two- dimensional theory. Despite these discrepancies however, it is expected that the final values of these coefficients will not be seriously modified since it has been shown by Jacobs et al. [5] that most of these terms which appear in the Korvin-Kroukovsky approach but not in the new approach are numerically small. It is hoped that in the near future these inconsistencies will be examined more carefully and their implications assessed on the basis of experimental data. Since most of the coefficients of the equations of motion depend on the theo- retically computed added mass and damping coefficients for two-dimensional cylinders, this matter will next be considered in some detail. The first solution of the potential problem of an infinite circular cylinder oscillating at zero for- ward speed in an ideal fluid was given by Ursell [28] and his results for added 333 Vassilopoulos and Mandel xp (x)N fen : xp x (x)a | ng + xpxX (X)N] - yovoiddy AysAoyNoIy-uTAIOY yoeoiddy man UOT}OW JO SUOCTJENb|Y JO S}UaTOTJJ90D Jo uosTIeduoD G OVAL 8 (x) 4 g PAH = (x) Ay ta031q 10g4 yUsTOT JZ }209 334 A New Appraisal of Strip Theory mass were assumed by Korvin-Kroukovsky [10] and Jacobs et al. [5] to apply to more general cylinders. However, different results were used in Refs. [10] and [5] for computing the damping coefficients: the first utilized an approximate ex- pression for A, whereas the second introduced and employed the graphical data computed by Grim in 1953 [29]. Since the publication of Ref. [30] which reviewed the state of art up to about 1955, the important problem of the oscillating cylin- der of arbitrary section has been examined and solved in greater detail, both theoretically and experimentally. For example, Grim [29], Tasai [31] and, more recently Porter [32] have extended the Ursell problem to more general cylin- ders, and have provided added mass and damping as a function of the frequency of oscillation. Damping coefficients for extreme V sections were also evaluated by Kaplan [33] using a Green's function technique, whereas TRG [34] presented their approximate method for evaluating these quantities. These theoretical studies were supplemented and verified by experimental work carried out by Tasai [31], Porter [32], and Paulling and Richardson [36] and Watson [35]. These studies were of course concerned with the small oscillatory motion of two-dimensional bodies where the effect of frequency on the distribution of damping and added mass, the effect of forward speed, and nonlinear effects are ignored. These important effects have been discussed on the basis of experi- ment in part by Golovato [37] and in part by Gerritsma [38], and Gerritsma and Beukelman [39] and others. Reference [39] has shown for example that the dis- tribution of damping along the length of a ship is appreciably affected by fre- quency and forward speed whereas the added mass distribution appears to be less affected by these parameters. Another very important point, which was anticipated from Newman's theoretical work [40] was the occurrence of negative sectional damping and added mass at certain speeds. Two-dimensional theory cannot of course predict such effects. Hence, strip theory fails to compute ex- actly the responses but more especially the bending moments in regular waves. The computer program described and used in Ref. [6] made use of a sub- routine which was based on more recent work by Grim, as outlined in Ref. [41]. His numerical results however appeared to be erroneous for certain combina- tions of sectional area coefficient and beam-to-draft ratio. This issue assumed great importance when disagreement was noted in the case of Model E as dis- cussed earlier in this paper. Furthermore, his results, as well as those of Tasai [31], are restricted to Lewis shape sections only. However, as far back as 1947, Prohaska [42] indicated that the definition of a ship section in terms of two parameters is unsufficient. This inadequacy has since been clearly demon- strated by Landweber and Macagno [43], in connection with high-frequency added mass calculations. The above points and the availability of a complete and exact analysis of the problem by Porter [32], launched a systematic examination of the problem which is currently still under way at M.I.T. by Porter and others. Some preliminary results of this work are herein included and discussed. Comparison of k, and A as calculated by two computer programs, one based on Grim [41] and another due to Porter [32], are shown in (a) Figs. 66 and 67 for semicircular cylinders of varying beam to draft ratio and, (b) Figs. 69 and 70 for the typical ship sec- Se Cee in Fig. 68. The latter figure and Table 3 are reproduced from Ref. [44]. 335 Vassilopoulos and Mandel PORTER ——--— GRIM Fig. 66 - k, versus § Table 3 Particulars of Ship Model Sections Full- Form Wide Vee Narrow Vee Bulb-Form Figures 66-70 indicate first of all that the computer algorithm of Porter is extremely accurate whereas that of Grim suffers from Severe numerical break- down, especially at high values of the non-dimensional frequency 336 A New Appraisal of Strip Theory Fig. 67 - A versus 5 221-249 O - 66 - 23 | 337 Vassilopoulos and Mandel WATERLINE MODEL 3 MODEL 1 (SEMI - CIRCLE) Fig. 68 - Model sections It is important to note, however, that in the important ship range of the frequency parameter (0-1.5), the disagreement is minimum. Generally speaking, in the case of the semicircular cylinders, Figs. 66 and 67, the free-surface correction factor, k,, aS computed by Grim is less accurate than A, the amplitude ratio. Also, for a given 5, the discrepancies for both k, and A increase with decreas- ing beam to draft ratio. This is particularly noticeable for the low beam to draft bulb-section (Model 5 of Fig. 68) as plotted in Fig. 70. This, in turn, causes underestimations of heave and pitch damping and consequently forces the strip theory to overestimate responses, particularly at resonance, as shown for example by Model E. However, it has not yet’been possible to examine, in de- tail, the overall effect these differences will have for a particular ship model in a given condition. The most important reasons why efforts are now being made to incorporate a version of Porter's work in our computer program are as follows: a. His solution of the theoretical problem is considered more exact and by far the most general to date. b. His numerical scheme is much more stable although, at present, more time-consuming than that of Grim. 338 A New Appraisal of Strip Theory / / / = SSS / ~ PORTER ——-—-— GRIM Fig. 69 - k, versus 6 c. His program can handle any arbitrary ship section which can be defined by either (1) an arbitrary number of parameters or, even better (2) a set of off- sets for the section [45]. d. Extension to the problem of horizontal oscillation can also be made for lateral ship motions. It follows from c. that Porter's program will discriminate between the added mass and damping of two sections of identical section coefficients and beam-draft ratio but differing in detailed section shape. Thus it should prove more flexible than programs tied to particular section families such as the Lewis or Landweber sections. CALCULATION OF EXCITATION FORCE AND MOMENT The second main category of the loads imposed on the ship hull are those due to wave action. As a result of the linearization of the problem, these forces and moments can be assumed to act independently on the ship which is moving at constant speed but is otherwise restrained from any translatory or angular displacements. Since there is no distinction, as far as hydrodynamic forces and moments are concerned, between a restrained ship in a vertically oscillating fluid and an 339 Vassilopoulos and Mandel MODEL 3 ae Fig. 70 - Aversus 6 oscillating ship in a stationary fluid, the excitation forces and moments can be determined by exactly the same arguments used in the calculation of the coeffi- cients of the equations. There are, however, two distinct points of difference which must be allowed for in the computation of the exciting loads: a. Whereas in the calculation of the coefficients of the equations of motion the total force and moment are obtained by summing up strip contributions for sections in identical flow, in the case of wave excitation loads consideration must be given to the distinct flow which each section sees when the wave pattern encounters it. In other words, differential exciting forces depend on the ship section properties as well as the local static and dynamic state of the wave. b. At every section and hence on the whole ship, an extra force and moment is brought about on account of the fact that the relative water flow at a given section involves a pressure gradient which is typical of gravity waves. This so- called ''Smith effect" is due to the orbital motion of the water particles and must be allowed for since the differential exciting forces at a given section depend on whether the section is instantaneously on a wave crest or trough. The best ap- proximate way of allowing for this effect is to consider in the calculations the static and dynamic state of an ''effective subsurface" rather than the actual wave surface. Havelock has suggested [46] that the effective subsurface is located at a mean draught equal to V/A, or (Cp/Cy)H, a result which is accurate for wall- sided ships. Since we shall compute the loads on the basis of two-dimensional 340 A New Appraisal of Strip Theory flow theory, the equivalent mean draft at a specific ship station, which belongs to an infinite cylinder, becomes o(x) H(x). An overall correction factor such as 277 er | a(x) HO] will therefore be employed in the calculation of the excitation force at a given section. It must be noted however that the Smith effect should, strictly speak- ing, be a single correction to the wave acceleration force only. Since the exact calculation of the total exciting force is a formidable hydro- dynamic problem, the following usual assumptions will be made in the approxi- mate computation: a. At each point on the submerged hull surface there is a pressure acting which is the same as the pressure that would occur at the corresponding point in the wave in the absence of the ship. This pressure is computed after the centripetal acceleration of the water particles has been accounted for (Smith effect). b. The wave geometry and dynamic state is not affected by the presence of the ship, i.e., any diffraction effects are neglected. Assumptions a. and b. constitute the well-known Froude-Kriloff hypothesis. c. The effect of the forward speed of the ship is neglected. . It is surmised that the differential heaving force as felt by the ship section depends on the instantaneous elevation, velocity and acceleration of the effective subsurface measured relative to the body coordinate system. Thus, e dZ cael: aa Z(h,h,h) exp - 2 t)| (36) where ((x) = o(x) H(x). For small motions, we can expand the function Z(h, h, h) in a Taylor series about the condition of no wave, i.e., h, = h, = h, = 0 and u =u, and retain only linear terms. Noting that Z(h,,h,,h,) = 0, we finally get, dZ, 8Z, OZ Ne ayaa age ae = = | (ae) text) + ae h(x, t) + a h(x,t) | exp = o(x) H(x) (37) - where the wave elevation is measured positive downwards. The subscript x de- notes that the derivatives (0Z./ch), etc., correspond to the section under con- sideration only. Since the coefficients of each term are readily computed as oZ eZ, eZ, Vassilopoulos and Mandel the total exciting force due to sinusoidal waves is simply obtained by summing up the individual contributions of each strip, i.e., hg ts +L/ 2 fo) iw A (oly Ioly lo, )) = { = dx = Zoe et (38) -L/2 * and the total pitching moment is given by Lies wiley & dZ iw M.(h,h,h, t) = J eS xdx = Moe et Ce (39) =L/2 It is interesting to compare the above expressions with those of Jacobs [14]. Following the same initial steps as Korvin-Kroukovsky [10] but pursuing a slightly different approach, Jacobs [14] modified and improved the excitation force expression as compared with the one given in Ref. [10]. In our notation, the formula as given by Jacobs [5] and as used in the existing computer program, reads as follows: Zo d 5 co Be = ae) h(x,t) + Ke mouy SH] h¢x,t) + u(x) hon} x exp F FT a(x) HC] 0 (40) Equation (40) differs from (37) in that the wave velocity term includes an extra pseudo-three-dimensional term which is furthermore speed dependent. The contribution of the latter term is small in comparison with the other terms and predicts a decrease of the exciting force and moment, a finding which, as dis- cussed by Vossers [47], contradicts that of Hanaoka. It is contended that the more rationally derived Eq. (37) will give almost similar results as Eq. (40) but this remains to be verified. As justification of using the cross-flow hypothesis in computing excitation loads, Fay [13] provides an intuitive criterion which requires that 2 wo g 2 Ilo The best criteria however of the success with which strip theory predicts the forcing function is the degree of correlation with experimental measurements and more sophisticated theoretical analyses. As far as the authors are aware, the only experimental data obtained with actual ship models is that of Jinnaka [48], Schultz [49], and Gerritsma [22], whereas Gersten [50] and Lee [51] meas- ured excitation forces and moments on mathematically defined bodies. Corre- lation between experimentally measured and theoretically computed exciting forces and moments have been presented by Vossers [47], Gersten [50], and Lee [51], but the theoretical expressions used for the exciting loads differed 342 A New Appraisal of Strip Theory from the one presented in Ref. [5]. These studies have shown that in general the Froude-Kriloff hypothesis (modified for Smith effect) supplemented by ap- proximate corrections for the body-wave interference provide reasonable pre- dictions of the excitation force and moment. The lack of severe dependence on speed has also been noted in these studies. To supplement these correlations, the results of a preliminary analysis are shown in Figs. 71 and 72. Theoretical forces and moments based on Jacobs' formula, Eq. (40), have been compared with the experimental values given by Gerritsma in Ref. [22] for an 8-foot, Series 60, C, = 0.60 model. The ampli- tudes of the heave exciting force compare more favorably than those of the pitching moment, although there are some discrepancies at low wavelengths. This finding seems to be in accord with Fay's statement [13] that the cross-flow hypothesis will be more valid for wavelengths equal to or greater than the ship length. Although results computed from Eq. (37) are not shown on Figs. 71 and ——— STRIP THEORY © @ EXPERIMENT Z, LBS — === SiRiP THEORY. O @ EXPERIMENT 0.75 1.00 1.25 1.50 1.75 NAL = Fig. 71 - Comparison of excitation force and moment amplitudes 343 Vassilopoulos and Mandel ——— STRIP THEORY © @ EXPERIMENT =——— STRIP THEORY O@ EXPERIMENT 0.75 1.00 1.25 1.50 1.75 A/L— Fig. 72 - Comparison of excitation force and moment amplitudes 72, as previously noted, it is expected that these expressions will not yield an- swers significantly different from Jacobs' Eq. (40). The discrepancies which are brought about by assuming that the body and wave do not interfere need also to be examined. Grim's [52] theoretical work, supplemented by Spens' [53] experimental work point out the considerable de- crease in wave elevation as the wave passes along the ship length as well as the presence of a bow-induced wave. There is no doubt that such interference ef- fects, especially in astern seas [53], will sensibly modify the theoretical exciting force and especially the pitching moment, which, at present, seem to be over- estimated. It is not yet known whether a convenient correction may be applied in Eq. (37) to allow for this discrepancy, but the matter will be considered more carefully in the future. The diffraction problem has also been investigated more recently by Neu- mann [54] on the basis of Haskind's theory. He presents a remarkably simple 344 A New Appraisal of Strip Theory relationship between the wave-induced force on a fixed body and the amplitude of the progressive wave caused by the motion of the body in still water. Al- though his analysis does not provide the phase between force and wave, his ex- pressions ought to be compared and evaluated on the basis of strip techniques. Using our notation, it can be shown that for a ship hull, his final formulations reduce to: 9 ,tL/2 Ze) = ale) fo Rex) noo ax ”) e -L/2 j and 2 +L/2 M, = Pp ae \ A(x) h(x) x dx. (42) . -L/2 Finally, three-dimensional corrections deserve comment. The work of Spens [53] and others has suggested that a three-dimensional correction to al- low for end effects, etc., tends to worsen agreement between theory and experi- ment. Further analysis on this point is needed, however, because there is re- cent experimental evidence at M.I.T. to suggest that neglect of three-dimensional effects may not be in order for certain ship forms. Provided that the other neglected effects are allowed for, it may well be that a three-dimensional cor- rection will improve agreement between theory and experiment. CONC LUSIONS 1. Following Abkowitz [12,26,27], the more rigorous development of the equations of motion shown in this report along with the more systematic and symbolic notation of the SNAME Bulletin 1-5 [25] lead more quickly and simply to an accurate definition of the various parts of the coefficients of the equations of motion than the Korvin-Kroukovsky approach. 2. The quantitative evaluation of the coefficients of the equations of motion using strip theory developed in this report leads to agreement with Korvin- Kroukovsky in the case of eight of the coefficients and disagreement in the case of four of the coefficients. 3. Figures 2-65 show that pitching and heaving amplitudes as well as phase angles as computed by Korvin-Kroukovsky's procedure using Grim's section damping and added mass [41] correlate reasonably well with existing experimen- tal data which however also include sources of possible error. 4. Substitution of the Porter method [32] for computing section damping and added mass should improve the discrimination amongst differing section shapes compared to Grim [41] and also removes the difficulties associated with oscil- lating nature of Grim's coefficients shown in Figs. 66, 67 and 69. 5. While it has been hypothesized that the correlation shown in Figs. 2-65 may be due to fortunate cancellation of substantial errors, it is not believed that 345 Vassilopoulos and Mandel the errors shown to exist in the Korvin-Kroukovsky procedure using Grim's section added mass and damping are large. This remains to be further investi- gated however. RECOMMENDATIONS While the present report represents a start, much more can be done to as- sess the reliability and usefulness of an improved strip theory. The following list of recommendations cover suggestions for correcting the work already ac- complished as well as suggestions for future work. 1. There is a strong need for phase angles to be uniformly and unambigu- ously defined. There are essentially three different ways for phase angles to be presented: a. Amplitude positive throughout and phase angles from 0°-360°. b. Amplitude positive throughout and phase angles +0-180°. c. Amplitude both positive and negative and phase angles from 0°-180°. The last was used by Vossers et al. [18,19] while the second way has been used in this report and in Ref. [6]. It is believed that the first way is the least am- biguous and that this should be used in the future. In the definition of phase, the maxima of response amplitudes should be referred to the maxima of the wave amplitude and not to the maxima of the wave slope as was done by Korvin- Kroukovsky. Furthermore, the reference point should be the mid-section of the ship or model rather than the longitudinal center of gravity since the former can be readily and precisely located. 2. The importance of the neglect of surge in the theory remains to be de- termined. The current work of Shen Wang at M.I.T. will help with the formula- tion of a system with the needed three degrees of freedom. 3. The assumption of wall sidedness as far as damping, added mass, and wave excitation is concerned is an important possible source of unrealism in the strip theory. The current work of Parissis [16] is important in this regard. Unfortunately, while success in coping with this problem should improve corre- lation between theory and experiment in regular waves, it will not be possible to incorporate this refinement in the prediction of statistical responses in random seas. The latter is strongly tied to a completely linear system. 4. Further refinements of strip theory should include the use of Porter [32] for computing section damping and added mass. 5. Correlations between the strip techniques and other theories for predict- ing ship motions should continue. For example, Fig. 73 shows a comparison between the non-dimensional pitch and heave amplitudes for a C, = 0.60, Series 60 model at zero speed using Grim's three-dimensional theory [55] and those predicted for Model D at zero speed by the program of Ref. [6]. It is seen that 346 A New Appraisal of Strip Theory 1.00 z,/h, AND 8)/kh, 0.2 —— B/D) WalElolRA? — 2 Die ORY, O 0.5 1.0 1.5 2.0 MES Fig. 73 - Comparison of motion am- plitudes of Model D at zero speed this agreement for heave is better than the agreement for pitch but that the com- parisons are reversed for the two motions. No further comment can be made on these comparisons at this time. 6. Because of the absence of a firm basis for assessing the accuracy of any method for predicting ship motions, efforts toward refinement of existing theo- ries and experimental techniques as well as the development of new theories and experimental techniques such as will be discussed by Davis and Zarnick at this Symposium should continue. In the meantime, in order to show more clearly than it has been shown in the past, the importance of ship motions to the process of selecting dimensions and hull shape for ships, to the earning power of ships and to their economical operation, the effort begun in Ref. [6] towards assessing the performance of ships in random seas will be continued at M.I.T. This work will perforce have to rely on the most workable tool currently available to the profession. In the authors' opinion, this is strip theory. ACKNOWLEDGMENTS The authors are deeply grateful to Professors M. A. Abkowitz and W. Porter of M.I.T. for their freely given advice and counsel during the preparation of this paper. Those familiar with the history of the development in the United States of the equations of motion will immediately recognize the guiding 347 Vassilopoulos and Mandel philosophy of Professor Abkowitz in the discussion of the equations of motion contained in this paper. Most of the material used in the analysis of added mass and damping was kindly provided by Professor Porter. Mrs. Barbara Allen of our Department at M.I.T. was most patient and co- operative in typing the manuscript of this paper. To her we extend our deep thanks. NOMENCLATURE English Letters a = coefficient of equation of motion A = coefficient of equation of motion >| II amplitude ratio b = coefficient of equation of motion B = breadth of ship or model B = coefficient of equation of motion B*, B(x) = station breadth of ship or model at designed waterline c = aconstant = 7/8 Q Il coefficient of equation of motion @ i coefficient of equation of motion @) wo ll block coefficient of ship or model Cp = prismatic coefficient of ship or model Cy = waterplane area coefficient jak ll coefficient of equation of motion o Il coefficient of equation of motion e = coefficient of equation of motion E = Coefficient of equation of motion Fr = Froude number _ F = total external force 348 A New Appraisal of Strip Theory gravitational acceleration coefficient of equation of motion moment vector of external forces coefficient of equation of motion center of gravity instantaneous wave elevation referred to absolute system instantaneous wave elevation referred to relative (moving) system instantaneous wave velocity referred to relative (moving) system instantaneous wave acceleration referred to relative (moving) system amplitude of sinusoidal wave (half wave-height) draft of ship or model angular momentum vector about G relative to fixed axes moments of inertia about x,y,z axes respectively rolling moment wave number, 27/ non-dimensional longitudinal radius of gyration high-frequency added mass coefficient of section (Lewis) low-frequency added mass correction factor (Grim-Porter) length of ship or model longitudinal distance of center of buoyancy from amidships longitudinal distance of center of erent from amidships mass of ship or model ; B?(x) sectional added mass = k,k, 7p = pitching moment wave exciting pitching moment 349 Vassilopoulos and Mandel amplitude of exciting pitching moment yawing moment sectional damping coefficient origin of body axes angular velocity of roll angular acceleration of roll angular velocity of pitch angular acceleration of pitch position vector of G relative to O angular acceleration of yaw angular velocity of yaw time longitudinal velocity component of origin of body axes relative to fixed axes longitudinal acceleration component velocity vector of G relative to fixed axes transverse velocity component of origin of body axes relative to fixed axes transverse acceleration component of origin of body axes rela- tive to fixed axes underwater volume of ship normal velocity component of origin of body axes relative to fixed axes normal acceleration component of origin of body axes relative to fixed axes longitudinal body axis or coordinate of a point relative to body axes fixed longitudinal axis or longitudinal coordinate of a point rela- tive to fixed axes 350 XG Greek Letters Ss oe A New Appraisal of Strip Theory longitudinal coordinate of center of gravity celetine to body axes longitudinal component of hydrodynamic force on body lateral component of hydrodynamic force on body transverse body axis or coordinate of a point relative to body axes transverse coordinate of center of gravity relative to body axes transverse body axis or coordinate of a point relative to body axes normal body axis or coordinate of a point relative to body axes fixed vertical axis or vertical coordinate of a point relative to fixed axes vertical coordinate of center of gravity relative to body axes heaving velocity of ship or model heaving acceleration of ship or model amplitude of heaving motion (for Figs. 2-65) vertical component of hydrodynamic force on body wave exciting heaving force amplitude of wave exciting heaving force non-dimensional frequency parameter heaving phase angle (lag) after wave displacement of ship or model pitching phase angle (lag) after wave pitch angle amplitude of pitching motion (for Figs. 2-65) wavelength water density 351 Vassilopoulos and Mandel o(x) = sectional area coefficient g = roll angle x = heading angle yw = yaw angle w, = frequency of encounter REFERENCES . Kriloff, A., "A New Theory of the Pitching Motions of Ships on Waves and the Stresses Produced by This Motion,"' Transactions of Institution of Naval Architects, Vol. 37, 1896, pp. 326-368 and plates 54-57. . Kriloff, A., ''A General Theory of the Oscillations of a Ship on Waves," and "On Stresses Experienced by a Ship in a Seaway," Transactions of Institu- tion of Naval Architects, Vol. 40, 1898, pp. 135-212 and plates 32 to 36. . Weinblum, G. and St. Denis, M., ''On the Motions of Ships at Sea,'' Transac- tions of SNAME, Vol. 58, 1950, pp. 184-231. . Korvin-Kroukovsky, B. V., "Theory of Seakeeping,'’ SNAME, New York, 1961. . Jacobs, W. R., Dalzell, J., and Lalangas, P., "Guide to Computational Pro- cedure for Analytical Evaluation of Ship Bending Moments in Regular Waves," Davidson Laboratory Report No. 791, October 1960, Stevens Insti- tute of Technology. . Vassilopoulos, L., ''The Analytical Prediction of Ship Performance in Ran- dom Seas, Including a New Correlation of Theoretical and Experimental Motions in Regular Waves,'' M.I.T., Department of Naval Architecture and Marine Engineering, February 1964. Cummins, W. E., 'The Impulse Response Function and Ship Motions," Schiffstechnic, H.47, B.9, June 1962 and David Taylor Model Basin Report 1661, October 1962. . Korvin-Kroukovsky, B. V. and Lewis, E. V., "Ship Motions in Regular and Irregular Seas,’ Technical Memorandum No. 106, July 1954, Experimental Towing Tank, Stevens Institute of Technology. . Korvin-Kroukovsky, B. V., "Investigation of Ship Motions in Regular Waves," Transactions of SNAME, Vol. 63, 1955, pp. 386-435. 352 10. 11. 12. 13. 14, 15. 16. ete 18. 19. 20. 21. 22. 23. A New Appraisal of Strip Theory Korvin-Kroukovsky, B. V., and Jacobs, W. R., ''Pitching and Heaving Mo- tions of a Ship in Regular Waves,"' Transactions SNAME, Vol. 65, 1957, pp. 590-632. Kaplan, P., "Application of Slender-Body Theory to Forces Acting on Sub- merged Bodies and Surface Ships in Regular Waves," Jour. of Ship Re- search, Vol. 1, No. 3, November 1957, pp. 40-49. Abkowitz, M. A., ''The Linearized Equations of Motion for the Pitching and Heaving of Ships,'' Proceedings, Symposium on the Behavior of Ships in a Seaway, Netherlands Ship Model Basin, Wageningen, 1957, pp. 178-189 and 883. Fay, J. A., ''The Motions and Internal Reactions of a Vessel in Regular Waves," Jour. of Ship Research, Vol. 1, No. 4, March 1958, pp. 5-13, 67. Jacobs, W. R., ''The Analytical Calculation of Ship Bending Moments in Regular Waves," Jour. of Ship Research, Vol. 2, No. 1, June 1958, pp. 20-29. Korvin-Kroukovsky, B. V., ''Theory of Ship Motions in Regular Head Seas," Notes of 4th Bi-Annual Seminar, Ship Behavior at Sea, Davidson Laboratory, Stevens Institute of Technology, January 1963, pp. 1-12. Parissis, G., ''The Effect of Hull Shape Non-Linearities on the Calculation of Heave and Pitch of a Ship,'' M.I.T., Department of Naval Architecture and Marine Engineering, Report No. 64-6, June, 1964 (to be published). Kerwin, J. E., ''Polynomial Surface Representation of Arbitrary Ship Forms," Jour. of Ship Research, Vol. 4, No. 1, June 1960. Vossers, G., Swaan, W. A., and Rijken, H., "Experiments with Series 60 Models in Waves," Transactions of SNAME, Vol. 68, 1960, pp. 364-450. Swaan, W. A. and Rijken, H., "Speed Loss at Sea as a Function of Longitu- dinal Weight Distribution," Transactions of NECIES, Vol. 79, 1962-63, pp. 165-188. Lewis, E. V. and Numata, E., "Ship Motions in Oblique Seas,'' Transactions SNAME, Vol. 68, 1960, pp. 510-547. Todd, F. A., "Some Further Experiments on Single Screw Ship Forms — Series 60,"" Transactions SNAME, Vol. 61, 1953, pp. 516-574. Gerritsma, J., "An Experimental Analysis of Ship Motions in Longitudinal Regular Waves," International Shipbuilding Progress, Vol. 5, No. 52, De- cember 1958, pp. 533-542. Gerritsma, J., "Shipmotions in Longitudinal Waves," International Ship- building Progress," Vol. 7, No. 66, February 1960, pp. 49-76. . 221-249 O - 66 - 24 353 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. Vassilopoulos and Mandel Mandel, P., ''Subcritical and Supercritical Operation of Ships in Waves,"' Journal of Ship Research, Vol. 4, No. 1, June 1960. "Nomenclature for Treating the Motions of a Submerged Body Through a Fluid,"' Bulletin No. 1-5, The Society of Naval Architects and Marine Engi- neers. Abkowitz, M. A., '1963 Lecture Notes on Ship Hydrodynamics,'' Hydro-Og, Aerodynamisk Laboratorium, Lyngby, Copenhagen, Denmark (to be pub- lished). Abkowitz, M. A., "The Dynamical Stability of Submarines,"’ David Taylor Model Basin, 1949 (limited publication). Ursell, F., 'On the Heaving Motions of a Circular Cylinder on the Surface of a Fluid,'' Quarterly Journal of Mechanics and Applied Mathematics, Vol. 2, June 1949, pp. 218-231. Grim, O., '"Bereehnung der durch Schwingungen eines Schiffskorper er- zeugten hydrodynamischen Krafte,"' Jahrbuch der Schiffbautechnischen Gessellschaft, 1953. Korvin-Kroukovsky, B. V., ''Brief Review of Ship Damping in Heaving and Pitching Oscillations,"' Stevens Institute of Technology, Experimental Tow- ing Tank Note No. 328, February 1955. Tasai, F., ''On the Damping Force and Added Mass of Ships Heaving and Pitching,"' Univ. of California, Inst. of Eng. Research, Series No. 82, Issue No. 15, July 1960. Porter, W., "Pressure Distributions, Added Mass, and Damping Coefficients for Cylinders Oscillating in a Free Surface," Univ. of California, Inst. of Eng. Research, Series No. 82, Issue No. 16, July 1960. Kaplan, P. and Jacobs, W., "Two-Dimensional Damping Coefficients from Thin Ship Theory" and "Theoretical Motions of Two Yacht Models in Regu- lar Head Seas on the Basis of Damping Coefficients Derived from Wide V-Forms,'' Davidson Laboratory Notes No. 586 and 593, June 1960. Kaplan, P. and Kotik, J., "Report on a Seminar on the Hydrodynamic Theory Associated with Ship Motion in Waves,'' Technical Research Group, Report 140-FR. Watson, T. C., ''Experimental Investigation of the Vertical Forces Acting on Prolate Spheroids in Sinusoidal Heave Motion," Univ. of California, Inst. of Eng. Research, Series No. 82, Issue No. 18, April 1961. Paulling, J. R. and Richardson, R. K., ''Measurement of Pressures, Forces, and Radiating Waves for Cylinders Oscillating in a Free Surface,"’ Univ. of Calif., Inst. of Eng. Research, Series No. 82, Issue No. 23, June 1962. 354 on. 38. 39. 40. 41. 42. 43. 44, 45. 46. 47. 48. 49. 50. A New Appraisal of Strip Theory Golovato, P., "The Forces and Moments on a Heaving Surface Ship," Jour. of Ship Research, Vol. 1, No. 1, April 1957, pp. 19-26. Gerritsma, J., ''Experimental Determination of Damping Added Mass and Added Moment of Inertia of a Shipmodel,"' Inter. Shipbuilding Progress, Vol. 4, No. 38, October 1957, pp. 505-519. Gerritsma, J. and Beukelman, W., ''Distribution of Damping and Added Mass Along the Length of a Shipmodel,"' Report No. 49S, Netherlands Re- search Centre T.N.O. for Shipbuilding and Navigation, March 1963. Newman, J. N., ''The Damping of an Oscillating Ellipsoid Near a Free Sur- face," Jour. of Ship Research, Vol. 5, No. 3, December 1961, p. 44. Grim, O., ''A Method for a More Precise Computation of Heaving and Pitch- ing Motions in Both Smooth Water and in Waves," Proc. of Third Symposium on Naval Hydrodynamics, Office of Naval Research, Dept. of the Navy, ACR-55, 1960, pp. 483-524. Prohaska, C. W., "Vibration Verticales de Navire,'' Assoc. Technique Mari- time et Aeronautique, Vol. 46, 1947, pp. 171-219. Landweber, L. and Macagno, M., ''Added Mass of a Rigid Prolate Spheroid Oscillating Horizontally in a Free Surface,"' Jour. of Ship Research, Vol. 3, No. 4, March 1960, pp. 30-36. Paulling, J. R. and Porter, W. R., "Analysis and Measurement of Pressure and Force on Heaving Cylinders in a Free Surface," Proc. of the Fourth U.S. National Congress of Applied Mechanics, 1962, p. 1369. Plant, J. B., "An Application of Linear Programming to the Problem of In- verting a Conformal Transformation," M.I.T., Department of Naval Archi- tecture and Marine Engineering, January 1964 (unpublished document). Havelock, T. H., ''Notes on the Theory of Heaving and Pitching,"’ Transac- tion of Institution of Naval Architects, Vol. 87, 1945, p. 109. Vossers, G., "Resistance, Propulsion and Steering of Ships - Part IIC - Behavior of Ships in Waves,"' The Technical Publishing Company, H. Stam N.V., Haarlem, The Netherlands, 1962. Jinnaka, T., ''SSome Experiments on the Exciting Forces of Waves on the Fixed Ship Model,"' J. Zosen Kiokai, Vol. 103, 1958, pp. 47-57. Schultz, F. H., Forces and Moments on a Restrained Model in Regular Waves,"' Colorado State Univ., Research Foundation, Report CER 61 EFS72, March 1961. Gersten, A., "A Comparison of Experimental and Theoretical Forces and Moments Acting on a Restrained Surface Ship in Regular Waves," Jour. of Ship Research, Vol. 6, No. 4, April 1963. 355 Vassilopoulos and Mandel 51. Lee, C. M., 'Heaving Forces and Pitching Moments on a Semisubmerged and Restrained Prolate Spheroid Proceeding in Regular Head Waves,"' Inst. of Engineering Research, Univ. of California, Report No. NA-64-2, Contract Nonr-222(93), January 1964. 52. Grim, O., "The Deformation of Regular Head and Following Waves by a Moving Ship," Stevens Institute of Technology, Davidson Laboratory, Tech- nical Translation by P. G. Spens and A. Winzer, November 1963. 53. Spens, G. P., "Experimental Measurements of the Deformation of Regular Head and Following Seas by a Ship Model," Stevens Institute of Technology, Davidson Laboratory Report No. 966, June 1963. 54. Newman, J. N., ''The Exciting Forces on Fixed Bodies in Waves," Jour. of Ship Research, Vol. 6, No. 3, December 1962. 55. Grim, O., "The Influence of the Main Parameters of the Ship Form on the Heaving and Pitching Motions in Waves,'' Hamburg Model Basin, H.S.V.A., Report No. 1253, September 1961. * * * DISCUSSION Winnifred R. Jacobs Stevens Institute of Technology Hoboken, New Jersey I am belatedly aware of your criticism of the Korvin-Kroukovsky linear theory of ship motions. I wasn't at Bergen and therefore missed your presen- tation and Dr. Kaplan's defense as well as the counter-attacks. Since I am equally responsible for what you consider erroneous in the anal- ysis, I should like to discuss your paper with you, in particular two statements which, I believe, epitomize your criticism. (I hope I am correct in not consid- ering as criticism the paragraph which states the fact that certain added mass and damping coefficients were used in one study, while different coefficients were uSed in other studies at Davidson Laboratory. Professor Korvin and, in- deed, everyone involved in this work at Davidson Laboratory have repeatedly said that when more suitable hydrodynamic coefficients are available, they will be used.) In one instance you say ''Table 2 shows that the expressions for four of the newly proposed coefficients do not agree with those derived by Korvin- Kroukovsky. The differences in the Korvin-Kroukovsky coefficients e(#,) , B(w,), C and E(w,) appear to be mainly due to an erroneous time differentiation of a fixed body coordinate with the result that 356 A New Appraisal of Strip Theory a. a factor of 2 appears in the velocity dependent terms of e(w,) and B(w,), and b. a pseudo-three-dimensional term is introduced in coefficients e(w,), B(w,), C and E(e,). "It would also appear that the introduction of terms dependent on the rate of change of added mass over the ship length is inconsistent with the use of two- dimensional theory. Despite these discrepancies, however, it is expected that the final values of these coefficients will not be seriously modified since it has been shown by Jacobs et al [5] that most of these terms which appear in the Korvin-Kroukovsky approach but not in the new approach are numerically small. It is hoped that in the near future these inconsistencies will be examined more carefully and their implications assessed on the basis of experimental data." Several pages later you say ''...In our notation, the formula [for the exci- tation force] as given by Jacobs [5] and as used in the existing computer pro- gram [at M.I.T.], reads as follows: Z ; * = | 28.80%) ln(o%5 16) ap Kou aed In((S5{E)) ar JUG) hos | dx OS Glx: x exp - = a(x) HC | 4 (40) Equation (40) differs from (37) [the new approach] in that the wave velocity term includes an extra pseudo-three-dimensional term which is furthermore speed dependent. The contribution of the latter term is small in comparison with the other terms and predicts a decrease of the exciting force and moment, a finding which, as discussed by Vossers [47], contradicts that of Hanaoka. It is contended that the more rationally derived Eq. (37) will give almost similar results as Eq. (40) but this remains to be verified." I should like to take up a few points. 1. It appears to me that this criticism boils down to one ingredient: we dif- ferentiated a ''fixed" coordinate é with respect to time and hence inevitably the "fixed" radius r of the circular section associated with €. The latter deriva- tive die Wounds: dr egy ddGuderah~yside would then give terms, dependent on a rate of change of r (and hence of added mass) over the ship length, and also speed-dependent. In our approach, the é-axis fixed in the ship is time-dependent with respect to the wave. Our strip method treated each ship section strip as if fixed ina frame of an animated cartoon with the strips changing from frame to frame and the frames changing from time to time. 357 Vassilopoulos and Mandel In support of your contention that our approach is incorrect, you cite the work of Professor Fay among others. May I quote Professor Fay's discussion of Korvin-Kroukovsky's 1955 SNA paper? In that paper the forces due to body motions are developed through the following equations: The potential Obi = (VO-z-£0)r cos a (37) and since che S dé ae = V tan B and ae V ’ the pressure = a ~: = GOs a (Vr + OV? tan B- Zr - zV tan G- @Vr - £6r -8EV tan /) . (38) The vertical force increment per unit length = 22 | P cos a da becomes = = (eZ rtv)és (0% rV? tan A) 6 - (22 r?) 3 - (Pane tan 6) 3 - (63 rv) 6 - (e3 rela - (aa tan 6). (39) Professor Fay said: "If 6 is positive when measured clockwise, z is posi- tive in the downward direction, and V positive for motion in the positive x-direction then Eq. (37) is correctly stated. However, dé/dt should equal -v, and terms (1) and (5) in (39) do not cancel but add. This term is the most im- portant coupling term in the equations of motions and exists even for a sym- metrical vessel.'' He also commented, with regard to the terms in dr/dt, that, since the method is a linear approximation, "the carrying of terms of higher order in subsequent equations does not seem justified." In the 1957 SNA paper by Korvin-Kroukovsky and myself, we corrected the sign of V, and reinstated the velocity-dependent terms, which had been omitted in the 1955 paper on the assumption that these terms in the potential theory de- velopment merely implied damping and could be replaced by damping terms de- termined on the basis of energy dissipation by waves, as a quid pro quo. A study of Haskind (1946) and Havelock (1955) confirmed what Fay had said in his discussion about the coupling terms. The Korvin-Kroukovsky approach now has values for the coefficients e(#,) and E(w,), as shown in your Table 2, which contain the identical dynamic coupling terms derived by Havelock for a long half-immersed spheroid and by Haskind for a thin 'Michell" ship. 358 A New Appraisal of Strip Theory 2. This brings me to the second point I wish to raise. Why is |e 4 oh ix Xx, "pseudo-three-dimensional,"’ whereas its equivalent - u, { .(x)dx is not? As shown in our 1957 SNA paper, if one integrates by parts (ae neGbe = | neayON(se)) = || p(x) dx L L L and therefore ~ | NOx) xx + Dug | ax) ox chal aia x dx e( @.) ~ Noo x dx + uy {Hoo dx which is equivalent to the value of e(w,) in your "new approach," as well as to Havelock's and Haskind's coefficient of ¢ in the heaving force equation. In the Korvin-Kroukovsky approach, the coefficient of z in the pitching moment equation is ~ [NO xdx + Us GEC) ee Ae.) dx - [xox x GES = ug J acx) dx the second term of which is missing in your "new approach," but is present in Havelock's and in Haskind's developments. Similarly, it can be shown that the Korvin-Kroukovsky coefficients B(w,) and C, after integrating by parts, become B(®,) = ike x2dx - Dug | ax) x dx “a ye xd = [xc x? dx ee | BOX) x? dx - 4, [ Noo x ax + we ao x dx o) i iH pa | Bx) x?dx - ug {NOx x dx = ug fucx) dx. 359 Vassilopoulos and Mandel These are the three coefficients which are different from yours. The difference is negligible in the case of B(w,) and small in the case of C. However, the sec- ond term of E is of the same order of magnitude as the first term. The reason for not substituting - [09 ax for cts in our definitions of the coefficients is that the unit force and moment coeffi- cients are required in the computations of bending moments. 3. You criticize the Jacobs formula for unit exciting force, given in your Eq. (40), because the coefficient of the damping component contains, in addition to N(x), a term du( x) o 6 6dx = {Ul a "pseudo-three-dimensional" term which predicts a decrease of the exciting force and moment as forward speed u, increases, whereas Hanaoka's calcula- tions, as shown in Vossers' articles, predict an increase. This criticism would be valid only if the damping coefficient N(x) were invariable with forward speed. However, N(x) is a function of speed through its dependence on frequency of en- counter, and itself contributes to the decrease in exciting force with speed. As you say, the contribution of the disputed term is small and your Eq. (37) which omits this term "will give almost similar results as Eq. (40) but this remains to be verified." 4. But why not verify it? Since the computer program at M.I.T. follows the computational procedure of Davidson Laboratory Report 791, it should be quite easy to drop the offending terms and test your new approach. If the Korvin-Kroukovsky approach is devoid of vitality, why keep flogging a dead horse ? DISCUSSION Martin A. Abkowitz Massachusetts Institute of Technology Cambridge, Massachusetts I should like to discuss specifically the nature of the various coefficients in the coupled linearized equations of motion for pitch and heave as tabulated in Table 2 of the paper. 360 A New Appraisal of Strip Theory In the column headed ''Coefficient" are the coefficients of the linear terms of each of the motion variables. On the left side of this column, the coefficients are merely expressed arbitrarily as letters in alphabetical sequence. On the right-hand side of this column, the coefficients are expressed in the nomencla- ture of Bulletin 1-5 of The Society of Naval Architects and Marine Engineers, which system is developed with reference to axes fixed in the ship, which pro- vides the advantage of centerline plane symmetry in any hydrodynamic calcula- tions. The appearance of double terms in the right-hand part of the column, arises from the rigorous treatment of transferring from axes oriented in the ship to axes (specifically heave) oriented relative to fixed space. The linear coefficients in this form are valid independent of any method one wishes to de- termine them — whether theoretically by strip theory, slender body theory, thin ship theory or by model experiments. Under the column designated "New Approach" is listed the formulation for calculating these coefficients by a ''pure strip theory"'—i.e., each section treated as a cylindrical section and completely independent of the shape of other ship sections. Since the terms are calculated by integrals over the various ship sections, in a geometry fixed in the ship, the forward speed effect on some of the coefficients very neatly falls in place, such as in the terms =un 2. = Uo [ Hoo ds since by strip method =4, = [ aco ax, SU, Zee ug [NCO dx since by strip method = Ze = i N(x) dx. In the column headed ''Korvin-Kroukovsky Approach" are listed formula- tions as attributed to the strip theory of Korvin-Kroukovsky. Perhaps a great deal of difficulty and confusion results from semantics in that what is often re- ferred to as Korvin-Kroukovsky strip theory is in reality not a pure strip theory, but a rather crude slender body theory which takes into account three- dimensional effects in a rather rough way. Nevertheless, because of the physi- cal realities of the situation, any method of calculation of the coefficients should be consistent with the terms listed in the right-hand side of the coefficient col- umn. Hence, the Korvin-Kroukovsky terms given below in the coefficient e(«,) should reduce to -u, Z. (or u, J (x) dx) aug | mx) dx + u, J¢ (2) Gk = up {HOO dx . 361 Vassilopoulos and Mandel It has been indicated by others, that integrating the expression on the left by parts will reduce it to the right-hand expression provided the sectional area curve goes to zero (continuously) at the ship ends. Since this is a requirement of slender body theory, the left-hand terms can be written in the simpler form of the right-hand term. Similarly, it can be shown that the two terms under the Korvin Approach for Coefficient ''B'', and indicated by the dotted block in the at- tached figure, reduce to the one term, indicated by the dotted block under ''New Approach."" Some ships, such as those with transom sterns, need not have sec- tional area curves which are zero at the stern end, hence the possibility of an error in Korvin Approach for this hull shape. On the other hand the Korvin Ap- proach gives a distribution of the effect along the length, which is desirable when bending moments are being considered. There are only two additional coefficients in the tabulations which take dif- ferent forms under ''New Approach" and ''Korvin Approach" — these are coeffi- cients C and E. For coefficient E, (or -M,), the Korvin approach has the addi- tional term SCS) x dx 2 dx as compared to the "New Approach" and this term reduces to -u, J u(x) dx or u,Z.. Since the Korvin approach is a slender-body theory involving some pseudo three-dimensional effects, it will be shown below that this additional term can result from three-dimensional considerations. As introduced by Korvin, coeffi- cient b (or -Z,) is expressed by {N(x) dx which is purely a frequency depend- ent effect (surface wave effect) since in potential theory, for a deeply submerged body b (or Z,), would be zero—i.e., no lift force with angle of attack in the ab- sence of circulation. Hence, in the attached table the term zero has been added to indicate the addition of a three-dimensional potential solution. If we include in the pure strip approach or ''New Approach" column, the other three- dimensional potential solutions in the appropriate terms, the following terms are added to the expressions in the ''New Approach" column: where - X. is the added mass for longitudinal acceleration. These additional terms appear in the attached table as encircled by a dotted line. The Z. terms are equivalent to the terms enclosed by dotted rectangles under the Korvin approach. However, we now find terms in X. in the "New Ap- proach" brought about by the rough three-dimensional correction based on the results of potential theory calculation. Since X. can be estimated for a given 362 A New Appraisal of Strip Theory 1 Xp X ——— PISS Goa {+ Xp Gon f Ae SS ier ag ae - e e ae eieenaraiie’ amreaaoain | Gao °n — xpx (x) | °nz -1Xp;X (x )N aa] | xp x aT en + eg +)ep x GEN Le----— — ee ee Ce ee = ee Ke YS yoeoiddy AysAoyNoIy-uUTAIOY aa M = P57 rat Ne 598) Cn f aN bs ~ n 5 CX-"Z) 2m +)xpx Gen [ °n - xXp;x (xa | 8 / a. cL ae ae ae i xp x (yr | &n -1xp;x Gon [ LL eoese> soseen See eset [a Se See eat Sy xpx Gon [ fhe eee ee See * UOTJOWN JO SUOT}eNbY JO S}JUSTOTJJo0D Jo uost1edwuo) G (981 (x do AytAaiIq 10q% Vassilopoulos and Mandel hull shape, using the ''New Approach" with the additional three-dimensional terms should be more realistic than the Korvin approach. Physically, the slen- der body assumption assumes such a large length-beam or length-diameter ra- tio that forward effects are neglected. The new approach corrected as indicated above, would hold better for the fuller ships. As an extreme, the deeply sub- merged sphere should have a coefficient E (or -M,) equal to zero; the corrected new approach would give zero for this case, whereas the Korvin slender body approach gives a relatively large quantity. (Of course a sphere significantly violates the slender body assumption.) It should be pointed out that the corrected coefficients, listed under 'New Approach" in the attached figure, have the symmetry required by the Timman- Newman analysis, i.e., d= D, e=E. * * * DISCUSSION O. Grim University of Hamburg Hamburg, Germany Coefficients for added mass and damping force for some sections are shown in the paper. They are computed using Porter's and my own method. The re- sults found by both methods are compared and discrepancies have been ascer- tained. However, these discrepancies appear not disturbing to me. The reason is very simple. The computer program used for my method was not designed for such a wide range of frequencies but only for the range important for the motions ina seaway. In the meantime the program has been supplemented which is valid for any arbitrary frequency and consequently the discrepancies have vanished. DISCUSSION William R. Porter Massachusetts Institute of Technology Cambridge, Massachusetts These comments will be relative to the calculation of added-mass and damping coefficients for two-dimensional cylinders. The numerical results for all elliptic cylinders and for Models 2, 3, and 4, obtained by the procedures used by the authors and attributed to Grim should agree with my results, because all 364 A New Appraisal of Strip Theory these forms can be uniquely defined by their beam/draft ratio and area coeffi- cient. Professor Grim has privately supplied to the original authors his values calculated by a different program, and my work is in much closer agreement with these later results. Model 5, however, cannot be defined by its beam/draft ratio and section area Coefficient alone. Therefore, calculations which define the cylinders by only these two parameters will not agree with more correct predictions. This is illustrated by the following figures. < Fig. 1 - Three shipform cylinders with the same beam/draft ratio; Models 5 and 5G have the same area coefficients; Model 5G is a Lewis form simi- MODEL 5G lar to Model 4 but slightly in more full MODEL 5 Figure 1 shows sections of three cylinders with the same beam/draft ratio. These are Model 4, Model 5, and a Model 5G which has the same area coefficient as Model 5. Model 5G can be described by its beam/draft ratio alone, Model 5 cannot. Figure 2 shows values of the waveheight ratio A for these three cylinders. The values attributed to Grim are taken from his values as subsequently re- ported to the authors. The results of Grim and my results show only small dif- ferences for Model 4. The results do not agree for Model 5; however, it is clear that my results for Model 5G would agree with Grim's Model 5 to. small differences. The difference between my values for Models 5 and 5G is due to the different vertical distribution of area. This difference is not one in theory alone as shown by the results of experiments with Models 4 and 5 as reported by Paulling and Porter in Ref. [44] or in Ref. [36] of the original paper. The conclusion is that two parameters alone are not sufficient to define the cylinder geometry. 365 Vassilopoulos and Mandel 0.5 o GRIM MODEL 4 oe GRIM MODEL 5 DI 1.0 2.0 8, NONDIMENSIONAL FREQUENCY Fig. 2 - Waveheight ratio A for Models 4, 5, and 5G * * * DISCUSSION A THE INFLUENCE OF THE ADDED MASS FORMULATION UPON THE COMPUTER MOTION PREDICTIONS Peter A. Gale Bureau of Ships Washington, D.C. To this discusser's knowledge, the significant differences between the Bu- reau of Ships computer program and the author's Massachusetts Institute of Technology (M.I.T.) program as of January 1964, are: first, the Bureau of Ships program is based upon ten station spaces while the M.I.T. program is flexible in this respect and it is believed that twenty station spaces are commonly used; second, the Bureau of Ships computer program uses the Prohaska added mass coefficients with Ursell's free surface corrections as presented in Davidson Laboratory Report No. 791 while the M.I.T. program uses Grim's 1959 added mass coefficients. Both programs use Grim's 1959 damping coefficients. In order to assess the influence of the added mass formulation upon the predicted ship motions, the motions of the DD 710 (this discusser's ship '"'A") were computed using both the Bureau of Ships program and the M.I.T. program with ten station spaces. The resulting motion predictions are plotted in Fig. 1 for a wave length to ship length ratio of 1.25. This figure gives an indication of the influence of the change in added mass formulation described above for a particular set of conditions. For other wave length to ship length ratios the in- fluence was found to be of the same or a lesser order of magnitude. 366 A New Appraisal of Strip Theory MW =1.25 ——O = Move. TEST —-—A = MIT CompuTeR 40 ° OV nore? 30 DD 710 ie AY HEAVE AMPLS. fA \a ——QO = MopEL TEST —-—A = MIT COMPUTER ——O= BusHIPS " oO | 40 5 a Vv, mudee 2a 120 100 DD 7IO PITCH PHASE ANGLES |ODEL TEST \T COMPUTER ——O=Busnips " DD 710 aipeees 140 fee sh__AES ——O = MODEL TesT —-_A = MIT COMPUTER 120 —O=BusHieps ” Vassilopoulos and Mandel DISCUSSION B THE PITCH AND HEAVE OF TEN SHIPS OF DESTROYER- PREDICTIONS COMPARED WITH MODEL TEST RESULTS COMPUTER PREDICTIONS COMPARED WITH MODEL TEST RESULTS Peter A. Gale Bureau of Ships Washington, D.C. NOTES 1. Regular wave model test results were collected for ten destroyer-like ships. The data for five of the ships (F-K) were classified. By coincidence, model test phase angle results were not available for these same five ships. Due to the above, data sources, ship identifications and dimensions, body plans, and phase angle comparisons are not presented for ships F-K. 2. For ships, F and G the longitudinal gyradius of the model was not known. In order to use the computer to predict the motions of these two ships, Ky was assumed to be 0.25L for both. These facts make the comparisons presented for ships F and G of dubious value. 3. The hull dimensions and coefficients presented in the Table of Ship Par- ticulars are those used for the computer motion calculations. In general they also apply to the model test hull forms. Ina few cases there are minor differ- ences between the forms model tested and those used for the motion computa- tions as, for example, when the model tested did not float on an even keel. Mo- tion computations were always made for the even keel case. 4. In the graphs, the circles connected by lines represent the computer calculations. The model test results are represented by symbols other than circles. Ship K was model tested in regular waves of several heights and all of the test results are presented necessitating the use of a different ordinate than used for the other plots. 5. The following reports were the sources of the model test data for ships A-E. a.. For ships A and B: "An Experimental Study of the Effect of Extreme Variations in Pro- portions and Form on Ship Model Behavior in Waves," by Numata and Lewis, ETT Report No. 643, December 1957. 368 A New Appraisal of Strip Theory b. For ships C, D, and E: "The Influence of Shipform and Length on the Behavior of Destroyer- Type Ships in Head and Beam Seas," by Muntjewerf, International Shipbuilding Progress, Vol. 10, No. 102, February 1963. 6. The computer program used to calculate the ship motions presented here was written in the Bureau of Ships and is based upon a theoretical method devel- oped by Korvin-Kroukovsky for computing the coupled pitch and heave of a sur- face ship in regular head waves. The step-by-step computational procedure followed by the computer is essentially that presented in Davidson Laboratory Report No. 791, "Guide to Computational Procedure for Analytical Evaluation of Ship Bending Moments in Regular Waves," by Jacobs, Dalzell, and Lalangas dated October 1960. The computer program uses the Prohaska added mass co- efficients with Ursell's free surface corrections and Grim's 1959 damping co- efficients, all published in D. L. Report No. 791. It is recognized that it would be more logical to use Grim's 1959 added mass and damping coefficients or perhaps even more recent data; this was not done for several practical reasons. The Bureau of Ships computer program divides the hull into ten station spaces for the computations. It has been found that the use of a greater number of station spaces has a negligible effect on the computed results. NOMENCLATURE y maximum single amplitude of pitching motion, Z maximum Single amplitude of heaving motion of ship's center of gravity, E phase lead of maximum pitch up measured with respect to the instant when the wave node preceding the wave crest is at the ship's longitu- dinal center of gravity location, 8 phase lead of maximum heave up defined as for pitch phase angle above, IN regular wave length, L waterline length of ship, h regular wave amplitude, 2h regular wave height (twice wave amplitude), Ky longitudinal radius of gyration of ship. 221-249 © - 66 - 25 369 Vassilopoulos and Mandel GH T9PONW dd 9poN 4 TepoN 0tL-dda yoynd young young pousy}sueT Sre[noyaeg dIys jo stqeL suOT}eOT UEP] ‘enyTeA peunssy,, T/A Raye Tl % 001 e(110°0)/V H/T H/d a/T MSLT TIM 0 V (3) ‘IM 0H (3) Im uo g (13) IMT 370 A New Appraisal of Strip Theory SHIP A IN i) SS SHIP B Vassilopoulos and Mandel ————— ORIGINAL DESTROYER ~—~ SHIP C 372 A New Appraisal of Strip Theory LENGTHENED DESTROYER ~ SHIP D a (HSS Ss 373 Vassilopoulos and Mandel SHIP E LENGTHENED DESTROYER WITH INCREASED BEAM ~s 374 A New Appraisal of Strip Theory SHIP A~ PITCH (AMPLITUDES) = O75 AY = 1.OO «| Tah = 48.0 4| h=48.0 y/ Von P = 4 6 aS ° (—) if>) 20 30 40 V, Knots V, KnoTs V, Knots 375 Vassilopoulos and Mandel SHIP Am HEAVE (AMPLITUDES) 376 A New Appraisal of Strip Theory SHIP A~ PITCH (PHASE ANGLES ) Lbh = 48.0 320 377 Vassilopoulos and Mandel SHIP A~ HEAVE CPHAsE ANGLES ) 378 A New Appraisal of Strip Theory Ssrinses —“FiPcr (AMPLITUDES) A_=075 Af, =1.00 379 Vassilopoulos and Mandel SHIP B ~~ HEAVE (AMPLITUDES ) A New Appraisal of Strip Theory SHIP B~ PITCH (PHASE ANGLES) y= 1.OO 381 Vassilopoulos and Mandel SHIP B ~ HEAVE (PHASE ANGLES ) M_=O.7S ULh =48.0 382 A New Appraisal of Strip Theory SHIP C ~ PITCH (AMPLITUDES ) M. =0.6 ph= 66.6 V, KNOTS V, KNOTS 383 Vassilopoulos and Mandel SHIP C ~ HEAVE (AMPLITU DES) WY = 0.6 Aj = o.3 A. Tah= 66.6 at ee fon= 44.4 KNOTS V, KNOTS AYf_= 1.50 lo|__eh= 267 | ° 10 20 30 40 ° Te) 20 30 <0 V, KNOTS V, KNOTS 384 A New Appraisal of Strip Theory SHIP C ~ PITCH ( PHASE ANGLES ) = 0.30 bsh= 44.4 V, KNOTS 221-249 O - 66 - 26 385 Vassilopoulos and Mandel SHIP C ~ HEAVE (PHASE ANGLES) My. = O.90 Lb 44.4 820 200 386 A New Appraisal of Strip Theory SHIP D ~— PITCH (AMPLITUDES ) M_= 0.50 \, =O75 Leh = 60.0 Lene 53.4 ° 10 20 30 40 ° lo 20 30 40 V4 KNOTS V, KNOTS 387 Vassilopoulos and Mandel SHIP D—~ HEAVE (AMPLITUDES ) = Ors an = 53.4 V, KNOTS 10 = |.25 = OO a a= 32.0 LEAST = V, KNOTS 388 A New Appraisal of Strip Theory SAP. D-~arPiiceH (PHASE ANGLES ) UL = 53.4 20 V, KNOTS 389 Vassilopoulos and Mandel SHIP D ~ HEAVE (PHASE ANGLES) 2a AM =075 X{_=1.00 Uhh = 53.4 Lbn=40.0 240 200 A New Appraisal of Strip Theory Sete. Ec — FIPCrH (am PL ITUDES ) Af_=0.50 LL,= 729.9 Vassilopoulos and Mandel SHIP E ~ HEAVE (AmPL ITUDES ) 392 A New Appraisal of Strip Theory SHIP E—~ PITCH (PHASE ANGLES ) 393 Vassilopoulos and Mandel SHIP E ~— HEAVE (PHASE ANGLES ) N= 1.00 =O25 LAn=534 AY =125 Lbn=320 394 A New Appraisal of Strip Theory SHIP F — PITCH “_= 1.01 Lfo.= 33.5 395 Vassilopoulos and Mandel SHIP F —HEAVE A= 1.01 (o) fo) ro} 20 30 40 V, KNOTS 2 Y xx > 6 i= wh a (fo) 1o 20 396 A New Appraisal of Strip Theory SHIP G —PITCH Af_=O75 LA." 53.3 */_= 1.00 U~n= 40.0 oO Te) 20 30 40 ° Io 20 30 40 V, KNOTS V, KNOTS 397 Vassilopoulos and Mandel SHIP G ~HEAVE = 1.00 LAL= 40.0 Af{_= 1.25 Lon® 32.0 (o) fe) 20 30 40 398 A New Appraisal of Strip Theory SHIP H ~ PITCH “= 1.00 —_» — 20 V, KNOTS 399 Vassilopoulos and Mandel SHIP H — HEAVE A 40.0 ¥_=1.25 U4, = 24.0 ° io 20 30 V,, KNOTS 400 A New Appraisal of Strip Theory SHIP J ~ PITCH %/_= 0.86 Be ee la Yah= 46% — LA,= 35.0 4 4 _* Vv DEG. Vv DEG. aia Sa | 2 sgt 2 1 ps A oO oO fo} 10 20 30 40 ro) 10 20 30 40 V, KNOTS V, KNOTS Af_= 1.7 221-249 O - 66 - 27 401 Vassilopoulos and Mandel SHIP J ~— HEAVE = 0.86 LAv= 467 ° 10 20 30 40 o 10 20 30 40 V, KNOTS V, KNOTS Af_=1.43 Vah= 28.0 402 A New Appraisal of Strip Theory SHIP K — PITCH M_=1.25 Lah Srmsot Mh. =2:OO@ Lin SYMBOL 30.8 9 403 Vassilopoulos and Mandel SHIP K ~— HEAVE ZS Ma Kee) Sey, SYMBOL | K 60.0 A eS 0.6|308 © A Vv e 404 A New Appraisal of Strip Theory REPLY TO THE DISCUSSION L. Vassilopoulos and P. Mandel Massachusetts Institute of Technology Cambridge, Massachusetts *Professor Grim has pointed out that the algorithm we have been using was originally intended to be valid only for the frequency range of waves which se- verely excites pitching and heaving. The new information which he has supplied to us privately has been used by Professor Porter to make the comparisons shown in his discussion, which for normal type ship sections show excellent agreement. There are two main reasons for pursuing comparisons between Professor Grim's work and that of Professor Porter. First, there is the natu- ral urge to make a comparison between two well-founded theoretical approaches to a question; especially in view of the fact that the first section of the paper still showed disagreement between theory and experiment for resonant condi- tions. In this regard Professor Porter's program does indicate higher damping in heave than the 1959 Grim data which was used in the first part of this paper. This would tend to reduce the gap between theory and experiment shown there. Secondly, Professor Porter's approach allows for the effect of changes in ship section shape which is important for sections found at the ends of the ship, whose contribution to pitch damping should be significant. Whether this refine- ment is of importance in the final answer as far as motion amplitudes are con- cerned we do not yet know. At the moment we would point out that Professor Grim's subroutine is very much faster than that of Professor Porter, but the latter program has not as yet been optimized with respect to time consumed in the machine. Mr. Gale's contribution supplements the objectives of the first part of the paper. His correlations are related to a family of destroyer forms and hence agreement appears better than in our results because of the wallsidedness of the ship sections in the vicinity of the designed waterline. In the M.I.T. pro- gram, a ship can be defined by any number of sections up to and including 20; nevertheless, it appears that computations using 10 sections yield approximately similar results. With respect to added mass computation, we prefer either the Grim or the Porter data to the Ursell- Prohaska data even though the differences according to Mr. Gale's calculations do not seem to be large. The comments of Professor Abkowitz are particularly welcome because he is an acknowledged leader and teacher in the United States in this field. A point on semantics was mentioned by Professor Abkowitz. The differences be- tween the approach of this paper and that of Korvin indicate that the newer ap- proach may be regarded as a "pure strip" theory, whereas the Korvin approach should properly be referred to as a "modified-slender body" theory. The first part of the paper demonstrates the practical utility of the Korvin-Kroukovsky and Jacobs theory. Indeed, this was our primary objective. The fact that we *See comments by Dyer on paper by Gerritsma and Beukelman. 405 Vassilopoulos and Mandel attempted to reinterprete the above theory in the second part was solely due to the difficulties explained in the previous paragraph. To Dr. Kaplan who has, we believe, in the past, offered explanations for the "erroneous time differentiation," the situation is very clear; to an outsider who attempts to trace back and forth the use of Galilean and non-Galilean coordinate systems in the derivation of the coefficients, the situation is not that clear. With the assistance of Professor Abkowitz, we developed the new approach with the hope that it would yield iden- tical results to the Korvin approach. We did not get identical results, but we did clarify several of the coefficients. With the additional corrections and ex- planations offered by Professor Abkowitz, the situation may be summarized as follows: If the added mass distribution for a given ship form is zero at the ends, then the new and Korvin-Kroukovsky approaches differ in two coefficients only, C and E. If the above assumption is not fulfilled, then they differ in four coeffi- cients, namely, e, B, C, and E. We would point out that for several kinds of ships the added mass at the stern is not zero, for example, destroyers, the latest aircraft carriers or even trawlers. Hence, added mass end-effects may be responsible for discrepancies between theory and experiment for these kinds of ships. Furthermore, the new approach as extended by Professor Abkowitz always satisfied the equalities indicated by the more sophisticated hydrodynamic analyses of Newman-Timman, whereas the Korvin-Kroukovsky approach does not. Finally, we believe that the new excitation term will be numerically as adequate as the Jacobs one, due to the small speed dependency. The authors wish to express their sincere thanks to all discussers. In this case it is not a cliché to say that each and every one of them made a significant contribution to the content of this paper. * * * 406 SOME TOPICS IN THE THEORY OF COUPLED SHIP MOTIONS J. Kotik and J. Lurye TRG Incorporated Melville, New York 1. INTRODUCTION In this paper we present several different results in the theory of ship mo- tions. Some of the results express certain physical quantities in terms of other such quantities, while the remaining results are in the direction of computing physical quantities by solving boundary value problems. The following of our results are of the first type: Kramers-Kronig relations with forward speed and cross-coupling. Impulse response in terms of force coefficient for simple harmonic motion. As work of the second type we present a numerical approach which seeks sim- plicity by avoiding integrations over curved surfaces and approximations to or representations of curved surfaces. These results already obtained are only a beginning, since they assume zero forward speed, but they are sufficiently promising to encourage us to extend them to include forward speed. 2. KRAMERS-KRONIG RELATIONS FOR HYDRODYNAMIC CROSS-COUPLING COEFFICIENTS AT FORWARD SPEED In this section we sketch the proof that the real and imaginary parts of the complex hydrodynamic cross-coupling coefficients are connected by the Kramers- Kronig relations* in the case of a submerged body having forward speed. We begin by defining these coefficients. Let the aforementioned body at first be at rest in a steady flow which (1) satisfies the usual normal velocity condition on the body surface, (2) satisfies the linearized free surface condition, and (3) becomes uniform with velocity -cx as x>+o, (Here x isa unit vector in the direction of the positive x axis.) This flow evidently represents forward motion of the body at speed c in the positive x direction. (The x and y axes are horizontal, the z axis is positive upwards, and the origin of the x,y,z coordinate system is at the center of gravity of the *See footnote after Eq. (2.2). 407 Kotik and Lurye body when at rest.) Now suppose the body executes a small time-harmonic mo- tion of angular frequency c in one of the six modes: surge, sway, heave, roll, pitch, or yaw. These modes are denoted respectively by the index i =1,2,...6 with i =1,2,3 representing translations parallel to the x,y,z axes respectively and i = 4,5,6 representing rotations about those axes. If F;,e~i7t is the com- plex hydrodynamic force or moment exerted by the fluid on the body in the jth mode when the body has a complex linear or angular velocity e-i7* in the ith mode with all other velocities zero, then the complex hydrodynamic cross- coupling coefficient H;; is defined by Hi ,(0) = -F,,(2) (Qian) where the dependence on frequency has been indicated. It is a familiar fact that a knowledge of the H;, together with the inertial and hydrostatic properties of the body suffices to determine the steady state re- sponse of the body to an arbitrary time-harmonic set of exciting forces or mo- ments applied simultaneously in all six modes. Writing I 7 Hey, (= Hy, ,() + Hy ;(¢) (QP) we now outline the proof that Hi ; and Hy F satisfy the Kramers-Kronig relations.* General Equations for Transient Problem Consider the transient disturbance that results when the body, initially at rest in the steady flow, is given at t = 0 a small displacement which is an ar- bitrary function of time in the ith mode. We characterize this displacement by a vector function of position and time 2,(x,y,z,t) defined only on the undis- placed body surface (call it S,), such that @,(x,y,z,t) is the displacement at time t in the ith mode of a body surface point whose coordinates were (x,y,z) at t= 0. Let 4,, u,, 4, be unit vectors in the x,y, and z directions respec- tively, x,(t), x,(t), x,(t) the instantaneous magnitudes of the translational displacements in the first three modes, and x,(t), x,(t), x,(t) the instantane- ous magnitudes of the angular displacements in the last three modes. Then (ib) SFC) eS Dg (2.3) Dien yaey) = SaCe) eae | 8 = 45. (2.4) ESerictly. only after certainterms have been subtracted from the Ble ae do the real and imaginary parts of the remainder satisfy the Kramers-Kronig rela- tions. See Eqs. (2.24) ff. 408 Coupled Ship Motions where F = xh + yliy + fy. (2.5) Note that as indicated, a, is independent of x,y,z for i =1,2,3. Note also that Eq. (2.4) is valid only for small x, (i = 4,5, 6). Now let ),(x,y,z,t) be the disturbance potential associated with the small displacement x,(t) in the ith mode only, where x,(t) = 0 for t <0. Then in addition to being a solution of Laplace's equation, ¥, also satisfies the following conditions: ow. O2W. 32. 2 32 mest oe Hi ap & Hi el eer 272005 St&>’ 0) (2.6) Oz 2 ot € oxot £ dx? Ow; oa; ee ‘ 2.7 on = - EE + GSTS -n (CGaye2) on Sa 2 We ( e ) In Eq. (2.7) [1], 1 is the unit normal to S, pointing into the fluid, 3/en is differentiation in the direction of n, and V,(x,y,z) is the velocity at (x,y,z) of the steady flow generated by the body at rest in the uniform stream. The two initial conditions on ¥,, applied at t = 0+ on the undisturbed free surface, are Wi(x,y,0,0+) = 0 (2.8) = ¥,(09.0,t)¢-04 = 0. (2.9) In case x,(0+) = 0, Eq. (2.8) follows from the fact that ¥, vanishes not only on z = 0 at t = 0+, but throughout the fluid. Equation (2.9) is then a con- sequence of Eq. (2.8) combined with the fact that the free surface elevation due to the body motion is zero at t = 0+. In case the body suddenly acquires a finite velocity at t = 0+, i.e., x,(0+) + 0, then ¥; vanishes on z = 0 at t = 0+, though not in general vanish- ing throughout the fluid. This follows from the equations of impulsively gener- ated motion [2] combined with the fact that the pressure is zero on the free sur- face. Equation (2.9) then follows as before. Now by modifying a procedure used by Cummins [3] we can write the follow- ing representation for the potential w;(x,y,z,t): t W(GEAva Zane) taxa (E)) PaCxany 12) +| KC TANG ees Yok, ce ah ae (2.10) -@ 409 Kotik and Lurye Here ¢;(x,y,z) iS a time-independent potential function satisfying the free surface condition $;(x,y,0) = 0 (21a) and the boundary condition od: “ = Su eae n iL = 2.12a sn (2x oToN WNC NSE gat Ly 3 ( ) Od. n 4 =< = = (in gumem ol S, 2 = 2.5, 6. (2.12b) ,;(*,y,z,t) is a potential function that satisfies the free surface condition, Kq. (2.6), for t >0, and the boundary condition Bee SRW (Vy gh onl BSAeN PAN=# ee (2.13a) fis = Ui eae 5D) et rs Ego KE, 8 (2.13b) ioe te SO," The initial conditions on y,; are ¥14(%, y,0,0+) = 0 (2.14) and a Vines sale eng =) eC nen (2.15) It can be verified by direct substitution that the function ¥,(x,y,z,t) defined by Eq. (2.10) does indeed satisfy Eqs. (2.6), (2.7), (2.8), and (2.9) when the func- tions ¢;(x,y,z) and y,;(x,y,z,t) satisfy Eqs. (2.11) through (2.15). We recall that @; appearing in Eq. (2.7) is given by Eq. (2.3) or (2.4). Duhamel's Principle We now suppose the body, initially at rest in the stream, to be given (at t = 0) a unit displacement in the ith mode. The fact that such a displacement is not small is irrelevant. Let the potential corresponding to the unit displace- ment be ¥,(x,y,z,t). Since in this case x,(t) = 8(t), it follows from Eq. (2.10) that *Note that ¢, has the dimensions of potential/velocity when i = 1, 2,3 and poten- tial x time when i = 4,5,6. 4,; has the dimensions potential/length when i = 1, 2,3 and potential/angle = potential when i = 4, 5,6. 410 Coupled Ship Motions W5(%y, 2, t) = d(t) P(X, Y,2) + H(t ) Wi4u(%,y,z,t) (2.16) where 5(t) is the Dirac delta function and H(t) the Heaviside unit function. Note that 5(t) has the dimension 1/T. Denote by p,(x,y,z,t) the linearized pressure arising from the unit dis- placement, the pressure being evaluated on the displaced surface of the body but expressed in terms of coordinates on the undisplaced surface S|. Then from the linearized form of Bernoulli's principle, we have Ne owe ae A P;(x,y,z,t) = Ape + VC nz) Wipe = a;(x,y,z,t) sau dsaohl (x,y,z) on S). (2217) In Eq. (2.17), the last term on the right corrects for the fact that coordi- nates on the undisplaces surface are used in expressing the pressure on the displaced surface. In that term a, has the forms of Eq. (2.3) or (2.4) with x;(t) = H(t), the Heaviside unit function. Let f,;(t) be the hydrodynamic force or moment on the body in the jth mode arising from the unit displacement applied at t = 0 to the body in the ith mode. Then a2 y= -| [Bicoy2 0 a-A, ds hee gl Beara 0) (2.18) ss ily Dees fij(t) = -| [Bieny.z.0 Fxa- i; 5s rita eee 5) (2.19) SS je =1 4h, Sato In Laplace's equation and in the Eqs. (2.6) through (2.9) satisfied by y,, the coefficients of ¥; are independent of time. From this it follows that if the unit displacement is applied at t = 7 instead of t = 0, the resulting force or mo- ment willbe f;;(t-7). Moreover, all the equations are linear. Thus we may invoke Duhamel's principle and write that f oo t), the force or moment in the jth mode corresponding to the velocity x,(t) in the ith mode, is given by t He) = i fe (Chat) x;(T) dr. (2.20) In particular, when x,(t) = H(t)e i?*, j zs -ior fat) = { i Cerda e dr 0 t eae J f(a es eed (2.21) 0 411 Kotik and Lurye From Eq. (2.21), we see that if F,;e-i7t is the complex steady state hydro- dynamic force or moment on the body in the jth mode corresponding to the steady state velocity e-1°* in the ith mode, then (oo) Pon = { feet dr’ = =i, , (2.22) ij 0 where f,,(t) is given by Eqs. (2.18) and (2.19) and the second equality in (2.22) comes from Gs: ((Patl))s Kramers-Kronig Relations If the integral in (2.22) converged suitably for all real c, then it would be an analytic function of c in the half plane Im o>0, vanishing aS o>, whence it would follow that H. ;;, and Hi. ;; Satisfy the Kramers-Kronig relations. Now, con- struction of f;,(7) from Eqs. (2.16) through (2.19) reveals that in fact, H,, is the sum of two types of functions of c, such that the real and imaginary mine of the first type satisfy the Kramers- -Kronig relations, while the functions of the second type are too singular either at > = 0 or co = o for the Kramers-Kronig relations to hold. On the other hand, the functions of the second type depend only on infinite frequency potentials and on the steady flow in the absence of oscillations, and may therefore be regarded as easier to calculate. Thus itis the less- known part of H;; that satisfies the Kramers- -Kronig relations. Specifically, when i,j = 1,2,3 we find by substituting from Eqs. (2.16), (2.17), and (2.18), into Eq. (2.22): {oo} 5H, (2) = J s(n) elt ar || 4, n- ji, ds 0 to} {oo} +f 8(T) ance VO; n: i; ds 0 + { Xr) Wh CS Yo 0D) eterar| [a “12; dS 0 S) I — gS ie) — XN iW) oo — + { V5? Wao BP) eer dr | [ae A, ds i 0 s wes ° J | Oo 412 Coupled Ship Motions After some manipulation this reduces to Hj ,(¢) = H; (2) = iop | | a; n+p; ds iS) ' e[ {¥, ‘7b, fifi, dS + p{ fe:A-A; 4s So So p ° ~ + B| | [A vow] n+; dS i Ss ° j (2.24) IL By & Although details are omitted, we have assumed in deriving Eq. (2.24) that the lim W,;(%, y,z;t) t7> © exists and is equal to the incremental steady flow associated with the body in its displaced position. In Eq. (2.24), the potential ¢,(x,y,z) is defined as Tim Wy (CX, Y¥5 Zt) nO and is therefore the infinite frequency potential satisfying the boundary condi- tion in Eq. (2.13) on s,. The real and imaginary parts of H;, satisfy the Kramers-Kronig relations: ¢ He (a. yoy = Bf RS ae (2.25) 1j a o'-o ©) f(a") sof Ter ahaa, ee f at aoe (2.26) 1j TT ol-o where the bar on the integral indicates the Cauchy principal value. Thus from a knowledge of either oe or Hae the other can be inferred, while as already mentioned, the remaining terms in Eq. (2.24) may be regarded as comparatively easy to calculate. For completeness, we include the expressions for H; ;, analogous to Eq. (2.24), for the remaining mode pairs. We have 413 Kotik and Lurye (ey iso | [oA +A, ds + p | [¥o-v8i 8-8, dS + ef fea-a, ds S) Ss > H; ; (2) = H; 5(7) = H; ;(¢) = ioo| [o; rxn-; 3 d5 S + e| [,- 94) FA “fi; _, dS + ef fax FxA-f,_,d8 S> ss Pp aN 2 ee = + ele VV; )| Txn Bj 3 ds 1 ie (2.28) ° J] = 4,5 he) = Tae iop | | 4: rxAn-f; 44S So + PI oe a stor + Oilens stain So 35 p ~ = ee ae + iz [ (is? -V(V2)| FxA-f;.,d8 i= 4,5, 6 (2.29) Si i=) 4 35,68 In all of these, the real and imaginary parts of H,,(c) satisfy Eqs. (2.25) and (2.26). We conclude with the following remarks: 1. The Kramers-Kronig relations imply that any symmetry property in i and j possessed by the element Hi is shared by Ai j and vice-versa. Thus one need only establish such a property for the real or imaginary part alone. 2. It is known [4] that a submerged body oscillating in a stream can for certain modes, frequency ranges, and speeds acquire energy from the stream as a result of the oscillation (negative damping). The question then naturally arises whether the Kramers-Kronig relations can still hold if over some part of the frequency range negative damping occurs. Highly tentative considerations indicate that there is at least a possibility of deriving a modified form of the Kramers-Kronig relations in the case of negative damping; however, no firm conclusions have been reached as yet. 414 Coupled Ship Motions 3. EXPRESSION OF THE IMPULSE RESPONSE IN TERMS OF ADDED-MASS AND DAMPING PARAMETERS In [5] it was pointed out that F(t), the hydrodynamic force exerted by the body when the body acceleration is 5(t), can be calculated from either the damping or added-mass parameter (for simple-harmonic oscillation) via the Kramers-Kronig relations followed by a Fourier transformation. We will now discuss this point further, including some observations on a later publication [6] which also treats transients and their relations to force parameters. Let us recall that according to Eq. (A-5) of [5] we have J Br ete et = Goin (a) = Tp [PA(o) + ip{(o)] (3.1) 0 F(t) = hydrodynamic heave force exerted by the body on the fluid, per unit step heave velocity of the body at t = 0, F(t) = 0 for t <0; p‘(c) = force parameter = p/ +ip4; P,(c) = added-mass parameter; pg(c) = damping parameter; oc = radian frequency of oscillation; 7+ = submerged (or any other) volume of the body for three-dimensional problems, and volume/unit length for two-dimensional problems. It follows that F(t) = = | p'(c) e i? t do 21 -@ foo} ts - =) [Pi(c) cos ot + pi(o) sin ot] do 0 = #fpue 76(t) +f [Api(c) cos ot + p4(o) sin ot] oo} (3.2) 0 where Apia) = Pie) = p.() - However, it is sufficient to know either p/(c) or p3(c), due to the Kramers- Kronig relations, and in fact those relations imply the following: 415 Kotik and Lurye F(t) TPP) (t) + 2 ro | p4(c) sin ot do (3.3) 0 INCE) = 7eiD-(@) Ce) + 2 re | [Ap*(o)] cos ot do (3.4) 0 (s(t) has dimensions T!). Note that the s-function acceleration of the body produces a 5-function hydrodynamic force having strength proportional to the added-mass parameter at infinite frequency. Heave at infinite frequency is uni- form translation of the double body in an infinite fluid. Note also that the two integrals in (3.3) and (3.4) are equal. This implies that J Ap,(a)do = 0, (3.5) 0 a useful fact which does not seem to have been observed previously. The relations Eqs. (3.2)- (3.4) are useful for direct calculation, when p/(c) and/or p,4(c) are known, exactly or approximately, and for finding asymptotic expansions as t>0,. For example, to find F(t) as t>~, we first write {oo} t : t i 3 U | De) Sim Cb de S = p(a) = +f = ae [PO] & 0 0 PONS Bera) (3.6) Now as stated in [5], for the heaving motion of a cylinder of arbitrary section, DAO) = (Bayer. (3.7) where 2a = width at the free surface and 7 is the submerged volume per unit length, so that for such a cylinder, we have from Eq. (3.3) Ca a Bacio F(t) TT ewaaTT CA Ga SATE AS, i¢ + © > (3.8) This hydrodynamic force per unit length exerted by the body on the fluid is downward if the 5-function acceleration is downward. For an arbitrary heaving three-dimensional body we have, as noted in [5], pi(o) = p,(Ka) = b,Ka + o(Ka), (3.9) 416 Coupled Ship Motions as Ka~> 0, with b, = Ti a3, a = VA./7 (3.10) where A, is the area in which the body intersects the free surface. Hence, at least formally, 2 ! (or oc) ~ b,a— P47) 8! = pic) 2b aofe) (3.11) Oh ae) De (CD a ANG EYE allasoc-0. After integrating by parts several times, we may write = ‘ , ee 1 “cos ot 33 ; J Pi(2) sin otdo = - Ra (2b, a/g) 2h 3 AE es ee A? SE Liotihe 3 = Sharon 2 A mg A OGIES) (3.12) g7 t3 as t>o. Therefore, from Eq. (3.6) the heave force exerted by an arbitrary body is 2 2 F(t) ~ 2, 2 oS Ln a ee ADs (Sets) 17 g t3 Tet 3 as t+. We See that this force exerted by the body on the fluid is upward if the 5-function acceleration is downward. We will now find the heave displacement, for large time, of a body released at zero velocity from hydrostatic disequilibrium. The equation of motion, for an arbitrary surface-piercing body, is t VAC ESP Ay (at) =| F(t - 7) ¥,(7) dr (3.14) 0 221-249 © - 66 - 28 417 Kotik and Lurye where y,(t) is the heave displacement measured with respect to the position of buoyant equilibrium. For three-dimensional bodies, M is the mass of the body, and A. its cross-section area in the free surface, while for two-dimensional bodies M is the mass per unit length of the body and A, its width in the free surface. Taking the Laplace transform of Eq. (3.14), introducing the initial conditions that y(t) = y,(0) at t = 0, y,(0) = 0, and converting Fourier transforms, we find for Y,(c), the Fourier transform of y(t), - iy_(0 [rTep' M] oe at7((O))), Cerilrr eho) (len) or (3.15) a = em noe (Ce) + il Separating real and imaginary parts, we can write Eq. (3.15) in the form Yo) = - iyg(0) | YRC) + iv/3(o)] (3.16) where the primes mean that -iy,(0) has been factored out as shown. Yj" and Y," have the following forms: a o(Tep,+M) [pgA, - o? (Top, +M)] - 097? p? py? (3.17) i 2 t [egA.- 0? (rep, +M)?] + 04 7? p? py? oTp* pi. gA ee) = Cie came | (3.18) i 2 f [eek = Sea ey ere pas Since p/(c) is an even function and pj(c) an odd function of c, one sees from Eqs. (3.17) and (3.18) that y/® is odd and Y;' is evenin c. It follows that upon taking the inverse Fourier transform of Eq. (3.15) we can write 0 @ Yok | Wee) cos ot - L(e) sin ot dco. (3.19) 0 Welt) = - We now use Eq. (3.19) together with Eqs. (3.17) and (3.18) to infer the as- ymptotic form of y,(t) aS t>«. This form depends on the behaviour of Y5*(7) and Y,®(c) in the neighborhood of «= 0. We treat the cases of two- and three- dimensional bodies separately. Two-Dimensional Bodies In this case [5] (3.20) | | = ) 7a Q q L } pi(c) = Coupled Ship Motions Z A From these equations combined with Eqs. (3.17) and (3.18) we infer that 12 YER Gai) Galllogitas, o>0 (3.22) ¥ Taye meeaele SPST Son (3.23) Incorporating these results into the integral of Eq. (3.19) we find, through integrating by parts, the following leading terms at large t: > A { VL (Ge) Con caecle = = 5 ‘ t > 0 (3.24) 0 et 3: A { Yi8(c) sin otdo = — , t > 0 (3.25) 0 et? whence Cn a Tn ge eed (Ones Secs (3.26) Yo ae Ns) we = = Be , : where a = A./2 is the half-width of the cylindrical body in the free surface. Equation (3.26) gives the large time behaviour of the heave displacement of a cylindrical body released at zero velocity from a position of hydrostatic dis- equilibrium. The expression on the far right of this equation agrees with that obtained by Ursell [6] for a half-submerged circular cylinder of radius a. How- ever we now See that this expression is valid for cylinders of arbitrary cross section having a width 2a in the free surface. Three-Dimensional Bodies For three-dimensional bodies, we have [5] greed one (3.27) Pao) = pA(0) - = —= a pico) = 5&0? | Se Incorporating these results into Eqs. (3.17) through (3.19), we find after a number of integrations by parts in (3.19) 419 Kotik and Lurye acu) = : t > ©. (3.29) A comparison of this expression with the corresponding one for cylindrical bodies, Eq. (3.26), shows that: 1. the approach to buoyant equilibrium in three dimensions is asymptotically faster than in two dimensions by a factor proportional to 1/t?, and 2. the approach to equilibrium in three dimensions is asymptotically from the side of the equilibrium position defined by the initial displacement; in two dimensions the approach is from the side opposite the initial displacement. It is our intention to present, in a future publication, calculations of tran- sient forces and displacements using Hi-Fi approximations* to p,(c). 4. NUMERICAL DETERMINATION OF HYDRODYNAMIC COUPLING COEFFICIENTS FROM VOLUMETRIC SINGULARITY DISTRIBUTIONS In this section we outline briefly a numerical scheme for calculating the hydrodynamic coupling coefficients H; ; (already defined in Sections 2, 3, and 4) for a fully or partially submerged body engaging in small time harmonic oscil- lations. Our computer program so far covers only the zero speed case, but its extension to forward speed should present no difficulty in principle; the chief additional complication would center on the calculation of the time-harmonic Green's function for a point source in a steady stream below a free surface. The idea of the method is to approximate the velocity potential exterior to the oscillating body by the potential of a time-harmonic finite set of singularities contained in the interior of the body surface. These singularities will usually be either sources or dipoles although higher order multipoles can also be used. The strengths of the singularities are determined by the requirement that the normal velocity they induce on the submerged portion of the undisplaced body surface, S,, should best approximate the actual normal velocity of S, ina cer- tain mean Square sense. Specifically, let P,(m = 1,...M) be the points where the M singularities of complex strength q, are located interior to S|, and let Pn = 1,...N) be a Set of points on S, with N>M. Let a, be the complex normal velocity at P" due to a singularity of unit strength at P,, the singularity potential satisfying the linearized free surface condition. Finally let v" by the actual complex normal velocity of S, at P" due to the oscillation. Then we seek to determine the q, so as to minimize the mean square expression oe 3 ne (4.1) m=1 Ht ce Ue n=1 *Examples are given in [5]. Owever, we plan to consider other types of approximation as well. 420 Coupled Ship Motions Note that the value of J when the q,, have their minimizing values, serves as a measure of the closeness with which the exact potential exterior to S, has been approximated. It is easily shown that the minimizing q,, satisfy the following set of linear algebraic equations: M Sg oe, te (4.2) m=1 where N Dim f a Aken ann (4.3) n=1 N é = Sapa. (4.4) n=1 where the asterisk denotes complex conjugate. Once the q,, are determined by solving Eq. (4.2), several methods are available for calculating the hydrodynamic forces and moments on the body and thereby the hydrodynamic coupling coefficients. Lagally's Method Cummins [7] has derived an extension of Lagally's theorem to time- dependent flows, which can be used to obtain the oscillatory hydrodynamic forces acting on the body. The calculation is exceedingly simple, requiring (for the linearized force in the case of small oscillations) a knowledge of the singu- larity strengths and locations and nothing else. (A simple summation over the singularities must be performed.) However, this method suffers from two limi- tations. One, it is applicable only to fully submerged bodies since the extension of Lagally's theorem to bodies that pierce the free surface does not yet seem to have been accomplished. Two, even for fully submerged bodies, Cummins' method gives only the forces and not the moments. Energy Method By considering the rate at which energy is radiated out to infinity, one can express the real parts of the complex cross-coupling coefficients for time- harmonic motions as a sum over the singularities. The terms in the sum in- volve the singularity strengths and certain potentials or potential gradients evaluated at the singularity locations. With this technique, the real parts of the coupling coefficients corresponding to both forces and moments can be obtained. Moreover the body need not be fully submerged. Finally, once the real parts of 421 Kotik and Lurye the coupling coefficients are determined as a function of the frequency, the imaginary parts can be calculated from the Kramers-Kronig relations. We quote the result for a distribution of sources: N; Ni R * * * His ee, TE indie Sindy aes eee (2-2) m=1 k=1 Here the q,,(m=1, ... N;) are the strengths of the sources at the points P,_, these sources generating the approximate motion in the ith mode, while the q;,(i=1, ...N;) have the same significance for the jth mode. Some of the points P;, and P,, may coincide. The function of position ¥(P,,,P,,) is the regular part of the Green's function G(P,,, P;,) Satisfying the free surface condition. Finally, pe is the fluid density and o the angular frequency of the oscillation. Pressure Integrals The most obvious way to arrive at the forces and moments on the body is to use the singularity strengths to obtain the pressure distribution on the sub- merged body surface and then form the appropriate pressure integrals over that surface. From a computational standpoint, it is extremely important to note that the integrations need not be carried out over the actual surface of the body. Rather, one can express each component of force or moment as an integral or combination of integrals over the plane domains defined by projecting the sub- merged part of the body surface onto each of the three coordinate planes. Thus only ordinary double integrals over plane regions need be computed. We conclude with the results of a preliminary numerical test. These re- sults were obtained by applying our procedure to the case of a prolate spheroid in an infinite fluid. The assumed motion of the spheroid was a small time- harmonic translation in the direction of its axis (surge). The thickness-to- length ratio was 1/8. For the singularity distribution, we chose a set of 45 axi- ally directed dipoles located on the axis of the spheroid. Having determined the dipole strengths in the manner already described, we then calculated the ampli- tude of the linearized time-harmonic pressure on the surface of the spheroid. Our results are shown in Figs. 1 and 2. Figure 1 is a plot of the normal- ized real amplitude of the time-harmonic dipole moment vs normalized axial distance. The normalized real amplitude is defined as 4/u,, where py is the real amplitude of the unnormalized dipole moment, and , is the amplitude of the dipole moment at the center of the spheroid. The normalized axial distance is x/a, where x is the distance from the center of the spheroid measured along its axis and a is the half-length of the spheroid. The solid curve represents the exact continuous distribution of dipole strength — this is known to be para- bolic for surge in an infinite fluid — while the two broken curves represent ap- proximations computed by our procedure. In both of the latter, a discrete dis- tribution of 45 equally spaced axial dipoles was assumed to lie between the foci. The two approximations differ in that the mean-square boundary condition in- volved 48 points on the spheroid surface in the one case and 96 points in the 422 0.03 0.02 Coupled Ship Motions LEGEND EXACT — — — 45 SINGULARITIES, 96 SURFACE POINTS — - — 45 SINGULARITIES, 48 SURFACE POINTS 1.0 0.8 THICKNESS _ 1 LENGTH 8 0.2 10) 0.2 0.4 0.6 0.8 10 oan a Fig. 1 - Normalized amplitude of dipole moment for surging prolate spheroid LEGEND —— EXACT -- —-- 45 SINGULARITIES, 200 SURFACE POINTS 45 SINGULARITIES, 96 SURFACE POINTS 45 SINGULARITIES, 48 SURFACE POINTS THICKNESS _ LENGTH Fig. 2 - Normalized amplitude of time-harmonic pressure on surface of surging prolate spheroid 423 8 Kotik and Lurye other. As one might have expected, the second approximation is somewhat bet- ter; however both are very close to the exact distribution. In Fig. 2 we have plotted the normalized real amplitude of the time-harmonic pressure on the surface of the spheroid vs normalized axial distance. (From symmetry, the pressure is obviously a function of the axial coordinate only.) The normalized real amplitude is defined as P/pcavV, where P is the real am- plitude of the unnormalized pressure, and v is the real amplitude of the sphe- roid velocity. The solid curve represents the exact pressure distribution, which in the case of surge in an infinite fluid, is known to be a linear function of the axial distance. As can be seen from their labels, two of the broken curves were calculated from the approximate dipole distributions of Fig. 1. The third pres- sure curve was obtained from a discrete distribution of 45 dipoles whose strengths were computed by applying the mean square boundary condition to a set of 200 points on the spheroid surface. Evidently it is only near the nose that the approximate pressures depart sensibly from the exact one, and even there the relative error is less than 15%. It is worth noting that neither the computation of the dipole strengths nor the subsequent pressure calculations exceeded 0.01 hr of IBM 7094 machine time for any one Case. REFERENCES 1. R. Timman and J. N. Newman, "The Coupled Damping Coefficients of a Symmetric Ship,'' Journal of Ship Research 5, pp. 1-7, March 1962. 2. H. Lamb, Hydrodynamics, 6th ed., Dover Publications, New York, pp. 10-11 (1945). 3. W.E. Cummins, The Impulse Response Function and Ship Motions, David Taylor Model Basin Report presented at the Symposium on Ship Theory, Institut fir Schiffbau der Universitat Hamburg, 25-27, January 1962. 4, J. N. Newman, "The Damping of an Oscillating Ellipsoid Near a Free Sur- face,'' Journal of Ship Research 5, pp. 44-58, December 1961. 5. J. Kotik and V. Mangulis, "On the Kramers-Kronig Relations for Ship Mo- tions,"’ International Shipbuilding Progress 9, pp. 361-367, September 1962. 6. F. Ursell, "The Decay of the Free Motion of a Floating Body," Journal of Fluid Mechanics 19, Part 2, pp. 305-319, June 1964. 7. W.E. Cummins, ''The Force and Moment on a Body in a Time-Varying Potential Flow,"' Journal of Ship Research 1, pp. 7-18, April 1957. 424 KNOWN AND UNKNOWN PROPERTIES OF THE TWO-DIMENSIONAL WAVE SPECTRUM AND ATTEMPTS TO FORECAST THE TWO-DIMENSIONAL WAVE SPECTRUM FOR THE NORTH ATLANTIC OCEAN Willard J. Pierson, Jr. New York University New York, New York ABSTRACT The two-dimensional wave spectrum has been estimated once by stereo photogrammetric techniques, and a number of times by buoys developed by the National Institute of Oceanography. The results obtained do not contradict each other. Some questions have recently been resolved and one remains unresolved. There do not appear to be spectral compo- nents in a pure wind sea traveling ina direction opposed to the wind. The theory relating wave number to frequency from linear considera- tions can be applied. Whether or not the spectrum is bi-modal as a function of direction for certain frequencies is not yet decided. A form for the directional spectrum of a fully developed wind sea is proposed. Under certain assumptions about the generation of wind seas attempts to forecast the two-dimensional spectrum at 519 points on the North Atlantic have been made. Verification of the forecasts against ob- served two-dimensional spectra are not possible. However, they verify fairly well in terms of significant height and against the observed fre- quency spectra and in terms of swell and wave decay. It appears that the forecasting procedure is fairly close to being correct. INTRODUCTION Nearly all of the papers at this symposium are concerned with the deter- ministic mathematics applicable to the analysis of the classical hydrodynamic problems that are concerns of the naval hydrodynamicist. However, one of the inputs to the problem of understanding the motions of marine craft at sea is essentially probabilistic in nature. The actual sequence of waves that will be met on a given cruise can never be predicted before the fact. To predict certain 425 Pierson features of the behavior of such a craft on a certain cruise for, say, the next 6 hours or, perhaps, even the next 24 hours, one must give up the deterministic world and predict the probabilities of certain events and statistics derivable from them. Such predictions can be quite refined statements, given sufficient knowledge and understanding of a number of factors. For example, it may be possible some day to make statements of the following kind: 1. Merchant ship design A is superior to merchant ship design B for cruises between New York and the English Channel because (1) if each ship were to follow the least time track route on each cruise for five years, merchant ship A would average five days two hours per crossing as opposed to six days one hour for ship B, (2) the bow of ship A would ship water 560 (+20) times (with a probability of 0.99) during the five year period and ship B would ship water 650 (+3) times (with the same probability) and (3) ship A would slam only 6 (+3) times (with a probability of 0.99) whereas B would slam 50 (+5) times. 2. Of five ships available for a rescue mission at a certain point, this par- ticular ship should move as quickly as possible to that point. It will arrive two hours (+20 minutes) sooner than the earliest of the other four ships. The sec- ond ship to send is such and such a ship as a safety factor or as a standby re- serve. 3. All ships in a certain part of the North Pacific will encounter seas in excess of the highest measured for the past decade beginning 18 hours from now and ending 30 hours from now. All possible safety precautions should be taken immediately. Predicted conditions for specific points in this area follow. Statements such as these will be possible when it is possible to describe the directional spectrum of the waves at every point on the ocean as a function of the winds over the ocean. The first statement can be made on the basis of the historical files of weather data. The second and third require the wind field to be forecasted a day or so into the future. It is therefore necessary to describe this directional spectrum in its infi- nite variety and to predict its form at future times. Strangely enough, this problem is, to a large extent, deterministic. As an analogy, to predict the vari- ance of a sample to be drawn from a normal population is not the same as to predict the actual values that would be drawn at random from a normal popula- tion with a known variance. In this particular problem, to predict the features of the directional spectrum that will be estimated from observations of waves in a particular area is not the same thing as to predict the exact form of the waves that will be observed in a particular area. The predicted spectrum in turn permits the determination of many wave and ship motion parameters such as the significant wave height, the average pitch motion, the number of slams and so on. One then assumes that the predicted parameters are those that will be the population parameters at the point of interest for the event of interest. These parameters are then estimated directly from observation, if possible, and compared with the prediction. The attempt is successful if the predicted and estimated values agree within the sampling variability of the estimate. 426 Two-Dimensional Wave Spectrum The purpose of this paper is to summarize what we now think we know about the directional spectrum of waves at sea and to discuss how we are trying to predict this directional spectrum at 519 points in the North Atlantic Ocean. FULLY DEVELOPED WIND SEAS If the wind blows with constant speed and direction for a long enough time Over an initially calm ocean area, if this ocean area is big enough, if no waves propagate into this area from outside it, and if the turbulent features of the wind do not change, then a fully developed wind sea should be observed over part of this area, and wave observations made in this fully developed wind sea should all be samples that have come from the same population. The spectral esti- mates, S(w,¢), made from these observations should display sampling variabil- ity in terms of departures from some unknown population spectrum S(w, 6). There are only a few available estimates of §(w,¢). There are, however, now available in useful form about 500 estimates of S(w) that were obtained from the analysis of waves recorded at a fixed point as a function of time. These estimates are given by Moskowitz, Pierson, and Mehr (1962,1963) and by Pickett (1962). Of these 500 spectral estimates, S(o), about 40 were found by Moskowitz (1964) to correspond to fully developed seas for winds from 20 to 40 knots. All of the others could not be simply defined by the wind speed measured at the time the wave record was made. As one example for winds near 20 knots, the waves are usually higher than those expected for a fully developed sea because components left over from previously higher winds and components from swell are present. Given some form for S(#,@) to describe the spectra of fully developed wind seas, One is therefore a long way from describing the spectrum that will be es- timated at a particular point at sea because the wind will not have been constant in speed and direction and waves from a distance may have propagated into the area. The spectrum for a fully developed wind sea is however a fundamental building block in attempts to describe the spectra that will occur in more com- plex situations. KNOWN PROPERTIES OF THE SPECTRUM Directional Spectra and Frequency Spectra The directional spectrum of waves can be thought of as being written in the form S(@,0) = S(#) [f(, 6)] (1) where f(#,@) in turn can be written as 427 Pierson f(w,0) = a + Dg [a,(2) cos nO + b,(#) sin né] . (2) n=1 It then follows that ih S(@, 6) dé > TA S(@) (3) and that 7 iI ira, @ ee > Ut I = (4) An attempt to describe S(#,6) correctly therefore implies that S(~) is correct. If spectra estimated from a time history at a point are not correctly described then surely directional spectra estimated from more complete data will not be correctly described. The Frequency Spectrum The book, Ocean Wave Spectra, describes a wide variety of proposed forms for S(w) as reviewed and summarized at the Easton Conference on Waves held in 1961. Based on an application of a theory given by Kitaigorodskii (1961), and by means of the results of Moskowitz (1964), Pierson and Moskowitz (1964) have proposed a new form for S(w). It is given by Eq. (5). 4 ee) (5) ’ aw? S(@) = where a = 8.10X10°3, 6 = 0.74 and «, = ¢/U. Here U is the wind speed meas- ured at 19.5 meters above the sea surface. The anemometer that measured U was at this elevation on the ship. This spectrum has many features that agree with other proposed spectra and an analysis of the effect of the variation of wind with height has reconciled many of the apparent discrepancies pointed out so strongly at the Easton con- ference. [I might add parenthetically that the spectrum proposed by my colleague, Dr. Neumann, is remarkably close to this one for winds near 30 knots. It is, however, seriously off for higher winds, and any design considerations based on the age due to Neumann for high winds should be re-evaluated (see Pierson, 1964). As the group to which this paper is addressed does not consist of those working on the problem of forecasting ocean waves, it is important to remark that the above spectral form represents the writer's opinion as to the best pres- ently available description of the frequency spectrum of a fully developed wind 428 Two-Dimensional Wave Spectrum sea. If one becomes seriously interested in using this spectrum for applications in naval hydrodynamics, he should check the opinions of those workers in this field who may not agree with this belief. Known Properties of S(«, @) With a handful of directional spectrum estimates available, it is not sur- prising that not much is known about S(w,¢). The available estimates have been given by Cote et al (1960), Longuet-Higgins et al (1963) (and in other publications describing the same data), Cartwright (1963), and Cartwright and Smith (1964). It was assumed in Cote et al that S(#,?) was zero outside of the range -7/2 < @ < 7/2 where 6 = 0 is the direction toward which the wind was blowing. There were good reasons for this assumption and the data bore them out, but it could not be proved that S(, 6) was zero outside the above range. Analyses by Longuet-Higgins et al (1963) were able to obtain the first five values in Eq. (2), that is 1/27, a,(#), b,(#), a,(#), and b,(#). The results suggested that S(«,) was not zero for spectral components traveling opposite to the wind. However, a new device, called a cloverleaf buoy, developed at the National Institute of Oceanography now yields a,(#) and b,(#). Indeed, that part of S(#,@) outside of -7/2 < @ < 7/2 is small. The preponderance of the available evidence now is that little or no spectral energy in a fully developed wind sea will be associated with spectral components traveling opposite to the wind. ; All of the available directional spectrum estimates also indicate that S(«,?) is more strongly peaked for low frequencies and that it broadens with increas- ing frequency. One possible explanation for this effect is contained in the theo- ries of Phillips (1957), which suggest that S(#,4) should have two peaks that move further apart with increasing frequency. None of the presently available directional spectrum estimates have the resolution and the degrees of freedom necessary to resolve the question of whether or not this bi-modal form occurs. An experiment could be designed to resolve this question by the combined use of both stereo-photogrammetric techniques and the latest buoy developed at the National Institute of Oceanography. For some applications of the power spectrum, it is desirable to be able to describe the sea surface as a function of distance instead of as a function of time ata point. This involves the transformation from an w,é representation to an ¢,m representation where 47+ m? = k?, k = w?/g, £= w? cos 6/g and m= w* sin 6/g. A discussion in Ocean Wave Spectra suggested that k did not seem to be given by «?/g, but since then Mr. Cartwright of the National Institute of Oceanography has informed me that subsequent analyses all verify this linear representation between wave number and frequency to within the present accu- racy of the available data. This result does not eliminate the problem com- pletely as nonlinear effects of a more subtle nature are present. It will be a long time before these nonlinear effects are completely understood. For information purposes, in our attempts to forecast waves for the North Atlantic, the form given by Cote et al (1960) has been used. In the notation of 429 Pierson this paper the function S(w,6) is given by Eq. (6) for -7/2 < @ < 7/2 and by zero otherwise. -(a/a@,)*/4 (6) -(w/o,)*/4 cos 26+0.032e rare) = [1 + (0.50+0.82¢ cos ae]. One should note that the Fourier series representation of S(w,¢) as in Eq. (2) would require many more terms to fulfill the zero otherwise condition that was assumed in the above expression. FORECASTING DIRECTIONAL WAVE SPECTRA At present, my colleagues and I are attempting to predict the directional wave spectrum, S(w,9), given the winds over the North Atlantic Ocean. If S(, @) is the directional wave spectrum, if one has the directions, 0, 7/6, 7/3, 7/2, 27/3, 57/6, 7, and so on to 27 with respect to north as zero, and if one has cer- tain frequencies, f,, f,,..., f,,, then our forecasting scheme attempts to pre- dict 180 numbers, one of which would be, for example, (i ine S(@, 6) | df. (7) f - 7/12 Stated another way, the directional spectrum is described by the variance con- tributions to fifteen frequency ranges for each of twelve direction intervals at each point. Based on some theory, and some empiricism — considerations too numerous to detail here* — we have started with observed winds over the North Atlantic ocean for December 9 to 17, 1955, December 11 to 27, 1959, and November 17 to 30, 1961. These winds have been described at each grid point of interest in the problem every six hours for each of the above periods. The winds are then used to predict these 180 numbers to define the directional spectrum at each grid point. No adjustments are made in the wave spectra and in the forecasting procedure for the entire period of the forecast. The output consists of numbers that describe the directional spectrum as defined above at each grid point. From them, we can get the frequency spec- trum by summing over direction and the significant height by summing these sums over frequency, taking the square root of the sum and multiplying it by 4. (The spectra discussed above are all in terms of variance, and the total volume under S(w,@) equals the variance of the wave motion.) The first indication of a good forecast is in the verification of the significant height. It tells one that the area under S(w) as predicted agrees favorably with the estimate of this area as obtained from an ocean wave record obtained as a time history at a point. *See, for example, Pierson and Tick (1964). 430 1 Wave Spectrum imensiona Two-D oTqUeTIV YITION Oy} ut ¢ drys toyyeaM ye aUIT} Jo UOTJOUNT eB Se JYySstoy JUeT}IUSTS 9Y} TOF sysedorTOj UO s4[nsea AaeUTWIITEIg - [ ‘3Iq 22939 92 990 S2 990 b2 990 €2 9290 22 99d 1g 990 02 '99q 61 99d gi 99d 2) 99d 91 99d S| 99q $|'99q £1 99q 2] 990 || 9280 4400 2! _ 00 @i__00__@i OORSc OO Ric EO OMe OO ic | OO Meee |e OO Mec Ie OO re OO ery OO cI OO Mee OO | (olor (olor oo 2i_00 = [2 te. — T T ur toe Gee NF rr ra pea Pa a 2 aS a oe T eT ea a a TS ny fas a g Ol Ins U\ . SI M7 NAN \ va \\\ 02 t \\ a \: 462 40¢ ce —dIHS daAuasao SNOILV907 LNIOd GIN9 -—ve8 fel Jop ---2@L a vone {1S (14) LHOISH 3AVM INIOd G1y9 LNVOISINDIS 6S6| 04d 431 Pierson For December 1955 and December 1959 these results were verified by the data provided by the British weather ships that are equipped with the Tucker shipborne wave recorder. For November 1961 verification is against the wave records obtained at Argus Island by the U.S. Naval Oceanographic Office. Figure 1 shows the significant wave height predicted at four points sur- rounding the British weather ship in December 1959 and the significant wave height obtained from the records obtained by the British weather ship. Four major cyclonic storms passed the weather ship during this period. The waves reached significant heights of 40 feet and decreased after each storm to signifi- cant heights near 15 feet. The predictions are quite good and although not shown the frequency spectra check quite well most of the time. Our other results are equally encouraging. The November forecasts were verified in a completely different oceanic area by means of records obtained by a different wave recording system. The data for the November forecasts were not used in developing the procedure, and, although again not shown here, the re- sults are quite good. The 180 numbers that describe the directional spectrum show a wide varia- tion of odd forms such as one would expect from sea plus swell, crossed seas, and swell. Most spectra cover a range of directions in excess of 180 degrees. Directional spectra cannot be verified as no data were taken to estimate them. However, the directional spectra cannot be too far off because it would be virtu- ally impossible to obtain the good results that have been obtained for the signifi- cant height and the frequency spectra if the directional spectra were wrong. If a fully developed sea should occur at a particular point, the numbers pre- dicted would be obtained by substituting Eqs. (5) and (6) into (7). Presently we are developing ways to process 300,000 ship reports so as to produce wind fields for fifteen months of weather data. Forecasts of the direc- tional spectra for these fifteen months will then be prepared. These results will be verified against frequency spectra already tabulated by Moskowitz, Pierson and Mehr. At that time, some statements can be made concerning the overall accuracy of our procedures. REFERENCES Cartwright, D. E. (1963): The use of directional spectra in studying the output of a wave recorder on a moving ship. Ocean Wave Spectra, Prentice-Hall, ine Cartwright, D. E., and N. D. Smith (1964): Buoy techniques for obtaining direc- tional wave spectra. Trans. 1964 Buoy Technology Symposium, Marine Technology Soc., Washington, D.C. 432 Two-Dimensional Wave Spectrum Cote et al (1960): The directional spectrum of a wind generated sea as deter- mined from data obtained by the Stereo Wave Observation Project. Meteor. Papers, vol. 2, no. 6, New York University. Longuet-Higgins, M. S., D. E. Cartwright, and N. D. Smith (1963): Observations of the directional spectrum of sea waves using the motions of a floating buoy. Ocean Wave Spectra, Prentice Hall, Inc. Moskowitz, L. (1964): Estimates of the power spectra for fully developed seas for wind speeds of 20 to 40 knots. Journal of Geophysical Research, vol. 69, no. 24, pp. 5161-5180. Moskowitz, L., W. J. Pierson, and E. Mehr (1962,1963): Wave spectra esti- mated from wave records obtained by the OWS Weather Explorer and the OWS Weather Reporter. (I) and (II). Technical report prepared for U.S. Naval Oceanographic Office under contract N62306-1042, New York Univer- sity, Research Division, School of Engineering and Science. Ocean Wave Spectra (1963): Proceedings of a conference at Easton, Maryland. Prentice Hall, Inc. Phillips, O. M. (1957): On the generation of waves by turbulent wind. J. Fluid Mech., 2, 417 445. Pickett, R. L. (1962): A series of wave power spectra. (Informal Manuscript Report No. 0-65-62.) U.S. Naval Oceanographic Office. Pierson, W. J. (1964): The interpretation of wave spectra in terms of the wind profile instead of the wind measured at a constant height. Jour. Geophys. Res., vol. 69, no. 24, pp. 5191-5204. Pierson, W. J., and L. Moskowitz (1964): A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorodskii. Jour. Geophys. Res., vol. 69, no. 24, pp. 5181-5190. Pierson, W. J., and L. J. Tick (1964): Wave spectra hindcasts and forecasts and their potential uses in military oceanography. First U.S. Navy Sympo- sium on Military Oceanography, June 1964. * * * 221-249 O - 66 - 29 433 Pierson DISCUSSION A. Silverleaf National Physical Laboratory Teddington, England Professor Pierson's application of mathematical technique to sea state studies has long been of the greatest value to those of us in Britain concerned with the performance of ships in waves. I am sure that most of us will agree that the two-dimensional or unidirectional wave spectrum is a "fundamental building block" which will aid further developments. However, in Britain we do not all agree with Professor Pierson's suggestion that the formula in (5) is the best for naval architecture purposes at the present time. An independent anal- ysis by Scott (Ref. A) on behalf of the British Towing Tank Panel suggests that it is not the best or even the most appropriate fit to the Moskowitz data. For example, Professor Pierson's relation between the frequency of the spectrum peak f, and the average wave period Ty is toe = Osta (eo) while that recommended by the B.T.T.P. is {Ae MORS OI cena estes e Consequently, an alternative formula has been proposed for use by the British towing tanks which are now carrying out experiments on models in irregular waves much more frequently than in the past, so that it has become urgently necessary to formulate a standard of sea spectra for such experiments. In his introduction Professor Pierson mentions three possible types of pre- diction of seakeeping performance. I suggest that only the first of these repre- sents the purpose of seakeeping research from the point of view of the ship de- signer and operator, who is primarily interested in knowing whether or not ship A will perform better than ship B for a particular purpose on a specified route. Professor Pierson suggests that the data necessary to make this type of predic- tion can be obtained from historical records and some current work in Britain is being devoted to just this approach. Statistical information about wind and wave conditions in the principal areas where ships operate is being analysed and processed using data collected from voluntary observing ships and recorded on punched cards at the Meteorological Office. At present data from 125 Mars- den squares have been grouped into 52 areas defining most of the principal shipping routes to give a detailed account of the likely sea conditions during all seasons of the year. A first report on this scheme has recently been issued (Ref. B) and it is intended to publish a complete compendium on ocean wave statistics within the next year or so. 434 Two-Dimensional Wave Spectrum REFERENCES A. Scott, J. R., A Darbyshire Type Spectrum Suitable as a Standard for Model Tests and as a Basis for Long Term Ship Prediction. B. Hogben, N., Lumb, F. E. and Cartwright, D. E., The Presentation of Wave Data from Voluntary Observing Ships. Ship Report 49, July 1964. X* * * REPLY TO THE DISCUSSION W. J. Pierson, Jr. New York University New York, New York Mr. Silverleaf states that the formula I gave in my paper may not be the best and proposes an alternate on the basis of work by Scott. It would be inter- esting to see if the subsets obtained by Scott would pass the test applied by Mr. Moskowitz to his data. On the other hand, it may be the best. For example, the ITTC has adopted a form quite similar to the form we have obtained at N.Y.U. It must be emphasized that the formula represents only fully developed wind seas as a function of the wind velocity. The documentation for our results is substantial and it forms a convincing total picture. Recent work of Kraus (1965) provides added support. Partially developed seas, dead seas, and swell all have spectra that differ from the form I gave. Whether meaningful averages of such spectra can be ob- tained is questionable, and I have expressed certain doubts in this connection in correspondence with Mr. Hogben. I believe that all of the examples given in my paper come within the domain of the naval architect. It is his responsibility to see that the ships he builds are so thoroughly understood that their performance in any given situation can be correctly described. Other inputs are needed from meteorology and oceanog- raphy, but in principle the problems posed differ only in degree and not in kind. Although not stated explicitly in my paper, each ship captain who receives such a warning should be thoroughly acquainted with the expected behavior of his ves- sel for the predicted extreme condition. Our work on waves would never have reached its present stage without the foresight of the National Institute of Oceanography in Great Britain. The routine collection of wave data by means of British weather ships and the Tucker ship- borne wave recorder has been the cornerstone of our work. 435 Pierson REFERENCE Kraus, E. B. (1965): The Influence of the oceans on atmospheric disturbances and circulations. Woods Hole Oceanographic Contribution No. 1576. * * * 436 Friday, September 11, 1964 Morning Session SHIP MOTIONS Chairman: F. H. Todd David Taylor Model Basin Washington, D.C. Force Pulse Testing of Ship Models W. E. Smith and W. E. Cummins, David Taylor Model Basin Washington, D.C. Deterministic Evaluation of Motions of Marine Craft in Irregular Seas John P. Breslin, Daniel Savitsky, and Stavros Tsakonas, - Stevens Institute of Technology, Hoboken, New Jersey Testing Ship Models in Transient Waves Lt. Cdr. M. C. Davis, USN, and Ernest E. Zarnick, David Taylor Model Basin, Washington, D.C. Prediction of Occurrence and Severity of Ship Slamming at Sea Michel K. Ochi, David Taylor Model Basin, Washington, D.C. The Influence of Freeboard on Wetness G. J. Goodrich, National Physical Laboratory, Teddington, England 437 Page 439 461 507 545 597 *. : peters Va : ‘ canada ik Oo ; A. marca ae 4 sarit trt a t di teotayds aso «aa : a Oaarte (is fe 298 has 1a aia aS Bea pied a, FORCE PULSE TESTING OF SHIP MODELS W. E. Smith and W. E. Cummins David Taylor Model Basin Washington, D.C. INTRODUCTION In a recent paper [1] one of the authors proposed that a useful and revealing way of treating oscillatory motions of a ship was to relate them to the transient response to an impulse. The response to an arbitrary excitation would be ex- hibited as a convolution integral over the past history of the excitation. The idea was hardly original, as this device is widely used in the discussion of linear systems. However, there seemed to be some reluctance by those working in the field to treat the ship response in this fashion. Most writers preferred to re- strict their attention to the frequency response function. There have been some exceptions to this trend, notably Fuchs and MacCamy in the discussion of the motions of a floating block [2], Dalzell in the treatment of destroyer motions in severe sea states [3], and the paper by Davis and Zarnick for the present symposium [4]. However, all of these are concerned with re- sponses to wave pulses or hypothetical wave impulses, and not the response to a force or moment impulse. The present paper is concerned with this latter prob- lem. As a matter of fact, the solutions to the two problems, the response to the wave pulse or impulse and the response to a force or moment excitation, com- plement each other very effectively. The first solution characterizes the total wave-ship system, while the second enables us to construct the equations of mo- tion and thus separate the effects of damping, added mass, coupling, and hydro- dynamic memory. When both solutions are known, the wave excitation can be determined, and one is then in a position to say not only what the ship does but why it does it. The designer then has clues as to how to make changes in the design in order to improve seakeeping qualities. One can discuss and even use the impulse response function without directly measuring it, as it is simply the Fourier transform of the frequency response function. If the latter is known for all frequencies, the impulse response func- tion can be computed. But to directly determine the frequency response function, one must measure the response to a set of frequencies at suitably close inter- vals over the whole frequency range in which there is significant response. The alternative approach is most attractive. That is, apply a known impulse or equivalent excitation to the model and observe the response. The frequency re- sponse function can then be computed, and we have replaced a time consuming and expensive test program requiring many runs with a single run. This paper is concerned with such measurements. 439 Smith and Cummins In principle, the experiment is beautifully simple. In practice, there are a number of difficulties to overcome. First and most obviously, we are dealing with a system with six degrees of freedom, and there is strong coupling among some of the modes of oscillation. A much more serious and subtle problem arises from the fact that we obtain the response of the ship for all frequencies from a small set of relatively short records. Thus, the desired information is highly compressed in the time scale. The resolution of this information re- quires records of very high quality and an analysis procedure which degrades the data as little as possible. Prior to the presentation of Ref. 1, experiments were performed to test this procedure as a practical tool. Declining oscillations were used instead of impulsive excitation, but most of the troubles encountered would be even more characteristic of the latter type of test. The measurement system was some- what superior to those typical of seakeeping work at that time. In the process of analysis it became quite clear that major improvements were necessary in order for the technique to be other than a curiosity. There were several sources of difficulty, and as the method of overcoming these are key factors in the present paper, they will be mentioned here. First is the question of accuracy. It is clear that when desired data is superimposed, the accuracy to which it can be separated is certainly no higher than the net ac- curacy of the system. The original system had an accuracy of perhaps 5 per- cent and this was not good enough. The second major difficulty was noise, as it is evident that the real objective is a high signal to noise ratio. By noise we mean here all unwanted disturbances such as wall reflections and true electrical noise. The input for the declining oscillation experiment is a step function, which is completely suitable theoretically, but has undesirable qualities practi- cally. These arise from the fact that the step function has harmonic content at all frequencies, all the way to infinity, and such an excitation not only causes the model to oscillate, but in addition it vibrates as a beam at its natural fre- quency. Further, all instruments, attachments, etc., are excited in their various natural frequencies. In consequence, the signal to noise ratio was well below that which is necessary. As the potential value of the transient experiment is great, much effort has been devoted to upgrading our measuremen.. and analysis system since these early tests. The present paper is a progress report on the present state of this program. The details will be discussed in the subsequent sections, but the most significant accomplishments will be mentioned here. The first is a technique of towing the model, rather than self propelling it. This is contrary to the current trend toward powered models for seakeeping work. However, we feel that this technique offers real advantages. Specifically, we measure all restraints on the model imposed by the towing, guidance, and excitation system. The sum of these is the net input to the model. Thus, towing gear inertias and frictions are of no concern, as their effects are included in the measured input. The second achievement is the use of an excitation pulse of controlled har- monic content. The technique is an analog of that used by Davis and Zarnick for 440 Force Pulse Testing of Ship Models generating wave pulses. An oscillatory excitation is imposed on the model by means of a variable speed drive which sweeps from the highest frequency de- sired down to the lowest frequencies which can be treated in our basin. Because of the method of generation, the very high frequency content of the step or im- pulse response is avoided. Because of the shape of the pulse, the separation into the various frequencies is achieved with good accuracy. And because of the length of the pulse, the intense concentration of the information is eased. The third achievement is a system of significantly improved absolute accu- racy, about two percent. The present limiting factor is the use of magnetic tape in the data handling path. It is possible that the use of tape can be avoided, with a further significant improvement. The last major advance has been the use of a new system for converting data from analog to digital form. This system has the capability of converting as many as 6,000 data spots per second, distributed among the various channels of data. It has been possible to sample the data at a rate of 30 spots per cycle of the highest frequency investigated. The net result of all these improvements has been a very high signal to noise ratio. In the range of greatest interest, the noise is 45 db below the sig- nal level. As a result, we have been able to characterize the model from fre- quencies so low that shallow water and wall effects become significant (in a basin 240 ft x 360 ft x 20 ft deep!) up to higher frequencies than any previously investigated. And this entire range was covered in a pair of runs lasting per- haps 50 seconds. The system has been in use only a short time, and we have much more to learn about it. The earlier, unreported tests produced a vast amount of infor- mation about how not to run the experiment. This time we have been more suc- cessful, but we have discovered a number of additional refinements which will be necessary before we can do all that we wish with the system. These proposed changes will be discussed in a later section. THE EXPERIMENT This initial experiment was primarily designed to provide an evaluation of pulse techniques as a method of obtaining the frequency response relationship between exciting forces and the motions of a ship. A Series 60 Block 0.60 ship form was oscillated in pitch and heave. All forces and responses were meas- ured and the damping and added mass terms in pitch and heave were computed. This permits a direct comparison with the results obtained by Gerritsma [5] for a Similar form. Experimental Details As the effect of surge upon heave and pitch is generally considered to be small, it was decided to restrict the analysis to these latter two modes only. 441 Smith and Cummins However, the towing system allowed small oscillations in surge, and it is planned to undertake a three-mode analysis at a later date. The towing system is as shown in Figs. 1 and 2. Such an arrangement per- mits the application of tow forces at the model center of gravity while permit- ting responses in all six degrees of freedom. Restoring forces in the surge and sway modes are provided by springs K, and K,. External forces in the heave, surge and sway modes are measured, using variable reluctance force gauges. The motions in the six degrees of freedom are measured by film type potenti- ometers mounted as indicated on the tow strut and excitation forces by force gauges mounted in the model. A VA Neos Ko Z PITCH FORCE GAGES POTENTIOME TER Ipae, — Ih - Pitch and heave experiment 442 Force Pulse Testing of Ship Models HEAVE —POTENTIOMETER WZ K, Ko PITCH FORCE GAGES POTENTIOMETER Fig. 2 - Heave experiment Excitation is provided by an electric motor — variable speed drive arrange- ment, with the forces transmitted to the model via a spring and cable. (Tests were conducted at speeds of Fr. = 0, 0.025, 0.05, 0.075, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35). Two series of tests were conducted: In one the model was excited in the heave mode only, and in the second the model was excited simultaneously in pitch and heave. In the heave test the excitor cable was attached to the heave staff as shown in Fig. 2. The six motions as well as forces along the heave, surge and sway axis were measured. In the pitch heave experiment, as shown in Fig. 1, an excitation cable was attached to the bow of the model through a fourth (excitation) force gauge. Measurements were the same as in the heave experi- ment, except for the addition of the excitation force gauge. For each test condi- tion, the frequency of the excitation force was varied manually, adjusting the 443 Smith and Cummins speed of the drive system from 0 to 3.5 cycles per second. The amplitude of the excitation eccentric was fixed at one inch. The frequency spectrum of the excitation signal is shown in Fig. 3. INCH POUNDS SONNOd PITCH—— HEAVE -— — w VL/g Fig. 3 - Excitation force spectrum The measurement system was as shown in Fig. 4. DTMB block gauges were used to measure forces and moments. High resolution film type potenti- ometers were used to measure the motion. All data was recorded simultane- ously on Sanborn strip chart recorders and FM analog magnetic tape. The in- strumentation system, exclusive of recorders, has an accuracy of 0.2 percent, a dynamic range of 60 db, and a frequency response which is essentially flat from 0 to 100 cycles. Phase shift between any two channels was held to less than one part in 10,000. The tape recorder, however, the system's weakest link, is accurate to only 1-1/2 to 2 percent, and its dynamic range is limited to 38 to 42 db. Test Procedures As previously mentioned, tests were conducted over a range of Froude numbers, from 0 to 0.4. For each test condition, the model and carriage were accelerated to the appropriate speed with the oscillator turned off. Care was taken to ensure that the model had reached a steady state condition and that all energy in the model-free surface memory had dissipated. When steady state conditions were established, the recorders were turned on and allowed to run 3 to 5 seconds before the excitation. Excitation was started with an initial fre- quency setting of 3.5 cycles per second, and was swept from 3.5 cps to 0 in 444 Force Pulse Testing of Ship Models HEAVE TRANSDUCER PITCH TRANSDUCER SANBORN STRIP CHART RECORDER TRANSDUCER SWAY TRANSDUCER HEAVE FORCE GAGE TRANSDUCER ee ve en FORCE leh CARRIER i AWALOG EXCITATION TAPE FORCE GAGE BALANCE AVECIEIER RECORDER UNIT SURGE FORCE GAGE i SWAY FORCE GAGE Fig. 4 - Analog measurement system about 20 seconds, care being taken to see that the excitor was stopped at zero amplitude. Recording continued until any remaining motions had ceased. Typi- cal recording time ranged from 28 to 48 seconds, depending on carriage speed. Test results were as shown in Fig. 5. It should be recognized that the towing structure, and thus the towing carriage, was used as a reference for all meas- urements. When the model was excited by a force restricted in bandwidth, with negli- gible energy above 5 cycles per second, the measured signals were excellent, with a dynamic range and accuracy limited only by the 40 db range of the analog tape recorder. However, when a relatively broadband signal was used, such as an impulse or step function, the model, strut and carriage structure were ex- cited at their own natural frequencies and produced oscillations which almost completely masked the motion responses of the model. DATA ANALYSIS One of the inherent disadvantages of a digital record is that it provides in- formation about the corresponding time function at the sampled instants only. Discrete samples completely define a continuous function, f(t), only if the func- tion is absolutely band-limited, and then only if the sampling frequency is at 445 Smith and Cummins PITCH “i me \ [| HEAVE x o7 my f| [| ——_ x \<—— 5 sec ——> HEAVE FORCE EXCITATION FORCE JANES —— x Fig. 5 - Test record from pitch experiment least twice the highest frequency, F, in the signal [6]. Further, the recovery of the original signal from the digital data is predicated on the use of an ideal filter. When dealing with empirical data and actual filters, this sampling theorem is of little use, as there is rarely an absolute limiting frequency, F, and filters cannot be built which are capable of cutting off perfectly above any assigned F. It is certain that data collected from a vibrating towing-carriage does not meet this condition. As in a practical case there are many additional considerations, such as the effects of filtering on the desired signal, aliasing, interpolation methods used for signal recovery, and limited availability of computer time, the selec- tion of a sampling rate required to provide 1 percent accuracy is anything but clear-cut. A more rigorous method of using the sampling theorem has long been needed but the mathematics for anything other than the ideal case is quite complex. Likewise, the obvious solution of increasing the sampling rate by orders of magnitude, while within the capabilities of the analog-digital converter, quickly becomes impractical from the standpoint of the increasing computer time necessary for each analysis. In order to select the proper sampling rate, an experiment was run in which typical samples of analog data were first digitized at 6,000 samples per channel per second. A harmonic analysis was performed and the complex spectra so obtained were used as analysis accuracy standards. The same data was then sampled and analyzed at successfully lower sampling rates until a difference approaching 1 percent was observed in the spectra. The sampling rate finally selected was 125 samples per channel per second. This sampling rate, coupled with an average run length of 48 seconds, produces approximately 6,000 data points per channel per test condition, or approximately 60,000 data points per test run. In performing the actual Fourier transformation to obtain the complex fre- quency response function, an additional problem must be considered. When 446 Force Pulse Testing of Ship Models model tests are conducted at forward speeds, quantities such as pitch, heave, and surge force are in general not zero, even in a steady state translation. The actual records of a transient pulse test, as analyzed, are necessarily truncated, and thus have the form of an oscillatory pulse superimposed upon a rectangular pulse of length equal to the record. See Fig. 6. The recorded signal can be con- sidered as part of function, CE) =e Pa Ce) -m Ga y Re 4 iby Rae yo 33 33 33 33°33 53 53 35 53°35 (Ons az ans yiRe = ObeUR | (ene aR] ibe Sheen 33 33 53 S38} 75) 8} 53 53 55 53 —~-S5 (Cane GANS) Ran BS ODad 4 GAGA GEE, eR ec = @ 33 33 53 33 °°53 53 53 55 DISmOL) (9) (Con = Lary Ray = On, 22 2 (Oo aay OER = © 35 35 33 35 33} 55 55 35 Sol) (Cae = Ol bns) Ran Oban Men (Ga. OAL) ee OR. =. 0 35 35 33 35 Ss} 55 55 35 SS) — 3h) (Che = GRAD) Ree OANA SI Ge eA) Rie abi = Sl 35 35 53 BG SS} 55 55 55 55° 55 (Can = G2 Ban) Ron ODay Ran + (Cane Gan.) Ran = OLR. = 6 35 35 53 3H 583 55 55 55 B33 Let Cc Ss Cc S R33 ~R3;, Rs ~R;; Re R R. R 33 33 35 35 R = (10) r Cc R;, -Ro, OR ~R, 5 Cc Ry, R53 R.. R,; Force Pulse Testing of Ship Models aa 33 35 35 obs; ob;. Kea (11) BOD) Behe! Sere = EE Clay oe lo wb wb 1 0 0 0 R°K = (12) 0 1 0 0 1 0 0 0 K = RL (13) 0 1 0) 0) and the coefficients in the two coupled equations of motion may be computed. TEST RESULTS As previously mentioned testing was done at a range of Froude numbers from Fr. = 0 to 0.35. However, due to a time limitation, only the cases for Fr. = 0, 0.15, and 0.30 are presented in this progress report. The results at Fr. = 0.15 and 0.30 are of course directly comparable with those obtained by Gerritsma. The responses obtained at zero speed were particularly good. In the non- dimensional frequency range 1 < w\/L/g < 6 there was very little scatter and the results were similar to those obtained by Gerritsma. Wall effects were significant only below w/L/g = 1.0 and then only at clearly defined multiples of tank width. Considerable scatter in the data did occur above 6.0 which can be traced to the dynamic range limitations inherent in an analog tape recorder. The principal damping and added mass terms obtained from this experiment are shown in Figs. 8 and 9. 453 Smith and Cummins —O— Gerritsma Fr-015 —O— Gerritsmo F,-015 —t—Transient Fr-00 —O)— Transient Fr-c 9 Transient Fr=0.15 —7— Transient Fr-0.15 Fig. 8a - Added mass —/\— Gerritsma —L\— Gerritsma —O— Transient —O— Transient » = BY, eer | Fig. 8b - Added mass 454 Force Pulse Testing of Ship Models —O— Gerritsma Fr=0.15 —O— Gerritsma Fr-Oj5 —O}— Transient Fr-0.0 —O— Transient Fr-0.0 —V7— Transient Fr=0.15 —\7— Transient Fr=0.15 —A— Gerritsma —_/\— Gerrltsma —O— Transient —O— Transient Fig. 9b - Damping 455 Smith and Cummins Similarly the forward speed cases were good over the same frequency range. There was, however, some evidence that coupling of surge into heave and pitch was significant at forward speeds and indicates the desirability of a three mode (pitch, heave and surge) analysis. The damping and added mass terms are shown along with those reported by Gerritsma. CONC LUSIONS It is apparent from the preliminary results that good experimental results can be obtained even at zero speed in a 240 by 360 ft tank. While wall effects do occur, they appear only as one or two sharply defined discontinuities in the re- sponse data. The results also indicate that, while a two mode pitch and heave test can be conducted, a further increase in accuracy should be obtained by an extension to the three mode analysis. While there is much to be done, especially in such areas as increasing the dynamic range of the instrumentation system and an extension to a 3 or 4 mode analysis, it is felt that these preliminary re- sults demonstrate not only the validity of the pulse testing technique but further show that satisfactory results are within the capability of modern instrumenta- tion and measurement systems. MODEL DETAILS Series 60 Parent Form L Length between perpendiculars 10.0 ft B Beam 1.35 ft H Draft 0.53 ft A Displacement 239.3 Ib C, Block coefficient 0.60 A Area of waterline plane 9.39 ft? (e Waterline coefficient 0.71 if Mass moment of inertia for pitch (in air) 557.89 Ib in./sec 2 r Radius of gyration 0.25L m Mass of model 7.439 slugs 456 Force Pulse Testing of Ship Models REFERENCES 1. Cummins, W. E., ''The Impulse Response Function and Ship Motions,"' Pre- sented at Symposium on Ship Theory at Institut fur Schiffbau der Universitat Hamburg, 25-27 Jan. 1962, Schiffstechnik H. 47 B. 9 (June 1962). 2. Fuchs, R. A. and MacCamy, R. C., ''Linear Theory of Ship Motion in Irreg- ular Waves," Institute of Engineering Research, Wave Research Laboratory T.R. Series 61, Issue 2, University of California, Berkeley, California (July 1953). 3. Dalzell, J. F., 'Cross-Spectral Analysis of Ship-Model Motions: A De- stroyer Model in Irregular Long-Crested Head Seas,'' Davidson Laboratory, Stevens Institute of Technology, Report 810 (Apr. 1962). 4. Davis, M. C., Cdr. USN and Zarnick, E. E., ''Testing Ship Models in Tran- sient Waves," To be presented at the Fifth Symposium on Naval Hydrody- namics, 10-12 Sept. 1964. 5. Gerritsma, J., 'Shipmotions in Longitudinal Waves," International Shipbuild- ing Progress, Vol. 7 (1960). 6. Hamming, R. W., "Numerical Methods for Scientists and Engineers," McGraw-Hill, New York (1962). * * * DISCUSSION OF FOUR PAPERS Leo Joseph Tick New York University University Heights, Bronx, New York Since the papers by Smith and Cummins; Breslin, Savitsky and Tsakonas; and Davis and Zarnick, concern themselves with testing, I will first combine my comments to these papers.* These papers and the oral discussion which fol- lowed devoted some time to the pros and cons of various methods for determin- ing the defining properties of systems (mostly linear). Unfortunately, the lack of a careful description of the logic of test procedures has served to add consid- erable confusion. With a hope (?) of providing some clarification, I start with brief discussions of ''test functions."" The test situation consists, as I see it, of some system, device, etc., whose input-output characteristics one wishes to determine. A test procedure is to be used to make the determination as distinct from an analytical one. To make the discussion simpler, suppose we restrict ourselves to linear systems. In this case the system is usually characterized by the transfer *pp. 439, 461, 507. 457 Smith and Cummins function or the impulse response; these being Fourier transforms of each other. The testing procedure then consists of driving the system with some function and measuring both the input and output and performing relevant calculations. The minimum requirement of a good test function is that it should be rich in the frequencies of interest. It will usually be the case that the calculations will involve some sort of division. If the numerator has an error component which is fairly uniform over the entire frequency range, then the ratio will be of lower quality for those val- ues at which the denominator is smallest. Since the denominator will consist of some characteristic of the input function, it is best that the denominator be fairly constant. With these characteristics in mind, let us now look at three possible classes of input functions: (1) sine waves; (2) deterministic functions like the ramp, or pulse, or step, and (3) stationary random input. The sine wave is of course the worst as far as frequency content is concerned since it has only one frequency. As a compensatory feature, the transfer function at this fre- quency is estimated by a very simple operation on the output and it can be esti- mated quite accurately since all we have to extract from the continuous record is just the amplitude and the phase angle. The effect of noise in the measuring system should be quite small. We also have a measure of linearity of the sys- tem from visual inspection of the output. This is a very expensive way to proceed since it requires a very large number of sine wave tests to complete the analysis of the entire frequency band. The pulse test function is a very convenient one because all frequencies are present in an equal amount. The calculations to be performed on the output are quite simple. To estimate the transfer function, one merely takes the Fourier transform of the response.* The drawback to this test function is that it may be difficult to generate as the Davis, etc., paper indicates. Now no system is exactly linear, but this observation should bother no one since it is sufficient for dealing with problems of nature that the systems are enough so for the purposes of its use. Therefore, one should try to test under conditions which are representative of the conditions of use. One would not test in very high waves. Similarly the fast rise time of a good pulse may activate nonlinear modes of response. Finally, the stationary random test function is in some way a mixture of the previous two. There is some sort of repetition, albeit, an average one, and a “poulsiness."" We may make the function broadband in frequencies (in an average sense), and its spectrum as flat as our generating methods will allow. As pointed out by Davis, the system will have to be brought into a steady state be- fore the relevant arithmetic may be performed on the output, and in case of *If I seem casual about this operation, Iam just reflecting the speakers. Let me assure you though that this is a numerical operation fraught with error es- pecially at the higher frequencies as we will then be taking differences ofa large number of numbers of approximately equal magnitude. Since experimen- tal data usually has low significance (numerically speaking) this is a real problem. 458 Force Pulse Testing of Ship Models model testing this may use up a Significant proportion of the available test time. The amount of usable test time may give rise to estimates having large sam- pling variability. The arithmetic processes involved here, though lengthy, are well understood and present no difficulty. One of the most important contributions of statistics to experimentation is the formalization of the notion and the provision of methodologies for having an experiment provide its own measure of its errors. This is done by setting the problem in a probabilistic framework by either assuming it into existence of manufacturing it by so-called "randomization" operations. Those of you who have read books or attended lectures on experimental de- sign, that branch of statistics concerned with these problems, will recall these points. This attribute of an experiment does not come free. The price for put- ting it into this framework is to blunt the precision of the results. All of these attributes and philosophy of statistical experimental design have their analogs in random test functions. If one uses a deterministic test function, a transfer function can be calculated whether it exists or the calculated one has any relationship to the true one. Verification is required from some other source, e.g., previous experiments, etc., before one can have confidence in the calculations. If these verifications are available, the experiment can be economic and precise. On the other hand, if the test function is embedded in a family of test func- tions in such a way as to make the statistical manipulation allowable, the ex- periment itself will provide a measure of its own error; and that will be the coherency function. It seems to me that this is worth something and, as men- tioned above, it does cost. What all this comes down to is simply that the choice of a test function is not a simple or clear one. A point which seems to have been neglected in Dr. Ochi's paper and the discussion following is why such a simple procedure works at all. After all, slamming is a very complicated process and yet the simplest of models appear to be producing excellent answers. Being the original instigator of this ap- proach, I think I can throw some light on this problem. Way back (I guess it is more than 7 years ago by now) when I was wondering if some model could not be constructed for slamming predictions, I was looking at some destroyer data. I do not quite remember who was with me at the time although I think it was Martin Bates, then of the then Bell Aircraft, who com- mented that if you put the data through a low-pass filter you could hardly tell from the resulting record that a slam had occurred. This indicated to me that slamming did not change the gross aspects of the motion and that a simple model, based on the occurrence of conditions which induce slamming, might serve to make an average occurrence prediction. This observation appears to have been justified. * * * 459 Be bi eh? ‘ 0 ante DETERMINISTIC EVALUATION OF MOTIONS OF MARINE CRAFT IN IRREGULAR SEAS John P. Breslin, Daniel Savitsky, and Stavros Tsakonas Stevens Institute of Technology Hoboken, New Jersey ABSTRACT The concepts of linear system analysis are applied to the coupled mo- tion of marine craft to illustrate in greater detail than previously pro- vided the procedures for obtaining their instantaneous response in arbitrary irregular long-crested waves. The solutions of the coupled equations of motion for heave and pitch in long-crested regular seas are examined to show how the ship-sea transfer function can be identi- fied when the wave is regarded as the input rather than the actual forces and moments. The theoretical expressions for the response to arbitrary forcing functions are next examined and shown to involve the inverse Fourier transform of the ship-sea transfer function and this is identified as the system impulsive response function. This function is convoluted with the given surface wave record to provide the instanta- neous response. The characteristics of these impulsive response func- tions are discussed in some detail and means for their determination from theory and experiments are outlined. Application of the procedures are made to exhibit the high accuracy of deterministically calculated motions derived from models of a de- stroyer, underwater body, and hydrofoil craft. Results of calculation of the bending moment of a surface ship model are also exhibited. It is concluded that the method can be applied to all features of marine craft responses attending irregular wave motion which satisfy the require- ments of linear systems. INTRODUCTION Operation of marine vehicles in irregular seas is a problem of serious concern to the naval architect. It is important that reliable analytical methods be available to predict the motions, acceleration, degree of deck wetting, etc., of these craft before they are constructed and put to sea. In 1953, two important papers on ship motions in irregular seas were pub- lished. St. Denis and Pierson [1] considered the statistical aspects of ship 461 Breslin, Savitsky, and Tsakonas motion and presented methods for determining the probabilistic behavior of a ship in a random sea. A second method of analysis, which is complementary to that of St. Denis and Pierson, was introduced by Fuchs and MacCamy [2]. This later method is not statistical, but deterministic; it is based, therefore, not on the knowledge of the statistical properties of the sea, but on that of the actual time history record of the sea surface. The statistical approach makes use of spectral analysis techniques and the characteristics of the ship response to random wave excitation are defined in terms of an energy spectrum based on frequency of wave encounter. This is the so-called transfer function method whose application has been successfully dem- onstrated by many researchers over the past years for a variety of marine craft, i.e., St. Denis and Pierson [1] and Dalzell [3] in the case of motion of dis- placement ships; Dalzell [4] for the case of ship bending moment; Savitsky [5] for submerged bodies in irregular waves; and Bernicker [6,7] for the case of fully wetted and super-ventilated surface piercing hydrofoil systems. The deterministic approach employs the concept of the impulsive response function, as given in linear analysis, to define the time history of ship motion in terms of the actual time history of the surface wave profile of the irregular sea. As the name implies, the impulsive response function describes the time history of the response of a given system when acted upon by an input consisting of a unit impulse at zero time. Superposition of these unit impulses to represent the actual wave excitation yields the total response of the system. Fuchs and Mac- Camy [2,8] first applied this technique for simple bodies in a random head sea. In recent years, the Davidson Laboratory, Stevens Institute of Technology, has investigated the application of this deterministic technique to predicting the random motions of a variety of marine vehicles, including displacement ships, hydrofoil craft, and submerged bodies in irregular waves. It is the purpose of this paper to present a review of the deterministic technique, to discuss its limitations, and to compare the results of the analytical studies conducted at Davidson Laboratory with experimental data. Some of these results have al- ready appeared in the published Davidson Laboratory reports, but will be sum- marized herein in an attempt to form a unified presentation. This work was sponsored by several bureaus of the U.S. Navy, including the Bureau of Ships, Bureau of Naval Weapons, and Office of Naval Research. The preparation of this paper was sponsored by the Davidson Laboratory, Stevens Institute of Technology. THEORETICAL FOUNDATIONS The linear theory of the motions of bodies in waves has been the subject of many papers and presentations in the past. As a result, there are several clear analyses of bodies in both regular and irregular waves, the latter case having been dealt with by spectral procedures. However, the deterministic or instan- taneous response of bodies in a given, nonuniform, temporally varying wave has not been given an entirely clear analysis beginning with the equations of motion. The procedures used thus far have treated the motion as the output of a linear system due to a wave input. This involves the identification of (for systems with 462 Evaluation of Motions of Marine Craft in Irregular Seas several degrees of freedom) what might be termed a "lumped" transfer function (or response to waves of discrete frequencies) and from this to calculate for- mally a correspondingly lumped impulsive response function which, when con- voluted with the given wave record, yields the instantaneous motion. Although this procedure has been shown to work exceedingly well (see section on applica- tions) questions arise as to the character of these impulsive response functions primarily because the analysis has not been sufficiently lucid. At the risk of appearing pedantic, the elementary theory is reexamined in the following pages in the hope of providing a firmer foundation for the more-or-less mechanical procedures used in arriving at the instantaneous response of a hull within, or upon, the surface of a long-crested sea which is arbitrarily specified.* Heaving and Pitching In or Under Regular Waves Korvin-Kroukovsky [9] (1955) was quick to realize that the combined heav- ing and pitching of responses of a ship are the solutions to a coupled pair of or- dinary second-order, linear differential equations with coefficients which vary with imposed frequency. Following his notation, the differential equations of motion are, for the case of simple harmonic forcing functions: az + bz + cz + d:6@ + 2d 2 Fiacnie (1) AQ + BO + CO+D-2+ Ez+ Gz = Mei? (2) where z is the heave displacement from equilibrium, @ is the pitch displacement from equilibrium, a,b,c are the virtual mass, damping and spring coefficients for pure heaving, d,e,f are corresponding cross coupling coefficients due to pitch, A,B,C are the virtual mass moment of inertia, damping and spring coefficients for pure pitching, D,E,G are corresponding cross coupling coefficients due to heave, F, and M, are the complex force and moment excitations for a regular wave of amplitude |7,| with the understanding that only the real parts of the right-hand sides of (1) and (2) are to be ulti- mately retained. (The dot notation is used to represent a total derivative with respect to time.) *The severity of the given wave trace (as a function of time) must be such as to permit application of linear system analysis. 463 Breslin, Savitsky, and Tsakonas The solutions of this pair of equations are written (again in Korvin- Kroukovsky's notation) in the compact complex variable form: i il a8) Rutt 3 aL ( eS Ore 8) bg MP - FOR iwt 4 Cen ( PS - QR Je @) where P(w) = -2w? + ibwte Q(w) = -d-w* + iew+ g (5) R(#) = -D: a? + iEwo+G S(w) = -Aw? + iBwic. Inspection of (3) and (4) shows that the response in heave and pitch are both linear combinations of the forces and moments and response or transfer func- tions of the body. Consider only the response in heave (pitch follows in com- plete analogy) which may be written AE) Stag dh a. (6) where E = ip ( S) ) iwt _ iot (7) f = === ]}) © = F, ®,() e and Zz = -M (see m fo} PS - QR —M, ®,(0) e*?*, (8) The complex functions ®, and ®, are called frequency response or transfer functions in heave per unit applied force and moment and are evaluated in terms of an amplitude and phase angle in the form: On = Lye) ee eS [Cay eS?) (9) where _ | S(@) |. _ |e) 10 A,(@) = ee ee) males (10) and, for brevity T(w) = PS- QR. As is well known, these unit response functions depend only on the body co- efficients themselves and not on the forcing functions. In what follows, it will be necessary to consider the forcing functions characteristics in some detail. 464 Evaluation of Motions of Marine Craft in Irregular Seas The forcing functions F, and M, for the case of bodies in regular waves are in general complicated functions of incident wave frequency. Conceptually they are secured by considering the in- and out-of-phase pressure distributions developed on the body when restrained from moving in the wave system. Thus, in general, they take the form at 3 3 Foe- © = a'(w) sat ib'(®) = t C7 (11) ; 37 3 Mie stu AUGa)) Ss iB‘(w) =e? C7 (12) where 7 is the wave or vertical fluid displacement at the body (which may be submerged) referenced to some arbitrary point on the body (often to the center of mass or amidships as a convention). For any regular progressive wave the vertical motion at any point ¢ is iwlolé wf g g iwt Ingle n(&, ©, t) Ing lec; Qre (13) where In, | is the amplitude of the wave at the surface and 2 iwlolg otf (14) g( Ory) =) ee oan The complex exciting force and moment (11) and (12) become, after use of (13), (Ce) = (= Gea (@) te tes (O) tS) eae ©) (15) Mee) = (G o2 AN(@)e tia B(@)) se \iimeie(@.c G). Thus it is seen that the force and moment acting on a body are both proportional to the wave amplitude on the surface and are arbitrarily phased to the body through the coordinates €,¢. For sake of brevity let the force and moment per unit of wave amplitude be written F eS Oa, 2, O) (16) In| and M, Ty UO eee (17) where f' and m’ are the complex polynomials 465 221-249 O - 66 - 31. Breslin, Savitsky, and Tsakonas £'(@)) = =2a"\(@) + dab '¢a) + ¢' (18) m'(w) = -@*A'(w) + iwB'(#) + C’. Equations (15) and (16) now allow one to express the response of the coupled system in terms of wave amplitude by inserting them in (7) and (8) and summing to yield a = [®() £'(«) - ©) m'(w)] g(a,€,0) e*** 2 (LOO On" ® iat 1 = ( T( @) g( QW, &, 4) e : ( 9) Thus one may recognize a lumped or effective frequency response function for heave (with freedom in pitch) for the ship-sea system per unit of wave amplitude as S(@) £'(e) - Q(e) m'(e) Ta) He, 216) 2 (20) ® 6 9( &., C) = ( This can be reduced to an amplitude function which depends on w and ¢ anda phase angle which depends upon w and é (or x); thus Dae) = Ia ye (21) and this is what is determined from either theory or from recorded responses of a model in regular waves. It is important to note that an arbitrariness is in- troduced into the phase by the reference system used or, what is the same thing, by the arbitrary definition of phase. Instantaneous Motion in Arbitrary Time-Varying Waves The equations for heave and pitch motion are the same as (1) and (2) for regular onset waves, but now the right-hand sides are functions of time explic- itly and are not functions of discrete frequencies. Thus Reesce and M,ei®t are replaced in Eqs. (1) and (2) by F(t) and M(t). The common procedure in solv- ing the equations in this case is to employ Fourier integral transforms which may be defined as follows: If z is the Fourier time transform of z(t), then Z(w) = | z(t) ee ie (22) -@ and the inverse transform is - 466 Evaluation of Motions of Marine Craft in Irregular Seas Z(t) = x | 2(w) et? * dw. (23) One then multiplies Eqs. (1) and (2) through by e-:*t and integrates over all time from -~ to + under the assumption of vanishing z, z, 6 and ¢ at +» and the satisfaction of integrability conditions by F(t) and M(t). The solution for heave is, aS an example, given by a Aad " S(a) + Oar ee a0)! -iwt Z= Z25+z, = x | FN F(w) e dw oe TN M(w) e dw (24) where one next replaces the transforms F and M by F °F ; A | = J a ante dr (25) M CMG) and, upon interchange of the orders of integration, obtains the familiar result in eel - i S(@) i(t-7T)o ¥ ele i i Q(@) i(t-T)o ON ro MO I te * (26) which leads to the definition of the kernel functions i zs S(@) iuw K -(u) = Oe T(2) e dw (27) 1 Q(@) 1u®@ K(4) = On The) e dw . (28) These are defined as the impulsive response function for the body in the fluid. It is to be noted that they are dependent only on the body coefficients obtained from either the impulsive response in calm water or from the response of the body to regular waves. The final expression for the heave is then ZS Lie ae Zn (29) with Ze = J IG) Us an) elie (30) Zee i M(7) K.Ct— 7) di (31) -@ 467 Breslin, Savitsky, and Tsakonas which simply states that the total heave response is the algebraic sum of two convolutions of the force and moment time histories with the appropriate impul- sive response functions. However, one does not have at his disposal the force and moment time his- tories, but rather only the wave input time history. It is, therefore, necessary to eliminate the explicit dependence of the result on F and M and to determine how one operates on the known (or given) wave record to determine the motions. For simplicity, the following development is applied only to part of the response z, to illustrate the procedure. It is noted from (16) that the normalized force at any discrete frequency is known and hence one can express the force as a function of time and the instan- taneous surface wave 7,(7) by convoluting the wave with the force transfer function, or F(t) = ef ongre | fa) e(w dt) el OPT datdr?. (82) Upon insertion of this into (30) and through the use of (27), one finds that this part of the response takes the form: es 1 ‘ : 1 . 1 iw’ (T-T ) 1 { S(@) eal (tira) Zia = (2)? { i if NaC tS) e ne © f im T(o) e dwdr. es Interchanging the order of integration and noting that the 7-integral is simply jo) I Shae <= 278 (w' = w) -@ where 5 is the Dirac delta function, yields es wo 1(@) =-@ eg = a i Ae ToS) { wee i f'(w') g(w')e '7 ° 8(w'-w) dw'dwdr' i oo wee a and, because of the delta function property the '-integral produces f'(w) g() eit’? to give 0 cy fee ) @(@,€& ©) S@) factor ; oe al No T al ae at ) dawdr (33) a completely similar result would be secured for the motion constituent z, so the total motion can be written 468 Evaluation of Motions of Marine Craft in Irregular Seas E62) se i No( T'S) le Ao ae g(,¢, C) et@(toT, ). dady’ ‘ ; (34) T= © The integrand is immediately recognized as the effective or lumped frequency response function for the ship-sea system defined by Eq. (20). Hence the w-integral is the effective impulsive heave response function for the ship-sea system: 3 1 ¥ /£'( 2») S = ‘(@ iuw z + | @ (@) ede (35) so that for both heave and pitch Z (eae) e Ke Gta es Sel) t= J TIE (Teens) i dr’ (36) CCE) areas Kee Ce Tabi Sone) where K,, is the impulsive response of the ship-sea system in pitch with free- dom in heave present. It is of interest to note that the same result for the response can be obtained by normalizing the forcing functions with respect to the vertical fluid displace- ment at depth. To illustrate this, consider the forcing function F(t) in this light and one can write ia@’lo'’lé Inde) = =) Ti Cet) J f'(w')e g ene (ter De dita (37) where it is to be noted that e-°''/® has been suppressed and the motion at depth “(7,€,¢6) is used. Then the component response in heave is a - S) iw(t-7T ZC tO = x| P(r) | ae Ce tdbed: rea fen’ 1 . a i J aOR aS) I ff. @@a)ee = Be RCE ome) ite uh ={00) > = 00 - © 469 Breslin, Savitsky, and Tsakonas and again i eee ae ie Q78(@' = w) and re) iw'lo'lé iwlwlé On i face) Q g oho ae (QM oe) cle” a Page" e g ee ee ‘ Then il (G0) . a 1 2(tiEQ = CEO) EL SS eee Now if one wishes to refer this to the wave at the surface ma 8€, Oy = 2 | He e9) | e G Ona Ta") Bln tslae™ « (39) The same integration procedure applies and the previous result is obtained, namely, 2 F C w G , dolols . ry (lain, S(a)_£'(e) xan HA. Gy G) = sib 7A 15, EO) yee en g g = dr. (40) Thus it is seen that the response calculated in terms of the subsurface motions as given by (38) is the same as that given by (40) when the subsurface motion is referred to that on the surface by Eq. (39). Evaluation of the Impulsive Response Function for Ships It is clear from Eq. (35) that the impulsive response function for coupled motion depends upon a knowledge of the response of the system to normalized forces and moments at discrete frequencies, i.e., one must know the frequency response operators, or what is called the transfer function. One may seek to evaluate ®,,(~) and ©,,(~) from theory alone or from experimental records of model responses in either regular or irregular waves. At present one may calculate the transfer functions from theory by using Grim's [10] methods for estimation of the body coefficients, eight of which are frequency dependent. Gerritsma's [11] recently completed work on determina- tion of the body coefficients has given strong support to the procedures used by Korvin-Kroukovsky and Jacobs [12] for ship motion calculations. It is to be noted that, in dealing with a "lumped" heave-pitch and pitch-heave response op- erators, it is also necessary to specify the normalized excitations as functions 470 Evaluation of Motions of Marine Craft in Irregular Seas of incident frequency. Integration required to obtain the impulsive response functions would undoubtedly have to be done by computer since the transfer functions with frequency-dependent coefficients are very complicated. An attractive alternative to theory is the use of experimentally determined responses of a model of the vessel to either regular or irregular waves. It is now a routine procedure to obtain from towing tank tests the amplitude response operators and their respective phases at selected values of frequency. In such tests the motions are related to wave measurements made by wave wire or other devices placed ahead of the model or abeam in time with amidships. It can, therefore, be appreciated that these transfer functions (obtained by graph- ing amplitude and phase response against incident frequency) are indeed depend- ent upon the location of the wave wire. It is often found that the impulsive re- sponse function derived from such data exhibits values other than zero for negative values of time t in distinction to completely mechanical or electrical systems for which it is known that K(t) = 0 for t <0. It is intuitively clear from physical concepts that the ship (or model) will respond to a wave before the crest (say) reaches the bow because of the spatial distribution of both the Ship and the pressure field of the wave. It will be shown in the following section how the extent of the part of K(t) for negative t can be reduced by judicious positioning of the wave measurement with respect to the model. In any event, it will be necessary to have some "future" information of the wave in order to compute the present time motion for all cases in which the vessel is of length comparable to the exciting waves. For those interested in applying this technique, it is appropriate to indicate in some detail how the impulsive response function for any mode of motion may be obtained from data obtained from a model in (a) regular waves and (b) irreg- ular waves. K(t) from Regular Wave Tests If one regards the regular sea motion (in a towing tank experiment) as the input a, = a, | sin ae and one records the output of any mode of the model motion as x(t) = [n,| A(#) sin [et - o(@)] where o(w) is the phase angle referenced to the wave and A(w) is the amplitude response of the model in a particular mode, then the transfer function or com- plex response function per unit amplitude of input is identified as Wey = hua e ey (41) Upon completion of a plot of A(w) and o(«) for enough discrete values of w so that smooth curves may be drawn to define both A(~) and 9(#), one may then find the impulsive response function by applying the operation 471 Breslin, Savitsky, and Tsakonas K(t) = x | O( w) gue dw = x | A(w) os [wt- p(w) ] da = s i A(@) cos [wt - 9(#)] dw + + | A(®) sin [ot - 9) | dw. Since K(t) must be a real function, the last integral must vanish identically for all values of t. This requires that 9(w) be an odd function of (and if it is not then something is in error in its determination!). Since A(w) must be an even function of w, then calculate K(t) from K(t) = 2] A(@) cos [ot - (| dao. (42) 0 It must be realized that by referencing the force and moment functions to the wave, an arbitrary phase angle is introduced or simply that the phase of the frequency response function is tied to the location of the wave measurement rel- ative to the body. To clarify this, suppose that the heave and pitch of a surface model is recorded at the center of mass and the wave is measured abreast of the center of mass. Then the derived impulsive response function will exhibit features peculiar to this reference point. It will, for example, have values dif- ferent from zero for negative time which then requires that the wave motion forward of amidships be known or, in effect, "future waves" are required in order to compute the response at the present time. If it is desired to reduce the extent of the negative time for which the empirically derived impulsive response function has nonzero values by, say, referring the motion to waves measured forward of the bow, it is necessary to shift the phase of the transfer function by the angle i («|«|x/g) where x is the amount of the horizontal shift (taking care to regard x as positive or negative) so that the modified transfer function be- comes Allene ee (43) O(w,x) = A(o) e [ Le | and the modified impulsive response function is ° Ble K(t,x) = al A(@) cos [. = @(@) = z i (44) 0 In addition, it will be necessary to convolute this shifted function K(x, t) with the wave record at the new point so that the 7 and the K are consistently referenced to the same point of measurement. If only the wave record abeam of the center of mass is available, this can also be shifted to the same point as is discussed in a later section. 472 Evaluation of Motions of Marine Craft in Irregular Seas K(t) from Irregular Wave Tests One method of obtaining the impulsive response function from motion rec- ords is to first calculate the frequency response function from the relation 7 ®(#) = 2Z(a) or @ i AC) 7 2Oe che ‘yep eceaae as n(t)e i?t dt and then to proceed to K(t) as indicated above. Another procedure is to use spectral analysis techniques which have been applied to the statistical analysis of ship motions. The spectrum of the model motion is given by the calculation T/ 2 Ixc@,)|° = =| [tint X(t) X(t-7) e Pare (46) =e [LE 0/0) and the wave height spectrum is similarly provided by 2 1 2 1 Nee -10_T E= 7(®,) = a J lim T J TCE) Cie = 7) Clie || © padace (47) -o {|T?®” =i) D The amplitude of the frequency response function is related to the motion and wave spectra by be) 48 Oca) | = See) and the phase o is calculated from Im W(@,) = -1 e ae eee] ae where vy is the cross-spectrum defined by ' Peciitinele eet -io 7 (50) (®.) = oe lim T MOE) Ce = 7) Ghee lr = ee = 40/7 2 473 Breslin, Savitsky, and Tsakonas The Effect of Shift of Wave Measurement on Reference Points Suppose that a wave system is moving from left to right in the direction of positive x or € and one has by measurement a knowledge of the waves as a function of time at x = €, and z = 0 and asks what the fluid motion is at some point x = € and z = -¢ "downwind" so € > €,. The answer can be secured by regarding the fluid as one following the linear system concept that at any dis- crete frequency: output = input x unit frequency response function and thence to deduce an impulsive response function which is then convoluted with the known wave record. However, a much more direct approach is to utilize the foregoing formulas for heave response by regarding the body to be shrunk to a point and thus indis- tinguishable from a fluid element. The heave frequency response function of the ship-sea system contracts to POMOC S Sse (51) Z zg g Dee and the vertical fluid motion at ¢,7 in terms of the surface motion 7t;&,,0) is 2(8EO = { n(7i€5,0) K,(t=73 €-€,, 6) dr (52) where the wave-induced impulsive response function is given by 8 LOVE eg) cn 3 Kanne uci ea = @ ei da, (53) = ©) This integral can be expressed (as shown in Appendix A) by Ky = a \Vs exp = Erfc (: au y=) (54) where R, indicates that only the real part is to be retained, ZAG CG aIGe),, Erfc is the complementary error function (which is tabulated for complex arguments). (The plus sign is applied for positive time u and the minus for negative u.) 474 Evaluation of Motions of Marine Craft in Irregular Seas Some reflection on the parametric dependence of K, on ¢ and ¢ imbedded in (54) will reveal that the action of the fluid between the two points is to filter the wave-induced motion as an inverse function of the complex ''distance" Z which means that the filtering effect depends on (£?+ ¢7)!/?2 and the "aspect" of the point defined by the angle tan~! (é/Z). Evaluation for ¢ = 0 yields the same result as given by Davis and Zarnick [14], viz., Kee, 6.0) = a feos (5 cat)?) i ccat)| + sin (Feo?) ee sot)]| (55) w ° 2 2 2 2 where at at a = Vaz ; Cat) = i) cos (5 12) dp ; SC siz) = J sin yw? dp. The functions C and S are Fresnel integrals which are tabulated. Curves of K,/a for ¢ = Oand ¢ = 50 feet are shown in Fig. 1. It is seen that the amount of future time wave record needed at €,,0 to compute the pres- ent time disturbance at €,¢ increases as one moves downward into the fluid. As € is made large with respect to €,, less and less future time record is re- quired as would be expected. For ¢ = 0 and large a or €-+€, Kyous E60) +2 a cos (F- Zc) which, being even in t, shows that both future and past information are equally weighted at €. The function K, as given by (54) collapses to the known, very simple, re- sult when one moves the point €,¢ under the point €,,0. Then the argument of the complementary error function becomes a pure imaginary and its value is then unity leaving _Bu (nn) 2 eee Oe Ee (56) Pa Al 5 2h A universal curve of K,/£ is plotted against Gu (or ft) in Fig. 2. It is seen that this function is symmetric, indicating that the motion at depth requires equal knowledge of future and past waves. These results allow one to handle the following problem. Suppose one has a system -K(t) for a particular mode of motion which has been derived from data in which the wave information was secured at £,0 and then one wishes to calcu- late the motions of a ship using wave records secured or assumed at some point €,. One may then do either of two operations: 475 Breslin, Savitsky, and Tsakonas 15 kKwitrerd) Fig. 1 - The impulsive response function for wave-induced fluid motion as a function of horizontal separation ¢ and for vertical distances ¢ = 0, 50 ft Fig. 2 - Heave impulsive response function for destroyer in head seas 476 Evaluation of Motions of Marine Craft in Irregular Seas (a) "Shift" the wave input from ¢, to é, so that it may then be convoluted with the K(t) determined for waves measured at ¢,; or (b) “Shift the K(t) from é, to é,. Step (a) is accomplished by convoluting the given wave record with kK, given by (55) and then convolve that result with the system k(t) to obtain the response. Step (b) is accomplished by modifying the transfer function from which K(t) is computed by the factor iwlo| eile e.g) (taking care to apply the correct sign to the exponent!) and thence to compute a new K(t) which can be convoluted with the given wave record. APPLICATIONS OF IMPULSIVE RESPONSE TECHNIQUE TO PREDICT SHIP MOTIONS IN IRREGULAR SEAS The previous sections of this paper have discussed the significance of the impulsive response function and have described its application in determining the time history of ship motions in irregular seas. During the past several years, the Davidson Laboratory has employed this technique to evaluate the mo- tions of a variety of marine craft operating in random seas. The results of these applications will be summarized and discussed. Displacement Ship in Head Seas In 1961, Fancev [13] used the impulsive response technique to determine the time history of heave and pitch motion of a destroyer model in irregular long-crested head seas. The model used in the experiments was the DD692 Class Destroyer (long hull). The full-scale ship is 392 ft long, has a beam of 40.83 ft and has a displacement of 3471 long tons in salt water. The model was tested in moderately high, irregular, long-crested head waves that had a broad energy spectrum. The average height of the waves was about 1/60 of the model length. Measurements were made of the wave elevation (at a constant distance forward of the model LCG), pitch angle, heave at the LCG and bending moment amidship. Dalzell [3] reported the results of these tests and, by the method of cross-spectral analysis, derived the transfer function of the destroyer for a wide range of speeds. It will be recalled from Eq. (21) that the transfer function @(«w) is written: O(w) = A(a) e 194%) = Pra) + iQ(w) where P(a) = A(@) cos [$(#)] (57) Q(#) = -A(#) sin [¢()] 477 Breslin, Savitsky, and Tsakonas and where A(w) is the amplitude function relating the wave amplitude to ampli- tude of ship motion for regular waves of a given frequency w. ¢(w) is the phase angle between the crest of a regular surface wave and the peak of the corre- sponding sinusoidal motion of the ship. Dalzell found that the transfer functions obtained in the cross-spectral analysis agreed very well with those obtained from tests of the DD692 in regular waves over a range of speed-length ratios from 0 to 1.25. The experimental values of the transfer function (A(w) and ¢(«)) for pitch and heave are summarized in Figs. 1 and 2 of Dalzell's paper. Fancev used the transfer function obtained by Dalzell to develop the impul- sive response function K(t) relating the motion of the destroyer at the LCG to the instantaneous wave profile recorded by the wave wire located ahead of the test model. As developed in a preceding section of this paper (Eq. (42), ex- panded): K(t) = al [P(a) cos at — Q(w) sin oot | dw. 0 Fancev performed this integration by a graphic numerical method and his re- sults are plotted in Figs. 3 and 4 of this paper showing the heave and pitch im- pulsive response functions of the destroyer at a Froude number of 0.187. Itis Fig. 3 - Pitch impulsive response function for destroyer in head seas 478 Evaluation of Motions of Marine Craft in Irregular Seas —80 Fig. 4 - Pitch and heave motions for destroyer in head seas predicted by impulsive response technique seen that the response functions are physically realizable, i.e., K(t) =~ 0 for t <0. If the surface wave probe were located at the LCG of the model, Fancev shows that the resultant response function would have values for t < 0 and hence be classified as physically nonrealizable. The physical explanation for this is that when the ship is long, relative to the wave length, the ship responds to the wave crest even before it reaches the bow and before this wave is re- corded by a wave probe located at the LCG; hence, in this case, the ship motion would always precede the arrival of the wave crest at the LCG. The heave and pitch time histories were computed on an IBM 1620 by eval- uating the convolution integrals [Eq. (36)], 0 MCE) = | Kee i nGtin@, Ga - © @ EXE) = l KS Gan Ce =m) nar —- © 479 Breslin, Savitsky, and Tsakonas where 7(t) is the time history of surface wave profile measured by the wave probe forward of the model. Figure 5 shows the results of the prediction of heave and pitch response to irregular seas. The continuous lines are tracings of the oscillograph records of heave motion and pitch motion obtained from the tests in Ref. 3. The circled points represent the results of convolving the im- pulsive responses of Figs. 3 and 4 with the surface wave time history. On the whole, agreement between observed and predicted responses is considered ex- cellent, hence validating the accuracy of the impulsive response technique in obtaining deterministic solutions. Submerged Bodies in Beam Seas The Davidson Laboratory has conducted an extensive series of model tests to determine the motions of a submerged, asymmetrically finned body at zero velocity when acted upon by regular and irregular long-crested waves approach- ing the body from various directions. In these tests, the motions of the sub- merged body were recorded in terms of the surface wave profile directly above the body. Response operators for heave, roll, and pitch motion in beam seas and head seas have been developed from these data by Savitsky and Lueders [5]. The response operators obtained from irregular wave tests were found to be in agreement with those obtained from tests in regular waves. The general con- clusion of this study was that, in regular beam seas, the heave and sway motions are those of a water particle at the center of gravity of the body. Also, in beam seas, the hydrodynamic roll moment is proportional to the wave slope at depth, (or equivalently to the inertia forces which vary as w2) and the roll motions are determined by using this wave slope, the natural roll frequency and damping of the body, and the usual dynamical equations of motion of a linear, single degree of freedom oscillator. Dalzell used the results of Ref. 5 to determine the impulsive response func- tion of the submerged body and to calculate the time history of heave and roll motions in irregular beam seas. Dalzell's results are rederived below following the theoretical procedures described in the previous section of this paper. It will be recalled that, in the present theoretical development, the kernel function of the wave system (due to shift of wave reference points) was separately de- veloped and then combined with the kernel function of the mechanical system to derive a so-called "system" impulsive response function. Since the test body was submerged, the wave characteristics at depth of submergence (¢), will be used as the input to the system. (It has been shown in the theoretical section that it is equivalent to referencing the output to the wave on the surface.) The relation between the measured regular surface wave pro- file and the orbital motions at depth exhibits a zero phase shift and an attenua- tion in amplitude of orbital motion given by the relation e-°*°/g. The kernel function relating surface wave profile to wave profile at depth is given in Eq. (56) which is reproduced below: : sets (58) K,(ti0,0) = > = a 480 0 =3 1e 2 yydap Jo uoT}OUN] & Se UOTJOW PINT} peonput-eaem Toyz uoTJOUNZ esuodsea oatstnduly - g ‘s1q 040934 3HL NO 3MIL | 23S / vz £2 zz 12 02 6! 8! di 91 S3HINI'2 23S ‘s SNOILOW G3L9I038¥d- $379ID HLIM S3NI7 G3HSVO SNOILOW G3G4093u - S3NI7 GIN0S 481 a) Evaluation of Motions of Marine Craft in Irregular Seas $338030 fi NOILOW HOLId (©) we £2 e2 12 02 6! 8! a) 23S G¥Y0534 3HL NO JAMIL 221-249 O - 66 - 32 Breslin, Savitsky, and Tsakonas This function (plotted in Fig. 2) is seen to be a symmetrical function of time and hence requires some future time knowledge of surface wave profile in order to predict the wave profile at depth. Using the above kernel in a convolution inte- gral will provide a time history of the orbital motions at depth 7(t) in terms of the time history of the irregular surface wave profile, 7,(t) immediately above the test point. me). = ) 167) F Vex ee cae (59) Predicted Heave Time Histories Considering the heave motion of the neutrally buoyant submerged body, it was shown in Ref. 5 that the body behaves in heave and sway like a water parti- cle at depth, i.e., it has identical amplitude A(w) = 1 and zero phase 9¢(w) =0, relative to a submerged wave particle. The heave transfer function of the sub- merged body (relative to water motion at depth) is then GEE) IND) ee NS ENG) ea = Gh, (60) The impulsive response function of this mechanical system is then written K(t) xy = x | @(@) euee d@ = =a a i@E cla = BCP) (61) where 36 is the Dirac delta function. Operating with the delta function ona bounded and continuous function f(t), it can be shown that {oo} ) iE) SG = EL cle = HES) (62) -@ Thus, convolving the body heave impulsive response function [Eq. (61)] with the wave motion at depth, Eq. (59), gives the time history of heave motion z(t) of the submerged body in terms of the surface wave profile (7,) to be (oa) _e(r-7')? we) = { J Te) =e ae dr’ 8(t = 7) dr so that Bet De ACE) = J nt’) 3 ore oe ee (63) The above integral was evaluated by Dalzell and resulted in a computed heave time history of submerged body. The analytical results are compared 482 Evaluation of Motions of Marine Craft in Irregular Seas WAVE ELEVATION TIME (CONTINUED), SECONDS KEY: OBSERVED TIME HISTORY OF WAVE AND RESPONSES POINTS PREDICTED FROM SURFACE WAVE TIME HISTORY Oo Fig. 6 - Deterministic prediction of roll and heave response for submerged body in irregular beam seas with experimental values in Fig. 6. It is seen that there is excellent agreement between computed and measured values of heave. Also included in Fig. 6 is the time history of the irregular surface wave profile. Predicted Roll Time Histories As shown in Ref. 5, the roll motions of a submerged body in regular beam seas are derivable from the equations of motion of a damped, linear, single de- gree of freedom oscillating system and are expressed by the following equation: 483 Breslin, Savitsky, and Tsakonas Og lor gr CRD wee (64) where © = maximum roll amplitude, w, = natural roll frequency of submerged body, w = wave frequency, 7, = Surface wave amplitude, h = depth to center of vertical fin on submerged body, and ¢' = damping ratio in roll. Since the wave orbital motion at depth h has an amplitude equal to o*h then the response amplitude, A(«),, defined as the ratio of maximum roll am- plitude to wave amplitude at depth is equal to: w2 A(@),, = ea 5 2 5 és (65) PG) The phase angle, 9, between passage of the wave crest over the submerged body and maximum roll amplitude of the submerged body can be derived from Ref. 5 to be: (66) The transfer function for the submerged body in roll is thus known (Fig. 7). O(a), = A(o), e' %” = Pla) + 10(a) where A(w) and 9(w) are given by Eqs. (65) and (66) above. Evaluating the real and imaginary parts of the transfer function results in the following: 484 Evaluation of Motions of Marine Craft in Irregular Seas PHASE DEFINITION: WAVE ELEVATION: 7) = cos wt No (e) ROLL ANGLE :@ = 6, cos (wt- 8) 8, PHASE, DEG. ROLL AMPLITUDE/WAVE AMPLITUDE, DEG./FT. w, ENCOUNTER FREQUENCY, RADIANS/SECOND Fig. 7 - Roll transfer function for submerged body in beam seas P(w) = A(w), cos [9()] : 2 = = a cos eins alo ung¢no Settt-iande yloviisias Yu shee Pie Siwonns oinkl~g , ul e ft a, ' ; BUOY GunURT Oe eameiac pe anit ¢ Fe LAS Tia. 1 j fm ‘ F A , A ‘ I Y rpe! + W ect Freres f i vS } i / , r it oe ? d 4 1 : 4 vy ;, Dicets { ids Roe scat Dice lies ¥ : ; ' i (SS ; , Ww . Ce TESTING SHIP MODELS IN TRANSIENT WAVES Lt. Cdr. M. C. Davis, USN and Ernest E. Zarnick David Taylor Model Basin Washington, D.C. ABSTRACT The seaworthiness characteristics of a ship design are often deter- mined by a series of model tests in regular waves. This report de- scribes a new model test procedure which makes use of a transient wave disturbance having energy distributed over all wave lengths of in- terest. With the use of this transient wave technique, the testing time required to characterize a model is reduced by an order of magnitude. In this report, the basic behavior of a unidirectional transient wave is discussed, and a simple Fourier transformation is developed in order to link wave height records measured at any two separated points along the path of such a wave. A particular form of transient wave, which is approximately sinusoidal with linearly varying frequency, has been used to test successfully a number of shipmodels. The results of these tests are presented and the practical problems in generating, measur- ing, and analyzing transient wave tests are discussed. INTRODUCTION The linear theory of ship motion in a random seaway has become generally accepted as a useful approximation to the actual nonlinear phenomena involved in ship-wave interaction. As outlined in the pioneering work of St. Denis and Pierson [1] the random sea surface can be visualized as a superposition of two- dimensional sinusoidal waves continuously distributed in amplitude, wave length, and direction. The total ship response in any degree of freedom is found from a . summation of the responses to each individual wave component. The primary role of a ship model testing facility, in providing information for quantitative full-scale motion prediction using this theory, is to measure the response of a model to sinusoidal waves of unit amplitude over the entire range of ship speed, wave length, and direction of encounter. The Harold E. Saunders Maneuvering and Seakeeping Facility, located at the David Taylor Model Basin, is admirably suited to conduct such an investigation in head and oblique waves. The scope of a complete model measurement program is without parallel in other engineering fields which perform frequency response testing of dynamic systems. A numerical example will illustrate the large number of tests which 507 Davis and Zarnick are required to characterize a model in regular waves. Assuming that the func- tions involved can be suitably approximated by tests at 10 wave lengths, 30- degree increments in direction from ahead to astern, and 5 speeds, 350 separate model tests are required, with measurement and analysis of a number of dy- namic variables on each test. A program such as this requires a major invest- ment of time and money, and any techniques which can be developed to abbrevi- ate the test time without technical compromise will reap high dividends. It is clear that relative wave direction and model speed must remain fixed for any one test. However, under these constant conditions, a series of experi- ments in varying wave lengths is nothing more than a frequency response de- termination of a linear dynamic system, a common experiment in systems in- vestigations for many engineering disciplines. The frequency response characteristics of linear systems may be measured in three fundamental ways, that is, using sinusoidal, random, or transient exci- tations. The first two techniques are commonly employed in testing ship mod- els. The latter technique, using a transient water wave, is the subject of this report. A transient wave will contain energy which, in general, is distributed over a range of wave lengths. Thus, a single motion test can yield information about the response of the ship at all wave lengths of interest. In the representative example just quoted, the number of tests required to characterize a ship design can be reduced from 350 to 35. In the following sections, the theoretical and practical aspects of transient wave testing are presented. First, the basic mathematics of linear systems analysis are outlined. Then, the analytic peculiarities of the ship-wave system are discussed, stressing the fact that a wave is not properly an "input" as the term is usually understood. Next, the complex subject of unidirectional water- wave transients is treated in a simplified fashion by developing an expression which relates various wave height time histories that might be recorded at var- ’ jous points along the path of the wave. A particular wave which arises quite naturally from this analysis is the one which would, in theory, produce an impulse of infinite height and zero duration at some measurement point. An approximation to this theoretical waveform is quite easy to generate in a seakeeping facility; it has been extensively used at DTMB for model testing because of its several attractive analytical properties. Three sets of model tests are reported herein to support the theory and to illustrate some of the practical problems, especially associated with the meas- urement of wave height, that can be anticipated in using transient waves to ex- cite ship motion. PRELIMINARY THEORY Mathematics of Transient Testing Suppose that a linear dynamic system (such as shown in Fig. 1) is under investigation, with an "input'’ x(t) as the independent excitation, and an "output" 508 Testing Ship Models in Transient Waves y(t) as the dependent or forced variable. If x (t) y (t) x(t) iS a sinusoidal signal at a particular hbur frequency, then, in general, y(t) will as- ymptotically approach a steady-state sinus- oidal response at the frequency. The ratio Fig. | - General linear of the output amplitude to the input amplitude SYSSER Seppo seas rie and the phase difference between output and input for all frequencies define the frequency response of this system, represented oy the complex transfer function G jw), where «@ is the frequency in radians per second. OUTPUT When the system G(jw) is at rest and a sudden transient x(t) is applied at t = 0, then some response y(t) will be measured, usually involving decaying transients. It is well known that a transient signal can be decomposed into a continuous distribution of infinitesimal sinusoidal components with the aid of the Fourier transform. For example, the Fourier transform X(j) of a particular input signal x(t) is given by the complex quantity oo MC ie®= = J dt x(t) e 3°* (1) - which represents the amplitude and phase of the incremental components at fre- quency ». Considering the output to be a summation of the response to each of the input frequencies, the well-known relation ¥(jo) = X(je) G(je) (2) gives a proper amplification and phase change to each of these components. To summarize, the frequency response of a system G( jw) can be found from a single transient experiment with input and output transforms X(jw) and Y(jw), respectively, with the relation Y(j@ G(jw) = = f (3) In the transient testing of ship models, the input x(t) is arbitrarily defined as the instantaneous amplitude of the undisturbed two-dimensional wave surface which would pass through the center of gravity of the model; see Fig. 2. The output y(t) is the time history of any one of the pertinent response variables, such as roll, pitch, or heave. The use of a water wave input is the key distinction between transient tests of ship models and those conducted, for example, in control systems analysis where often a voltage is available for easy introduction of input transients. Visualization of the Ship-Wave System The fact that the wave height referenced to the center of gravity of the model is defined as the input can lead to great mathematical difficulties. For 509 Davis and Zarnick LCG Undisturbed Free Surface Wave Height Reference Fig. 2 - Wave height defined as an input signal example, in defining a linear system in the time domain it is conventional to use a unit impulse as a standard input, which causes an "impulse response"' whose Fourier transform is identical to the frequency response function. This is seen from Eq. (2), where ¥(jo2) = G(jo) (4) because the transform X(j) of a unit impulse is unity. Since it is impossible for a physical system to look into the future, to "laugh before it's tickled,'' the impulse response must be zero prior to t = 0 (when the impulse arrives). However, in the ship motion problem there is no reason to believe that the inverse transform of an experimental frequency response will exist only in positive time. In fact, as will be shown later in this report, an "impulse" of wave height observed at t = 0 at the center of gravity of the model would be caused by the contraction of a wave train, which had previously passed along the forward part of the ship, producing force on the hull and resultant mo- tion in negative time. Thus, a more accurate description of the phenomena in- volved would be the very general configuration pictured in Fig. 3, where uni- directional wave height and ship motion are both viewed as responses to some undefined initial excitation, the mechanism which produces the waves. However, since wave height and ship motion are completely related in the sense that they respond to the single cause, it is proper to consider that ship motion can be re- lated to wave height by the frequency relation WAVE WAVE HEIGHT SYSTEM x ( t ) H (jw) ULTIMATE CAUSE SHIP MOTION y(t) Fig. 3 - Representation of wave height and ship mo- tion as "'effects'' rather than ''cause'' and "effect"! 510 Testing Ship Models in Transient Waves H,( i) ¥( jo) = x jo) | = «cian (5) 1 where G(jw) is not properly the response of a physical system but the ratio of the frequency response of two physical systems. With this philosophical restriction in mind, we will continue to call wave height an input and ship motion an output, but at no time will the intervening system be required to have the characteristics of a real physical system. Wave Transients The study of transient waves on a free surface is an advanced top in hydro- dynamic theory, but it is amenable to the ''systems" approach if linear wave be- havior is assumed. Consider first that all waves are traveling in the same di- rection on a surface of infinite extent and in a fluid of infinite depth. Suppose further that a wave disturbance of finite energy per unit crest length has been traveling for all time and is observed at a single stationary point x,. The wave height 7(x,,t) may be expressed in terms of its Fourier transform by the re- lation: {oo} Mx,,t) = x] dw N(x,,0) ef?*. (6) - @ Following the technique of Stoker [2], the complex quantity N(x,,) is vis- ualized as the infinitesimal wave component with frequency +a. This wave at any instant of time extends over the entire plane and at any one point persists for all time. At another point x,, which is a distance x along the direction of wave travel from x,, this same infinitesimal wave is observed but with a phase lag of #?x/g radians, according to linearized wave theory. That is, the time history at x, is given by Mat) = ae J de Noy 0) oo 18!2!*78 oles (7) where the absolute value of w is used in the phase operator to ensure that the quantity w) e Jelolx/g (8) is conjugate with N(x,,-.). This property is necessary in order that a Fourier transform represent a real function of time. The operator ec /°!'®'*/ can thus be viewed as the "transfer function" of water, or the frequency response function which relates wave heights measured at two points separated by a distance x in the direction of travel. 511 Davis and Zarnick To illustrate this result, suppose that Wyo 8) = CES OE « (9) The Fourier transform of this wave height is given by N(x,,®) = m[H.(@-@,) + uo(wt o,)| (10) where ,(-) represents the unit impulse. The transform of 7(x,,t) is given by N(xp,) = m[n,(@-o,) + p(ore,)] ee, (11) We have for 7(x,,t), then, the relation il ‘ -jololx/ jot MEae i) = =| dw 71 [Ho @ ~ &) + Ho(@+ @,) | enya a toPeey aD : 1 =JO 3/% Oot a e e cos (wt = ae x/¢) (12) which is a well-known result from linear wave theory. The remarks of the previous section apply to wave height pairs in the sense that the latter are both "effects" rather than "cause" and "effect."" However, with knowledge of wave height at one point, the corresponding time history at another point can be determined by convolving the first wave height with the in- verse transform of e~j°'®'*/& as is well known from linear systems analysis. This inverse transform is computed in the Appendix and yields the "weighting function" or "impulse response" of water w(T) = a eas > at F [2° Tee | + a stim aE [2° Tel 7 (13) T ir 10/2) ata las (14) and C(aT) = i dm cos 5 m? : S(aT) = J dm sin lk m2. (15) 0 0 This weighting function is shown in Fig. 4, where 512 Testing Ship Models in Transient Waves w(T) % Vg/7x cos (g- = a) (16) as ar becomes large. This function can be heuristically interpreted as a linear frequency sweep which looks back into the past of the wave height signal being processed and detects those frequency components that have gone by at a past time which would influence present wave height at a distance x in the direction of the wave. This is motivated by the convolution integral ™(X5,t) = | dr wer) ene t=) (17) -@ which is the time domain equivalent of N(x,,@) = N(x,,@) auiolaeger (18) a=\%ory Fig. 4 - Weighting function of unidirectional waves in water Suppose we ask the physically ridiculous but mathematically interesting question: "What signal would be observed at x, if a unit "impulse" of wave height were recorded at x,?"'' A unit impulse is described as a signal which is zero except at an instant in time where it has infinite amplitude, such that the integral over this point has a value of unity. The Fourier transform of a unit impulse is 1.0. Solving Eq. (18), we find that N(x,,@) = soe reH (19) This can be readily shown to have an inverse transform which is equivalent to that shown in Fig. 4 except for a reversal of the time variable. 221-249 O - 66 - 34 513 Davis and Zarnick Thus, if we observe a transient wave in the water which initially has a very high frequency (associated with slow velocity) and if this frequency linearly de- creases toward zero with constant amplitude and tapers off as in Fig. 4 with time reversed, then at some point in space and time a very large wave would be created for a brief instant, assuming that linear wave theory holds. This phe- nomenon can be viewed as the simultaneous meeting of a large number of wave components whose individual speeds and starting times were properly adjusted so that the faster traveling waves were behind but catching up with the slower ones. For the purposes of wave generation in a model-testing facility, it is mani- festly impossible to provide a sinusoidal wave at infinite frequency. However, it is certainly possible to generate a wave train which has a frequency varying linearly from the highest desired value toward zero. Such waves have been the backbone of the Model Basin transient studies and will be described more fully along with experimental results in the following sections. Briefly, the linear theory appears to hold quite well, and in the early exploratory studies very large peaked waves — approximating the impulse — were formed, although they were limited by cresting and other nonlinear mechanisms. An important property associated with a transient water wave is that the magnitude of the Fourier transform remain constant regardless of where it is observed and when the origin in time is fixed; i.e., the water transfer function is solely a phase operator. For a pair of moving probes separated by a fixed distance x, the same frequency response relation is applicable. However, transforms of wave height measured at nonzero speed are computed using the frequency-of-encounter time scale, where each wave length component corre- sponding to a stationary frequency w is measured at the frequency. @®. = @ + oa (20) where v is the speed of the wave probes against the direction of wave travel. If a Fourier transform of a transient is computed for one wave height measure- ment, the companion wave measurement in the direction of wave travel will have the same magnitude at each frequency but the phase will be shifted by e-}2!'°!*/8 where «w is the stationary frequency of the wave component concerned. For waves traveling in the same direction as the wave probes, ambiguities exist, and special techniques, which are beyond the scope of this report, must be em- ployed. The preceding treatment of transient water waves does not follow a conven- tional path in that spatial effects are suppressed and initial conditions or excit- ing forces on the water are not considered. If an instrument measures unidirec- tional wave height at some point for all time, then the time history at any other point is readily estimated through transform techniques. Even though a wave- maker may be generating the transient wave in the testing basin, the height- measuring probe and the ship model considered the wave to be one that has been traveling forever on an infinite surface. 514 Testing Ship Models in Transient Waves TEST TECHNIQUES Outline of Model Testing Technique Using Transient Waves Transient testing with a model in ahead waves is accomplished quite easily. As currently conducted, the first waves to be generated are slowly progressing high-frequency waves. When these first waves have traversed a portion of the test basin, the model is brought up to speed in calm water and measurement of all dynamic variables is commenced. The wave train passes, induces motion, and then the water and model return to the quiescent condition where recording is stopped. Each time-record is used to compute a Fourier transform from a common time base. The wave height transform must be corrected to the location of the model center of gravity by the transfer function e-i*!*!x/£, where x is the dis- tance to the ahead wave probe and » is the wave frequency. The ratio of motion transform to corrected wave height transform defines the transfer function, amplitude and phase for that motion. Programming for Transient Wave Generation The wavemaking system in the Harold E. Saunders Maneuvering and Sea- keeping Facility, described recently by Brownell [3], is quite well suited for the generation of transient waves. Eight electrohydraulic servo systems can be used to control the flow of air to and from domes along the shorter side of the basin, thus imparting energy to the water which travels away in the form of waves. These servo systems can be driven in unison by an electrical signal from either a low-frequency sine wave generator or a tape recorder. The actuator servo system has proved to be a considerable improvement over the previous electromechanical arrangement for wave generation, which provided a constant-amplitude variable-frequency, sinusoidal excitation to the water. The actuator system can allow independent control of both amplitude and frequency, or it can introduce transient or random disturbances of more general form. Random wave generation has been described recently in Ref. 4. The transient waveform which has been used to date is characterized by a linearly decreasing frequency, starting at the highest frequency of interest for model testing — nominally 1.0 cps. The electrical signal that produces these waves is recorded on magnetic tape by the crude but effective procedure of linearly decreasing the frequency of a low-frequency sinusoidal source. The frequency response characteristics of the basin, relating wave height to actuator motion, are such that frequency components near 0.4 to 0.5 cps are greatly amplified. Two modes of transient waves have been used — one has the program amplitudes weighted so as to counteract this frequency behavior; the other has a constant amplitude with varying frequency. 315 Davis and Zarnick Fourier Analysis of Recorded Transients The Fourier transform of a transient record is defined by the mathematical relation F(jo) = i dt f(t) cos ot - j { dt f(t) sin ot (21) and is readily accomplished by digital computation or by special-purpose de- vices designed for this application. For this exploratory investigation, it was decided to use a particular analog computer configuration which has interesting properties. A single channel is shown in Fig. 5 where conventional analog computer symbols are used. As de- scribed in Ref. 5, this undamped resonant circuit is driven by a transient input and oscillates as t > at an amplitude corresponding to the magnitude of the transform of the input transient at the particular frequency w and with a phase corresponding to the phase of the transform. A number of similar computer configurations all adjusted to the same frequency, were driven simultaneously by the tape-recorded transients resulting from a particular experiment in order to maintain a common time base for phase measurements. TRANSIENT STEADY STATE OSCILLATION AMPLITUDE =| F(jw)| PHASE =2F (jw) Fig. 5 - Transient analyzer configuration on analog computer The use of this scheme for transient analysis was motivated by considera- tions of accessibility and operator control of the computations. However, for mass handling of transient records on an assembly-line basis, other techniques will be employed. TEST RESULTS Tests of three different models are described in this report. Emphasis is placed on a correlation of transient wave results with regular wave results rather than on a complete description of the characteristics of any particular hull form. A chronological description is used to indicate the problems encoun- tered and the progress obtained to date. 516 Testing Ship Models in Transient Waves Test Series 1 The first transient tests were conducted in December 1962 using Model 4941, a 13-ft model of a C4-S-A3 Mariner hull which had been previously tested in regular waves. Pitch and heave in ahead waves were measured as well as wave height with a sonic wave probe mounted approximately 12 ft abeam of the center of gravity of the model. The philosophy of wave generation during this series of tests was to create a wave which would contract and "peak," as described in an earlier section, at a point somewhere near the model. Figure 6 shows the records taken during a WAVE HEIGHT BF Su pha uli: T = <= Amplitude and Phase of Computer Output Proportional to Fourier Transform of Input at f = 1/T rem eee i hele iat dk eal ede oe Analog Computer Output Fig. 6 - Measured transients for Model 4949 and typical analog computer fre- quency response measurement 517 Davis and Zarnick typical test where the wave transient reached a peak height at the location of the stationary model. Along with these signals, the outputs of the analog computer circuitry (described in the previous section) are displayed for a particular fre- quency of analysis. These sinusoidal outputs have magnitudes which are essentially equal to the magnitudes of the Fourier transforms of the respective signals up to that point. Although the variation of these amplitudes at the end of the transient has been a vexing problem with this method of transient analysis, sources of test error have been uncovered such as waves reflected from the sloping beach. Another practical difficulty encountered is also seen in the transient wave height record where the energy in the water does not decay rapidly after pas- sage of the main signal. Fortunately, much of this disturbance is above the fre- quency range of interest. A zero-speed transient test was analyzed at many different frequencies using the relative lull at the end of the passage of the main wave as the defined end of the transient. Figures 7 and 8 display the resulting heave and pitch fre- quency responses; they show good agreement with those of the regular wave 4 REGULAR WAVES © TRANSIENT TEST Heave Response fo) 2 4 6 8 1.0 Frequency ir cycles per second Fig. 7 - Heave response for Model 4941 at zero speed in head waves; transient test compared with regu- lar wave test results 518 Testing Ship Models in Transient Waves o LEGEND s 4 REGULAR WAVES r © TRANSIENT TEST a ” © © ° © a=) £ © 7) c °o a 4 c £ & a fe) 2 4 6 8 1.0 Frequency in cycles per second Fig. 8 - Pitch response for Model 4941 at zero speed in head waves; transient test compared with regular wave test results tests. The only severe deviation is a sharp peak near 0.35 cps in both responses; it was caused by an unexplained sharp null in the measured wave height trans- form. In Fig. 9 the heave/pitch ratio is plotted for the zero-speed case. Although heave/pitch ratio plays no part in prediction of motion, it is a ratio which has as much significance as any other in the isolated transient experiment, since all three dynamic quantities are ''effects,"' as previously discussed. This ratio has the virtue of being independent of errors in wave height measurement, and has provided a useful index of the accuracy of the motion measurements throughout the series of transient tests. A forward speed transient test was conducted at a speed corresponding to a Froude number of 0.09. Unfortunately, heave calibration was lost and only pitch results are valid, as given in Fig. 10. These results show a moderate agree- ment with the regular wave pitch response. In analyzing the results of these initial tests, it was felt that the transient technique had demonstrated considerable promise but that there were many pos- sibilities for improvement in testing techniques. 519 Heave-— Pitch Ratio ininches per degree Davis and Zarnick LEGEND 4 REGULAR WAVES © TRANSIENT TEST Frequency in cycles per second Fig. 9 - Heave/pitch ratio for Model 4941 at zero speed in head waves; transient test compared with regular wave test results inch in degrees per Pitch Response LEGEND 4 REGULAR WAVES © TRANSIENT TEST fo) 2 4 6 8 1.0 Frequency in cycles per second Fig. 10 - Pitch response for Model 4941 in heave waves at a Froude number of 0.09; transient test compared with regu- lar wave test results 520 Testing Ship Models in Transient Waves First of all, it was recognized that there is no particular virtue in having a model at the point of contraction of the wave transient, since the amplitude of an ideal wave transform is invariant with position. In fact, there is a strong pos- sibility that nonlinear water or model behavior would be accentuated near the point of highest wave amplitude and that surge modulation effects would be sig- nificant. In addition, passing through a longer program before it coalesced would mean that more controlled energy could be imparted to the wave excita- tion, which, other things being equal, would lessen the effects of extraneous noise. And finally, the practical virtue of not having to conduct a precisely timed meeting of model and wave is still another motivation for altering and ex- tending the duration of the wave transient. A second major source of error was believed to be in the wave height meas- urement. Besides the previously mentioned wave reflections from the beach, there was a considerable possibility that waves generated by the model were being picked up by the side wave probe. Although there are some advantages in measuring a wave signal at its geometric reference point, they seem to be out- weighed by the readily demonstrated fact that waves are generated much more efficiently by the model in the abeam direction rather than in the ahead direction. Test Series 2 A second series of transient tests was conducted during January 1963; in these tests an attempt was made to profit from the lessons learned in the initial tests. The model selected was a Series 60, Block 0.60 hull form, Model 4606. Heave and pitch were measured in ahead waves. Attention was focused on the zero-speed case in order to minimize the number of factors affecting the ex- periment. During these tests, wave height was measured with two sonic probes. One was placed in the same location as in the previous tests, approximately 12 ft abeam of the model center of gravity, and the other was located 19 ft 2 in. ahead of the center of gravity. The wave program employed was considerably longer than that used in the previous test series. The excitation signals from both Series 1 and 2 are dis- played for comparison in Fig. 11, which shows that the duration of the control voltage for the second test was doubled, that is, raised from 40 to 80 seconds. A typical transient test recording of this series is shown in Fig. 12. The wave height and motions are seen to be of a form considerably different from that observed during the first tests; they resemble the varying frequency and amplitude characteristics that would be predicted by theory. One immediately obvious result is that the side wave measurement contains a peculiar null in its envelope which is not present in the ahead wave measurement, an anomaly which is most likely due to model-created waves generated to the side. In Figs. 13 and 14, respectively, frequency response operators obtained from four transient tests in heave and pitch are compared with regular wave measurements made during the same series of tests. Agreement seems to be 521 Davis and Zarnick ANNI 10 Sec. +} <= Excitation for First Series of Tests a ACU MEAWSATAUATATATATAVAUAVAVAVAVAVAVA AVA WAU Om 10 Sec. > —<=— Excitation for Second Series of Tests Fig. 11 - Transient voltage excitation to wavemaking system during first and second series of model tests MODEL 4606 SERIES 60 RUN 6 PROGRAM 12 PITCH ZERO SPEED HEAVE FWD WAVE HEIGHT SIDE WAVE HEIGHT rrr —wvj)i)\)\ \ \\ WAN Venema Fig. 12 - Typical transient recording taken during second series of model tests 522 Testing Ship Models in Transient Waves 1.4 LEGEND © REGULAR WAVES 1.2 + TRANSIENT RUN 4 4& TRANSIENT RUN 6 O TRANSIENT RUN 3 © TRANSIENT RUN 5 1.0 ae ay 8 O Fig. 13 - Heave response for Model 4606 at zero speed in head waves; transient test com- pared with regular wave test results Heave Response % 2 4 6 8 1.0 Frequency in cycles per second quite good over most of the frequency range. The solid curves, which are also shown on these plots, were computed on the basis of slender-body hydrodynamic theory by Newman [6] for a roughly similar hull, Series 60, Block 0.70. The Newman computation neglects everything but buoyancy. For many models tested at the David Taylor Model Basin, the resonant frequencies of heave and pitch are sufficiently high so that in a zero speed test, the wave length components that produce significant pitch and heave motions act at frequencies which are considerably below resonance. The resulting motions are essentially a wave force measurement or the response of the ship without ship dynamics. The comparison between computed and measured zero speed response is very impressive. A detailed frequency analysis was conducted for one test shown in Figs. 13 and 14, and wave height transform amplitudes were computed for the same wave program measured without the model in the water. This was done in order to remove the hypothesized effect of model generated waves. A comparison of analyzed wave heights measured under these various conditions is presented in Fig. 15 which shows a large percentage variation in the high frequency range. The heave/pitch ratio for these tests is shown in Fig. 16. The ratio demon- strates a considerable consistency except for some of the regular wave values. These deviations, together with Fig. 13, lead to a strong suspicion that the heave measurements during the regular wave tests at 0.4 and 0.55 cps were low. The results of this second series of transient tests added further experi- mental support for the utility of transient waves in ship model testing. In most cases, frequency response functions could be estimated with about 10 percent 523 Davis and Zarnick LEGEND © REGULAR WAVES © TRANSIENT RUN 4 4 TRANSIENT RUN 6 z= O TRANSIENT RUN 3 rr) S TRANSIENT RUN 5 i= Ps a ” a o o a vo £ @ 7) i| le Fig. 17 - Excitation voltage program for third series of transient tests Regular and transient tests were conducted at zero speed and at a model speed corresponding to a Froude number of 0.14. The transfer function plots — heave, pitch, and heave/pitch — are presented in Figs. 18, 19, and 20, respec- tively, for zero speed. The corresponding phase data are presented in Fig. 21. The agreement between the regular wave tests and the transient tests is impres- sive. Of special interest is the heave/pitch ratio, which, of course, is independ- ent of wave-height measurement error. The agreement between regular and transient wave tests shown in these figures presents the strongest indication of the potential accuracy of the transient technique, and incidentally, of the linear- ity of motion response of a ship in waves. Note also that the pitch transfer function in Fig. 19 demonstrates, convincingly through a close-spaced series of regular wave tests, the lack of smoothness of pitch response when examined in detail. 526 Hea ve Response inch in degrees per Pitch Response Testing Ship Models in Transient Waves | ok ean LEGEND A REGULAR WAVES ~ © TRANSIENT TEST — NEWMAN'S THEORY BE ee orl al | J = ro = = LC - } a OS 1A 5 Finale | 5 eo b ——= a et bas ene ees (e) Fe. 4 6 .8 1.0 1.2 Frequency in cycles per second Fig. 18 - Heave response for Model 4889 at zero speed in head waves; transient test compared with regular wave test results LEGEND A REGULAR WAVES ~ © TRANSIENT TEST — NEWMAN'S THEORY fo) 2 4 6 8 10 Frequency in cycles per second Fig. 19 - Pitch response for Model 4889 at zero speed in head waves; transient test compared with regular wave test results 527 Davis and Zarnick Fig. 20 - Heave/pitch ratio for Model 4889 at zero speed in head waves; transient test com- pared with regular wave test results 4A REGULAR WAVES © TRANSIENT TEST — NEWMAN'S THEORY 4. ee l 2 4 6 8 10 Heave-Pitch Ratio ininches per degree Frequency in cycles per second The ahead speed results are presented in Figs. 22, 23, 24, and 25 where again good correlation is noted. The appearance of transient wave records taken during this series is some- what different from that of earlier test records. Figure 26 shows the zero- speed test with a wave height record which appears to be distorted in contrast with the smooth quasi-sinusoidal wave behavior presented earlier in Fig. 12. This is a result of one of the techniques which was used to vary the transform amplitude of the input voltage to the wavemakers, a varying frequency Sweep rate which caused a nonuniform deformation of the wave shape. The progressive transformation of this wave shape is shown in Fig. 27, which presents measurements taken on the program at three locations. For contrast, similar measurements for the transient of the second series of tests are given in Fig. 28, where the contraction of the shape is much more orderly. In both figures, waves were measured without a model in the water. Another interesting test performed during this series employed a human transient generator. To investigate the degree of corruption of wave height re- cording by model-generated waves, the model was made to oscillate in pitch by manually forcing the bow over a range of frequencies. Generated waves were measured by the two forward wave probes and analyzed for their Fourier con- tent, along with the motion record. The transfer functions, wave height/pitch motion, are given in Fig. 29, where for the nearer probe an average wave height of 1 in. results for a pitch motion of 5 deg over the high-frequency band. The effect of heave is neglected. 528 Testing Ship Models in Transient Waves Fig. 21 - Phase angle be- tween pitch-wave height, heave-wave height, and pitch-heave at zero speed; transient test compared with regular wave test results Heave Response 221-249 O - 66 - 35 8 Frequency in cycles per second in degrees Phase Angle 120 Frequency LEGEND & REGULAR WAVES © TRANSIENT TEST 529 LEGEND 4 REGULAR WAVES © TRANSIENT TEST — NEWMAN'S THEORY Heave — Wave Height Pitch — Wave Height a Lee Pitch-Heave 6 O: 0.8 1.0 in'cyc les per second Fig. 22 - Heave response for Model 4889 in head waves at a Froude number of 0.14; transient test compared with regular wave test results inch in degrees per Pitch Response Davis and Zarnick LEGEND 4 REGULAR WAVES O TRANSIENT TEST Frequency in cycles per second Fig. 24 - Heave/pitch ratio for Model 4889 in head waves ata Froude number of 0.14; tran- sient test compared with regu- lar wave test results Heave- Pitch Ratio ininches per degree 530 Fig. 23 - Pitch response for Model 4889 in head waves at a Froude number of 0.14; transient test compared with regular wave test results LEGEND ~ & REGULAR WAVES o TRANSIENT TEST 2 4 6 8 10 Frequency in cycles per second Testing Ship Models in Transient Waves ‘_ O Shao 7S) BD To in LEGEND a 4 REGULAR WAVES 180- © TRANSIENT TEST ” a Vy o ® 140 a) = © 100 OR p = Woh | 4O Pitch— Wave Height < 200 = : Pare 4 = A 160 ry = oOo, R 120 = = a 5 Ao Ola alake ay Pitch-Heave Ee 2 4 6 8 1.0 Frequency in cycles per second Fig. 25 - Phase angle between pitch-wave height, heave-wave height and pitch-heave at a Froude number of 0.14; transient test compared with regular wave test results The results of this final series of tests show clearly that the transient tech- nique is a usable tool for the investigation of ship response to waves; however, further improvement is possible. When considering (1) the very close agree- ment between transient and regular heave/pitch ratios observed in Figs. 20 and 24 and (2) the variation in the other frequency response estimates that is due solely to choice of forward or after wave height probe, an unwavering finger of suspicion points to the measurement of the dynamic wave disturbance. Figure 30 compares the wave height transform of the zero-speed transient, with that of the forward speed run properly transferred to the frequency scale of the stationary measurement. The general agreement is quite good, but the difference resulting from measurements made only 8 ft apart on the same wave — with forward and after probes — is puzzling. To weigh the possibility of this difference being associated with model- generated waves, Fig. 31 displays the same wave height transform at zero speed compared with that of a similar measurement made under identical conditions except that the model was not in the water. The agreement is not impressive, 531 Davis and Zarnick FWD WH “Vv —_—_—EOee"* iN I | ee vw VN fie LP LO eae Fig. 26 - Record of transient test conducted with Model 4889 at zero speed in head waves 104 Feet from Wavemaker 184 Feet from Wavemaker he WA ye) |) Po 260 Feet from Wavemaker Fig. 27 - Transient waveform measured at various locations in basin; program used in third series of test 532 degree in inches per per unit Pitch Wave Height Testing Ship Models in Transient Waves ~$aerrrrnrrrrrr wn 80 Feet from Wavemaker 160 Feet from Wavemaker 272 Feet from Wavemaker | Fig. 28 - Transient waveform measured at various locations in basin; program used in second series of test LEGEND PROBE #1 —-—-— PROBE ¥2 2 a) 4 6 8 Frequency in cycles per second Fig. 29 - Model generated wave height/forced ratio; effect of motion generated waves on forward wave height measurement 533 Davis and Zarnick 100 LEGEND ZERO SPEED Ses AE Nee he PROBE # 1 Bt th eee he PROBE # 2 PROBED a 2 J 2 3 a ‘SW Chuet Sco Frequency in cycles per second Fig. 30 - Comparison of transient wave height spectra measured during zero and forward speed runs with Model 4889 and the wave transforms of the latter measurement do not correlate well, for- ward probe with after probe. The remaining sources of error in the wave height transform estimate have obviously not been isolated by these tests. AREAS OF CONTINUING DEVELOPMENT EFFORT General The understanding and counteracting of the various factors leading to an in- accurate wave height measurement will be undertaken as the major effort in de- veloping further the capability of the transient wave test. A series of tests in waves, with and without a model, is planned in order to investigate the error contributions from (1) nonlinear water dynamics, (2) nonlinear measurement by the wave height probe, (3) spatial variations in basin waves, (4) side or end re- flections, (5) residual waves after passage of the main wave, (6) faulty Fourier analysis, (7) electronic interaction between adjacent wave probes, and (8) model- generated waves. A second major goal will be the development of a program for a transient wave with linear sweep rate, constant amplitude over the frequency range of in- terest, and smooth starting and ending, so as to minimize initial and terminal transients and the extraneous "noise" at the end of the wave train. 534 Testing Ship Models in Transient Waves LEGEND MODEL IN WATER —— —-— PROBE *| MODEL OUT OF WATER PROBE #| PROBE *2 | 2 | @ a) 6 7 .B6 .9 10 Frequency in cycles per second Fig. 31 - Comparison of transient wave height spectra measured with and without model in water at zero speed To assist in these investigations and to speed up data analysis, digital com- puter programs are being written for Fourier transform computations. An im- portant part of these programs will be the ''smoothing" of the transient records prior to transformation, or the multiplication of all time histories by a quantity which is unity over the duration of the test and eases to zero at the beginning and end of the test. This smoothing, standard in spectral analysis, will consid- erably reduce the effect of residual noise in the water near the end of a run. After more experience has been gained in forward speed runs, a critical analysis must be made to determine the effect of surge variations in transient analysis. This very knotty theoretical problem might require conducting wave transient tests at reduced wave height levels in order to avoid the time distor- tion of motion records. The particular problems associated with tests in astern waves must be re- solved. Unfortunately, a given frequency measured in the water with respect to the moving model can originate from astern waves at three different wave lengths. This ambiguity may force the use of transient waves with energy in more limited frequency bands, the use of multiple wave probes to make use of phase information, or both. Special Use of the Transient Testing Technique In one interesting use of a transient test, the variance of all motions ina given unidirectional random seaway can be found directly without need for 535 Davis and Zarnick frequency analysis. To show how this can be done, consider the equation which relates mean Square motion m? to the wave power spectral density at the fre- quency of encounter 0) and to the applicable transfer function G(~): 3 Pad J do @(a) |G(a)|’. (22) In a transient test, the integral square motion is given by i dt mt) = x | den nia) | cosy | (23) where N() is the Fourier transform of the measured wave height, using a well- known relation from linear systems theory. Thus, if N(«)? is programmed to be equal to the wave height spectrum (#), we have Random Transient sits 2 24 ma? = | Ghe sme). (2) Simple analog data processing, conducted during the model test, would square and integrate the motions of interest and yield, at the end of the run, voltages proportional to motion variances in the defined random seaway. Although such a simplified scheme of data processing would not extract much of the significant information available from a transient test, it is con- ceivable that there might be occasions when a very fast answer to the seaworthi- ness characteristics of a ship form in a given seaway would be required. One example would be a search for the worst combination of speed and heading. GENERAL APPLICATIONS OF THE TRANSIENT TECHNIQUE The method for producing a transient water wave described in this report was developed in order to take account of the behavior of waves on a free sur- face. The form of this wave, however, would appear to have considerable prom- ise for applications in many linear systems investigations. Conventionally, a pulse-like transient is used for system excitation. The frequency range of the excitation is determined primarily by the concentration of the transient about a single point in time, and the amount of signal needed to produce measurable response is a function of the amplitude levels of the pulse. As a consequence, to faithfully reproduce a rapidly changing signal, measure- ment requirements are severe and the probability of nonlinear behavior of the system under test is high because of the large inputs often required to override the effects of measurement noise in the recorded output signal. 536 Testing Ship Models in Transient Waves The use of a signal with linearly varying frequency and constant amplitude as an input signal removes these strong drawbacks to the pulse technique, how- ever. Since it has constant amplitude, the input can be constrained to lie within the linear range. With proper choice of sweep rate and starting frequency, the controlled signals can have any desired energy level in each frequency band and thus defeat to a great extent the effects of random measurement noise. Another strong advantage in the use of transients with a linear sweep rate is that in many cases the entire transfer function of a system can be obtained by cursory analysis of the transient records alone. If the rate of change of fre- quency and amplitude is slow enough, the signals involved behave very much like sinusoidal waves. From the integration method of stationary phase [2], it can be shown that the amplitude of the transform at frequency w of such a signal is equal to one-half of the single amplitude of the signal at the apparent local fre- quency « divided by the square root of the rate of frequency change (in cps/sec). Such a computation was performed for the transient test shown in Fig. 13 for the heave measurement. The results of this computation for the heave transfer function are shown in Fig. 32. This figure compares the measurement and cal- culation techniques and shows that there is good agreement among them. LEGEND 4 REGULAR WAVES © STATIONARY PHASE © FOURIER TRANSFORM Heave Response Frequency in cycles per second Fig. 32 - Comparison of methods of analysis of heave response operators 537 Davis and Zarnick SUMMARY AND CONCLUSIONS The complete determination of the response of a ship in regular waves in- volves a large and expensive testing program. When the transient wave tech- nique is used properly, the total testing time can be reduced by an order of magnitude. In a theoretical discussion of ship dynamics, it has been stated that the usual systems representation of ship motion as an "output" and wave height as an "input" is a misconception; both dynamic quantities are "'output."’ Water wave transients traveling in a single direction can be readily analyzed with the use of Fourier transforms; the transforms of two measurements on a single wave transient are related by the so-called "transfer function" of water, e Je!!X/® | where x is the distance separating the measurement points in the direction of wave travel. The wave used at the Model Basin for transient testing has a continuously increasing wave length which results in an intense concentration of wave energy for a short period of its travel. Model transient tests are commenced in calm water, then passed through a wave having energy in all frequencies of interest, and eventually returned to the smooth water condition. The transfer function for a particular motion variable is found for all frequencies by dividing the Fou- rier transform of the motion transient by that of the wave height record, refer- enced to the model center of gravity. With a suitably tailored wave transient, mean square motion levels in a particular random seaway can be found immedi- ately by squaring and integrating the motions during a transient test. Transient tests conducted on three models in ahead waves have verified the theory presented. Close agreement between transient and regular wave tests has been obtained for heave and pitch motions at zero and forward speed. The major difficulty encountered has been in the generation and measurement of waves, where further refinements and research are proposed. Digital programs are being written for the bulk processing of transient records. Finally, the technique of using a transient excitation which is a linear fre- quency sweep is an original contribution to general linear systems analysis; it has virtues of linearity and noise-immunity, the capability of determining fre- quency response of a system by visual inspection of the transient records. Appendix WEIGHTING FUNCTION OR IMPULSE RESPONSE OF WATER It has been shown in this report that the operator e7!®!°'*/® is the fre- quency response function that relates wave height measured at two points sepa- rated by a distance x in the direction of travel. The weighting function w(7) can be determined by taking the inverse Fourier transform of the frequency re- sponse function: 538 Testing Ship Models in Transient Waves +o wir) = py e Ielolx/e ee ae (A-1) Deny -o Expanding into trigonometric terms and noting the symmetrical properties of the function, we have after simplification w(T) = = cos (-w* x/g twr)dw = +] cos (w* x/g -ar) dw. (A-2) 0 0 Employing the technique used by Lamb [7] we let and peal Oger oO = ° ax1/2 These terms are substituted into Eq. (A-2), yielding (oe) Ty 2 w- = mul cos (62-07) dl ew/2 e Pe Seales { cos (62-07) dC + { cos (672-07) dC x see 0 ihe eee p % Sh gD cos co” ) cos erat + | cos C2dZ x -o 0 -o 0 0 o + sino? | sin (2d + | sin vail| : (A-3) We can make the following identification: 0 o i cos (7 dl = I cos C2dl = To 7 CUD -o 0 2 0 o { sin (*dC = somos dG a a2 EOL) = Toy 0 |o| Davis and Zarnick | Soe | cos (dt = Lya7a 0 0 where (2 bh C(/) : J cos (5 mu | dp ; S() = ) sin (57) du. 0 0 Substituting the above into Eq. (A-3), yields | Ww = (fe) {cone E + yc leree)| + [sin o?] Ee ls S) (v7 xl} For simplification, let 1/2 te g che lal ; Then GS = Wif2® % ar and w(aT) = a ie | [2 ial es) + [sing a \2* ial sn} . (A-4) For large values of aor c, going back to Eq. (A-3), we find that w(aT) &% a | [cos 5 a2 7? + sin Fat] 2 = l= cos (ee = x) af 22> 0) o (A-5) REFERENCES 1. St. Denis, M. and Pierson, W. J., "On the Motions of Ships in Confused Seas,'' Transactions, The Society of Naval Architects and Marine Engineers, Vol. 61 (1953). 2. Stoker, J. J.,""Water Waves,"'Interscience Publishers, Inc., New York (1957). 540 Testing Ship Models in Transient Waves 3. Brownell, W. F., "Two New Hydromechanics Research Facilities at the David Taylor Model Basin,'' Presented before Chesapeake Section, The So- ciety of Naval Architects and Marine Engineers (Dec 1962). Also David Taylor Model Basin Report 1690 (Dec 1962). 4. Davis, M. C., "Simulation of a Long Crested Gaussian Seaway,'' David Taylor Model Basin Report 1755 (Mar 1963). 5. Huskey, H. D., et al., ''Computation of Fourier Integrals without Analog Multipliers,'' Computer Handbook, McGraw Hill Book Co., Inc., New York (1962). 6. Newman, J. N., ''A Slender Body Theory for Ship Oscillations in Waves," In Preparation for Journal of Fluid Mechanics. 7. Lamb, H., "Hydrodynamics," Sixth Edition, Dover Publications (1945). * * * DISCUSSION E. V. Laitone University of California Berkeley, California Since the linearized equations of motion for either the pitch or the roll of a ship in regular waves can be written as Gar eG) So a ose (1) therefore there is a distinct advantage in running a series of model tests in regular waves. This advantage over the pulse or transient-type wave test oc- curs because Eq. (1) represents a circle in the velocity amplitude and phase plane as shown in Fig. 1 : A |e| - aE cos Os : (2) Consequently the departure of the experimental points from a perfect circle for varying values of the regular wave frequency («) directly indicates either the nonlinear effects, or the dependence of b or k upon o, for constant values of A,. Similarly varying A, and repeating the tests for different values of « illustrates the nonlinear dependence of b or k upon the amplitude (A,) of the regular wave. A transient wave test could not so easily pin-point the wave fre- quencies or amplitudes that correspond to a breakdown of the assumed linearity which results in Eq. (1). 541 Davis and Zarnick @ = > (Vb? + 4k - b), $.=4 é Phase angle of 6 = ?. = Vie" oe =u Figure 1 In addition various other simple relations can be derived for determining b or k quickly. For example when the phase angle of the response ¢ is exactly 90° with respect to the regular wave at frequency w, then A A =./iz . (BN se ag EGY (3) o = Jk MD tea tere Similarly if the phase angle of @ is exactly 45° then A A |e] = : eee (4) V2 (k-o?) = 2 (bw) * * * REPLY TO THE DISCUSSION M. C. Davis and E. E. Zarnick David Taylor Model Basin Washington, D.C. Mr. E. V. Laitone is apparently unaware of some of the more recent devel- opments in the area of ship motions. It has been known for some time that second-order differential equations do not properly describe ship motions, par- ticularly in the pitch and heave modes which we have concerned ourselves with in our paper. This problem has been glossed over in the past by the use of fre- quency dependent coefficients. The tests in waves are further complicated by the fact that the wave height (referenced at the center of gravity) is defined as the input to the system which results in a particular amplitude phase relation- ship exclusive of any dynamics effects. Consequently, these data would not be 542 Testing Ship Models in Transient Waves expected to describe a circle if plotted in the velocity phase plane as suggested by Mr. Laitone, and furthermore, the departure from such a circle would shed no light as to the nature of the problem. A better understanding of the problem can be obtained by separating the effects of added mass, damping, cross cou- pling and hydrodynamic memory by use of the integro-differential equations of motion developed by Dr. Cummins. This information, along with knowledge of the wave excitation, should provide us with information as to why a ship per- forms the way it does as well as possible changes in design to improve her sea- keeping qualities. We would like to apologize to Dr. Leo Tick* if we have in any way contrib- uted to his confusion. We are also very grateful for his pedagogical dissertation on the basic philosophy of testing. Of special note is his recollection of pro- found personal conversations (with someone whose name he can't quite remem- ber). Apparently the choice of a suitable test function is not clear to him. This is understandable. The choice of a test function or procedure depends upon many factors. We may be less philosophical and more pragmatic; this choice in the final analysis depends upon whether or not it serves the purpose intended, i.e., to provide an efficient and economical means of obtaining a reliable meas- urement of the transfer functions. We believe that the transient test technique meets these objectives. We have provided both theoretical and experimental evidence to support our claims. It is anyone's prerogative to accept or reject them. *See discussion by Tick (p. 457) on paper by Smith and Cummins. 543 ay hea is aa ously, ae afta ars ek ge igh peda binew! stonis:a tous” imesh ged SEE gargs edt te suibicdesabaa rated A cenaketey ty Pate pe 2 ai! LOO Reais [Ria past be he Bouts, mer Spistte reaaT ae, ed o-agolteupe, Lslide’ pote ofweaioel Dh Reece pads) Sher kre Tg aghalwortut fit Pega Sien' tiene cee wal amebteenitz) iC 1. * sce CE 5 Oi a az ‘eober (eer tri Sir” eine ng Heald hoe tan eoycynaeyr : eg saial a pega ieee Bh Nan wa won, } asine betes cn oath ‘i ee Jeitxapch iw. 4G: wy sere me lf HH, unt (hs ‘ Y ae org. ein ee evi » bah Hf foMdistalts (eo igen tsi be ae Foy ified) a Ne me 1. fee wae: Steere aeel et ed i me ayoee gr? arin f Be} Sh, at : 8 j REDS : 4 are: r sth Ni ; é ‘ ipt anh em? Fm Res SE alll a, alft: vid oi mh > ot te ij Om) tess ape nie 2 es Soon y if ( t ¢ oy 4 \ ia ey mi y i Ni iy q Mii peasy s Git 5 F : f / TL St ¥ siti i a iety)' Ot : ts > 3; iT: ty r) ust ae r ; : sb Ops bua Rice tee TA ered Doe bi? ARCM a LA bare 4 ‘ ‘ ' : ; j 3 oh ane piG at APN: Rey SO A cee + aaah Shih Sane ee Die Nh nd i 4 bd | “ % 4 | ny ea, Sind, Pipe (Es 8 ANE SEG BOS FO SN ALR EEE "ES EI hi TO mi = Talnerdt rere hae Deduomert hod festvary avid 6 .eev ia R | ara ; oer ‘ : ¢ f sf perl Tt ped ow F ie ; / ? ‘ i { Listy , ris Sy és ; f tis 1) wy Pay Ion TOA T Cee k my r ’ aah oe Ls 4 r% 3+ ee — ¥ } aay Ms Pe if a volre yo) ‘ Mut) | PREDICTION OF OCCURRENCE AND SEVERITY OF SHIP SLAMMING AT SEA Michel K. Ochi David Taylor Model Basin Washington, D.C. ABSTRACT Basic properties which govern ship slamming in rough seas are dis- cussed from theoretical consideration. Specifically, the probability of occurrence of slamming, magnitude of impact pressure associated with slamming, and time interval between successive slams are studied from a statistical approach, and formulae are derived for the predic- tion of these events. The prediction method is also applied to the prob- lem of deck wetness caused by shipping of green water at sea. Theo- retical results are compared with those obtained in experiments conducted on a MARINER model in rough seas. INTRODUCTION When a ship navigates at certain speeds in rough seas she frequently expe- riences slamming at which time the forward bottom sustains large forces re- sulting from the impact. Slamming occurs at random at sea. The severity of slamming and time in- terval between two successive slams are also at random. Sometimes a ship may slam successively with varying intensity; while again no slamming may occur for a relatively long period of time and then suddenly a severe slam oc- curs. In statistical terms slamming is a random phenomenon, and the severity and time interval between successive slams are random variables. For this random phenomenon, only one study has appeared in the literature as of this date. This study made by Tick concerns the prediction of the rate of occurrence of slamming at sea [1]. The purpose of the present paper is to develop a method for predicting the probability of occurrence and severity of slamming, and the time interval be- tween successive slams in rough seas. Specifically, it is the intent of this paper to predict the following: (a) Probability of occurrence of slamming for given conditions, such as for a given sea state, course angle, loading condition, etc. 221-249 O - 66 - 36. 545 Ochi (b) Probability distribution of impact pressure associated with slamming, and magnitudes of the average one-third and one-tenth highest impact pressures. (c) Probability distribution of the time interval between successive slams and between two severe slams. Probability that a time, T (or more), elapses between severe slams. (d) Probability of occurrence and severity of deck wetness caused by ship- ping of green water, i.e., application of the theory to the deck wetness problem. The above subjects are evaluated theoretically, and the results are com- pared with statistically analyzed experimental results obtained in tests con- ducted on a 13-ft MARINER model. PREDICTION OF OCCURRENCE OF SLAMMING Basic Concept Prediction of the occurrence of slamming is made from two viewpoints: one being the prediction of slamming occurrence per cycle of wave encounter, the other being that per unit time. The question pertaining to how many times a ship will slam during a certain period of time belongs to the latter prediction. The basic concepts used for development of the theory for these two predictions are, however, essentially the same. First, the conditions leading to slamming will be discussed. Szebehely has shown that three conditions should exist for slamming to occur [2]. They are: (a) bow (forefoot) emergence, (b) certain magnitude of vertical relative velocity between ship bow and wave, and (c) unfavorable phase between bow motion and wave motion. The present author has also arrived at the same conclusion through his tests [3]. Tick considered three conditions in the development of his theory for predicting the number of slams per unit time. These are: (a) bow emergence, (b) relative velocity, and (c) angle between keel line and wave surface at the instant of impact [1]. All of the above conditions were inferred from results of model experiments conducted in regular waves. The question then arises as to whether or not these are necessary and sufficient conditions leading to slamming in rough seas also. To answer this question, data obtained from slamming tests conducted in irreg- ular waves were carefully analyzed, and two conditions leading to slamming in rough seas were obtained. They are: (a) bow (forefoot) emergence, and (b) a certain magnitude of relative velocity between wave and ship bow. In other words, the probability of occurrence of slamming is the joint probability that the bow emerges and that the relative velocity exceeds a certain magnitude at the instant of reentry. Bow emergence is prerequisite for slamming. Results of the tests revealed that slamming never occurred without bow emergence. This was found to be true irrespective of sea state, ship speed, course angle or loading condition [4]. However, bow emergence is not a sufficient condition for slamming. There 546 Prediction of Ship Slamming at Sea were many cases during the tests in which no appreciable impact pressure was imparted to the ship bottom even though the forefoot emerged from the water surface. It was found that a certain magnitude of relative velocity between wave and ship bow (hereafter referred to as the threshold velocity) was required to induce slamming. The threshold velocity is a critical relative velocity between ship bow and waves below which slamming does not occur. Although little information is available concerning the magnitude of the threshold velocity associated with slamming, the magnitude was evaluated from various available sources [3-5], and the results are tabulated in Table 1. For convenience, the values have been converted to those for a 520-ft ship for comparison with the MARINER. As can be seen in the table, the values have been obtained for various test conditions. Nevertheless, the magnitudes of the threshold velocity are nearly constant with an average of 12 ft/sec. To determine the threshold velocity for the cargo ves- sels (U- and V-Form) listed in Table 1, ship speed was increased until the ship started to slam in the given regular waves (\/L = 1, h/\ = 1/30). The speeds for which slamming first appeared were 10.4 and 11.9 knots for the U- and V-Form, respectively. The relative velocities evaluated for these speeds were taken as the threshold velocities. For higher ship speeds slamming was severe, and hence the relative velocities between wave and the ship bow for these speeds could not be considered as the threshold velocity. Note that the threshold ve- locity is the minimum velocity which causes slamming. In regular wave tests conducted on a high speed craft listed in the table, an immersion sensing element was fixed to the model at Station 2. Hence, the rel- ative motion between wave and the bow was directly measured, and the relative velocity was obtained by differentiation. It is of great interest to mention that the magnitude of the threshold velocity evaluated from the MARINER tests in irregular waves is very close to that evaluated for other types of vessels tested in regular waves. For evaluation of the threshold velocity for the MARINER the data obtained in a severe Sea State 7 at a ship speed of 10 knots were analyzed [4]. Since the wave measuring device was located 9.83 feet (410 feet full scale) ahead of the model in these tests, one assumption was introduced in the analysis. That is, waves measured at the lo- cation of the wave probe would maintain their form until they reached the ship bow. With this assumption, the magnitude of relative velocity at the instant the ship slammed was evaluated from simultaneous records of pressure, ship mo- tion, bow vertical acceleration and wave. Figure 1 shows the relationship be- tween relative velocity and impact pressure measured at 0.10 L aft of the for- ward perpendicular. As can be seen in the figure, no impact pressure is observed for a relative velocity less than 12 ft/sec. On the basis of the above finding, it is considered appropriate to take 12 ft/sec as the threshold velocity associated with slamming for a 520-ft ship. The reader's attention is called to the fact that this magnitude of threshold velocity cannot be used universally. For a ship of different length, the above given value should be modified accord- ing to the Froude scaling law. 547 Ochi Table 1 Threshold Velocity for Various Types of Ships (Values are Converted to Those for a 520-ft Vessel) High Speed ae Cargo Type of Ship ) | (V-Form) LIBERTY | MARINER ee Block coefficient 0.741 0.741 0.733 0.624 Draft Light Light Light Light Waves Regular | Regular Regular Irregular W/L 1.00 1.00 0.91 Severe Sea State 7 h/A 1/30 1/30 1/16.7 Ship speed 10.4 11.9 10 10.0 (knots) (Estimated) Location where 0.053 L 0.093 L FP 0.10 L the threshold aft of FP | aft of FP aft of FP velocity is evaluated Threshold velocity (ft/sec) Reference Unpublished In connection with other proposed conditions leading to slamming (such as unfavorable phase between bow and wave motion and angle between keel and wave), it is mentioned that these are included in the two required conditions found from the present tests. For example, the phase changes from time to time in irregular waves, and it is apparent that the largest relative velocities are associated with the out of phase motions. Thus, it may be concluded that bow emergence and threshold velocity are the only conditions prerequisite to ship slamming. It is noted here that the occurrence of impact pressure at Station 2 (0.1 L aft of the forward perpendicular) was used as a criterion for slamming. The justification of this statement is given in Ref. 4. Probability of Occurrence of Slamming per Cycle of Wave Encounter Let w be the wave displacement and b the bow (forefoot) displacement from their respective at rest (zero) positions (see Fig. 2). Upward displacement is 548 Prediction of Ship Slamming at Sea Fig. | - Pressure on the keel plate as a function of impact velocity (MARINER, Station 2, light draft, ship speed 10 knots, moderate Sea State 7) in PSI Pressure Relative Velocity in Ft./Sec BOW (FOREFOOT) MOTION Fig. 2 - Explanatory sketch of bow emergence taken as positive. The distance between these two zero-lines is equal to the ship draft, H, at a specific location, x (in this example, Station 2), for which the probability of slamming is evaluated. Note that this draft is not necessarily the design draft. Next, let r = b-w; then the relative motion, r, must always be positive and greater than H when bow emergence occurs. 549 Ochi For a better understanding of the relationship between slamming and rela- tive motion, Fig. 3 was prepared. Figure 3(a) is an explanatory figure showing time history of relative motion. At the instant of a slam as the bow re-enters the water, the relative motion r(t) must be positive and equal to H. The rela- tive velocity z(t) at this instant is negative and its absolute value must be greater than the threshold velocity, r,. The above condition is given on the phase-plane diagram shown in Fig. 3(b). RELATIVE MOTION (+r) RELATIVE MOTION (+r) RELATIVE VELOCITY (+r) (b) Fig. 3 - Explanatory sketch of time history of relative motion and phase- plane diagram 550 Prediction of Ship Slamming at Sea The relative motion is considered as a random variable having a narrow- band normal distribution with zero mean, since the relative motion is a combi- nation of pitch, heave, and wave motions, all of which have narrow-band normal distribution with zero mean. The relative motion is expressed by the following formula r(t) = r.(t)-cos {wt + Eac ty} (1) r,(t) = amplitude of the envelope of the relative motion, r? w. = expected frequency = o. /o €, = Slowly varying phase angle, o? = variance of relative motion, o. = variance of relative velocity. It is noted that the relation «, = o./c, holds since a narrow-band normal process with zero mean is considered. Assuming that r, and <, are small for a narrow-band normal process, the following equation is derived from Eq. (1). =a re = r24+——. (2) % Now, the probability density function of r,(t) is a Rayleigh distribution. Since slamming occurs only when the relative motion is positive, the probability density function of the positive r,(t) can be written by Dr Manne (3) f(r5) = e ; Tae Oe Note that Eq. (3) represents the probability density function of the cross points on the OA-line in Fig. 3(b), and that the parameter, R‘, involved in the equation is not eight times but is twice the variance of the relative motion. Hence R/ is equal to the cumulative energy density, i.e., the area under the en- ergy spectrum, E, using the St. Denis- Pierson definition of the spectrum. From Eqs. (2) and (3), 2 r? ct ees ®o (4) £(1 5) Ss : As was mentioned earlier, slamming occurs when the relative velocity ex- ceeds the threshold velocity at the instant of reentry, i.e., r = H, and r > r,. dd1 Ochi In the phase-plane diagram shown in Fig. 3(b), slamming occurs whenever the circle crosses the line DC. Thus, the probability of occurrence of slamming is given by Pyooly WSilanh =. ol fe = iH, ie S ina} 2 32 ‘ ae S) where H = draft at the ship bow, r, = threshold relative velocity, R{ = twice the variance of relative motion, R. = twice the variance of relative velocity = R’«?. As can be seen in Eq. (5), it is necessary to evaluate the variances of rela- tive motion and velocity for estimation of the probability. The application of the superposition principle by using the response amplitude operators may be valid to evaluate the variances even for conditions severe enough to induce slamming. The justification of this statement will be given in the next section in which a comparison between the predicted and measured probability of occurrence of slamming are shown. The variances of relative motion and velocity at an arbitrary point along the ship length can be approximately estimated from irregular wave tests also. The method for evaluating the variances for this case is discussed in Appendix 1. Number of Slams per Unit Time The number of slamming occurrences per unit time is essentially an appli- cation of the problem of the expected number of zero crossings per unit time. The theory on the zero-crossing problem was first developed by Rice [6], and later applied to ship slamming by Tick [1]. Therefore, the development of the 552 Prediction of Ship Slamming at Sea theory will not be described here, but the formula which meets our require- ments (r = H, |r| >t,) is given instead. The number of slams per unit time, N, is given by H2 r2 The definitions of R’, R';, H, and +, are the same as those used in Eq. (5). It is noted that Eqs. (5) and (6) are related by the formula for the expected pe- riod, T,, for a narrow-band random variable having a normal distribution with zero mean. nN. = ae = 27; Yous R Te ee =r (7) Table 2 shows the predicted probability of occurrence of slamming per cy- cle of wave encounter and the number of slams in a 30-minute operation of the MARINER for various conditions. Included also in the table are the experimen- tal values observed in tests conducted on a 13-ft model [4]. To evaluate the predicted values, the response amplitude operators of the relative motion at Station 2 were obtained for various course angles and ship drafts by conducting tests in regular waves, and the superposition technique was used for estimating the variance of the relative motion. The variances of relative velocity were obtained from the energy spectra of the relative motion [7]. Examples of the response amplitude operators of the relative motion and the computed energy spectra of relative motion and velocity are shown in Fig. 4. As can be seen in Table 2, the predicted values show satisfactory agree- ment with the observed values; there being approximately a 10 to 15 percent discrepancy, except for moderate and full draft conditions. For the deep draft condition, however, the discrepancy of 25 percent is not surprising since the probability is small. Thus, the application of superposition principle for evalu- ation of relative motion and velocity appears to be adequate to obtain realistic engineering estimates of the probability of occurrence of ship slamming at Sea. It is of interest to discuss the effects of course angle and loading condition on the probability of occurrence of slamming. It was found experimentally that the probability decreases with increase of course angle and with increase of loading. In other words, the probability of slamming is highest when a ship navigates in head seas at light draft condition [4]. The occurrence of slamming becomes less with increasing course angle because both the relative motion and velocity between wave and ship bow significantly decrease as can be seen in Table 2. For example, the computed R‘ and R. (twice the variances of relative motion and velocity, respectively) for a 45 degree course angle both decrease to 60 percent of their values in head seas. On the other hand, the occurrence of slamming becomes less with increase of loading primarily because ship draft deepens and thereby bow emergence is less frequent. As can be seen in Table 2, 553 VIVO G8¢e°0 peaArasqg peyoIpeid uoTJv.19dO 9}NUIU OF B UI SWIRTS JO TequINN c80°0 8610 GST 0 862°0 €€E°0 SEéTO poAaresqo 990°0 09T'0 9710 eccan & 90€°0 8¢c1T 0 pe} oTpetd I9}uNODUe BACM JO a[DAD Aad SWIRTS Jo A}ITIGQeqoIg 3) TLt L LT @ UOT}ES 7e JJeId 8&2 cos aay 9} 819POT WSTT WUSTT qUSTT UOT}IPUOD SUTPvOT (eioun) (UANIMVIN (EER ei) 9)) a[sue ssanoyg (35) FUSToYy aAemM JUBITIIUSTS (Ss.noy) uoTyeanp pulM (s}oux) AIDOTOA PUTA 9781S vas ) swie[S Jo zequInN pue AjI[IQeqoIg peAIESqO pUue pe}0Ipertg jo uoSsTreduIOD @ PIAB.L 554 Spectrum in Ft? - Sec [Response Amplitude Operator] Spectrum in Ft” - Sec Spectrum in Ft? / Sec Prediction of Ship Slamming at Sea SPECTRUM (SEVERE SEA STATE 7) WAVE BRe > we ie) can Be eitis RELATIVE MOTION SPECTRUM Vaan. RELATIVE VELOCITY SPECTRUM Sar en eee ne CM Ps ta eau 1200 800 a aie in |/Sec Fig. 4 - Energy spectra of relative motion and relative velocity by applying the superposition technique (MARINER, light draft, severe Sea State 7, head seas) 955 Ochi the computed R‘ decreases only slightly with increase of loading. However, the probability of bow emergence is an exponential function of the square of the draft at the bow [Eq. (5)], and thereby the probability decreases drastically with increase of the draft. For a better understanding of the above statement, Fig. 5 was prepared to show the computed probability of slamming as well as the probability of bow emergence and the probability that the relative velocity exceeds the threshold velocity for the MARINER in head seas of a moderate Sea State 7 at a ship speed of 10 knots. The probability of occurrence of slamming, is, of course, the prod- uct of the other two probabilities. It is clear in the figure that the probability of bow emergence, Prob {r >H}, is responsible for the rapid decrease in the prob- ability of slamming. Fig. 5 - Probabilities of occurrence of slamming and bow emergence, and probability that the relative velocity exceeds the threshold velocity Probobility Cargo Loading in Percent Moderate PREDICTION OF SEVERITY OF SLAMMING Ship slamming is always accompanied by an impact pressure on the flat bottom, and the magnitude of the pressure is indicative of the severity of slam- ming. The impact pressure is approximately proportional to the square of the magnitude of relative velocity at the instant of impact as was shown in Fig. 1. The same conclusion was obtained from results of tests conducted in regular waves [3]. Hence, this basic relation of the impact pressure and relative veloc- ity will be considered in the development of the theory. Prior to a discussion on the prediction of slamming severity, a statistical consideration of the magni- tude of relative velocity will be given. 556 Prediction of Ship Slamming at Sea Prediction of the Magnitude of Relative Velocity Between Wave and Ship Bow In order to predict the magnitude of relative velocity between wave and the ship bow, the probability density function of the relative velocity associated with slamming must be established. In other words, the probability density function of the cross points on the DC-line shown in Fig. 3(b) should be obtained. Al- though the relative velocity associated with slamming was defined as negative, the sign will be changed hereafter for convenience. Since slamming occurs when the relative motion is equal to H, let r = H in Eq. (2). Then, oP ee wee, (8) Consider the probability density function of r, when r, is greater than H; namely, consider the probability density function of the cross points on the BA- line shown in Fig. 3(b). The result is De eee (9) f(r,) = aS s ; ceo RS From Eqs. (8) and (9), es A) = he eu r>0. (19) U ie Thus, the probability density function of the cross points on the BC-line in Fig. 3(b), neglecting the sign of the relative velocity, is the Rayleigh distribution. Next consider the probability density function of the cross points on the DC-line in the figure, since it is necessary to consider the threshold velocity, r,, for slamming. Then, the probability density function of the relative velocity for slamming is given by ape lit pee? er: ea Nad a (r) = R! e ’ PZ 1 se where R! = twice the variance of relative velocity, rt, = threshold velocity. 557 Ochi Thus, the probability density function of the relative velocity associated with slamming is a truncated Rayleigh distribution. The truncation should be made at the threshold velocity, r,, which is a function of a ship length as was mentioned earlier. From the probability density function given in Eq. (11), the average of one- third highest (significant), r,,,, and one-tenth highest, r values of the relative velocity can be obtained as follows: 1/10? ed a 2 TT. (173) R! a R! - (12) cpa =e 2 Fapae 5 + y7R. {+- (fv) iP where : i 1 = ts ~ R. log 3 Pipe Ty 5 08 3 u t? Die ae | e 2 de V27 ee Cups) x Re | R, rae (13) Pa 10 e T1710 & + y7R. 1 -©® R’ Evang r where F PRAHEG F 1 sa = a log 7 - The derivation of these formulae is given in Appendix 2. A comparison between theoretical probability density function and the his- togram of the relative velocity obtained from tests conducted on a MARINER model is shown in Fig. 6 (values are converted to those for full scale). The ex- ample shown in the figure is for tests conducted in a severe Sea State 7 ata 10-knot ship speed, the same condition as was shown in Fig. 1. As can be seen in Fig. 6, the prediction curve agrees well with the observed histogram. Also, the average of the one-third and one-tenth highest values calculated by Eqs. (12) and (13), respectively, agree well with the measured values. Prediction of the Magnitude of Impact Pressure Associated with Slamming It was shown earlier that the impact pressure associated with slamming is approximately proportional to the square of the relative velocity and that the 558 Prediction of Ship Slamming at Sea Averoge of Average of 1/3 Highest 1/10 Highest in Ft./Sec. in Ft./Sec. Predicted 27. 336 Truncated Rayleigh Distribution Ft./Sec. Percent Original Rayleigh Distribution Relative Velocity in Ft./Sec. Threshold Velocity ¢2!2.0 Ft/Sec. Fig. 6 - Comparison between sample histogram and the truncated Rayleigh distribution for relative velocity (se- vere Sea State 7, ship speed 10 knots, light draft) probability distribution of the relative velocity follows a truncated Rayleigh dis- tribution. From these two conditions, the probability density function of the im- pact pressure can be derived. Let the impact pressure associated with slamming, p, be expressed by joe ee Cine (14) where (@) I constant dependent upon the ship section shape, 4. I relative velocity. From Eqs. (11) and (14) and with the aid of the transformation theorem on random variables, the following truncated exponential probability density func- tion can be derived for the impact pressure associated with slamming 559 (15) where impact pressure = 2Cr?, uo) i p, = threshold pressure = 2Cr2. The probability that an impact pressure exceeds a certain magnitude, p., per cycle of wave encounter can be obtained 1 o Ray 25 De) Prod {9 29) = | i(p) cb = & : Pp cele (16) Pp fo} It is of importance to note here that Eq. (16) is a conditional probability; namely, it represents the probability that an impact pressure exceeds a certain magnitude given that a slam occurred. Hence, the probability that an impact pressure exceeds a certain magnitude in a given sea state and at a given ship speed is the product of the two probabilities given by Eqs. (5) and (16). Also, the problem concerning how many times an impact pressure exceeds a certain magnitude in a prescribed ship operation time can be obtained by multiplying the operation time by the product of Eqs. (6) and (16). The averages of one-third highest, p,,, and one-tenth highest, p,,,, pres- sures are given by the following formulae: yes (2c aa f D210 R:) (17) rer (2c 32 = 8.20 Ri). (18) Derivation of Eqs. (17) and (18) are given in Appendix 2. Figure 7 shows a comparison between the theoretical probability density function and the histogram of impact pressure obtained at 0.1 L aft of the for- ward perpendicular of the MARINER in a severe Sea State 7 at a 10-knot ship speed. The value 2c = 0.086, determined from Fig. 1, was used in the calcula- tion. Included in the figure are the predicted average of the one-third and one- tenth highest pressures calculated by Eqs. (17) and (18) as well as the observed values. As can be seen in the figure, the theoretical density function is trun- cated at 12.4 psi due to the threshold relative velocity. Although pressures lower than 12.4 psi were actually observed a few times during the tests, reason- able agreement between theoretical and experimental results can be seen in the figure. The discrepancy is of the order of 10 percent for the average of the one-third highest, and 20 percent for the average of the one-tenth highest values. 560 Prediction of Ship Slamming at Sea Average of Average of 1/3 Highest 1/10 Highest in PSI Predicted 67.5 Percent /PSI Threshold Pressure in PSI Pressure , Px 12.4 PSI Fig. 7 - Comparison between experimentally obtained histogram of slamming pressure and predicted probability density function (severe Sea State 7, ship speed 10 knots, light draft) Comparison between theory and experiment were made for two additional cases; namely for moderate and mild Sea State 7, at a 10-knot ship speed. The results are shown in Figs. 8 and 9, respectively. Two histograms are shown in Fig. 9; one obtained from a 30-minute observation in a mild Sea State 7, while the other was obtained from a 70-minute observation in the same sea state. Al- though some discrepancy between the experimental histogram and the theoretical probability density function can be seen in Figs. 8 and 9, good agreement was obtained between the predicted and observed averages of one-third and one-tenth highest values in these two cases. It is noted here that a discrepancy between the experimental histogram and the theoretical probability density function is noticeable in the neighborhood of the threshold pressure. The discrepancy for these marginal conditions might be attributed to the actual angle between wave and keel. For higher relative ve- locity, however, the angle would not be expected to have a strong influence upon the magnitude of impact pressure. It is of interest to mention that the probability density function of the im- pact pressure given by Eq. (15) can also be applied for any course angle or loading condition. Figure 10 shows a comparison between the experimental 221-249 O - 66 - 37 561 Ochi Average of Average of 1/3 Highest 1/10 Highest in PSI In PSI Predicted 55.4 79.9 Measured 51.6 Percent /PSI| Threshold Pressure in PSI Fig. 8 - Comparison between experimentally obtained histogram of slamming pressure and predicted probability density function (moderate Sea State 7, ship speed 10 knots, light draft) histograms and the predicted probability density functions for various course angles in a moderate Sea State 7, at a 10-knot ship speed. Figure 11 shows a similar comparison for various loading conditions. The prediction curves were established by using the values listed in Table 2, and a threshold velocity of 12 ft/sec. Satisfactory agreement between the prediction curve and the experi- mental histogram can be seen in these figures. Based on these results, it is concluded that the impact pressure associated with slamming follows a trun- cated exponential probability law. PREDICTION OF THE TIME INTERVAL BETWEEN SLAMS Prediction of the Time Interval Between Successive Slams For prediction of the time interval between successive slams, the following question must first be answered: is the slamming phenomenon a sequence of events occurring in time according to the Poisson process? If the occurrence 562 Prediction of Ship Slamming at Sea Average of Average of 1/3 Highest IO Highest in PSI in PSI Predicted 386 Measured 3911 Experiment —— For 208 Cycles of Encounter ---- For 468 Cycles of Encounter Percent /PSI Threshold Pressure in PSI Pressure 12.4 PSI Fig. 9 - Comparison between experimentally obtained histogram of slamming pressure and predicted probability density function (mild Sea State 7, ship speed 10 knots, light draft) of slamming is considered as a Poisson process, then the time interval between successive slams is a random variable which must follow an exponential proba- bility law theoretically [8]. In order to obtain an answer to the above question and thereby to determine the probability density function for the time interval between successive slams, a sample of the time history of slamming obtained in tests conducted on a MAR- INER model will be shown. Figure 12 shows the time history of slamming pressure (converted to full scale) measured at 0.1 L aft of the forward perpendicular in a severe Sea State 7 at a 10-knot ship speed [4]. The ship was in light draft condition; specifically, 40 percent of cargo loading. A total of 84 slams were observed during 203 cy- cles of wave encounter in a 31 min-7 sec observation. It is noted that the sam- ple shown in the figure is the composite of four records taken in the tests. Hence, there exists three points of discontinuity as marked in the figure. Al- though the tests were carefully conducted, there is a possibility that several wave encounters and a small amount of time were lost at these discontinuities. The vertical line marked in the figure indicates a slam whose pressure magni- tude is proportional to the height of the line. The black circles indicate wave encounters without slamming. 563 Percent/PSI Ochi COURSE ANGLE HEAD SEA Prob. [Slom]= 0.333 Pressure in PSI Fig. 10 - Sample histograms and the predicted probability density functions for impact pressure observed at 0.10 L aft of FP for various course angles (moderate Sea State 7, ship speed 10 knots, light draft) 564 Prediction of Ship Slamming at Sea LOADING CONDITION LIGHT Prob, [Som] = 0.333 —— tee, as MODERATE Prob. [Slom] = 0.198 Percent /PS| FULL Prob. [Stom] = 0.088 Pressure in PSI Fig. 11 - Sample histograms and the predicted probability density functions of impact pressure observed at 0.10 L aft of FP for various loading conditions (moderate Sea State 7, ship speed 10 knots, head seas) 565 (2 9781g Vag e1eAeS ‘sjouy QO] peeds diys ‘3e1p IYyst{) Surururets yo Aroysty awry, - ZT] “S17 ONS SLANIW IE Ochi cI bl €l él i} NNY GNe LYVIS SLANIN. S ZLONIN BLNNIW € BLONIW 2 3LONIA | SWiL ——— Luvs J1wds 3yNss3add 566 Prediction of Ship Slamming at Sea As can be seen in the figure, the shortest time interval between two succes- sive slams is 7.7 sec, a value very close to the natural pitching period of 7.6 sec. Although periods shorter than the natural pitching period were observed between two wave encounters, no slamming was observed for these cases. Hence, it may safely be assumed that the natural pitching period is the minimum time interval between two successive slams. Figure 13 was prepared to verify that slamming is a sequence of events oc- curring in time following a Poisson process. In preparation of this figure, the number of slams occurring during 20 sec intervals was counted from the time history (Fig. 12), and the experimental frequency for each number was obtained. To determine the Poisson distribution curve, the expected value (mean) of slams for every 20 sec was computed from the frequency. By using this value (0.89), the Poisson distribution was obtained by the following formula: PROmt,. (23) Ss m- 1 s t e dt | F(m) mt y The constant m in the above equation was given in Eq. (21), and m is not always an integer. Hence, the denominator in Eq. (23) cannot be expressed by a practically usable formula. However, the integration can be evaluated as fol- lows: Let N.t = Z/2, and obtain the probability density function of a random variable Z. Then, the denominator of Eq. (23) is equivalent to SCRE ae f = a PN Baz, ZZ, (24) mZ where mZ, = 2mN,t,. The above integral is the probability integral of the incomplete gamma function and a table is available for this integration [9]. The integral values for various m, N,, and t, appropriate for full scale ships were taken from Ref. 9, and are shown in Fig. 16(a). The probability that a time T, or more, elapses before the next severe slam occurs can readily be obtained from Eq. (23). That is, 570 Prediction of Ship Slamming at Sea I'(m) T Prob {t > T} = ————______ (25) No -N_t | Ss elie e s dt I'(m) mt» The integral value of the numerator in the above equation for various m, N., and T, appropriate for full scale ships are given in Fig. 16(b). A numerical example of Eq. (23) will be given as follows: Consider the MARINER to be operating at light draft condition (40 percent of cargo loading) at a 10-knot speed in a severe Sea State 7. We will evaluate the probability density of the time interval between two severe slams for which an impact pres- sure of 50 psi or greater will be applied at the location 0.10 L aft of the forward perpendicular. In this case, we have 2C = 0.086 psi-sec 2/ft2, Pp, = 50 psi, SLC SPUR rors R’ = 605 ft? (see Table 2), R! = 305 ft?/sec? (see Table 2), N. = 0.0435 1/sec [by Eq. (6)], sie orSee 4.19 [by Eq. (21)]. 3 iH} 71 Ochi By using these values and Eq. (23) the time interval between two severe slams was evaluated, and the results are shown in Fig. 17. Included also in the figure is the experimentally obtained histogram. On the basis of the agreement between experimental and theoretical results, it is concluded that the time in- terval between two severe slams follows a truncated gamma probability law. Percent /Sec 120 Time in Sec Truncated at 31.8 Sec Fig. 17 - Sample histogram and the predicted probability density function for time interval between two severe slams (severe Sea State 7, ship speed 10 knots, light draft) APPLICATION OF THE PREDICTION METHOD TO THE DECK WETNESS PROBLEM Prediction of Probability of Occurrence of Deck Wetness The problem of probability of occurrence of deck wetness due to shipping of green water can be treated in a manner similar to that for slamming. However, two differences in the treatment of these phenomena must be considered. These are: (1) The bow emergence and threshold velocity are the required conditions leading to slamming, while the bow submergence is the condition leading to deck wetness. (2) The reference location along the ship length for which the proba- bility should be considered is 0.1 L aft of the forward perpendicular for slam- ming, and the forward perpendicular for deck wetness. Since deck wetness is caused by the green water flowing over the deck from the top of the stem, it is proper to consider the forward perpendicular as a reference point. Justification for selection of the reference point of 0.1 L aft of the forward perpendicular for slamming is given in Ref. 4. With the above two considerations, the probability of occurrence of deck wetness can be obtained from Eq. (5), by substituting D (freeboard at the forward perpendicular) for H (draft at Station 2), and by letting t=O ebhatas. 572 Prediction of Ship Slamming at Sea (26) ald Prob {Deck Wetness} = Prob {r > D} = e * Oo ll freeboard at FP, R’ = twice the variance of relative motion at FP. It is noted that R/{ in the above equation has a different value from that in Eq. (5), since the relative motion between wave and ship bow at the forward perpendicular is considered for this case. The number of occurrences of deck wetness per unit time, N,, is given by : D aie ates (27) No 2 ee eee WwW QT Re , Table 3 shows comparisons between predicted and observed probability of occurrence of deck wetness per cycle of wave encounter and number of deck wetnesses in a 30-minute operation of the MARINER in a moderate Sea State 7 at a 10-knot speed. Variance of the relative motion at the forward perpendicular used in the computation of the probability was evaluated by the method given in Appendix 1. Although satisfactory agreement between the predicted and observed values can be seen in Table 3 for full loading condition, agreement for moderate and light loading conditions is poor. However, this is not surprising since only 12 occurrences were observed for the moderate and 4 occurrences for the light load condition as compared to 34 occurrences for full draft condition. It is noted that a comparison of the predicted value with the observed value which was obtained from a small number of samples is not statistically proper. How- ever, the comparison is included in the table to provide some indication of how significantly the probability decreases with decrease of loading condition. It is of interest to discuss the effect of freeboard forward on the probability of occurrence of deck wetness. Newton, based on his experimental study on a destroyer-type vessel, concluded that the freeboard forward had a most impor- tant influence on the degree of wetness [10]. Newton's conclusion derived from tests in regular waves is valid in irregular waves also since the probability of occurrence of deck wetness decreases significantly with increase of freeboard forward [see Eq. (26)] and since the severity of wetness also decreases as will be seen later in Eq. (30). As a practical example of the effect of freeboard forward on the probability of occurrence of deck wetness per cycle of wave encounter, Fig. 18 was pre- pared. The figure shows the probability of deck wetness of the MARINER for various heights of freeboard forward. The probability was computed for a 10- knot speed in a moderate Sea State 7 for full load condition. The actual height of the freeboard forward on the MARINER is 36.7 feet. As can be seen in Fig. 18, if the freeboard were increased by 10 percent, the probability of deck 073 Ochi wetness would decrease by 32 percent. Conversely, if the freeboard were de- creased by 10 percent, the probability would increase by 42 percent. Table 3 Comparison of Predicted and Observed Probability and Number of Deck Wetnesses (MARINER) Wind velocity (knots) Wind duration (hours) = —=-— Significant wave — Ss O12 = =< height (ft) Course angle Freeboard forward (at FP) (ft) R! at FP (ft?) Number of deck wetnesses in a 30-minute operation a a A Prediction of Severity of Deck Wetness As was mentioned earlier, the pressure associated with slamming is of the impact type and is proportional to the square of the relative velocity between wave and ship bow at the instant of impact. The pressure associated with deck wetness, on the other hand, is not an impact type and approximately corresponds to a static pressure due to the head of water flowing over the deck. Thus in the 574 Prediction of Ship Slamming at Sea Probability Freeboard at FP in Fi. Fig. 18 - Effect of freeboard forward on the probability of occurrence of deck wetness (moderate Sea State 7, ship speed 10 knots, full draft) derivation of the probability density function for the pressure due to green water, the following conditions will be considered. These are: (1) magnitude of relative motion is greater than the freeboard forward (bow submergence condition) and (2) magnitude of peak pressure during one cycle of deck wetness is equal to the static water-head corresponding to the difference between the maximum value of relative motion and the freeboard forward. Now, the double amplitude distribution of the relative motion follows the Rayleigh probability law. Since deck wetness occurs only when the bow is sub- merging, the relative motion in one direction is taken instead of the peak-to- peak value. Then, analogous to Eq. (9), the probability density function of the amplitude of the relative motion r,, when r, is greater than the freeboard for- ward, D, is given by 1 Dee gn ace (28) EGpS) = R! e ’ it 2 D . uF It is convenient to express the above formula in terms of pressure units (psi). For this, let q, = r,/a and q, = D/a. Here, a = 2.32 ft/psiif r, and D are expressed in the foot-unit. Then, the probability density function given in Eq. (28) becomes 575 Ochi Qa2 R! (aig = al. ) (29) (Cl) = ar Gee - Since q, in the above equation is the pressure corresponding to the peak of the relative motion, and q, is that corresponding to the freeboard forward, the pressure due to the green water on the deck q, is the difference between them. Thus, the probability density function of the pressure due to green water can be derived from Eq. (29): 2 a Dee a a) (30) #(G) = R! (q+q,) e qr 0 Tt q = pressure due to green water on the deck (psi), w H ' = twice the variance of relative motion between wave and ship bow (ft?), Gx D/a (psi), D = freeboard at the ship bow (ft), a = constant = 2.32 (ft/psi). Equation (30) is essentially a truncated Rayleigh distribution. However the base line is shifted, so it may be considered as a modified Rayleigh distribution. The average of the one-third highest (significant), q, 73, and one-tenth high- est q,,,) pressures are given by the following formulae respectively: 2 2 . 8 Dee? = 2 9 . ar (aliparcr) ch) Ri Ri ds aD Gy a3 Gipy + 7 aie «: 1- 0 aie (Cle * Ge) Tt (31) where R! id oy E 1\_ Gaya = 7\f GE eS (108 2 | qx ®(u) - J e 2 a 576 Prediction of Ship Slamming at Sea a2 2 a = aa Cdyunoie ie) da) Ri aay le 2a2 Giyio = Wl ayjyye * ame ea 1-0 {7 (41/19 + 90] Tr where ‘ Petes 1 TAF te ay Ne log\io) - qs: The derivation of the above formulae is the same as that for the average of the one-third highest and one-tenth highest slamming pressures. Figure 19 shows a comparison of the theoretical probability density function of pressure experienced on deck due to green water with an experimental histo- gram. The experimental histogram was obtained from tests on the MARINER operating at a 10-knot speed in a moderate Sea State 7. Included also in the fig- ure are the averages of the one-third and one-tenth highest pressures. Another comparison between theory and experiment was made for a high speed research ship form and the result is shown in Fig. 20. This form is one of the Series 64 family having a block coefficient of 0.45. The freeboard at the Average of Average of 1/3 Highest I/10 Highest in PSI in PSI Predicted 7.2 10.6 Measured 59 oe Ree AN: Percent /PSI Pressure in PSI Fig. 19 - Histogram of pressure ex- petienced “ons the deck due to preen water (MARINER, moderate Sea State 7, ship speed 10 knots, full draft) 221-249 O - 66 - 38_ O77 Ochi Average of Average of 1/3 Highest I/\10 Highest in PSI 6.7 Meosured 5.7 6.4 Percent /PSI Pressure in PSI Fig. 20 - Histogram of pressure ex- perienced on deck due to green water (high speed research ship, Sea State 6, ship speed 20 knots, design draft) forward perpendicular is 23.7 ft. Tests were made in a head Sea State 6, at 20- knot ship speed [11]. (All values have been converted to those for a 400-ft ship.) In these tests, 36 deck wetnesses were observed in 236 wave encounters, hence the probability of deck wetness per cycle of wave encounter was 0.153. For computing the pressures by Eqs. (30) through (32), the variance of the relative motion was estimated from Eq. (26) by using the above probability. On the basis of the reasonable agreement between theory and experiment shown in Figs. 19 and 20, it may be concluded that the pressure associated with green water on the deck follows a modified Rayleigh probability law. CONCLUSIONS A theoretical study was made to predict the probability of occurrence and severity of ship slamming, and the time interval between successive slams in rough seas. The theory was also applied to the deck wetness problem. The theoretical results were compared with experimental results obtained from tests conducted on a 13-ft MARINER model. On the basis of the results of this study, the following conclusions are drawn: 1. The linear theory of superposition of ship motion in waves may be used to obtain realistic engineering estimates of frequency and intensity of slamming 578 Prediction of Ship Slamming at Sea and green water. For the MARINER, the predictions are valid at least up toa severe Sea State 7, ship speed 10 knots. 2. The conditions leading to ship slamming in rough seas are bow emer- gence and a certain magnitude of relative velocity between wave and ship bow (threshold velocity). It is considered appropriate to take 12 ft/sec as the threshold velocity for a 520-ft ship. For a ship of different length, the above given value should be modified according to the Froude scaling law. 3. Probability of occurrence of slamming decreases with increase of course angle from head seas because both the relative motion and relative ve- locity decrease with increasing course angle. The probability of occurrence of slamming decreases with increase of loading condition primarily because the probability of bow emergence significantly decreases with increasing draft. 4. Relative velocity between wave and ship bow at the instant of slamming follows a truncated Rayleigh probability law. Truncation should be made at the threshold velocity. 5. Impact pressure applied to a ship's forward bottom when slamming oc- curs follows a truncated exponential probability law. Truncation should be made for the pressure induced by the threshold velocity. The law appears to be valid for any course angle and loading condition. 6. Time interval between successive slams follows a truncated exponential probability law. Truncation should be made at the natural pitching period of the ship. 7. The time interval between two severe slams follows a truncated gamma probability law. 8. The probability of occurrence of deck wetness is simply the probability of bow submergence. It is an exponential function of relative motion between wave and ship bow and the freeboard forward. The probability decreases sig- nificantly with increase of freeboard forward. 9. Pressure associated with deck wetness follows a modified Rayleigh probability law. ACKNOWLEDGMENTS The author wishes to express his appreciation to Dr. W. E. Cummins for the valuable discussions and his encouragement received during the course of this project. Thanks are also due to Lt. Cdr. M. C. Davis (USN) for his techni- cal advice. The assistance of Mrs. S. R. Zoomstein and Mr. J. A. Kallio in carrying out the numerical calculations is gratefully acknowledged. 579 10. 11. Ochi REFERENCES Tick, L. J., ''Certain Probabilities Associated with Bow Submergence and Ship Slamming in Irregular Seas,"’ Journal of Ship Research, Vol. 2, No. 1 (1958) Szebehely, V. G. and Todd, M. A., "Ship Slamming in Head Seas,'' David Taylor Model Basin Report 913 (1955) Ochi, K., 'Model Experiments on Ship Strength and Slamming in Regular Waves," Transactions, Society of Naval Architects and Marine Engineers, Vol. 66 (1958) Ochi, M. K., "Extreme Behavior of a Ship in Rough Seas — Slamming and Shipping of Green Water,'' Paper to be presented before the Annual Meeting of the Society of Naval Architects and Marine Engineers (1964) Szebehely, V. G. and Lum, S. M., ''Model Experiments on Slamming of a Liberty Ship in Head Seas,'' David Taylor Model Basin Report 914 (1955) Rice, 8. O., 'Mathematical Analysis of Random Noise," Bell System Tech. Journal, Vol. 23 (1944) and Vol. 24 (1945) Cartwright, D. E., "On the Vertical Motions of a Ship in Sea Waves," Inter- national Shipbuilding Progress, Vol. 5, No. 52 (1958) Parzen, E., 'Stochastic Processes,'' Holden-Day, Inc., San Francisco, U.S.A. National Bureau of Standards, ''Handbook of Mathematical Functions,"' Ap- plied Mathematics Series 55, U.S. Department of Commerce (1964) Newton, R. N., 'Wetness Related to Freeboard and Flare,'' Transactions, Royal Institution of Naval Architects, Vol. 101 (1959) Sheehan, J. M., ''Model Tests of a Series 64 Hull Form in Regular, Irregu- lar, and Transient Waves," David Taylor Model Basin Report (In prepara- tion) Appendix 1 METHOD OF EVALUATION OF VARIANCES OF RELATIVE MOTION AND VELOCITY BETWEEN WAVE AND SHIP BOW The relative motion and velocity between wave and ship bow at a specific location along the ship length can be obtained from model experiments if an im- mersion sensing element is fixed to the model at the longitudinal position of in- terest. By this method, tests in regular waves provide the response amplitude 580 Prediction of Ship Slamming at Sea operator of relative motion at this location. Then, by applying the superposition principle, the energy spectra of the relative motion and the velocity and thereby the variances for a given sea state can be obtained. That is E PP es Cesc: C= a = 8 xcs dw, (A.1) P= 2 [a2 0(0,) do, where o” = variance of relative motion, ao” = variance of relative velocity, E. = cumulative energy density of relative motion, i.e., the area under the relative motion spectrum, ® (w,) = energy density of relative motion, w, = frequency. For a constant speed test it is possible to obtain the response amplitude operator of the relative motion by installation of an accelerometer in the model at the location of interest, and a wave-height probe on the carriage so that it is in line with the accelerometer. The above two methods are the direct methods for obtaining the relative motion and velocity at a specific location. It is necessary in practice, however, to evaluate the variances of relative motion and velocity at arbitrary points along the ship length for a given sea. For this, the response amplitude operators of relative motion at the points of interest may be evaluated from the pitch, heave, and wave motions including their respective phases. Another approximate method to estimate the variances of relative motion and velocity at arbitrary points is to use the correlation co- efficients if the variances of vertical motion and/or acceleration are known at two points along the ship length. The method is as follows: The variance of the relative motion at an arbitrary point along the ship length is given by oe tes a 15 oe = Dit er Oke (A.2) q Il E variance of relative motion between wave and ship bow at point x, oy = variance of wave motion, o, = variance of vertical motion at point x, Pw = correlation coefficient between wave and vertical motion at point x. 581 Ochi The variance of wave motion, c,?, is simply determined from the energy spectrum for a given sea state. Variance of vertical motion at an arbitrary point, X, can be evaluated by the following formulae if the variances of motion at two different points along the ship length, o? and o,” are known. D2) [RN x-b\ fa-x a-x\? 2 me (==?) as 204, (22) (2=*) vai3 | (2-3) sat Soe x,a,b = distances between points X, A, and B from the aft perpendicular (see Fig. 21), fo) II variance of vertical motion at point A, o, = variance of vertical motion at point B, correlation coefficient of vertical motion at two different points, A and B. Thus, the relative motion at arbitrary point along ship length can be ob- tained from Eqs. (A.2) and (A.3). However, two correlation coefficients, p,, and p,,, involved in these equations must be determined experimentally. The correlation coefficient, o,,, can be obtained by the following formula with the aid of auto and cross-spectral analysis of the vertical motions at points A and B. 2 2 — Yay Cie ide) = Clos aides) (A.4) PED oo Ts hee J ®,,(@,) do, J O,(w,) do, where Cip(@.) = energy density of cospectrum, i.e., energy density of the real part of the cross-spectrum of vertical motions at points A and B, Q.p(@.) = energy density of quadrature spectrum, i.e., energy density of the imaginary part of the cross-spectrum of vertical motions at points A and B, ®, 4(e) energy density of the auto-spectrum of vertical motion at point A, %,,(#,) = energy density of the auto-spectrum of vertical motion at point B. In the above formula, the definition of the variance and covariance given by St. Denis and Pierson was used. If the acceleration is measured instead of the vertical motion at one point (say, point A), Eq. (A.3) is still valid, since the ac- celeration spectrum can easily be converted to the motion spectrum. The fol- lowing relations are used in Eq. (A.4) in this case. 582 Prediction of Ship Slamming at Sea C.5( a) = ee 2 Ge (®,) We ab 1 Sh A Gete hai esa bu Ce) (A.5) We a 1 ® ale) = aa ®....(@,) . aw aa e The value of the correlation coefficient, »,,,, depends on the relative posi- tion of the two points A and B. As will be shown later in Table 4, if point A is located near the ship bow and point B is located near the midship, the correla- tion coefficient is very small for conditions severe for slamming. This means that the motions at these points (ship bow and midship) are statistically almost uncorrelated, and thereby the second term of Eq. (A.3) can be neglected prac- tically. The correlation coefficient, o,,, can be obtained by a formula similar to that for the coefficient o,,. That is, Pe ey CC) nO One.) can (A.6) OBS oe Cig ees e JO (@,) dw, J O,.(@.) do, C,x(®.) = energy density of cospectrum, i.e., energy density of the real part of the cross-spectrum of wave and vertical ship motion, where Qyx(@-) = energy density of quadrature spectrum, i.e., energy density of the imaginary part of the cross-spectrum of wave and vertical ship motion, ®,,(%.) = energy density of the auto-spectrum of wave, ®,,.(®,) = energy density of the auto-spectrum of motion. If the wave is measured not at the same location at which the bow motion is measured but at a certain distance ahead of the model (as is illustrated in Fig. 21), then the following phase correction due to the distance between wave probe and point X is required in the evaluation of the cross-spectrum _ ies (A.7) where ®,,(®,) = cross spectrum between wave and vertical ship motion at point x, 583 ®_ (#,) S) Ochi Fig. 21 - Explanatory sketch of distances a, b, x, etc. cross spectrum between wave and vertical ship motion measured at two different points, Ww and xX, respectively, distance between points W and Xx. If this correction is included, the correlation coefficient between wave and ship motion at points x becomes: where (So. (o,) cos (o- 0) dae) + (2 (a) Sitiod (er= 2?) do.) (A.8) J ®__(@,) dw, J Of) dae Ja (el = {e,,c@e)} + 12,,¢e)f energy density of cospectrum between wave and vertical ship motion measured at two different points, W and x, energy density of quadrature spectrum between wave and vertical ship motion measured at two different points W and x, Gaim © {2 (fc. (of, energy density of auto-spectrum of wave measured at point W, energy density of auto-spectrum of vertical ship motion meas- ured at point x, w*S/g, encounter period with wave = w + (V/g)w?, wave period, ship speed. 584 Prediction of Ship Slamming at Sea Table 4 Values of Correlation Coefficients (MARINER, Light Draft) eee Mila? | Moderate 7 Between 0.034 L aft of FP and CG Between 0.1 L aft 0.12 of FP and CG Correlation coefficient of vertical velocity, p ab Between 0.034 L 0.03 0.05 0.02 aft of FP and CG Between 0.1 L aft mas 0.05 of FP and CG Correlation coefficient of relative velocity between wave and ship, p.. .36 At 0.1 L aft of FP (0.42) 0.36 (0.40) (0.37) (0.37) Note: Values in parentheses are those estimated by the interpolation. In the case when acceleration is measured instead of vertical ship motion, a modification similar to that given in Eq. (A.5) is required. That is, ® (w,) = - a ®_..(@,) (A.9) wx w wx 585 Ochi where ®_ (w,) = energy density of cross-spectrum between wave and vertical bi ship motion, ®_.(w,) = energy density of cross-spectrum between wave and vertical eh acceleration. The variance of relative velocity between wave and ship bow can be obtained by the same procedure as that for the relative motion. Numerical examples o the evaluated correlation coefficients, o,,, p,, (for relative motion) and p. (for relative velocity) are tabulated in Table 4. These were evaluated fon “losin. results obtained on MARINER in Sea State 7. As can be seen in Table 4, the correlation coefficients ,, and p., are very Small in this case, since point A is located near the forward penpenanine and point B is located at the center of gravity. From this table, the coefficients required for evaluating the relative mo- tion and velocity at an arbitrary point along the ship length can be estimated by either interpolation or extrapolation. Appendix 2 DERIVATION OF THE AVERAGE OF THE HIGHEST ONE-THIRD AND HIGHEST ONE-TENTH VALUES FOR THE TRUNCATED RAY- LEIGH AND EXPONENTIAL PROBABILITY DENSITY FUNCTIONS (A) TRUNCATED RAYLEIGH PROBABILITY DENSITY FUNCTION It was mentioned in the text that the probability of the relative velocity be- tween wave and ship bow follows a truncated Rayleigh probability law. The probability density function in this case is given by Eq. (11) in the text. That is, (A.10) f(r) ane aa J 5 eo ie The average of the one-third highest values for this probability density function is evaluated as follows: Let r,,, be the lower limit of the one-third highest values of relative velocity. Then Prob {r > yet = { f(t) dr = =: (A.11) 586 Prediction of Ship Slamming at Sea From Eqs. (A.10) and (A.11) Ff) = eee late (A.12) Next, let the average of the one-third highest values be ty /3» 2nd consider their moment about the origin of the probability density function. Then +2 Ty, 1% b «tg Ll oe a ce ie R. 3 71/3 2) TED SE ie trae i} re dr _ SS w saa 4 a — w where é (oa ®(u) = — | ee ee (A.14) PH hs ee Thus 22 where r,,, is given in Eq. (A.12). The above equation gives the average of the one-third highest values of the relative velocity for the truncated Rayleigh distribution. Similarly, the average of the one-tenth highest of the relative velocity for the truncated Rayleigh distribution is given by fe: (T1/10) - Re res: 5 (A.16) Tiyio = We Fayag © * mR. 18 (av r where : : ; 1 A.17 Tagen = rot = i (10g =) : ( ) Ochi Suppose that the distribution is not truncated and that the double amplitude is considered instead of the single amplitude; then, +, = 0 and R! = 4£. (where E, = area under the spectrum for the relative velocity). In this case, we have from Eqs. (A.15) and (A.16) ita 2.83 VE. (A.18) fg = SECO Widh iP These are well known formulae for the averages of the one-third highest and one-tenth highest double amplitudes of the ordinary Rayleigh distribution. (B) TRUNCATED EXPONENTIAL PROBABILITY DENSITY FUNCTION As was given by Eq. (15) in the text, the truncated exponential probability density function may be expressed as RES ee 1 2CR: P-Py (A.19) where p = pressure = 2Cr?, p, = truncated pressure = 2c7,’, Riel, 2 r C = constant. Then, the lower limit of the one-third highest values, p,,,, can be obtained from the following relation: foo) PED DP > Day, = | f(p) dp = =. (A.20) Pi/3 Hence Pi73 = Px De 2CR: (102 5) O (A.21) Next, let the average of the one-third highest pressures be p,,,, and take the moment about the origin of the density function. That is, 588 Prediction of Ship Slamming at Sea foo) 1 ~ Bet ahs | p f(p) dp. (A. 22) P1173 From Eqs. (A.19), (A.21) and (A.22) the average of the highest one-third values becomes ae h 1 PD yaaa ert 2CR . (1 - log 4) HE (:2 + 2.10 R:) (A.23) Similarly, the average of the highest one-tenth values is ~ ' 1 Pi/i10 = Pp, + 2CR. (1 - log i) 2G (# + 3.30 R:) (A. 24) * * * DISCUSSION G. Aertssen University of Gent Gent, Belgium The first look at this paper gives the impression that it is a remarkable example of the truncated exponential probability law applied to the study of slamming and deck wetness from model results. Were it not that there is much more in it for the naval architect it would not have deserved much attention. There is a difficulty in carrying out slamming experiments on models be- cause the rigidity of the model cannot be easily scaled up to the rigidity of the ship. Giving the relation impact pressure, relative velocity, the author however gives —I think for the first time — the means to correlate his model results with full scale. His threshold velocity is 12 ft/sec and if I modify this value, ac- cording to the Froude scaling law, to a cargo ship of 480 ft I obtain a threshold velocity of 11.5 ft/sec which according to the author's relation transforms to our impact pressure of 11 psi. I am interested in this cargo ship of 480 ft be- cause last winter I made a westbound crossing of the North Atlantic in very se- vere weather on board such a ship which was instrumented by the Centre Belge de Recherches Navales. There were on board a shipborne wave recorder, strain gages, ship motion recorders, 2 pressure transducers in the keelplate, 589 Ochi etc. When weather was worsening, shocks were felt but the impacts on the fore- body did not induce any reaction among ship's officers until at a certain moment they mentioned in the log book: "le navire travaille et fatigue,'' the ship works and there is fatigue. At this moment the whipping stresses in the main deck stringerplate amidships were 0.5 t per sq in. and the impact pressure on the pressure transducer located at 0.15 Lpp from FP was about 10 psi. The ship was in nearly full-loaded condition and the location of the pressure transducer was not exactly the same as the location 0.1 Lpp indicated by Dr. Ochi. Unfortu- nately I have no impact data of this ship in light-loaded condition hitherto, but the nice agreement between the threshold of whipping stresses and impact pres- sure established on our cargo ship in nearly full-loaded condition and the threshold of velocity established by Dr. Ochi is certainly an encouragement to believe in his prediction of slamming from model results. This prediction of slamming is very well presented in Table 2 for a Mariner ship. Looking at the number of slams in a 30 minute operation there are ina moderate Sea State 7 only 12 slams in full-loaded against 60 in light-loaded condition in head waves and they are again reduced when the captain changes course 45 degrees. This might indeed give the picture of what happens on the bottom at the forebody and the danger of damage there. But modern cargo liners are longitudinally framed and often reinforced in the forebody beyond classifica- tion requirements, so today bottom damage is more seldom stated after a cross- ing in severe weather condition. Whipping stresses however are excited in the main girder and they might increase to a certain extent the longitudinal bending stresses and be a source of fatigue. Therefore I think that perhaps more than the number of slams these whipping stresses ought to be considered. At each slam there is a vibration in the ship main girder and an initial whipping stress. Summing up these initial whipping stresses for let us say again a 30 minute op- eration and dividing by the number of low cycle stress oscillations a slam num- ber is obtained which might as well give the intensity of the effect of slamming on the hull girder. Establishing this slam number, whipping stresses less than 0.4 t per sq in. were ignored. I had these whipping stresses measured in a sea state about the mild 7 Beaufort of Dr. Ochi's paper, once in light-loaded condi- tion on a cargo ship of 446 ft in waves ey, 1o = 27 ft at 12.5 knots, on another occasion in nearly full-loaded condition on a cargo ship of 480 ft in waves H,,/,) = 33 ft at 9 knots, and in this nearly full-loaded condition the whipping stresses and the slam number representing their intensity were larger than in light-loaded condition. In light-loaded condition the severe slams are heard like a gun shot whereas in full-loaded condition they are more like far-off thun- der. In light-loaded condition the slams are more conspicuous and captains are keen to reduce speed. That is perhaps one of the reasons why the slam number is not larger in light-loaded than in full-loaded condition. As long as not too much green water is shipped the captain of a full-loaded cargo ship goes ahead in high waves and modern cargo ships with a long forecastle and a fair fore freeboard maintain a good speed in these high waves. And here I should like to ask Dr. Ochi why he has taken the same speed of 10 knots for his comparison light-loaded and full-loaded? Has he any informa- tion as to what extent captains of Mariners accept 60 slams, i.e., 2 slams every minute, in light-loaded condition in waves of 31 ft significant height? As a rule captains of cargo ships of 10,000 tons deadweight and 16 knots service speed do not accept these waves at a speed of 10 knots, when in light-loaded condition. 590 Prediction of Ship Slamming at Sea DISCUSSION E. V. Lewis Webb Institute of Naval Architecture Glen Cove, Long Island, New York This paper is of particular significance because it attempts to establish criteria for the occurrence of slamming. Such criteria have been badly needed in connection with the calculation of ship performance in irregular seas by the method of superposition. The criteria will make possible, for example, the de- termination of the speed at which slamming would become serious — or the pre- diction of comparative slamming characteristics of alternative ship designs. It is hoped that for completeness the work will be continued to allow for the effect of section shape on the critical vertical velocity for slamming — and also to allow for the effect of form and fullness on the fore and aft location of the critical section. The equations for various probabilities in evaluating performance in irreg- ular seas will be very useful. It should be pointed out that the probabilities are based on assumed stationary conditions — constant ship speed and heading, as well as steady sea conditions. Hence, the equations must be used with caution. For in the case of the full-scale ship at sea, the shipmaster is certain to change course or speed if slamming becomes serious, so that conditions would not re- main stationary. Another point is in regard to the assumption in the paper that the pressure of water on deck is purely static. It would be expected that there would be con- siderable dynamic effect associated with the aftward velocity of the water. * * * DISCUSSION W. A. Swaan Netherlands Ship Model Basin Wageningen, Netherlands In the course of the last 10 years the possibilities of applying the super- position theory or the problem of ship motions in irregular seas have covered an increasing range of phenomena. At first only ship motions were-considered, subsequently the superposition methods for resistance and power were evalu- ated and checked by experiments. The results presented in this paper here cover the final gap, that is the relative motions at the bow with the associated 591 Ochi problems of slamming and wetness. The test results leave no doubt about the possibility to apply these methods to ship predictions from now on with full con- fidence. In his explanation about the basic concept the author distinguished two problems; that is the prediction of the probability of slamming per cycle and the prediction of the number of cycles per unit time. I would like to make a minor remark on both points. In Appendix 1 it is mentioned that it is possible to determine the relative motion at the bow using an accelerometer on the model and a wave probe in front of it. This seems to be a method containing some uncertainties. In the first place it will be necessary to know the smooth water level at the station which is considered critical for slamming. In Eq. (5) of the paper this is assumed to be equivalent to the draft. This may be true for a vessel like the 'Mariner" at a speed of only 10 knots but at higher Froude numbers a significant difference may be found because of the smooth water bow wave system of the ship. The bow wave of a ship usually de- creases the probability of slamming and increases the wetness. The second objection against the use of a wave height transducer in front of the model is that the bow of a pitching and heaving ship creates an oscillating bow wave which will affect the variance of the relative motions. This will be somewhat less important for fine ships than for full ships. It is therefore concluded that the only reliable way to measure the relative motion is to do so at the critical station which is used for the determination of the probability of slamming per cycle. The second remark concerns the use of the second moment of the spectrum in order to obtain the expected number of zero upcrossings. Our experience indicates that the quotient of the spectrum area and the first moment gives a better approximation to the number of zero upcrossings. This is only of impor- tance when the spectrum is not narrow because otherwise the two methods yield the same result. The sea spectrum, however, is not always narrow, for instance when it is desired to simulate a Neumann spectrum which has a width of E = 0.815. Using the first moment of the spectrum will result in the prediction of less slams per unit time as is shown in the Table 1 where results are shown from some relative bow motion and wave height measurements. The width of the spectrum was estimated with the quotient of the number of maxima and the number of zero upcrossings. According to the results in Table1, Eq. (7) from the paper is more accurate in predicting the number of maxima than in predicting the number of zero upcrossings. 592 Prediction of Ship Slamming at Sea Table 1 Zero Up- Maxima Crossings per per Unit Time Unit Time (%) Relative Motion 103 105 102 101 * * * DISCUSSION L. Vassilopoulos Massachusetts Institute of Technology Cambridge, Massachusetts For those involved in seakeeping research the present paper is a very wel- come contribution for it deals with the two most important phenomena that dic- tate the speed which a high-powered fine ship can sustain in rough water opera- tion, namely slamming and wetness occurrence. At the same time the probabilistic methods presented and verified in Dr. Ochi's paper provide further useful tools for a realistic evaluation of the importance of seaworthiness in ship design. Although the author's analysis and verification was performed only for a Mariner model at moderate speeds, there is no reason to believe that a similar approach would be invalid for other conventional ship forms and at slightly more severe conditions. Of particular interest are the conclusions with regard to the actual mechanisms of slamming and wetness phenomena and the necessary and sufficient conditions which must prevail for their occurrence. It is particularly 221-249 O - 66 - 39 593 Ochi encouraging that in the case of slamming the number of critical factors has been reduced from three in regular seas to two in irregular seas. This favorable result overcomes otherwise unsurmountable calculation difficulties. The formula that Dr. Ochi has developed for the probability of slamming rests on the assumption that the relative motion of an arbitrary ship point is a narrow-band Gaussian process. Although the satisfactory correlation of meas- ured and predicted results which Dr. Ochi shows suggests that this appears to be the case, it must be stressed that one cannot a priori assume that the relative motion will indeed be a narrow-band process because the wave motion is not always a narrow-band process, except perhaps for severe sea conditions. Fur- ther, the sum of two narrow-band processes need not necessarily be a narrow- band process itself. Any absolute ship response, however, such as bow motion for example, can safely be regarded as a narrow-band process since the wave motion is mostly wide-band and the ship-system is strongly resonant. The next step in Dr. Ochi's analysis follows the approach employed in other engineering fields in that attention is focused on the envelope of the time function rather than its amplitude. In this connection I would like to point out that Eq. (2) can indeed be regarded as the definition of the envelope and which, stated other- wise, essentially regards |r,(t)| as the instantaneous radius of the image point on the phase plane diagram of Fig. 3(b). Dealing with the envelope rather than with the actual amplitude turns out to be very convenient for we can immediately obtain a closed form expression for the probability of slamming, such as Eq. (5). I cannot precisely follow the steps leading to (5), but I assume that Dr. Ochi multiplies the integrated Rayleigh probability density functions for the relative motion and the relative velocity. This is, of course, permissible since both processes are Gaussian and hence linearly as well as statistically independent. The author employs the nomenclature "probability of slamming per cycle of wave encounter.'' For a narrow-band process one may perhaps speak of cycles in an extended sense and even then the precise meaning of cycle is not very clear. But for a wide-band process, like the wave motion record, is it really possible to identify a cycle of wave encounter ? Also, Fig. 3 seems to indicate that slamming only occurs when r = H and r > r*. Is it not more correct to say that slamming can occur as long as r > H and provided that the relative ve- locity has assumed at least its threshold value ? The paper deals with the wetness problem in a similar and more simplified way and thus provides prediction methods for the propeller emergence problem also. The author has obtained a fascinating result with regard to the distribu- tion of slamming occurrences. It seems to me that utilization of the exponential distribution of the time intervals between slams together with the expected num- ber of slams per unit time as developed by Tick can be used to provide an an- swer in a statistical sense of the average sustained speed for a given ship. Has the author perhaps examined whether the wetness phenomenon is also a sequence of events which are Poisson distributed ? In conclusion, I would like to raise one further point which was so strongly mentioned by Professor Weinblum in his paper presented during the First 594 Prediction of Ship Slamming at Sea Symposium on Naval Hydrodynamics ten years ago: Is it not true that the time has come for a scientific evaluation of the freeboard problem of a ship on the basis of wetness considerations? It would seem that Dr. Ochi's paper as well as that of Mr. Goodrich in this Symposium both provide essential evidence that we are properly equipped to undertake such an investigation. * * * REPLY TO THE DISCUSSION Michel K. Ochi David Taylor Model Basin Washington, D.C. Professor Lewis mentioned that the probabilities presented in this paper are based on assumed stationary conditions, i.e., ship speed, heading as well as sea conditions are constant. The assumption of stationary conditions, however, is considered to be a proper approach in the analysis. Since voluntary reduc- tion of speed or change of course angle are entirely dependent on the personal judgment of ship operators, it is appropriate not to include human elements in establishing the statistical rules. He also discussed that the aftward velocity of the green water would have a considerable dynamic effect on the pressure on the deck. The pressure on the deck reported in this paper is the vertical component of green water flowing over the deck from the top of the stem. Pressure records obtained in the ex- periments have shown that pressure normal to the deck is not an impact type and that the pressure magnitude approximately corresponds to the static water head experienced at the stem. Judging from these results there is no reason to believe that consideration of the dynamic effect of the aftward velocity is neces- sary for the vertical pressure on the deck. This consideration is of course necessary for the horizontal component (aftward direction) of pressure on the deck, since the green water would crash at the front face of the deck super- structure with considerable velocity. Mr. Swaan remarked that the bow wave of a ship usually decreases the probability of slamming and increases the wetness. For this reason, he said most reliable way to obtain the relative motion is to measure it at the location considered. Consideration will be given to his remarks in future experiments by the author. Professor Aertssen asked why the same speed of 10 knots was used for comparison of frequency of occurrence of slamming for light and full draft *See discussion by Pierson to paper by Ogilvie and discussion by Tick to paper by Cummins and Smith. 595 Ochi conditions. This is due to the following reason: that is, if different speeds are used for comparison, two factors (speed and loading condition) both of which significantly affect the frequency of occurrence of slamming are involved in the results, and hence we cannot identify which factor had the greatest effect on the frequency. For example, the result of full scale trials introduced by Professor Aertssen shows that the slam number for full loading condition is higher than that for light loading condition. However, we cannot conclude from this result that full loading is more severe than the light loading, since the speed was higher for the full load than for the light load condition. It is also noted that the slam number as defined by Professor Aertssen is expressed in terms of whip- ping stress. This automatically includes the ship mass effect. In other words, even if the ship motions are the same for two different drafts, whipping stresses are quite different since the dynamic characteristics are entirely different. Thus, we cannot identify which factor increased the slam number for full load- ing. Thus, in order to obtain the effect of loading condition the same speed was used for evaluating the frequency of occurrence of slamming for light and full draft so that the difference in slamming rate could be attributed to the differ- ence of ship motion characteristics. Mr. Vassilopoulos questioned whether or not the relative motion between wave and ship bow is a narrow-band Gaussian process. It cannot be said, of course, that the relative motion is a sharp narrow-band Gaussian process as is frequently observed in strongly resonant vibratory systems. However, the fol- lowing table may provide some information on this subject. Domain of Significant Expected Frequency for Energy in the Ob- Narrow-Band Gaussian Process, /RVR™ served Spectrum of Relative Motion Severe 7 i 0.68 to 0.78 Moderate 7 : 0.65 to 0.75 Mild 7 : 0.68 to 0.75 The above table pertains to a ship speed of 10 knots and light draft condi- tion. Since the expected frequencies lie in the domains of significant energy in the observed spectra, it may be said that the relative motion can be treated as a narrow-band Gaussian process. Mr. Vassilopoulos pointed out that the condition r > H be used instead of r = H in Eq. (5) of the paper. Although the final result is the same for both conditions, r > H is the correct expression. The author agrees with Mr. Vassilopoulos' opinion that the deck wetness condition should be considered in the freeboard requirement. The author would like to continue further studies of the effect of section shape on the magnitude of threshold velocity as was suggested by Professor Lewis, although the values obtained on five different ships have shown fairly consistent values. 096 THE INFLUENCE OF FREEBOARD ON WETNESS G. J. Goodrich National Physical Laboratory Teddington, England ABSTRACT Model experiments in regular waves and probability theory have been used to predict the probability of occurrence of wetness at the fore- end of a ship of given type. Calculations made for ships of different fullness have suggested that the frequency of occurrence of wetness varies with block coefficient as well as with length for a given free- board ratio. INTRODUCTION The prediction of the probability of occurrence of wetness from model ex- periments in regular waves has been attempted by Newton [1] using statistical sea data to represent full scale conditions. Newton's work suggested that for a given freeboard ratio a 200 ft ship would be drier than say a 400 ft ship under North Atlantic conditions. This general conclusion seemed contrary to what would be expected and consideration was given to the possibility of using model data and probability theory to predict the probability of occurrence of wetness for ships of different fullness and length. The intention of the present paper is not to provide detailed design informa- tion but to indicate a method of analysis which could be used for specific design studies and to show the trend of the variation of wetness with ship length and block coefficient. WETNESS DEFINITION When considering the prediction of the probability of occurrence of wetness it is sufficient to say that if the motion of the bow relative to the water surface is such that the water rises above the deck level at the fore end, then the prob- ability of wetness exists. No attempt is made to say how wet the deck will be, nor to what height the water will rise above the deck. 597 Goodrich Fig. la - Response curves for constant A/L 0.70C, ; "20 FROUDE NUMBER 12 A. Fig. lb - Response curves for constant speed 0.70C, MODEL DATA The most systematic model data available at present are those of Vossers and Swaan [2] and these have been used in the present analysis. Measurements were made of the relative bow motions of a series of models and the response curves presented as the ratio of the relative bow motion to wave height on a base of block coefficient and for a range of speeds. Cross curves have been derived of the relative bow motion to wave height ratio for constant wave lengths to a base of Froude Number. Some account has been taken of the loss in speed due to wave action by assuming a loss in speed curve for each model. The responses have then been obtained from the cross curves for the speed corresponding to the particular Beaufort scale being considered. Typical response curves are given in Figs. la and 1b for the 0.70 c, form. 598 The Influence of Freeboard on Wetness 40 30 20 H SIGNIFICANT WAVE HEIGHT (Fe) re) 10 20 30 40 50 W WIND SPEED (tenots ) Fig. 2 - Significant wave height vs wind speed REPRESENTATION OF THE SEA Sea spectra are needed in the analysis in order to obtain motion response spectra and a modified form of the Darbyshire formulation has been used. The curve of significant wave height against wind speed shown in Fig. 2 was used and the three Darbyshire spectra are shown in Fig. 3. The equation of the Darbyshire spectrum is ‘ 1/2 Hs) (Gi ff) ( dif = 23.9 exp) = | aces GES ae OO se = SH, , jan} iS) I ae) eS i spectral ordinate, Hh ll frequency, h Il frequency of the peak value of the spectrum, 1.65H. 2) Il This latter value of H,,, is that derived by Darbyshire from his analysis. The spectrum in this form cannot be combined directly with response operators which are expressed in terms of wave length to ship length ratios, nor in fre- quencies of encounter. If the response curves for one ship speed and for varying 399 Goodrich 8,000 7,000 6,000 5,000 BEAUFORT 9 BEAUFORT 7 2,000 BEAUFORT 5 1,000 0-02 0:04 0-06 0:08 0:10 0-12 0-14 Fig. 3 - Sea spectra used in the analysis wave length are used they can be combined with a spectrum transformed from the frequency base to a wave length base. The transformation is: 2 [r(d)]? = BE 1 /fk. ee aS) df Wa Oy Oa df \H and includes the change from the energy expressed in terms of wave height, to the energy in terms of wave amplitude. It must be appreciated that although a unique curve of significant wave height versus wind speed has been used, wide variations of wave height exist in practice for a given wind speed. It is assumed that using this ''mean curve" and deriving the resulting response spectra results in mean values of the root mean square response for a given wind speed or Beaufort Number. 600 The Influence of Freeboard on Wetness METHOD OF ANALYSIS A number of gross assumptions have been made in the analysis as follows: (a) It has been assumed that for the extreme motions the conditions remain linear. The model experiments were carried out for a constant height-ship length ratio of 1/50. (b) It has been assumed that the motion is regular about the mean still water draught of the ship. (c) The head sea case only has been considered with no spreading of the wave spectra. (d) For comparative purposes it has been assumed that the ships are in the head sea condition 100% of the time. Other assumptions made in the analysis will be stated later. By combining the response curves such as in Fig. 1 with the sea spectra given in Fig. 3, the response spectra are obtained and by integration of these spectra, the mean Square response is derived Seer en le@oll ae The derived curves of root mean square response amplitude S_ for a range of Beaufort numbers are shown in Fig. 4 for 0.70 C, ships of 200, 400 and 600 ft lengths. (e) 2 Gr 6 8 10 BEAUFORT NUMBER Fig. 4 - Root mean square response for constant ship lengths 0.70 C, 601 Goodrich It has been assumed that the short term distribution of the variation of rela- tive vertical motion of the bow will have a Rayleigh distribution. With this dis- tribution the probability of exceeding a specific value of relative bow motion S, is 2 2 -S; /S., e In order to obtain the long-term distribution of S, a weighting factor for weather distribution must be included. As was stated earlier no weighting factor has been included in this analysis to take account of variations in the sea direction. The probability of exceeding a specific value of S, is therefore: : 2 Que die CSE) SIP 5 J J where P. is the weighting factor for the general weather probability distribution. The weather distribution used is given below over the range of weather groups 1 to 5. Group Beaufort Number Distribution % 1 0-3 52.0 2 4-5 29.0 3 6-7 15.0 4 8-9 3.9 5 10-11 0.5 The mean value of S,, for each group has been used in the calculation of Q,, with values of S,; of 10, 20 and 30 ft for all lengths of ships. From the calculated values of Q; for specific values of S,; probability curves can be drawn such as in Fig. 5. If freeboard at the fore perpendicular is substituted for S, then these curves show the probability of the water rising above the freeboard. A non- dimensional freeboard ratio can be used, (defined as the ratio of the freeboard at the fore perpendicular to the ship length) rather than absolute freeboard and the results for the 0.60, 0.70 and 0.80 C, ships are given in terms of this ratio in Figs. 6, 7 and 8. The curves for the 0.80 Cy, ships include lengths of up to 1000 ft since there is a growing interest in the behaviour of bulk cargo carriers of such lengths. Figures 9, 10 and 11 show the freeboard ratio required for various ship lengths for equal probability of wetness. DISCUSSION OF RESULTS The results show that for equal probability of occurrence the freeboard ratio decreases with increasing ship length. The results for the 0.60 and 0.80 C, ships are Similar but the analysis shows that the 0.70 C, ships require a greater free- board. This result is a direct consequence of the higher responses obtained for 602 30 (F*) 1 Oo FREEBOARD fe) 0-001 % 0-14 O12 0:10 0:08 0:06 FREEBOARD RATIO 0:04 0:02 The Influence of Freeboard on Wetness oe 6 % Gre. a = 6 Ro) (e) Ax lex 4 fe) Ke 0:01% 0-1% 1% iG Ye PROBABILITY Fig. 5 - Freeboard vs probability of wetness for constant ship lengths 0.70 c, 0-01% O1% 1% 10% % PROBABILITY Fig. 6 - Freeboard ratio vs probability of wetness for constant ship lengths 0.60Cc, 603 100% 100% FREEBOARD RATIO FREEBOARD RATIO 0-12 0:08 0:06 0:04 0:02 . 0:001% 0-14 O12 010 0-08 0:06 0:04 0:02 ) 0-001% Goodrich 0-:01% O-1% 1% 10% Ye PROBABILITY Fig. 7 - Freeboard ratio vs probability of wetness for constant ship lengths 0.70 Cc, hed s 5 (@) aw: L +00 as Le 800 be Ls 1,009 0:01% 01% 1% 10% Yo PROBABILITY Fig. 8 - Freeboard ratio vs probability of wetness for constant ship lengths 0.80 c, 604 100% 100% The Influence of Freeboard on Wetness O:12 0:12 0:10 0:10 0:08 0:08 oO Q — E q ao oe 0:06 Q 0:06 Q oc a < < O 1@) a ry ul uJ WwW rw O04 fk 0-04 ite 0-02 0:02 fe) (e) fe) 200 400 600 fe) 200 +00 600 SHIP LENGTH (Fe) SHIP LENGTH (Ft) Fig. 9 - Curves of freeboard Fig. 10- Curves of freeboard ratio for constant probability ratio for constant probability of wetness 0.60 C, of wetness 0.70 c, the 0.70 C, model tests. In Fig. 11, the slope of the lines of freeboard ratio for constant probability of occurrence of wetness indicate that for ship lengths in excess of 600 ft a constant freeboard gives equal probability. The question arises as to what is an acceptable level of probability of wet- ness. At this stage it is difficult to say what is acceptable but ships which are known to be good sea ships could be plotted in the diagrams in order to see what level of probability would be expected for them. It is the intention to run models of the 0.60, 0.70 and 0.80 block coefficient in irregular wave systems to check the number of times wetness occurs in a given train of waves. The system of generating irregular waves in the Ship Division's No. 3 Tank is such that the scale of the spectrum is easily modified. A constant length model can therefore be used with varying scale of spectrum to simulate different ship lengths. ACKNOWLEDGMENT This work has been carried out as part of the research programme of the National Physical Laboratory, and the paper is published by permission of the Director of the Laboratory. 605 Goodrich 0-10 0:08 0-06 FREEBOARD RATIO 0-04 0-02 ° 200 400 600 800 1,000 SHIP LENGTH (Fe) Fig. 11 - Curves of freeboard ratio for constant probability of wetness 0.80 Cc, REFERENCES 1. Newton, R. N., ''Wetness Related to Freeboard and Flare," Royal Institution of Naval Architects, Vol. 102, 1960. 2. Vossers, G., Swaan, W. A., and Rijken, H., Experiments with Series 60 Models in Waves," Society of Naval Architects & Marine Engineers, Vol. 68, 1960. DISCUSSION E. V. Lewis Webb Institute of Naval Architecture Glen Cove, Long Island, New York This paper shows how available techniques for predicting ship behavior in any particular sea condition—as described in my own paper—can be significantly extended by considering representative sea spectra of different levels of severity. Then, with the help of probability theory, long-term predictions can be made of quantities such as frequency of deck immersion forward. This approach provides a rational basis for establishing standards of bow freeboard. 606 The Influence of Freeboard on Wetness One question arises regarding ship speeds in the calculations. It would be of interest to know what speeds were assumed for each ship and each sea spec- trum, since the wetness certainly depends on speed. * * * DISCUSSION R. F. Lofft Admiralty Experimental Works Gasport, England As one who was concerned with Newton's original paper on wetness, Iam pleased to see this work being developed and extended in Goodrich's paper. Both papers point to the need for more wave data, and the need for caution in interpreting results based on present sparse data. In Newton's paper, the wave information was taken from Darbyshire's tables of frequency of occurrence of waves of given length and height, published in 1955. These were the dominant waves, and shorter or longer waves which were present simultaneously were ignored. This may account for some empha- sis On waves around 500-700 ft long, and so to an underestimate of wetness of smaller ships, in particular. On the other hand, the Darbyshire spectra on which Goodrich's work is based, relates specifically to local wind-generated seas, and excludes swell waves, which may affect larger ships. This paper therefore may give a some- what optimistic picture of the wetness of the longer ships, as in Fig. 11. Clearly we cannot obtain reliable estimates of wetness until more complete and reliable data are available on sea spectra and their frequency of occurrence. It should be pointed out that the 'wetness'' derived by Goodrich corresponds approximately to the very wet condition as defined by Newton. It is not uncom- mon for ships to be under spray, i.e., Newton's wet condition, without the bow becoming immersed. Finally, plottings of the form of Figs. 9-11 are purely comparative. It means nothing to the mariner, or to the designer, to be told that a particular ship has a probability of wetness of 0.01%. Studies of this nature must be asso- ciated closely with sea experience to be meaningful. If plottings of this type were prepared for existing ships of known good or bad reputations for wetness, as advocated by Newton, then perhaps an equivalence could be established be- tween the estimated probability of wetness and a degree of wetness which could be regarded as acceptable in practice. 607 > he 5 ‘ ¢ Lua = : VSR ‘ pase ae Pe Fa pe fini jes ‘ Glee ne erdoac | "i of. hau fi “enmity ind sald af shonge cube sill tle esette ac =9og@ B68 dane birs ide doeeal hayes & Siow ubougqe leiw Rage ao phnawoh yinisi199 zaoniow wnt ‘ ; ve he 4 o 4 * OA ¥ « e Re Ba i a \ A the ; Par, ; kel ‘ 3 ii ; P iw my > j 4 ‘ : 42 i ae ; : . - ‘ A 7 ic PAY ® i . , G 4 ! d e vf i one ; or . “ . “ ionod vl } By i et oe reideyd . at So folic hei? a Sy aG , Y bee ~ rte , , U S; f 4 _ : { +, = —— as y 5 ~ = mur? t ry Ae - f PETAR) : Ties fi Ae » » ’ | W. Suey rin Pigs } j 1 ’ de { ie 35% ; a f , i i i a 2 4 i ‘ rf A wast Ue ye ry 4 \ — ae ve str : 4 + ay é bi Ae 1 ' . f * 71h an 5 ' f i] * / ur ' ‘ a) a ae WAY ’ | Ce tee se. ne P.2 . a anal Pe Ate Wi Friday, September 11, 1964 Afternoon Session SHIP MOTIONS Chairman: C. Falkemo Chalmers Tekniska Hogskola Goteborg, Sweden Page Hydrofoil Motions in a Random Seaway 611 B. V. Davis and G. L. Oates, De Havilland Aircraft of Canada, Limited, Downsview, Ontario, Canada The Behavior of a Ground Effect Machine Over Smooth Water and Over Waves 691 W. A. Swaan and R. Wahab, Netherlands Ship Model Basin, Wageningen, Netherlands Behavior of Unusual Ship Forms 717 E. M. Uram and E. Numata, Stevens Institute of Technology, Hoboken, New Jersey A Survey of Ship Motion Stabilization 747 Alfred J. Giddings, Bureau of Ships, Washington, D.C., and Raymond Wermter, David Taylor Model Basin, Washington, D.C. A Vortex Theory for the Maneuvering Ship 815 Roger Brard, Bassin d'Essais des Carenes de la Marine, Paris, France 221-249 O - 66 - 40 609 \ , Poona TA ‘ ™ “wee? _ ot 4 | \ ¥ ? AA iist4 | en : f ttatD 7 7 sri \ ° + sia 4 ta - . if : | | a s ' Kes aT | ~ ic a0 fy UT iY) a | ¥ + cf 6 hs ~ Ag’ ’ ' ' » i 7 is { ; * uly ~ | | ‘ A . : ™ ™ - } , ; 715+ saa] itz J Ler D. | a k Pe ie iy ar < ‘ cya uM » aye ri ery i y or ' t (wor ef ' . | 4 = ps bh aineoh ee bs | voice y 3 HYDROFOIL MOTIONS IN A RANDOM SEAWAY B. V. Davis and G. L. Oates De Havilland Aircraft of Canada, Limited Downsview, Ontario, Canada INTRODUCTION This paper outlines the analog simulations and the complementary model test programmes conducted by De Havilland (Canada) during the design of the 200 ton FHE-400 Hydrofoil Ship for the Royal Canadian Navy. The equations of motion required to describe the motions of the hydrofoil are discussed in detail, together with the simulation of the equations and the seaway forcing functions. The model trials are also discussed and it is demon- strated that good correlation has been achieved between predicted and actual behaviour of a 1/4 scale model of the FHE-400 and between simulated and actual seaways. The achievement of satisfactory dynamic stability requires an iterative design procedure similar to that followed in aircraft design, first to establish steady-state requirements and then to examine the dynamic behaviour. When examining the hydrofoil system in a seaway, it is necessary to consider hydro- dynamic and structural requirements in order to develop a balanced and practi- cal design. This is illustrated in Fig. 1. Initial studies can be carried out using simplified equations with calculated derivatives, as only "broad" outlines are required. Subsequent studies have to be performed in greater detail as more accurate information becomes available from calculations and model trials data. The initial studies should show up any major shortcomings in the design. Some modifications are likely to result from the initial simulations. Once a reasonable foil configuration has been derived, then extensive model trials should be conducted and the results used for further and more accurate dynamic stability studies. Sophisticated equations are then required to take account of all significant nonlinearities. Because of the complex nature of both the ship and random seaway simula- tion, model trials are necessary to verify theoretical predictions. While towing tank trials of foil units are necessary to measure resistance and to provide foil derivatives, it is even more important to evaluate "seagoing'' models, prefer- ably manned, in order to measure response ina scale seaway. By comparing measured response with the mathematical model, the validity of the simulation can be established. 611 Davis and Oates HISTORICAL NOTE The design study and stability analysis reviewed in this paper commenced in October 1960 and has led to the current contract to design and build a 200 ton development prototype ship known as the FHE-400, for the Royal Canadian Navy. The initiative came from the Canadian Defence Research Board, following many years of surface piercing foil system development at the Naval Research Establishment, Halifax, Nova Scotia. In 1959 N.R.E. published a report which considered the feasibility of a 200 ton ship based upon a canard arrangement, of fixed surface piercing foils. N.R.E. recognized the advantages of a canard arrangement in reducing head sea accelerations and improving stability in following seas. In addition, they fore- saw the need to develop a foil design method to provide optimum foil angle of attack range in high sea states. Further, N.R.E. emphasized the value of de- signing a foil system to provide maximum damping in the hullborne mode of op- eration which is particularly important in a military search mode. Encouraged by the technical interest of other NATO navies, the Canadian Government agreed with N.R.E.'s contention that a thorough design study should be made and awarded a contract to De Havilland (Canada) in 1960. The work statement drawn up by the Defence Research Board in consultation with the R.C.N., laid down the parameters to be considered. These included the development of design methods for foils, response characteristics in random seas and the performance to be achieved. N.R.E. supported the programme with their 3-1/2 ton experimental test craft and an experienced trials team to con- duct sea trials of the foil system developed by De Havilland. The trials con- ducted from 1961 to date have substantiated the predictions made by N.R.E. in 1959. This paper discusses the design and stability studies and the supporting N.R.E. trials of the RX craft fitted with a representative foil system, 1/4 scale full size. DESIGN METHOD As there are many factors to be considered in relation to the dynamic sta- bility it is helpful to have a clear picture of the relation of this study to the other design parameters. Once the basic role has been decided upon, the required performance, range, load carrying capacity and approximate craft size can be determined; the latter of course, will be dictated to some extent by the sea state in which the craft will have to operate, as the hull will have to clear all but the larger waves. Parametric studies have to be carried out to determine the optimum configura- tion and size to meet the design requirements. These parameters then dictate the foil areas that are necessary to support the craft throughout the required foilborne speed range. Foil section thicknesses and section types are dictated 612 Hydrofoil Motions in a Random Seaway by the maximum design speed and by structural stiffness. In this respect there is some conflict between hydrodynamic requirements for the thinnest possible foil section, to avoid cavitation, and structural requirements for the thickest possible section to avoid divergence and flutter. In some instances the maxi- mum speed may well be decided by stiffness of the foil elements, as sections below a certain thickness may suffer from hydroelastic problems. This mini- mum thickness may not be sufficiently low to allow cavitation free operation at the maximum design speed and a physical limit will be placed on the maximum attainable speed. Stability is also adversely affected by cavitation. Foil loads, however, are effectively limited by cavitation, which is beneficial in this respect. When the hydrodynamics, hydrostatics, hydroelastics, structural integrity, power and machinery requirement, operational roles, accommodation spaces, etc., have been considered then the initial stages in the design of a practical hydrofoil craft will have been completed. At this stage the dynamic stability and the operational environment of the craft have to be considered in some de- tail. Foilborne seakeeping in rough water is of paramount importance since the craft must be stable under all sea conditions and must have acceptable response characteristics from the standpoint of human tolerance to motion. Some factors influencing craft motions are foil taper ratios (for surface piercing foils), rate of change of lift with angle of attack and rate of change of lift with immersion depth. The foil system should be insensitive to angle of attack changes (i.e., low C,,) to reduce the effect of wave orbital velocities but should be relatively sensitive to changes in immersion depth (C,,) to control foil broaching and hull slamming. The ideal response would be with the craft platforming all waves below those which would cause broaching or slamming and contouring all larger waves. In practice this ideal is not attainable and the craft motions are between platforming and contouring for all significant waves. To obtain the above characteristics some compromise is necessary. A low C,, usually implies a low aspect ratio (Fig. 2) and this yields a low lift-drag ratio which is detrimental to performance. For maximum performance the foil system should have the highest possible L/D ratio. A good compromise in this respect can be achieved with a canard system, in which 80-90% of the total lift is provided by the main foil. The bow foil supplies only 10-20% of the total lift; therefore its L/D ratio can be relatively low without contributing an unaccept- ably high drag to the total. Thus the bow foil can be optimised to produce mini- mum motions resulting in relatively small angle of attack excursions at the main foil. The main foil can then be designed to have a high aspect ratio (low drag) without incurring unacceptably high accelerations at the craft c.g. In practice the main foil 0C,/da and 0C,/och lift-curve slopes have to be optimised to produce satisfactory performance in both head and following seas. However, the values so obtained do not differ greatly from those desirable for best per- formance. The bow foil unit is optimised to produce minimum motions in a seaway and to a large extent, controls the natural frequency of the craft in pitch and gives adequate separation between the craft natural frequency and the domi- nant frequencies of encounter in head seas which produce significant inputs of energy to the craft (Figs. 3, 4, 19 and 20). It is considered that fully cavitating bow foil sections are necessary to provide the required characteristics in a surface piercing system. These sections give low-lift curve slopes and are not 613 Davis and Oates subject to large lift changes due to changing from fully wetted to fully or partially cavitating flow. The bow foil system can be designed to provide a low 0C,/da and the optimum 0C,/¢h. Some of the interrelated problems to be examined and solved are listed below: 1. Hydrodynamics — (Foil section design, cavitation suppression, ventilation effects, hydrodynamic loads, performance predictions, etc.) 2. Hydrostatics 3. Hydroelastics — (To date there is no accurate and proven method for predicting flutter speeds of surface piercing or cavitating foils and much re- search still needs to be done.) 4. Dynamic Stability — (The stability equations had to be developed together with a method for simulating the random seaway.) 5. Structural Integrity — (Lightweight structures of FAIS QWES stiffness are difficult to design and required sophisticated analysis.) 6. Materials — (High strength materials had to be found for the foils and random fatigue studies conducted. Coatings had to be developed to help guard against corrosion and erosion.) 7. Transmission Design — (As with many other hydrofoil problems this is practically at the current limit of the "state of the art" in gear technology be- cause of the high torque and low weight requirement.) THEORETICAL EQUATIONS OF MOTION Hydrofoil Ship Simulation Two methods of simulating the motions of a surface piercing hydrofoil in a random seaway have been derived and both methods were used in the design of the hydrofoil under consideration. The first method is based on the normal aircraft equations, in which sets of partial derivatives representing the sum of various force or moment contribu- tions are used to simulate the craft dynamics. The second method differs from the first in that the various forces at the craft centre of gravity are obtained by summing the forces developed by each foil element. Moments at the c.g. are the product of these elemental forces and their respective moment arms about the c.g. The first series of studies to broadly define the hydrofoil was carried out in calm water using linear equations of the aircraft type suitably modified to account for free surface effects. Small perturbations were assumed and a se- ries of partial derivatives was calculated for the complete hydrofoil. These 614 Hydrofoil Motions in a Random Seaway equations were sufficiently accurate for the initial studies, but proved to be in- adequate when more detailed information became available and a more accurate simulation was required. Varying coefficients had to be introduced. All of the derivatives are functions of immersion depth and second order derivatives had to be introduced to account for some of the more nonlinear functions. This re- sults in a set of complicated equations. In fact, each variable has to be written in the form of a Taylor series and linearisation of even the second order terms can lead to significant errors, particularly in the roll derivatives. These equations became very cumbersome, difficult to mechanize on the computer and still had significant inaccuracies in the roll terms. Because of the complex analog computer set-up required and the inaccuracies that were still present in the nonlinear ''Taylor Series" equations, the so-called "explicit variable" method of simulation was developed, in order to simplify the compu- tation and to achieve greater coherence in the derivation of the longitudinal and lateral equations for the surface piercing hydrofoil. Each derivative is a func- tion of immersion depth, which in turn is a function of heave, pitch and wave ef- fects, all of which are derived from the longitudinal equations. In the explicit variable simulation, this coherence can be achieved because all forces are de- rived from two parameters; the lift-curve slope for a given foil element, and the total angle of attack on that element due to all motions about the craft centre of gravity. The development of these equations from Euler's basic equations of motion is outlined below for the axis convention of Fig. (i). (forward) V C ct S) CDT Ww a a Downward) Assume a rigid body with Oxy as a plane of symmetry. Figure (i) Assume a rigid body with 0xz as a plane of symmetry. Euler's equations are: Linear Motions and Forces mCU + OW oURV)) = xe meet (1) 615 Davis and Oates m(V + RU - PW) = Y + mg cos @ sin © m(W + PV - QU) = Z+ mg cos @ cos © Angular Motions and Moments AP - ER + QR(C-B) - EPQ = L BQ + RP(A-C) + P?-R2=M EP + CR + PQ(B- A) + EQR = N Velocities Along Space Axes x. = U cos ® cos ¥ + V(sin ® sin © cos WY - cos ® + W(cos ® sin © sin WY - sin © We = Ucos © sin ¥ + V(sin © sin © sin YW + cos ® + Wicos © sin © sin YW + sin © z, = -U sin ®+ V sin ® cos © + W cos © cos O Relations between Angular Velocities P = @- Wsin® 0) = © cos © + WY cos @ sin © R = Wcos @ cos ®- @ sin © @ = Q cos ®- R sin © ® P + Q sin ® tan @ + R cos © tan © y (Q sin ® + R cos ®) sec ©. sin cos cos cos ¥) Me) ¥) ¥) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) All of the dynamic relationships that are necessary to investigate the mo- tions of a body in response to impressed forces and moments are given in the above equations. These equations are general and are accurate for motions of any magnitude. The hydrofoil motions, however, are relatively limited, in which case small angle approximations can be made for this craft without any signifi- cant loss of accuracy, thus the equation can be simplified. This simplification can be accomplished by writing all of the equations in terms of their deviations from a fixed or reference condition with the exception of the craft forward speed (u) and heading angle (¥), which are subject to large changes. Small approxi- mations cannot be applied to them. The parameters in the deviation equations will be denoted by lower case letters. Reference values will be denoted by the suffix zero. Thus (U,V,W) (P,Q,R) (@,0,¥) are redefined as 616 Hydrofoil Motions in a Random Seaway U= (uu, +u) P= (p,7P) Oe Gert), etc. The hydrofoil reference condition will be with the axes of the craft hori- zontal and with the craft travelling symmetrically in the Ox direction. Thus y, is usually put equal to 0. However, any arbitrary value may be assigned without affecting the equations of motion. Making small angle approximations and substituting the perturbation varia- bles in the foregoing equations we have Euler's equations for small perturba- tions. Linear Motions and Forces mu = X- mg @ (16) miv + (ot uw) cl = ¥ + me @ (17) mlw- (u,+u)q] = Z+ meg (18) Angular Motions and Moments Ap - Er = L (19) Bq = M (20) Cr - Ep = N (21) Velocities Along Space Axes ee (OR ew) os Ui yn chin (22) ¥, = (u,+u) sin y+ v cos p (23) Zo = (Ut Ow (24) Relations between Angular Velocities p=¢ (25) q=@ (26) eae (27) Davis and Oates The above equations could be simplified further if the reference forces and moments were to be subtracted from the basic equation. However, hydrodynamic forces and moments are more readily derived in terms of their full values rather than in changes from the reference condition. It is more convenient, therefore, to leave the equations in this form. Since the full values for the forces and moments have been left in the equa- tions the craft probably will not be in a trimmed condition at the reference con- dition but will stabilize out at some other attitude. Some caution is necessary when considering craft centre of gravity height above the sea surface. It is necessary to translate velocities into space axes before integrating to derive position. For example w may be integrated directly to give w the velocity of the craft along the instantaneous, or current direction of the craft Oz axis, but to find the c.g. height, velocities must be converted to space axes. z, is the parameter that is to be integrated in this case. Consider the craft to be moving in the x,z, plane with a constant velocity V directed along 0,x, away from a set of space fixed axes 0.x.y,z. which was co- incident with the moving axis system Ogxpypzp at time t = 0. Attime t, let O,Xg make an angle ¢ with 0.x, [Fig. (ii)] — oa — — ~—_ —-—-— Fixed Axes. Figure (ii) The components of 0, relative to 0, are x. = i Gos @ (28) Ss z_=-V sin @. (29) Ss The acceleration in the z, direction is obtained by differentiating z, thus b= of sin Os cos 06: (30) Ss however, V = 0 as V is stated to be constant. Therefore 618 Hydrofoil Motions in a Random Seaway Zz, = -Vcos 06. (31) Consider now a point where the velocity of O, is parallel to 0.x, then 4=0 and z. = - vé. Thus the body possesses an acceleration in the O.z, direction (centrifugal force) due to an impressed force, however, no acceleration is evi- dent in the body axis component z,. In the moving axis system (by definition) there is never any component of velocity (v) along 0,z, that is z, = 0, hence Zp = 0- Note that no definition of displacement of 0, is given with respect to its own axis. Distances quoted in body axes merely serve to locate parts of the body with respect to 0,. To obtain displacements, velocity components in space fixed axes must be integrated. Euler's equations take all of the above effects into account, but to ensure that the results are interpreted correctly, it is recommended that the results be transformed into components with respect to space fixed axes. The reverse also applies and care must be taken when applying external forces to the craft. These have to be correctly resolved into craft axes before substitution into the equation of motion. The Normalised Equations In order to compare craft of various sizes it is convenient to normalise the various parameters in the equations of motion. If this is not done, then for two craft which were similar in design but different in scale size, a different trans- formation law would exist between most of the sets of equivalent parameters re- lating to the two craft. This comparison of results obtained for the two craft would require recognition of how each variable should scale in relation to changes in craft scale size. Scaling for the analog computer is also complicated if the equations are not normalised. Computers operate within rather a limited volt- age range so that changing the scale size of the simulated craft would involve changing the voltage scaling levels within the computer for most of the problem parameters. It is convenient, therefore, to make the parameters more or less independent of scale size. Satisfactory normalising can be accomplished by dividing each parameter by a reference value of that parameter to produce a set of nondimensional vari- ables, the reference values being selected according to the scale size or per- formance of the craft. Because differentiation in the equations is with respect to time, it is necessary also to scale the time variable. Four reference parameters are required for the hydrofoil equations repre- senting combinations of length, mass and time. They are: p = fluid mass density (slugs/cu ft), s = reference length (usually semi-span of selected foil in feet), 619 Davis and Oates S) fe} reference area (usually the area of the selected foil in sq ft), V, = reference velocity (ft/sec). The specific parameters required representing mass, time, and force are obtained from products or divisions of the four standard parameters. All inertias Ue sese A xX 5 PS, 5° = oS s Development of the Normalised Equations As stated earlier the forces and moments impressed on the craft are non- linear functions of craft position and motion. Expressions for these forces and moments in terms of hydrodynamic derivatives are subject to significant errors unless high order derivatives are used. Therefore all forces are derived for each foil element as functions of angle of attack and immersion depth. These are the simplest functions which can describe adequately the forces developed by each foil. Thus for example, the foil lift C, at a depth h is Cy,(h)a, where a is the total angle of attack on the foil element due to all motions. That is a total = (a,+a,) radians where a, = the reference angle of attack for a given foil in calm water, a, = the "dynamic" contribution to angle of attack and represents the angle between the longitudinal axis and the free stream direction, and the foil element normalised immersion depth. ‘oF §) ll h and a are derived as follows. Consider a foil element at a longitudinal dis- tance x and lateral distance y from the craft c.g. Then it can be shown that heoiy = [ho + Ah, ,. - x6 + yb ~ fi, cos ¢l = [hj + Ah, ,. - x8 + 9¢- hy] (42) (assuming cos ¢ x 1.0), where hei, = the reference immersion depth of the foil [Fig. No. (iii)], Ah. ee the perturbation about the reference height due to heave of the com- plete craft, 620 Hydrofoil Motions in a Random Seaway die g. = { (W- b14+a9] af, (43) 2) iH} the change in water height from datum at the foil (Note: h,, is given in space coordinates and has to be resolved into craft coordinates), D Il pitch angle of the boat, and ase Il roll angle of the boat. Note: for small angles the following assumptions can be made Sin pa > radians Cos p= ho Figure (111) The expression for angle of attack is basically the differential of the above equation. By = Gi aE (We a - x9 + yp- w, cos ¢) = a,+ (w- x0 + yh - w,) for fixed forward velocity. 621 Davis and Oates For a varying forward velocity a = ae ils: Qo ey) + aq(1+a—u.) (45) this expression is derived as follows. Consider the instantaneous velocity Wea = w= Uh where u, = reference or steady-state velocity, u = the perturbation of craft velocity, and u,, = the horizontal component of wave orbital velocity. Now V; Pepe =a [e} where a u a We wR, = Uy (the normalised velocity perturbation). Thus u a an ar a 6 = a. + == (ws Bde) Se WA = Si) (w - xO + an = W) an = oy. a = Ob. 3 ————e (46) Cie uu) Now steady lift L = 1/2 eU/S, ~ f(a,h) and unsteady lift = 1/2 o(U,+ u)?2Sx f(a,h). Normalising is based on U, the steady-state reference velocity thus C,, is normalised with respect to 1/2 9U,’S and a = w/(U, tu). Therefore we have to consider the variation in the product V,* =: 622 Hydrofoil Motions ina Random Seaway 2 os Zs 2 oq Vy, @ 205 Gh wu.) [e. *$ 4 | Ci uu) 2 A a 2 A A = Ue a(1 +a-a,)° 1 Us agl+u- WL) 6 (47) For small perturbations in G and u, the u? terms can be neglected thus Voto iy el orca eae e Ue aeen= any (48) Thus C= “oe DaGar) —ie oltera Qala) (49) 2 In the drag terms there is the expression Ve a. This becomes a 2 We a? = Ghai oe A Cie wu) U, [a,(1+a-Gw) + ag)? Sideslip angles are obtained in a similar manner to give haem cane (50) for fixed speed and Bie ene = eg (ae u- Oy) for varying speed. Only the expression for thrust remains to be derived before the final equa- tions can be written. Two effects need to be considered: (1) the effect of changes in throttle setting and (2) the effect of changes in water velocity at the propeller. Let k be the increment of thrust horsepower available for a given change in throttle setting, then the effect of a change in throttle setting can be expressed as Hie aii (eleralcayie (51) If thrust is assumed to be a function of local water velocity at the propeller when the throttle setting is constant, then an expression of the following form is derived. 623 Davis and Oates pats eyes Sa eee (52) (1+u-u,)" Combining the above effects we have it T = (1+k) ————. = 1, | |. (53) (1+u-a,)" @+u=a,)2 If we assume small perturbations (the above expressions will not hold for large speed fluctuations) expand the R.H.S. of the equation and neglect all but the first term in the binomial expansion a A = 1)u? (i w)P = Cis soit) pp eee 2! We have T= Ci eikyiil = m= tI (54) and T i fates = €(1+k)M=mu=t,)) (55) iO a D5 The complete set of equations may now be written in the following normal- ised form: Normalised Euler Equations for Small Perturbations Linear Motions and Forces Duin _ oe eo G8 (56) 7 Vo Se A @ A ide f = gulv+ y(1tay) = S92 PFE LG 4 (57) BPN mA a Li De OCG oe eR : (58) 1 V 2 S L, ia o ~o Angular Motions and Moments a a OB Rolling moment LyyP > Ld ze 1 (59) 2 7 PV S55 624 Hydrofoil Motions in a Random Seaway a Pitching moment 2 iE ere as (60) ViVi ae = 1 2 7 PVo S58 ad ia eC bcs Yawing moment ew - lee a GnyCrea ne ee (61) 7 PVo 5,5 Velocities Along Space Axes K = (1+) cos W - v sin w (62) ¥, = (140) sin Wy + 0 cos (63) Zz. = - (1+0)6 + @. (64) Equations (56) to (61) are required for the six degrees of freedom of the craft. Equations (62) to (64) are the relations between craft and space axes and are required for relating sea motion to craft motion. Hydrofoil Equations Basic Equations isis aul - A140] =-C) -G (65) Sue 2a OC, = (Cy = (Cp cy | (66) Pitch iOS Ce ce Cee) by (67) Sideforce Duly + TOM = Cy + Cq é (68) Roll Loos Gps Lol + (iyy-i,,) Ov (69) Yaw Fe ieee meme ee (Gaon (70) 221-249 O - 66 - 41 625 Davis and Oates Expanded Equations The expanded equations, including the expressions for the basic forces and variations due to speed perturbations are as follows: Heave NS es a eG aes Cie ne ee) nS [2c | cos Mery [EL gy Peder) | cos Try ~ &,- (71) Surge Dh aC Ce (1+k)[1-nca-a,y] x uel rs Rapes 2 = Cyc ry(hp)[1 + 244 - Gycpy)| -K (hp)[o,¢py(1 + O- akg) + Gace | ten) x ee 2 pats D 2 SrelGe ts 2G - yey) “Kp, (hed [ao¢ey(1+ = fycey) * Pace) A Res ~ Ahern 2 a Cyc y(hr)[1 + 208 Gye) ]- Kp, , Per) [2ocny(1 + 8 - Byer) * Faqry| oy # ns ae 2 S Coc (hg) | £G= ace) Kp, (ery) So¢ny(t 4 G-Gyery)t cau = Cp! (72) Pitch oe = XcF) [ee Mere |t Xe) [r,.,(Bed@e) ] — Xp) [e eas | cos Mery + XcR) (et (PR) (| cos eR) ( a ( + Cie CLs NE) [1-m(G- Gy 6] ZT) apy ear ans ey iy dean) i ~ : = - Sl aco Ra 73 CD ny ceorar (iF) 2c) (dragy + Gn (h) + Ca Cie) + Gaz - teal ew (73) 626 Hydrofoil Motions in a Random Seaway Sideslip Quiv + W(1ltu))] = Cy apy ORB R) 4 Cy i cy BeBe) Saves (h, ja ] sin P +[q (hp)a ] sin P fee Ty Achy (L) Lippe CD (R) + Cy ¢. (74) Roll ie ® = - 2c) [Cyan MSc) |- 2(e) ie cm Se YL) [hPL cL) | ~ 9¢R) [he ay CPeRy Aer) | “F [oe ces (ts. ».) cae sesie| a [Co gy (FcR) (D.P.))2¢R)(D.P. | + iw + (iyy-i,,) Ov- (75) Yaw ~ iz2¥ = XR) [ey 4) (Pr ocr) ] CS) Levac (©) DR cceiealy, ©) (15x 7 yy) Care must be exercised when applying these equations to a particular hydro- foil as it is possible to overestimate some effects such as damping in roll. If side forces, for example, are assumed to be derived from equivalent vertical struts which in fact have an appreciable dihedral angle then the roll damping may be overestimated. In some instances due to foil geometry the local velocity vector ¢z may be along the foil element and not normal to the foil as is implied in the above equations. 627 Davis and Oates Forces and Moments In the case of the hydrofoil craft, if the foil system is considered as a whole, then the foil derivatives are usually functions of more than two dependent variables. However, if the foil system is divided into a number of elements, then for a particular foil element the lift, drag and side forces are a function of two variables only, the immersion depth and the angle of attack. These varia- bles can be obtained at any instant, for a given foil element, and the forces along the three body axes continuously computed. The moments about the craft c.g. are then given by the product of these forces and their moment arms about the c.g., the net forces and moments at the craft c.g. being obtained by a summation of the forces acting on the individual foil elements. These net forces and mo- ments when divided by the appropriate inertia coefficients then produce the lin- ear and angular accelerations that are required in the basic Euler equations of motion. Derivation of Forces The basis for computation of the forces acting on a given foil or strut ele- ment is the lift-curve slope (C_,) together with the angle of attack on that ele- ment. The lift coefficient (C,) developed being a product of C_, anda. The lift-curve slope is a function of immersion depth (h) and aspect ratio (A) which also is a function of immersion depth when the foil element is surface piercing and is readily obtainable from Refs. 12, 17, 18, 19, 20 and 21. The angle of at- tack experienced by a foil is due to pitch and roll, yaw and heave rates together with wave orbital velocities, the actual angle of attack at any instant being de- pendent upon the free stream velocity and the respective distances from the craft c.g. in the x, y and z directions. peor Immersed Area Figure (iv) 628 Hydrofoil Motions ina Random Seaway For example consider a point on a surface piercing foil element as shown in Fig. (iv). The velocity normal to the foil element is ype [= (Gin Boa > A) = Vw. cos fp - Ww Silja vo) Spiior JG + (w- xO - V,, Sin@ = wy, cos @) cos!’ | (ft/sec) . (77) 1 1 The angle of attack is The sideslip angle is b= ae Wa 1 tonite: oo ; : (78) + - (vtxp-zd- V,,. coSP-w,, sing) sin I+ (w-x9+ydt Vy, Sing-w, cos¢) a U. 1 for small perturbations sin ¢ +0 and cos $1.0, U; ~U,. Therefore 7 es ae Pa Gm “a Son See Gr Aeoi1 = % AS ae oe a Eo I 4 ome es a <> i and but U, = V, therefore Therefore Oy. ae [- (v + oe eee ey sald r+ w- oan ae W,,) cos Ipale (79) Coyle o7 Thus for left- and right-hand foil elements 629 Davis and Oates 5 A A ~ Ary = aa a + (v4 Xp ye yet) sin Try -(w- Xp 9 + YoLye- Bey) £08 Mey x (< + XcRyP- ZRyP- or) sin Very + (- XR t YR)? a cos Per) It will be seen from the foregoing that it is convenient to produce the net angle of attack normal to the foil element. C,, can be obtained for vertical forces or for forces normal to the foil. It is logical therefore to derive the lift normal to an element and to then resolve this into vertical and horizontal com- ponents to produce the lift and side forces respectively. In this manner any di- hedral angle (I) and roll angle (¢) can be taken into account in the computation of forces Derivation of Moments The moments about the craft centre of gravity are dependent upon the foil element loading distribution and thus the centre of pressure location relative to the c.g. For a surface piercing foil the loading varies with immersion depth and therefore spanwise c.p. has to be derived as a function of the foil immersion depth (h). A linear variation with h is usually sufficiently accurate. Chordwise c.p. movement is usually a negligible percentage of the distance from the foil element c.p. to the craft c.g. and can be assumed fixed at say the foil quarter chord position. REGULAR AND RANDOM SEAWAY CALCULATIONS Regular Seas Simulated regular seas are an important aid to hydrofoil craft design. They are easy to produce and much useful data can be obtained. For example, mag- nification factors can be determined for a realistic range of amplitude for each significant frequency as shown in Figs. 3 to 6. The regular seas to be simulated are usually decided by the frequency of encounter range which gives a significant energy input to the craft (Figs. 9 and 11). Once the frequency of encounter is known then it is a simple process to derive the other parameters that are necessary for the sinusoidal wave simula- tion. The pertinent expressions for gravity waves are as given below. Frequency of Encounter aw’ = Inf! (rad/sec) (1) 2 (rad/sec). (2) V @+—@ g 630 Hydrofoil Motions in a Random Seaway Wave Frequency w= 2nf (rad/sec). (3) Wave Length CHEE GEES: (4) ee Wave Orbital Velocity win tarTZ =WZiawn Wit/see) (5) orbital Velocities. Figure (v) Sinusoidal waves can be simulated by a second order system = 74, = 4 a Zo (6) such that Vi Zo cos wt (7) where t = time in seconds. The horizontal and vertical components of the orbital velocities with these waves are given by the following expressions: Vertical Component Ww, = - 02, sin ot. (8) Horizontal Component Uy = +@Z, cos ot. (9) 631 Davis and Oates The vertical component w, has a phase angle ©, = 90° relative to the wave amplitude; the horizontal component u,, has a phase angle ®, = 0° and is in phase with the waves. When there is more than one foil then there will be a phase lag between the forward and rear foil units. If we denote the forward and rear foils by the sub- scripts (F) and (R) respectively and the phase lag is ® then we have the general expression based on oe= a PadilanS 2 where L = distance between front and rear foils (feet), cos (wt-®) = cos wt cos ® + sin at sin O (10) ® Sin (@t-®) = w sin wt cos ®- wcos wt sin ®O. (11) The equations for the wave and the orbital velocity components at the front and rear foils can now be written, viz: Wave Amplitude ZF) = 2. cos at (22) ZR) = Z. cos (at - ® Ry) = 4. GOR BE COS DR) fA. Bille) @se Slo) DR): (23) Vertical Component of Orbital Velocity W = SOL Sitio eae (24) Snes = +oZ , sin (BIE = ey) = +a(Z, sin wt cos PR) = lb, cos wt sin PR)? F (25) Horizontal Component of Orbital Velocity u = +@Z, cos at (26) aoe = wi COS (at — PR) = 0(Z, cos wt cos DR) cy ZS UI tas pl Pry): (27) The computer block diagram for simulating the above Seaway is given in Fig. 7 for a front foil and a main foil, with the main foil split into three ele- ments, left foil, centre foil, and right foil, denoted by the subscripts (L), (c), and (Rr) respectively. 632 Hydrofoil Motions ina Random Seaway Random Seas At the beginning of the hydrofoil stability study, it was recognized that ex- clusive use of regular sinusoidal seas as forcing functions might be misleading, since they are hardly representative of actual seaway conditions. It was de- cided, therefore, to simulate a random seaway based on a mathematical model which is used successfully for wave forecasting purposes. The following subsections are contributed by E. R. Case (De Havilland Staff Engineer) who was responsible for the original analysis and simulation of the random seaway for the hydrofoil study, and the subsequent spectral and statis- tical analysis of the computer and trials results. The random seaway The most obvious feature of a seaway is the almost complete lack of any consistent order or pattern to the wave motion, an observation which led to con- sideration of the seaway as a random process. By assuming further that the process was Gaussian, Pierson [23,24] derived a mathematical model based on a Fourier representation of random noise due to Rice [26], and the propagation properties of deep-water gravity waves. About the same time, Longuet- Higgins [22], using the Gaussian assumption, and the results of Rice's paper, derived the statistical distribution of wave heights for wave forecasting purposes. The remaining quantity required to complete the description of the Seaway as a ran- dom process was the power spectrum, which was supplied by Neumann [27] on the assumption that the wave energy varied as the fifth power of the generating wind velocity. These results were successfully incorporated in a book published by the United States Navy [23] on practical methods of wave forecasting. On the basis of the above, the Pierson representation and the Neumann spectrum were assumed to characterize a seaway with sufficient accuracy for the purposes of the stability study. It was assumed further that a Neumann wind speed of 22 knots corresponds to a Sea State Five. A typical estimate of a seaway surface elevation probability distribution function is shown in Fig. 8. The linearity indicates normality out to over four standard deviations, which validates the Gaussian assumption for engineering purposes. Attention was restricted to the consideration of ''sea'' waves, which, as dis- tinct from ''swell'' waves, exist within a storm generating area due to the action of the local winds. Attention was further restricted to a seaway which had reached the fully-developed state, where a state of equilibrium exists in the in- terchange of energy between the waves and the wind. The fully-developed sea state is reached only when the generating wind has blown over a sufficient fetch and time duration [23], and can be considered a stationary, ergodic random process. The Neumann spectrum applies only to the fully-developed seaway [28], and takes the form in the one-dimensional case, for f > 0 633 Davis and Oates © ii Hone eo DD) = sae (ft) 2/cps (28) where f is the wave frequency in cycles per second, v is the generating wind speed in knots, and c, and c, are constants. A typical wave elevation spectrum is shown in Fig. 9. Implicit in the description of the seaway as a stationary Gaussian random process is the assumption that the instantaneous surface elevation at any point results from the superposition of an infinite number of small sinusoidal compo- nents of different frequency, phase and direction of propagation. Analytically, the wave elevation can be expressed as a stochastic integral of the form A) = | cos [wt - (w)] 20,(f) df (29) 0 where w= 27f and (w) is a randomly chosen phase angle uniformly distributed in the range (0,27). While this is not integrable in the ordinary sense, it can be expressed in the form of a Fourier sum (see St. Denis and Pierson). This representation can be extended to include the effects of distance by using the wave equation for transverse wave motion for each sinusoidal compo- nent. Thus, if x is the distance measured in the direction of the wind from a fixed point on the earth, the wave elevation can be expressed by Z(t,x) = j cos (at - Ox - f) /20,(f) df (30) (0) where Q = wavenumber = o/c = 27/), > Il wavelength in feet, and c = crest speed (wave celerity) in knots. If each of the small sinusoidal components is assumed to propagate as a gravity wave, then, in addition, The validity of this assumption has been confirmed by the general success of the wave forecasting methods based on Pierson's theory. The wave elevation can then be expressed by 634 Hydrofoil Motions in a Random Seaway ZGts a= | cos (at - “ x - ) V20,(f) df . (31) 0 Equation (31) can be differentiated to give what can be assumed to represent the vertical component of the water particle orbital velocity. Thus, ies) W843) = Z(t, x) = ai sin (ut - S x4) /28,F) df (32) 0 where the spectral density for the vertical velocity is given by Dif) — Comic De (ata) The hydrofoil ship in the random seaway The random seaway can be considered as a disturbance input to the hydro- foil craft. These inputs induce motions, which are not present in calm water, and which result from a combination of wave elevation, orbital velocities and the forward velocity of the craft. If the reference coordinate system is chosen fixed to the hydrofoil ship, then the effects of the seaway and craft velocities can be combined together to produce wave elevation and orbital velocity forcing functions which are functions of craft speed. This is accomplished by trans- forming the original seaway spectra by a change of variable to produce new spectra which are functions of frequency of encounter. To illustrate briefly, consider the coordinate systems as illustrated in Fig. 10. The moving coordinate system is designated by primes. The coordinate transformation is then given by Kea Ke Vit} Gand™= Zs Z- (33) where Vv, the ship speed, is defined to be negative in head seas. Substituting Eq. (33) in Eq. (31) gives . aw? 7 At,x') = | cos (at = = x'- 6) 20589 df! (34) 0 where the frequency of encounter w' is given by wt = oa? = Onf" (35) and the "transformed" wave elevation spectrum by fet 2s Of y= cr SOV. (36) @ Davis and Oates an expression which is easily derived from the fact that the mean square wave elevation is unchanged by the coordinate transformation. The orbital velocity expressions are transformed in a similar fashion, and, along with Eq. (34), formed the basis for the simulation. An example of the effect of the transfor- mation on wave elevation spectra is given in Fig. 11. Simulation of the random seaway The basic method used for simulating the random seaway is common in analogue computer practice, and involves the use of suitable linear filters to shape the output of a random noise generator to obtain signals with the desired power spectra. The simulation was done entirely in moving coordinates, and thus all spectra were functions of frequency of encounter. In addition, the sim- ulation was done for constant craft velocity only, since varying velocity would require filters with changing characteristic frequencies and consequent extrav- agant use of analogue computer components. Head, following and beam seas were Simulated for both the quarter and full scale hydrofoil craft for speeds of 25 and 50 knots, respectively. The starting point for the simulation was the vertical velocity spectrum since, in sea coordinates at least, wave elevation is obtained therefrom by an integration rather than a differentiation. In moving coordinates, wave elevation is obtained from vertical velocity by a "transformed" integration, the charac- teristics of which can be derived by considering the frequency response function of an integrator as a function of frequency of encounter. The frequency re- sponse of an integrator in sea or fixed coordinates is Gea (37) jT7@ where » is the sea frequency. Using (35), the frequency transformation given in terms of frequency of encounter is il # hee te g 2V/¢2 (38) Substituting (38) in (37) will give the frequency response of a ''transformed" in- tegrator, thus TAG) a = ee (39) ir [1+ V1 - (4V/g) @'] It can be seen that I'(jw') has the same 90° phase lag for all frequencies as the ordinary integrator, but that the magnitude is quite complicated. While (39) is obviously non-realizable in the strict sense, it can be approximated over a range of frequencies by a combination of minimum and non-minimum phase net- works. The procedure was to first approximate the magnitude without regard to phase with a combination of first and second order filters, and then correct the 636 Hydrofoil Motions ina Random Seaway overall phase to approximate 90° over the frequency range by all-pass networks. Typical head and following sea transformed integrator frequency response char- acteristics are shown in Fig. 12. A block diagram of the head sea simulation is shown in Fig. 13. Notice that the transformed integration involved two all-pass filters, the difference in phase between them being such that the phase angle between w’ and z’ is 90°. Figure 14 shows the filter arrangement for following seas. Following seas present special problems since the transformed wave elevation spectrum can contain both following and head components for certain craft velocities; and in- deed also a steady value for that component whose crest velocity is equal to the craft velocity. Simulation for such a condition is clearly impossible, since the transformed integrator and foil separation filters would have to be approximated over an infinite number of decades in frequency. When the craft velocity is high enough, however, all significant frequencies become head components and a simulation is feasible. The simulation is similar to that of the head sea except for the all-pass filters which are required to supply the constant component of foil separation phase shift required. It should be noted that simulation of the effect of the separation between the foils cannot be accomplished by a Padé approximation to a pure delay. The de- lay is distributed, and has a phase characteristic proportional to the square of the frequency. The foil separation filter required two second order all-pass filters to approximate the transformed phase shift over the significant frequency range of the vertical velocity spectrum. A second method of simulation, using a number of superimposed sinusoids of appropriate amplitude and frequency, was used for simulating following seas at the lower ship speed. The method was unsuitable for the other cases, how- ever, because of excessive demands on computing equipment to give a sufficient number of components to approximate a normal distribution. ANALOG COMPUTER SIMULATION TECHNIQUES Analog Simulation of the Equations of Motion As mentioned previously the lift-curve slope is the basis for computation of all lift forces active on the foils. This is simulated on a function generator in the analog computer. The diode function generator creates a sequence of straight lines that are connected together to form the desired function. Obvi- ously if a large number of segments are used then the function will be generated more accurately than if just a few points are selected. In practice about 8 or 9 "break points" will simulate most lift curves with sufficient accuracy. For ex- ample consider the following lift curve [Fig. (vi)]: 637 Davis and Oates 5 4 3 a. 0G, 2 per (Yotts) fadsan t o as 5 76 bo 20 6o 40 sok Getts) (a) (b) Figure (vi) An arbitrary voltage scaling of 80 volts/per unit h and 10 volts per unit C,, is assumed. The input to the function generator will be 80 h volts which gives an output of 10 C_, volts. This voltage is then fed into one channel of an elec- tronic multiplier and multiplied by («,+a,) to give C, as avoltage. C, is then subsequently summed with other voltage variables in the dynamic equations. If C, is equal to the weight of the craft (CL,) then the heave equation for example will be in balance and the output of the vertical acceleration integrator will be zero. This is of course an oversimplified example, but it does illustrate the basic procedure on the computer. A simplified circuit for the heave equation is shown in Fig. 15. Cavitation Cavitation and its effect on the craft dynamics is very important and must be simulated if a realistic representation of the hydrofoil motions is to be ob- tained from the computer. Cavitation gives rise to nonlinearities in the lift- curve for a given foil element. A typical example is shown in Fig. (vii). The angles of attack at which partial cavitation and eventually full cavitation occurs are a function of cavitation number and thus speed. The step in the curve and the C,; at which the slope changes are a function of the lift-curve slope (C,,) which in turn is a function of immersion depth. The lift on a cavitating hydro- foil is obviously a complicated function to simulate. However, a reasonable ap- proximation can be made by simulating the lift as shown in Fig. 16 to produce the curve of Fig. (vii) b. 638 Hydrofoil Motions in a Random Seaway region of - ae ts ly rtial cavitation i C. [ee GRE i + "I Lully Vier Angle of Attahy ' eaytcating Hae teed es lmmersion depth a. bass Linear Icét-evrve slope is a function oth (igo i 4 We + oc These ang les are Functions of velocity. * @avitation Actual Lift, Simulated Lift, (a) (b) Figure (vii) Ventilation Cavitation is unlikely to occur on foil elements at the slower foilborne speeds unless the foil angle of attack is very large (>10°). However, ventilation, which has a similar effect, can occur at any speed when a foil is surface pierc- ing or is Close to the surface. The effects of ventilation have been simulated on the analog computer, but this proved to be an extremely complex problem requiring a large number of computing elements. The criterion for ventilation of a given foil element may be either the angle of attack (a) or the lift coefficient (C,). a was used in our simulation. In terms of a, at a given speed it was assumed that there exists a fixed value of a, (ey say) for which ventilation must occur if a> a,. Similarly, there exists an a stop (c,), for which ventilation, if occurring, will stop when a - 200k ()# 200 bas 2] YP vai re 1 > ©) 1 Boh f Sm $a] ® z] Z Suot. > +200 ws, + & cu 5 a 00 50 au, Fig. 7 - Typical sine wave generator circuit 660 1334 - LHDI3H JAVM Z- 99.9 99.8 Hydrofoil Motions in a Random Seaway PROBABILITY OF WAVE EXCEEDING A GIVEN VALUE - % 99 98 95 90 80 70 60 50 40 30 20 10 5 2 1 0.5 0.5 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 PROBABILITY OF WAVE BEING LESS THAN A GIVEN VALUE - % Fig. 8 - Probability distribution function -- Sea State 5 661 99.8 99.9 99.99 SPECTRAL DENSITY - (Ft2/cps) ®.(F) Davis and Oates WIND SPEED 22 KNOTS FULLY DEVELOPED SEA FREQUENCY - (cps) Fig. 9 - Wave elevation spectrum 662 Hydrofoil Motions in a Random Seaway Fixed z Coordinate System Moving Coordinate System x=x!+Vt Fig. 10 - Coordinate transformation 663 &, (f) (Fr? sec) POWER SPECTRAL DENSITY -.2 FOLLOWING COMPONENTS Davis and Oates SEA STATE FIVE - 22 KNOTS WIND SPEED HEAD SEA : SOLID FOLLOWING SEA - DASHED BOAT SPEED = Vv KNOTS 0 2 4 6 8 1.0 1.2 HEAD COMPONENTS FREQUENCY - (f or f cps) Fig. 11 - Transformed wave elevation spectra 664 MAGNITUDE T'( jo) 0! Hydrofoil Motions ina Random Seaway a ied rhe eked Senirmvaeeeeial lh atae | ene a H+tH orth aes Hf me in TEE et] ty Saas soeees ttt | “0 SESE eiSsessees Beebe te tei eH aiSS SSeS iia Seas aEAAEEE et bifie sets Senses Sia s assed fit tid Saeed atid aa aaee sm eas rane cnt ge Magesd alll nw "a Suet ef meme eset ts HUaEaTt Fea SSSeHaE SReaeaanatitiu Htllao bape lil ad apet MH mSRSETEE i maa nea Soe aos ieee rere Po ae Boles ibeessereriaeiisetel SESroririiiiaissee SEs ra eee ole Tern cles eases) nee ie: see use vee Seg Se ee el sme aS 2 ane 2S scale a ae ill Serer oe Se HEE Paes fi He Bese: aa = anes ee Mises fae aM ey a a i Frequency of Encounter f' - ae Seni Fig. 12 - Typical transformed integrator frequency response FE a EF LiEeE Fy ian HEE 665 w - Vertical Velocity z - Wave Elevation F - Front Foil M- Main Foil Primes - In moving coordinates Davis and Oates Noise idea All-pass Generator srecity Filter Filter Transformed Integrator Magnitude Filter Filter Foil Separation Filter Transformed Integrator Magnitude Filter Filter Fig. 13 - Head sea simulation Vertical Velocity Filter Noise Generator w - Vertical Velocity z - Wave Elevation F - Front Foil M - Main Foil Primes - In moving coordinates Fig. All-pass Filter Transformed Integrator Filter All-pass Filter Foil Separation Filter Transformed Integrator Filter 14 - Following sea simulation 666 All-pass All-pass Filter All-pass All-pass Filter All-pass Filter All-pass Filter All-pass Filter WIE 'F we =M “MM Hydrofoil Motions in a Random Seaway (es (Constant fer a given reference velecity) 100Vvelts. >) Speed dependent potentiometers Maia Foil lmmersion De ptt. Main Foil Lift, A = 4 hu) é Tapat from 28 pite! aA ry Aa 4 hy (wave amplitude he Inpat From Seaway, SUMMING DIODE FUNCTION AMPLIFIER GENERATOR INTEGRATING Giu3.) &] | AMPLIFIER 100 volts od, e a A Bilisest pee 2 NOTE: Speed dependent Potentiometers dictate either a change in craft speed or a change in computer scaling. pite cqpa lon A 4 Wie (Input from Sears a, s ae 5 Ue @dianc. A w 2 U. Cadiane. All of the equations are built up in this manner with the various inputs feeding into the summing amplifiers and integrators etc.. Fig. 15 - Simplified block diagram of heave equation 667 Davis and Oates Creat h; [e4 Cy wibh ap cavitation “Variation in C, with of at depth h; av. FUNCTION INDICIAL DISTANCE TRAVELLED, S’, SEMI-CHORDS Fig. 17 - Exponential approximation to indicial function 668 Hydrofoil Motions in a Random Seaway 9 PEAK TO PEAK AT C.G. PEAK TO PEAK AT C.G. HEAVE AMPLITUDE PEAK TO PEAK (FEET) HEAVE AMPLITUDE PEAK TO PEAK (FEET) PITCH AMPLITUDE PEAK TO PEAK (DEGREES) PITCH AMPLITUDE 2 FT FOLLOWING SEAS He RESPONSE TO VARIOUS C_ DELAY FUNCTIONS AND TO q@ TYPE VIRTUAL INERTIA fi fea se NO Cc, DELAY NO.1= EXPL A+EXPL 3 NO.2= EXPLB L.F.:.7 NO.3 = EXPL B=1.F.=.5 CL DELAYS (IN ORDER) LIMIT OF CHANGE WITH | C, DELAY 2.0 3.0 4.0 100 WAVE LENGTH Fig. 18 - RX craft -- longitudinal heave/pitch only 669 o; (F) WAVE ELEVATION SPECTRAL DENSITY F+2/CPS pee SEAWAY WAVE SURFACE ELEVATION SPE Davis and Oates AND: | RMS WAVE HEIGHT £4 AVERAGE WAVE HEIGHT AVGE. HIGHEST 1/3 es ae AVGE. HIGHEST 1/10 RMS VALUE =O = .62 FT THUS Ez=207 = .77 Ft? MEASURED DURING SEA TRIAL No. 1 TRANSFORMED WAVE ELEVATION SP FOR RX CRAFT IN HEAD SEA AT 25 KNOTS ECTRUM f - WAVE FREQUENCY OR f' CPS - FREQUENCY OF ENCOUNTER Fig. 19 - RX sea trial No. 1 wave elevation spectra 670 I ., (F) VERTICAL VELOCITY SPECTRAL DENSITY - (Ft/Sec)2/CPS Hydrofoil Motions in a Random Seaway 1.8 fake NR Ss y S| [il Ine] ras eee ee THUS E,,2207=1.4 Ft?/Sec * anu lle Mite Lea RMS VERT. VELOCITY = AVGE » ” =z AVGE HIGHEST 1/3 ” ” 1/10 TRANSFORMED VERTICAL VELOCITY SPECTRUM Xo FOR RX CRAFT IN HEAD SEA AT 25 KNOTS f- SEA FREQUENCY OR f'- FREQUENCY OF ENCOUNTER Fig. 20 - RX sea trial No. 1 vertical velocity spectra as derived from measured wave ele- vation spectrum 671 FEET2/CPS TRANSFORMED WAVE ELEVATION SPECTRAL DENSITY Davis and Oates RX SEA TRIAL NO.1 TRANSFORMED WAVE ELEVATION SPECTRUM SIMULATED QUARTER - SCALE SEA STATE FIVE TRANSFORMED NEUMANN SPECTRUM E=.77 tt2 FREQUENCY OF ENCOUNTER f' - CPS Fig. 21 - Comparison of transformed wave elevation spectra 672 2 TRANSFORMED VERTICAL VELOCITY SPECTRAL DENSITY - (FT/SEC) / CPS 221-249 O - 66 - 44 Hydrofoil Motions in a Random Seaway RX SEA TRIAL NO.1 RMS=.85 ft/sec FREQUENCY OF ENCOUNTER es al pees tee VERTICAL VELOCITY SPECTRUM SIMULATED QUARTER SCALE S.S. 5 TRANSFORMED VERTICAL VELOCITY SPECTRUM. RMS= 1.0 ft/sec 2.4 2.8 - f' CPS Fig. 22 - Comparison of transformed vertical velocity spectra 673 7 CPS 2 (DEGREES) PITCH ANGLE SPECTRAL DENSITY Davis and Oates FREQUENCY OF ENCOUNTER VP EPS Fig. 23 - Comparison of pitch angle spectra 674 2 SPECTRAL DENSITY 4g /CPS Hydrofoil Motions in a Random Seaway COMPUTER PREDICTION Eee nea ka Hel ia Ee, eee) FS eS Os Ce AS a ee FREQUENCY OF ENCOUNTER — f'—CPS Fig. 24 - Comparison of center of gravity vertical acceleration spectra 675 (UNITS)? / CPS SPECTRAL DENSITY Davis and Oates RX TRIAL RESULT (SEA TRIAL FREQUENCY OF ENCOUNTER f'- CPS Fig. 25 - Comparison of main foil lift coefficient spectra 676 Hydrofoil Motions in a Random Seaway 10-- WAVE VERTICAL ORBITAL VELOCITY 9, Ft/sec. (BOW) wave 7) VERTICAL ORBITAL VELOCITY Fr./Sec. 9 (MAIN) Time Seale: - 4 Divisions 1 Sec Fig. 26 - Sample computer traces: R200 at 50 knots head sea wave traces --State 5 random sea forcing functions 677 Davis and Oates (BOW) = 1o- BRR 10- i | WAVE = HEIGHT : on HBS ior Seelice oe Se ies Ft. [Fel eg [ia se | =e ae i ee Se (BOW) [=| Be eel eee ee ee ed es Ee SS 10- =| ee| el Ba bee ee ee sil Hn nl Cc WAVE (== SSS >= SSS = VERTICAL === SSIES FES ORBITAL «= 23) =a SESS Ss VELOCITY [EAS Ae=S = Pees Fifsec, (lees | Eee (MAIN) BaB-s abe Pa SFE dae ooo ee se ET jGeeaae eee! a EE ES] Ba Sass ESS =| WAVE — HEIGHT ee : =e == ASS =| F= Bad ES SSS: jene 2a eer + EAN = eee es Or ai eee ES Bee ee= = EESe= 22S SeSS Se seSeleSsSee. PS EES SSeS SS =a 22S ESE! 10 PEE PE SE ESE Baa rer SEES) Fr. (MAIN) Time Scale; - 4 Divisions 1 Sec Fig. 27 - Sample computer traces: R200 at 50 knots following sea wave traces -- State 5 random sea forcing functions 678 Hydrofoil Motions in a Random Seaway 10- WAVE VERTICAL ORBITAL MERSIN Ft./sec, (MAIN FOIL LEFT SIDE) 10 ORBITAL VELOCITY Ft./sec. Q- (MAINFOIL LEFT SIDE) WAVE HEIGHT Fr. e (MAIN FOIL? LEFT SIDE) wave = ‘10 VERTICAL ORBITAL — VELOCITY Time Scale: - 4 Divisions — 1 Sec Fig. 28 - Sample computer traces: R200 at 50 knots beam sea wave traces --State 5 random sea forcing functions 679 Davis and Oates 10 REF. WAVE HEIGHT ae Fe, i oo fe (FRONT) z= IAA ————/ VERTICAL ACCELN Ft/Sec? 9, e (BOW FOIL) Time Scale : - 4 Divisions =1 Sec Fig. 29 - Sample computer traces: head sea --R200 response at 50 knots to a State 5 random sea 680 Hydrofoil Motions in a Random Seaway MOTION = AT GG. LE} Ft. 0 ae — = Ze a= +] ===5 Es ———s === TOTAL i a || VERTICAL Eps === 5 ACCELN := SSS AT C.G. = ===> Ft./sec? Sea 0 fee. Tai a ae FS = 2== = = ES === FE ee ed es ae = == PITCH ; eS ANGLE PAA Eee Degrees Fa EES =*. = = = =I ab = SURGE = VELOCITY INCREMENT Fr./Sec, 0 20 Time Seale: - 4 Divisions = 1 Sec Fig. 30 - Sample computer traces: head sea --R200 response at 50 knots to a State 5 random sea 681 10 REF. WAVE HEIGHT. Fr. (BOW 10: PITCH ACCELN Degrees. Per Sec. 2 VERTICAL ACCELN Ft./Sec> (BOW) Davis and Oates Fig. 31 - Sample computer traces: itt ts = Time Scale : - 4 Divisions = 1 Sec following sea -- R200 response at 50 knots to a State 5 random sea 682 Hydrofoil Motions in a Random Seaway VERTICAL ACCELN AT C.G. Ft./sec.? TOTAL EOGUUNANOAIUERAL ES = = Bes St HUH ES z 8 il SURGE VELOCITY INCREMENT Ft./Sec. Time Scale 1 - 4 Divisions = 1 Sec Fig. 32 - Sample computer traces: following sea --R200 response at 50 knots to a State 5 random sea 683 Davis and Oates cn f —=r Linn oni Time Scale: - 4 Divisions =1 Sec Fig. 33 - Sample computer traces: beam sea — R200 response at 50 knots to a State 5 random sea 684 Hydrofoil Motions in a Random Seaway SMOTA o[Tjord pue url - HE ‘BIT 685 Davis and Oates Baste wss 686 Hydrofoil Motions in a Random Seaway _ ASW HYDR ae MAID Figure 36 687 Davis and Oates DISCUSSION H. D. Ranzenhofer Grumman Aircraft Engineering Corporation Bethpage, Long Island, New York Generally, the paper is excellent, in that it presents a full and detailed pic- ture of De Havilland's work on the FHE-400 hydrofoil ship, in terms of the ap- proaches used and the results obtained. The thoroughness of the work is attested to, in part, by the extensive use of both fixed and free models in the development. The analysis in some respects, parallels that used at the Grumman Aircraft Engineering Corporation in our work in the hydrofoil field. It is interesting to note that the authors' conclusions as to the complexity of the craft equations of motion when using the method of small perturbations are identical to ours. Another item of significance is the use of the surge degree of freedom in the analog computer program. From the results obtained, it may be inferred that, for the foilborne cruising conditions, craft forward velocity can be assumed constant, thus eliminating the surge equation. For our work in the design and development of hydrofoil autopilots, this assumption was made. However, the surge equation is useful in studying takeoff and landing performance if hull lift and drag terms are included. A possible limiting factor here is the amount of analog equipment available; it was found in our work that the addition of the surge equation and associated terms required a 50% increase in a five degree of freedom analog program. A point of criticism is the omission of the lift and drag equations from the discussion of the approach using forces and moments. These forces comprise the major portions of the total force and moment terms, X, Y, Z, L, M, and N in Eqs. (1) through (6) and in our opinion would be of interest to others in this area. The frequency response charts form a valuable basis for performance com- parisons with other hydrofoil craft, but only if identical wave length-to-height ratios are used for all craft, or if wave lengths are normalized to foil base or another suitable craft parameter. 688 Hydrofoil Motions in a Random Seaway DISCUSSION A. Silverleaf National Physics Laboratory Teddington, England This paper is probably the most thorough account yet available of the over- all development of the design of a seagoing hydrofoil ship of unorthodox and ad- vanced foil configuration. Among the many significant points which it r‘ ises is the clear indication that fixed surface-piercing foils may yet have an important and useful role to play in such craft in spite of many recent statements to the contrary. The authors have naturally emphasised the value of analogue computer studies in investigating the motions in a seaway of a craft of this kind. It is, of course, important to simulate correctly the performance characteristics of ventilated foils in such analogue calculations, particularly if the motions of the craft may cause ventilation to be intermittent, alternating with short periods during which the foils are either fully or partly wetted, in which case their force characteristics will be very different. Some of the early experiments at N.P.L. with a 1/8 scale skeleton model of the complete craft, free to heave and pitch, showed that intermittent ventilation of the bow foil unit could occur in certain sea conditions. In these circumstances there were disturbing differences between the analogue computer calculations of the craft motions and those measured on the large model in the high speed tow- ing tank at Feltham. However, when steps were taken by the authors to ensure that ventilation was continuous, the motions of the model were very considerably improved and there was then good agreement with the calculated values. This episode well illustrates the need to simulate the correct physical conditions in any computer calculations; if the hydrodynamics are incorrectly reproduced it is unlikely that useful conclusions will be obtained. The authors have pointed out that many of the model experiments have been carried out at N.P.L.; as mentioned in Table 1 these include not only measure- ments to determine hydrodynamic performance but some very unusual experi- ments to investigate hydroelastic characteristics. All these experiments have been and are being made as one aspect of a most interesting three-part approach; analogue computer studies in Toronto, trials with a manned craft in Halifax, and model experiments at Feltham have proceeded simultaneously and in parallel. It is I think fair to say that, particularly during the early development stages, each of these three approaches identified and resolved problems which at first sight appeared more than daunting. This comprehensive and thorough attack emphasises the need for such procedures if advanced high speed marine craft are to be successfully developed. 221-249 O - 66 - 45 689 i guss sia ni THE BEHAVIOUR OF A GROUND EFFECT MACHINE OVER SMOOTH WATER AND OVER WAVES W. A. Swaan and R. Wahab Netherlands Ship Model Basin Wageningen, Netherlands ABSTRACT The results of tests on the over water behaviour of a flying model of a Ground Effect Machine are given and discussed. Over smooth water the effect of a variation in water depth was investigated. Over waves the variables were the wave length, height and direction, and the rise height. The effect of side wind was also considered. INTRODUCTION The information presently available on the behaviour of ground effect ma- chines over water is rather limited. See Refs. [1] through [8]. It is based on the experience gained with a few man-carrying prototypes and a number of model tests. The model experiments were in general conducted with a fixed model with the object to determine the static forces and the effect of the air cushion on the water surface. Additional information has been obtained now with a dynamically scaled free model of the SKMR-I "'Hydroskimmer" in a model basin where various wave patterns were simulated. In order to avoid telemetering problems and energy storage in the model a towing carriage was used. The maximum carriage speed in the seakeeping basin of the N.S.M.B. is about 15 ft sec-1. The service speed and size of the SKMR-I is such that a very small model would be required to use this carriage for the whole prototype speed range. In view of these problems it was decided to construct a 5 ft long model of the SKMR-I with a weight of about 22 lbs. Equivalent speeds up to 35 knots could be attained with a model of this size. The N.S.M.B. was only concerned with the model tests, which have been re- ported in Refs. [9] and [10], and not with the design of SKMR-I itself. In this paper some of the most characteristic data are discussed. 691 Swaan and Wahab GENERAL CONSIDERATIONS The model of the G.E.M. was tested in the Seakeeping Laboratory and in the Shallow Water Laboratory of the N.S.M.B. It would have been desirable to use a model with six degrees of freedom but that requires an autopilot in order to keep the model in its track. Because of weight considerations it was decided to be content with only three degrees of freedom for the model. In terms normally used in naval architecture they are: heave, pitch and roll. This restriction is only of importance for the behaviour of the model in oblique seas. In order to compensate for the lack of information caused by restricting some of the motions the exciting forces were measured for surge and sway to- gether with the yawing moment. The vehicle considered here proceeds over a free water surface, therefore the Froude number (V//gL) must be the same for model and prototype in order to equalize the scale factors for inertia forces and gravity forces. The same rule must be applied in order to simulate the dynamic OEGIDSINSS of the air cushion as shown by Tulin [4]. The Reynolds number is of importance in order to take into account the effect of viscosity. This has some effect on the flow around the vehicle in for- ward flight and for the behaviour of the jets. The fact that this Reynolds number is different for model and prototype will not be of importance for the frictional resistance because this will be small in comparison with the total drag. The jet flow will be highly turbulent in the actual vehicle. Because the Reynolds number of the model jets exceeds the theoretical critical value of 5500 it is justified to assume the model jets to be turbulent as well. It is therefore expected that the flow around the model will be to a large extent similar to that around the prototype. Another aspect which had to be considered is the effect of surface tension which was clearly visible in the amount of spray generated by the jets. The surface tension will be primarily of importance for the energy needed for main- taining height. This aspect was not included in the program and therefore it is considered to be of minor importance that the Weber number necessary to en- sure similarity with respect to surface tension, is not the same for model and prototype. DESCRIPTION OF THE MODEL The 1/14 scale model of the SKMR-I ''Hydroskimmer" was made of plywood, aluminum and plastic foam in order to provide for sufficient stiffness and strength combined with light weight. A general arrangement plan of the model is given in Fig. 1. 692 Behaviour of a Ground Effect Machine SIDE VIEW A BOTTOM VIEW WITH RIGID JET EXITS WITH FLEXIBLE TRUNKS a AAI tS SN ‘ VJ SECTION A-A SECTION. A-A secTION B-B SECTION B-B SEcTION C-C SECTION C-C Fig. 1 - General arrangement Table I gives some of the principal characteristics of the 'SKMR-I." It was equipped with four independently controlled cushion fans, driven by synchronous electric motors. The number of revolutions of the cushion fans could be adjusted by changing the frequency of the alternating current supplied to the motors. The cushion fans of the model were designed independently of the fans in the actual G.E.M. Therefore there is no relation at all between the numbers of rev- olution per unit time mentioned in this paper and thevalues for the actual vehicle. They should be considered as a parameter representing the power absorption. The model was tested with two different bottom configurations. The first one had rigid jet exits, in the second one the jet exits consisted of flexible trunks. Both are illustrated in Fig. 1. The flexible trunks were manufactured of a plastic covered fabric. The shape was maintained by means of air pressure provided by the jets. Propulsion screws, nozzles and control devices were not 693 Swaan and Wahab Table 1 Model with Rigid Model with Flex- Jet Exits ible Trunks Length, over all 65.3 ft 65.3 ft Length, air cushion 57.15 ft 57.15 ft Total weight 58490 Ibs 62500 #£Ibs forward of centre Centre of gravity 5.08 ft 5.50 ft 0.53 ft 0.53 ft nozzle intersection Longitudinal radius of 16.19 ft 7 Aah gyration Longitudinal mass moment 476400 659300 _—s ft ‘Ibs sec? of inertia Transverse radius of 876 ft 8.92 ft gyration Transverse mass moment 139500 ft lbs sec 2 154600 _—s ft-lbs sec? of inertia simulated. The. forward speed of the model was provided by the towing carriage. The connection between the model and the towing carriage consisted of an appa- ratus which left the model free to heave, pitch and roll. It enabled the measure- ment of these motions by means of potentiometers. The resistance, lateral force and yawing moment were determined by means of strain gauge balances. Model and towing apparatus are shown in Figs. 2 and 3. above flat bottom During the tests the weight distribution corresponded with the conditions of the actual which is shown in Table 1. THE TEST PROGRAM The purpose of the investigations was to get an insight into the behaviour and the forces working on the vehicle when proceeding over smooth water of various depths and over waves. The tests conducted may be divided into the following categories. 694 Behaviour of a Ground Effect Machine Fig. 2 - Model and towing apparatus In the first instance the static and dynamic properties were investigated on the model hovering over smooth water and over land. For this purpose the static stability, the reactions of the model to an impulse and the relation between the rise height and the number of revolutions of the cushion fans were deter- mined. These tests were performed both on the model with flexible trunks and with rigid jet exits. Over the water the model appeared to be liable to self- induced roll pitch and heave oscillations when fitted with rigid jet exits. It was felt that the tendency to self-induced rolling would be inconvenient in an operational GEM. Moreover this phenomenon could obscure the effect of oblique waves on the motions. Therefore it was tried to improve the over water hovering behaviour of the model with rigid jet exits by minor structural changes in the bottom con- figuration by modifying the central jet effectiveness and the directions of the Side jets. The model was tested also with varied radii of gyration. None of these modifications improved the roll behaviour very much. The increment of the radius of gyration, needed for a substantial reduction of the roll amplitude, was beyond the possibilities of practical application. 695 Swaan and Wahab Fig. 3 - Model flying over shallow water The tests with the model flying over smooth water were conducted with the rigid jet exits of original shape. The model fitted with flexible trunks was liable to self-induced roll oscilla- tions only and the amplitudes were smaller compared with the model fitted with rigid jet exits. For this reason the tests in waves were executed with flexible trunks only. Flying over smooth water the resistance and the motions of the model were investigated as a function of the water depth. The behaviour of the vehicle proceeding over waves was investigated on the model flying in several directions over regular deep water waves of various lengths and one height. The effect of variations in the wave height, rise height and trim was investigated for wave directions and wave lengths which appeared to be the worst for the model. Most of the tests were conducted on the model having zero trim at zero speed, with the four cushion fans adjusted as to keep the differences between the numbers of revolutions of the fans as small as possible. In the trimmed condi- tion the numbers of revolutions of the fore and aft cushion fans differed in order to give 1 ft difference in height between the bow and stern. The numbers of 696 Behaviour of a Ground Effect Machine revolutions given in the diagrams are average values of the four motors, scaled up for the actual vehicle. Finally the model was tested flying over beam seas with a 15 knots wind coming from the same direction as the waves. This wind was generated by some fans mounted on the towing carriage. The number of revolutions and the direc- tion of these fans were adjusted in such a way that at 28 knots the resultant wind speed and direction had the correct values. The speed range in which the vehicle was investigated was limited by the maximum speed of the towing carriages of the Seakeeping Laboratory and of the Shallow Water Laboratory. They enabled measurements up to speeds corre- sponding to 35 and 22 knots respectively. Unfortunately these are considerably lower than the maximum speed of the actual vehicle. THE RECORDED DATA The Figs. 4 through 14 are graphical representations of the most charac- teristic data recorded. The given values apply to the actual vehicle. Motions, forces and moments are in general characterized by a mean value and a periodic oscillation round that mean. The periodic oscillations are shown as double amplitudes. The mean values are given as the difference with respect to the stationary condition with the cushion fans off. In Fig. 5 the number of points determined during the hovering tests did not justify the fairing of curves. Therefore the actual test results are indicated. The curves in the other figures are the result of fairing or cross fairing. The number of points available for fairing depended on the investigated speed range. Over a speed range from 0 to 35 knots generally about 12 runs at various speeds were made. For a speed range from 20 to 35 knots about 6 points were con- sidered to be sufficient. In general the test results appeared to be reproducible in a satisfactory manner. However, the lateral force and yawing moment showed a rather large scatter. This was caused by torsional vibration in the towing apparatus. The natural frequencies of this instrument combined with the model were in effect not high enough for the wave experiments, especially at higher speeds. The vertical motion of the centre of gravity is designated as heave. The mean value (rise height) over land is the distance between the ground and the flat bottom. Over water it is just the difference in height with respect to the floating condition in still water. The mean pitch angle (trim) is considered positive with the bow down. Roll is positive when starboard side is down. The wave direction was defined as the angle between the velocity vectors of the vehicle and the waves, positive when counterclockwise. The motions are shown in degrees and inches. The forces are given in metric tons (2205 lbs) which are about equal to long tons (2240 lbs). 697 Swaan and Wahab WITH RIGID JET EXITS WITH FLEXIBLE TRUNKS HEAVE HEAVE | 10 10 va) uo x 2 0 ——- 9 z 5 10 5 10 a} SECS. SECS = qt Ww xe -10 10 PITCH PITCH 1) 5 5 Ww wu (3 (¥) wu a oy ac = ° 2 5) 10 5 10 q SECS SECS 32 ) l= a -5 =) - ROLL ROLL 5 ) ra) i) x (0) Ww Sj Zz W (©) 4 o 5 10 cS SECS = | (o) x OVER LAND ——— OVER WATER / 2 RIGHTING MOMENT WEIGHT OF GEM RIGHTING ARM IN FT RIGHTING ARM IN FT Ss 10 HEEL ANGLE IN LEGREES 5 10 HEEL ANGLE IN DEGREES Fig. 4. Motion extinction and transverse stability curves, 2580 fan revolutions per minute 698 peoeds o190z ye ANOTAeYIg - G ‘SIT 3LANIW Y3d SNOILMIOASY NVs — > — 3LANIW Yad SNOILNIOAZY NWS = --- Behaviour of a Ground Effect Machine ‘S335 NI GOIW3Sd 1108 —- OOSE OOOE 00Gz2 OOSE OOOE 3q0NLINdWwY 318nNo0d 00Sz2e OL 3aNLidwv Ena Jeg Po en a Oz doluad ae ->° dolwad a ols SSIONV 17104 SSTONV 1104 NN ee ae — ( t Z 3qNLidwy 37aNoa SSTIONV HOlLId C 3qnindwy 37eNoa aqaninawy aianoac —~-— ~~ a om = Srarn oe Fo 02 LHOISH 3Si4 NV3W Y3lvVM Y3AO0 ——— Of GNv1 43AO man N 72 —-AHOISH «3514 NVaW Or SAV3H SYNNYL 318IX313 HLIM SiIX3 L3F GIOld HLIM D1ONV 1108 — $3348°30 Ni 4930 HDLId—™ - S3HONI Ni SAV3H~— 699 Swaan and Wahab Accelerations are given with the acceleration due to gravity (32.18 ft/sec”) as unit. They were measured at the bow and stern of the model, in the longitu- dinal plane of symmetry. The resistance over waves was only determined as an average value. HOVERING PERFORMANCE When hovering over land the model provided with flexible trunks or rigid jet exits was stable in pitch, roll and heave. The motion extinction curves are given in Fig. 4. Because of the rapid extinction it is difficult to draw definite conclusions. However, the results indicate that the heave and pitch motions were well damped. The roll damping may be qualified as fair. Over water, the hovering behaviour of the model provided with rigid jet exits was characterized by a sustained roll and heave oscillation, apparently caused by a dynamic unstability. The rolling developed fairly slowly. It took about two cycles to double the amplitude. The model appeared also to be dy- namically unstable in pitch, but to a less degree than in roll. The most remarkable phenomenon found during the tests was that the model with rigid jet exits had two modes of motion, one of which always prevailed. Which of the two dominated during a test depended partly on the initial disturb- ances to which the model was subjected. The model might roll considerably while pitching slightly or it might pitch considerably while rolling was only moderate. At the lower hovering heights the model showed a preference for the mode of motion in which rolling was dominant. The Figures refer to this con- dition. The behaviour described here is illustrated in Fig. 5. It was found that if the centre of gravity of the model was fixed at the same mean rise height which the model had when it was free to heave, the roll motion remained. This gave rise to the supposition that the origin of the roll motion could be explained by considering the uncoupled equation of motion. When the roll angle is indicated by 9, this equation is: Mo + No + Bo = O. Because the roll damping was not too large, the roll period may be approximated by 27 /WB. The coefficient B is a measure for the static stability. The measurements indicate that the value of B is larger for the vehicle hovering over water than over land. This is in contradiction with the experiences of Kuhn, Carter and Schade [5]. The natural roll periods over land and over water were almost equal. This leads to the conclusion that the virtual moment of inertia if the model hov- ering over water was larger than hovering over land, which is acceptable. The origin of the roll motion could possibly be explained by a non-linearity in the damping coefficient N, caused by the presence of the free water surface under the air cushion. A complete investigation into the cause of the dynamical 700 Behaviour of a Ground Effect Machine unstability would require the execution of forced roll and heave experiments over a range of frequencies and using various base pressures. Such an investi- gation was not included in the present research. When fitted with flexible trunks the model in the over water hovering condi- tion only suffered from self-induced roll oscillations, with amplitudes smaller than when the jet exits were rigid. The heave and pitch damping seem to have increased also, in spite of the enlargement of the air cushion by means of flexi- ble trunks. Comparison of the extinction curves over land and over water learns that the heave and pitch damping is larger over land than over water. The natural periods of these motions were smaller over land. FLYING BEHAVIOUR OVER SMOOTH WATER AND OVER LAND The behaviour of the model with both rigid jet exits and flexible trunks was quite satisfactory over land. It was dynamically stable in roll, pitch and heave. It skimmed smoothly over the ground with no appreciable change of trim at speeds up to 20 knots. Flying over smooth deep water, rolling decayed with increasing speed. The motion returned above the hump speed and it decayed again with further increase of the forward speed. In shallow water the picture was the same. This behav- iour is shown in the Figs. 6 through 9. The resistance curves had their highest hump at speeds between 10 and 12 knots, corresponding with Froude numbers between 0.40 and 0.45. These are speeds for which also the highest specific wave resistances of ship hulls are found. Apparently the water depth did not largely affect the speed where the re- sistance showed the highest hump. It affected primarily only the height of the hump. BEHAVIOUR OF THE MODEL PROCEEDING OVER REGULAR WAVES The natural periods of the pitch, heave and roll motions at zero speed lie between 1.8 and 2.5 seconds. It is reasonable to assume that these quantities do not change much with increasing speed. So the speed range and simulated wave- lengths assure that in many cases the period of encounter was equal to the natu- ral period. Figure 10 shows only slight humps in the curves of the pitch and heave am- plitudes. This indicates that these motions were well damped. The curves of the roll amplitude have a hump at the speed for which the period of encounter is expected to be about equal to the natural period. This picture of the dynamic properties is in accordance with that obtained from the motion extinction curves of the model hovering over smooth water. In these conditions the motions are 701 RESISTANCE IN METRIC TONS IN INCHES HEAVE PITCH ANGLE IN DEGREES = 40 30 20 10 Swaan and Wahab RESISTANCE MEAN WITH FLEXIBLE TRUNKS —-— -—_ WITH RIGID JET EXITS HEAVE MEAN RISE HEIGHT PITCH MEAN TRIM ANGLE 10 20 30 SPEED IN KNOTS Fig. 6 - Flying behaviour in smooth deep water, 2580 fan revolutions per minute 702 40 HEAVE IN INCHES ROLL ANGLE IN DEGREES ROLL PERIOD IN secs. Behaviour of a Ground Effect Machine HEAVE DOUBLE AMPLITUDE WITH FLEXIBLE TRUNKS —-—-— WITH RIGID JET EXITS ROLL ANGLES DOUBLE AMPLITUDE 10 20 30 40 —_=—SPEED IN KNOTS Fig. 7 - Flying behaviour in smooth deep water, 2580 fan revolutions per minute 703 IN METRIC TONS ———= RESISTANCE HEAVE ININCHES PITCH ANGLE IN DEGREES Swaan and Wahab RESISTANCE MEAN WATER DEPTH O feet 24 12. HEAVE MEAN RISE HEIGHT PITCH MEAN TRIM ANGLE — SPEED IN KNOTS Fig. 8 - Flying behaviour over land and in smooth shallow water, 2580 fan revolutions per minute 704 HEAVE IN INCHES IN DEGREES ROLL ANGLE ROLL PERIOD IN secs Fig. 9 - Flying behaviour over land and in smooth shallow water, 221-249 O - 66 - 46 Behaviour of a Ground Effect Machine HEAVE DOUBLE AMPLITUDE ROLL ANGLES DOUBLE AMPLITUDE WATER DEPTH ROLL PERIOD 5 10 15 SPEED !N KNOTS 2580 fan revolutions per minute 705 O feet ” 20 Swaan and Wahab S1ONX NI G33dS ——— Oe (er4 Ol usolL wKey4 SSS SSS WO PueSZL'Or isv NOIlVY3S1S95V —— wWole USZt NVv3aw 3qnNiNdwy 378nod —— 39404 IWwaaLVvI PueOge;s 02 Woz wsor Worl wWSZ you goldsd Zanindwy 318noa SSTONV 1104 dn—— 2° —=umop 5 NI NOlwd37399V ° ° fo) N SNOL DINL3W NI 35403 TWd3lv1 S33Y930 NI SIONV TON =—— $295 NI GOIdad eaynurut ted suotjnjTOAst ueT Y8GgZ SLONY Ni G33dS OE 02 OL WOle PUEGZL “Om GYVMYOS NOllVes1395V NVv3aW SINVLSISSY SHION3S1 SAVM 11V YO3 1H9O13H 3SIN NVaW 3AV3H dn—— © ——uMmop 6 NI NOILvd37350V SNOL SINLSW Ni 3DNViSIS38 =——_ ‘JooF Z JUSTOY BAeM {(COS MOG) GET UOTJIETIp sAeM fsSoAeM IeTNBer UT INoTAeYysg - OT “S8tqT SLONY Ni G33dS SNOL’ 13 Ni MvA 30 LNSWOW OE oz ol ) oe ee ae Mae TT --————_. — —-—~SHLON3T SAVM 11v HO4 NV3W a WSL Tey worl SZ wole |S 3qNiiidwy a1snoad MvA JO LNSWOW Paice ie leg fe] < 2S == cS os be 2. Deo UD -— ~7~-~ SHLONST SAVM WIV yYOs NVSW 777 | 0) wold wkey4 yor wsoL aqNLidWwy 318nod SAISNV HOLId SSHONI NI 3AV3H US. ZL7HLON|ST SAVM 3q0NLNdwy 318Nnod 3AV3H OL Ov os $334930 Ni 31DNV HOlid S3HONI NI 3AV3H 706 Behaviour of a Ground Effect Machine to a large extent proportional to the exciting force and moment for heave and pitch respectively. So the largest pitch amplitudes were expected in waves of about the air cushion length. The experiments showed that the largest ampli- tudes occurred when the waves were slightly longer. HEAVE IN INCHES RESISTANCE IN METRIC TONS MOMENT OF YAW IN FT.TONS -25 With regard to the wave direction, it was found that pitch, resistance and accelerations were the largest in head seas as appears from Fig. 11. Conceiv- ably, the roll motion, lateral force and yawing moment were maximum in beam seas. The worst condition for the model appeared therefore to be a bow sea prdehis MEAN RISE HEIGHT -——. SSeS SS —— HEAVE PITCH DOUBLE AMPLITUDE DOUBLE AMPLITUDE ALL WAVE LENGTHS MEAN TRIM ANGLE mapper a7 £ FOR ALL WAVE LENGTHS PITCH ANGLE IN DEGREES. L fo} LATERAL FORCE DOUBLE AMPLITUDE RESISTANCE MEAN a ———_ WAVE _ LENGTH 70 ft 105 f& 140 ft 175 ft 210 f& LATERAL FORCE IN METRIC TONS MOMENT OF YAW ACCELERATION FORWARD DOUBLE AMPLITUDE up MEAN FOR ALLWAVE LENGTHS —=~S~S~STS ACCELERATION IN g down——-_ 5 i] ou) 45 30 135 180 te) 45 90 135 180 WAVE DIRECTION IN DEGREES WAVE DIRECTION IN DEGREES Fig. 11 - Effect of wave direction at 30 knots, wave height 2 feet, 2580 fan revolutions per minute 707 Swaan and Wahab because the resistance and vertical motions were still considerable while com- bined with the lateral force and yawing moment occurring in oblique seas. The effect of a variation in wave height is shown in Fig. 12. It indicates that an increment of the wave height increased the motion amplitudes and re- sistance about proportionally. At higher speeds, however, the amplitudes of the lateral force, yawing moment and accelerations forward increased more than proportionally. The accelerations aft were hardly affected by the wave height. The effect of the wave height on the mean values of the lateral force and yawing moment was small in comparison with that on the amplitudes. For small differ- ences the rise height increased with increasing wave height. If the increments exceeded a certain value the rise height remained constant. HEAVE HEAVE DOUBLE AMPLITUDE MINE RISE HEIGHT ROLL ANGLES DOUBLE AMPLITUDE 6 ft 2and 3ft [o) WAVE HEIGHT 6ft 6 and 3 cae PERIODE 2-3.and 6ft —o. (o) WAVE LENGTH 70ft e-—— . x 105 ft -——— — HEAVE IN INCHES --e— HEAVE IN INCHES $ WITH 15 KNOTS SIDE WIND PITCH ANGLES DOUBLE AMPLITUDE RESISTANCE LATERAL FORCE MEAN a“ DOUBLE AMPLITYDE =— LATERAL FORCE IN METRIC TONS ——PERIODIN secs —=ROLL ANGLE IN DEGREES a) r4 ro) = O Ing E Ww = Zz WW oO z dq e a n W a ——<— PITCH ANGLE IN DEGREES a 2-3and 6ft MOMENT OF YAW ACCELERATION FORWARD ACCELERATION AFT DOUBLE AMPLITUDE ACCELERATION IN g = MOMENT OF YAW IN FT TONS ACCELERATION IN g down ——-- 09 —--—up eS SSS 2-3and 6ft 20 30 20 30 20 30 —— —=— SPEED IN KNOTS — --— — SPEED IN KNOTS ———— SPEED IN KNOTS Fig. 12 - Effect of wave height and side wind, wave direction 90° (beam sea), 2580 fan revolutions per minute 708 ————= HEAVE IN INCHES —.. --=— PITCH ANGLE IN DEGREES —- —— MOMENT OF YAW IN FT TONS Behaviour of a Ground Effect Machine Figure 13 learns that the resistance, the pitch and heave amplitudes de- creased when the rise height was increased by means of higher fan revolutions. The accelerations were therefore expected to be lower at larger flying heights. This appeared to be true except for the accelerations forward in bow seas. When the cushion fans were adjusted in such a way that at zero speed the vehicle was trimmed by head or by stern, the character of the behaviour did not change very much. Comparison of Fig. 14 with Fig. 10 shows that heave and pitch amplitudes are lower in both trimmed conditions than at even keel. At speeds over 30 knots the resistance increased with trim by the head and decreased when the vehicle was trimmed by the stern. At lower speeds this was reversed. HEAVE HEAVE DOUBLE AMPLITUDE MEAN RISE HEIGHT FOr 105 ft 70 -105ft —_ = SSS ROLL ANGLES DOUBLE AMPLITUDE WAVE LE NGTH-105ft —— = -— 2580 FAN REVOLUTIONS PER MINUTE — 3210 . PERIOD 195 ft SS — SS ae 7O0ft —-——— HEAVE IN INCHES — PERIOD IN secs —ROLL ANGLE IN DEGREES) LATERAL FORCE DOUBLE AMPLITUDE ~--—--=— LATERAL FORCE IN METRIC TONS 1 fo} to starboard a) z re) e o a - rt) = Zz ey 8) Zz x w 7) i) 4 MOMENT OF YAW ACCELERATION FORWARD ACCELERATION AFT DOUBLE AMPLITUDE ACCELERATION IN g ACCELERATION IN g down ——— 0 ——uPp —— ‘70 -105tt — SPEED IN KNOTS -- ~— SPEED IN KNOTS —-— -=— SPEED IN KNOTS Fig. 13 - Effect of rise height. Wave direction 135° (bow sea), wave height 2 feet. 709 Swaan and Wahab HEAVE HEAVE DOUBLE AMPLITUDE MEAN RISE HEIGHT ROLL ANGLES DOUBLE AMPLITUDE TRIMMED BY HEAD TRIMMED BY STERN WAVE LENGTH 105ft ~— ——— — HEAVE IN INCHES = HEAVE IN INCHES PERIOD PITCH ANGLES RESISTANCE DOUBLE AMPLITUDE MEAN LATERAL FORCE DOUBLE AMPLITUDE fostt MEAN — MEAN TRIM ANGLE a a and 70ft Oft --— PITCH ANGLE IN DEGREES m1) z re) - 2 a - Ww = z i) ©) z z z uw ie) Qo © < % = z Ge he Ww WwW ui = it rm 3 8 9 | : : -25 20 30 20 30 20 30 ———. = SPEED IN KNOTS —-__—== SPEED IN KNOTS SPEED IN KNOTS Fig. 14 - Effect of trim on the behaviour in waves. Wave direction 135° (bow sea), wave height 2 feet, 2580 fan revolutions per minute The effect of a beam sea combined with a side wind was investigated for a speed of 28 knots only. The measured data are given in Fig. 13. It shows that the effect of a side wind was very small. CONC LUSION With regard to the behaviour of the vehicle the following general conclusions may be drawn. The design with rigid jet exits proved to be dynamically unstable over water especially in regard to rolling. Minor changes in the jet exit arrangement could not remove this difficulty. The installation of flexible trunks, however, improved 710 Behaviour of a Ground Effect Machine the behaviour considerably. Although it is expected that the dynamical unstabil- ity is caused by non-linear damping this could not be established with certainty. The weight of the model was such that the base pressure during the experiments was higher than in the actual design condition. The resistance over smooth water showed a maximum in the speed range between 9 and 14 knots depending on the water depth. The highest resistance hump in shallow water was about 50% higher than on deep water. The pitch and heave motions of the model proceeding in regular waves showed the character of well damped systems. The behaviour was apparently worst in bow seas of about vehicle length or somewhat longer. With increased cushion power and resulting larger rise heights the motions and resistance showed a tendency to decrease. At high speeds the resistance could be reduced considerably by trimming the vehicle by stern. In this condition the motions decreased as well. A side wind seemed to have only a minor effect on the behaviour of the vehicle. The measured data did not show many unexplainable trends and in general the results could be reproduced within reasonable limits. An exception must be made for the yawing moment and sway amplitudes which records were rather blurred by high frequency noise. REFERENCES 1. Fuller, F. L., 'Gravity Wave Drag Theory for the Waterborne Ground Effect Vehicle,"" Grumman Aircraft Engineering Corp., Research Note RN-111, June 1959. 2. Hirsch, A. E., ''The Hovering Performance of a Two Dimensional Ground Effect Machine Over Water,'' Symposium on Ground Effect Phenomena, Princeton University, Princeton, 1959. 3. Mack, L. R., Theoretical and Experimental Research on Annular Jets Over Land and Over Water,'' Symposium on Ground Effect Phenomena, Princeton University, Princeton, 1959. 4. Tulin, M. P., "On the Vertical! Motions of Edge Jet Vehicles,'' Symposium on Ground Effect Phenomena, Princeton University, Princeton, 1959. Ds Kuhn, R. E., Carter, A. W. and Schade, R. O., "Over Water Aspects of Ground Effect Vehicles," Institute of Aeronautical Sciences, paper no. 60-14, New York, 1960. 6. Lin, J. D., 'Dynamic Behaviour of Ground Effect Machines Over Waves," Journal of Ship Research, Vol. 6, number 4, April 1963. 711 Swaan and Wahab 7. Wiegel, R. L., c.s., "Research on Annular Nozzle Type Ground Effect Ma- chine Operating Over Water"; 'Operation Over Waves,'' Hydraulic Engi- neering Laboratory, University of California, Berkeley, 1963. 8. Cumming, J. D., "Research in Annular Nozzle Type Ground Effect Machine Operating Over Water"; ''Water Surface Configuration,"' Hydraulic Engi- neering Laboratory, University of California, Berkeley, 1963. 9. ''Tests with a Flying Model of a Ground Effect Machine SKMR-I 'Hydro- skimmer','' Seakeeping Test Report no. 121, Netherlands Ship Model Basin, August 1963. 10. "Tests with a Model of a Ground Effect Machine SKMR I 'Hydroskimmer' Flying Over Regular Waves,"' Seakeeping Test Report no. 157. Netherlands Ship Model Basin, July 1964. * * * DISCUSSION W. A. Crago Saunders-Roe Division of Westland Aircraft Limited Wight, England For reasons of commercial security it is relatively rare for practical data obtained on models of full scale hovercraft to be published and I personally would like to say that I was, therefore, very pleased to see this excellent paper by Mr. W. A. Swaan and Mr. R. Wahab. This is all the more interesting to me because in the Saunders-Roe Tanks we spend a fair proportion of our time testing hovercraft. We now have full scale area model test results for the N1, N2, N3 and 5 full scale variants of the N5 (these are craft ranging in weight from 7 to 37 tons) and, with this background, I can confirm that the type of test reported in Mr. Swaan's paper, using dynamically scaled free models, can give results which correlate acceptably well with data obtained in the full scale regime, although as a result of our experience we would prefer not to use a model quite so small as the N.S.M.B. Hydroskimmer because of scale effects in the jets. The N.S.M.B. model test philosophy, in which the propulsion propellers are not represented is, I feel, acceptable except in cases where the propellers can affect the flow into the fan intakes. The way in which the fan intake flow is rep- resented is important because the intake flow geometry determines the moment arm at which the momentum drag acts. This moment affects the crafts running trim and this in turn affects the drag. Correct representation of the intake flow is thus essential. 712 Behaviour of a Ground Effect Machine Mr. Swaan's paper presents results obtained in regular seas. We have found tests in such an environment of only limited usefulness and it is now our practice to run all hovercraft tests whether for ourselves or customers in ir- regular seas having energy spectra based on a Darbyshire formulation modified to make it represent the coastal and local conditions with which we have to deal in our full scale trials. We have found good agreement between the derived amplitude response operators and the accelerations obtained from irregular sea tests and regular sea tests up to sea conditions associated with a wind of Beaufort force 5 but be- yond this, I feel that irregular sea testing is essential because of the non- linearity of response. A further point which is perhaps worth mentioning is that we have used mo- tion extinction curves similar to those given in Mr. Swaan's paper, together with the static characteristics and a fairly simple wave impact theory as inputs to an analogue computer. In this work we have found that we can predict the model motion and accelerations and the effect of pitch control systems, steering sys- tem failures, etc., to a very acceptable degree of accuracy at least in regular waves and irregular waves up to a Beaufort force 5 wind. (This work has been published by my colleague, Mr. J. Stafford, as a paper read before the British Society of Instrument Technology in June, 1963 and ap- pears in their transactions). The Wageningen data shows that the value of V//gL at which the hump pitch- ing attitude occurs decreases as water depth decreases. The way in which it does this agrees precisely with our own findings and the way in which the cor- responding resistance varies is also generally similar. In connection with the presentation of the data, I would like to see values given for the momentum drag. This would facilitate analysis and comparison with other data. In closing I would like to make a further small criticism. It is a pity the test points have not been shown in Figs. 6 to 9. I feel the curves have been over-smoothed and that there should be almost a discontinuity for example in the heave curves at Froude numbers between 0.3 and 0.6 depending on water depth. 713 Swaan and Wahab DISCUSSION R. F. Lofft Admiralty Experimental Works Gasport, England This paper represents a useful addition to the published literature on the behaviour of hovercraft over water. It illustrates the difficulty of testing models of such light, high-speed craft in normal ship model tanks. It is gener- ally impossible to fit equipment in the model to measure all the motions of in- terest, and the arrangement adopted by the authors to permit heave, pitch and roll, and to restrain the model in yaw, sway and surge seems a reasonable compromise. Turning now to several points of detail: (1) The two diagrams at the bottom of Fig. 4 show that, with rigid jets, the righting arm is much greater over water than over land, while with flexible jets, the difference is much less. No reason can be seen for this and it would be in- teresting to have the authors' comments. (2) The results of the shallow water tests in Fig. 8 show a marked peak in the resistance curve at 12 ft depth. It is interesting to note that this occurs at the critical speed for this depth, viz. 11.7 knots. The same does not appear to be true for the other two depths tested, at which the critical speeds are 4.7 knots for 2 ft depth and 19 knots for 32 ft. (3) Figure 10 gives the results of tests in waves with flexible trunks, in which the mean rise height is given as 25-30 inches. This is nearly 10 inches less than the corresponding figures for still water, from Fig. 6. This is some- what surprising, since the wave tests show the mean rise height to be independ- ent of wave length, and one would therefore expect it to be about the same as in still water. (4) The authors suggest that the maximum pitch amplitude would be ex- pected to occur in waves of about air cushion length. This is true of normal ship speeds; but at higher speeds, e.g., in model tests of fast planing craft, it has been found that maximum pitching generally occurs in waves of 2 - 2-1/2 times model length. This is consistent with the results in Figures 10 and 11 in which the greatest pitch occurred in waves 105 ft long - nearly twice the air cushion length. 714 Behaviour of a Ground Effect Machine REPLY TO THE DISCUSSION W. A. Swaan and R. Wahab Netherlands Ship Model Basin Wageningen, Netherlands It is gratifying to have Mr. Crago's comment on the test results because of his experience both with models and actual hovercrafts. The small size of the model was necessary because of the maximum available carriage speed other- wise it would certainly have been desirable to use a bigger model. The use of regular waves was selected in order to obtain an impression about the transfer functions. Moreover the wave generator in the Seakeeping Laboratory is not suitable for the generation of irregular long crested oblique waves. It is true however that the use of irregular waves is to be preferred in many respects when predictions have to be made for the performance in a given area where the sea conditions are known. Because the air flow for the jet system was not measured the momentum drag can not be determined from the experiments. However Fig. 8 gives the total resistance when flying over concrete. Because of the negligible trim, resistance must be mainly momentum drag. In regard to Mr. Lofft's question about the effect of the flexible trunks on the difference between the stability over land and over water it must be re- marked that the cause of this phenomenon can only be found when measurements are taken of both pressure and water surface shape under the vehicle. The en- ergy loss caused by the smooth water surface waves is not only compensated by the resistance of the GEM but also by the air cushion. This can be shown by the fact that no "wave resistance" will be found notwithstanding the visibility of surface waves if the GEM is kept horizontal, provided that the air flow is kept constant. Therefore it is the opinion of the authors that coincidence of maxi- mum resistance with the critical speed is not physically necessary. In the conclusions it is mentioned that pitch angles are the largest with waves of about vehicle length or somewhat larger while Mr. Lofft notices that the diagrams show a maximum at about double the air cushion length. However, if the system had no damping, the maximum pitch angles would occur at reso- nance; that is a wave length of 175 ft at the speed of 30 knots in bow seas. Because the maximum under these conditions occurs in much shorter waves it is clear that damping is rather large. Therefore the pitch angles are much more determined by the wave moment than by the frequency of encounter. The remark about the effect of wave length or pitch must be considered in this light, although it is admitted that the expression "about air cushion length" was stretched somewhat too far. 715 rg : RM i 3 ae a ; oP \ - i 1 ay -~ ee i a a , he . , a i - 17 Te, ; : ade + ‘. i aa Wy eae open é : Le . i r CORY Tes vy, ; I “ he a4, ; ; pe 4 iy i; < - os if i ; iy +) * ' D de WP at her 2 ; rae I as Fi AY | ey : \ f . 4 F rf * ay uss x ‘ rr 4 vi ’ ts ’ ae ee i ev i a » ; * 7 are t se ¥ 4 , : ’ t 1h hy. rt hy i tone wih. 8 To F Pry ; et te 4 , ‘ee y 4 t n i i ae - } 1 ), " Li y , : , 5 1 / : ve ra j ‘ i : A ‘ fA i, at H } j Ny i bs o } v 5 " a ‘ vw a 5 i } ee had ai c eal bi « yh a f ‘ig fu eet ; ; i by ~) : iy ’ } a 4 4d é a ea ae Bip ae uy ‘ Ne i Lary 7 ut , 4 “Y is : wy ‘ * ; : y y " ; f *] : } ; j , } ni Md ! , } i / i 7 id ae ii i : : ) y gael ¥ be : } t "i i } ON 1 ' nN =, ~ Hh 7 fu i ® A t ‘ 7 y 4 =) " i i ' \ t it j 7 f uy si! i ; ‘ + ae : bit . BEHAVIOR OF UNUSUAL SHIP FORMS E. M. Uram and E. Numata Stevens Institute of Technology Hoboken, New Jersey INTRODUCTION Four years ago, almost to the exact date, the Third Naval Hydrodynamics Symposium, constituted and attended by many of the gentlemen participating in this symposium was held at Scheveningen in the Netherlands; a short skip, gur- gle, or flight from here, depending upon which unusual high performance ship one chose to use. Almost the entire symposium was devoted to discussions on the nature and problems associated with high performance ships. The papers of Mr. Owen Oakley [1] of the United States Navy Bureau of Ships, Dr. Van Manen [1] of the Netherlands Ship Model Basin, and Dr. Breslin and Professor Lewis [1] of our own laboratory, at that time pointed out many of the design and operational problems attendent with the unusual ships shown in Fig. 1. Their relative power and seakeeping behavior were discussed based upon reasonable analytical estimates, or very limited experiments. A substantial number of the other papers presented at that symposium and at other subsequent meetings of- fered information concerning the performance and limitations of hydrofoil craft, planing craft, and GEM's. We will not reiterate these arguments at this time, but just point out that the prime objective is the attainment of high speeds in rough seas while maintaining reasonable horsepower requirements and reducing the well known severity of motions in seas at high speeds. We will be concerned in this discussion primarily with unusual surface or sub-surface vessels in the 3,000 ton displacement class at speeds in the vicinity of 40 to 50 knots, although we will discuss the behavior of these ships over the entire speed range. The design philosophy of unusual form surface ships such as the Large Bulb Ship (Escort Research Vessel) and the Semi-Submerged Ship (Decks Awash Ship) is such as to change the pitching and heaving periods so that the ship will operate in sub-critical or super-critical zones of operation as de- fined by Professors Lewis [2] and Mandel [3]; operation conditions in which the ship is not in resonance with the encountered wave system. The shallow running submersibles like the Shark Form and Semi-Submarine take advantage of the at- tenuation with depth of wave system effects. However, the single surface pierc- ing strut of these ships makes them unacceptable from a stability and control point of view. The Hydrofoil Semi-Submarine, Fig. 2, is a design conceived by the senior author [4] affording inherent stability in this ship type. The stability referred to is defined as the ability of the vessel with controls fixed to seek and return to its initial trim, depth, and course after being disturbed from these conditions. TQly) Uram and Numata ee cee A rere le [PLANING HULL > ROAR DESTROYER ENGTHENED DESTROYER SHARK FORM Se AD SEMI- SUBMARIN Fig. 1 - Ship forms for high speed at sea The Third Symposium was a major impetus which propagated substantial investigative work into the performance of these unusual ship forms. The Bu- reau of Ships andthe Office of Naval Research supported an extensive program to obtain information concerning the performance of these unusual ship forms. A substantial investigation of the characteristics of the Shark Form was con- ducted at the Massachusetts Institute of Technology by Professor Mandel [5] and his associates in a relatively low speed range. It gives us pleasure to say that a substantial amount of work was done at our own Davidson Laboratory on the Semi-Submerged Ship [6], the Large Bulb Ship [7] and the Hydrofoil Semi- Submarine [8]. Professor Mandel [9], acting as a consultant and member of the Panel on Naval Vehicles of the National Academy of Sciences' Committee on Undersea Warfare, published a primarily analytical, exhaustive comparative study of novel ship types from which we will draw from time to time. It is of general interest to note at this point that Professor Mandel's study of endurance and pay load indicates that the pay loads for all of the ship types that we will 718 Behavior of Unusual Ship Forms Fig. 2 - Hydrofoil semi- submarine discuss today in the 3,000 ton class are very much competitive in the 2,000 mile endurance range at a cruise speed of 20 knots. It is mainly our purpose in this paper to present comparisons of the several unusual form ships based upon experimental information accumulated to date. First we will take up the powering characteristics of these ships in both calm and regular sea conditions and then go on to the motion characteristics under these same conditions. It is of interest to point out that most of the results we will present on the Hydrofoil Semi-Submarine and Large Bulb Ship are relatively new and have not been discussed widely. Therefore, we will dwell in some de- tail on some of the characteristics of these two particular ships. SPEED AND POWER BEHAVIOR The resistance characteristics of the Large Bulb Ship with the forward bulb in various positions was investigated in the course of the study. As shown in Fig. 3, the results for residual resistance are given for the various bulb posi- tions, as well as for the bare hull, and it is seen that the most forward bulb position results in substantial residual resistance reductions from the bare hull as well as the other bulb positions over the speed range. It was this for- ward bulb position that was used during the remainder of the study on ship mo- tions. In order to establish the existence of an optimum form for the semi- submarine hull, a study was made of streamlined body of revolution character- istics in which it was discovered that in the high speed range, Froude number in the vicinity of unity, the residual resistance coefficient of such bodies run- ning near the surface can be considered to be approximately 25% of the deeply submerged frictional resistance coefficient. Therefore, it was necessary only 719 R, Las TON 4 Uram and Numata BARE HULL =a BULB AFT BULB FORWARD BULB AFT POS, BULB INTER, GosaNG BULB FwO, — POS. /LWL Fig. 3 - Residual resistance comparison for various large bulb ship configurations 720 Behavior of Unusual Ship Forms to study the deep submergence frictional characteristics in order to determine whether there indeed exists an optimum form in the design Froude number range. Figure 4 indicates for such bodies the specific horsepower (EHP per ton of displacement) as a function of speed, fineness ratio and body length. Since for each set of fineness ratio curves the velocity is constant and the abcissa used is body length, it is possible to associate a Froude number with that velocity and body length. Therefore, a Froude number scale is superim- posed on the abcissa of the figure. We see from Fig. 4 that there does indeed exist for, say, a 3,000 ton vessel an optimum fineness ratio of 5. This was used in the design of the Hydrofoil Semi-Submarine. We will dwell a little further on the Hydrofoil Semi-Submarine in order to acquire a proper interpretation of the information to be presented subsequently for comparison with the other ships. A substantial part of the tests performed L/D=5 2X ARBITRARY SCALE BODY LENGTH L, FT 270 250 230 210 190 170 160 FROUDE NUMBER SPECIFIC HORSEPOWER, EHP/A 45 KNOTS a —— A=3000 TONS BODY LENGTH L, FT. 270 230 0 190 170 150 ARBITRARY SCALE 08 0.9 1.0 1d FROUDE NUMBER Fig. 4 - Specific horsepower (streamline bodies of revolution at deep submergence) 221-249 O - 66 - 47 721 Uram and Numata on the Hydrofoil Semi-Submarine were such that the ship was free to surge, pitch and heave; the variable ballast, hydrofoil flap and stem plane angles being set for an equilibrium ship trim attitude at the design depth and run speed. During the runs the model exhibited excellent stability and sought its own running equilibrium trim condition for the speed of the run. Therefore, the ship depth and trim attitude, in many cases was different from design conditions or from those used in the tests conducted under restrained motion conditions. Figure 5 shows the pitch and heave equilibrium attitudes of the Hydrofoil Semi-Submarine in calm water and we see that the assumed trim angle of the vessel varied around the design equilibrium trim angle of zero degrees. We See, also, that the submergence depth of the vessel varied around the design depth of approxi- mately 1.5 diameters below the surface. Figure 6 shows the corresponding calm water total resistance coefficient as a function of Froude number. Also shown for comparison are results obtained from the restricted motion tests at the design depth for various trim angles. The calm water resistance coeffi- cient plot is, therefore, quite realistic; representing what might actually be en- countered under operational conditions while the other curves give much lower resistance coefficients under absolutely ideal conditions. The calm water re- sistance coefficient was used in the calculations of horsepower requirements. Figure 7 depicts the mean equilibrium attitudes of the Hydrofoil Semi- Submarine in regular waves. Not all of the data are presented here, but enough are presented to give an idea of the range of conditions encountered. Figure 8 and Fig. 9 show the total resistance coefficient as a function of Froud number for this ship under regular following waves. We see that there apparently is no discernible difference in the drag coefficient with respect to the height of the wave system in 1.0 L waves, whereas in wave lengths twice the ship length a sub- stantial difference exists between the resistance coefficients for different wave heights. Further, spotted onto these figures is the calm water resistance curve. In both figures we see that the resistance coefficient in regular following waves is higher than the calm water resistance, particularly for the wave height to ship length ratio of 1/22.5. These resistance coefficients and resistance coefficients taken from Van- Mater's [7] data for the Large Bulb Ship, Lewis' and Odenbrett's [6] for the Semi-Submerged Ship and Davidson Laboratory data for a conventional destroyer were used to calculate the horsepower requirements for the calm water and vari- ous regular sea conditions. The standard calculation method for EHP was em- ployed with the exception that a 30% increase in the Schoenherr skin friction co- efficient was applied to the Hydrofoil Semi-Submarine to account for the skin friction contribution of the main hydrofoil system and stern planes. This is reasonable and in keeping with knowledge of the additional frictional resistance experienced in normal submarines due to the sail, fair water, and stern planes. Figure 10 gives an EHP comparison of the various unusual form ships and the conventional destroyer in calm water. We see that up to 30 knots the power of the Hydrofoil Semi-Submarine is substantially higher than the other three ships, whose powers are comparable, because the Semi-Submarine experiences its maximum wavemaking resistance in this speed range. Between 30 and 40 knots all three unusual ship forms are better than the conventional destroyer. At 40 knots and above the Hydrofoil Semi-Submarine is substantially better than the 722 TRIM ANGLE SUBMERGENCE DEP TH/MAXIMUM DIAMETER Behavior of Unusual Ship Forms iT: FROUDE NUMBER 0.2 0.4 0.6 0.8 1.0 to) 10 20 30 40 SHIP SPEED, KNOTS Fig. 5 - Pitch and heave equilibrium attitudes calm water (hydrofoil semi-submarine) 723 50 Uram and Numata d/0= 1.445 HYDROFOIL O.028L AFT MIDSHIPS 3 C, X10 FROUDE NUMBER Fig. 6 - Cy) vs Froude number motion tests in calm water (hydrofoil semi-submarine) 724 MEAN TRIM ANGLE MEAN SUBMERGENCE DEPTH/MAXIMUM DIAMETER Behavior of Unusual Ship Forms 4 @ 100 45.0 © 100 45.0 4100225 A 100 225 fo) fo) 20 30 40 SHIP SPEED, KNOTS Fig. 7 - Mean pitch and heave attitudes regular waves (hydrofoil semi-submarine) 725 ee i 1.6 Ee. We 15 kA eke Baste 4 eMnSlY Be el 0 Lor 1.4 Ol O AHEAD FOLLOWING A d/L L/h Wile merit 13 ) FROUDE NUMBER 0.4 0.6 0.8 3 Cy X10 Uram and Numata O h/t =!/22.5 O h/L = |/45.0 Fig. 8 - D FOLLOWING SEAS 0.6 FROUDE NUMBER vs Froude number regular 1.0 L waves (hydrofoil semi-submarine) 726 3 Cp X10 Behavior of Unusual Ship Forms O-h/L:= 1722.5 O-h/L:= 1/45.0 FOLLOWING SEAS 0.6 FROUDE NUMBER Fig. 9 - C, vs Froude number regular 2.0 L waves (hydrofoil semi-submarine) 727 Uram and Numata LARGE BULB SHIP 4=3358LT i 5) DESTROYER B= 2844 TONS EFFECTIVE HORSEPOWER X10 SEMI SUBMERGED SHIP HYDROFOIL =2770LT SEMI-SUBMARINE L 10 20 30 40 50 60 SHIP SPEED, KNOTS Fig. 10 - Effective horsepower comparison calm water other ships; the conventional destroyer is next best, followed by the Semi- Submerged Ship and the Large Bulb Ship, in that order. The Shark Form would be higher than all of the ships over the entire speed range. Figures 11 and 12 present an EHP comparison in regular waves. As in calm water, the Hydrofoil Semi-Submarine shows to best advantage at speeds 728 ="3 EFFECTIVE HORSEPOWER xX !0 Behavior of Unusual Ship Forms LARGE BULB SHIP 4=3358LT SEM! SUBMERGED SHIP 4 = 3835 LT DESTROYER HYDROFOIL SEMI — SUBMARINE SHIP SPEED, KNOTS Fig. 11 - Effective horsepower comparison regular 1.0 L waves 729 EFFECTIVE HORSEPOWER x 10° Uram and Numata | ! y LARGE BULB SHIP | QO = 3358 LT 50 SEM! SUBMERGED SHIP A= 3835LT HYDROFOIL SEMI- SUBMARINE A=2720 LT 30 a DESTROYER A= 3358LT 20 — (ie) - ai ah 0 r | (0) 10 20 30 40 50 SHIP SPEED, KNOTS Fig. 12 - Effective horsepower comparison regular 2.0 L waves above 35 knots, while the Large Bulb Ship generally has an advantage at speeds up to 35 knots. MOTIONS BEHAVIOR Professor Mandel's calculations [9] on the critical speed zones of operation for various sea states having a Neumann spectra are reproduced, in part, in 730 Behavior of Unusual Ship Forms Figure 13; critical speed zones correspond to severe motions, wet \decks, and slamming while sub and supercritical zones correspond to very moderate mo- tions and intermediate zones to motions intermediate between the two extreme conditions. A complete analysis of this figure is given in Mandel's paper. It is of interest for our purposes to examine the major differences in zone extent pre- dicted for these unusual ships and, further, to remark that these results of an idealized analysis are supported to a large extent by the available regular sea data. The destroyer is seen to be sub or supercritical in all following seas, while the Semi-Submarine enjoys these conditions for all ahead seas and following LENGTH OF LONGEST SIGNIFICANT WAVE uw = a > n a WwW wn ASTERN SEAS SHIP SPEED (KNOTS) AHEAD SEAS ————o CRITICAL ZONE INTERMEDIATE ZONE SUB OR SUPER CRITICAL ZONE PITCH — —— HEAVE Fig. 13 - Operation zones in rough seas for several unusual ships 731 Uram and Numata seas up to about 15 knots. It is interesting that the analysis correctly infers increased heave activity for the Large Bulb Ship. The figure indicates that re- duced motions at high speed can be expected from all the unusual ship forms. For particular speed ranges, Mandel investigated the extent of each zone area relative to the entire plot area for a given speed range. Figure 14 was so constructed and gives a more direct comparison of the various ships although the probability of a sea state occurrence is not included. It is seen that the Semi-Submarine is superior in the narrow, low and high speed ranges of 0-20 knots and 40 to 50 knots as well as over the entire speed range, 0-50 knots. Motions data, mostly in regular seas, for these ships, the pertinent dimen- sions of which are given in Tables 1, 2 and 3, have been obtained at the Davidson Laboratory. Figure 15 shows results for regular 1.0 L head waves. Substantial pitch reductions are realized with both the Large Bulb and Semi-Submerged ships above 10 knots and heave reductions realized above 20 knots. The detun- ing transfers the severe motions to the low speeds, as predicted. Figure 16 shows that in 2.0 L waves pitch reductions are obtained above 20 knots but both surface type unusual ship forms encounter more severe heave motions over the speed range than does the destroyer. Results in irregular seas for the Large Bulb Ship, Figure 17, show pitch reductions at high speed but substantial heave increases are incurred. 0-20 KNOTS 40-50 KNOTS 0-50 KNOTS SLENDER SHIP CATAMARAN DESTROYER LARGE BULB SHIP SEMI SUBMERGED SHIP. SHARK FORM SEMI SUBMARINE (0) 20 40 60 80 100 O 20 40 60 80 100 O 20 40 60 80 100 PERCENT PERCENT PERCENT CRITICAL ZONE INTERMEDIATE ZONE SUB OR SUPER CRITICAL ZONE Ea Fig. 14 - Extent of operation zones relative to entire operation range, op- eration range: State 2 to 7 head and following seas, 0 to 50 knots 732 Behavior of Unusual Ship Forms Table 1 Hydrofoil Semi-Submarine Principal Data —(A= 45)—> Model Length Maximum Diameter, ft. Displacement L.C.B. (fwd of midship), ft. V.C.B. (above body axis), in. V.C.G. (above body axis), in. G.B., in. Radius of Gyration (Long.) Surface Area, sq. ft. oo ONFONrFwWaA4 ONworsa Hydrofoil (NACA 16-009) Chord, ft. (20% flap) Span (wetted @ d/D = 1.5), ft. Aspect Ratio (planform) Dihedral Angle, degrees Area (wetted), sq. ft. Horizontal Stern Plane (NACA 16-009) Chord Ft. Span, ft. Aspect Ratio Area (wetted), sq. ft. Vertical Stern Planes (NACA 16-009) Chord, ft. Top Bottom Midspan, ft. Top Bottom Area (wetted), Top Bottom Aspect Ratio Top Bottom 733 Uram and Numata 69 L'T TLVE LI ITLvé Iaf£014S89q drys peszeuiqng-twieg eyeq eTeog-TIng syuv} yeod u1a}s pue MOQ SutpooTy Aq padastyoe ‘yyerp deep yx o1jyer aTeog ‘das ‘potzed sAvay Ternyen IMTA /AL o1yer yy8ueT-potsreg -9a8 ‘Ay ‘potzad Suryoud jernjen uolje1AS JO SNIPE. TeutpnyisuoT A/V ‘JWeto1yJa00 vare-poyjem &/@ JUSTITJJOOO ouvTd-19zeM ¥8 g(00T/TM1)/V dseq ‘al «(OL bP Tewi.tou ‘Te,0], “AI ~09°CT Tewltou “sqing ‘aT ~OG'TE yeutrou ‘TN yusUIaDeTdSIq yusws0KeTdstp deseq qusUIa0eTCSIp Teul.tou ‘q{tnq JO wI0}}0gq quowieoKe{dsIp Teur.rou ‘TINY "yy “Weaiq "yy ‘wee "73 ‘QING Yove Jo ‘wig "7 ‘QINq Yove Jo yYoueT "3 CIMT aH NOVA dtys T2POIn qing-o31e 7 qiIng-os1e7 eyed [epoW Z MAeL, dIISTIIJOVILYD 734 Behavior of Unusual Ship Forms Table 3 Shark Form Principal Data Length, ft. Maximum Diameter, ft. Displacement (Total) Radius of Gyration (Long.) Wetted Surface (Total) sq. ft. Pitch Period, sec. Heave Period, sec. Strut Length, ft. Strut Maximum Beam, ft. Hull Prismatic Coefficient Strut Prismatic Coefficient Hull and Strut Offsets see reference 5 SRS Ss Sis ADANMOAUNHEAMNS o|w 2|5 SEMI— SUBMERGED SHIP s|a AT DEEP DRAFT, 4=3835 LT WAVE HT=L/48 alliin DESTROYER WAVE SLOPE = |.87° a|z A=3471LT = x|* o|x< a == LARGE-BULB SHIP O=3471LT SHIP SPEED, KNOTS SEMI-SUBMERGED SHIP AT DEEP DRAFT. A=3835LT LARGE- BULB SHIP 4 =3471LT DESTROYER =< ae LT ~ Pa HEAVE AMPLITUDE WAVE AMP! ITUDE fe) bo) 10 15 20 25 30 35 40 45 so SHIP SPEED, KNOTS Fig. 15 - Pitching and heaving motions in regular 1.0 L waves 735 PITCH AMPLITUDE / MAXIMUM WAVE SLOPE HEAVE AMPLITUDE / WAVE AMPLITUDE Uram and Numata WAVE HT=L/48 WAVE SLOPE=3.75° LARGE BULB SHIP A=3471LT DESTROYER A=3471 LT SEMI SUBMERGED SHIP AT DEEP DRAFT, A=3835LT 10 20 30 40 50 SHIP SPEED (KNOTS) SEMI SUBMERGED SHIP LARGE BULB SHIP AT DEEP DRAFT A=3835LT A=3471 LT DESTROYER A =347) LT 10 20 30 40 50 SHIP SPEED (KNOTS) Fig. 16 - Pitching and heaving motions in regular 2.0 L waves 736 Behavior of Unusual Ship Forms OS Sb seos Te[NdetIAt ul ‘Sq’ Fo suotjour Sutaeoy pue Suryoyid oseaoay - PT “317 Ov ¥3A081S30 S¢ Of 149° E8E =7M7 Diidve2 V 13 97E=10M7) 1129227 (SLONN) G33dS dIHS Se Oe S| Ol S fo) S Ol S| —=— Sv3S 0V3H Sv3S SNiIMO1703 ————— dIHS 91N8-39uv7 ad = ie} > o ° c o (f= m > = v - 4 [= o m o m (>) WNYLISdS ADY3N3 SAVM UV TIWIS ¥ ONIAVH AVMVSS V NI JINVNYOIY3d JO SNOILIIOSYd JUV 3Y¥v 00 404 SSNTVA S3AVM NNVL BVINS3Yy! NI SANTVA G3YNSV3W 3yV 7u3A0u81S 30 dIHS 81N8-39YvI HOS S17NS3auY 2 14 092@*HLON31 SAVM 39VY3AV ‘dJIHS @1NE-39yv1 14396" 1H 3AVM 39VY3SAV | *S3LON 13‘ 30NLINdWY 379N0d 3AV3H 737 221-249 O - 66 - 48 Uram and Numata PITCH AMPLITUDE / MAXIMUM WAVE SLOPE HEAVE AMPLITUDE / WAVE HEIGHT WAVE LENGTH/ MODEL LENGTH L, /L Fig. 18 - Shark form motions Shark Form motion behavior with varying wave length in following regular seas, obtained at Massachusetts Institute of Technology, over a very limited speed range, 0 to about 10 knots (F = 0.30) is given in Fig. 18. Although, as Mandel points out, the behavior for F = 0.1 is suspect, the behavior at the other Froude numbers gives insight as to influence of wave length on the motions for a given operating speed. The pitch response of the Hydrofoil Semi-Submarine in 1.0 L and 2.0 L regu- lar waves is presented in Figs. 19 and 20, while the heave response is given in 738 O-h/L=1/45.0 @-h/_=1/225 PITCH DOUBLE AMPLITUDE/ MAXIMUM WAVE SLOPE Behavior of Unusual Ship Forms HYDROFOIL, SEMI SUBMARINE FOLLOWING SEA 4.0) @-h/L= 1/225 O-h/L=1/45.0 PITCH DOUBLE AMPLITUDE/ MAXIMUM WAVE SLOPE nN ° 50 40 30 20 Fig. 19 - Pitching motion in regular 1.0 L waves FOLLOWING SEA Fig. 20 - Pitching motion in regular 2.0 L waves LARGE BULB SHIP 4-347) OVERTAKING SEA SHIP SPEED (KNOTS) SEMI SUBMERGED HULL ™ >~ — DESTROYER Fa i Sos | A=3471LT a DEEP DRAFT 4=3835 40 AHEAD SEA | HYDROFOIL SEMI SUBMARINE 10 fe) OVERTAKING SEA SHIP SPEED (KNOTS) fm / i i | » DESTROYER } Jp A=3471 LT =i | | a LARGE BULB SHIP =e | A =3471LT Ft | SEM! SUBMERGED SHIP DEEP DRAFT A=3835 LT 20 AHEAD SEA 30 40 Figs. 21 and 22 for 1.0 L and 2.0 L waves, respectively. An interesting comment is in order here concerning the nature of the response. As will be noted, the re- sponse data in ahead seas of the other three ships also shown on the figures for comparison, have similar characteristics in that the response reaches a peak under critical conditions and then falls off. However, the response of the Hydro- foil Semi-Submarine contains two peaks instead of one. It was found that for the condition where the ship speed exactly equalled the wave celerity in overtaking 739 HEAVE DOUBLE AMPLITUDE/ WAVE DOUBLE AMPLITUDE Uram and Numata QO-h/L=1/450 SEM! SUBMERGED SHIP O-h/L=1/225 DEEP DRAFT DESTROYER A=3471LT | LARGE BULB SHIP A = 3471 LT ‘ie : 20 10 20 30 FOLLOWING SEA OVERTAKING SEA AHEAD SEA SHIP SPEED (KNOTS ) Fig. 21 - Heaving motion in regular 1.0 L waves seas, the ship "locked in" with the wave pattern and experienced no pitch or heav- ing motions. It is unfortunate, or fortunate, depending upon how one views the situation, that the model natural frequency and wave-exciting frequency, as well as the model speed and wave celerity correspondence occurred roughly at the same operating condition. In irregular seas, this condition can be expected to occur, but would be of importance only if the wave with celerity correspondence is that wave having a major contribution to the ship excitation. Figures 19 and 20 indicate that the pitch response of the Hydrofoil Semi- Submarine is indeed critical, as theory predicts, in following seas in the veloc- ity range between 10 and 25 knots. The pitch amplitude response is somewhat larger than the destroyer, but only slightly larger than the Large Bulb Ship and the Semi-Submerged Ship in their respective critical ranges. In 2.0 L waves, the Hydrofoil Semi-Submarine in its critical region has a substantially higher pitch response to the wave system than any of the other three ships. However, it must be pointed out that, whereas in surface vessels very little can be done to control or alleviate the situation because of their inherent very large longi- tudinal metacentric height and large wave exciting moments, such is not the case for the Hydrofoil Semi-Submarine. The very small longitudinal metacentric 740 Behavior of Unusual Ship Forms height and exciting moments of this type ship afford a great advantage. It would be no problem, with a relatively simple control system, to activate the main foil flaps or the stern plane to counter these motions. Quite possibly, control by manual adjustments of the control surfaces may only be required as the motion picture records indicate the pitch frequency to be quite low. Figures 21 and 22 present the heave response for 1.0 L and 2.0 L regular waves. Beyond doubt, it is seen that the heave characteristics of the Hydrofoil Semi-Submarine are far superior to any of the ships with which it is compared. Since this ship has an extremely small water plane area, its natural frequency in heave relative to the excitation from the wave system would be very near zero (practically that of a submarine). The heave characteristics, therefore, are more dictated by the hydrodynamic forces resulting from the pitch variations of the ship, its change in proximity to the free surface and the effect of the wave system on vertical force and pitching moment induced upon the ship due to its proximity to the surface, as forcing functions. As indicated, through prudent design this ship concept can be made quite stable relative to the hydrodynamic forces and moments on the ship and large heave motions can be avoided. O-h/L=1/45.0 SEMI SUBMERGED SHIP DEEP DRAFT Q=3835LT 1.2 LARGE BULB SHIP QO =3471LT DESTROYER A=3471 LT 0.8 0.6 HY DROFOIL SEMISUBMARINE A) 0 0 £\ Ad ° a goo jo 50 40 30 20 10 fo) 10 20 30 FOLLOWING SEA OVERTAKING SEA AHEAD SEA SHIP SPEED (KNOTS) 0.4 HEAVE DOUBLE AMPLITUDE/ WAVE DOUBLE AMPLITUDE 0.2 Fig. 22 - Heaving motion in regular 2.0 L waves 741 HEAVE DOUBLE AMPLITUDE / SHIP DIAMETER HEAVE DOUBLE AMPLITUDE/ SHIP DIAMETER 0.08 004 002 50 0.08 0.06 0.04 0.02 50 40 30 FOLLOWING SEA Uram and Numata 2 fal al PS SHIP SPEED (KNOTS) 10 — | 5 | | | | | 0 = 20 30 AHEAD SEA Fig. 23 - Heave double amplitude/ship diameter in 1.0 L waves (hydrofoil semi-submarine) FOLLOWING SEA SHIP SPEED (KNOTS) 10 AHEAD SEA 20 Fig. 24 - Heave double amplitude/ship diameter in 2.0 L waves (hydrofoil semi-submarine) 742 e 30 Behavior of Unusual Ship Forms DESTROYER Oe ed me FA ca ae La —-—- if grea sai i SHIP | OL X L/4B REGULAR WAVES LARGE- BULB SHIP SHIP SPEED, KNOTS NOTE: PLOTTED VALUES OF ACCELERATION FOR DESTROYER AND SEMI-SUBMERGED SHIP OBTAINED BY DIFFERENTIATION OF REPORTED HEAVE AMPLITUDES. LBS ACCELERATIONS ARE MEASURED VALUES. BOW-DOWN ACCEL.(g's) BOW-UP ACCEL. LWL 4 LARGE-BULB SHIP 346FT 3471LT SEMI-SUBMERGED SHIP 383FT 3835LT DESTROYER 3836FT 347ILT ine ya DESTROYER soe oe — -—-= ——_— a es Pe LARGE- BULB SHIP SEMI-SUBMERGED SHIP 2.0L X L/48 REGULAR WAVES BOW-DOWN ACCEL. (g's) BOW-UP ACCEL. Fig. 25 - Comparison of heave (C.G.) accelerations in regular waves Finally, in Figs. 23 and 24, it is of interest to show the heave amplitude re- lationship to the ship diameter for the Hydrofoil Semi-Submarine since it is of importance that this ship traverse a limited corridor in the vertical plane. These figures show that for either wave condition and for various wave heights the ship rarely can be expected to traverse, above or below its equilibrium run- ning depth, distances greater than 3% of the ship's diameter. 743 Uram and Numata Part of the acceleration data obtained by VanMater [7] is shown in Fig. 25. It is seen that in 1.0 L regular waves the unusual surface ship forms are supe- rior to the destroyer, while in 2.0 L regular waves they offer greater acceler- ations. No acceleration data was obtained for the Hydrofoil Semi-Submarine, but, as we will see, a study of the motion picture records indicates the pitching motions are of low frequency, resulting in relatively low pitch accelerations. It has been our pleasant task to attempt to collect and summarize the work done on unusual ship forms and realizing the inadequacy imposed by space and time limitations, we hope we have furnished an up-to-date balance sheet to aid those interested in possible application of the unusual ship form concepts. REFERENCES 1. Third Symposium on Naval Hydrodynamics, Office of Naval Research ACR- 65, September 1960 2. Lewis, E. V., "Ship Speeds in Irregular Seas,'' SNAME Volume 63, 1955, pp. 134-174 3. Mandel, P., ''Subcritical and Supercritical Operation of Ships in Waves," SNAME Journal of Ship Research, Volume 4, No. 1, June 1960 4, Uram, E. M., "Research on High Speed Ship Forms," 4th Seminar on Ship Behavior at Sea, June 1962, Davidson Laboratory TM 136, January 1963 5. Mandel, P., ''The Potential of Semi-Submerged Ships in Rough Water Oper - ation,'’ New England Section SNAME March 1960. (See also Sahlgren, J. A., "The Performance of a High Speed Displacement Ship in Critical and Super- critical Operation", SM Thesis, MIT, 1959) 6. Lewis, E. V. and Odenbrett, C., Preliminary Evaluation of a Semi-Sub- merged Ship for High Speed Operation in Rough Seas,"' SNAME Journal of Ship Research, Volume 3, No. 4, March 1960 7. VanMater, P. R., ''Preliminary Evaluation of a Large-Bulb Ship for High Speed Operation in Smooth Water and Rough Seas,"' Davidson Laboratory Report 834, May 1964 8. Uram, E. M., ''Study of the Design and Behavior of a Hydrofoil Semi- Submarine,'' Davidson Laboratory Report 1023, August 1964 9. Mandel, P., ''A Comparative Evaluation of Novel Ship Types,'' SNAME Trans. Volume 70, 1962 744 Behavior of Unusual Ship Forms DISCUSSION E. V. Lewis Webb Institute of Naval Architecture Glen Cove, Long Island, New York This paper presents results of an interesting and important investigation into a possible means of obtaining higher speed in an air-breathing near-surface ship. This ship shows superior resistance characteristics in comparison with other ships at speeds above 35 knots, in both smooth and rough water. The ingenious hydrofoil strut design provides an excellent solution to the problem of stability in a vertical plane, enabling the craft to maintain constant depth below the surface. An interesting feature of this hydrofoil semi-submarine is its motions in waves. Its long natural periods of heaving and pitching result in "supercritical" conditions of operation for all head seas. It is only in astern seas that large motions are experienced, and since the periods of encounter are very long, ac- celerations must be low. It thus appears clear that such a craft would be an excellent complement to a more conventional type of surface craft: the former would be able to make high speed in head seas and the latter in astern seas. Operational studies should be made to evaluate the effectiveness of pairs of such ships working together in A.S.W. and other naval missions. 745 Tain Vieaeariae, eal i vey Wh. dee y i Jib ue " hue , my ae ms i ii ha WV a On ; een Ne Rene, | yt Ae ayy) Alas Me i i ice i an gin ni Ae Ot eet ., yn Tn f an i Y i] i, ’ er ii i ried CTs i ~ i i al { i are! ip ALF sy) fl i Dany yh i ‘ i ies] iff i i Pal j wi a} i 1) ia) r We \ i , une 4 it : ‘ Pala , 1 . 4 \ i , : i LAY 4 i ty ir : : ioe Me al WUE vor ee ea] ron ‘ beet , eit Mt iy CD are ee: mre a's ’ f i Ny, te ‘ Or rar, heer J ’ { Was } ( a eh tat, : ‘ rae h v4 ‘ah , bea PUT He i , é " b wivree al Ie Bred ; " ¥ el Mey ee : “al i tu i vt i : i or b A 4 y io \ hk er ly iy r aR DMM io wt et Le RD f i) ‘ oa ae, ve ob VN a at a ee a iA \ + hei i ipha j ; ) bait Mat GY i SU ' t ib { 4, Cub : i i } i ‘ y ‘ i ray f i 4 il k } : i ' 4 “ - , i OH ) i # j Wee kh j r ‘ \ j j s ay ‘ < Ts ni & fi cd 7 7 ) aot j h i / a i i eras th ; i ; ; si 1 ‘ i i i : j WAY i i i ' j hy i 7 i aed i 2 7" 1 : lah _ re iy H iach ity! Pn A SURVEY OF SHIP MOTION STABILIZATION Alfred J. Giddings Bureau of Ships Washington, D.C. and Raymond Wermter David Taylor Model Basin Washington, D.C. ABSTRACT A brief historical review of significant developments in stabilization is presented. Some recent investigations in roll are discussed followed by a survey of the progress and potentialities of pitch stabilization. The important differences between pitch and roll stabilization are ex- amined, and the reasons for the greater difficulty of the former are discussed. Since pitching, relative to rolling is not a sharply tuned resonant phenomenon, large magnitude moments are needed to develop appreciable effects. Model test results are presented to indicate the degree of stabilization possible and the vibration problem associated with bow fin installations is examined. The effects of configuration, platform area and aspect ratio are also mentioned. INTRODUCTION Stabilization of ship motions can be considered in a very broad sense, or in a narrow sense. In the broadest sense, consideration should be given to static stability, motion amplitude and controllability in each of the six degrees of free- dom of rigid body ship motions. A more narrow view might consider only the limiting or prevention of one of the motions. It is the aim of this paper to strike a middle ground, recognizing that there are significant and undesirable motions in all six degrees of freedom, but expanding only on those of particular interest. It is advisable to define what is meant by "stabilization" in this paper. By this is meant the deliberate limiting of a ship motion caused by waves, which motion is otherwise stable. With this definition, automatic steering of a direc- tionally unstable ship, or of a stable ship in calm waters are only of passing in- terest while the more obvious cases of pitch and roll ''stabilization" in a seaway are of definite interest. A cursory examination of the literature on ship motions and stabilization reveals some interesting trends. The principal interest of those writing on 747 Giddings and Wermter prediction of ship motions has been in the longitudinal plane, while in contrast, the writers on motion stabilization have been more interested in the lateral mo- tions. This may well be due to the almost linear and seemingly manageable na- ture of pitch and heave motions which attracts theorists away from nonlinear rolling and turning problems, while the highly commercial nature of roll and course stabilization has attracted inventors and engineers. The paper will provide a brief survey of the state of the art in stabilizing the motions of translation, course keeping, roll stabilization and pitch stabiliza- tion. The latter of these is to be the subject of a more elaborate discussion. SURVEY OF THE ART Translatory Motion Stabilization The deliberate stabilization of translatory motions of conventional ships has very little technology or theory to survey. While it could be said that mooring problems are problems of stabilization and control of lateral translation, the process of mooring is more an art in the classical sense than in the scientific sense. Shipboard devices which affect lateral motion directly include bow thrusters, vertical axis propellers and right angle drives. The application of these devices to conventional ships has been for purposes other than "'stabilization,''as de- fined in this paper. In the case of submarines, hydrofoils, and ground effect machines, the con- trol of vertical translation has received a great deal of attention, but this subject would warrant an extensive survey of itself, beyond the scope of the paper. It should be recognized that the physics of ships is such that pitch and heave, yaw and sway, and roll and sway are so strongly coupled that control or stabilization of the angular partner of each couple inevitably affects the other. The effect on translatory motion is a by-product of the angular stabilization, rather than a deliberately sought objective. Translatory accelerations on the order of one-tenth of gravity are not unu- sual. In order to have significant direct effects on such motions, control forces on the order of 5 to 10 percent of the ship weight would be needed. Generation of such forces by direct means is impracticable. Yaw Stabilization Stabilization or control of yaw is the most ancient of stabilization problems. It is actually not vital that a ship be stable in yaw, but it is vital that it be con- trollable. Provision of adequate stability and controllability for ships is such an obvious necessity, that years of tradition and experience provide useful de- sign rules. References 1 and 2 provide useful information on the selection and 748 A Survey of Ship Motion Stabilization design of rudders. The many Naval Architecture text books also offer practical methods leading to design of directionally stable and controllable ships. There have been analyses of the forces and moments in yaw exerted by a seaway, especially Refs. 3, 4, 5, and 6. The inherent stability on course of ships is discussed in Ref. 7, and the automatic control of directionally unstable ships is treated in Ref. 8. The general subject of automatic steering control is treated in Refs. 9 and 10. Additional references onthe subject are [11] and [12]. In general, yaw stabilization or course-keeping has been in the province of commercial developments. The devices and methods used are largely proprie- tary, and their success is evident from their widespread use. Even without automatic systems, the control of a well designed ship in a seaway is well within the capabilities of skilled men. Roll Stabilization General Discussion All ships are stable in roll in that a properly loaded intact ship will not capsize, so that roll stabilization is really roll angle limitation. In contrast to the yaw case, as long as rolling is a stable motion, it need not be "controllable." There is an extensive background of experience on the control of roll in a sea- way. The subject has fascinated inventors since steamships were invented, and the general subject is dominated by inventions. A glance through the patent office files on roll stabilization reveals not only the bad drafting favored by pat- ent attorneys, but evidence of the highly imaginative approaches generated by the problem. The roll stabilizers can be divided into two major groups, internal to the ship and external. Each of these can be further divided into active and passive types. Table 1 categorizes stabilizers from a mechanical point of view. Chad- wick [13] offered a more elegant and complete categorization based on the dy- namics involved. Bilge Keels The earliest deliberate roll damping devices were bilge keels, fitted to steamships to make up for the roll damping lost when the sails were removed. References 1 and 14 present curves of bilge keel size as a function of ship size, based upon experience with ships in the past. References 15 and 16 present ex- perimental results on the forces acting on bilge-keel-like plates oscillating in water. It is rare that an occasion requiring more than rule of thumb design of bilge keels will arise, but when such a case is at hand, analysis of bilge keel forces can be carried out using simple concepts and data such as that cited. Bilge keels can be counted on to increase hull damping in roll by 50 to 100 percent. This will result in 25 to 50 percent reduction in roll. It must be re- membered however, that the principal advantage of bilge keels is found at low speeds. As ship speed is increased the hull damping increases proportionately 749 Giddings and Wermter Table 1 Classification of Anti-Roll Devices Percent of Percent of Active, | Active Tanks 1/2 to 1 Fins, Flapped up to 85 and Plain 1/4 to 1 percent average | "Sperry" Gyroscopes | 2 to 3 stabili- zation Moving Weights 1 to 2 Passive, | Frahm Tanks 1/2 to 1-1/2 | Bilge Keels 1/2 to1 about 50 percent | Free Surface Tanks 1/2 to 1-1/2 | ''Fishermans" | average Keels 1 stabili- zation "Schlick" Gyroscopes Fin Keels 1 Staysails 1/2 to 1 more than the bilge keel damping. There is a small increase in hull inertia due to bilge keels, but the principal effect is increased hull damping. Historically, bilge keels have been discussed in the literature by White [17] and Spear [18], followed by many individual] model test reports on specific designs, too numer- ous to mention. Certain fishing craft are reported to use a technique for roll reduction while adrift. Booms are rigged out over each side, and lines carrying weighted drogues are lowered into the water. As the boat is rolled, it tends to pull up one of the drogues which provides damping, while the other drogue sinks. The ubiquitous staysail is also used for roll damping by boats throughout the world. There may well be other unique and homely devices used on boats in specific instances. Anti-Rolling Fins Anti-rolling fins have had a relatively long history. References 59 and 60 are among the earliest references to this form of roll stabilization. Chadwick - [13] gives a good historical view of fin stabilization, and Bell [21] discusses the history of fin controls. In general, progress in fin stabilization has been char- acterized by a series of inventions, each limited more by the state of the art in automatic controls rather than in hydrodynamics or mechanical engineering. Only within the past fifteen to twenty years has it been possible to design and analyze fully automatic controllers through straightforward engineering, rather than through inventive inspiration and insight. The current state of the art in fin stabilizers is shown by Chadwick [13], DuCane [22] and Flipse [23]. The latter commendably frank reference, along 750 A Survey of Ship Motion Stabilization with Ref. 24 discuss actual performance at sea of specific installations, while Chadwick, Bell, and DuCane deal with control theory and design. Various types of fins are used by the different manufacturers. There are articulated-flap fins and simple fins, both tapered and untapered. The range of aspect ratios selected depends on the method of retraction, or lack of retraction. The hydrodynamic design of fins [25] is influenced strongly by maximum lift co- efficient as limited by cavitation and the free surface, while lift curve slope and low drag considerations are not very important. Unsteady effects on lift slope are not significant, even considering the high tilt rates required. There is evi- dence [26] that the maximum lift coefficient is augmented by unsteady effects, but use of this phenomenon in design is not widespread. Unsteady effects must be included in the computation of tilting torque. The torque loads proportional to acceleration and velocity are significant, and if not allowed for, the slow re- sponse of the fins to control system orders could be embarrassing. In those cases where flapped fins have been specified by designers, cavita- tion must have been a principal consideration. For merchant ships, wherein cruising speed and full speed are nearly the same, the design speed for the fins is relatively high. To economize on fin size, the desired stabilization capacity is provided with the fins producing nearly their maximum lift coefficient. Under these conditions, the more uniform pressure distribution on flapped fins is bene- ficial. For warships, or ships having a cruising speed much less than maximum, the design condition for the fins is not as severe. Large lift coefficients are re- quired only at cruising speed, and the fin angle is limited at higher speeds to maintain the stress level in the stock. At speeds where cavitation would be a concern, only small lift coefficients are required. For this reason, plain fins may be used. The elimination of the flap actuator and flap hinge structure may in turn save enough weight to compensate for the increased area of a plain fin. Stabilizer fins are usually located in pairs, port and starboard. If more than one pair is to be installed, the downwash effects of the forward fins upon the flow to the after fins must be considered. Most modern roll control systems use combinations of roll angle, roll ve- locity and roll acceleration to generate ordered fin angles. The "Denny Brown" types order fin angle [21] while the Sperry type orders fin lift [13,23] using a deflection gage within the fin to feedback the lift. Earlier control systems used much simpler concepts, having been analyzed and designed to deal with regular waves. As experience with real seas and the ability to analyze random seas has accumulated, more sophisticated controls have been adopted [21]. It is possible to design control systems to minimize any of several parame- ters of the motion. The most common index of performance is roll angle reduc- tion, although roll velocity or acceleration could be the factor of most interest. References 27 and 28 are two of many papers in the control system literature which discuss designing to minimize various energy criteria. Minimum energy demand on the stabilizer, or minimum energy of residual motions are but two of the possibilities. How much benefit such refined techniques might give to ship motion reduction remains to be seen. 751 Giddings and Wermter Model tests of anti-rolling fins, either alone or on ship models, have not been reported in the open literature. As more and more model tanks develop the ability to generate random model seas, perhaps greater use of ship model testing to prove out design concepts will result. Factors such as the interaction of bilge keels and stabilizer fins, yaw-heel, and the interaction of a passive tank stabilizer with active fin stabilizers could be examined on model scale. Even without wavemakers, model tests using rotating eccentric weights or other roll moment devices can be of use in examining the hydrodynamics of stabilizers. Stabilizing Tanks Anti-rolling tanks have had a checkered history. Since Froude's first spec- ulations [29] a great variety of tank installations have been tried, with different degrees of success. Until recent times, the most successful of these were Frahm tanks [30] either cross-connected within the ship, or with port and star- board tanks open to the sea. More recently, passive tanks with free-surfaced cross connections have been successful [31]. Active tank stabilizers have not had a successful past, but the future looks brighter. A series of reports by Chadwick [20,32] analyze the dynamics of both active and passive anti-roll tanks. This analysis for passive tanks was extended slightly in Ref. 31. Blagoveschensky [33] presents a simplified analysis for passive tanks open to the sea. Hydronautics, Inc. under the sponsorship of the Bureau of Ships is currently conducting a theoretical study of active anti-roll tanks. This study will once again reanalyze the equations of roll motion as pre- sented by Chadwick to insure that all significant nonlinear terms are properly included. Pumping rate specifications and tank design criteria will be estab- lished and it is hoped that sufficient information will be generated to permit a successful design. The recent success of the free surface type passive tanks compared to the narrow acceptance of Frahm tanks is due to several factors. The high internal damping due to wavemaking in free surface tanks makes precision of design less demanding than for Frahm tanks. The tank response curves are flatter, and highly nonlinear in a fashion kind to the designer. The recent trend for ship de- sign to be controlled more by volume than weight has also made it easier for the owner to accept the weight of tank stabilizers. Application of theory to describe the action of Frahm tanks was shown to be fairly successful (Chadwick [20]) in that assumption of linear damping within the tank gave fair approximations to the model test results. Little agreement has been found for free surface tanks. The theory developed in Ref. 31 included a provision for equivalent nonlinear tank damping evident from model test results. In addition, the U-tube analogy for computation of the natural frequency of free surface tanks has been shown to be somewhat inaccurate. Reference 58 presents some corrections, based upon basic shallow water wave theory compared with experimental results. An additional comparison is presented here. As derived in Refs. 30 and 31, the natural frequency of oscillation of the fluid in a U-tube can be found from 752 A Survey of Ship Motion Stabilization anor y= (1) where w, = tank frequency, by U-tube analogy, and S = effective length of the U-tube. i A, Ss | = ds (2) 0 where A. is the area (constant) of free surface in one "wing tank" of the U-tube, A is the cross section area of the U-tube normal to the U-tube centerline, S is the girth-like coordinate along the centerline, and L is the total "girth length" of the centerline. In the case of a free surface tank, the U-tube analogy is applied by assuming that the U-tube centerline is as shown in Fig. la. Reference 34 presents an approximate solution for the natural frequency of a tank of the configuration shown in Fig. 1b, WD, = a= cain (3) be S! Sh where w, = tank frequency, by ''exact" theory, S' = "effective beam of tank," and — 2) ll fluid depth. The effective beam of the given tank configuration is shown by Lamb to be (4) Relating the two methods of calculating frequency; 2(1+$) coth (7=2) (5) iL, TT 221-249 O - 66 - 49 753 Giddings and Wermter (b) Fig. 1 - Typical tank configuration where Be Oise BL, a = h/B T= \S/Bee Applying Eq. (2) to the computation of 7 for the configuration shown yields: il T\2 1 r G8) §) 2 1 z Doreen a a B 2 5) 4 - ws 4r? Table 2 indicates that the U-tube analogy can be used, with appropriate care, as an approximation to a more precise theory, at least for this particular family of tank geometrics. 754 A Survey of Ship Motion Stabilization Table 2 Comparison of a U-Tube Analogy with Theory r/B «w(U-tube) co( Lamb) Active Internal Systems This paper will not elaborate on moving weight or gyroscope systems. It can be said that moving weights have an attraction due to the possible high den- sity of such an installation. It may be that the effective density of a moving weight system, after including the volume needed for the operating machinery, necessary to move the weights and safety devices, would be about the same as that for a tank system. The additional property of a solid weight system that there is no loss in hydrostatic stability when at rest is aiso attractive. How- ever, there is no active research or development known to the authors in this field. Reference 35, besides being interesting reading, contains a good discus- sion of the first successful moving weight installation. Gyroscopes, both passive and active have been installed in many ships in the past. References 36 and 37 discuss early installations. The dynamics of both types of gyroscopes are analyzed by Diemel[38]. Their great weight and the engineering difficulties of highly loaded bearings have limited their accept- ance. The most recent installations of gyrostabilizers was in POLARIS subma- rines, where they performed well enough, but changes in operational concepts caused their removal. Recent Applications A limited number of naval installations have been completed or studied since the recent paper by Vasta, et al. [31]. These have usually all been in an area requiring stabilized gun, launching, or search platforms. Oceanographic research ships have all been considered for the installation of passive tank sys- tems in recent years. The recent studies conducted on passive anti-roll tanks and active anti-roll fins will be discussed. The results of full scale trials and/or model tests will be presented. Interpretation of full scale tests requires care. To quote Pierson [39], "The surface of the sea is a mess."" This complicates the roll records. It is rare that the statistical properties of the sea remain static long enough to com- plete the schedule of trials necessary for a good evaluation. The trial analysis therefore demands a good deal of judgment on the part of both the analyzers and the readers, especially without good measurements of sea conditions. USNS ELTANIN — Passive Tanks. The USNS ELTANIN (TAK 270) was con- verted from a cargo ship to a scientific research ship of 3330 tons displacement. 755 Giddings and Wermter Anti-roll tanks were installed in late 1961 and full scale sea trials were con- ducted by the David Taylor Model Basin in December 1961 [40]. Figure 2 shows a photograph of the ship and indicates the general tank location while Fig. 3 pre- sents a schematic sketch of the tanks. These tanks displace 75 tons when filled with 6.5 feet of water. The trials were conducted over a three-day period and Figs. 4 and 5 show the measured sea spectra. (b) External view of tanks looking forward from bridge _ Fig. 2 - Location of tanks on the ship 756 A Survey of Ship Motion Stabilization FORWARD ® PRESSURE PRESSURE GAGE —t GAGE SHIP AND TANK ¢ (a) Top view DESIGN WATER DEPTH(5'-6") TANK BOTTOM AT OILEVEL 18'-2" (TRIAL VALUE ) ot SHIP CG (b) Side view Fig. 3 - Sketch of principal dimensions of tank system Tank tuning experiments were conducted over the three-day period in the several seas encountered and while the results of these experiments as plotted in Fig. 6 indicate that an optimum water depth had not been achieved, it would appear that 6.5 feet of water yields a reasonable operating condition. Figure 7 presents the effect of sea angle encounter. The sea spectra curves presented in Fig. 5 indicate extreme variations in sea conditions and consequently, the test results presented by Fig. 7 cannot be interpreted as indicative of representative trends without extensive interpolation between sea spectra. 757 Giddings and Wermter 7 LEGEND RUN NO. DATE 19 DEC 61 19 DEC 61 21 DEC 6I 21 DEC 61 SIGNIFICANT WAVE HEIGHT (FEET) 42 5.1 73 9.6 (Ft)? (Ft)? bu, 0.2 03 04 05 0.6 O07 08 0.9 1.0 We IN RAD/SEC Fig. 4 - Sea spectra as measured during trials of first and third days 40 : a LEGEND \ RUNNO. DATE= WaVE HEIGH: \ (FEET) iN — 8 20DEC6I 4.4 Vale ue Weeeonodeon 21 20 DEC6I 6.8 i | \ ee 22 20DEC6I(NIGHT) 84 3 20 : => 05 o6 07 08 0.9 We RAD/SEC Fig. 5 - Sea spectra as measured during trials of second day 758 SIGNIFICANT ROLL ANGLE , DOUBLE AMPLITUDE IN DEGREES A Survey of Ship Motion Stabilization SYMBOL RUN NO 6,5,4,3 16,1411 24 ,23 29,28,27,26 DATE 19 DEC 6! 20 DEC6I 20 DEC 6! (NIGHT) 21 DEC 6I ALL RUNS MADE AT 60RPM fe} | 2 3 4 WATER DEPTH IN FEET OF ENCOUNTER DEGREES 90 135 90 90 Fig. 6 - Effect of various water levels on tank effectiveness 759 SIGNIFICANT ROLL ANGLE , DOUBLE AMPLITUDE IN DEGREES 6.0 Giddings and Wermter 8,9,10,11 z 21,17,18,16 TANK DATE WATER DEPTH 20 DEC6I 65 FT 20 DEC 61 EMPTY ALL RUNS MADE AT 60 RPM HEAD SEAS O DEG a —_ 45 90 -SEA ENCOUNTER ANGLE IN DEGREES Fig. 7 - Effect of sea encounter angle on 760 135 tank effectiveness A Survey of Ship Motion Stabilization Finally, Fig. 8 indicates the effect of speed on roll stabilization. An inter- esting and contrary effect is the increased roll amplitude with increase in per- centage roll stabilization with increased speed. One would expect the natural hull damping to increase in the unstabilized condition and the percentage of roll stabilization to decrease with increased speed. The changing sea environment SYMBOL D 12,10,13 20 DEC 6! | 19,18, 20 20 DEC 6I ° SIGNIFICANT ROLL ANGLE , DOUBLE AMPLITUDE IN DEGREES io) 100 160 SHIP SPEED IN RPM Fig. 8 - Effect of ship speed on tank effectiveness 761 Giddings and Wermter must again be suspected. Foster [40] continues to explain that a considerable directional spectra must have existed in the confused beam sea and as speed in- creased the frequency of encounter with these directional components approached the natural roll period, resulting in increased roll response. USNS GILLISS — Passive Tanks. The USNS GILLISS is a 209-foot, 1200-ton oceanographic research ship and is one of a large class of such ships. The de- sign specifications of the ship limited the displacement to the stated value. Maximum length was maintained consistent with the displacement to provide as much work space as possible. The ship was fitted with anti-roll tanks consist- ing of two wing tanks with an open channel crossover and fixed entrance nozzles. Full scale sea trials were conducted by the David Taylor Model Basin in December 1963. Figure 9 presents the results of the tank tuning experiments. These tests were conducted on two separate days in both beam and quartering seas. These curves indicate a very well defined trend toward an optimum water depth of 3.5 feet. It should be noted that the tank effectiveness can be decreased by the addition of too much water. Whether this is due to poor tuning or the limiting of tank fluid transfer due to overhead clearance is not clear. O 4 DEC 63 BEAM SEAS SHIP SPEED 2 KTS S 14 16 DEC 63 QUARTERING SEAS SHIP SPEED 4 K: xq Ww oa ° 12 4 ¢ Ww oa a 0 2) uw 10 a © LJ a ” =e ] —] G a O i ¢ J 6 () () re) Ns a o 4 Ww < m 4 S =? 2 (o) 10 20 30 40 50 60 70 NOMINAL WATER HEIGHT (INCHES) IN TANK Fig. 9 - Effect of various water levels on passive anti-roll tank effectiveness 762 A Survey of Ship Motion Stabilization Figure 10 shows the effect over various angles of encounter with the sea- way, in both the stabilized and unstabilized condition. While these curves again indicate the general effectiveness of the tanks, they also indicate that the chang- ing environment makes it impossible to draw definitive design conclusions. 4 6 DEC TANK LEVEL =39" O6DEC « O7DEC " 07 DEC « AVERAGE ROLL ANGLE IN DEGREES (PEAK TO PEAK) HEAD BOW BEAM QUARTER FOLLOWING SEA DIRECTION Fig. 10 - Effect of sea direction on passive anti-roll tank effectiveness Bench tank tests and model tests conducted in irregular seas at the David- son Laboratory [41] showed that significant stabilization was possible, as much as 90 percent at resonance. Figure 11 shows the roll amplitude operator indi- cating this result. It is further concluded that once tanks are tuned to damp out the narrow frequency band of roll response, rolling at resonance is limited to amplitudes approximately equal to the maximum surface wave slope. Finally, as might be expected, bilge keels had small effect in further reducing the roll of a ship already stabilized by a passive tank system. 763 Giddings and Wermter ROLL RESPONSE 9 BOW SEAS (135°) FROUDE NO, = 0,08 WITH OUT BILGE KEELS 8-— STANKS INOPERATIVE O TANKS OPERATIVE (ru) a (eo) D 8 > = > a4 T _ fo) a 3 2 U r l LJ a Se poo 7 m = adn 0 Messe l 2 3 4 5 6 7 MODEL FREQ OF ENCOUNTER RAD/SEC (0) 0.2 0.4 0.6 0.8 1.0 1.2 SHIP FREQ OF ENCOUNTER, We, RAD /SEC (0) 0.5 1.5 2.0 1.0 TUNING FACTOR, Ag . Fig. 11 - Effectiveness of passive anti-rolling tanks In continuing the model study on the AGOR class of ship, the Davidson Lab- oratory conducted experiments in regular waves to determine speed and wave height effects [42]. This study concluded that an increase in the height of regu- lar beam waves decreases the effectiveness of the tank system and the peak of the unstabilized roll response moves to a slightly lower frequency. Figure 12 shows a comparison of the various roll responses derived from experiments conducted by Refs. 41 and 42. The speed study indicated the obvious result of an increase in roll damping with increased speed for the unstabilized ship condition and an increase in the stabilized roll amplitudes (decreasing tank effectiveness with increased speed). ARIS-3 Passive Tanks. The ARIS-3 is a design for a 496-foot Advanced Range Instrumentation Ship of 13,600 tons displacement. Bench tests were con- ducted at the David Taylor Model Basin on a 1/20-scale model passive tank. In addition to determining the depth of water required for properly tuned operation, 764 A Survey of Ship Motion Stabilization FROUDE NO. = 0.08 ROLL IN IRREGULAR SEAS TANKS INOPERATIVE 9 WITHOUT BILGE KEELS FROM REF | MEAN WAVE HGT=3.4FT ROLL IN REGULAR SEAS TANKS INOPERATIVE WITHOUT BILGE KEELS 6 FROM REF | TANKS INOPERATIVE WITH BILGE KEELS MEAN WAVE HGT=3.8 FT iss IN REGULAR SEAS ROLL / WAVE SLOPE a ROLL IN REG SEAS TANKS OPERATIVE . | WITH BILGE KEELS ae MEAN WAVE HGT=5.2 FT Gearee 3 4 5 6 MODEL FREQ OF ENCOUNTER RAD/SEC ee ee ee ee ee ee ee eS ees fe) 0.2 0.4 0.6 0.8 1.0 1.2 SHIP FREQ OF ENCOUNTER, We , RAD/SEC Fig. 12 - Roll response of AGOR in irregular and regular beam seas with different types of stabilization 3 different nozzle shapes were investigated to determine damping effects. Fig- ure 13 indicates the general tank arrangement while Fig. 14 shows the various nozzle configurations. The results of these experiments, Fig. 15, indicated that nozzle configura- tion ''A''gave the most favorable dynamic characteristics based on the more de- sirable moment produced. Figure 16 shows the variation of phase angle be- tween moment and roll angle with roll frequency and indicates no appreciable advantage between nozzles. The effect of water depth is shown in Figs. 17 and 18. Generally speaking, low water depth would be more advantageous at low frequencies, higher water levels at the midfrequency range with no appreciable effect for either water depth at high frequencies. 765 Giddings and Wermter HALF BREADTH 35'-0" WATER LEVEL _ Fig. 13 - ARIS-3 anti-roll tank configuration NOZZLE "A" NOZZLE "B" NOZZLE "Cc" Fig. 14 - Various nozzle configurations tested with ARIS-3 anti-roll tank 766 Model Moment in inch-pounds A Survey of Ship Motion Stabilization 100 an 13.33 O A NOZZLE QO B NOZZLE C NOZZLE : mee = pate dai 40 5.33 20 2.67 ‘ = Q 0 0.100 0.200 0.300 0.400 0.500 0.600 Model Roll Frequency in cycles per second 0 0.0224 0.0447 0.0671 0.0895 0.1119 0.1342 Ship Roll Frequency in cycles per second Fig. 15 - Variation of tank moment with roll frequency for three different nozzles 767 Ship Moment in foot-pounds ( x 10°) Phase Angle in degrees of lag 180 140 120 80 > o 2 fo) SSN NO Be aes PI Giddings and Wermter ee ee eee see abe ele | rede 9 a aes LEGEND O A NOZZLE O 8B NOZZLE 4 C NOZZLE aaean a ! A ee 0.100 0.200 0.300 0.400 0.500 0.600 Model Roll Frequency in cycles per second 0.0224 0.0447 0.0671 0.0895 0.1119 0.1342 Ship Roll Frequency in cycles per second Fig. 16 - Variation of phase angle between moment and roll angle with roll frequency for three different nozzles 768 Tank Fluid Angle in degrees Le) > 12] nO APE eer peas M i) = A Survey of Ship Motion Stabilization LEGEND thy pia epee ak Babak Unmnie. ry Wa 6 ? a a Be Bs = 0 0.100 0.200 0.300 0.400 0.500 Model Roll Frequency in cycles per second 0 0.0224 0.0447 0.0671 6.0895 0.1119 Ship Roll Frequency in cycles per second Fig. 17 - Variation of tank moment with roll frequency for three different water depths 221-249 O - 66 - 50 769 O 3.5 FOOT WATER HEIGHT ~- & 4.5 FOOT WATER HEIGHT O 5.5 FOOT WATER HEIGHT 0.600 0.1342 Phase Angle in degrees of lag Giddings and Wermter FOOT WATER HEIGHT FOOT WATER HEIGHT FOOT WATER HEIGHT 0.100 0.200 0.300 0.400 0.500 Model Roll Frequency in cycles per second 0.0224 0.0447 0.0671 0.0895 0.1119 Ship Roll Frequency in cycles per second Fig. 18 - Variation of phase angle between moment and roll angle with roll frequency for three different water depths 770 A Survey of Ship Motion Stabilization This study further attempted to compute damping assuming that the tank- ship system could be described by a second order differential equation with either linear or quadratic damping terms. It was indicated that while the quad- ratic damping might be a reasonable representation in the vicinity of tank reso- nance, damping was shown to be a much more complicated phenomenon that can be treated with present knowledge. The results of the above as yet unpublished work of Finkel led to the design of a tank system which was installed in a 1/38.15-scale model. Tests were con- ducted in regular waves and indicated that roll would be reduced by as much as 55 percent in a beam Sea at a speed of 7 knots, Fig. 19. These predictions could not be extended to the irregular sea condition because of the nonlinearities in- volved in the roll phenomenon. AVT-7 — Passive Tank. The AVT-7 is a planned conversion from the CVL 48 and is a 683 foot hull of 18,760 tons. Model tests were again conducted on a 1/19 scale model tank. This tank was installed below the roll axis of the ship. The tank was again oscillated over the frequency range with various water depths; the tank configuration is shown in Fig. 20. Tank moment versus frequency is Note: Head Sea is Zero Degree Ship Heading 45 Without Tank ae 7 Knots Double Amplitude of Roll in Degrees 90 Ship Heading in Degrees: Fig. 19 - Variation of maximum roll angle with ship heading for a wave steepness of 1/50 771 Giddings and Wermter PLAWE OF oy mmETRY a sae 35.50 =[- = TAS) fe Sa le x a o Fig. 20 - Symmetrical plan view of 1/19 scale model tank 772 A Survey of Ship Motion Stabilization shown in Fig. 21 while the variation of phase angle between moment and roll is shown in Fig. 22. The general conclusions arrived at from these experiments are in agreement with those reached in previous tests. There appears to be no unexpected adverse effect from putting the tanks below the roll axis. MAX, ROLL ANGLE 2° WATER LEVEL _S3FT 0 4FT O5FT MODEL TANK MOMENT IN POUND-INCHES 0.1 0.2 0.3 0.4 0.5 0.6 MODEL ROLL FREQ IN CPS .0229 .0459 .0688 0918 A147 1376 Fig. 21 - Tank moment as a function of roll frequency 3, 4, and 5 foot water levels at 2 degrees roll amplitude USS BRONSTEIN — Active Fins. The USS BRONSTEIN (DE 1037) is a 350- foot ASW vessel of 2500 tons displacement. This ship is fitted with active anti- rolling fins that are fixed in the out-rigged position. In other respects this in- stallation is similar to that of the USS GYATT [43]. Forced roll experiments were performed on this ship during the sea trials conducted early in 1964. Figure 23 shows the comparison of the stabilized and unstabilized roll quenching capability. Figure 24 compares the roll angle enve- lopes for the stabilized and unstabilized conditions and it may be seen that the damping of the stabilized curve is approximately 3 times that of the unstabilized curve. 773 120 100 80 60 PHASE ANGLE IN DEGREES 20 ROLL ANGLE IN DEGREES Giddings and MAXIMUM ROLL ANGLE 2° WATER LEVEL {o) 0.1 0.2 0.3 0.4 0.5 MODEL ROLL FREQ IN CPS Wermter Fig. 22 - Phase lag of tank moment relative to roll for 3, 4, and 5 foot water levels at 2degrees roll amplitude [Sgt | tg ef eal fo) .0229 .0459 .0688 0918 147 SHIP ROLL FREQ IN CPS 1376 USS BRONSTEIN DE 1037 FREE ROLL TIME HISTORY —— UNSTABILIZED ---— STABILIZED 25 KNOTS FIN ANGLE & II° 0 5 10 15 20 TIME IN SECONDS Fig. 23 - Comparison of stabilized and unstabilized roll 774 A Survey of Ship Motion Stabilization QUALITATIVE COMPARISON OF FICTIONAL ROLL ANGLE ENVELOPES FOR STABILIZED AND UNSTABILIZED CONDITIONS FOR SHIPS EQUIPPED WITH ACTIVE ANTI-ROLL FINS UNSTABILIZED eg STABILIZED (COMPASS ISLAND) STABILIZED (BRONSTEIN & GYATT) Zia ROLL ANGLE ENVELOPE AMPLITUDE DIMENSIONLESS TIME VTn Fig. 24 - Comparison of roll angle envelopes The application of stabilizer fins to this class of ship is the first since those previously reported on GYATT, COMPASS ISLAND and OBSERVATION ISLAND [31]. The apparent success of all installations would seem to indicate that more attention should be given to this area of stabilization. It should be mentioned that the tests presently under discussion were conducted at a ship speed of above 20 knots. There appears to be an obvious advantage to using ac- tive fin roll stabilizers on high speed ships. Current Studies. Under sponsorship of the David Taylor Model Basin, the Southwest Research Institute is conducting a continuing study of ship roll stabi- lization tanks [44,45]. This program provides for four related studies: (a) The- oretical tank damping characteristics; (b) experimental tank damping character- istics; (c) extended theory of ship-tank systems; and (d) application to design. After progressing in phase (a) and (b) for a period of time it became appar- ent that the study was hampered by a lack of a physical understanding of the tank fluid behavior. Finkel also discovered this in his work on ARIS-3 as did Motora and Lalangas [41]. To illustrate the point, Fig. 25 shows comparisons of several experimental approaches. The lack of agreement is startling. Addi- tional experimentation is indicated and a nonlinear model must be discovered. Pitch Stabilization Pitch stabilization has received a moderate degree of attention in recent years in both theoretical and experimental studies but as yet these studies have not resulted in a successful full scale installation. The problems of reducing pitch are quite different from those of roll stabilization. Pitch is already con- siderably dampened by the ship's hull. This of course means that large forces 775 Giddings and Wermter PHASE LAG PRESENT THEORY WITH FREQUENCY DEPENDENT DAMPING | | PRESENT THEORY WITH CONSTANT DAMPING (8 =.4287) | MECHANICAL MODEL (8 = 0.396) | MECHANICAL MODEL (8 = 0.519) sah (PHASES AS FOR h/B= 0.05625 5=0.396) Q,/B = 0.25 Z/B =- 0.0917 bn/B= 0.2292 10,000 |M|/pgB* (0) 4 8 1.2 14 1.6 1.8 (w/w)? Fig. 25 - Comparison of various mathematical models with experiment must be involved in any further magnification of this damping. It is quite im- practical to generate these large forces by means of internal devices generally associated with roll stabilization, i.e., tanks, gyros, moving weights. To further complicate the problem, the pitch phenomenon is not resonance dominated as is roll, that is to say, the roll spectrum is sharply tuned while pitch responds to a broad range of frequencies. Thus, all attempts at pitch stabilization have been through the use of exter- nal devices, capable of sustaining the large forces involved. These devices have been fixed bow fins, moveable stern fins and to a lesser degree moveable bow fins and fixed stern fins. Only the fixed bow fins have been installed on ships, these installations being made on the RYNDAM of the Holland-American Line and the American ship COMPASS ISLAND. While considerable pitch reduction was achieved in both cases, a severe horizontal bow vibration associated with 776 A Survey of Ship Motion Stabilization the fin installation caused their removal. This vibration problem has been the subject of much of the investigation conducted on bow fins in recent years; it has also caused the virtual abandonment of these devices as pitch stabilizers. The highlights of the work conducted in this area will be the subject of this section. Mention will also be made of some recent experiments not as yet re- ported in the literature. Fixed Bow Fins In 1956, Pournaras [46] fitted a set of fins to a Series 60, Block 60 model. The fins were flat plates with a planform area 2.5 percent of the load waterplane area. The leading edge was swept back to reduce tip load and thereby decrease the root-bending moment. The fins were also fitted with tip fences. From the limited tests conducted it was observed that pitching motion was considerably reduced, the speed range was extended, much less green water was shipped over the bow, and forefoot emergence was eliminated. Of most significance, however, Pournaras noted that on the downward stroke, sheets of water were forced around the leading and trailing edge of the fins, closed in over the upper surface and formed a whirl near the water surface as the two sheets met. Removal of the tip fence caused the formation of a third sheet and added to the problem. In a subsequent study, Pournaras [47] tested four different fin configurations on a model of a MARINER class ship. In addition to varying planform, some of the fins were slotted and others had through holes in an attempt to destroy the sheet vorteces. Figure 26 shows these various fin arrangements. All configu- rations were fitted with fences with the exception of fin 3. Figure 27 shows a summary of test results obtained and indicates that while substantial pitch reductions were obtained, configuration variation had very little effect. The major conclusions of this study are summarized as follows: 1. Fins operate most effectively near the synchronous range and have little effect at higher or lower frequencies. 2. Fins have little effect on the phase lag of heave and pitch but it should be remembered that a slight change in phase could have a marked effect on relative motions. 3. Area of fin planform has little effect on motions. 4. The loadings caused by the vorticity effect can be lessened by deeper submergence, greater fin span, tip fences and relief mechanisms such as slots and holes. Next, Abkowitz [48] conducted a comprehensive study on the effect of bow anti-pitching fins on ship motions. This study included a discussion on the na- ture of pitch damping in addition to presenting some experimental results of tests conducted on a Series 60 Block 60 model, an aircraft carrier model and a destroyer model. All indicated good pitch reduction trends and good agreement with theoretical calculations. It was again concluded that the major effect was produced at resonance. 777 Giddings and Wermter Fin Number | Root Chord 24' Fillet Fairing to Hull Fin Plane of Symmetry at 3 ft. above Base Line Fin Area Waterplane Area Fin Area: 620 sq. ft. = 0.022 eo ic’ | Tip Chord Fin Number 2 Fin Plane of Symmetry at 3 ft. above Base Line Fin Area Waierplane Area = 0.022 Fin Number 2s — Obtained from Fin Number 2, by cutting 5ft. off each tip, P&S. Tips off Centerline: 14 ft. Fin Area =0.014 Fin Area: 400 sq.ft. Waterplane Area ~° Fin Number 2h — Obtained from Fin Number 2s, by drilling two 12 ft. diameter holes 2ft. aft of leading edge of fwd fin and two | ft. diameter holes 2 ft. fwd of trailing edge of aft fin. Area and span of fins not changed. Fig. 26a - Plan views of anti-pitching fins 1 and 2 778 UW, Dimensionless Pitch Amplitude A Survey of Ship Motion Stabilization Fin Number 4 Fins separated by 1 ft. Upper surface of fin tangent to baseline. Fin Area: 640 sq ft. Fin Area = 0.023 Waterplane Area Fin Number 4h Obtained from Fin Number 4, by drilling five 1-ft diameter holes on each fin as indicated in sketch. Fin 3 — Same as Fin Number 1 with 30° dihedral angle. Fig. 26b - Views of anti-pitching fin 4 and description of anti-pitching fin 3 Froude Number, F, Froude Number, F, Fig. 27a - Dimensionless pitch amplitudes 779 Z, Dimensionless Heave Amplitude 6 , Phase Lag of Heave Referred to Pitch in degrees Giddings and Wermter With Fins fo) 0.10 020 Froude Number, F, Froude Number, F, Fig. 27b - Dimensionless heave amplitudes With Fins 0.10 0.20 Froude Number, F, Froude Number, F, Fig. 27c - Phase lag of heave referred to pitch 780 0.30 A Survey of Ship Motion Stabilization Abkowitz concluded that the horizontal vibration at the bow was due to the large angle of attack on the fin during the downward motion giving rise to a low pressure on the upper-fin surface. Just before the downward stroke begins (fin near surface) the fin is near the surface and the low pressure area causes a suction and possible ventilation. As the fin goes deeper, the bubble collapses causing a large pressure impact. He further concludes that since port and star- board fins do not ventilate uniformly, a differential pressure impact situation is created. He is at variance with the conclusions of Pournaras in that tip fences will be harmful rather than beneficial since they increase aspect ratio which in turn will increase the low pressure on the upper surface and lower the angle of attack at which the breakdown occurs. Abkowitz further conducted some comparative experiments between trumpet shaped fins and hydrofoil fins located at the keel and concluded that hydrofoil fins would produce less horizontal vibration and when separation did occur, only small horizontal vibrations were produced. In 1959, Becker and Duffy [49] presented the results of full scale sea trials conducted on the COMPASS ISLAND. It was concluded in this study that pitch was reduced but the magnitude of reduction could not be properly established because of lack of an exact "without fin" correlation condition. Vertical and transverse vibrations were excited in the ship, very seriously in heavy seas. While it was possible to calculate the fundamental mode of the vertical hull stresses (+11,000 psi) the transverse vibration was not measured. This was unfortunate since it was apparent that the transverse mode was excited at a more moderate sea state than the vertical, and all previous studies have indi- cated the transverse mode to be the probable problem area. Stefun [50] continued the effort by conducting an interesting experimental study investigating the effect of planform area and aspect ratio. Table 3 pre- sents a summary of the various fin configurations. This study again reaches the general conclusion that fins are effective mostly at near synchronous condi- tion and again significant pitch reductions were achieved. Aspect ratio is indi- cated to be a significant parameter. Pitch reduction is 30 percent less for a fin with an aspect ratio of 0.5 compared to a fin with an aspect ratio of 2.0. It was further indicated that while an increase in planform area was helpful in achiev- ing increased pitch reduction this increase was not in direct proportion to the increase in this area. The fins decreased ship resistance in waves of between 75 and 120 percent of the ship length. Heave was increased in long waves while in intermediate waves, heave was increased at low speed and decreased at high speed. Finally, this study indicated that the use of tip fences reduced pitch by an additional 5 percent. This is the result of apparent increased aspect ratio due to the addition of tip fences. It should be noted that this particular study used tip fences for the reasons stated by Pournaras [10,11]. 781 Giddings and Wermter Table 3 Fin Particulars : Plan Area Aspect 1 2.0 Tip Fences (length, width) 0.5 In 1962, Stefun and Schwartz [51] presented the results of a study conducted to determine the effects of various bow fin configurations on hull vibrations. These tests were conducted on an aircraft carrier model using 16 different fin configurations as shown in Fig. 28. All tests were conducted at one wavelength (A/h = 40, 30, and 24). The results of this study are presented in Fig. 29 asa figure of merit. This figure is defined to mean that for any fin, N Pitch Reduction (Fin N)/Pitch Reduction (Fin 1) _ Fieuke of Meee Vibration Level (Fin N)/Vibration Level (Fin 1) — ac i While none of the fins completely eliminated transverse hull vibrations, considerable improvement was indicated by several configurations. Aspect ra- tio, tip fences or increased depth of submergence showed improvements. Holes, dihedral angle and swept edges showed less improvement and while annular fins indicated promise, the test results showed much more research would be re- quired before their entire nature would be understood. In 1961, Ochi [52] conducted a very complete hydroelastic study on a ship equipped with an anti-pitching fin. In addition to forwarding an explanation as to the mechanism of the induced vibration the study also discussed the effect of the fin location, size and configuration. While there is general agreement that the induced vibration is caused by a cavity collapse (or cavity collapse plus fin slam in the case of shallow draft), the study differs as to the cause of the mechanism inducing the vibration. The premise forwarded here is that rather than the vibration being purely horizontal in nature, it is initially a torsional vibration. If the natural frequencies of both the torsional and horizontal modes correspond, the vibration is severe. 782 A Survey of Ship Motion Stabilization suotyeinstjuoo uty Suryoqtd-tyue snore, - 97 ‘317 *QUl| [294 aAoge payunow ¢ uly “|espayip aaljedau Bap-Gz yim | uly “ul O"g = WyBuaq “ul O° = Jajawerg rsiapurjAd sejnoig "jespayip aalyisod Bap-¢z WIM | ly WT Pasiaaal » uly *yuaosad 0g Aq aouassawagns jo yldap aseasout 0} yniys YIM | Uy (8)1 Base quejdajem jo Juadiad py = ealy “Ul By = PIOYD Wawixey “ul 976 = Ueds WnhwixeW *saouay diy yyim (q) | U4 (P)1 (pasiaaas ¢ uly) ‘plemso) -daams 3ap-¢z yim Zz uly “ul O° = JuBiay aouay *saouay diy yim | uly (9)1 000 "yoeq -daams 3ap-sz yim z uly “S904 Ja}aweIp-"UI-G°9 ySia 10) ydaoxa | uly se awes (4) 1 “jespayip aarjesau 3ap-¢z yim Z uly *SajOy Jayaweip-"ul-Q° | (e)1 INO} 10) Jdaoxa | uly Se awes es Op = Oey JIadsy ease auejdiajem jo juadiad py = ealy “UL 9°Z = ploy ‘ul 9°6 = Ueds 0'2 = oney yoadsy Pale auejdiajem j0 juadiad pp = ealy “UL bE = ploy “ul grg = ULdS bh Ey Ps ba im Ee MalA URI ulg Mal, pug MalA URld ene 783 PRR ER ROP I OOP a OF OOF ey ny eee M3IA NV 1d Giddings and Wermter [| SSS WH LEAS {| S SAA Se SESS SN SNA SSS CSAS NSN NS NS NSS ESQ WSSsSsg \ SAAS SSS SSS ie Lis YAN. RS SON RSS SSS ANS SNS SSS Ss SOON SSS swag SSS > SAN ANY RSS ~S “NS SVAN BSE SSS SSsssan ‘ Vas FIN FROUDE 0.30 = N FIN 4(0) GG «FROUDE O UZZ777777B FROUDE 0.15 UZ a < - V4 & FA 3 (a) FIN V(f) SSsSx SSS DSSS SSS SSS SSS WA FIN MOO MAME Ni AAA Aas FIN (cc) SOX MX mroe NSN NN NN FIN ESS PrN NNN) FIN SMVSSd VOM’ HF 4 DATA 3 Peed ' ° ° ° ° ° = ¢ Ue) a = LIS3W JO 3Y8NSIS 784 5 (a) 5 4 3 2 (0) 1 (9) 1 (e) 1(d) 1(b) 1() Fig. 29 - Qualitative evaluation of fin performance A Survey of Ship Motion Stabilization Many other investigators have also made the point that the time differential of port and starboard collapse also add to the severity of the vibration. Ochi goes further and states that if this time differential is the same as the loading time, the vibration will be magnified. It is further stated that if the fins remain below the water surface within some limit (8 feet for the MARINER), the cavity will not form and thus vibration will not occur. Some other general conclusions are as follows: 1. Maximum pitch reduction is achieved by fins located in the forward 10 percent of the ship. 2. More severe vibration is induced by fins in these forward locations. 3. Planform area and pitch reduction are linearly proportional for the MARINER hull. 4, Vibration increases with fin area and with violence of pitching. 5. Fins with properly designed holes can be as effective as solid fins in re- ducing pitch and are superior in reducing vibration. In selecting an optimum fin location, Ochi presents an interesting compari- son of parameters which is reproduced here as Table 4. The values underlined with the double line are considered to be acceptable design values and of course only the location with all parameters underlined will be optimum. Table 4 Optimum Fin Location for Various Parameters at a 15-Knot Ship Speed Location of Fin Aft of FP Resistance Pitching Heave Bow Vertical Acceleration Slamming Induced Vibration 221-249 O - 66 - 51 785 Giddings and Wermter Fixed Fins — Partially Activated Brief mention will be made here of various suggestions and/or studies that have been forwarded to retard flow separation over a fixed bow fin. They are termed partially activated because some means of flap or flow control would have to be provided for their proper operation. In his paper, Abkowitz [48] mentions a study conducted at MIT wherein boundary layer suction was used to control the pressure on the suction side of the fin. While the angle at which breakdown occurred was increased it was not prevented at large pitch angles. Stefun and Schwartz [51] recommend further study in the use of moveable trailing edge flaps as devices to retard the onset of stale. Along this same gen- eral line, Goodman and Kaplan [53] have recently proposed the use of a jet- flapped hydrofoil as an anti-pitching fin. This device would take advantage of the existence of two pressure peaks (leading and trailing edge) causing the for- ward peak to be smaller than for a conventional foil for the same loading. This initial theoretical study indicated the foil did not separate. It was concluded therefore that the jet-flapped foil would be cavitation limited and for reasons of the lower initial pressure peak considerably more lift would be produced before cavitation occurred. The authors of this preliminary work plan additional ef- forts in this area. Activated Bow Fins The authors were unable to find any experimental work dealing with acti- vated bow fins. Abkowitz [48] discusses this type of fin from the point of view of automatic control. With pitching motions as the control the fin angle would be additionally increased over an already large angle caused by the large amplitude of pitch. This method of control would therefore hasten the onset of ventilation. This leads to the concept of negative control, that is when the pressure on the foil reached a certain point, the foil angle would be decreased and thereby re- tard the onset of ventilation. The results of a computer study at MIT comparing this type of activated bow fin with a fixed fin indicated no difference in either method. Abkowitz concludes that there is little to recommend the use of an ac- tive bow fin. Fixed Stern Fins Abkowitz [48] makes mention of the use of fixed stern fins and concludes that they would be much less effective than bow fins. In addition to the obvious disadvantages of operating in the ship's boundary layer, the stern fin would probably increase the excitation due to waves and the pitch damping effect would also be reduced. The force applied to a stern fin would also produce less mo- ment than a bow fin since the apparent pitch axis is generally aft of amidships. In his experiments, Ochi [52] fitted a stern fin of equal area to the bow fin to the MARINER model. It was also found that the stern fin was much less ef- fective than the bow fin in reducing pitch. Vibration was not a problem although 786 A Survey of Ship Motion Stabilization maximum vibration still occurred at the bow. A combination of fixed stern fin and bow fin was also much less effective than a properly designed bow fin. Activated Stern Fins Spens [54,55] conducted model experiments on a MARINER class ship fitted with activated stern fins and tested these fins operating singly and in combina- tion with fixed bow fins. The stern fins were NACA 0012 airfoil section of 22 foot span and 14.75 foot chord. These fins were fitted forward of the propeller as shown in Fig. 30. The fixed bow fins were similar to those fitted to the COMPASS ISLAND and are shown in Fig. 31. After first conducting forced oscillation tests in calm water to determine basic fin characteristics, controlled tests were conducted in both regular and irregular waves. Table 5 presents the results of the regular wave tests while Table 6 shows those test results obtained in irregular seas (fully developed Newmann spectrum for 26 knot wind). These results indicate that the oscillating stern fins alone perform as effectively as fixed bow fins alone. There further appeared to be an additive benefit to using both sets of fins. It can also be seen (a) Model as towed Fig. 30 - Stern fins (Continued) 787 Giddings and Wermter (b) Model equipped for self-propulsion Fig. 30 - Stern fins that while the bow fins increase in effectiveness with increased wave height (or increased angle of attack before the breakdown), stern fins produce a rather constant result irrespective of wave height. Table 5 Effect of Oscillating Stern Fins With and Without Fixed Bow Fins in Regular Waves Wavelength Wave height Pitch without fins, deg (double amplitude) Pitch reduction by stern fins oscillating +25° Pitch reduction by oscil- lating stern fins together with fixed bow fins, deg - 788 A Survey of Ship Motion Stabilization LONGITUDINAL SECTIONS OUTBOARD ELEVATION Nes oaceaes & SHOWING GATE LOOKING AFT Fig. 31 - Bow fins 789 Giddings and Wermter Table 6 Effect of Fins in Irregular Waves Bow and Stern Fins Together Average of 1/3 largest pitches = P,,,, deg (double amplitude) Reduction in P,,, Average pitch = P,,, deg (double amplitude) Reduction in P,, Gersten and Cox performed additional work with activated stern fins at the David Taylor Model Basin. The results of these experiments are as yet unpub- lished. The tests were conducted on a model of the DE 1040 fitted with a pump- jet. The aft end of the pumpjet was further fitted with an oscillating flap in ad- dition to an upper and lower flap in the shroud. An automatic control loop using pitch and pitch rate as control parameters was incorporated in these tests. Experiments were conducted in calm water to determine which of several flap configurations could produce the largest pitching moment. Comparative tests with and without a flap fitted to the pumpjet were further conducted in ir- regular seas. As might be expected the flap arrangement with the largest total area produced the greatest calm water pitching (41.2 sq ft of flaps in pumpjet shroud plus 95.6 sq ft of flap C fitted to stern of pumpjet). Table 7 shows the preliminary results of the tests conducted in irregular waves. The table indi- cates that some pitch reduction was achieved in each case. These test data are undergoing complete analysis and a report should be issued soon. The decrease in fin effectiveness for increased wave height is an unexpected result. , Miscellaneous One other area appears worthy of mention although it does not fall within any of the above categories. This is a technique of effectively reducing water- plane area by the use of open tanks. Linearized equations of motion for such a system are given in the Appendix. Results of such tests will be reported by Gersten in a forthcoming Taylor Model Basin Report. The work was performed on an oddly formed special purpose type naval vessel. A stern tank with sides open to the sea was fitted to this ship which had unusually bad pitching charac- teristics. Results of these tests indicated that while maximum motions were not reduced through the use of the tank, these maximums were transferred to much lower speeds. This would permit the ship to operate effectively in the design speed range. 790 A Survey of Ship Motion Stabilization Table 7 Preliminary Results of Activated Stern Fin Tests Ship Percent Pitch Sea State Speed Reduction Based (knots) on Pitch (rms) Middle 5 Middle Middle Middle Middle Middle Middle Middle CONCLUSIONS AND RECOMMENDATIONS It should be stated that a design capability exists to produce successful in- stallations of roll stabilization devices in ships. In the case of passive tanks, however, much remains to be learned of the nonlinear behavior of the tank-ship system. It has been indicated that many experimenters have concluded that the basic lack of a physical understanding of the behavior of the tank fluid will pre- vent further progress in this field. The knowledge required will probably only be gained through the proper simulation of a nonlinear model. Southwest Re- search will continue their efforts in this area and additional work is planned at the Taylor Model Basin. Model and full-scale experiments will continue to be important design tools in this area until more theory is understood, even though both methods also have limitations. The continually changing nature of a seaway makes the collection of definitive design information during full-scale sea trials an extremely difficult task. While the capability for measuring sea spectra is increasing, proper ac- count cannot be taken of the directional components of their effects on frequen- cies of encounter. Since the roll phenomenon may be nonlinear it is additionally difficult to properly normalize test data collected in this changing environment. Extreme care must be exercised when design information is extracted from full scale experiments. While the various forcing functions can be controlled to a high degree during model experiments, scale effects and nonlinearities continue to complicate this approach. However, there are a large number of projects currently in progress aimed at providing an understanding of these scale effects and an insight into 791 Giddings and Wermter the basic nature of ship roll. It is felt that as nonlinear model tank simulations are achieved and the model problem areas cited above are rationalized, the model experiment will provide the most definitive design information. Model experiments should also be conducted to provide design information for active fins and to evaluate the performance of existing designs. The Taylor Model Basin is currently designing such an experiment to evaluate the fin per- formance on a new class of destroyer escort. The same control device previ- ously used during the activated stern fin experiments will be adapted to these tests. Suitable control parameters of roll angle, roll velocity and/or roll accel- eration will be selected. Additional experiments should be conducted on pitch stabilizing devices. In at least two areas cited there is discrepancy as to the effect of aspect ratio. These discrepancies might more fully be understood if the nature of varying aspect ratio were more closely examined. Stefun [50] and Stefun and Schwartz [51] vary aspect ratio independent of area. In both cases, the fin with larger span (increased aspect ratio) perform more reasonably in reduction of both pitch and vibration. Abkowitz [48] contends, however, that increased aspect ratio will have the effect of increasing low pressure on the upper surface and enhances the onset of breakdown. Laminar separation on the model may be the cause of varying test results in this area. It would appear that for many naval applications, the use of pitch stabiliza- tion devices would definitely be in order. In addition to the common arguments in favor of stabilizing pitch for reasons such as stable radar sonar, or fire con- trol platforms, Spens [55] makes one other valid point. He relates a pitch re- duction to a possible decrease in freeboard and/or forefoot depth. When one considers the design difficulty associated with increasing freeboard or draft of smaller vessels such as destroyers any freeboard decrease would be a decided design advantage. Increased depth and proper configuration design are the most important parameters to consider in bow fin design. Since the maximum depth of fin is a parameter not easily changed, lift control devices would have important design application. Additional model tests should be conducted to find proper design criteria and to clarify the hydrodynamics of the phenomena. In this respect, the following areas should be investigated: 1. Moveable flaps and jet flaps. 2. Activated bow fins using the pressure on the suction face as a control parameter. 3. Additional boundary layer control studies. 4, Additional investigation on parameters effecting relative bow motions and the subsequent effect on performance of bow anti-pitching fins. 5. Additional investigation of activated stern fins. In addition to conven- tional devices already described, items such as ring control surfaces around the propeller might be investigated applicably. 6. Investigation of scale effect on model results such as Reynolds number, near surface effects, etc. * 792 A Survey of Ship Motion Stabilization Appendix EQUATIONS OF MOTION FOR ANTI-PITCHING TANK With the recent interest in passive anti-roll tanks, it is of interest to spec- ulate on anti-pitching tanks. If, for instance, the forepeak tank of a ship were somewhat enlarged and fitted with large openings at the bottom, the surging of water in and out as the ship and waves interact might reduce pitching. Refer- ence 56 reports on one special case in this regard. The tank involved was in the stern of a slender double-ended ship form. Not enough data is presented to compare with and without tanks, but the influence of the size of the tank openings on the pitch period is presented. Figure 32 shows, in schematic form, the geometry of a bow tank open to the sea at the bottom. The equations of motion, assuming uncoupled pitch and heave for the "'unstabilized"' ship, have been derived using the Lagrangian formulation. Linearizing assumptions include all the usual ones in regard to small motions and linear damping. The tank geometry is assumed to be such that the area of the tank free surface does not change as the tank water level changes. It is also assumed that added mass terms and other coefficients of the ship are constants. Zz Fig. 32 - Sketch of coordinate system 793 Giddings and Wermter The equations are: I,yh + Byd + kaye + I 4 n2 + Ty,h + kph = ik 44K, X Tin? $I 2 + Biz tk, 2+ 1, ht kj = kgncet len ple, ziti GAB whl iikgh latKyles nave iwt Kk Zz zz lo © pitch amplitude radians, positive bow up heave amplitude feet, positive up change in tank water level from equilibrium, feet, positive up pitch inertia of ship, tank mass and ''added mass" heave inertia of ship, tank mass and ''added mass" tank mass = PAS A, = area of tank free surface Oo A S = i == ds A -H H = draft to bottom of tank A = cross section (waterplane) area of tank at any vertical location e = mass density of seawater linear damping coefficients in pitch, heave and of tank water motion pitch and heave "'stiffnesses" excluding the tank free surface effect pgA,, "tank" stiffness pr ¥ ¥ = tank volume 4 distance of tank center of gravity from ship center of gravity, positive forward 794 A Survey of Ship Motion Stabilization ee ey: Koh = pg 4A, K, = wave "effectiveness term"' in pitch [57], the result of integrating the static wave profile over the hull length to determine pitching moment K, = wave "effectiveness" term in heave [57] K, =e ?""/*, attenuation of wave height to keel 7, = Wave amplitude ® = wave frequency. The equations are symmetrical, and made somewhat more manageable in that several of the cross coupling terms are equal. The equations can be rewritten by defining several natural frequencies and coupling coefficients. Dividing all equations by k,, 2 co) Za. Za. F areaoae: ee Antes PEN Se ae ae ne le Se @ (22) w@ oN r r rv to zh zp-d : ee oe 7] es SOs Z Lege ese Sue le Ora Zz 2 Za Nee i ne Nob i (e+ 2) t t t tPt No — P+ r,o+ BAN A= GA By ABN CR aC (=) « S ee it wo? t oe W, t hit He. where Z h L == 3 a=—, 4 4 2 = Ko yee Oo 7 3 ’ Bee , od t k k A w2 = ——_ es She ons Uae 1 795 Giddings and Wermter When the tank parameters are set equal to zero, the equations reduce to the fa- miliar simple equations for uncoupled pitch and heave. To find a solution, assume that oe lwt P= Pre o) lot “= ly. @ 4 iwt a=a_e In principle, given all the coefficients, these equations could be solved for pitch as a function of wave amplitude, and a response operator derived. REFERENCES 1. Mandel, P., 'Some Hydrodynamic Aspects of Appendage Design,"' Society of Naval Architects and Marine Engineers, Vol. 61, 1953 796 10. 11. 12. NB, 14. 15. 16. A Survey of Ship Motion Stabilization Taplin, A., ''Notes on Rudder Design Practice,"' First Symposium on Ship Maneuverability, David Taylor Model Basin Report 1461, May 1960 . Wahab, R. and Swaan, W. A., ''Coursekeeping and Broaching of Ships in Following Seas,"' Journal of Ship Research, Vol. 7, No. 4, Apr 1964 Davidson, K. S. M., ''The Steering of Ships in Following Seas,'' Stevens In- stitute of Technology, Experimental Towing Tank, TM 94, Apr 1960 DuCane, P. and Goodrich, G. J., ''The Following Sea, Broaching and Surg- ing,"’ Royal Institute of Naval Architects, Vol. 104, No. 2, Apr 1962 . Grim, O., 'Surging Motion and Broaching Tendencies in a Severe Irregular Sea, " Davidson Laboratory Report 929, Nov 1962 Dieudeonne, Jean, "Collected French Papers on the Stability of Route of Ships at Sea, 1949-1950,"' David Taylor Model Basin Translation 246, Jan 1953 Strandhagen, A. G., "Preliminary Analysis of Manual and Automatic Steer- ing of a Directionally Unstable Model which Exhibits a Loop,"' David Taylor Model Basin Report 1280, Oct 1958 Schiff, L. and Gimprich, M., "Automatic Steering of Ships by Automatic Control,'' Society of Naval Architects and Marine Engineers, Vol. 57, 1959 Nomoto, Kensaku, "Directional Stability of Automatically Steered Ships with Particular Reference to Their Bad Performance in Rough Sea," First Sym- posium on Ship Maneuverability, David Taylor Model Basin Report 1461, May 1960 Sutherland, W. H. and Korvin-Kroukovsky, ''Some Notes on the Directional Stability and Control of Ships in Rough Seas," Stevens Institute of Technol- ogy, Experimental Towing Tank, Note No. 91, Oct 1948 Davidson, K. S. M. and Schiff, L., ''Turning and Course Keeping Qualities," Society of Naval Architects and Marine Engineers, Vol. 54, 1946 Chadwick, J. H., ''On the Stabilization of Roll,'"' Society of Naval Architects and Marine Engineers, Vol. 63 (1955) Heller, S. R., Discussion of "Some Hydrodynamic Aspects of Appendage Design," Ref. 1 Martin, M., ''Roll Damping Due to Bilge Keels,'' University of lowa, Nov 1958 Keulegan, G. H. and Carpenter, L. H., "Forces on Cylinders and Plates in an Oscillating Fluid,"' Journal of Research, National Bureau of Standards, Vol. 60, No. 5, May 1958 197 Ur(- 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. Giddings and Wermter White, W. H., ''The Qualities and Performances of Recent First Class Bat- tleships,'' Institute of Naval Architects ; Spear, L., "Bilge Keels and Rolling Experiments USS OREGON," Society of Naval Architects and Marine Engineers, 1898 Minorsky, N., ''Problems of Anti Rolling Stabilization of Ships by the Acti- vated Tank Method,"" American Society of Naval Engineers, Vol. 47, 1935 Chadwick, J. H. and Klotter, K., "On the Dynamics of Anti-Rolling Tanks," Stanford University, Feb 1953 and Forshungshefte fur Schiffstechnik, 1955 Bell, J., "Ship Stabilization, Controls and Computation," Institute of Naval Architects, 1957 DuCane, P. and Dadd, R. H., ''Control of Roll-Damping System," First Sym- posium on Ship Maneuverability Flipse, J. E., "Stabilizer Performance on the SS MARIPOSA and SS MON- TEREY,"' Society of Naval Architects and Marine Engineers, Vol. 65, 1957 Wallace, W., ''Experiences in the Stabilization of Ships," Institution of En- gineers and Shipbuilders in Scotland, Vol. 98, 1955 Chadwick, J. H., ''The Anti-Roll Stabilization of Ships by Means of Activated Fins,'' TR No. 2, Stanford University, Feb 1953 DuCane, P., Discussion on "Stabilizer Performance on the SS MARIPOSA and SS MONTEREY," Ref. 23 Avramescu, A., ''A New Comprehensive Integral Criterion for Optimaliza- tion of Automatic Control Systems," from Rev. Elecltotechn. et Energ (RPR) Vol. A7, No. 1, 1962 Aref'ev, B. A., "Use of the Integral Criterion in Certain Control Problems," Isvestia Vysshikh Vchebnykh Zavedeniy Priborostroyeniye, Vol. 6, No. 1, 1963 Froude, W., ''On the Rolling of Ships," Institute of Naval Architects, 1862 Frahm, H., ''Results of Trials of the Anti-Rolling Tanks at Sea," Institute of Naval Architects, Vol. 53, 1911 Vasta, J., et al., "Roll Stabilization by Means of Passive Tanks," Society of Naval Architects and Marine Engineers, Vol. 69, 1961 Chadwick, J. H. and Morris, A. J., ''The Anti Roll Stabilization of Ships by Means of Activated Tanks," Stanford University (in four parts) Dec 1950, Jan 1951, Jul 1951, Mar 1951 798 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44, 45. 46. 47. A Survey of Ship Motion Stabilization Blagoveshchensky, S. N., ''Theory of Ship Motions,'' Dover Publications, New York, 1962 Lamb, H., "Hydrodynamics," Sixth Edition, Dover Publications, 1945 Thornycroft, J. I., "Steadying Vessels at Sea'’ (Moving Weight System), In- stitute of Naval Architects, 1892 Jackson, P. R., "The Stabilization of Ships by Means of Gyroscopes" (Sperry System), Institute of Naval Architects, 1920 White, W., 'Experiments with Dr. Schlicks Gyroapparatus for Steadying Ships," Institute of Naval Architects, 1907 Deimel, R. F., "Mechanics of the Gyroscope,"' The MacMillan Co., 1929, Dover Publications, 1950 Pierson, Willard J., Jr., "Ocean Waves," International Science and Tech- nology, No. 30, Jun 1964 Foster, John J., 'Preliminary Evaluation of Passive Roll Stabilization Tanks Installed Aboard the USNS ELTANIN (TAK 270),'' David Taylor Model Basin Report 1622, May 1962 Motora, Seiza and Lalangas, Petros A., "Experimental Study of a Passive Anti-Rolling Tank Installation in a Ship Model Running in Oblique Seas," Davidson Laboratory Report R-961, May 1963 Lalangas, Petros A., ''Effect of Speed and Wave Height on the Rolling of a Ship Model with Passive Anti-Roll Tanks,"" Davidson Laboratory Report LR-1018, Mar 1964 Zarnick, E. E., "USS GYATT (DDG 712) Anti-Rolling Fin Evaluation in a State 4 Sea,'' David Taylor Model Basin Report 1182, Jan 1958 Dalzell, John F. and Wen-Hwa Chu, ''Studies of Ship Roll Stabilization Tanks," Southwest Research Institute Quarterly Progress Report Project No. 55- 1268-2, Sep 1963 Dalzell, John F., Continuation of Studies of Ship Roll Stabilization Tanks," Southwest Research Institute Proposal 2-3175, Mar 1964 Pourmaras, Ulysses A., "Pitch Reduction with Fixed Bow Fins on a Model of the Series 60, 0.60 Block Coefficient,'' David Taylor Model Basin Report 1061, Oct 1956 Pourmaras, Ulysses A., "A Study of the Sea Behavior of a MARINER-Class Ship Equipped with Antipitching Bow Fins,"' David Taylor Model Basin Re- port 1084, Oct 1958 799 48. 49. 50. D1. 52. D3. 04. D0. 56. D7. 08. 59. 60. Giddings and Wermter Abkowitz, Martin A., ''The Effect of Antipitching Fins on Ship Motions," Transactions of the Society of Naval Architects and Marine Engineers, Vol. 67, 1959 Becker, Louis A. and Duffy, Donald J., "Strength of Antipitching Fins and Ship Motions Measured on USS COMPASS ISLAND (EAG 153),"' David Taylor Model Basin Report 1282, Apr 1959 Stefun, George P., ''Model Experiments with Fixed Bow Antipitching Fins," Journal of Ship Research, Vol. 3, No. 2, Oct 1959 Stefun, George P. and Schwartz, F. M., "Effect on Hull Vibrations of Various Bow Anti-Pitching Fin Configurations,'' David Taylor Model Basin Report 1659, Nov 1962 Ochi, Kazuo M., ''Hydroelastic Study of a Ship Equipped with an Antipitching Fin,'' Transactions of the Society of Naval Architects and Marine Engineers, Vol. 69, 1961 Goodman, Theodore R. and Kaplan, Paul, ''Feasibility of Employing Jet Flapped Hydrofoils as Ship Anti-Pitching Fins,"’ Oceanics Inc. Report No. 63-08, Dec 1963 Spens, Paul G., ''Research on the Reduction of Pitching Motion of Ships by Means of Controllable Fins,'' Davidson Laboratory Report No. 733, Dec 1958 Spens, Paul G., ''Research on the Reduction of Pitching Motions of Ships by Controllable Fins," Davidson Laboratory Report No. 913, Nov 1962 Lalangas, P., Van Mater, P. R., and Marks, W., "Interim Report on Model Experiments on Escort Research Vessel in Waves," Stevens Institute of Technology Report 814, Nov 1960 Giddings, A. J., "Engineering Estimates of Ship Motions,"' Bureau of Ships Associate of Senior Engineers, 1964 (to be published by American Society of Naval Engineers) Giddings, A. J., "Progress in Tank Stabilizers,"’ Fourth Seminar on Ship Behavior at Sea, Stevens Institute, Jun 1962 Motora, S., "Steadying Device of the Rolling of Ships," U. S. Patent, 1925 Allan, J. F., ''The Stabilization of Ships by Activated Fins," Institution of Naval Architects, Vol. 87, 1945 800 A Survey of Ship Motion Stabilization DISCUSSION Peter DuCane Vosper Limited Portsmouth, England The authors of this interesting paper mention that model tests of anti- rolling fins either alone or on ship models have not been reported. However, we at Vosper have, in fact, carried out quite a number with what we consider to be a useful degree of success so far as the actual results are concerned. We claim that in the case of the nonretracting low aspect ratio fin we can produce a fin section which can be equally, or more, effective than the flapped fin for the same area. Without entering unduly into details it could be mentioned that in many cases it is clear that the greatest percentage of roll reduction does not of necessity in- dicate the most comfortable condition so far as roll amplitude is concerned. Quite small rolling amplitudes in certain complex wave patterns can give a most disappointing result on the basis of roll reduction with fins on against fins off. However, these cases do not really matter to the passenger and there is probably still quite a possibility of saving power in the operation of these fins by area reduction. The fin sizes can be substantially reduced without much loss of effective performance in their true capacity asroll dampers if, instead of 3°-4° being aimed for, say 6°-7° double amplitude is aimed for under the same conditions — no passenger could reasonably complain at this. At the same time I believe it is a short sighted policy to ignore the possi- bility of, and in fact reported occurrence of, quite large rolls in ''stabilised" ships under certain circumstances. The situation is, of course, that the activated fin is not in truth a stabiliser, or if it is ordered to act as one by an amplitude signal in the control system it is a very poor stabiliser. By far the most important function of an anti-rolling activated fin is to act as a damper controlled from a velocity signal. As the fins are usually designed on an empirical basis to cause a heel of 5 °-7° when at full incidence and full cruising speed it can well be understood that this means little in restoring effect when considering a roll induced by yaw at practically no frequency when leading up to conditions of broaching such as are met by even the largest liners in a quartering Sea. While acting as a damper a sluggish fin movement can cause an important phase lag leading towards the case where the fin is helping the roll. I do not say for one minute this is a normal state of affairs but in nearly all installations this can happen despite the fact that most of the time the fitting of anti-rolling 221-249 O - 66 - 52 801 Giddings and Wermter fins is highly effective and universally popular. This is where the acceleration term can help by incorporating an element of phase advance and getting things moving in plenty of time. Incidentally we as a firm always point out that good as they may be fins do not, of necessity, reduce the incidence of sea sickness and it is a somewhat dangerous policy to say they do because it is probably more the pitching accel- erations which cause the trouble. It is probably time that "stabilising'"’ devices were offered subject to per- formance specifications as the present method of advertising the optimum re- duction percentage under ideal conditions, or at least specially selected condi- tions is meaningless and misleading. The difficulty here, of course, is in specifying, in a meaningful way, the seaway in which the performance specified should be achieved and furthermore in recording the nature of the actual sea in which the performance is achieved. Though without first hand experience it must surely be a somewhat sobering thought that at the very low frequencies experienced in quartering seas in the Western Ocean there is quite a likelihood, if not certainty, that the water in any passive tank will provide an unstabilising moment just at the wrong time. Again I thank the authors. DISCUSSION John F. Dalzell Southwest Research Institute San Antonio, Texas The discussor would, in all sincerity, like to compliment the authors on one of the most straightforward and informative papers on the subject to come his way in some time. The authors' summary of the work on passive anti-roll tanks at Southwest Research Institute is adequate and exactly to the point. We have recently submitted a draft Technical Report* summarizing our efforts in this field which concludes, as did the authors, that a nonlinear model must be dis- covered before any significant gain over present design methods can be fore- seen, and that experiments will continue to play a large part in passive anti-roll *Studies of Ship Roll Stabilization Tanks, Technical Report No. 1, Contract NONR 3926(00), by John F. Dalzell, Wen-Hwa Chu, J. Everett Modisette, Southwest Research Institute, August 1964. 802 A Study of Ship Motion Stabilization tank investigations. We feel that the tank scale effect problem must be further explored if passive anti-rolling tanks are to continue to be installed in seakeep- ing basin ship models. We have attained better agreement than that shown in Fig. 25 of the paper between our current weakly nonlinear theory and other ex- perimental data. This better agreement is, however, not sufficiently good for practical use. One further remark is perhaps justified and that is that detailed space-time mappings of the free surface in a tank indicate that the fluid seldom behaves in a fashion similar to either that in a U-tube or to a first mode stand- ing wave. Evidently, considerable additional effort on the fluid dynamics of the free-surface passive anti-rolling tank will be necessary. DISCUSSION S. Motora University of Tokyo Tokyo, Japan I would like to make some short comments on the anti-pitching tank. As Mr. Giddings has mentioned, the idea is to put openings at the bottom of fore or aft peak tanks to let sea water come in and out in a 90 degree phase lag behind the pitching motion resulting in a reduction of pitching motion. This problem was initiated by the Technical Research Laboratory of Hitachi Shipbuilding Co. and was published in the fall, last year. In that paper, move- ment of the water level in open tanks, installed at the bow and the stern, is ana- lysed theoretically, and the pitching angle of a ship in regular waves, affected by such tanks, is calculated. A model experiment with a model of a passenger ship was conducted to check the calculation. Two tanks were installed; one at the bow and one at the stern. The total water plane area of the tanks was 25 percent of the ship's water plane area. Results are as shown in Fig. 1, where a is the area of the openings at the bottom of the tanks and A is the waterplane area of the tanks. About 45 percent reduction at the maximum was attained. I treated the same problem and dealt mainly with the fundamental charac- teristics of tanks with openings under the waterline. At first, let us consider a tank with a vertical wall. In Fig. 2, suppose a tank has openings of area a. Free surface area of the tank is A, the depth of the openings is h,, heaving of the tank is z, and elevation of tank water is <. 803 Giddings and Wermter soces WITHOUT TANKS —— WITH TANKS MAGNIFICATION FACTOR OF PITCHING ----- WITHOUT TANKS x Ke —— WITH TANKS MAGNIFICATION FACTOR OF PITCHING Figure 1 Then the equation of motion will be written as Eq. 1 in Fig.1. It is noted that the excitation is modulated by a function which becomes zero when «@, = Vg/h,. It means that tank water does not move at all at this frequency regardless of amount of heave. Therefore, this frequency will be called as zero response fre- quency. From this it can be seen that h, should be chosen so that w, does not coin- cide with the natural frequency of pitching. Considering the average pitching period, it can be easily seen that h, should not be too large. On the other hand, the resonant frequency of the tank water level is also vh,/g which is the same as the zero-response frequency. Therefore, in this case of wall sided tanks, the response of tank water is very small and will not be effective. ; 804 A Study of Ship Motion Stabilization ee ala) eee Glen) W. =\-- ZERO RESPONSE FREQUENCY t fi NATURAL FREQUENCY OF TANK WATER LEVEL Bupune 2 In Fig. 3, the magnification factor of the response of the tank water level is plotted against the frequency. Two solid lines show the solution of the Eq. 1 for A/a = 4.17 and 6.52. It can be seen that the smaller the openings, the less re- sponse. Plots are made of the experimental values. There are some disagree- ments with the theory, but, if the damping coefficient is doubled, i.e., the effec- tive area of the openings is reduced to 7/10, the theoretical values agree very well with the experimental data. AMPLITUDE OF THE TANK WATER 0.5 805 Giddings and Wermter To avoid the defect that zero-response frequency coincides to the resonant frequency of the tank water, a flared tank was studied. In this case, as shown in Fig. 4, the inertia term changes somewhat and h, becomes h/ in this case. Since h{ > h, for normal flare, resonant frequency does not coincide with the zero frequency and becomes nearer to the ship's natural pitching frequency. Therefore the effectiveness of the tank will be improved. TANK WITH FLARED WALLS Hingtg, 6+ (ASE + 98 = (fu 9)zZe -----(2) haa ad Re Figure 4 However, the amount of the flare will not be chosen arbitrarily. If ducts of certain length ¢ are attached to the openings, the equation of motion will be written as Eq. 3 in Fig. 5. In general is called hydraulic length. The longer and narrower the duct, the longer the hydraulic length and the smaller the resonant frequency. Therefore it will be possible to bring the resonant frequency of tank water to equality with the pitching frequency, and to make it quite different from the zero-response frequency. A 2m model of a catamaran was provided with fore open tank and tested in waves. The waterplane area of the tank is 5 percent of the total waterplane area. 806 A Survey of Ship Motion Stabilization (Ag+ Hon ty) Et (G)t 96 = (howe a)ze™ V—_—__~——_— fio 2 t A fe = HYDRAULIC LENGTH = ie ax 4% Figure 5 807 TANK WITH DUCTED HOLES. Giddings and Wermter —o— WITHOUT TANK --A— WITH FLARED TANK --x-- WITH DUCTED TANK In Fig. 6, solid lines show the pitching magnification factor when the tank was blocked. Broken lines show the results with a flared tank and with simple holes. Chain lines show the results with a flared tank and with ducted openings. About a 20 percent reduction at the maximum was attained with a ducted tank. 808 A Survey of Ship Motion Stabilization DISCUSSION E. Numata Stevens Institute of Technology Hoboken, New Jersey Davidson Laboratory is pleased to have been associated with the USN Bu- reau of Ships and DTMB in experimental model research on anti-pitching fins and passive anti-rolling tanks since 1956. One of the earliest, although subsid- lary, investigations conducted at DL concerned the magnitude of the influence of fixed bow anti-pitching fins on the longitudinal midship bending moment of the COMPASS ISLAND. It was found that the fins had no adverse effect on hull bend- ing moment. In connection with stern anti-pitching fins, an analytical study for DTMB at Davidson Laboratory showed that in head seas at wave and ship speed conditions bracketing synchronous pitching motion, the hydrodynamic angle of attack of fixed stern fins is very small. Thus stern fins must be activated to be effective, producing a stabilizing moment and decrease in pitch angle which are propor- tional primarily to their amplitude of oscillation and relatively independent of the pitching amplitude. We found this to be true also in the case of oscillating fins astern of a pump jet propeller on a destroyer escort model tested for East- ern Research Group several years ago. This characteristic of a fixed number of degrees reduction may explain why in the author's Table 7 the percentage pitch reduction decreases as sea state and pitch angle increase. In connection with full scale evaluation trials of passive anti-rolling tanks, it seems to me that instead of vainly hoping for ideal wave conditions of unvary- ing severity and direction, it might be better to conduct trials in the calm seas one usually finds when searching for rough water. Rolling excitation could be provided by some form of portable oscillating weight device. Since most naval and oceanographic vessels fitted with passive tanks are of modest size with reasonable metacentric heights, it should not be too great an engineering prob- lem to design and assemble a device whose oscillation frequency can be varied while providing sufficient roll exciting moment to give a static heel of about 2°. Thus a frequency response could be obtained for the ship with and without the passive tanks operating. The omission of sway excitation would be a necessary but not totally undesirable condition. 809 Giddings and Wermter DISCUSSION K. C. Ripley John J, McMullen Associates, Inc. Washington, D.C. I desire to comment on the figure of 90 percent as the reduction of roll at ships resonance, that was mentioned by Mr. Giddings in connection with one of the slides of his talk, namely, the slide showing Fig. 11 of the paper. The design of passive tanks of the particular ship to which Fig. 11 refers is one with which Iam familiar. Up until late 1960, I had been employed for 25 years with the U.S. Bureau of Ships, and during this time designed a number of passive anti-rolling tanks, one of which was for the Oceanographic Research Vessel, AGOR. This is the vessel that by model test in an irregular, bow sea showed the reduction of roll at ships resonance of the 90 percent. It is my opinion that the foregoing, reported roll reduction is real, and can be accepted as representative of what would have been obtained by the same or similar test performed full-scale at sea. This opinion just expressed is based on personal experience obtained at sea with a merchant ship. This ship tested at sea was fitted with bilge keels, and was tested for the stabilizer tanks opera- tive, and inoperative. The sea was a quartering sea. The reduction of roll at ships resonance was found to be 85 percent. How is it possible for the reduction of roll at ships resonance to be as large as 85 to 90 percent when the test is conducted in an irregular sea, whether the sea be model-scale, or full-scale? When the reduction of roll is found for an- other condition of test, namely, for a bench model type of test, the reduction of roll is not as great. In this latter type of test, the roll response is that for pure, steady-state, forced roll. In the former type of test, nothing closely resembling steady roll is ever obtained, and what might appear to be forced roll is in real- ity an interaction between the instant to instant values of stored energy of roll of the ship, and the instant to instant values of input of energy of roll from the sea. It is well known that ships at sea tend to roll at or near ships resonance almost irrespective of the frequencies of excitation existing in the sea. We all know that this is what actually happens in roll at sea, but then we are all prone to forget what the actual situation at sea is, in order that we may treat the in- stant to instant roll as representing steady-state forced roll. It is true that after a ship has been well stabilized against roll, the ship will behave more like one the roll of which is pure forced roll. Before the ship has been well stabilized against roll, however, the ship will be rolling more often at ships resonance than otherwise would be the case. It is clear that a roll reduction at ships res- onance as great as 85 to 90 percent when the determination is by test at sea is both reasonable, and comprehensible. A part of the roll reduction is from hav- ing a less amount of energy tending to roll the ship at the ships natural frequency, and a part of the roll reduction is from allowing less forced roll of the ship when the roll can be treated as more nearly resembling pure, forced roll. 810 A Survey of Ship Motion Stabilization I want to compliment the authors for an interesting and informative paper. The paper by being a survey is a mine of information on a wide range of topics, having to do with ship motion stabilization. DISCUSSION A. Silverleaf National Physical Laboratory Teddington, England This interesting survey is almost as surprising for its omissions as for the topics which it discusses at some length. For instance, it is more than surpris- ing to find no reference to the paper by the late J. F. Allan, ''The Stabilisation of Ships by Activated Fins,'' Transactions of the Institution of Naval Architects, 1945, Vol. 87, which was the first published account of the modern development of the type of roll stabiliser still that most commonly used and adopted. The authors' suggestion that the design of activated fin stabilisers has developed in an unscientific manner is completely contrary to the facts. Activated fin sta- bilisers of the type now known as the Denny-Brown-A.E.G. have been continu- ously developed for the past 25 years by a skillful and systematic combination of theory, model experiment and full scale practice. This applies not only to the activating and control mechanisms but also to the basic hydrodynamic design of the fins themselves, for which in 1942 I developed an inverse Theodersen pro- cedure for designing foil shapes with delayed cavitation characteristics. Similar methods were being independently and simultaneously developed for aerofoil sections and have produced among other things the well known "flat top"’ sections. The authors’ doubts about the value of roll stabilisers of this type were certainly not endorsed by the crews of the ships of the Royal Navy fitted with such stabi- lisers during the Second World War; in many cases they were the only ships able to offer any effective defence against air attack because they provided a reason- ably stable firing platform. The authors' discussion of roll stabilisers of the passive tank type is of great interest to us at N.P.L., where such stabilisers have been designed for some time. It is our growing opinion that a wide variety of shapes and configu- rations can be effectively used for this purpose, and indeed it is almost true to say that only a very good man can design a really bad system. Dr. Kaplan's reference to activated pitch stabilisers revives interesting memories for me. Almost seven years ago Mr. Goodrich and I took out a provisional patent for just such a stabiliser, incorporating a jet flap, but allowed it to lapse because we found great difficulty in producing a system of reasonable overall mechanical and hydrodynamic efficiency. Naturally we shall be most interested in this new attempt to exploit this attractive idea. However, I might venture a word of 811 Giddings and Wermter caution. While many devices, including bulbous bows and fins, show a reduction in pitch in regular waves, this improvement is often not shown in terms of sig- nificant motions in irregular waves. General experiments in regular waves are not carried out in long enough wave lengths in which these devices can show un- desirable response characteristics; experiments in irregular wave systems in- clude the responses to such wave lengths. REPLY TO THE DISCUSSION Alfred J. Giddings Bureau of Ships Washington, D.C. and Raymond Wermter David Taylor Model Basin Washington, D.C. The authors' are pleased with the response to the paper. Mr. Ripley's re- marks are appreciated, as coming from one who re-initiated the interest in passive tank stabilization. The additional information on anti-pitching tanks presented by S. Motora is especially interesting. Continued work on this line may well lead to much im- proved seakeeping, at least for special ships. Mr. Dalzell's recent work on the details of anti-roll tank dynamics is somewhat discouraging in that the nonlinearities inherent in the phenomenon are confirmed. The simplified analyses that have sufficed for design in the past, must be replaced by more elegant processes to realize the full potential of passive tanks. The state of the art in fin stabilization as discussed by Commander DuCane continues to advance. The reduction of design fin capacity as advocated by the Commander, is not endorsed by the authors. There may be cases, for ships with very long roll periods, wherein the fin capacity is not so readily taxed, but for most ships, saturation would defeat the value of the fins. It is agreed that the circumstance associated with the occasional very large roll should be clari- fied. Fin effectiveness in a following sea is reduced by the orbital velocity of the water, and by the difficulty of designing a control system to cope with low fre- quency disturbances as well as the more usual frequencies. 812 A Survey of Ship Motion Stabilization The literature having to do with seasickness substantiates Commander Du- Cane's statement as to the motion that is the principal cause. In addition to the literature, personal experience leads to this conclusion. In regard to the specification of performance for fin stabilizers, it might be possible to test the performance of the control system by pre-programming the fins on one side of the ship to a certain time history of fin angle, and having the control system and the fins on the other side stabilize. The authors' must apologize to Mr. Silverleaf for the apparent omission of reference to Mr. Allan's work. This reference was inadvertently omitted in the typing of the manuscript. An errata sheet was issued correcting this oversight prior to the meeting but was not available in time for distribution. The authors' conclusions on the inventive approach to design of active anti- roll fins was based on published literature. It is apparent from Mr. Silverleaf's remarks that a great deal of unpublished scientific work has been performed in this area. The reference to additional model work in this design area was made specifically with respect to activating ship model fins, that is to say model in- vestigations of the entire control loop. To the authors' knowledge, little work has been done in this area. The authors agree with Mr. Silverleaf's views on the design of passive tanks. The authors are grateful for Mr. Numata's interesting observations and supplemental comments. His proposal for inducing roll by a moveable weight system is an interesting one and is quite parallel to the present scheme of forc- ing roll with active fins and determining roll quenching ability. We would have to determine the amount of weight required for such a system and the required frequency responses of the control system before we could evaluate the practi- cality of applying such a scheme to practice. 813 lS pera one senate ” Ww Ne gn a a pm . ities Ta vt Ty , M4 na bes re neha avec fee shmilwe estan ob a 4 ghd )sa + PEN \j jae py Hud’! Phe TARE wat “aff Ate lithe Seay cay oy PONE, SUR RC e N e US an aay we baie A ioee oe Nn AE PRE TaD aT EER ibirscsenentede i iy (ithe “a alae , } t th sy Feiner vines nly pareagoraiiog th Bel yng VAM th non Rites ett ky ane en rr Span ba} , aa > bas isn" cy * wets ryt + As j f 7 ») byt uh ifs i é We Oo tela LAL ’ iy yt Pye Poni % ¢ apy | ' thy ey te 4 \ 4 vi ai y | mig ‘ ‘ iy Sa i i Lt ya i / Ai 4 i! \ uf om ' } ane ' : i A 4 pers , i ae A ’ Pupil \ i be ae di , a : ' v ¥ ° * pd i Mi Ta ey = iy Mu : * ry oe he i ' \ j t ( { rd i ‘5 t { F i i >? a, ‘ ge { y I \ ¥ i rea Mj i iio i i ; ) i i ai t { A VORTEX THEORY FOR THE MANEUVERING SHIP Roger Brard Bassin d'Essais des Carenes de la Marine Paris, France FOREWORD The present text differs on many points from the draft which was prepared for the 5th Symposium on Naval Hydrodynamics held at Bergen (10-12 September 1964). Firstly it appeared necessary to correct many misprints and also omis- sions which made the reading difficult. Moreover it was useful to explain with more details the theoretical views which lead to the introduction of a delayed circulation around’'a maneuvering submerged body. The line of thought is unchanged, but some results are presented with a greater precision. The new paragraph on some experimental results (par. 16) shows that some "apparent coefficients'' may be found increasing and not decreasing when the re- duced frequency increases. That seems to mean that the effects of the terms in o/et in the equations of the motion may be higher than in the case of a wing of infinite aspect ratio. The effects of the wake on the stern planes are confirmed to be very high. INTRODUCTION The work to be done in the naval hydrodynamic field in order to solve the problems related to the unsteady motion of the ship is often a very difficult one. A mathematical model of the physical phenomena has to be found. That requires various compromises. For, if the equations, which the mathematical model leads to, were too complicated in regard to the possibilities of an effective treat- ment, no real improvement would have been obtained. That is undoubtedly why the equations of the classical ship hydrodynamics are differential and of the second order. Nevertheless, in some cases, such equations are not suitable at all, and the modern ship hydrodynamics must often consider other classes of equations. It is, for instance, admitted that the equa- tions which govern the rolling, heaving and pitching motions of a surface ship on 815 Brard irregular waves are integro-differential equations of the Volterra's type [1]. That is already true even on regular seas, because the waves generated by the ship have to be added to the incident waves. The problem which the present paper is devoted to is that of the maneu- vering ship. For this problem, a ''classical"’ theory already exists. That is, the quasi- steady motion theory. It is admitted that, with the exception of the effects of the so-called ''added masses," the hydrodynamic forces exerted on the maneuvering ship are identical to those found for a steady motion with the same angles of attack and the same linear and angular velocities. That leads to a set of differential equations of the second order. This set is rather complicated in the case of a submerged body in an infinite fluid because the number of the degrees of freedom is high. Moreover, the lin- ear approximation is most often insufficient. Consequently, the equations con- tain many, many terms. As the theory is unable to yield them, it is necessary to resort to an empirical determination of their numerical values. When the equations are written, it is necessary to solve them by using analog computers. And the work is not finished by this time. The empirical determination of the coefficients of the equations would have been practically impossible if the mo- tion had not been split in its components; then the results so obtained must be gathered. That is not so easy since the equations are not linear. Consequently a comparison between the calculated motion and the real motion of the model or of the full scale ship must be undertaken. Finally, the precise study of the maneuvering qualities of a ship, especially of a submarine, requires a great deal of work. Therefore, the idea that the quasi-steady motion theory might be too simple is attractive to very few. That is, however, the question about which the author of this paper has tried to make up his mind. The starting point of the present investigation is that the hydrodynamic set of forces exerted on a maneuvering ship is partly due to some circulation around the ship. If so, this circulation around the body generates a vortex wake since the circulation along a closed fluid circuit is null. And the vortex wake is what prevents the equations to be purely differential. As in the Karman-Sears theory of the unsteady motion on an airfoil of infinite aspect ratio [2], we shall expect to deal with Volterra's integro-differential equations. Consequently the forces in the real motion and those calculated by using the quasi-steady motion theory must differ from one another, no circulating being able to take instantaneously the value relating to the steady motion. This starting point needs some comments. For the quasi-steady motion theory does not preclude some circulation. In- deed, this circulation cannot come from the set of forces deduced by Lagrange's method from the kinetic energy of the absolute motion of the fluid surrounding the body: it is assumed that this motion depends upon a velocity potential reg- ular at the infinity. But, if some circulation exists in the steady motion, we shall find it in the equations expressing the quasi-steady motion theory. 816 A Vortex Theory for the Maneuvering Ship That is the case, because the lift is not null. For instance, when the lift component on the z-axis is opposite to the direction of this axis, the mean pressure on the upperside of the body is smaller than on the lowerside; on the contrary, the mean velocity is greater on the upperside. And the circulation around the body along closed circuits parallel to the (x,y)-plane is necessarily non-null. The same reasoning holds in the case of a maneuvering surface ship, the lift being now in a horizontal plane. In 1950, one of our assistants has calcu- lated a distribution of free and bound vortices for a thin surface ship in a steady turning motion and obtained by this way some results which help under- standing several phenomena unexplained to this time (see [3] and also [4)). Some authors [5-7] probably have ideas quite similar to the one expressed above. But they are principally interested in the configuration of the vortex wake and in the mechanism of the transport into the wake of the vorticity which originates in the boundary layer. Such a line of thought is the best from a Ssci- entific point of view. Unfortunately, such a study is very difficult and will not lead rapidly to results that the naval architects may easily use. That is why we have chosen here another way. A mathematical model of the vortex shedding has to be defined. Preferably it has to be flexible enough to be adaptable to the various hull forms we encoun- tered in the practice. Consequently, this model is not made for giving all the means necessary for a complete calculation, in each case, of the hydrodynamic set of forces in steady and unsteady motions. In return, it has to yield the gen- eral form of the expression of this set, and also, to supply a criterion which permit to decide whether, according to the experimental results, the differences between the quasi-steady forces and the real forces are negligible or not. The present paper gives a first answer to this problem. Section I defines a mathematical model of the wake vortex and leads to the Volterra's integro-differential equations which govern, in an unsteady motion, the circulation and the forces exerted on the body. Attention is drawn —as in [8]—to the pressure distribution on the hull, and also to the effects on the stern planes and rudders of the wake generated by the submerged body itself. Section II shows that in a harmonic forced motion, the forces differ from those given by the quasi-steady motion theory. Some experimental results show that there is a possibility to estimate the magnitude of the errors involved in the quasi-steady theory. Some of them are small. Others are significant. Section III is devoted to possible further developments of our present views. It is shown that tests in various steady and harmonic forced motions are able to yield all the unknown coefficients and functions found in the so-called "true" equations of the free quasi-rectilinear motions. Unfortunately, other motions are of great interest too, those which require non-linear equations. In these cases, the technique of the steady and harmonic forced motions is unable, in its present state, to supply all the necessary information. Moreover, the "true" equations are more complicated than those of the quasi-steady theory and lead, 221-249 O - 66 - 53 817 Brard not in principle, but in fact, to non-negligible difficulties even in the field where the equations are linear. The first answer given here is therefore faulty. The conclusions of this paper will probably not satisfy fully the naval architects. It is hoped, however, that the ideas developed here may be of some practical interest. I. THE FORCES EXERTED ON A SUBMERGED BODY MOVING IN AN INFINITE FLUID 1. Notations Let 0O'(x',y',z') be adextrorsum set of fixed axis. The z'-axis is vertical and positive downwards. We consider also a dextrorsum set of axis O(x,y,z) attached to the sub- merged body. When the body is in a normal position (that is, when the heel and trim are null), the z axis is vertical and positive downwards. The x axis is going from the stern to the bow. 0 is in the middle trans- verse section. Generally, the body is symmetrical with respect to the (z,x) plane. The coordinates of O referred to the fixed axis are €,7, ¢. In order to define the position of the body we introduce firstly a set of axis O(x,,y,,2,) having its origin at O, but with the axis Ox,,Oy,,0z, parallel to the axis O’x',O'y',O’z'’ respectively. We consider three non-Eulerian angles y,6,¢ (Fig. 1). y is the head angle. By a y-rotation about the z,-axis, the x,-axis comes in the (z,x) plane on an axis Ox,; by this rotation, the y,-axis comes on an axis Oy,. The z,-axis coincides with the z-axis. 6 is the trim angle. By a ¢-rotation around the y,-axis, the x,-axis comes on the x,-axis; by the same rotation, the y,-axis andthe z,-axis come, respec- tively on axis Oy, and Oz,. The x,-axis coincides with the x-axis. ¢ is the heel angle. By a ¢-rotation about the x-axis, the y,-axis and the z,-ax1S come, respectively, on the y-axis and on the z-axis. The absolute velocity of 0 is V,, of components é,7, © on the fixed axis. The components of V, on the x,y, z-axis are respectively u,v,w. © is the heaving velocity; the derivatives y, 0,¢ are, respectively, the head- ing velocity, the pitching velocity and the rolling velocity. 818 A Vortex Theory for the Maneuvering Ship ae Figure l Between the unit vectors i,,,i,,,i,, and the unit vectors i,, i, i, we have the relations deduced from the following table: Bio -sinwcos $+ cosw sin@ sing | siny sin d+ cos wWsin cos ¢ poe aoa 819 (1) Brard Moreover, the components p,q,r of the angular velocity of the body on the moving axis are = Sah oie 2 @ = Ocos¢ + Wcos 6 sind, r = cos 6cosé-@ sind. (2) Let G be the center of gravity of the body. Let 5m be the mass of a small volume Sw which coordinates with respect to axis parallel to the x,y, z-axis, but having their origin at G, are x",y",z". When the body is symmetrical with re- spect to the (z,x)-plane, the moments of inertia of the body are Te, = S(y"4 4 2") om , I S (a4 # oe” 2) om 5 te S(t fy) om (3) Mg = 202 8 jai o Let » be the specific mass of the fluid, and » the mean density of the body with respect to the fluid. We introduce dimensionless coefficients by the formulae: ty Slee ey = Ure, A = ete, yy = lL wx, (4) where L is the length of the body along the x-axis. When G is not at 0, its coordinates are Lé,,0,L¢,. We assume that ¢¢, %, have negligible squares and products. Otherwise, the moments of inertia of the body about the (x,y, z)-axis would be 2 2 t 2 1 = OWLS ECxi + SG) > 15 =pWL x); [| s | (5) Lal | = OWL? u(x + be) ee pw u(x, , + CGSq) - The set of the absolute forces has a general resultant ¥ and a resultant moment { referred about the origin 0 of the axis attached to the body. One has: SRP OXat ey Naeem Scie tal ici (6) In this paper, we don't consider the relative forces, that is the forces in the set of axis attached to the body. 2. Some Particular Motions Motions Parallel to the (z, x)-Plan — The y-component v+rx-pz of the absolute velocity of any point attached to the body is null. Consequently Go = Ou 10), p= On. (1) Therefore p= sin 6= Olg g tg 8, q = (2) 820 A Vortex Theory for the Maneuvering Ship When ¢=0, one has Wy =- 6 tg d/cos 6 = 0, w = constant. The motion is also parallel to a vertical plane. Motions Parallel to the (x,y)-Plane — The z-component w+ py - qx of the absolute velocity of any point attached to the body is null. Consequently Ww = (0) . jo) = Oy | = (0) « Therefore d= W sin 6, 6=-\ cos 6 tg ¢, genet (3) This motion is parallel to the horizontal plane when ¢ and @ are simultaneously equal to zero. Quasi-Rectilinear Motions Parallel to the x-Axis —In the caSe, we substi- tute U+u for u. One considers that u Vv WwW Ten ea (4) are small. B being the breadth of the body, Hog 8 ls ka ie Urals Chul SU? a (5) are small too. The square and products of ratios (4) and (5) are negligible. 3. Vortices Attached to a Body on Steady Motion in its (z,x)-Plane It is well known that a submerged body may be considered as equivalent to a distribution of bound vortices when no wake exists and to a distribution of free and bound vortices when a wake is shed. On the other hand, it is well known, too, that a closed filament vortex is equivalent to a distribution of doublets. To write the expressions of the forces generated by such a distribution of vortices or doublets, it is helpful to bear in mind the main aspects of the theory. 3.1. Bound Vortices are Equivalent to a Submerged Body in a Perfect Fluid As a matter of fact, when the fluid is quite perfect, the absolute motion of the fluid may be considered as generated by a distribution of vortices located on the hull when the angular velocity Q is equal to zero, on the hull and inside the body, when Q + 0. 821 Brard This possibility comes from the property of the vector W(M), having its origin in M, and defined by the formula where VW), which has its origin in ,, is continuous with respect to the point 1 which describes the volume 1; its first derivatives also are assumed to be con- tinuous; moreover div Wyn) = Taking for the space ©, exterior to the body, and for Wj.) the absolute velocity V, - - grad ©, of the fluid, then using the equation curl curl W = grad div W - VW, (1) where 2 2 V = oF + oF 3 Ox? oy? oz? we obtain V.(M), when M is in 1 curl Sey Wert eee aaah ee eo . Tee CUE 7M (/) aene uM (H) = le s S 0, when M is in QO. : Q, being the volume inside the body. Taking now for © the volume 2, , and for Wj ) the absolute velocity of yu considered as at rest with respect to the body, we have Wu) = V,(u) with Ve(u) = U+QAO0u, curl Vz = 20, AV dQ. - curl ‘ J= = zee dS(jz) + II) curl V, a ch 0 when M is in 2 1 nV_( /) - e? + ras grad \) rie ads@D= V, when M is in on i and By addition, we find, with V, = V,-V, = the relative velocity of the fluid, 822 A Vortex Theory for the Maneuvering Ship nV, V_(M) when M is in 2_, i curl {\) Sas} {7% 20 4.09} Seo > F (2) V(M) when M is in QO; : since V,n = Vjn on S. Let us consider now the surface of the hull as covered by a very thin bound- ary layer (of thickness 5); we see that a vortex equal to (1/5) nAV, on the mean surface S(m) between the internal face S, and the external face S, of the bound- ary layer, and to 2Q in ,, generates an absolute fluid motion which has the fol- lowing properties: outside the body, the motion is identical to that of the fluid; inside the body, the fluid is at rest with respect to the body. 3.2. A Distribution of Doublets is Equivalent to a Distribution of Bound Vortices When the Angular Velocity is Equal to Zero When Q=0, we have a velocity potential in 9, and in {); , which may be re- garded as due to doublets normal to the hull: 1 gl al CD) = ae {| Dig) eS eae Ts ds(m) , V =~ grad 0, ;| S 3 (3) a U in QO; : n being the unit vector normal to the hull and positive outwards. Therefore, m, and m, being on the normal n(m) to S(m), m; on S;, m, onS,, we have > Yom) = :- { Yo(m') a =a dS(m') = f(m) (4) Ss with fo) ge U |iieeex((mi) Tes Si( Jit) & i,z(m,)] + constant. (5) Equation (5) is Fredholm's equation of the 2nd kind relative to an interior Dirichlet's Problem. For any value C of the constant in the right member of (5), the solution of (4) is unique. One has ®(m,) - O(m,) = ~¥o(m) - Therefore when another constant Cc’ is substituted for C, ®,(m,) is changed in ®,(m,) + C'-C. Hence the motion of the fluid outside the body does not depend upon the value of the constant C. Let us assume this constant chosen in such a way that y,(m) = 0 on the forebody. Let Cm) be the rings normal to V ro (m)s o, the arc of this ring, the EES of their orthogonal trajectories C, (o! =0 at the forebody, >0 sare i,. i, the unit vectors tangent to C, and to € e. respectively, the directions of these vectors being those of do, >0, do; >0, and these directions themselves being chosen in such a way thation = A cihia 7 823 Brard The flux of the vortex T, = (1/5) nAVr, inside the small area sdo; normal to C, is equal to dy, and is constant between two rings C,,C; of abscissae o,, Gade son ee a(Seenisie: 2): The rings C,(m) are the curves y,(m) = constant. Moreover, on S,, V.,do, = dy.. When Q-0, v=0, that is when we have a motion parallel to the (z,x)-plane with no angular velocity, it is convenient for what follows to write Definition of 5;,5 ,Se —p Definition of the bound vortex To(= const, along Cy ). and of the density X%, (m) of doublets normal to the hull when ; 1°) there is no wake, 2°) the angular velocity is null, Figure 2 824 A Vortex Theory for the Maneuvering Ship Dy(M) = Gy o(M) + Dy o(M) + Oy CM) Tp (6) with 5 9 (M) =e aa il) VYoo(m) i + dS(m) , S m (7) aa ®,(M) = - = ill Yo x(m) o— ee dS(m) . S m Functions y,,(m) and y,,(m) are solutions of the same Fredholm's equation of the 2nd kind, with a right member equal to Ux(m,)+C, for y,, and to UzG@mey GC, ior “y, 4: The potentials ,,,®,, may be regarded as generated by bound vortices t,, and t,,- The rings on which the vortices ty, are lying are the curves 70 = constant. If do,, is the distance between two rings 7% = constant, doy being positive downstream, i,, the unit vector tangent to S and normal to the ring, i,, the unit vector tangent to the ring, with i,, = nAi,,, one has 00 00? ., 4970 to = n Aigo do,, . (8) A similar formula gives the vortex t,,. 3.3. Case When the Angular Velocity is not Equal to Zero (Fig. 3) Let us assume now that u/U = 0, w/U = 0, La/U + 0. The absolute velocity potential is L ie ®)(M) = ®)9(M) + ®, = , when M is in 2,. (9) IDWS 825 Brard There is no velocity potential in 2; since q+0, and curl y, = 2qiy- But we may write: -grad ©,,(M) = +: curl All: — dy ee Beller =i 40 ibe of Mt aoe (10) We have to define t,,(m) (Fig. 3). In order to do that, let us consider on S the rings c, and c, located in the planes of abscissae x and x+dx, (dx<0). Let m',m, be two points on c,, m'‘ being on the starboard side, m, on the portside, with z(m,) = z(m’). A, origin of the are c, on c,, is chosen on the upper arc of the contour @ along which the planes tangent to S are parallel to i oa LN being on c, and on the lower arc of @, we consider a point m on the arc m‘A'm,. Let doj(m'), do,(m) be the distances measured on S, at m’ and at m, respectively, between c, and c,. These dis- tances are considered as positive. Moreover, i,(m) is the unit vector tangent to c,; i,(m) is positive with respect to the x-axis. The arc doy > 0 has a di- rection identical to this of i,. Now we define at m an element of bound vortex dt, ,(m) = i,(m)dt'(m) by the condition that v 1 2U ° 1 dt (m)(édo,) | = + i 1y(n do, doy) : Consequently, the filament vortex which intensity is equal to AY. f = a 1 dx dz(m‘) on the segment mim’ and to dt,,(m)(Sdo,)_ on the arc m‘A’m,, is closed and this intensity is constant along the filament vortex. It is the flux of the vortex (2U/L)i, through the small area (do,do,), on S. The total vortex at m has an intensity given by 20 ty o(m)(i,odoy) = ) | i(m") n(m’) dox¢n" | clen(am )) (11) A This vortex is equal to zero at A It constitutes with the vortices (2U/L)i, located in ,, a family of closed filament vortices having a constant intensity along their length. Consequently, the vector dQ, Vo .(M) = x curl {dh op dS(m) + he 1 ae a (12) 826 A Vortex Theory for the Maneuvering Ship satisfies the condition 0 in Q,, curl V),(M) 2U aT ly in oF ° Consequently, V;,(M) depends in 2, upon a velocity potential. Let ©; ,(M) be this potential. One has Vo.(M) = -grad ®,(M) in Q,. In 2,;, V,, does not depend on a velocity potential; but viocm) - Ui AOM 02 rt y depends on a velocity potential ¢: 1 Une P -grad ¢(M) + Vj.(M) = L i, AOM in 0,. Let us now consider the velocity potential ®;,(M) defined in 2; and in %, by the distribution of doublets y'(m) on § so that u i d O(N) - ii} AO) ei + dS(m) , (13) S m with aga 1 eee é 1 Hae ee ay (m) a J) (in ») ch, ie dS(m') = -¢(m,;) + constant . One has ®5.(M) = $(M) + constant in 0); . Hence, one has u“ 1 U e > -grad 0),(M) = -grad ¢(M) = -V,,(M) + = i, AOM in 0, . Therefore F , Ue -grad ®),(m;) + Vj.(m,;) = L y AOm; on S, . When M passes through the boundary layer, from m, to m,, the normal com- . . 1 . . ponent of grad ;, is continuous. The normal component of V,, is continuous, too. Consequently 827 Brard - grad [®,(m,) + ©) ,(m,)] n(m,) = [- grad ©) ,(m,) + Vg,(m.)] m(m,) =I SI [i, AOm,] n(m,) - (14) Therefore ®;,+,, fulfils on S, the same condition as the wanted potential ® Because these two potentials are regular in ©, and at the infinity, one has 02° ®,,(M) = ),(M) + ®),(M) + constant in 0,. (15) This equation defines t,,(m); one has t,(m) = = {-grad Dy o((Lil.)) = at AOm,} An(m,) - (16) This solution does not depend upon the choice of the rings c, since the potential ®,, is perfectly defined (with the exception of an additive constant), by the con- dition on S,. 3.4. The Vortex Distribution When the Fluid is not Quite Perfect In this case a vortex wake exists. Let us assume firstly that WwW bo = Poot P1 (17) where WwW Ww W Gy = Soot Moos Coa = “ony ® “Oo: In these expressions ®,,, and ©, ,(w/U) are the solutions obtained in par. 3.2. The potential ¥,, has to be added to ®,, when a wake already exists for w/U = 0; the potential ¥, ,(w/U) has to be added to ®, ,(w/U) when a wake exists for w/U + 0. Figure 4 suggests that the wake is made of free filament vortices shed along a not necessarily closed line @,,. @,, iS approximately in the (x,y)- plane. For reasons of generality, we consider a closed line @, which contains the arc @,,- Onthe arc @, - @, no vortex is shed. It is possible to consider ¥ = ¥,, + ¥,,(w/U) as generated by two families fi, £/ of free and bound vortices (Fig. 5). 828 A Vortex Theory for the Maneuvering Ship ean SI\\ RRM mm tees ANTS CBDR Figure 4 Figure 5 A vortex of the f;-family is lying on a closed contour made itself of two arcs; one is P'm’P starting from P’ on the port side of @,,, and going to P, on the starboard side of @,,. m' is on the upper side S’ of S, the other arc is made of the streamline of the relative motion starting from P and going to the infinity downstream, and of the similar streamline starting from P’, but de- scribed in the opposite direction. The intensity of the filament vortex above is dy, - A vortex of the f,j-family is similar to the previous one; but the arc Pm’P’ has to be substituted for P'm’P, m" being on the lower side S” of S; moreover the streamlines starting from P and P’ are described from P’ to the infinity downstream, and from the infinity downstream to P. The intensity of the fila- ment vortex is dy. The intensity of the free vortex resulting from the addition (fj) + (fj) is oI, = shA ache nulls 1 eo 1 (18) The arcs P'm'P and Pm"P’ are orthogonal to the contribution of ¥, in the total relative velocity on S,. 829 Brard It is possible also to consider a free vortex of the f,-family as made of three vortices of intensity dyj;: vortex (i) on the arc P'm'P and the segment PP’; vortex (ii) on the arc PKP’ of @,, (K in the (z,x)-plane), and on the segment P’P; and vortex (iii) on the arc P’KP of @,,, and on the stream lines of the rela- tive motion starting from P and P’ and leading to the infinite downstream. Vortex (i) is equivalent to a distribution of normal doublets on the part S'(P) of S' behind the arc P'm’P and on the part >,(P) of the surface *,(P) generated by the segment PP’ when P and P’ describe @,,; Vortex (ii) is equivalent to a distribution of normal doublets on the surface E(C2)5 Vortex (iii) is equivalent to a distribution of doublets on the part =(P) of the wake which edges are the arc P’KP, and the streamlines starting from P and l7goreN JP Because the distributions of doublets on =,(P) are equal and opposite, the contribution in ¥, of the vortices (i), (ii), (iii) is due only to the doublets dis- tributed on S‘(P) and on 5(P). A similar reasoning may be repeated for a vortex of the f o~tamily. Finally, the contribution in ¥, of the vortices just considered is ay, = chy, + av, (19) with Dayal ached els dX, = - ge il oi d3() x dP, (P) , P) ) (19" 1 dé 4 1 a ge iio, Rte feral cay oe | cae ua 2 ee a ( 0 ar J) do aM 0 In these formulae, the unit vector n is normal to =, and therefore, approx- , : 2 H ee imately identical to i,; the unit vectors n,.,n,. are positive outwards. The total potential x, is therefore given by 0 XC¥p) 1 ge d Xo(M) = - 7A, PoPidyp fa aw CECH» (20) 01 a where P describes the arc P|K-P,, P; and P, being the extremities of @,,, and x(yp) the abscissa of P on @,,. The coordinates of u on = are é,y,. 01? On the other hand, the total potential x}(M) , generated by doublets on S, may be written Z 830 A Vortex Theory for the Maneuvering Ship Xi(M) = - ygcm) Go asm). (21) a0 4 mM Seas! Of course WwW Xo(M) = Xoo + Xo4 iw Xj, being due to [,, and x,,(w/U) to I,,(w/U), and, similarly: SCI) = oa aaa os Pee NG Oth The When u/U, Lad/U + 0, we have: r u Ww Lg Xo(M) = Noon Papen e eooM HO? ay” ’ t ' , Uu 1 Ww 1 Lq Xo(M) = X90 + Xoo me Mote y 02 ay? (22) Sentai Wie Gs iis: AY oot Lq Yo (M) Yoo + oo oA 024° with We evae tke Pot Gua Oty 2) (23) Now let us assume that x,, is known. Since ¥,, must satisfy the condition Yj 9(M;) = constant, when M, is in ;, the density ¥)9(m) on S is given by the Fredholm's equation of the 2nd kind: 1 1 ' d 1 ee x 9 Yoo(m) = An if Yoo(m ) dn! ae dS(m ) 5; =X oo (mM; ) = —Xo9(™) ’ (24) m, and m being on the same normal to S and infinitely close to one another. The solution of Eq. (24) is: Vogal) st 2X yg Cnl) = ff A(m,m,) Xgo(m,) dS(m,) , (25) S where A(m,m,) is the "solving nucleus" of the Fredholm's equation. We observe that x,,.(m) is discontinuous when M is crossing through =; the discontinuity is: Decne) eat cx CM are Gray with M’M’ = en,» Ce0) 831 Brard But, when M is in the vicinity of P, as P is on an edge of =, the discontinuity is the half of the previous one. Consequently, Eq. (25) gives Mal ) - Manin ) = ate ae (m',m” infinitely close to P) , (26) which was easy to foresee. We have yet to determine !,,(P). In order to do that, we need to know a condition which must be satisfied on @,,- Let us assume, as a first approximation, that > may be regarded as nearly parallel to the (x,y)-plane even in the vicinity of @,,. In this case, the condition = ane) oe) = = EO (27) pb BL may be expressed rather easily. It is a singular Fredholm's equation of the first kind which yields the unknown function [,,(P). Similar reasonings may be repeated for [,, and !,,, and finally, the prob- lem consisting in the determination of the wake is, in principle, solved, at least, under the condition that the @,,-line is known. The latter problem, of course, depends upon the mechanism which governs the transport into the wake of the vorticity which originates in the boundary layer. For the present moment, if a complete, explicit solution had to be given, it would be necessary to consider the @, ,-line as supplied by the experiment. In the considerations above, we don't take into account the tendency of the free vortices to wind around themselves and to form two vortices only at some distance from the body. This question would be of importance. But, on this paper, we mainly need to have an idea on the structure of the Various potentials which sum gives the motions of the fluid outside the body. We note finally that Bu Wo(m.) = ~[%oo(ms) + Yool™) yt Yorl™s) yt Fox) FG |- (28) 4. Case of an Unsteady Quasi-Rectilinear Motion Parallel to the (z,x)-Plane Let t’ = Ut/L be the reduced time (L = length of the body). We assume that the components of the absolute velocity of the origin of the axis attached to the body and the absolute angular velocity (of components p,q,r on these axis) satisfy the following conditions: 832 A Vortex Theory for the Maneuvering Ship 7 =) J) Sn = 0) for -m < t’ < +o, \f = GeO), ovis Tawi, U = constant , G) Ms 0 forst<0 N= £466) fort 9 20n t (1) (5). eMOncor st. <0. geet a(t) toret’ 20); Gan = 0) for to 0, a vortex generated in a small interval (7', 7'+dr'), with 0<7'<7r'+dr’ (t') between the arc P’KP of @,,, and the arc PP_,P’,P’. Obviously the two distributions on >,(P) are equal and opposite. Hence the vortex d,d_,J'\(P,7') lying at the time t’ on the arc P‘m’,P and on the contour PP_,P’,P‘ is equivalent to the sum of the distributions of normal doublets on Su(E 7a andsone>¢P te aac An identical reasoning applied to vortex d,d_,I{(P,7') lying, at the time t’, on the arc Pm’,P’' and on the arc P'P’,P_,P shows finally that, we have to deal with two distributions of doublets, say on S‘(P,7') and S"(P,7") and on (Rte i): dla Gly U2, t") S Gedo ¥a(@L te") & Ghali K(Pot’) (3) with ; ; il 0 mn LG 1 1 Ga Gly Watt )= aad. | f Ti Cot aa, PEvs (m') S’ CP oe") “ Ul d 1 “W “ + il) esa )) aac ae dS‘(m") (3') S"(P,7') m m’M d,d Mer eo Lad ewe a agp) ples Moe) = = ye eae en Wear) anti MCP, Ela gel )) d X,(M,t") = - + |) o4(1) day aw 2209 (4) A Vortex Theory for the Maneuvering Ship Let £',7' be the absolute abscissa and ordinate of » on >t’). The density o,(,) on the area dé‘d7’' is the effect of the free vortices shed during the interval (0,7') along the arcs PP, and P'P;, with yi=7' (or yp, = 7’ when 7’ 0. In the case ''a,," one has: AGE = Gee) on and so on. Equation (13) yields YP, X'(n") Cot) 52 ‘ 5 [ van J = G(npyag’ = { *an’ | = ee Pee fo Ty y(n") 3"! 3237)" 1 1 Where hs(7.7 )=0 when 7 (Gas) = 1, CS) wit). 838 A Vortex Theory for the Maneuvering Ship Substituting in Eq. (16), we get: Yp | [= fr(s) v(7')| = hal S) {= Bap Up( ip) ven )} dn! Yp! Yp 1 / ‘ / = =| 1 [2 ay u,( 7p) A) )| Gli) Ypv 1 P Putting JB Gara as | v,(7')dn" , Yp! we obtain E uy(m) [EZ An(s) Pap] F508) = =a coun Tp) » p n or HG) aos ART ahs WO S ye An(s) bea Therefore Way ea eave OE GE) Pp is known. Of course, for t' = +, the motion is steady, and consequently Pe Cah) =D ene p This equation must hold for any value of 7’ in the range Tp! between the two arcs deduced from @,, by the translations -i,l(t'-7') and -i,L(t'-7'-dr') and the two streamlines of the relative motion coming from P and P’. Function ia t') is the solution of a singular Volterra's equation of the first kind t! | i (ro) ICE ae eer? = i 0 Putting 7’ =\t’, t’-7' = (1-A)t’, this equation becomes: 1 en | pve] iC =e1eh = 1, 10) what implies as DMEM JCih= Nyell = Ores) (21) for t’ very small. 840 A Vortex Theory for the Maneuvering Ship So Eq. (20) is quite analogous to the equation which yields the circulation around an airfoil of infinite span in case "'a,." Equation (20) is not convenient for numerical calculations. But, if the nucleus H(t’) was really known, as in the case of the airfoil of infinite aspect ratio, it would be possible to solve it after some transformations. Equation (20) is equivalent to ‘ / t pel t (2) J at" - 6)ae | F*(7') H(@- 7')dr’ - | ace’ - ado = [ A(r’)dr' , 0 0 0 0 or to t t | F*(7') i A(t‘ - @) wo-ya8 dr’ =| A(r')dr’ . ) t 0) T The nucleus in the brackets is 1 (etary | AlGl= y(t HACE id= KCE (22) 0 If we choose A(t’) in such a way that K(0) = 1, what implies only Act") = O{F*(t')} (23) for t’ small, we obtain, deriving with respect to t’: t ‘ F*(t') +| F*(7r’) K(t'-7')dr’ = A(t’), (24) 0 what is the wanted form of (20). Now consider the density y{(m,t‘) of the distribution of doublets on the hull. Since ) ok t pea fo) Ae ENE ) i =e 1(m) ’ when t’ is small, and m close to @,,, one has Gis) = OCD \e (25) This result is compatible with the conditions 841 Brard Ce) CaN) = CYS 0, Ger 0+) (m', m" infinitely close to P), which lead to Yea OH) = Va Ut) = 0. We set y(t, €") = 7 (i, OF) + Sy3(m,t') (26) with Syi(m,0+) = 0, Sy3(m,0+) = 0(1). (27) ea The variation of 5y7(m,t') between (0,t’) is partly due to the fact that yi(m,t') depends upon the distribution of the arcs P'm{,P, P’m;,P on the hull, distribution which is variable with t'. For t'=0+, these arcs are concentrated in the vicinity of @,,. Now, consider the case '"'b,,"" when (w/U),, is, for t' >0, an arbitrarily given function. For (20), we have to substitute: t U J a(t Se") BCR or de® = cd ie (28) 0 the general solution of which is: t t ney = re LEU Pee ae 0 when (w/U),, is continuous for t' >0. One has Pape’) = Cala) BEY « (30) Because of (8) the density y,(m,t‘) of the distribution of doublets on the hull is: : t! Vid, = \ IB Ce Bl aioe =a: year ¢ (31) 0 That gives, in the case ''a,,"' the expression already written above: 842 A Vortex Theory for the Maneuvering Ship t Uy yi(m, t') (w),. = (),, F¥(r') Hy ,(m, t!- 7! )dr! [vi(m,0) + dy{(m,t')] , and, in the case "'b,": yi(m,t") = w), [vi(m, 0+) + Syi(m,t')] ean iia sles ee 2 (a), yi(m,0+) + (z),. Syz(m, t') t : d WwW * , ‘ Ul bso =) dr’, + J aa (u),, Saye Gulye $5 re | YC er when (w/U),, is continuous for t’ >0. (33) If (w/U),, has discontinuities of the first kind for t’ >0, one has the general formulae: Me UN ener canon F(t!) a (3), Te ee and t / t tC) t fi ia y,(m,t ) = (),. y3(m, 0+) + J (o),., var syi(m, (c= an Nolr og 0 In the case of the quasi-steady motion, we would have a circulation WwW 1 * = ’ We (o),, Loner G2) 5 Vana iris and a density of doublets i), 2010 = (B),, Bitmors + Stems). That leads in the case ''a,," to the deficiencies 843 (29") (33') Brard t U i (a) rm ) ) F*(7') [H(t ’= 7’) = 1) dr’, 0+ a and fn), ovine = (FY [eracmtey erica] When (w/U),: is arbitrarily given, the differences are: (34) C) / ‘ ‘ Ay,(m, t") al . Sy5(m, +) cr | (a), a7 Sy3(m, Esa jer’ « i 0 In the cases "b,,"’ when (w/U),: = 0, (La/U),, =0, (u/U),:=£,,(t ) for t’ >0, and in the case "b,,'' when (u/U),; = 0, (w/U),,=0 and (Lq/U),. =f,,(t’), we have similar formulae. In the general case, £,,(t ), £,,(# ). f,,0, we get: INGnety n= » ran") \ fone’) = in(t <7 \clr” (35) ‘ t i ) C) t fh / {osc ) ¥;, Cm, 0+) +f tga ) Sai Oval 2 = 7 \Gr” 0 W(iit; - )) Mes 0. Hydrodynamic Forces Due to the Velocity Potential (case of par. 4) 5.1. Definition of the Hydrodynamic Forces When w/U and Lq/U are small, there are no strong eddies due to separation. Therefore the set of forces acting on the body is purely the sum of the follow- ing sets: (i) (Fs) due to gravity (weight of the body, hydrostatic pressures), (ii) (¥,) due to the inertia of the body, 844 A Vortex Theory for the Maneuvering Ship (iii) ($,) due to viscosity (friction or, more exactly, viscous drag), (iv) (F,) due to the velocity potential of the absolute motion, (v) (¥,) due to the propeller, and (vi) forces due to the system —if any — which reduces the freedom of the body or generates its forced motion. The set of hydrodynamic forces is $,. We assume here that the body is not fitted with planes and fins (see par. 6). For what follows, it is helpful to separate the set of forces (F,) into two additive parts, $, and $', (¥,) existing alone when there is no wake, while (f') is the contribution of the wake. It is possible to obtain rigorously this result by starting from the contribu- tion in the absolute momentum of the fluid of each part of ¢ (see par. 5.3). However, we will firstly proceed using an approximate expression of the hydro- dynamic pressure p(m,,t') on S,. The velocity potential ¢(M,t') is, at time t’+dt’, when M is at rest with respect to the fixed axis: Oi dt) =O (Gay jai, t Hdt) — o(x—u, dt’, y-vpdt', z-wedt’, t'+dt’), (1) where ug, vg, we are the components of the absolute velocity V,(M,t') of the point attached to the body which, at t’', coincides with M. Consequently: og" : og ; ; : lis) ee (bt ) - Va(M,t') grad f(M,t') . (2) The hydrodynamic pressure is given by: U 1 3 1 3 (DGbe )=D 5) = os (HE y= eC Ys (3) where V(M,t’) is the absolute velocity of the fluid at M. V_ being its relative velocity at the same point, one has 1 7 i) ! 1 / = GLE sae) = Sf (mM, t » + CVEY wet 7 (Mt) (4) 1 : _ o¢ : ee : ee : ; PON ene ae (TE ee ns (Me i Vin (at ae (4') Since |V/V,| is generally small, we could neglect the last term in the right member of (4) and write 1 risk petty Atl | op pee t ) = ip, + [-v, grad a). - e’ 845 Brard This expression being linear with respect to ¢, we would obtain 1 1 1 ; ey; 1 @ P(m,e.t ) = 7p a + p Pi(m,..t at Te (m,,t ) (5) with 1 . 3 - jp Dalle ) = aid Ve grad | | o(me) + » Dry CMe) digneat | ; k=0 (6) 2 1 C) pP (m..t’) = E - VE grad | [oo(m,) + » Hm, t")| : k=0 p,(m,,t’) generates the set of forces ($,) when the fluid is quite perfect, say when there is no wake; p‘(m,,t’) gives the contribution of the wake in the hydrodynamic forces. In order to use the density of the distribution of normal doublets on S and the equation When) = =MeCmhe") 4 it is, however, easier to consider the streamlines C of the relative motion on S,. These streamlines are the orthogonal trajectories of the curves y = con- stant, where y is the total density of the normal doublets on S. Because all the components of y are small with respect to »,,, that is, with respect to the density of normal doublets which generate ,,, we may consider that the unit vector i; tangent at m, to the streamline @ passing through m, at t’ is practi- cally independent of t'. We choose i; positive downstream and also the ele- ment of are do’ on @. Consequently, the relative velocity V(m,,t’) is approxi- mately given by Wallen E) = VV Cisn) thal.) A) 2 Pina de [tootme) v a Po (Me) Gem rs [etal ae: 3 : Star [YooCme) + De w(m,,t | Hence, ~ FV, (mt) is the sum of the three following terms: 846 A Vortex Theory for the Maneuvering Ship 2 2 1 2 1 fc) i ef f 2Vr, == {2 [o(me) + yb o),(m,) Gs ] + [Yeis] 3 ; k=0 ii | N|e (>) Ale =a a= o ° ZS 3 0 — + Me» +e = oN 3 Q) ct a zs ey i} (t) k =0 (7) a = 3 ; - ie De ®, ,(m,) fox) | x = [Yootme) + om w(met')| ; k=0 k=0 2 3 3 : = 3Ve =) 2 ®)9(m,) + veil | x 557 [Mootme) + De me | “ k=0 Since (b) is negligible, we obtain 1 C) : ‘eae i me Pi(m,, t’) = 2/h Dp .(M,) Foxtt”) + 5 VE (Ciel e= 2 V,,(m.t’), k=0 1 ! ! ’ 3 3 Ui 1 Q ae Gnisnit) = i era Wiha) a7 Ve(m.,t ) 1,(m,) 500 (8) ; : - {evi a ss weet} k=0 In par. 5.4, we give the expression of the set of forces (F,) due to p;- Now we consider the set of forces ¥' due to p‘(m,,t’). 0.2. The Set of Absolute Forces Due to the Wake Let us assume, for instance, that we are in the case ''b," of par. 4, when (u/U),+, (La/U),, are identically null, while (w/U),, is, for t’ >0, an arbitrarily given function of t’. According to a result of par. 4, the density of the normal doublets on S due LOM Gn t )) 1S: t , , oe W * W * , ili, Ww * LI ie Ul t vim) = (GF) mor) + (FG) Same 4 = (a), Syi(m, t'-7')dr’ (9) when (w/U),. is continuous for t‘ >0. The pressure pj(m,,t’) due to ¥,(m,,t')=-y,(m,t’) is given by 847 Brard 1 7] U U C / e7 ) 3 UT mm Picter t )= acs SD We (m,t’) + [Vetmat ) 1,(m,) + 5! Yoo(me) | ae y,(m,t ) or by so ee UH pc p p,(m,,t ) zm L bie.oy dt! ce o7 ) ic) W * W * ' + vei: ale Ae 0] ae ad y,(m, 0+ ) ais aa oy ,(m, t ) O+ Stn! a : (10) Let us consider firstly the particular case 'a,,'' when Om f° SO, 1Le0 Ba Cm &7) Oo, be the expression of pi(m,,t') in this case. Equation (10) gives: 1 1* 1 U * ] er C) C t 5 Os (Hot ) == Lot OV (imy (&)) $F vei: + So 59| Xe [yi(m, 0+) + Sy4(m, t )] Therefore, one has 1 = 1 1* eu 3 3 * * 3 Dons) = a Pa (id HE) = || gt ae 55! ®5 9 Agi [aa Or)) + Sy3(m, +) ] = ((7.3)} Now consider the difference in the case "b,,'' when (w/U),, iS an arbi- trarily given function, between the pressure p,,(m.)(w/U),, in the quasi-steady motion and the pressure pj(m,,t‘) in the real motion. Assuming that (w/U),, is continuous for t’ >0, we have: 5 Dani a - Z Pi(m,, t') = 4 Api(m,,t") , (12) \ t! where Z Apy(me,t) = 5 py) (mt) + ZA'Pi(met"), (12") 848 A Vortex Theory for the Maneuvering Ship with il : U d Ww iv (Gap) 1 eed * oe uh p P, (m,,t ) = iC. y,(m, 0+) dt’ (a), , (13) and 1 HH 1 a e7 eC) Q * W WwW 1 3 See ) = eae + 557 50| aa {s5cm, +0 (u),, = (7) 5y7(m, t ) = || a a) Sy7(m, eo ryar'h (0) Tn W ) * A d Ww * ! ! ! z{G).. Sere enon) | dr’ lol SAO ern ee \ oy 0 Integrating by parts the integrals in the right member of (14), we get: 1 inf ! ov 3 3 * Uo * D Asp, (m_.t ) = ne {| vei: bad o,0| Sau oy,(m,+®) + 7 oe sicm.0+)} w { vss: ml )2-ts}am weet = == El, + 7 ®) 7 7 7 y,(n, t Tata i This equation holds when (w/U),, has jumps for t'>0. Let us introduce now the difference IM Soy 6) = Sym, +©) - Sy3(m, t’) : (15) We set 1 a: o7 cc) tc) se U 0 ! * t R(m, t ) = (tee ats Aaw 50) Ez L ey A dy,(m,t ) . (16) We obtain ot C) ) * U oa * R(m,0) = (V.is + ao2 0) ano oy (im, +2) + ¥ ava dy ,(m,0) ’ R(m,+@) = 0. So Eqs. (14') and (14) give respectively: t ‘ 3 mie = A'py(m,,t’) = (7) R(m,0) + i] (Fae seems a Shdreak ene) te 0 7! 3 221-249 O - 66 - 55 849 5 AP (me, t") : (5) R(m,t") + ‘\ (3) Ram, @!= 7 ae”. (17) O+ 0 . U 7! We note that the first term in R(m,0) comes from the fact that the circula- tion around the body is null at t’=0+. It leads to a deficiency of the pressure. But the second term acts in the opposite direction. In the quasi-steady motion, the resultant force and the resultant moment referred about the origin of the moving axis are respectively for (a). = ~ Gi), J} eS Sages (18) I,, (a). ee (a), ) Po ,(m,) Om, An(m,) dS(m,) . (19) In the real motion, the resultant force and the resultant moment are, re- spectively, Sees Sis (5) OG ACen, (20) Gen) Te ee = a (21) with, for instance: nq d in ta . | as *\ R(m,0 W(t!) =~ J (m) n(m) \().., (m,0) on + J (o)., a R(m, ti 7'yar’| : ) The force Sat ‘) and the moment es : Ut ‘) act in order to increase the ap- parent inertia of the body and will be included in the final formulae in the set of forces ¥, (see par. 5.4). They do not exist in the theory of the thin airfoil of infinite aspect ratio. On the contrary, the force A’¥{(t') and the moment A'I,(t') are quite analogous with those found in this theory. Similar formulae can be obtained when (u/U),: + 0, (w/U),: = 0, (La/U),. =0 for t'>0 (case "b,"') or when u/U(t')=0, (w/U),: = 0, (La/U),: + 0 (case 'b,"'). The total set of forces due to the wake in the most general case is the sum of those found in the three cases "b,," ""b,"’ and "'b,." 850 A Vortex Theory for the Maneuvering Ship 5.3. Another Expression of the Set of Forces Due to the Velocity Potential The absolute momentum due to the distribution of normal doublets asi 222 = an d2(p) dn, uM on a small surface d2(u) is pd2(u) n(-) , it acts through point u. Consequently, the total set of hydrodynamic forces exerted on the body may be obtained by starting from the variations during the interval (t’', t'+dt’) of the absolute momentum due to the distributions of normal doublets on S and on the wake surface. These momentums are additive. Consequently, the set of forces (¥,) comes from the velocity potential: 2 ® = ®,(m,) + ye eGo) ence ain k=0 while the set of forces (¥') comes from the velocity potential: 2 Woe Ua Gi eo Van@are k=0 Here, we deal only with the set of forces (¥'). The set (F,) will be ob- tained in par. 5.4 by another way using the absolute kinetic energy due to ©. Let us consider the case ''b,."' At time t' the absolute momentum of the fluid is OFGE)) —wO Cea O7Gh)- Q,(t') is due to the doublets distributed on the wake and Q/(t') to those dis- tributed on s. The general resultant of the hydrodynamic forces due to ¥, is d dt’ sae SS |e (Q,+Q))- One has 851 Brard X'(m',t’) Q(t’) = om, J. Poitn’ dan’ | F (y)dé" On Cap geo) clbe t/ = pi,L id M5 ay yal” J F,(7')dr' 01 O+ with me = Ne) a EL ie omar when (w/U),, is continuous for t'>0. Similarly, De") Sp f y,(m, t') m(m) dS(m) S) with t! d * * t Ww * Y] U U y(m,t') = (i), yi(m,0+) + (i), syj@me) + | ose (G) ) oyiaiee 0 T Hence Fad oy = -pt a (a), {J y,(m,0+) n(m) dS(m) s t Uy fp nb d Bn ‘ ; ea ee 0 oP 01 W U o * 1 = (i), Tr at! J) dy, (m, t ) n(m) dS(m) t! Uae x“) il Home ay p L ‘ dr’ (i pe dg ¢ ot’ oy ,(m, t y ) n(m) dS(m) 0 In the case ''a,,"" when (w/U),; = (w/U),, + 0 for t‘'>0, one has, at t’ >0, a general resultant 852 A Vortex Theory for the Maneuvering Ship Fe lg) a) fei, we Py (1! )dn! xF*(t") Cc és : -p ¥ J a dy,(m,t') n(m) ssim| ; Consequently, in the quasi-steady motion, we have a general resultant: Fei(t') = (i), [-ri.u J Fo(n'3a'| (23) 01 The difference between this resultant and the resultant in the real motion is AS P(Cer een Sie (i), =F '(t'). One has AFI) = FO 4") + AY where n(2) t zs U nal w * Sige (Ce De = 620 eee (a), il) ¥,(m,0+) n(m) dS(m) . Moreover: 1g t — * : ' ’ WwW i W * ' NSiCe y= ei,u { My 1(n')dy (a), (G),. P(t) 01 (eu d = an (5) : (et 7 47" | 0 TT + oy (a) | a dy3(m, t') n(m') dS(m) S) d Ww 4 ) @ : ; ah aan I) cra] meen maar ascm)| : Rete Z(t), Z, ,(w/U),. be the components on the z-axis of the forces above. We put: 23 (t),, = ~ 2 eavtas (F) (24) Brard where A is the projected area of the body on the (x,y) -plane. The first bracket in the expression of A’¥)(t') is t v (yates fof) rete yl ae’ O+ p. 0+ t/ Ww * Ww Cee a * 1 1 1 - FL, =u ‘| Glesn 2oe Gio le The second bracket is el (iT) Ladle St Sy, (m, 0) n(m) dS(m) el (5 Ee a (a 2 8y3(m, t'-7')dr' The new expressions hold even when (w/U),, iS not continuous for t’ > 0. Hence, putting y= aramid Is a a in a sO ROU EY oer (25) we have ZAG = Toe a (a). = age) = mE EP + IMC” )) 5 (26) with i Nee) = ral The ll Vo *(m, 0+) n(m) 1, dS(m) , (27) and t! A'Z(t") = - 5 pAUPa, 2), £(0) + | oe a Syke } (28) 0 OF In the last formula One be ate, {J a Sy%(m,0) m(m) i, dS(m) <1, ; (25') p( +0) = In the quasi-steady motion, the component on the x-axis of the resultant force is null, but not in the real motion. One has 854 A Vortex Theory for the Maneuvering Ship 1(i) BXCE Diy a Xa ey Yay (29) with rca) = 433 (o) a y,(m, 0+) n(m) i, dS(m) , (30) and pyeeE)) = -F ah i} = Syt(m,t') m(m) i, dS(m) s ce i poets mae >| dr’ (a), de I} ot’ oy,(m, t -7 ) mm) 1,(m) asm) 0 This expression may be written: Se oe se t! 0 opal with ' C * i 5 w(t’) = ie J) mal dy (m,t’) n(m) 1, dS(m) . (32) The first formula (25') shows that, at t'=0+, the deficiency is, in case "a,,'' less than 1. (i (i) We have, moreover, components Z, VEE, Xo sot manda AuZaGt a, ANOCICE Ore Although y{(m,0+) = 0(1), the component Zs (04) is small, because firstly, yi(m,0+) has significant values only when m is close Lou. and ‘secondly, in this case, n(m) is nearly normal to i,. The components x a GE. ) and -A'xX;(t") are small too, for the first of the previous reasons, and also because the projected area of the body on the (y,z)-plane is small with respect to A. Now, let us consider the moment of the momentum. It is WeGt oon Wate ist WaGt.) with 855 Brard Oe et | ele ae @ 0 01 Wie y= 2 il) lie yi(m, 0+) + (i), Syi(m, t") Ss t! d Ww ¥ ieee F ; olny oe on yar’ O'mAn(m) dS(m) . The resulting moment is v I ° T 7 U d Uy t W 1 \ CG re AGS ees ee aren (wae) ae We) 2 Because é'(y) is independent of t’, we have t! U * U d * U U U M(t") = pu J, Py") x x(n") x dy" a. ye | sel ar" on O+ U d WwW * ° ° oP (#) If vicmony [eom moma, = xm) n¢mi,] 48(m) te S eas (G), J) _ Syi(m,t") [z(m) n(m)i, - x(m) n(m)i,] dS(m) Tl ar {ff Oye ((imy te = ar” ) (AC) n(m)i,- x(m) n(m)i, | dS(m) + th S +pU J) (i), yi(m,O+) + (ct), Sy*(m,t') ! D cle o— Qu. = |a. —_——— fen] a a (r)., 76s tear] i,n(m) dS(m) . It is assumed here that the moving axis coincide at time t’ with the fixed axis, and the moment is referred about the latter. The last term comes from the derivative A a‘mAn(m) . dt In the quasi-steady motion, the moment is 856 A Vortex Theory for the Maneuvering Ship Mo (u)., ; {ou J My 07") x x9") x dn’ + pU il} [vi(m, 0+) + 8y3(m,+)] i, n(m) ssn} (z),, . (83) S The difference between the two moments is amie) = Mos (B),, BOC + AMC (94) me Ce = “og aa lel ul, lees (m, 0+) [z(m) n(m) 1, —~x(m) n(m) i z |] dS(m) . (35) In order to get A'm{(t'), we will reason as above. The contributions of the first terms in m,(t') and in M,,(w/U),. lead to t ‘ / , U WwW * 1 d W ul * ia ! 1 eu ryan’) xen) xan'f (Hf) CBC || reaca (sy oa Bia (Gt: ryder’ Ol 0+ The second term in Ji, ,(w/U),,, and the last term in M{(t') give: t f d * * 1 t 1 ° + | aa (a), Bric - ayfem, try] er fi mcm dS(m) . O+ Lastly, the terms in the 3rd and 4th lines in the expression of WI’ ict give the contribution: t U WwW C) * 4 d Ww ee 4 Het , ae I (ee 7 dy,(m, t") “Al ae (a), E dy3(m, t'-7 |e} x {z(m) n(m) i, — x(m) n(m) i, } dS(m) . Integrating by parts, we obtain 857 Brard AVI GES) = ou | ry 1¢n') x(n')d7' {(8)., hee co @ o1 0 96 + pU { n(m)i, dS(m) Nel [ Sy4(m, +) - 8y4(m,0) | S f(a), & lotus + ten alae Oo + PL J) bm n(m)i, - x(m) n(m)i ,|S(m) ie so 4m, 0) WwW oF * 1 ] ! +] (a) | oy,(m, t re 0 Let ms (8), = Foatutes (B),, (36) be the moment in the quasi-steady motion. We have 2h iJ Pyscn’) x¢n' dd’ + ff Ly{cmory + By4(m.4ey] 4, mcm) | a O01 : Then, putting 1 I 2 / i ‘ * if p(t) - maim {J Caan) Cp yey aCe) yy + il} [Syi(m, +) - Syi(m,t')] m(m) i, dS(m) iS) x i) ae Byi(m,t") [2(m) m(m)i, ~ x(m) n(m)i, | say pe S we have 858 A Vortex Theory for the Maneuvering Ship ~ (0) = 2 7 J Wore exC mayank + ] Sy4(m, +) n(m)i, dS(m) ALUa , @ L 01 (38') il] si Sy4(m,0) [z(m) n(m)i, - x(m) neni J0500} S) I =) DP (HS) and t Uy ATECE a= + pALU? aj (@),. ¢'(0) +f (a) 4 | od | | 0 Te This last formulae holds even when (w/U),, is discontinuous for t’>0. The expression of the deficiency A’m{(t') and the expression of ee it 3) could be subject to comments similar to those made above about A’Zit’), Moe ywand. 7, e), Xi 9 Ceo). In the following paragraphs, the effects of TA yi, x oe Ne ) and mi‘#(t') will be included in the contributions of the accelerations in the set of forces due to the pressure p,. In the first draft of this paper, we gave an affirmative answer to the follow- ing question: is the deficiency A’$, fixed with respect to the body? But in fact the proof given was not valid. In the case of an airfoil of infinite aspect ratio, it is possible to show start- ing either from the momentum of the fluid Q or from the pressure p'‘(m,,t’), that the deficiency of the lift and its moment are actually proportional to one another, the ratio being independent of the time.* We think that it is true also in the present case. But the proof would require a finer analysis that the one given above, although the latter is sufficient in order to yield the structure of the main formulae. In return, however, we have, in cases "b," and "b,, Vacant peas SHie (a, ME Ct a limiine ayant Se =. CHIPS) Es Ces) imine oO) © being the linear and homogeneous functional defined by ny eo) = Grlinae ae ye) (Geue ima: For this reason, it seems that the three functions ¢,v,¢' introduced here are suitable also in the cases ''b," and "b,'' as in the case ''pb,.'' We will admit this fact, at least for the sake of simplifying the writing. For instance, we will have: *See, respectively, (2) and (8). 859 t Z(t") = ~ 5 pAU? a, {(3),. gO) + | (a) so ace! oar} ; 0 oF (41) zy") = - 2 eavta, {(48) eco) +f (48) 2 acer arf 0 and So on. In par. 8, the notations used here will be slightly modified. We will use a,a’ instead of a,,a; and b,b’ instead of a,,a). 5.4. Set of Hydrodynamic Forces Due to p, and p‘“?) When the body is symmetrical with respect to the (z,x)-plane, the kinetic energy of the fluid in the absolute motion due to the potential u Ww Lq v\ Lr Lp ®oo+ %0 (i), *%: (i), #82 ce lo),. * % ae P80 E) is Dye = oll CU ee) a aS at em oF Dw (Ue - OL [¥35wa + v,5(Utu)q + Yy,vP + Vogvr] +1L2 LA? + We cle + Nair = = 2A,3Pr]} ; where W is the volume of the body. The components of the set of forces X,, Y,, Z;, £;, M,;, Nl, are given by the Lagrangian expressions: oa a (eee a) is dt o(U+u) OW ml ovy : ee ee i 7 dt Ba NY Oy Oe or Sayaka Writing before the accelerations »,,...for y,, ..., in order to take into account the forces due to the pressures p'‘:), and assuming that u/U, w/U, lq U_ are of negligible squares and products, we get, in the case of a motion parallel to the (z,x)-plane: (with 2 i 1(i) teeny one and so on): 860 A Vortex Theory for the Maneuvering Ship preety 2 2W Lq , Lw pein pom icta Z,+Z; = @ PAU AL {es eur M32 PIS V36 Tae? se pete al reel Ea y yh EY Se el gn bea (42) i +X, = % PAU” Ap FietO ieeema ee Geo Miva steharer) eal U2 , fe 2 2W u Ww ty a eq M,+Mz = yee AL {urs + 2uy3 ue (43 - My) Ur p88 y2 Tae wee Tee As is well known, the force is null in a steady motion, when Lq/U = 0, but not the moment if the body is not symmetrical with respect to the (x, y)-plane (u,,; +0). When q}0, the force and the moment are not null, even when the motion is steady. 6. Sets of Forces on the Diving Planes and Fins, Effect of the Wake The diving planes and fins contribute to the forces ¥, (par. 5.4) and Fa (par. 7). We assume here that the expressions of these forces take into ac- count the effect of the diving planes and fins. But we have yet to introduce the effects of the lift, moment and drag due to the diving planes and fins. We neglect here the history of their motion because the length of their chord is small with respect to the length of the body itself. Let Lé,, 0, L¢, the coordinates of the axis 0, of a diving plane B, Ly, the ordinate of the center of the lifting surface, and o, its area. In 7, and oy, the fin associated with the plane is assumed to be included. When the motion is steady and parallel to the x-axis (w= 0, q = 0), the wake may induce on the plane a velocity which component on the z-axis is wy, = ajpU. The components of the relative incident velocity are =U), (0). ag RU - Let £ be the angle of the plane. 6-0 when the lift due to the previous inci- dent velocity is null; 6>0, when the lift generated by the plane has a negative component on the z-axis. In the most general quasi-steady motion, the relative incident velocity is ee alla Brard The four last terms in the z-component are due to the velocities induced by the wake on the plane; a,, and a,, are positive dimensionless coefficients. Conse- quently, in such a quasi-steady motion; the effective angle of attack is a= oe Gl a 2 eli ala cece Let us assume that we are in the case "a,"’: Gece 4 el Oy AAO mae ef t' W pe 1 Bs Ww 1 (a), =@0 tor t’ <@, = che Q vor 12 2s At t‘=0+, the velocity induced by the wake is null; at t'=+~, itis a,p,(w/U),,.- ANG ie 2 Oy ie set a8 (qj) where ¢,, is an increasing function equal to zero at t’= 0+, and to 1 at t'=+0. In the general case, when (w/U),, is a given function f,,(t’), the velocity in- duced by the wake generated by f,,(t’) is ele") « : 0 ta ead ' d ' ! ! i Pe) f : ; (i), emt) + J a? (i), Prn(t 747 =} foil7") 37 Pin(t = 7dr 0 O+ oe t U 2 | CE) Gnade =F" yeir’ - 0 Finally, when Gy = Se > al = ee a, Seca the effective angle of attack becomes t! 2 [f= BA: (5), = (2) =| aR fan(n sone enicems ‘ (1) 0 k=0 The square of the incident velocity is ‘Lq vfiea(Z 02) Sap 2) : a t f ( U t t B ( 862 A Vortex Theory for the Maneuvering Ship be the characteristic coefficients of the plane fitted etic; cy and ~< to the body. F The absolute set of forces due to the plane, referred to the axis attached to the body, is Xp = - 5 POR? {1 Phe (i), fap ) a} cy (1+---) wes Oe fe = pag? {1 + 2 (ale + 2 = ta | cy Be ta > 5 A opLv? {1 toe (u),., + 2 (3), ‘sf [ Tp ex, belo It = 5 POBLU? {1 + a oF ae) Ss |} oe Bae, 7) SISA Sa Be Mg = 5 ep opLU ola ee | [mcs ] = But a set of diving plane is generally made of two parts, symmetrical with Consequently, £, and Jig are null. The non-null respect to the (z,x)-plane components of the set of forces are *s 1 u > P OR UC, eal Bee 7 fe 1 2 u Lq Bee De ee (|| bse 2 (aoe 2 T),. L : | WwW q i ; Bae : ie (3) 2] 2 20x FoR(t ) Paes L (3) Bes porate” [tata + ntig Sm) lft + 2 (), © 2 Cet), Is 3 + [opcy Saar i. (33) és | 2 ve (Sar, ~ eag]| 2 Ao, fox (t |} AM, k=0 863 Brard with t U 2 1 5 AZp = 3 papU% ey, | 2 ao, {fox(t”) - J Fan )) dyg(t'-1'yar’} ' 2 t! x [2 Bon (Foxct”> = J eon T ) dgce'-7'yar'I | é k=0 0 Obviously, the delayed circulation around the body gives -AZ{(0+) <0, -AZ3R(0+) <0, -AM{p(0+) <0, -AM,p(0+) <0, and leads to an increase of the effi- ciency of the plane. Because c,, is great with respect to a, (see par. 5), the effect of a diving plane located near the stern of the body may be very important and shall not be neglected. 7. Other Sets of Forces Exerted on the Body (case of Par. 4) The constituents of the total set of forces were encountered at the beginning of par. 5. In par. 5 and 6, we studied the forces due to the velocity potential of the absolute motion. Let us now consider the other constituents. 7.1. Forces Due to Gravity Let Lé,,0,L¢2, be the coordinates of the center of gravity of the body, Lé.,0, L¢, those of the center of the volume, » the density of the body with re- spect to the fluid. We assume that the square of the angle of trim @ is negligible. The components on the axis attached to the body of the forces and moment due to gravity are he = = 8 WL yes We = 0, fon = 28 WL 1) 5 | (1) @ 20, MW, = oth pga Ge t Gee, We = O- 7.2. Forces of Inertia for the Body Let pWL?x,, PWL2ux,, pWL? ux, be the moments of inertia of the body with respect to axis acting through G and parallel to the axis 0(x,y,z). Let pWL?yx,, be the rectangular moment of inertia due to the product zx. 864 A Vortex Theory for the Maneuvering Ship Assuming that u/U, v/U, w/L, Lp/U, Lq/U, Lr/U have negligible squares and products and the same for é,, (,, the general expression of the set of forces of inertia is: 1 > 2W ibe ie Leh = = — — - —_- +t — ae seat. | Es U # ec | (2) 1 > 2W jij Lv 2p een ae = 5 PALU® AL L- | i SG a U2 Cg a 1 U2 13 U2 ’ 1 2W Lu Lw L2qg ji, = Sahu == - — + — €,.-X, —|, eS Ae AL Le y2 CG U2 fel 2 y2 1 » 2W Lp Lv gis L?p ee Us pe |- # GSO eh les yo) In the present case, the motion is parallel to the (z,x)-plane. Then, taking into account the formulae (42), par. 5.4, we have: 1 2W Lq _. Lw PER ’ Lag Z.+Z; = 5 e AU ar (an Alan el lS ee a AO) ama 1 Ww Lq hw Peghiies L2q Bs 5 COU a l-113 Pata era) ue I) een Catan! Ca) meg (3) 1 2W u w ; Lw M+; = 3 PALU® No Ee ity 2443 wi (43 4D) wo (V3, +H&g) U2 : Li L2q + (¥15 -HSq@) ao (A, +X) wee 7.3. Viscous Drag and Propeller We assume, for simplification, that the viscous drag may be expressed by a c, anda c_-coefficient. On the other hand, we assume that the thrust of the propeller is T, the suction coefficient t, we neglect the torque due to the propeller. 221-249 O - 66 - 56 865 Brard Finally, when the motion is parallel to the (z,x)-plane, we have a Set of forces Oo. on Oz: - = p AU? (c+ 2 Be, )s rat] @ 5. r= 7 ; B = p AU? 2 (4a) on Ox: oS NUE (c 5 2B + (l=) || o ; D ze b\ Sy The moment about the y-axis is 1 2 = oR \. 4 - 9 PALU Cra TH ox, B te MG |) (4b) In these formulae, symbol * indicates that we may have to deal with several sets of planes. 8. Final Formulae (case of par. 4: quasi-rectilinear motion parallel to the (z,x)-plane) © Nature of Deal) = the Forces 2/(3 eeu ) 3 Quasi-Steady Motion + Classical Forces of Inertia i 2W gL 4 2W gL = = nad ae = (n= 154) ra =a [G46 = ce) = (Le = %e) 4] Propeller, fe fer op friction (div- Egitie x = e [-c.+ Pe t7- | [cat ® FE engin 7] ing planes - included) F.+F, + FC) W Lq q OW Lq , Lw u =Hayns: ae pass + = + uae (diving planes | AL U/,! ' AL | “33 y fis U2 AL #134 24)3 (5) (H3- y) U included) Lu fy L2q =(4+ 1) gat (is HS) La} Circulation around the body (diving planes included) Diving planes (effect of the wake generated by the body not included) Diving planes (effect of the wake due to the body) 0 ( Continued) 866 A Vortex Theory for the Maneuvering Ship Nature of 1 , .. /0 , ?) : n/(5 “ALU?) Effect of the Delayed Circulation -aa{(3),, vor fi), deere] | -asflG) ere fF (B),, eer =7 947] Circulation around the body a4((¢) 0) + re ) ike'-ryar| t T 4 tind ), mor f (42), dee'-rvar'| -»'[(8),, #0» f (8) dar ar'] Diving planes 9. More General Quasi-Rectilinear Motions One of the reasons why we are interested in the motions parallel to the (z,x)-plane of symmetry of the hull, is that they are also parallel to the vertical plane. As a consequence, if we know which forces are exerted on the body in forced motions parallel to the (z,x)-plane, we are able to determine the free, natural motions in the vertical plane. Motions parallel to the horizontal plane are also of a great importance. But their approach is much more complicated. Let us consider the equations: Ge (UE SA Cusp iewe MDM =udn aelays Omi is Wen eel Coe When the motion is parallel to the horizontal plane, £=0. But the components normal to the (z,x)-plane of the hydrodynamic force and of the force of inertia of the body do not act through the same point. Consequently, they generate a £-component for the resulting moment; p and ¢ cannot keep null values. Even if ¢=0, the final motion is not parallel to the (x,y)-plane, since w and q are different from zero. 867 Brard A motion parallel to the horizontal plane is generally impossible when the diving planes are at a zero angle. For instance, the steady motions are, in this case, helicoidal motions around a vertical axis. Let us assume that u,w,q are null. The arc @’ along which the free vortices are shed depends upon the form of the hull. 1°) We consider firstly the case when there are, in the (z,x)-plane, no singularities, appendages and so on, which constrain this arc to be located in this plane. a) p-0. Because of the symmetry of the hull with respect to the (z,x)-plane, the arc @’ is in the (z,x)-plane, or in a plane parallel to it. Con- sequently, when the motion is quasi-rectilinear and parallel to the x-axis, the wake surface is approximately parallel to the (z,x)-plane. The previous rea- sonings for the motions parallel to the (z,x)-plane hold in the present case, provided v/U and Lr/U are respectively substituted for w/U and Lq/U. b) p+0. It is possible that an U-shaped free vortex shed during the small interval (7', r'+d7’) have, at t’', an orientation with respect to the axis attached to the body different from the orientation at 7+'+d7’. In this case the summation at t’ of the effects of the free vortices shed during the intervals (r', r'+dr') cannot be carried out on the same manner as in the case of a mo- tion parallel to the (z,x)-plane. This case occurs, for instance, for a body of revolution with respect to the x-axis. 2°) Let us assume that there are in the (z,x)-plane singularities so that the arc @’ is in this plane. a) p=0. This case is quite similar to this of par. 1°) a). It is the simplest from the point of view considered in this paper. b) p+0. In this case, if |¢| is relatively great, the wake surface is not a plane; it is more or less helicoidal. The nuclei found in the integrals which yield the effect of the wake depend not only upon t‘- 7’, but also on 7’, and the expression of the hydrodynamic forces due to the wake is much more complicated than in the case when the motion is parallel to the (z,x)-plane. 3°) It occurs very often that the singularities mentioned above (par. 2°) exist only on the upperside of the body, and not on the lower side, or inversely. In this case, when p =0, the wake surface is inclined with respect to the (z,x)- plane. Consequently, the velocities induced by the wake on the body itself and on the rudder and planes generate necessarily forces which components on the z-axis are not null (see Fig. 8). If, for instance, Lr/U =0, v/U>0, and if, moreover, the upper arc of @’ is located in the (z,x)-plane, because of the singularities of the hull, but not its lower arc, this latter is located on the portside of the hull; the wake surface induces on the body and on the diving planes located on the stern velocities 868 A Vortex Theory for the Maneuvering Ship Ipaifeqbineey ts) which components on the z-axis are >0. Consequently, the variation of Z and of mM are both >0. This effect is independent of the sign of v/U. It leads to a per- turbation of the motion in the vertical plane even when the angle of heel is null. 4°) When the six parameters u/U, w/U, Lq/U, v/U, Lr/U, Lp/U depend upon the time, is it possible to add the effect of the wake due to the three first of them and the effects of the wake due to the three others? The velocities due to one of the wakes may act on the configuration of the other wake. Nevertheless, when the dissymmetry of the body with respect to the (x,y)-plane is not too strong, and, when, moreover, we are in the case of par. 2°) with moderate angles of heel, the velocities due to the wake parallel to the (x,y)-plane are nearly parallel to the wake due to the variations of v/U and Lr/U and inversely. So it is possible, in a first approximation, to obtain the set of hydrodynamic forces exerted on the body in a quasi-rectilinear motion parallel to the x-axis by adding the sets of forces separately found for motions parallel to the (z,x)- plane and for motions parallel to the (x,y)-plane. In paragraphs 10, 11, 12, we restrict our analysis, for reason of simplicity, to the cases when such an addition is allowed (however, in par. 12.4, we will consider a more general case). We set ii) as Ieee ee Say ' Lr 1 B Lp B 1 lin fo fost ay og > aCe =F tes and assume, in par. 10, 11, 12, that these functions have negligible squares and products. 869 Brard 10. The Absolute Forces Due to the Velocity Potential (case of par. 9) The velocity potential in the absolute motion is 5 5 P(M,t') = ®jo(M) + D> Oy (M) fy, (t’) + DY (M.t’). k=0 k=0 Potentials © are those which are found alone when the fluid is quite perfect. Potentials ¥, yield the effect of the wake. This formula involves the hypothesis at the end of par. 9. However, in the present paragraph, we don't consider the effects of the wake on the appendages (see par. 11). 10.1 Effects on the Wake on the Body Itself The contribution of f,.(t’) in V,(M) is Vop(M,t') = -i, pz(M) + i, py(M) . (V,n),, 1S generally very small when mison S. For this reason, we neglect here the contribution of f,.. As seen in par. 5.3, the wake has an effect on the apparent forces of inertia. This effect will be taken into account in par. 10.2. For reasons of simplicity, we will change slightly the notations of par. 5.3. We use here the following symbols: a for the lift due to w/U; b for the lift due to Lq/U; a’ for the moment due to w/U; b’ for the moment due to Lq/U; (ays ae) and (b,,b/), are substituted respectively for (a,a‘) and (b,b’) in the terms coming either from v/U or from Lr/U. Moreover, we assume, as in par. 5, that a set of three functions ¢, y, ¢' is sufficient for yielding all the effects of the delayed circulation when f,,,f,, are null; and in the same way that a similar set ¢,,¥,,¢, gives this effect when fam tgarta, Gece maul Lastly, we admit that the £-component of the moment, when f,,,f,, are different from zero, may be expressed by means of two coefficients m,m’. Finally, that leads, for quasi-steady motions, to the set of forces he = 0, Ceara onaa tt la, (5). ee ar (1) (Cont.) Le Shel 5 AU? [20 # 8 a), joe (a), mae ce 870 A Vortex Theory for the Maneuvering Ship Vv ; er p ALU? | ma, (o) + m‘b, ail , dole no, = Fone? [esos (i), +2 (8). 2» GB)]. atpall 2 fap ' (=) Ny, = 3 e ALU a; (5). a nla | The effect of the delayed circulation gives a system of forces where DS = MC = ANZ = B i k t pau ¥ xcs wo) + [ elie.) jee" ar'| k=0 hole 0 k=3 0 4 3: ‘i a [foxtt”) ~ (0) +| Hye) Jace —oy dr] ie cet rar" | ( ro fC) acre f CE) ayer ar’) el< i | ie) ee S NS) ——— » — (=||S| ©- Ss one (jo) SS + iS) ct T 4 |) i. £(0) + fy a) fearon a'] als Tr 1 De, (ae) p,(0) + ie ( iue'-ryan']f. 871 (1) (2) (3) (Cont.) Brard t t AM = 5p ALU? e te). $'(0) + J Fale Hee 2 ar" 0 T yy! io dO) + if =| eerryar']t, 0 T (3) t! A a eae 2 ! Ww ! Vv wa ee at 1 AN = 5 p ALU {3 (a). $1(0) al (5), bi(t'-7 bar] lL iD U r U r 10 UJ 1 ty apa (e) AO) || Fal, Pe ei ) ar" | 10.2. Effects of Potentials © These effects give a system of forces (X;,Y,,...) which expression will be added to the expression of the forces of inertia of the body in par. 12. 11. Forces Due to the Appendages (case of par. 9) We assume here that the contribution of the appendages in the forces $, and $, will be included in the expressions of the latter (see par. 12). We consider now the effects of the various lifts generated by appendages as diving planes and fins, rudder and aileron, sail. 11.1. Diving Planes and Fins The set of forces on diving planes and fins was studied in par. 6, but with the restriction that The absolute velocity of the axis 0, of a diving plane B is Vis) = 7, WW we lbel Gall ar ay hye re Se Gall ae a, il Ie Sal. The absolute velocity of the center of the lifting surface is Vig. = nls wt Le Ga ibe Gell iylv Pir Gp Lp Gel + 1, lwp lp ae] laa In order to obtain simpler formulae, we will assume that the effect of the com- ponent of y, parallel to the span is negligible (in fact, some corrections would be necessary; the lift decreases, particularly on the part of the plane which is in the hydrodynamical shadow of the hull). According to this hypothesis, the 872 A Vortex Theory for the Maneuvering Ship velocity induced by the wake due to f,,,f,,,f,, has no effect on the diving plane. The square of the effective velocity is consequently v2 E 0, (i), we Gal, ene: (5) ns]: The component on the z-axis of the effective incident velocity is, in the quasi- steady motion: ir, - fo(t')&gy t+ fos(t')7B- 2oB- op foo(t’) ~ ip foi(t’) - arp foa(t')}- The plane is at a zero angle £6 when the lift generated is null, the motion being parallel to the x-axis. Consequently, in the quasi-steady motion, the effective angle 8, is 2 (B= ie eect) SAE Sa ee its |= » Bie lite) 5 k=0 In the real motion, the velocity induced by the wake is 2 to Yy ane Fo(0%9 Genet.) | aS Roi) dhp(t’-7') ar" k=0 0 2 t! : = ye “hn || fan@ Oct =a dig : k=0 0 and the effective angle of attack is £3 = 8,-AS,, with 2 @u AyB. =) = » arp [foxct) - { LeU) dale =—7') ar] é 0 k=0 Consequently, the set of hydrodynamic forces acting on the diving plane and fin B is: 1 Xp = - 7 PU? c, [1+ foo + 2foote ~ 2fo 475] » Yp = 0 ’ = 1 2 ae te = = 5 PoBU ou Le er Bese = lating 2 x E + (£91 ~ fooSp) + fos 7p - ei aupfor 7 2B. ’ k=0 873 Brard Oe ee ; 2 x E + (f9,~ fo95p) + fos7p - B arpfon ~ MB, | k=0 Mg = - 5 PTR LU? CBC x, [He 2foo + 2fo9%R - 2fo4 7B | No] Re oa lius [ée ore = Cm |L 2 + DN a at Din Ge Be 2f 5 478 | 2 x [os (fo17~ foodp) + fost - Dd apfon ~ a, | : k=0 1 Mg = 7 PvBLU*c, [1 PAN a th ign Ge = 2f 9 47B | Tp - Taking into account the fact that a diving plane is made of two parts sym- metrical with respect to the (z,x)-plane, and neglecting the squares and prod- ucts of the functions f,,, we get: Cation 2 wl = Xp = 7 PoBU Co [1 +2 (a). +2 U ae , 2 x eae i ol hs a: int} AzZp k=0 tee Ferra fal). 0008) J ” Thy = (eee 1 pop, LU? CBCx, E ny AF a 2 (=a) oe | @P Gy 2 a (oe k=0 A Vortex Theory for the Maneuvering Ship / 2 t , : ' 1 y 1 1 1 ‘Zan = 5 Pop U? Cy re » ang fox ) || Eisai Pratt am )en k k=0 0 (2) 2 € 4 1 > ' - 7hoB LU (ERcL. ~ Cn.) Do ake fox We | k=0 0 Me = fou(7) ype! r")ar" | , 11.2. Rudders, Ailerons, and Other Appendages Located in the (z,x)-Plane Let Lé,,0,0, be the coordinates of the axis 0, of a rudder A; Lé,,0,Ll, those of the center of the lifting surface, which area (aileron included) is c,. The absolute velocity of the center of the lifting surface is Vig = i, ((U+ u) + Lal, | = i,[v ap RE LpZq| + i,|[w - Lag, | - We assume that the components of the incident velocity parallel to the span (hence, to i,) have no effect (this assumption is similar to the hypothesis already made about the diving planes and motivates similar comments). According to this assumption, the velocity induced by the wake generated by PrGuOiGmianct. ) and 5(t ) has notetfiect on the rudder: (he square of the effective velocity is UZ E +2 (o) oo (2) Hel The component on the y-axis of the effective incident velocity is, in the quasi- steady motion: oH {(§) y fog(t Eq - EC ENDS as Aza fo3(t’) = aeafoat Ot. t! The rudder is at a zero angle « when the rudder is in the (z,x)-plane; ais >0 when the lift has on the y-axis a negative component. In the quasi-steady motion, the effective angle of attack is 4 dg SB [TCE BN ae ene Dg | en nen ece 3) & k=3 In the real motion, the velocity induced by the wake has on the y-axis a component 875 Brard Uy 4 t ss ana )foxcor) Dy a(t’) + J fan iT ppe(@) dya(t!=7) dr" | k=3 0 4 ex = » 4) ion ® ) dea(t'-7') dr’ ‘ k=3 0 and the effective angle is a! = a,-Aa,, with One has t ‘ 4 Ma = = ye aA [fox - | ont’) dale’ =2') ar’ ‘ = 0 k=3 Py a(0) =05 Ger) = 15 is monotonic. set of hydrodynamic forces due to the rudder (and aileron) is: _1 2 u bg 5 ec,U ee, [2 7 (a), + (Te) cane eh _i 2 La =| 4 as |: © Cigg © “gaSn = Ugg Sn) = ye ana fo, 7 Aa, | , k=3 4 x E HOG” Mans muoso = yy aA “oni da, |. 876 A Vortex Theory for the Maneuvering Ship In these formulae, c,,, cx, and c,, are the characteristic coefficients of the rudder (and aileron, if any) or of the sail. Finally, one has: Be ak 2 u ea PS i GPA Cy F ee (a). allen z 1 2 u Lq Ya = a eye cL { E aD) (a), +2 ce ta + [fa + CB) ee FB). ta] 2 aan tance po ome t t t k=3 Zine = 0: oA = 5 PeALU2 ey Ca ‘ [ ee (Ce oh a). | (3) Vv Lr Lp : [an . el ea=!GEL, oo PUG }- Ae, Hie = 7 oon CA E +2 (5), v2 fae in] ta = = Bora Basey-eod {2 [2 +2 +2 Cf 4 lS), + Fa l= us| 4 ara fo, (t }- AN, with 1 : ih ays Zee Ae cy De aA fot’) -| fo, (7) Hatt! —1) ar" | k=3 , 1 : ‘ Ae, = 1 og LU? Sn, Ga SS BA tone) | Four") dealt! 7") dr’ , (4) k=3 0 4 t! 1 t / 5 ’ i t AN, = 5 Po LU? [faci ,~ em] 2 aA [fox ) -| f4(7') Pyea(t - 7°) dr 0 k=3 877 Brard When the rudder is made of two parts symmetrical with respect to the (x, y)-planes, the term in ¢, are vanishing, but not those in Cnet In this case, one has: 1 X, = ~7PoaU?cy, [142 | |[[5 _—— + es A Yae= - Fo, U2 Cian {e[t+2 al ss (i), “(eal = ee foxtt)} ~ AYa, k=3 4 ex= eraboten, fo[t+2(8) .] +08). °C@) ta] Zi sea foxt9-Aa dy k=3 4 delete |e lee 2 ALA BOO) = AM § k=3 AN, = as above. Of course, this simplification is impossible in the case of such appendages as Sails. 11.3. Some Comments About the Previous Formulae 1°) At the beginning of a maneuvering, the delay in the growth of the circulation around the body increases the efficiency of the rudder and diving planes. 2) Cm, and cp, have been taken >0 when, ina steady motion, the torque about the axis tends to bring back the rudder or the diving plane to zero angle. 878 A Vortex Theory for the Maneuvering Ship 12. Other Sets of Forces Acting on the Body (case of par. 9) 12.1. Gravity The set of forces is 2W eb nigh 2 SE yi Bees AUG a 1)6] , ve = Lea? oe eG 1) al Boe we AL y2 1 QW gL Le =p AUC ee ely; Sor 1 vA (1) 1 Qw el Ly = > pALU? AL U2 [(GuGz = C4] ) W gL Me = 5 ALU? = = Cube €.)8 - (ig - 201 ib Me = 5 PALU? a G#se-6¢] 4 12.2. Friction, Propeller Here the thrust T of the propeller (1/2) pAU’7 and the moment about the y-axis iS (1/2) pALU*A7. t is the suction coefficient. One has 1 Xen) g CAU eL + a1 -£))| Year ae Os 1 Zip = pone a r(1-t)]é, nag 2 Leet = 7 PALU Ags 2 Meer = deatu? [-C,4 a7]. Ne,p = O- 12.3. Forces of Inertia ¥. for the Body, Forces Due to Potentials » and to Pressures p'‘‘') The components of the general resultant on the (x,y, z) -axis are: 879 Brard xX x Xe fan a AU2 2W = Lq Uj Lu 1 ers sHy= o (e AL Pe yp CE ED) saa Pag =m Lee Lee + (145 - 2g) aa ee). 1 1 2W Lp L Y.+¥,+¥j = 5 eAU? Mu, 8 T = (EE ae (H+ ha) => (3) ; L’?p : UA ls (144+ Hg) ye + (Mag = BE ) ae igh 2W » ht ele, ; IL” | Z.+Z,+Z, = 5 PAU? a, MT cut ny Richa a (wens) Bhs Of, t reg ESE. In these formulae the squares and product of ¢,,é, are neglected. Lastly, the components of the resultant moment are: 1 2W Vv L i ¢2, $e = 5 pALU? AL {rss U (157 #%e) A Lp Wee + Okt pls) - (A, +ux,) — P+ (Aig + HXy3) ae ae 2W u Ww Mo +I, + M5 = ae PALU? AL {ous + 2H, ut (2g = [Ba) U (4) ‘ et 1 Li U AG + (V5 ~ Hg) ae (135 + Hq) a ~ (A +EX>2) a ; 1 2W Vv Lp WL 3 itl, 4 = 3 PALU? Be a=) U t (Pont Pag = We) U Vinee are L?p L?r + (V6 “Hig S44 tH) 2 = (Ag +E 8) eo : 12.4. Case When the Wake Generated in a Motion Parallel to the (x, y)-Plane is not Parallel tothe (z,x)-Plane We assume that, because of singularities, appendages, and so on, on the upperside of the hull, one of the two arcs which constitute the arc @’ along which the free vortices are shed is in the (z,x)-plane, is on the upperside of the hull. The other arc of @' is on the portside when 1.4 oh 880 A Vortex Theory for the Maneuvering Ship for the values of x inferior to the abscissa of the axis of the gyration here con- sidered (this abscissa is normally positive for the natural gyrations, and, con- sequently, for the forced gyrations which are not too different from the natural ones). The component on the z-axis of the velocity induced by the wake is always positive, whatever the sign of v/U + rx/U may be. It seems that it is a quadratic form of the arguments v/U and rx/U, or more exactly of the arguments: t! t! d Vv d (=) / r i erase ae ie / ‘ = 5 Peat Sie = d ‘ | dirt Gal ' eS aaa [ Gur NU Pal EIS 0 Li 0 th when (v/U),- and (Lr/U),, are continuous for t‘>0. Functions ¢,(x,t’),¢,(x,t’) are null for t'=0, and their limits, for t'=+, are finite and positive. This assumption leads to introduce new functions ¢,(t’), ¢,(t'), $;(t'); ¢,(t') null for t’=0, equal to 1 for t’'=+, and to add to the previous compo- nents of the hydrodynamic forces exerted on the body itself, the components t! t! ~5Z = 5 pAv? 5,|| (5) OSI die = 20)| (a) EON Sart air’ SU 0 % 0 i (5) 1 ic ih -8M = 5 ALU? eal ical dt! dr’ e4 | (E) TCO SO Year 0° 0 i 0 ue where c,, c3, c,, c, are positive dimensionless coefficients. There is also an effect on the diving planes located at the stern. The effec- tive angle of attack 8. = 8,-A8, (cf. par. 11) becomes t U aan | (a). bi(t'-7') dr’ 0 Bi = B,-46,-8f,, with 8B, = - Saf GA) ae va|. 0 T In this formula, a,, and a,, are positive dimensionless coefficients, and Ben e,(t ) are nullier t= 0) andvequal to) ior t) >+0. 13. Other Motions of Practical Interest Previously we restricted our analysis to the ''quasi-rectilinear'' motions. But there are other motions of great interest, and particularly, the change of depth and the change of head. 221-249 O - 66 - 57 881 Brard In many circumstances, the angles 6 and a are not small. Consequently, the variations of u/U, w/U, Lq/U, v/U, Lr/U and Lp/U may be great. In such circumstances, the equations of the "quasi-steady motions" are the same as above, at least when the steady effects of the wake are neglected. How- ever, many coefficients which are found in the set of forces (F,) are unknown. Generally, the theory is unable to yield them. It is necessary to resort to ex- periments. But tests on models themselves require special and complicated instrumentation because of the high number of degrees of freedom and, conse- quently, because of the number of the coefficients which are to be determined. If we now consider the effects of the wake, we encounter difficulties which we partly emphasized in par. 9. When the nuclei found in the integral equations of the motions are functions of t'- 7’ only, they can be deduced, as we will see in Section II of this paper, from measurements made with small harmonic forced motions. But, when these nuclei are functions not only of t'- 7’, but also of 7’, the problem is much more intricate. The main difficulties are of two types. The first is due to the fact that, for certain forms of hull, there is no rea- son why the free vortices should be shed along lines attached to the body (for instance, that is the case of a submerged body of revolution, the complication, in this case, being due to the fact that the axis of revolution is not always of revolution for the distribution of the masses inside the body). The second is due to the curvature of the trajectory described by the origin of the axis attached to the body, and also, to the roll motion. Obviously, the velocities induced by the wake are no more given by the formulae above. That does not mean that there are no possibilities to investigate this prob- lem with some chance of success, but, before undertaking such a research, it is desirable to check whether the effects of the wake are or not of importance. That is why, in the next section, we study the effects of the wake in har- monic forced motions in the (z,x)-plane. We will see that these effects are not negligible, at least for some coefficients. That will give a lead for fruitful researches. fl. STEADY AND HARMONIC FORCED MOTIONS PARALLEL TO THE (z,x)-PLANE 14. Definition of These Motions — Set of Forces Acting on the Body 14.1. Steady Motions, Purely Heaving Motions, Purely Pitching Motions The fixed axis 0’z’', O’x’ are in the (z,x)-plane. The z’'-axis is vertical and positive downwards. The x’-axis is horizontal. The absolute coordinates of the origin O of the axis attached to the body are {,é. 882 A Vortex Theory for the Maneuvering Ship We assume here that The angle of trim is 6 = (0'x',Ox). absolute velocity of 0 is ora constant. 6? is assumed to be negligible. The (1) (2) VO)! See ese iwt 1.(U+u). 2° Ghe x dt Z a One has w= feos o+ E sings P+ Fe, vru=- esing+ Boos os Sore. Therefore w= Seve ue, Hee The drift angle is P= (OES) 2 e+e The angular velocity is ; a0 Ge oot lore (w/U)?, (Lq/U)? and |(w/U) (Lq/U)| are assumed to be negligible. A. Steady motions The steady motions here considered are defined by the conditions: 6 = constant , = =aOr Consequently, we have: 6 eas, wit = acd} eneh OlSeconstant = ye U U The set of forces have the components on the axis attached to the body: TAK eM ’ B. Purely heaving motions In these motions @ = constant. 883 (3) Brard Let 0'z,,0'x, be fixed axis respectively directed as the z-axis and the x-axis. Let ¢,,£, be the coordinates of 0 in the new Set of axis. We have Cre Ga Gin een Se SiGe In the purely heaving motions here considered, one has: 1 a ; 7 = OG this product is therefore negligible. Moreover is a Sinusoidal function of the time. Therefore, we set: Ge = cle(@) + Wet + Zt Zy cos wt (Z = constant, z,>0, 0 = constant = By) « and obtain 2 (*) cos at. (4) In these motions, the components of the set of forces are: Zs Daw az ae So) hres (6d) tay cos (63) 5} X= X + X,_ cos att X, cos (be Sh (5) 2 m= M+ t + fee ( Hyctaewias hy cos (« a The subscript 1 is relative to the components in phase with the motion; the subscript 2 is relative to those which are out of phase with the motion. C. Purely pitching motions In these motions ¢ is a sinusoidal function of the time: ¢ = 6+ 6, cosoat, (* — constant, @5 >0);7 moreover, (0'z,, Ox; being fixed axis, with (2 [z7)e—ses the ordinate ¢} = ¢¢ + ( is also a sinusoidal function of the time chosen in such a way that the angle of drift be a constant. Consequently, the purely pitching motions here considered are defined by the conditions 884 A Vortex Theory for the Maneuvering Ship Z => ; ; RaW. Rs +. Ue, eA 7 90= 0+ OG, cos at, (6, 70), ee ae - -= [= —* cos (at - 5), (6) II o I The set of forces has the components: — TT Ly =A ee: 2, cos at + Z,, cos (at oye X= Xt Se coeeea ee eeaedal arti, (7) Se . 7 Ne= M+ Ml, cos ot + I, cos (at + 5). 14.2. The Set of Forces in a Steady Motion Using results of par. 8, and substituting © for w/U, we get: L o- o = pave | 2H at (C= ID) SF |- (+22 Cx. + 5 c,,)+71- ty] 2 hole “3 Suge ae 7 Ci) eae = 2 oat. [=( -18] +|- (c Re eae i ae: )era-t| ft: x Wooo AMS ; | lI | ie) > (| iS) (pS Ble | (8) hole pALU? {2 =) Reach. Clg oe] + > we - F (Et, ~ Sag) 5} 14.3. The Set of Forces in a Purely Heaving Motion We have to substitute 6 for 94, 885 for w/U, and for Lw/U? in formulae of par. 8. We obtain, for instance: (given by the first formula 8) wht 2W ale °B L Zi= Et 5 pau? | 2 (we /) 2 cos wt - & » 7 ey (1 a1p)| a cos (at aye (which comes also from the contribution of the quasi-steady motion); t SS el 2 OL TT oL (2 ' ) at i 20 Z= is eee ptt pat t pec ps = al = z+ boauta |.4(0) yp cos (a+) + U cos |G t +5 b(t yar} & 0 (which gives the effect on the body itself of the delayed circulation around the body); (which gives the effect on the diving planes and fins of the delayed circulation around the body). Because the harmonic motion is assumed to be perfectly established the interval (0,t’) is infinitely wide, and consequently, the lower limits in the integrals above must be taken equal to -~. But we have t foo) J cos [4 po Balog) ie" =| cos EB Colma) + 5 | 7') dr! -~0 QO . = cos wt xg (=) + cos (ot + Z)¢ (<4) 0 (9) (20) = J WT") cos (= 7) alr”, g(=") a Or") sin (= r') dr’ (10) 0 are the cosine and sine Fourier transforms of the derivative #t’). 886 A Vortex Theory for the Maneuvering Ship Finally, we obtain =) 1 Ww NGG Zn, cos wt = 5 PAU AL (Ht H3) +a @L U tare + 2 “B me U Pay Os at 2, A wide s OL U iL, Cc CW) arth (11) 2, cos (at +3) = 5 PAU? {- a E -~$(0) - ‘(4)| cal) = fh date cos (Ee eel B) =f bxer sin (Em) or a 0 are the cosine and sine Fourier transforms of the derivative ¢,,(t'). A quite similar reasoning leads to: 1 (OL le SW NN CE Ss (aah 2 xn cos at = my PAU AL 41372 aL Tr) eco ONE 5 (13) aye a ea ; (at\| | ot 20 1 Xn, cos leo) = 7 PAU aco) + £3 (2) we cos (ot +5). where foe) foe) CE) = fdr eden cil [der ain (rar a 0 0 Similarly, we obtain ai MOT or » & (@L/U) Mh cos at = 5 ALU {BY cos, + nig) - 2 yn io op € 1 _(@L/U) cae Zo (15) ACE ou ofcne)) 4m men 7U Win eco es (ont) Brard My, COS (xt +5 = 4 pat? { 24 (a, = fig) PB” [1- 4'¢0)- f (+)| (or B oL i, 2@ + 2 oe (Sen. em.) ie Bue Fh ale La sw (# i 3) 8) foo) i & = J $'(7") cos (= r') Gliese (+) = i wr") sin (ot | dr’. (16) : 0 14.4. The Set of Forces in a Purely Pitching Motion In formulae of par. 8, we have to substitute: = a L @+ @, cos at for 0, @ fice 0, Ecos lone S| foe = L\? q = (=) cos wt for ae ; That gives: [Z = expression given by (8) |, See on mel 7B “B : Zo, cos ot = om | 5 [ext ec, +2 ex, + 71 t)| Fe cote o oH (8 (17) + 2B, santan (4)/(4)} OB)? 8, con 7 1 2W wL °R 25, G28 (t +5] = J pau? | 28 ((O" [4,) = 15 [1-405 - f (4) | - > Xz CU aa +2 =e co [int aopfop (+) se @, cos (ot + a FanlE) = fo dase’) cos (B') ar’, exp E) = [dan (B) sin (a 0 0 Similarly, we obtain: [x = expression given by (8) ], 888 A Vortex Theory for the Maneuvering Ship = 4 paye {2H 4-1) /fob)’ Xo, cos) at = 7 PAU | AL 2 (Cfoi= al)) U ~ BE cig ater — bai (E/E) |B)? 2% 08 ot (18) Xo, cos (xt + =) = 4 pau? | fa Pa3 2 a|ucoy toate (2t)) } a 6, cos (at + =) ; Lastly, we get: [i = expression given by (8)], + a’ (Y/R) ~ 25 (encng co) tones ()/E)} x (4) a5 cos at , (19) TT 1 1 ’ , (@L oR My, Cos (ot 4) = 1 pau? {p ja-¢ (Q)) = sf (¢)| + 2 mh (SB°L~ Cm,) 2£lp oF oL 1 (Seo. > Sma) leat a5 f op ()|} « & oe, cos (at + 3) - 15. Interpretation of Experiment Results (Steady and Harmonic Forced Motions) 15.1. Steady Forced Motions By testing a submerged body in various steady motions — without and with diving planes, without and with propeller —it is possible [see par. 14.2, Eq. (8)] to get the numerical values of the following coefficients: Pao (ey Lee Brard 15.2. "True" and ''Apparent" Coefficients Now, let us consider the expressions of the forces in harmonic forced motions. We define the ''apparent" coefficients of the motion by the following formulae: a) From par. 14, Eq. (11), we deduce: @L QW _ 2W g ail AL (pet fe = A (Ji ae fies) ae Zi Pye (without planes), .( eal 2W 2W ; U is} 1B \U : AL (ee a) = TG (bat pee) Fe Ta + 2 car eT 418 See (with planes), (2) Oe se a Bay =e i= @(O) = i U (without planes), or antes a [1-40 =i (et) +2 Ble 1- ayn fip (:4)] (with planes). b) The first equation of Eq. (13), par. 14, gives: ir ow ay ONG ee ENT (3) AD ig o5 AG eis = aU c) From par. 14, Eq. (15), it follows: QW o OW - , _, (@b OL : AL (ag) ese) Say (Man tugg)ta g (e\/(e) (without planes), 2W “s OW is ae 2 , 9 f@le L AL (98,44 7 MSs) = AL (Y¥3,+Heég) +a g eye Cpe a oL oL : + > TR (Gs &,.* Sa) ainfip (=) /(S) (with planes) , 4 2W ' 2W ' ? nel ap (P3 M1) + app ~ ap (43° M1) 2 1L=@'(Q) = i (+ (without planes) , AL (Big = (SD) 3 lees = a (y= my) +2'[1-20)- f/ (4) ] oR aL é + 2 a (SB8L.> me) 1+a,pfip (+) (with planes) . 890 2W AL Ww AL A Vortex Theory for the Maneuvering Ship d) From par. 14, Eq. (17), we get: 2W ' @L)\ | [wL q Gia cit. + WEG) = AL (¥3,+HSq) — beg aries (without planes) , 2W ' eo wL wL, ate + Eq) AL (Uae HSE) bg (=) /() a8 wL\ /{aL - 2 a om aop kop (2) /(st) (with planes) , (5) W L : Ga ie) = Deon = = aa ee y= Is [1-#(0) = (5)| (without eens) : 2W wL GPs) = Bap, = a Cut my) = b [1-#00) - (| O; + 2 ~ cum cpt Blas is (=) (with planes) , e) The first equation of Eq. (18) gives: 2W 2W ' ,{@L L 2H oils) = A elepto wel PEER). o f) Lastly, Eqs. (19), par. 14, yield: ue ' ; U : = OF en 1 UX 2) = An (CONG alae) am Io oL/U (without planes) , « (3) 2W DS ' U RUE oe Oe my Ea) kere Ib, oR 0B (=) =2 — (GaG =6 A» “apoer. (watide jollemeas)). A (BCL, ~ my) 22B aL /U (7) ion | = |p" [1-9'¢0) - f' ()] (without planes) , lor | abs [1-4"(0) -f! eal Bt 4p i (4)| (with planes). | M Q >| oy Uy wo fo) J w | fo) w a | fae | Uy 891 Brard Equations (2) to (7) show that the apparent" coefficients are constant if, and only if, the effects of the wake are negligible. When this is not the case, they depend upon the "reduced frequency." According to the quasi-steady theory, the harmonic forced motions should yield the coefficients which are found in the ($,)-set of forces. In fact, they may yield the effects of the wake and allow to check whether these effects shall be taken into account for practical purposes. 15.3. The Behaviour of the "Apparent Coefficients" Let us consider again the functions f, f’, f,, fips @ 2» S19 Sp: We have admitted (par. 5) that an unique set of functions ¢,y,¢' is sufficient in order to define the wake due to the body. We admit, here, for the same rea- son, that the functions ¢,,,¢,, (and ¢,,) are identical. Such an assumption is presently not essential, since, in principle, the in- terest of tests on a body in harmonic forced motions is to supply these func- tions. But, the discussion which follows will be easier. Firstly, we may observe that, for instance: ioe) AO) = | Agri ar? = ace = G0) = =eK0)) - 0 Consequently, any "apparent" coefficient involved in an "out phase" force or moment has, when @L/U > 0, a limit equal to its "true'’ value. For instance [see (2) |: lim a oat reas wL/U> 0 and so on. On the contrary: oOL () = O hol ee Ee ee OTE Tevicn Therefore, when Higa ah aia topes) gia = PlG pak UN Tee In many cases, these five coefficients will be either nearly constant or small, and the calculation of the g-functions will be perhaps without practical interest. Similar reasoning based on the results of the harmonic tests with planes would show the possibility to obtain 9. Practically all the unknown coefficients and functions may be determined, except those which are connected with the variations of u/U. In its present state, our Planar Motion Mechanism is unable to yield them, because no sinus- oidal motion parallel to the x-axis is possible. But it is to see that the system could be modified for that purpose, if necessary. 19.2. Motions in the (x,y)-Plane From pars. 10-12, we could deduce, for this family of motions, formulae similar to those of par. 8, and we could show, in the same manner as in 19.1, that harmonic forced motions in the (x, y)-plane give also the numerical values of the coefficients and functions which are needed to write the equations of the motion in the (x,y)-plane, or more generally, of any motion, provided the ex- pressions of the forces are additive. 20. Effects of the Non-Linearity and Other Sources of Errors 20.1. Non-Linearity The so-called "true" equations are true only in the linear field. The non- linearity may affect many points of the semi-theoretical views explained in this paper. Some of them are related to the part of the quasi-steady motions theory which we use in our formulae. Some others concern specifically the structure of the wake and the method used for taking its effects into account. 1°) Because submerged bodies are generally very poor lifting surfaces, the coefficients a, b, a’, b’, for the motions in the (z,x)-planes, a,, b,, aj, b}, for the motions in the (x,y) -plane, are not really constant. A question would be to know whether it is possible to substitute for their expressions versus the drift angles <« or 5 such expression as 2° for a5+28|8| or for ad + 283, We have also to observe that our integrals t a a | rae @ cee er eae 0 become 901 He en els te iW U W w\° J [ele +e@), 0 2°) The non-linearity is also to be taken into account when the mo- tions are not really quasi-rectilinear. From a practical point of view, this is very serious Since, in many cases, the trajectory of the origin O of the moving axis is not a straight line. In such a case, the nuclei depend upon +r’ and t’-7’, and not upon t'- 7‘ only. Consequently, we encounter here a new problem, which consists in the empirical determination of the new function ¢(7', t’-7') which have to be substituted for ¢(t'- 7’). A similar circumstance happens when the heel becomes great even if the trajectory of O is nearly a straight line, for, in this case, the wake cannot be considered as a plane surface but is an helicoidal surface. The first phase in a change of heading would be different and the wake due to a gyration in the verti- cal plane could have a severe effect on the trim. 3°) The non-linearity may affect also the scale effect since the coef- ficients a, b, ..., depend upon the Reynolds' number. This cause of error exists also in the quasi-static theory, but it has no effect on the functions Bao nrc 20.2. Other Causes of Errors They are the effects of the free surface and those of the walls and of the bottom of the tank. In order to get accurate measurements of the sets of forces, it is necessary to operate at a sufficiently high speed. But U becomes great and the range of values of wL/U which is accessible becomes narrow. In order to increase this range, one may be obliged to operate sometimes beyond the critical speed U//gH, where H is the depth of the tank, and sometimes below. On the other hand, the coefficient V//g?, where ¢ is the depth of 0 may be great and consequently, the waves generated by the model may be not negligible at all. Lastly because the range of values of is not very wide (from 1.1 to 3.27) it may be necessary to work at various values of wL/U~ U?/gL in order to keep constant the values of «L/U and, consequently, the changes of the wave patterns which results from that, may lead to errors about the true effect of the reduced frequency. That means that experiments conceived in order to determine the functions fp ,-.-,, require a very caution approach. 21. The Solving of the True Equations Generally, one admits that, when the forces acting on the model are known, the equations of the maneuvering ship may be solved by analog computers. Such computers are most often fitted with curve-plotters, and it is possible to get the curves which give the motion of the body following a given maneuver as a function of time. 902 A Vortex Theory for the Maneuvering Ship However, if we deal with integral equations, the conventional analog com- puters are no more Sufficient. Bigger computers, analog, digital, or hybrid, are in fact necessary and the work to be undertaken to study all the possible interesting cases becomes really huge... 22. General Conclusions 22.1. As already stated in the Introduction, this paper is devoted to the effects of the circulation around a ship on the set of hydrodynamic forces exerted on her. As a matter of fact, the subject is restricted to the case of a submerged body in an infinite fluid. But the case of a surface ship is similar, apart of the fact that the free surface effects have to be taken into account. 22.2. Our mathematical model is defined in Section I, pars. 3 and 4. We start from the possibility to substitute for a submerged body moving in a perfect fluid an equivalent distribution of bound vortices on its hull (and inside the volume interior to the body when the angular velocity is not identically null). Consequently, a motion is defined in the whole space; the fluid interior to the body is at rest with respect to the latter. Then we introduce a new family of bound and free vortices in order to get a wake. This new family has to be added to the first. We consider firstly the case of a small motion with one degree of freedom around an uniform motion of velocity U parallel to the x-axis. This small mo- tion is aSsumed to be parallel to the (z,x)-plane of symmetry. Neglecting the deflection of the fluid due to the reaction of the body on the fluid or, which is equivalent, to the velocities induced by the vortices on themselves, we admit that the free vortices are at rest with respect to the fixed axis. They are lying on U-shaped arcs which are nearly located on planes parallel to the (x,y) - plane; because the angle of attack (or the reduced angular velocity) is small, these arcs are approximately located on a wake surface attached to the body along a line which is assumed to be known (given by experience for each body), and which acts as the trailing edge of a lifting surface. The bound vortices as- sociated to these free vortices are distributed on the hull. The total distribution fulfils the condition that the circulation around a closed fluid arc is equal to zero. The total potential equivalent to the free and bound vortices of the second family induces a velocity which is null inside the body and which, outside the body, is tangent to the external face of the hull and to the surface of the wake. It is shown that these conditions lead, when the motion is unsteady, to a formula which gives the circulation in term of the circulation in the quasi-steady motion. This ex- pression is a convolution function t d Gta) el IE (ae) 0 dk Gite las Saeed a7 af + ot ’ where t’ is the reduced time t'=Ut/L, L being the length of the body, and t' = 0 the time at the beginning of the unsteady motion. 903 Brard Then we have to add the effects of a motion with three degrees of freedom u/U, w/U, Lq/U, the total motion being still parallel to the (z,x)-plane. In order to do that, we admit that these three parameters have negligible squares and products, and, consequently, that the wake surface is practically the same as in the previous case. 22.3. In par. 5, we introduce the hydrodynamic forces due to the total velocity potential of the absolute motion of the fluid. For the sake of simplifi- cation, we assume here that the body is not fitted with planes or fins. We con- sider firstly the distribution of the pressure on the hull. It is shown that it is the sum of three terms. One is due to the velocity potential when the fluid is quite perfect, that is, when the wake is not taken into account. The two other terms are due to the wake. The first of them is generated by a local Kutta- Joukowsky or gyroscopic effect; the second is due to the partial derivative with respect to the time. A second method, more rapid, gives the total force and moment starting from the absolute momentum of the fluid and from its moment with respect to the fixed axis. In order to do that, we substitute to the vortices a distribution of doublets normal to the hull and to the wake. We show that it is possible to express the difference between the set of forces yielded by the quasi- steady motion theory and the real set of forces in terms of convolution functions: t U ~ | fear (AD) bel =r) de k 0 for the lift, t! 0 ye | fox) W(t! =!) dr! k for the drag, tl 9 es | f(T) b(t <7") dr’ k for the moment; f,,(t’), k=0,1,2, are the arbitrarily given functions (i). Gi) Ga). We call "deficiencies" these differences. In fact, each deficiency is made of two terms, one of them is really a deficiency, because it is due to the fact that the circulation is unable to take instantaneously the value relating to the quasi- steady motion; but the second one which is due to the partial derivative 3/3t, acts in the opposite sense. The three functions ¢,y,¢' are probably the same whatever k may be; but we did not prove that rigorously. Moreover we cannot prove that they are proportional to one another (which is the case for an airfoil of infinite aspect ratio). From a theoretical point of view, something is lacking there but, from a practical point of view, that is without great importance, be- cause these functions may be obtained through experiments. 904 A Vortex Theory for the Maneuvering Ship In par. 6, we consider the forces acting on the planes and fins. We neglect the effect of the history of their own motion. But we take into account the effect of the velocity due to the wake generated by the body itself and show that it acts so as to increase the efficiency of these appendages at the beginning of a maneuver. Paragraph 7 is devoted to the other set of forces acting on the body (fric- tion, gravity, inertia, ...), and par. 8 gives the total expression of the forces when the motion is parallel to the (z,x)-plane. 22.4. Paragraphs 9-12 are devoted to motions not parallel to the (x,y)- plane. In par. 9, we explain the difficulties we have encountered in this task. They are partly due to the fact that in the most general case, the field of vortices may be different from the sum of those which we deal with when the number of degrees of freedom is smaller. For instance, at a given instant t’, perhaps the free vortices are generally shed along a line only and not, simul- taneously, along the two lines which are respectively related to the components of the motion parallel to the (z,x)-plane, and to its components parallel to the (x,y)-plane. Nevertheless, after a discussion, we admit that such an addition is possible in some cases of great importance from a practical point of view, when the perturbations are small. Consequently, we obtain final formula simi- lar to those of par. 8. But it is necessary to consider, that in some cases, par- ticularly when the angle of heel is great, or when the body turns with a small radius of gyration, the nuclei found in the integral expressions of the forces and moments depend not only upon the difference t'-7', but also upon +’ (see jozie, USNs 22.5. In Section II (pars. 14-16), we examine the case of steady and har- monic forced motions in the (z,x)-plane. Such a study leads to consider the differences between the case of the quasi-steady motion theory and the theory developed in the previous paragraphs. In both cases, it is possible to express the lift, the drag and the moment in phase with the motion in terms which are proportional to the square of the re- duced frequency «L/U, and the lift, the drag and the moment outphase with re- spect to the motion, in terms which are proportional to the reduced frequency itself. But, if we use the quasi-steady motion theory, we find that the coeffi- cients before (#L/U)? or oL/U are constants; on the contrary, if we take into account the delayed circulation, they depend upon the reduced frequency. That leads to define ''apparent'' coefficients. Those related to the outphase forces and moments, have their limits, for «L/U->0, equal to the "true" coef- ficients; the other are equal to their true values only for large values of «L/U. Consequently, tests carried out in harmonic forced motions give the pos- sibility to decide whether the effects of the wake are of importance, or may be neglected. Experiments showed that some of the apparent coefficients, those which are not mixed with terms of inertia of the body or with term coming from the rotation of the axis attached to the body, have important relative variations. Experiments show also that the effects of the wake on the stern diving planes and fins are very high. 905 Brard 22.6. In Section II, pars. 17-21, we examine some possible further devel- opments. The equations of motions nearly parallel to the x-axis involve unknown coefficients and functions. The unknown coefficients are those we find in the steady forced motions, the ''ladded masses" and the terms which come from the rotation of the axis attached to the body. The unknown functions are due to the wake; they are the Fourier transforms of the part of the apparent coefficients which depend upon the reduced frequency in the harmonic forced motions. Con- sequently tests in steady and in harmonic forced motion yield, in principle, all the unknown coefficients and functions which are necessary for writing the equa- tions of such motions, although these motions are not harmonic. The equations so obtained are not differential, but integro-differential. Consequently, they are more complicated than the differential equations intro- duced by using the classical static derivatives, that is, the theory of the quasi- steady motions. For the naval architects, this new aspect of the problem is somewhat unpleasant and it would be of interest to check whether the errors from the classical treatment of the problem are great or not. Probably they are not negligible in the transient motions. But, until now, we have had no pos- sibility to compare the two families of solutions. Moreover, some people may consider as negligible differences which are important to the eyes of some others. In any case, we think that the views developed in this paper may explain some interesting particularities of the transient motions, because they call the attention to phenomena which prediction would be impossible according to the classical equations. Even if it is finally found that the differences between the solutions of the classical equations and those of the integro-differential equa- tions are not very high, it is of interest to discern why. From this point of view, we think that harmonic forced motion tests are useful, because the results so obtained lead to understand better how the term coming from the partial derivative may partially cancel those coming from the delayed circulation or inversely. Some points are yet to be emphasized. Firstly, tests in steady and har- monic forced motions require much care, because of the possible free surface effect in an ordinary tank. Moreover, it is possible that the planar motion mechanisms are not perfectly adapted for systematic research about such motions. For instance, the range of the possible amplitudes and frequencies is probably too narrow. For a point of importance would be to study the limits of the linear field. That means that the planar motion mechanisms, which interest has been many times emphasized, do not enable us to solve all the problems involved in the maneuvering qualities of a submerged body. Tests in a steering tank with a rotating arm are certainly necessary in order to explore motions of great amplitude and gyrations at a very large angle of rudder, as was previously the case for researches about the maneuverability of the surface ships. The planar motion mechanisms give new means; but the latter do not replace the previous facilities. It is even allowed to deem that it is necessary to explore the maneu- vering qualities of submerged bodies by using free models as it is already done in the case of surface ships. _ 906 A Vortex Theory for the Maneuvering Ship 22.7. Now we come back to the mathematical model which has been the starting point of the present paper. Obviously such a model could give too many remarks and criticisms. For instance, we consider as a fact that a wake exists, but the real structure of the wake is in connection with the mechanism of the transport into this wake of the vorticity which originates in the boundary layer. Certainly the approximation of the quasi-perfect fluid is not a refined one. The present theory does not stand on the same refined level as the theory of the wings of finite span. Many improvements would be desirable from a sci- entific point of view. But, for practical purposes, we have for the time being to admit semi-empirical theories. The quasi-steady motion theory which until now has been the only one practically used, is also a semi-empirical theory. The most important point in this paper is the following: In practice, have we or have we not to take into consideration the facts disclosed by harmonic forced motions tests? We don't answer this question. But we sincerely hope that it is worth putting. REFERENCES 1. W.E. Cummins, ''The Impulse Response Function and Ship Motions, D.T.M.B. Report 1661, Oct. 1962. 2. Th. von Karman and W. R. Sears, "Airfoil Theory for Non-Uniform Mo- tions,'' Journal of the Aeronautical Sciences, Vol. 5, No. 10, Aug. 1938. 3. BP. Casal, Sur les Qualités Evolutives des Navires" (Thesis, 1950). 4. R. Brard, ''Maneuvering of Ships in Deep Water in Shallow Water and in Canals,"" SNAME, Vol. 59, pp. 229-257, 1951. do. §. B. Spangler, A. H. Sacks, J. N. Nielsen, ''The Effect of Flow Separation from the Hull on the Stability of a High-Speed Submarine," Vidya Report No. 107, Aug. 1963, for O.N.R. and D.T.M.B. 6. E. J. Rodgers, "Vorticity Generation of a Body of Revolution at an Angle of Attack,'' Paper No. 64-FE.S (read before the American Society of Mechani- cal Engineers), Philadelphia, May 1964. 7. M. Sevik, "Lift on an Oscillating Body of Revolution,’ Am. Inst. of Aeron. and Astronautics Journal, pp. 302-306, Feb. 1964. 8. R. Brard, 'Mouvements Plans Non Permanents d'un Profil Déformable," Bulletin de 1'Association Technique Maritime et Aéronautique, Paris, 1963. 907 Brard DISCUSSION Nils H. Norrbin Swedish State Shipbuilding Experimental Tank Goteborg, Sweden Admiral Brard has presented a fine piece of work on the mathematics of a changing system of lifting vortices on bodies in transient and periodic motions, accepting the physical picture of a flow separating along lines more or less parallel to the body axis and producing a downwash over almost the full length and width of the after body. When the body changes its attitude the interference between the vortex wake and after body, and fins, does also change, and this interference must be depend- ent on the time history of the motion. The physical picture also brings with it the concept of a certain time required before a change of boundary flow condi- tions develops into a change of separation and vortex wake, thus complicating the dependence of the history of a transient motion, or of the frequency of a periodic one. The present speaker fails to see to which extent the results of the oscillator experiments quoted by the author are in any quantitative support of this theory. To the speaker again, the flow separation parallel to the axis as mentioned is more associated either with surface ship forms with spontaneous separation, or with bodies of revolution at angles of attack no longer small. For the body of revolution at a small angle of attack, on the contrary, the Nonweiler theory sug- gests separation to occur much further aft and along the contour of a plane almost at right angles to the axis. The vortex wake then covers a narrow region of the after body only, and the circumferential flow will also be more rapidly adjusted to the boundary conditions, thus extending the domain of practically frequency independent derivatives. This might explain why ordinary differential equations with constant coefficients are seemingly sufficient to predict the nor- mal motion of a fair shaped submarine, but it would be interesting to hear of the authors experience of such predictions. DISCUSSION A. J. Vosper Admiralty Experimental Works Gasport, England The mathematical presentation in Admiral Brard's paper is welcomed as a laudable attempt to calculate the forces on a submerged body. This isa 908 A Vortex Theory for the Maneuvering Ship problem which has defeated many people in the past, so that the outcome of the author's work is awaited with interest. I imagine that few will quarrel with the general principle expressed in the paper, that the motion of a ship or submarine depends on the past history of its motion. However, because of the insuperable difficulties involved in any other approach, the use of quasi-static derivatives has been widely accepted as a suitable approximation, since they were first introduced by G. H. Bryan in 1911. Professor W. J. Duncan later attempted to justify the use of quasi-static deriva- tives, and concluded that for the kinds of motion occurring in stability investiga- tions of aircraft flight, the use of constant derivatives was justified. However, he admitted that the influence of the frequency parameter had been neglected, apart from its consideration in the studies of flutter of control surfaces. It is not therefore surprising that in the submarine field, for which the the- ories from the aircraft world were adapted, a quasi-static approach has been used to consider motions well beyond the range of the small deviations for which it was derived. However, one cannot ignore the not inconsiderable argu- ment that in submerged body work good correlation has been achieved between theory and practice. To this extent one can reasonably claim that the end has justified the means. From this point of view, which is all-important to the practising naval architect, the introduction of a considerably more complex representation of the problem seems unnecessary. However, the case of a surface ship in a disturbed sea is entirely different and there may here be greater justification for the author's approach. Comparison of data obtained by rotating arm and planar motion mechanism will undoubtedly help to throw light on this problem, and the I.T.T.C. Maneu- verability Committee by sponsoring a series of international cooperative tests using the ''Mariner"' Class form, will eventually obtain data which may help to answer Admiral Brard's question. Finally, I must admit to some lingering doubts about the basic concept of the planar motion mechanism. If Admiral Brard will permit, I would like to re- phrase Question 1 on page 29: "Is it possible to deduce from the forces and moments measured in a harmonic forced motion, the true forces and moments experienced by a ship or Submarine in a motion which is rarely harmonic ?" REFERENCES 1. G.H. Bryan, "Stability in Aviation,'' MacMillan, 1911. 2. W.J.Duncan, ''Some Notes on Aerodynamic Derivatives,''R. & M. 2115 (1945). 3. W. J. Duncan, "Control and Stability of Aircraft,’ C.U.P. (1952). 909 Brard REPLY TO THE DISCUSSION BY NORRBIN Roger Brard Bassin d'Essais des Carenes de la Marine Paris, France Dr. Norrbin has drawn the attention to the mathematical model which I choose as a Starting point. I agree that the choice is a difficult one and, indeed, I have hesitated for a fairly long time before deciding. That is why I wrote that the NACA model only ''suggests" the physical picture of a surface wake limited by lines more or less parallel to the x-axis. Asa matter of fact, the length and the shape of the arc along which the separation occurs depends strongly upon the hull form. For instance, for a thin surface ship, this arc is practically the keel line and the maximum of the density of the free and bound vortices is located near the bow. For a body of revolution, the arc depends also upon the angle of attack. But it does not seem to me that the final structure of the formulae giving the expression of the forces exerted on the ship or on the submerged body in an unsteady motion strongly depends upon these circumstances. I hope that, in the field of linearity, at least when the body is moving in its centerplane, the forces are always given by convolution functions. The behaviour of these functions may differ when the form of the hull changes, and their calculation should be very intricate. My purpose was only to give means in order to get these functions starting from experimental results and not through mathematical calculations. When the motion is not parallel to the centerplane of the body, the form of the hull is of still greater importance. It is my intention to insist on this ques- tion in the final text of the paper. For instance, in the case of a nonsymmetric body with respect to the (x, y)-plane, strong forces along the z-direction and moment about the x-axis may appear. The intensity of these force and moment strongly depends on the form. Presently, the theory does not permit to predict which hydrodynamic forces are exerted on the body whatever its form may be. But it leads to a method for deducing these forces from these measured in particular motions, the steady and harmonic forced motions. The theory also indicated that the wake has a great influence on the forces acting on the stern planes and fins. Dr. Norrbin said that the surface of the body on which the wake acts is of small area when the body is of revolution and when the angle of attack is small. He expressed the opinion that it might explain why ordinary differential equations with constant coefficients are seemingly sufficient. I have indicated in the paper that, from my point of view, this question is presently not solved. It is quite evident that the coefficients which depend upon the added masses or which con- tain terms due to the rotation of the axis are much less sensitive to the history of the motion than the others. For this reason the history of the motion should act mainly on the lift coefficient due to the angle of drift. 910 A Vortex Theory for the Maneuvering Ship REPLY TO THE DISCUSSION BY VOSPER Roger Brard Bassin d'Essais des Carenes de la Marine Paris, France From a practical point of view, the use of quasi-static derivatives is prob- ably justified for solving the problems concerning the stability of steady mo- tions. Of course, as indicated in Section 5.2 of the paper, the ''equation in s''is different, at least in principle, whether the history of the past motion is taken into account or not. But the stability of the motion depends mainly upon the siens of the real parts of the roots of the equation which are in the vicinity of zero, and the signs seem to be very little affected by the history of the motion. In some cases, it is possible to observe motions of surface ships, such as harmonic variation of the heading, the rudder angle being constant and equal to zero, for which complete explanation seemingly requires consideration of the history of the past motion. In the case of a surface ship, I believe that we generally do not need a very accurate theory to predict the motion of the ship, and, therefore, to introduce in the calculation the effects of the history of the motion. But, I am not sure that these effects are not of importance in the case of a submerged body. You state that in submerged body work a good correlation has been achieved between prac- tice and theory (that is the classical theory, without correction for taking into account the vortex wake generated by the body). I personally have no knowledge of results of comparisons between theory and experiments on models or on full- scale submerged bodies, which permit to conclude in a way or in the other. That iS why I should be very grateful to you if you could give me more precise infor- mations about this point. I was surprised to find that our experiments carried out with the Planar Motion Mechanism show a great influence of the reduced frequency. Before getting these results, I considered the phenomenon as possible, but Idid not be- lieve that it could be of such great importance. However, I should like to re- mind you of the fact that the coefficients are not equally affected. It would be interesting to compare calculated motions with constant coefficients and with variable coefficients. But I have had no time to do it yet. I have also some doubts about the possibility to deduce the time forces and moments acting on a ship or on a submarine from the forces measured in har- monic motion by use of a planar motion mechanism. But, perhaps the reasons behind our lines of thought are not identical. You seem to consider that your doubts are justified by the fact that the actual motion of a surface ship or a sub- marine is seldom harmonic; I rather consider that the actual motions of a ship Or a Submarine are often outside the linear field, and therefore, that the inver- sion of the Fourier integral becomes either impossible or meaningless. Ot Pel a Pani ve Re a Hane nvnatntiel it ie math nie a en nen ey ie * vie esi hres an wat weaons Bis & BAw " . . f i oY 7 Cipite: Ach ty Vy , ee bol a Le Ne ‘ 1 7 : ' } “0 AY j meet “ nie A AIA oy ola oar i ert pe eh ie SAS ; 5 ; ¥ ‘ ¥ 4 ae ee its Ab: Stats Oran a -eed ea ei7ty ire nee en GR Raa rir ¥ me sie oe} a hat Lay nt y 2 ‘ Pe Ae su i ie i i ‘ ; i \ € ee - ‘ ‘ x i ; A Snes r er Carer me Rall Pa eee eves fontantng j Verey sf = eR Cg Cee te Fie “y 5 PLAS ‘eu? iout (A 4 i S/R TEAS. 4 Lo A { i A 2 eee , i ¥ a % fag : v sa Uh i f + ry Y pi 2 rf a) Jr {ced ‘ h 8 ard yeaee ns i We Sh i pint ; a f ; sia ag i rag ae r ' Fy Mf e : i TEN rth iT e yy Hs pare ps (hh = ‘1 i Rp . ees has rey a tneee) { vo eas eel phi 2 ” Re al J ees eid } eo! oe awe <> : SAY 6 trie fyb! ' ree ; ti wy sor i { i Avy Sava i rt! : f; ar Gaceetree) etic A ee, OLED Le) r i. wipe et, han yh Lari eel wa Fi arte wei 0 ate thy ott mae iy abet Le Armee Saturday, September 12, 1964 Morning Session DRAG REDUCTION Chairman: C. Prohaska Hydro-Og Aerodynamisk Laboratorium Lyngby, Denmark Page The Reduction of Skin Friction Drag SS J. L. Lumley, The Pennsylvania State University, University Park, Pennsylvania The Effect of Additives on Fluid Friction 947 J. W. Hoyt and A. G. Fabula, U.S. Naval Ordnance Test Station, Pasadena, California An Experimental Investigation of the Effect of Additives 975 Injected into the Boundary Layer of an Underwater Body W. M. Vogel and A. M. Patterson, Pacific Naval Laboratory, Victoria, British Columbia, Canada An Experimental Study of Drag Reduction by Suction Through 1001 Circumferential Slots on a Buoyantly-Propelled, Axi- Symmetric Body Barnes W. McCormick, Jr., The Pennsylvania State University, University Park, Pennsylvania 221-249 O - 66 - 59 913 Ee apeteri sig oe eRe nee sel youll | if MON ONGIR OANA THE REDUCTION OF SKIN FRICTION DRAG J. L. Lumley The Pennsylvania State University University Park, Pennsylvania ABSTRACT A survey andanalysis is presented of the various principles which have been suggested to reduce the skin friction drag; a description of some of the techniques for the application of these principles and experimen- tal results are given. INTRODUCTION The majority of the drag of a properly streamlined underwater vehicle is skin friction drag resulting from the excessive momentum transport of the tur- bulent boundary layer. All techniques which have been suggested for the reduc- tion of skin friction drag act to reduce this transport by altering, or preventing the formation of, the turbulent boundary layer. Few of the techniques which we will describe are supported by complimentary experimental and theoretical in- formation; for some, only theory exists; for others only experiment; for a few, there is both, but in conflict. I will try to present here the principles so far as they are known, the results where they are available, and attempt to explain the discrepancies. CONVENTIONAL TECHNIQUES General Considerations Most of the drag reduction techniques which have been suggested involve the stabilization of the laminar boundary layer, and these will be referred to as con- ventional techniques. The boundary layers in question are always thin relative to some relevant length, and are usually considered, for an examination of stability, as plane parallel flows without inflection points. In the discussion of the various stabilization techniques which follows, sev- eral things must be borne in mind. First, from the work of Klebanoff, Tidstrom and Sargent (1962) it is clear that transition can be caused in a laminar boundary layer at any Reynolds number by a sufficiently violent disturbance—of the order Oils Lumley of ten percent of the free stream velocity. It must be anticipated that a laminar boundary layer can be successfully stabilized only in the absence of large dis- turbances, since once transition has occurred, few stabilization techniques could be expected to have the capacity to reestablish laminar flow downstream of a dis- turbance.“ Disturbances appear either at the boundary, or in the free stream; consequently, great care must usually be exercised to make both as free of dis- turbances as possible. The only kind of stabilization that appears to be possible, then, is stabilization to small disturbances; that is, the preventing of small dis- turbances from growing to be big ones. This is what is customarily meant by stabilization. There are, generally, two types of small disturbances to which laminar boundary layers are unstable. One consists of progressive waves; these are known as Tollmien-Schlichting waves (Lin (1955)). The other consists of stream- wise standing vortices; these are known as Taylor-Goertler vortices (see Lin (1955), p. 96). The Taylor-Goertler type of instability only appears where there is concave curvature in the streamwise direction or where a surface is heated with a liquid flow above it in a gravitational field (Goertler (1959)). However, the condition on the curvature in order to assure the appearance of Tollmien- Schlichting waves before Taylor-Goertler ones is quite stringent, and probably few supposedly flat plates satisfy it. Practically without exception, the analyses which indicate a possibility of stabilization have reference only to Tollmien- Schlichting waves. In addition, most of these analyses have reference only to progressive waves in the streamwise direction. While for the ordinary boundary layer Squire's theorem (Lin 1955)) assures us that such waves become unstable first, in some of the situations under discussion, we may not have such assur- ance. While nearly all of the suggested techniques attempt to control the growth rate of the streamwise progressive waves, at least one (Kramer (1962a)) attempts to prevent the development of three-dimensionality, which appears to be (Kleb- anoff, Tidstrom & Sargent (1962)) a necessary prelude to transition, while another (Kramer (1962b)) attempts to control what appears to be a secondary instability associated with the developing three-dimensionality (Klebanoff, Tidstrom & Sar- gent (1962)). There are distinct differences between discussions of stability on two di- mensional bodies and on bodies of revolution. If the diameter of the body is in- creasing, two conflicting effects are felt. In the first, an increase in diameter means that the boundary layer must be spread over an ever widening area, pro- moting thinning and altering the profile (much as suction does). It might be ex- pected that this would delay instability beyond the point to which it is already delayed by the favorable pressure gradient usually present on the forward part of a body of revolution. In the second, cross-stream vorticity is being stretched, which, due to the associated increase in intensity, should result in an earlier occurrence of Tollmien-Schlichting instability. There is evidence (Groth (1957)) to indicate that the stretching dominates. The picture is complicated further, how- ever, by the possibility of Taylor-Goertler instabilities in the concave flow near the stagnation point (Goertler (1955), Goertler-Witting (1958)), and by the stretch- ing (and intensification) of vorticity which may be present in the free stream. *Although probably most will reestablish laminar flow if the disturbance is removed. , 916 The Reduction of Skin Friction Drag Finally, it should be mentioned that, even if the boundary layer can be sta- bilized in the absence of large disturbances, the wake cannot. The turbulent wake is known (Townsend (1956)) to be subject to large-scale, unsteady organized motions of the character of instabilities, which on a sphere interact strongly with the boundary layer and are responsible for the wandering of a rising free balloon of small size (Scoggins-private communication). It is possible that these motions, which are present in the wake of a streamlined body also, can disturb the supposedly stabilized boundary layer there. Change of Profile Of the various stabilization techniques* (see Fig. 1), the first method we will discuss is the alteration of the velocity profile to a more stable one. Roughly speaking, the stability of a profile is increased by an increase of the curvature of the profile, since the lower critical Reynolds number above which small dis- turbances will grow is monotonic with curvature of the profile at the critical layer. The critical layer is that layer at which the wave velocity and fluid ve- locities are equal (Lin (1955)). A more exact way of correlating this change in profile is through a shape parameter. DRAG REDUCTION IN LIQUIDS TO THE TURBULENT LAYER STABILIZATION OF LAMINAR LAYER TO SMALL DISTURBANCES COMPLIANT | | VISCO-ELASTIC INCREASING BOUNDARY NON-NEWTONIAN! |CURVATURE OF ADDITIVE PROFILE CONSTANT VARIABLE FLUID FLUID PROPERTIES PROPERTIES PRESSURE SUCTION HEATING NON — WALL LAYER GRADIENT WALL NEWTONIAN OF ADDITIVE LOW pz FLUID DISTRIBUTED DISCRETE INJECTION FILM CAVITATION SUBLIMATION CHEMICAL BOILING REACTION Pass | Fig. 1 - Techniques for the stabilization of the laminar layer to small disturbances *Specific citations will not in general be given; reference should be made to the appropriate section of the bibliography. 917 H = 8%/@ Fig. 2 - Effect of profile change expressed in terms of the ratio of displacement thickness to momentum thickness. (from Lin (1955)). Figure 2 shows the lower critical Reynolds number versus a shape param- eter, the ratio of displacement to momentum thickness. This parameter assumes the value unity for a "Square" profile, and increases as the rise to free stream velocity becomes more gentle, reaching a value of roughly 3.5 at separation. While the curve in Fig. 2 was computed specifically for profiles in the presence of pressure gradients and heat transfer at the surface, it is only a slight gen- eralization to speculate that the same curve will describe, at least qualitatively, the effect of other conditions which work principally through a change in profile. In describing these stabilization methods, it should be remembered that some of them, especially suction, in addition to changing the profile (in a direc- tion to increase the lower critical Reynolds number) may also prevent boundary layer growth, if applied with sufficient intensity. Thus a boundary layer per- mitted to grow will eventually reach its critical Reynolds number (based on 918 The Reduction of Skin Friction Drag thickness) no matter how delayed by a change in shape. A boundary layer whose growth is prevented may never reach its critical Reynolds number. The ways in which the profile can be altered can be placed in two categories, depending on whether constant or variable fluid properties are necessary. Under the heading of methods which work with constant fluid properties we can include pressure gradients and suction. The suction may be either distributed, or it may be through discrete slots (see particularly the work of Pfenninger et.al.). Dis- crete slots are satisfactory so long as the boundary layer is caught by the next downstream slot before disturbances have time to grow to a significant extent. Among the methods that involve variable fluid properties, most are dependent on a variation of the ordinary viscosity 1. An increase of » with distance from the wall increases the curvature. The viscosity » can be varied in several ways: in water it can be changed by heating the wall, a film of a different fluid can be placed next to the wall, such as a gas film or a liquid with a lower » —such a film being produced by injection, film boiling, cavitation, sublimation or chemical reaction. Finally, an additive could be placed in the boundary layer so that the fluid becomes non-Newtonian, in particular ''shear-thinning"; then the high shear near the wall will mean a lower vu there and the » will increase with distance from the wall. It should be mentioned that there is some disagreement as to whether the low » fluid film should be considered primarily as a stabilization technique; this seems to be largely a matter of taste, and I have taken the posi- tion that if it did not stabilize, it would not work, since the low » fluid would be mixed with the high. Flexible Boundary To the best of my knowledge there are only two methods that do not depend on changing the profile; the first of these is the stabilization of the laminar boundary layer by a compliant boundary. This does not damp the disturbances; as a matter of fact, it is a result of the theory (Betchov (1959), Benjamin (1960) Boggs & Tokita (1960), Landahl (1962)) that damping in the wall is in general destabilizing. Rather, the compliant boundary acts to change the phase rela- tions between the pressure and the velocity in the neighborhood of the wall, re- sulting in an alteration of the Reynolds stresses there, and changing the energy budget of a disturbance. While a passive wall in general changes the lower critical Reynolds number, Betchov (1958) has shown that an active wall may be expected to eliminate it entirely. In a tenuously related investigation Wu (1959), has shown that a suitable active wall can propel. In Fig. 3 are shown the phase relations induced at the surface by a visco- elastic material, together with the phase relations corresponding to a small dis- turbance in the region between the inner viscous layer and the critical layer of the laminar boundary layer over a rigid surface. If the former are added to the latter as a first order approximation, the influence on the disturbance Reynolds stress may be seen. 919 Lumley SPRINGY LOSSY LOSSY GAINY OOWNSTREAM UPSTREAM MASSY GAINY BOUNDARY CONDITIONS FOR SEMI — INFINITE LINEAR VISCO-—ELASTIC SOLID LOSSY DOWNSTREAM UPSTREAM GAINY 6 SPRINGY LOSSY | GAINY — iV] P ov MASSY PHASE RELATIONS PHASE RELATIONS IN IN VISCOUS REGION VISCOUS REGION OVER OVER RIGID SURFACE FLEXIBLE WALL SPRINGY GOOD MASSY BAD LATERAL RESPONSE BAD Fig. 3 - Small disturbance phase relations in the laminar boundary layer between the critical layer and the inner viscous layer: first order modifica- tion by flexible wall. Non-Newtonian Additive The second method not dependent on a change of profile (Giles 1964) depends on the use of a non-Newtonian additive of viscoelastic character. One may ex- pect that if the apparent viscosity to a temporally sinusoidal simple shear in- creases with frequency, then the flow would be more stable to progressive waves, since the history of a material point involved in such a wave is unsteady. The opposite case is of greater interest for real fluids, and a recent analysis (Wen (1963)) indicates a destabilizing effect, but that may be because a model was used that is not materially objective. Drawbacks of the Conventional Techniques Most of these techniques are, or have been, under experimental investigation by various groups and individuals and show some prospects of success, but in 920 The Reduction of Skin Friction Drag most there are difficulties. Many of these difficulties are related to kinds of in- stability other than those considered in the analysis which suggested the experi- ment. For instance, with a gas film one has an interfacial instability of the Kelvin-Helmholtz (Lamb (1945)) type. With a heated wall one has a gravitational instability due to the density differences, which can be shown to be analogous to the instability on a wall concave in the streamwise direction (Goertler (1959), Kirchgaessner (1962)). The boundary layer over a flexible surface is subject to two types of instability not present in the boundary layer over a rigid surface (Benjamin (1963)). Furthermore, the difficulties mentioned earlier relative to freedom from disturbances, both at the surface and in the free stream, are not easily overcome. It should be remembered that natural transition due to the growth of small disturbances seldom occurs earlier than a length Reynolds number of 10°. Simply by removing all the disturbances this figure can be in- creased by a factor of about twenty-five, but a limit is reached in this direction. It has been suggested by Betchov (1960) that this limit is due to amplified mole- cular agitation. To achieve a substantial reduction in drag, the length Reynolds number must be increased at least an order of magnitude beyond this. Further- more, at these high Reynolds numbers the laminar boundary layer is very thin. The requirements on smoothness and in general the tolerances on construction of the surface are proportional to the inverse of the ''Reynolds number per foot,"' and are extremely stringent. If all other disturbances are removed, the velocity field associated with a sound field can disturb the boundary layer, particularly in a gas-liquid combination. This velocity field in a liquid is ordinarily much smaller for a given sound pressure level than in a gas (by the ratio of the values of the product of density and speed of sound), but if there is a gas-liquid inter- face this does not appear to be true. Considered from all points of view it seems desirable to examine the possibility of altering a turbulent boundary layer so as to reduce the drag. If this can be done then all the difficulties mentioned above are eliminated. NONCONVENTIONAL TECHNIQUES General Considerations Several approaches have been suggested by means of which the turbulent boundary layer may be altered. In order to understand how these may work, it is necessary to recall to mind the physical principles which govern the normal turbulent boundary layer. For simplicity, let us consider the boundary layer with zero pressure gradient. These principles are (cf, Townsend (1956)): 1. Reynolds number similarity: that the turbulence, once fully established, is predominantly inertial in the energy containing range (that part of the spec- trum responsible for drag and heat transfer); i.e.,—that the structure of these eddies is essentially independent of viscosity. 2. The ''Law of the Wall''—that there is a layer of turbulent fluid near the wall that has no characteristic length scale other than distance to the wall, and that this layer has a single characteristic velocity, and therefore a universal structure. By the first principle, this layer is independent of viscosity, so that the Reynolds stress is constant. Defining the characteristic velocity .* as the 921 Lumley root of the kinematic Reynolds stress, and noting that mean velocity differences in the layer also must scale with »*, we have yu’ /u*=1/K a universal constant, which gives immediately the familiar logarithmic law p/* = 1/K In y/y, , where y, is a constant yet to be determined. 3. The viscous sublayer—that there is a layer of fluid next to the wall in which dissipation is dominant, in the sense that no disturbance can be in equi- librium there without energy transfer into the layer. The profile of mean ve- locity there is nearly linear, since the stress is constant, and production of tur- bulent energy is not important. Phenomena seem superposable in this region (Sternberg (1961)) since the nonlinear convective-production terms are not sig- nificant. The thickness of the layer is fixed by the Reynolds number based on thickness. If we set R= y* »*/v as the Reynolds number based on thickness where the sublayer profile, i/* = yu*/v, meets the logarithmic profile (roughly 12.6 in a normal boundary layer) then we can write ra YE fi ie R- — InR = Hse = =P In 4. "Law of the Wake''—in the outer part of the layer, it is assumed that the profile is similar when referred to local length and velocity scales— [1 - U]/p*= f(y/8) which of course also involves Reynolds number similarity. If it is as- sumed that there is a region of overlap with the law of the wall, then we obtain the familiar drag law This is the relationship which must be changed if the effect on drag of the tur- bulent boundary layer is to be changed. Change of Viscosity Let us now consider ways in which the familiar drag relationship can be changed (see Fig. 4). The simplest which comes to mind is a change of viscosity. This will not change the structure outside the viscous sublayer, since that was dominated by inertia. Therefore it does not matter whether the change in vis- cosity extends to the fluid outside the sublayer. A change in viscosity will not change the value of R so long as the change is uniform in the sublayer. Hence, any mechanism which changes the viscosity in the viscous sublayer will produce a turbulent boundary layer indistinguishable from a normal turbulent boundary layer at a different length Reynolds number. Since drag is only a weak function of length Reynolds number, this is not a particularly effective way to change drag. The viscosity in the viscous sublayer might be reduced by heating the wall (in a liquid). 922 The Reduction of Skin Friction Drag ORAG REDUCTION IN LIQUIDS TO THE LAMINAR LAYER ORAG REDUCTION WITH TURBULENT LAYER SAME STRUCTURE CHANGE STRUCTURE CHANGE VISCOSITY ONLY BEATING: WALL VIOLATE WALL VIOLATE REYNOLDS CHANGE SUBLAYER SIMILARITY NUMBER SIMILARITY THICKNESS REYNOLDS NUMBER COMPLIANT PARTICLES VISCO—ELASTIC PARTICLES] | COMPLIANT]| vISCO-ELASTIC ADDITIVE ADDITIVE PARTICLES & FIBERS Fig. 4 - Techniques to alter the structure of the turbulent boundary layer The Nonuniform Sublayer Some question may be raised about the possibility of having a nonuniform value of » through the sublayer; it is not difficult to show that for a non- Newtonian fluid of arbitrary constitutive equation in a zero-pressure gradient boundary-layer the viscous shear stress near the wall is constant to terms of third degree in distance from the wall, so that there will be a region of uniform strain rate and hence uniform viscosity. Relations will, of course, not be simple at the outer edge of the sublayer, but it seems unlikely that the picture developed above, which ignores this transition region between sublayer and inertial region (on the grounds that the transition is in a thin layer (Townsend (1956)) can be far wrong. A hot wall (in a liquid) can produce a temperature (and hence viscosity) variation in the sublayer, if the heat flux is large enough, and a mechanism by which this could reduce drag has been suggested by G. B. Schubauer (private communication). The ratio of the temperature drop in the sublayer to that through the boundary layer is given to first order by (using a two-layer model) (AT) oRu*/U0 sublayer (Al etindesy layer Peake (ea) /U At moderate Reynolds numbers (in liquids), most of this drop takes place in the sublayer, and we may take the temperature (and hence the viscosity) at the outer 923 Lumley edge of the sublayer as being essentially the free stream value. Then the shear at the outer edge will be nearly that without heating at the same wall stress. If we take the thickness Reynolds number as being determined largely by the shear at the outer edge, then this will be essentially the same (although it may increase somewhat due to the favorable curvature of the profile) so that the thickness will be essentially the same. Thus the whole effect will appear from outside the sub- layer as a Slip at the wall of value (to first order) Ryu* Av 5 ; Bas Assuming self-preservation, negligible laminar length, and large Reynolds num- ber, we may obtain the effect on * of a change in » by this mechanism: yw Cu jw l® we ae Do which is of the order of 1/4 at moderate Reynolds number. The sensitivity of viscosity to temperature in liquids suggests that (at ordinary pressures) changes in «* by 20% may be possible before boiling occurs. It should be mentioned in passing that surface heating in a gravitational field may produce secondary motions which will only increase the momentum trans- port and the drag. Change in the Wall Layer A slightly more sophisticated way in which the principles outlined above could be violated is by a change in the "law of the wall.'' This could be done by the introduction either of a length scale or of a velocity scale. These are essen- tially equivalent, since a height can be defined at which the mean velocity equals the velocity scale selected. Thus a new parameter is introduced, say the ratio of «* to the new velocity scale. A simple way in which this can be done is by coating the wall with a nonrigid material having a Rayleigh wave speed below the free-stream speed. Then convected fluctuating pressure fields can exchange energy with the wall in the same manner as described by Phillips (1955) for the generation of ocean surface waves by turbulent wind. It is not obvious a priori why such an interchange should necessarily result in a reduction of drag. The random wave motion of the surface would necessarily be associated with dissi- pation of energy in the surface so that the simple existence of such an interaction would only increase the total dissipation, if it did not drastically alter the struc- ture of the boundary layer so as to reduce the dissipation in the fluid. Again, we have, as before for the laminar layer, that damping in the wall material is prob- ably detrimental, and it seems likely that we will not achieve favorable effects unless the damping in the surface material is considerably smaller than that in the fluid. This is the case for an air boundary layer over water, and P. A. Shepphard (private communication) has observed drag reduction in such bound- ary layers. Unfortunately, it is more difficult to find wall materials of viscosity lower than water. 924 The Reduction of Skin Friction Drag Reynolds Number Similarity Another way in which the boundary layer may be attacked is through the principle of ''Reynolds number similarity.'' Violating this principle is not a straight-forward matter. For instance, if the fluid viscosity is increased, there will be no important change (other than the slow increase in drag associated with the weak dependence on length Reynolds number) until the dissipative and energy- containing scales are nearly equal, at which point the turbulence can no longer extract energy from the mean motion at a sufficient rate to maintain itself, and the flow will become laminar. This would, of course, result in a drag reduction, but falls more properly in the realm of stabilization. One might suggest using a non-Newtonian medium which is shear-thinning. If indeed it behaved as though it had a simple shear-dependent viscosity (Lumley (1964)) it would change noth- ing. In the high-shear viscous sublayer, its viscosity might be expected to be nearly the value of the solvent; in any event, R would remain unchanged. If the flow outside the sublayer were turbulent, then it would be inertia dominated, and nothing would be changed. Only by increasing the effective viscosity outside the Sublayer until the layer became laminar could a change be made, and again this falls more properly under the heading of stabilization. Evidently, in order to in- fluence Reynolds number similarity, it is necessary to have a material whose constitutive equation is such that terms in the energy equation, arising from that part of the stress which is not a pressure, are appreciable in the energy contain- ing range of wave numbers, without being dissipative in character, so as not to turn the turbulence off. That is, they must be non-negligible in the energy con- taining range of wave numbers without being viscous in character. There is both theoretical (Lumley (1964)) and experimental (see particularly Fabula (1963)) support for the conclusion that only a material having viscoelastic properties can behave in this manner, although the exact mechanism is not understood. Particles and Fibers There has been reliable observation of drag reduction in flows containing particles and fibers. Although this effect is described as ''damping"' the turbu- lence, the intensities are observed to increase (Elata, Ippen (1961)). From the principle of Reynolds number similarity, we know that a simple change in the mechanism of dissipation, so long as the flow remained turbulent, would be un- likely to change the turbulent structure, since this is determined by inertia. There is a known interaction of suspended particles with the viscous sublayer, which will be described below, but if the observed drag reduction does not arise from this source, then it seems likely that it is due to a violation of Reynolds number similarity by the introduction of other length and time scales. Depend- ing on the ratios of these scales to others in the flow, this may also be regarded as a violation of the law of the wall, of course, since particles having relatively small length or time scales may leave the outer part of the flow unaffected, be- ginning to exert an influence only as the scales of the energy containing eddies shrink to corresponding size as the wall is approached. Length scales may be introduced in a very direct way by long fibers, while velocity scales may be in- troduced by the settling velocity (in a gravitational field), and time scales by the characteristic time of the particles (the response time to a step function in rela- tive velocity). The mechanism associated with this latter may be similar to 925 Lumley viscoelasticity, since a particle of long characteristic time in a flow of short time scale will tend to remain motionless as the flow sloshes past it. Thus an unsteady fluid motion will be more dissipative than a steady one, as in a visco- elastic fluid having an effective viscosity increasing with the frequency of a temporally sinusoidal simple shear.* The particles will tend to store energy associated with steady, organized motions (steady from a Lagrangian viewpoint) and to oppose unsteady motion. This was probably first mentioned by Saffman (1962). See also Hino (1963). The presence of particles, or colloidal suspensions, can of course, be even more effective if the suspended material tends to combine with itself to form elastic structures capable of resisting small shear. This is possible with Bentonite, and may explain observations in flocculated thoria (Eissenberg and Bogue (1963)) and in flows of fine aqueous suspensions of wax-laden oil droplets. The suspended material then behaves somewhat as a Bingham- Plastic and need not depend on a long time constant to make unsteady motions of the fluid more dissipative than steady ones at low shear. Changing the Sublayer Finally, we may change the boundary layer by changing R. The effect of a small change inR at constant Ux/v is given by obtained by differentiating the expression for drag, assuming self-preservation, negligible laminar length, and indefinitely large Reynolds number. At the value of R associated with the normal turbulent boundary layer, this is negative, and of the order of one half. R may be changed in a number of ways. If a viscoelastic medium is used, the effective viscosity of which in a temporally sinusoidal simple shear increases with frequency, we may expect that a disturbance which is unsteady (from the Lagrangian viewpoint) will be more dissipative than would be indicated by the viscosity at the steady shear rate. Since R (based on the steady state viscosity) is determined by that thickness below which all disturbances must import energy, we may expect R to be increased.* Ina similar way, particles may be intro- duced in the sublayer. If their time scale is large they also will make unsteady motions more dissipative and thus increase R. If they can form elastic struc- tures, like flocculated thoria, (Eissenberg & Bogue (1963)), the effect is even more pronounced. If the time and length scales are such that the energy con- taining eddies in the turbulent flow outside the sublayer are unaffected, then the familiar “law of the wall" will remain, K will be unaffected, but the logarithmic part of the profile will be displaced upward. This effect is illustrated by Fig. 5, the mean velocity profile in a flow containing a low concentration of flocculated thoria, reproduced from Eissenberg and Bogue (1963). *But real viscoelastic media appear to display the opposite behavior. 926 The Reduction of Skin Friction Drag Fig. 5 - Velocity profile in the wall layer in flow of flocculated thoria, from Eissenberg and Bogue (1963). Nondimensionalization by shear velocity with small empirical correction c. Solid line is Newtonian pro- file. Another method of changing R, suggested by G. F. Wislicenus (private com- munication) is to change the boundary condition, by making the surface flexible. Again, the action of such a surface depends in a detailed way on changing the phase relationships, and thus the Reynolds stress. To make this distinct from the violation of the law of the wall mentioned above, we must have the wave speed in the wall well above the free-stream speed. A detailed analysis based on energy considerations (Lumley and McMahon (1964)) shows that the situation is rather complicated, due to the fact that, although over a rigid wall no small disturbance can extract energy from a linear profile fast enough to maintain it- self, while some large ones can, this is no longer true over certain flexible walls. Thus, while the wali changes the energy budget of large disturbances, it also provides a mechanism* by which small disturbances can extract energy. In Fig. 6 are shown the phase relationships calculated for small disturbances. It can be seen that, for this wall material, there is always a wave whose speed is such that it can extract energy. Evidently only a wall which is prevented from moving laterally is worth examining. Conclusions This outline has surely not exhausted the possibilities of changing (or elim- inating) momentum transport in a turbulent boundary layer. For example, we have not discussed the possibility of influencing transition by oscillations of the *Similar to Rayleigh wave propagation inthe wall--the class B waves of Benjamin (1963). 927 Lumley SPRINGY LOSSY ©) @ LOSSY GAINY DOWNSTREAM UPSTREAM MASSY GAINY BOUNDARY CONDITIONS FOR SEMI — INFINITE LINEAR VISCO-ELASTIC SOLID DOWN ' A STREAM | U <=, -— | ne ae GAINY rer ip cia c> SPRINGY A v A Vv v LOSSY <@—'—_® GAINY PHASE RELATIONS IN BOUNDARY LAYER OVER RIGID SURFACE siasey BOUNDARY LAYER WITH FLEXIBLE WALL SPRINGY /GOOD MASSY/BAD LOSSY /BAD GAINY/GOOD LATERAL RESPONSE BAD Fig. 6 - Small disturbance phase relations in a vis- cous region near the wall: first-order modification by flexible wall. surface (Miller) & Fejer (1964)).* Detailed, qualitative experimental data on any technique is relatively sparse, aS Sparse, Say, aS equally detailed theory. Prob- ably most is known about and greatest success has been achieved with suction through slots and the Toms phenomenon. The mechanism of the former is clear, though the mechanism of the latter is far from being so. If another speculation may be added to a growing list, it seems quite possible that we may learn more about the ordinary turbulent boundary layer by examining the effects of various changes; it is at least clear that there are interesting areas here for investiga- tion. *Nor have we discussed blowing, and other means of artifically thickening the turbulent boundary layer, since it does not seem obvious that one can recover the work done to thicken the layer. 928 The Reduction of Skin Friction Drag ACKNOWLEDGMENTS Many of the ideas embodied in this paper have grown from, and benefitted by, discussions with workers in the field; it is a pleasure to acknowledge this necessarily rather diffuse debt, although I take full responsibility for the form of the ideas as presented here. Iam particularly grateful to my colleagues Drs. Hoult and Margolis for numerous comments and suggestions. The bibli- ography is not intended to be exhaustive, and represents a very personal selec- tion from among those papers of which I was aware, and those which were brought to my attention by workers in this area; for this latter I am grateful. I am grateful to the author involved and to the Cambridge University Press and the American Institute of Chemical Engineers for permission to reproduce fig- ures two and six respectively. REFERENCES 1. General Betchov, R. B. (1959), ''Simplified Analysis of Boundary Layer Oscillation," Rept. No. E529174, Douglas Aircraft Co. Betchov, R. B. (1960), ''Thermal Agitation in Turbulence," STL/TR-60-0000 AE 279, Physical Research Lab., Space Technology Lab., Inc. Brown, W. B. (1955), "Extension of the Exact Solution of the Orr-Sommerfeld Stability Equation to Reynolds Numbers of 4000,"" Northrop Corp., Norair Div., R NAI-58-73 (BLC-78). Brown, W. B. (1959), ''Numerical Calculation of the Stability of Cross- Flow Profiles in Laminar Boundary Layers on a Rotating Disc and on a Sweptback Wing and an Exact Calculation of the Stability of the Blasius Profile,’ Northrop Corp., Norair Div., R NAI-59-5 (BLC-117). Granville, P. S. (1963), "Effect of Fluid Injection on Drag of Flat Plates at High Reynolds Number," Inst. Shipbldg. Progress, V. 10, No. 101, p. 30-33, Jan. 1963. Gray, W. E. and Rullam, P. (1950), ''Comparison of Flight and Tunnel Meas- urements of Transition on a Highly Finished Wing (King Cobra) RAE TN 2383." Goertler, H. (1955), ''Dreidimensionale Instabilitant der ebenen Staupunkt- stroemung gegenueber wirbelartigen Stoerungen, 50 Jahre Grenzschichtfor- schung,"' (H. Goertler, W. Tollmien, ed.) Vieweg, Braunschweig. p. 304. Goertler, H. and Witting, H. (1958), ''Theorie der sekundaeren instabilitaet der laminaren Grenzschichten, Boundary Layer Research,'' (H. Goertler, ed.) Springer-Verlag, Berlin, p. 77. 221-249 O - 66 - 60 929 Lumley Goertler, H. (1959), "Ueber eine Analogie zwischen der Instabilitaeten laminarer Grenzschicht-Stroemungen an konkoven Waenden und an erwarmten Waenden,"' Ingeneur - Archiv 28 (Grammel Anniversary Issue) p. 71. Groth, E. E. (1957), ''Boundary Layer Transition on Bodies of Revolution," Northrop Corp., Norair Div., R NAI-57-1162 (BLC-100). Klebanoff, P. S., Tidstrom, K. D. and Sargent, L. M. (1961), The Three- Dimensional Nature of Boundary Layer Instability," J. Fluid Mech. 12. Kirchgaessner, K. (1962), ''Einige Beispiele Zur Stabilitaets Theorie von Stroemungen an konkovenund erwarmten waender,'' Ingenienr-Archiv 31, p. 115. Kuethe, A. M. (1957), "On the Stability of Flow in the Boundary Layer Near the Nose of a Blunt Body,"' Rand Corp., R. RM-1972. Kulin, G. and Pao, Y. H. (1961), "Boundary Layer Stability on Blunt Nosed Bodies of Revolution,"’ TROO7-1, Hydronautics, Inc. Lamb, H. (1945), ''Hydrodynamics,"' Dover Publications, New York. Lin, C. C. (1955), ''The Theory of Hydrodynamic Stability," Cambridge, The University Press. Miller, J. A. and Fejer, A. A. (1964), ''Transition Phenomena In Oscillating Boundary Layer Flows," Journal of Fluid Mechanics, 18, p. 438-448. Raetz, G. S. (1957), "A Method of Calculating Three-Dimensional Laminar Boundary Layers of Steady Compressible Flows,"' Northrop Corp., Norair Div., R. NAI-58-73 (BLC-114). Siegel, R. and Shapiro, A. H. (1953), ''The Effect of Heating on Boundary Layer Transition for Liquid Flow in a Tube,"’ ASME Paper No. 53-A-178. Sternberg, J. (1962), ''A Theory for the Viscous Sublayer of a Turbulent Flow," J. Fluid Mech., 13, p. 241. Townsend, A. A. (1956), "The Structure of Turbulent Shear Flow,'' Cam- bridge, The University Press. 2. Suction Anon (1952), ''Experiments on Distributed Suction Through a Rough Porous Surface,'' Cambridge Aero. Lab., Aero. Res. Council, C. P. 84. Bacon, J. W., Jr., et. al. (1959), ''Experiments on a 30° Swept 12% Thick Symmetrical Laminar Suction Wing in the 5 ft. by 7 ft. University of Michigan Tunnel,'' Northrop Corp., Norair Div., R. NOR-59-328 (BLC-119). 930 The Reduction of Skin Friction Drag Braslow, Burrows, Tetervin, Visconti (1951), "Studies of Area Suction for the Control of the Laminar Boundary Layer on an NACA G4A010 Airfoil,'’ NACA R 1025. Burrows, D. L. and Schwartzberg, M. A. (1952), ''Experimental Investigation of an NACA G4A010 Airfoil Section with 46 Suction Slots on Each Surface for Con- trol of Laminar Boundary Layer,'"’ NACA TN 2644. Carmichael, B. et.al. (1957), ''Low Drag Boundary Layer Suction Experi- ments in Flight on the Wing Glove of an F-94A Airplane. Phase IV - Suction Through 81 Slots Between 8% and 95% Chord,'' Northrop Corp., Norair Div., R. NAI-57-1025 (BLC-102). Dannenberg, R. E. and Wesberg, J. A. (1952), ''Section Characteristics of a 10.5% Thick Airfoil With Area Suction as Affected by Chordwise Distribution of Permeability,"", NACA TN E847. Gault, S. E. (1958), "An Experimental Investigation of Boundary Layer Con- trol for Drag Reduction of a Swept Wing Section at Low Speed and High Reynolds Number,'' NASA TN D-320. Goldsmith, J. (1956), 'Investigation of the Flow in a Tube with Laminar Suc- tion Through 80 Rows of Closely Spaced Holes,"' Northrop Corp., Norair Div., R. NAI-56-293 (BLC-86). Goldsmith, J. (1958), ''Preliminary Experiments on the Maintenance of Laminar Flow by Means of Suction in the Region of a Wing Leading Edge and Fuselage Juncture,"' Northrop Corp., Norair Div., R. NAI-58-249 (BLC-106). Goldsmith, J. (1959), ''Experiments with Laminar Flow Near the Juncture of a Fuselage and Wing Trailing Edge,'' Northrop Corp., Norair Div., R. NOR-59- 306, (BLC-120). Gross, L. W. (1962), ''Investigation of a Peichardt Body of Revolution with Low Drag Suction in the Norair 7 x 10 ft. Wind Tunnel,"' Northrop Corp., Norair Div., R. NOR-62-126 (BLC-143). Groth, E. E. et.al. (1957), "Low Drag Boundary Layer Suction Experiments in Flight on the Wing Glove of an F-94A Airplane, Phase II - Suction Through 69 Slots,'' Northrop Corp., Norair Div., R. NAI-57-318 (BLC-94). Groth, E. E. (1958), "Low Speed Wing Tunnel Experiments on a Body of Rev- olution with Low Drag Boundary Layer Suction,'' Northrop Corp., Norair Div., R. NAI-58-335 (BLC-107). Jones, and Head, (1951), ''The Reduction of Drag by Distributed Suction,'' Proc. 3rd Anglo-American Aeronautical Conf. Brighton. Lang, T. G., and Brooks, J. D., (1959), "Control of Torpedo Boundary Layers by Suction,'"' U.S.N.O.T.S. NAVORD 6536. 931 Lumley Libby, P. A., Kaufman, L. and Harrington, R. P. (1952), 'An Experimental Investigation of the Isothermal Laminar Boundary Layer on a Porous Flat Plate," J. Aero. Sci., 19, p. 127. Loftin, L. K. and Horton, E. A. (1952), "Experimental Investigation of Bound- ary Layer Suction Through Slots to Obtain Extensive Laminar Boundary Layers on a 15% Thick Airfoil Section," NACA RM L52D02. Parkhurst, R. C. (1955), '"Recent British Work on Methods of Boundary Layer Control,"" Proc. Symp. "Boundary Layer Effects in Aerodynamics," Nat. Phys. Lab., Teddington. Pfenninger, W. (1953), "Experiments with a 15% Thick Slotted Laminar- Suction Wing Model in the TDT Tunnel at NACA Langley Field," AFTR 59082 WADC. Pfenninger, W. et.al. (1954), 'Investigation of Laminar Flow in a Tube at High Reynolds Numbers and Low Turbulence With Boundary Layer Suction Through 80 Slots,'' Northrop Corp., Norair Div., R BLC-53. Pfenninger, W. et.al. (1957), "Experiments on a 30° Swept 12% Thick Sym- metrical Laminar Suction Wing in the 5 ft. by 7 ft. Michigan Tunnel,’ Northrop Corp., Norair Div., R. NAI-57-317 (BLC-93). Raetz, G. S. (1953), "The Incompressible Laminar Boundary Layer on an Infinitely Long Swept Suction Wing with a Few Different Pressure and Suction Distribution,'' Northrop Corp., Norair Div., R. (BLC-14). Raetz, G. S. (1953), "The Incompressible Laminar Boundary Layer on an Infinitely Long Swept Wing with Continuous Suction from the 0.37 Chord Line to the Trailing Edge,'' Northrop Corp. Norair Div., R. (BLC-25). Raetz, G. S. (1953), "A Method of Calculating the Incompressible Laminar Boundary Layer on Infinitely Long Swept Suction Wings Adaptable to Small- Capacity Automatic Digital Computers," Northrop Corp., Norair Div., R. (BLC-11). Raetz, G. S. (1954), 'The Incompressible Laminar Boundary Layer ona Typical Tapered Swept Suction Wing,'' Northrop Corp., Norair Div., R. (BLC-21). Schlichting, H. (1949), ''An Approximate Method for Calculation of the Laminar Boundary Layer with Suction for Bodies of Arbitrary Shape,'"’ NACA TN 1216. von Doenhoff, A. E. and Loftin, L. K., Jr. (1949), "Present Status of Re- search on Boundary Layer Control," J. Aero. Sci., 16, p. 729. 932 The Reduction of Skin Friction Drag 3. Gas Films Afify, E. (1960). "On the Hydrodynamic Stability of a Gas-Water Interface," TRW ST M 142, Thompson-Ramo-Woolridge, Inc. Aifify, E. (1959), "The Closed Analytic Solution of the Water Boundary Layer with a Gas Film on a Flat Plate,'’ TRW-STM-83, Thompson, Ramo- Wooldridge, Inc. Bradfield, W. S., Barkdoll, R. O. and Byrne, J. T. (1962), "Some Effects of Boiling on Hydrodynamic Drag," International Journal of Heat and Mass Trans- fer 5, 615. Bullock, R. D. (1962), ''An Experimental Investigation of the Use of Air to Reduce the Viscous Drag of a Flat Plate in Water,'' B. M. E. Thesis, University of Delaware. Cess, R. D., Sparrow, E. M. (1961a), "Film Boiling in a Forced Convection Boundary Layer Flow," J. of Heat Transfer, 83, p. 370. Cess, R. D., Sparrow, E. M. (1961b), "Subcooled Forced-Convection Film Boiling on a Flat Plate," J. of Heat Transfer, 83, p. 377. Eichenberger, H. P. and Offutt, J. D. (1961), "About Gas Film Drag Reduc- tion,'’ ASME Paper 61-WA-218. Johnson, V. E. and Ramick, T. A. (1961), 'Drag Coefficients of Parabolic Bodies of Revolution Operating at Zero Cavitation Number and Zero Angle of Yaw,'’ NASA TR R-86. Krasnoff, E. (1962), "Equilibrium Gas-Water Boundary Layers,'' TRW ER 5101 Thompson-Ramo-Wooldridge, Inc. Krasnoff, E. (1959), 'Momentum Integral Approximation of a Gas-Water Boundary Layer ona Flat Plate,''TRWSTM-78 Thompson-Ramo-Wooldridge, Inc. Krasnoff, E., et. al. (1961), "Preliminary Investigations of a Gas Film Hydrofoil,'’ TRW ER 4317 Thompson-Ramo-Wooldridge, Inc. Maeder, P. and Krosnoff, E. (1959), "Preliminary Study of Water Boundary Layer Stabilization by Gas Injection'' TRW-STM-47, Thompson-Ramo-Wool- dridge, Inc. Offutt, J. D. (1962), "Gas Film Boundary Layer Stabilization Feasibility Demonstration," Final Report, ER-5110, Thompson-Ramo-Wooldridge, Inc. Offutt, J. (1959), "Gas Film Distribution on a Torpedo Body with No Suc- tion TRW STM-128."' Thompson-Ramo-Wooldridge, Inc. Pao, Y. H. (1961), "Viscous Flow Along a Surface with Gas Lubrication," TR-007-2, Hydronautics, Inc. 933 Lumley Sparrow, E. M., et.al. (1962), "A Two Phase Boundary Layer and its Drag Reduction Characteristics," TR ASME, J. Appl. Mech., p. 408. Stefan, H. G. and Anderson, A. G. (1964), ''Cavity Formation and Associated Drag in Supercavitating Flow Over Wedges in a Boundary Layer,'' Rept. No. 69, U. of Minnesota, St. Anthony Falls Hydraulic Lab. Stoller, H. M. (1963), ''Experimental Investigation of Gas Lubricated Water Boundary Layers,'' TR-007-3, Hydronautics, Inc. Wilson, M. B. (1964), ''An Experimental Study of a Cavity-Running Body," TR-007-4, Hydronautics, Inc. 4. Flexible Surfaces Becker, E. ( ), ''The Laminar Incompressible Boundary Layer Over a Plane Wall Deformed by Traveling Waves,'' Deutsche Versuchsanstalt fur Luft- fahrt, Muelheim on der Ruhr. Benjamin, T. B. (1963), ''The Threefold Classification of Unstable Disturb- ances in Flexible Surfaces Bounding Inviscid Flows," J. of Fluid Mechanics, 16. p. 436. \ Benjamin, T. B. (1960), "Effects of a Flexible Boundary on Hydrodynamic Stability," J. of Fluid Mechanics, 9, 513. Boggs, F. W. and Frey, H. R. (1961),'' The Effect of a Lamiflo Coating on a Small Planning Hull having Zero Dead Rise,'' U. S. Rubber Co. Res. & Dev. Rep. Boggs, F. W., Frey, H. R. and Hahn, E. R. (1962), "Construction and Testing of Coatings for Drag Reduction,'' U. S. Rubber Co. Boggs, F. W. and Tokita, N. (1960), ''A Theory of the Stability of Laminar Flow Along Compliant Plates,'"' Proc. 3rd Symp. Naval Hydrodynamics. Galway, R. D. (1963), ''An Investigation into the Possibility of Laminar Boundary Layer Stabilization Using Flexible Surfaces,'' M. Sc. Thesis, Queen's Univ., Belfast, Ireland. Gregory, N. (1960), "The Present Position of Research on the Maintenance of Laminar Flow Over Flexible Surfaces,'' ARC. 22, 293-F.M. 3014-Perf. 1939. Gregory, N. and Love, E. M. (1961), ''Progress Report on an Experiment on the Effect of Surface Flexibility on the Stability of Laminar Flow,'’ ARC Rept. 23314, F. M. 31136, Pef. 3053. Harris, F. D. and Price, J. F. (1962), "Effect of a Flexible Wall on the Sta- bility of a Poiseuille Flow,'' Physics of Fluids 5, p. 365. 934 The Reduction of Skin Friction Drag Haines, F. D. and Price, J. F. ( ), "Stability of Plane Poiseuille Flow Between Flexible Walls,"' Proc. 4th U. S. National Congress Appl. Mech. Kane, J. (1963), ''The Propagation of Rayleigh Waves Past a Fluid Loaded Boundary,'' J. Math. Phys. 41, p. 179. Karplus, H. B. (1963), ''Turbulent Flow Transition Near Solid and Flexible Boundaries," Ill. Inst. Tech. Rept. No. IITRI 1205-4. Kramer, M. O. (1962), ''Boundary Layer Stabilization by Distributed Damp- ing,’ Naval Engineers Journal, 74, p. 341-8. Kramer, M. O. (1962), ''Material Requirements for Boundary Layer Sta- bilizing Coatings—Water Application," Rand Corp. Memo., No. RM-3018-PR. Kramer, M. O. (1961), ''Dolphin's Secret," J. Am. Soc. Naval Eng's. 73, p. 103. Kramer, M. O. (1962b), "Speculative Consideration on High-Frequency In- stability of the Laminar Boundary Layer and its Effect on the Design of Stabiliz- ing Coatings,'’ Memo. RM-3284-PR, Rand Corp. Kramer, M. O. (1960), "Boundary Layer Stabilization by Distributed Damp- ing,'' J. Am. Soc. Naval Eng's. 72. Kramer, M. O. (1957), "Boundary Layer Stabilization by Distributed Damping," J. Aer. Sci. 24. Landahl, M. T. (1962), ''On the Stability of a Laminar Incompressible Bound- ary Layer over a Flexible Surface," J. F. Mech. 13, p. 608. Lang, T. G. (1963), ''Porpoises, Whales and Fish: Comparison of Predicted and Observed Speeds,'' Nav. Eng'rs. J. 75. Lang, T. G., Daybell, D. (1963), ''Porpoise Performance Tests Conducted in a Seawater Tank," U.S.N.O.T.S. NAVWEPS 8060. Laufer, J. and Maestrello, T. (1963), ''The Turbulent Boundary Layer Over a Flexible Surface,'' Document D6-9708 Boeing Co. Lumley, J. L. and McMahon, J. F. (1964), ''Stability of Plane Couette Flow Over an Admissive Boundary." In preparation. Mercer, A. G. (1962), ''Turbulent Boundary Layer Flow Over a Flat Plate Vibrating with Transverse Standing Waves," U. Minn., St. Anthony Falls Hydr. Lab. Tech. Paper 41, Ser. B. Nonweiler, T. R. F. (1961), ''Qualitative Solutions of the Stability Equation for a Boundary Layer in Contact with Various Forms of Flexible Surface," ARC. 22, 670-FM3071. 935 Lumley Phillips, O. M. (1957), 'On the Generation of Waves by Turbulent Wind," J. Fluid Mech. 2, p. 417. Rae, W. J. and Moore, F. K. (1960), ''On the Inviscid Stability of the Bound- ary Layer ona Flexible Wall,"’ Cornell Aero Lab. Rep. No. AF-1285-A-5. Ritter, H., Messum, T. T. (1964), ''Water Tunnel Measurement of Turbulent Skin Friction on Six Different Compliant Surfaces of 1 ft. length," ARL/G/N9, Admiralty Res. Lab. Ritter, H. and Porteous, J. S. (1963), ''Water Tunnel Measurements of Skin Friction on a Compliant Coating,'’ ARL/G/N8, Admiralty Res. Lab. Smith, L. L. (1963), ''An Experiment on Turbulent Flow in a Pipe witha Flexible Wall,'' Masters Thesis, U. of Wash., Seattle, College of Engineering. Williams, J. E. Ffowcs (1964), "Reynold Stress Near a Flexible Surface Responding to Unsteady Air Flow," Bolt, Beranek & Newman, Report No. 1138. Wu, T. Y. (1961), "Swimming of a Waving Plate," J. of Fluid Mech. 10. p. 321-345. 5. Non-Newtonian Fluids Anon (1963), ''The Effects of One Non-Newtonian Additive on the Speed of a MSB Minesweeper,'' Westco Research, Dallas, Texas. Bayless, L. E. (1960), ''The Anomalous Viscosity of Blood. Flow Properties of Blood,"' ed. Copley and Stainsby, Pergamon, p. 29. Crawford, H. R., Pruitt, G. T. (1962), "Rheology and Drag Reduction of Some Dilute Polymer Solutions," U.S.N.O.T.S. Rept. N60530-6898. Davies, C. N. (1949), "Discussion of Papers by Oldroyd and Toms," Proc. Int'l. Cong. Rheo. III. p. 47. Dever, C. D., Harbour, R. J. and Seifert, W. F. (Assignors to Dow Chemical Co.) (1962), "Method of Decreasing Friction Loss in Flowing Fluids," U.S. Patent No. 3,023,760. Dodge, D. W. and Metzner, A. B. (1960), ''Turbulent Flow of Non-Newtonian Systems,"’ Am. Inst. Chem. Eng. 5. p. 189. Eisenberg, D. M. and Bogue, D. C. (1963), "Velocity Profiles of Thoria Sus- pensions in Turbulent Pipe Flow," A.I.Ch.E. Reprint, Symposium on Non- Newtonian Fluid Mechanics I. Fabula, A. G. (1963), ''The Toms Phenomenon in the Turbulent Flow of Very Dilute Polymer Solutions,"’ Proc. 4th Int. Cong. Rheo. Brown. 936 The Reduction of Skin Friction Drag Fabula, A. G., Hoyt, J. W. and Crawford, H. R. (1963), ''Turbulent- Flow Characteristics of Dilute Aqueous Solutions of High Polymers,"' Bull. Amer. Phys. Soc. 8. p. 430. Fabula, A. G., Hoyt, J. W. and Green, J. H. (1964), ''Drag Reduction Through the Use of Additive Fluds,"' U.S.N.O.T.S., NAVWEPS 8434. Gadd, G. E. (1963), ''The Effect on the Turbulent Boundary Layer of Adding Guar Gum to the Water in Which a Disk Rotates,"' Ship T.M., 42 National Physical Lab., Ship Div. Giles, W. B. (1964), ''Laminar Viscoelastic Boundary Layers with Rough- ness,'' G. E. Advanced Technology Labs., Rept. No. 64GL51. Granville, P. S. (1962), ''The Frictional Resistance and Boundary Layer of Flat Plates in Non-Newtonian Fluids,"' J. Ship Res., 6, p. 43. Hamill, P. A. (1964), "A Preliminary Experiment on the Effect of Additives on Hydrodynamic Drag,'' National Research Council of Canada, Div. of Mech. Eng'g., Ship Sect. Lab. Memo MTB65. Hanks, R. W. (1963), ''Laminar-Turbulent Transition for Fluids with Yield Stress," A.I.Ch.E.J. V. 9, No. 3, p. 306-309. Hoyt, T. W. and Fabula, A. G. (1963), ''Frictional Resistance in Towing Tanks,'' Proc. 10th International Towing Tank Conf. Lindgren, E. Rune (1959), ''Liquid Flow in Tubes III: Characteristic Data of the Transition Process," Arkivfor Fysik 16, p. 101. Lumley, J. L. (1964), ''Turbulence in Non-Newtonian Fluids,"' Phys. of Fluids, 7, p. 339. Lummus, J. L., Randall, B. V. (1964), ''Development of Drilling Fluid Fric- tion Additives for Project Mohole,'' Pan American Petroleum Corporation, Re- search Department, F64-P-54/295-3(4) Job #3918. Merrill, E.W., Mickley, H. S. and Ram, A. (1962), "Degradation of Polymers in Solution Induced by Turbulence and Droplet Formation,"' J. Polymer Sci. 62. $109. Oldroyd, J. G. (1948), "A Suggested Method for Detecting Wall-Effects in Turbulent Flow Through Tubes," Proc. Int'l. Cong. Rheo. II. p. 130. Pruitt, G. T., Crawford, H. R. (1963), ''Drag Reduction Rheology and Capil- lary End Effects of Some Dilute Polymer Solutions,"' U.S.N.O.T.S. Rept. 60530- 8250. Ripken, J. F. and Pilch, M. (1963), "Studies of the Reduction of Pipe Friction with the Non-Newtonian Additive CMC," Univ. of Minnesota, St. Anthony Falls Hydraulic Res. Lab. Tech. Rep. #42, Series B. 937 Lumley Savins, J. G. (1963), ''Drag Reduction Characteristics of Solutions of Macro- molecules in Turbulent Pipe Flow,'' A.I.Ch.E. Preprint, Symposium on Non- Newtonian Fluid Mechanics I, Houston, Texas. Shaver, R. G. and Merrill, E. W. (1959), ''Turbulent Flow of Pseudoplastic Polymer Solutions in Straight Cylindrical Tubes,"' A.I.Ch.E. Journal 5, p. 181. Thurston, S. and Jones, R. D. (1964), "Experimental Model Studies of Non- Newtonian Soluble Coatings for Drag Reduction,'' AIAA Bulletin 1 (May), p. 225 (abstract only). Toms, B. A. (1949),'' Some Observations on the Flow of Linear Polymer Solutions Through Straight Tubes at Large Reynolds Numbers," Proc. Int'l Cong. Rheo. II, p. 135. Wen, K. S. (1963), "On the Stability of a Laminar Boundary Layer for a Maxwellian Fluid,'' General Electric Space Sciences Laboratory Tech. Inf. Series R63SD102. 6. Particles and Fibers Anon (1963), "Effect of Stationary Fibers on Drag Reduction,"’ Westco Re- search. Bugliarello, G. and Daily, J. W. (1961). Tech. Assoc. Pul. Paper, Ind. 44 p. 881. Chien, N. (1956), ''The Present Status of Research on Sediment Transport," Transactions ASCE, 121, p. 883. Daily, J. W. and Chu, T. K. (1961), "Rigid Particle Suspensions in Turbulent Shear Flow: Some Concentration Effects,'' Tech. Rept. No. 48, MIT Hydro. Lab. Einstein, H. A. and Chien, N. (1952), "Second Approximation of the Suspended Load Theory," 47, 2, Univ. of Calif. Elata, C. and Ippen, A. T. (1961), "The Dynamics of Open Channel Flow with Suspensions of Neutrally Buoyant Particles,'' MFT Hydro. Lab. Tech. Rep. 45. Hino, M. (1963), "Turbulent Flow with Suspended Particles,"’ J. Hydraulics Div. ASCE 89, p. 161. Ismail, H. M. (1952), "Turbulent Transfer Mechanism and Suspended Sedi- ment in Closed Channel," Transactions, ASCE, 117, p. 409. Murota, A. (1953), "On the Relation Between the Concentration of Suspended Sediment and the Velocity Distribution of Water Flow," J. Japan Soc. Civil Engi- neers, 38, p. 8. 938 The Reduction of Skin Friction Drag Nino, M. (1962), "Changes in Turbulent Structures of Flow with Suspension of Solid Particles,'' Proceedings, 7th Conf. on Hydr. Res., Hydr. Committee, Japan Soc. Civil Engrs., p. 49. Saffman, P. G. (1962), "Flow of a Dusty Gas Between Rotating Cylinders," Nature, 193, p. 463. Shimura, H. (1957), ''On the Characters of the Water Flow Containing Sus- pended Sediment,"' Transactions, J. Soc. Civil Engineers 46. Sproull, W. T. (1961), ''Viscosity of Dusty Gasses,"’ Nature, 190, p. 976. Tsubaki, T. (1955), ''On the Effects of Suspended Sediment on Flow Charac- teristics,'' J. Japan Soc. Civil Eng's. 40. Vanoni, V. A. (1946), ''Transportation of Suspended Sediment by Water," Transactions, ASCE, 111, p. 67. Vanoni, V. A. and Nomicos, G. N. (1961), "Resistance Properties of Sedi- ment Laden Streams," Transactions, ASCE, 126, p. 1140. * * *K DISCUSSION S. K. F. Karlsson Brown University Providence, Rhode Island The following comments refer to the effects on a fluid flow by a non- Newtonian additive, which have been discussed to some extent by Professor Lumley and which appear to offer possibilities for considerable reduction in skin friction in turbulent boundary layers. For a visco-inelastic, shear thinning (Reiner-Rivlin) fluid, Lumley con- cluded (Phys. Fluids, Vol. 7, No. 3, March 1964) that turbulent transport effects in an existing turbulent flow would be no different from that in Newtonian fluids. However, it seems that this does not exclude the possibility that even in sucha simple non-Newtonian fluid the development of instabilities both in the laminar and turbulent boundary layers may well be substantially altered, resulting in considerable changes of the overall boundary layer skin friction. We have started some laminar stability experiments with such a non- Newtonian fluid in rotating Couette motion at Brown University recently, and although our geometry is different from that of the boundary layer, the results may still be of some interest here. Our fluid is a suspension of Milling Yellow, a dye stuff, in distilled water. Peebles and co-workers at the University of Tennessee have studied its properties extensively (e.g., A. E. Hirsch and F. N. 939 Lumley Peebles, The Flow of a Non-Newtonian Fluid in a Diverging Duct, experimental; Department of Engineering Mechanics Report, August 1964, University of Tennes- see, Knoxville, Tennessee) and they found it to be a shear-thinning, visco-inelastic fluid. Figure 1 shows the viscosity variation with shear rate of a particular sample of Milling Yellow as computed from data obtained with a capillary viscometer. 50 40 —> C. IP 30 VISCOSITY 20 (0) iKeXe) 200 300 400 500 600 SHEAR RATE y sec! —> Fig. 1 - Rheological data for milling yellow: 1.608% concentration The stability experiment performed is the well-known Taylor experiment in which one studies the motion of the fluid in the gap between two concentric cylin- ders rotating at different speeds. In our experiment the outer cylinder is sta- tionary and only the inner one rotates. Because of the shear rate dependence of the viscosity the tangential velocity profile in the gap is different from that of a Newtonian fluid. Figure 2 shows a comparison between the two for identical boundary conditions at the inner (R = 3.14 cm) and the outer cylinder (R = 3.49), obtained by computation using the experimentally determined viscosity. 940 The Reduction of Skin Friction Drag NON-NEWTONJAN (MILLING YELLOW: 1.608% CONCENTRATION) ——— NEWTONIAN cm/sec —> U 3.1 3.2 5h5) 3.4 3.5 R cm > MELOGCII YC PIRORILES Fig. 2 - Velocity profiles In our experiments we have made use of the fact that Milling Yellow sus- pensions are doubly refractive when subjected to a shearing motion. Thus, the flow field has been observed using standard birefringence techniques. So far we have measured three different quantities as functions of concen- tration of the additive: 1. "critical" or "neutral stability" speed for the primary (Couette) motion, i.e., the rotation rate at which Taylor cells first appear; 2. the rotation rate when the Taylor cells first become unstable. This in- stability consists of a sinusoidal deformation of the cells, making the heretofore steady flow time dependent; 3. cell width of the primary cells, thus obtaining the wave number of the perturbation that is most unstable. The results from these measurements appear in Figs. 3, 4 and 5. The crit- ical velocity is given in terms of the Taylor number which is the significant non- dimensional quantity for this problem: 941 2 2) 7 = 27 fas 1- 7? yp? where R, i= —— fiael C= 1K I R 2 In computing T we have used the value of viscosity, », which corresponds to the average shear stress in the gap. With this, somewhat arbitrary, choice of the viscosity the primary motion of the non-Newtonian fluid appears less stable than its Newtonian counterpart. (Fig. 3.) O 1.3 14 1S 16 % CONCENTRATION OF MILLING YELLOW — Figure 3 With respect to the second time dependent mode of instability, however, the non-Newtonian fluid is relatively more stable as can be seen in Fig. 4, showing the ratio between rotation rates for the appearance of the secondary and primary (Taylor) instabilities. Thus we have the seemingly somewhat contradictory re- sult that the primary motion is less stable in the non-Newtonian fluid, whereas once the instability has occurred the resulting motion is relatively more stable, when compared to a Newtonian fluid. Finally, Fig. 5 shows the variation with concentration of the Taylor cell width, normalized with the gap width between the cylinders. This plot is 942 The Reduction of Skin Friction Drag 1,28 ie) 13 14 1.5 1.6 % CONCENTRATION OF MILLING YELLOW — Figure 4 1.8 1.6 F|z 2|e ale = ails 2 is e) 1.2 1.0 a See ce eas es Diva leee, Hed Cobh ee trees o) 13 1.4 1.5 1.6 % CONCENTRATION OF MILLING YELLOW —> Figure 5 943 Lumley particularly interesting because it does not depend on our choice of viscosity. It exhibits a distinct and consistent variation of this parameter with concentration. Hence it is clear that the non-Newtonian character of this fluid has a direct effect on the stability of its motion. Possibly this effect is a result of shear- induced normal stresses or anisotropy in the relation between stress and rate of strain, which is implied by the fact that the fluid is birefringent under shear. * * * DISCUSSION A BASIC THEORY THAT COULD EXPLAIN DRAG REDUCTION IN A FLOW CARRYING ADDITIVES — A. Cemal Eringen Purdue University Lafayette, Indiana Lumley [1], Hoyt and Fabula [2], and Vogel and Patterson [3] gave excellent experimental demonstrations of the phenomena of drag reduction by minute amount of additives to fluid surrounding a moving object. We do not possess as yet a theory explaining this phenomena. Classical Stokesian fluids do not contain a mechanism which could provide the desired mathematical treatment. In fact, I do not believe that even the modern theories of visco-elastic fluids [4] can throw light into this phenomena. Quite by accident, a new theory, ‘Simple Micro- fluids," introduced by Eringen [5], in a different context, seems to have just the proper mechanism for this purpose. The theory of simple micro-fluids requires that we determine nineteen un- knowns p, i, =i. >» “%, and v, by solving nineteen partial differential equa- tions given in [5] subject to appropriate boundary and initial conditions. Here P, ipms Y_, and v, are respectively the mass density, the micro-inertia, the gyration tensor and the velocity vector. The micro-inertia i,,, provides a mechanism for the inertial anisotropy. Roughly speaking, it is similar to the inertia tensor of rigid dynamics. The gyration tensor provides a mechanism for the local micromotions and small vortices. The present theory is shown [5], [6] to contain the celebrated Navier-Stokes Theory of fluid dynamics and the theory of anisotropic fluids. A theory of tur- bulence based on this theory is as yet lacking. Some sample calculations made are indicative of the above mentioned drag reductions. However, presently this work is too naive for publication and the possible application of the theory of simple micro-fluids to the problem of drag reduction by additives is brought to your attention as a conjecture. 944 The Reduction of Skin Friction Drag REFERENCES 1. ''The Reduction of Skin Friction Drag" 2. "The Effect of Additives on Fluid Friction" 3. ‘An Experimental Investigation of the Effect of Additives Injected into the Boundary Layer of Underwater Bodies" 4. A.C. Eringen, Nonlinear Theory of Continuous Media, McGraw-Hill, New York, 1962. 5. "Simple Micro-fluids,'' International Journal of Engineering Science, Vol. 2, No. 2, 1964. 6. A. C. Eringen, ''Mechanics of Micromorphic Materials,"" ONR Technical Re- port No. 26, April 1964, presented at the XIth International Congress of Ap- plied Mechanics, Munich, Germany, and scheduled for publication in the Proceedings. DISCUSSION Alan Kistler Yale University New Haven, Connecticut Professor Lumley has given an apt summary of the various proposals for reducing the skin friction on objects moving through a liquid. Since the motiva- tion for studying these methods is to find a way to reduce the total drag of an object, a few words about the rest of the drag problem for a submerged object might be appropriate. The neglected component (pressure drag) is associated with separation of the boundary layer. A technique that either increases or de- creases the friction drag could have the opposite effect on the pressure drag. The change of sphere drag with transition is the best known example. All of the suggestions for affecting the friction could affect the separation either by chang- ing the rate of momentum transport across the free Shear layer or by changing the location of the separation point. Sufficiently detailed measurements of the pressure distribution about realistic shapes should be taken in order to evaluate and understand what is occurring when a particular drag reduction technique is being tested. Aeronautical experience has shown that most drag reduction schemes that depend on the delay of transition, with the possible exception of boundary layer suction, do not work well outside of the wind tunnel. Surface roughness, wake interaction, and cross flow all work against laminar flow. For this reason, it 221-240 O - 66 - 61 945 Lumley appears likely that the techniques that change the structure of the turbulent boundary layer offer the most promise. The limits of what can be done with these techniques have still to be determined, however. * * * REPLY TO THE DISCUSSION J. L. Lumley The Pennsylvania State University University Park, Pennsylvania I wish to thank Professors Karlsson, Kistler and Eringen for their helpful comments. I feel that in particular the preliminary data presented by Karlsson indicates the caution with which one must use one's intuition in this very difficult problem. The contribution by Eringen will be somewhat more difficult to assess until turbulence dynamics have been worked out using the constitutive relations he suggests. Since the turbulence dynamics of non-Newtonian media in general are not understood, it is difficult to say whether constitutive relations fitting within the framework of the simple fluid of Noll* are adequate, or whether a locally orientable medium such as that proposed is required. The comments of Kistler seem particularly germane to the paper of Vogel and Patterson and sug- eest caution in the interpretation of their measurements in the near wake. * * * *Noll. W., Archiv. Rat. Mech. Anal. 2, (1958), 197. 946 THE EFFECT OF ADDITIVES ON FLUID FRICTION J. W. Hoyt and A. G. Fabula U.S. Naval Ordnance Test Station Pasadena, California INTRODUCTION It is now well established that very small concentrations of many natural and synthetic high-polymer substances have the property of reducing the turbu- lent friction drag of the liquid in which they are suspended or dissolved. Be- cause of the many immediate possible applications of such an effect, current interest is high. The earliest published data showing turbulent-flow friction reductions in dilute polymer solutions appear to be those of B. A. Toms [1] who studied poly- methylmethacrylate in chlorobenzene. Flow of ''thickened gasoline" was the subject of a U.S. Patent in 1949 [2]. Work with aqueous solutions of polymers was reported simultaneously by Shaver & Merrill [3] and Dodge & Metzner [4] both of whom used sodium carboxymethylcellulose as the friction-reducing ma- terial. The technique has found commercial use in oil-field applications [5, 6]. Because the earlier workers in the field attributed the friction-reduction phenomenon to "non-Newtonian" fluid properties, the term has become synony- mous with the effect. However, one purpose of this paper is to show that the turbulent-friction reduction effect can be observed (indeed, becomes most prominent) at polymer concentrations at which the solutions are Newtonian by conventional viscometry. Further, it will be shown that polymer additives can be effective in reducing the turbulent friction in concentrations of as little as a few weight parts per million (wppm). Although the exact mechanism of the effect is not shown, general rules as to the type of material likely to be effective can be developed, and predictions can be made of the maximum polymer effectiveness in several simple flow situ- ations. It is believed that the generalizations formulated here apply to all sol- vent fluids, but the experimental work has concentrated on aqueous solutions. EXPERIMENTS WITH ROTATING DISKS Simply because the apparatus happened to be on hand, early work in Pasa- dena was performed on a large-scale rotating disk facility. This equipment (Fig. 1) consists of a 3785 liter water tank in which a 45.7 cm diameter risk is rotated by a d-c electricmotor at such a speed that turbulent flow extends over 947 Hoyt and Fabula a major portion of the disk. Disk speed and torque are measured using various concentrations of polymer additives in the tank. It can be reasoned that most of the torque is developed near the outer disk edge, so that torque reduction is es- sentially equivalent to friction reduction. Thus these terms are used inter- changeably hereafter. An example of the type of data obtained using this apparatus is given in Fig. 2. The polymer additive used here is guar gum, the refined endosperm of Cyam- opsis tetragonolobus, a plant grown commercially in India, Pakistan, and the United States for food and industrial purposes.! At constant rotative speed, ad- dition of the polymer produces immediate lowering of the torque until at concen- trations of 300-400 wppm the torque has been reduced to between 30 and 40 per- cent of its pure water value. As the concentration is further increased, the torque is increased somewhat, which can be attributed to the increased viscosity of the solution. Much more striking results can be obtained using the synthetic polymer poly(ethylene oxide) which is commercially available in four different molecular ~ weight distributions.* Figure 3 shows data taken with the 45.7 cm diameter disk at 40 rev/sec for the four molecular weights of the same chemical. As molecu- lar weight is increased, the material becomes more effective, and Fig. 3 shows that 70% torque or friction reduction may be obtained with less than 100 wppm of additive, using the highest molecular weight material. Similar tests have been made using a wide variety of natural and synthetic polymers, with the results shown on Table I, where the weight parts per million to achieve a friction reduction of 35% (half way between no effect and the maxi- mum of about 70% observed on this facility at 40 rev/sec) are listed together with the molecular weight of the polymer. From the table, it appears that at least three significant parameters affect the ability of a polymer to lower the turbulent frictional resistance of the fluid in which it is dissolved: linearity, molecular weight, and solubility. Linearity The striking thing about the most effective polymers is that they are "'long- chain" materials having an essentially unbranched structure. The chemical for- mulas of guar and poly(ethylene oxide) (Fig. 4) indicate this characteristic, and a photograph of a model of a segment of the poly(ethylene oxide) molecule fur- ther illustrates the thread-like appearance of the material. While the exact configuration of these molecules in solution is poorly under- stood, calculations indicate approximate length-to-diameter ratios of from 350 to 500 for guar, and from 22,000 to 165,000 for poly(ethylene oxide) of 6 million molecular weight depending on the helix model selected, if we ignore, for the Divine guar gum used in these experiments was ''Westco J-2 FP" supplied by the Western Company, Research Division, 1171 Empire Central, Dallas, Texas. 2Supplied by Union Carbide Corp., 270 Park Ave., New York, New York. 948 The Effect of Additives on Fluid Friction Table 1 Comparative Friction- Reduction Effectiveness of Water-Soluble Polymer Additives Measured With the Rotating-Disk Facility Additive Caran aMilon eae Notable Characteristics Guar gum, w, s (J-2FP)© 60 (0) Straight chain molecule with single-membered side branches Locust bean gum, m 260 Orod Similar to guar but with (260)¢ fewer side branches, caus- ing reduced solubility and less hydrogen bonding Carrageenan or Irish 650 0.1 - 0.8 | Strongly charged anionic moss, M (Stamere NK) | (420) polyelectrolyte Gum Karaya, m 780 OF5 Highly branched molecule; relatively insoluble; acidic Gum Arabic, b Ineff. 0.24 - 1 Highly branched molecule Amylose, s (Superlose) Ineff. 2 (55 Linear chain molecule; ret- rogrades rapidly Amylopectin, s (Ramalin G) Ineff. 12; Highly branched molecule Hydroxyethyl cellulose, u (Cellosize QP-15000) 220 athe Nonionic; formed by addition (Cellosize QP-30000) 220 spake of ethylene oxide to cellu- (Cellosize QP-50000) 160 MR lose; has side branches of various lengths Sodium Carboxymethyl- cellulose, h (CMC 7HSP) 400 OU2F ONT. [Mes Poly(ethylene oxide), u (Polyox WSR-35) 70 0.2 Very water soluble; no bio- (Polyox WSR-205) 44 0.6 logical oxygen demand; ap- (Polyox WSR-301) ily 4 parently an unbranched (Polyox Coagulant) 12 5 molecule with unusual af- finity for water Polyacrylamide, d (Separan NP 10) 26 1 Nonionic (Separan NP 20) 25 2 Nonionic (Separan AP 30) 29 2-3 Anionic Polyhall-27, s 130 ee ciaen wna) a feeeee are 949 Hoyt and Fabula Table 1 (Continued) Additive Cha Seo IL Sea: 19 Notable Characteristics Polyvinylpyrrolidone, f (K30) (K90) Polyvinyl alcohol, e (Elvanol 51-05) E 0.032 (Elvanol 72-60) - O17 = 0722 Silicone, u (L-531) Polyacrylic acid, g (Goodrite 773x020 B-3) (Goodrite K-702) (Goodrite K-714) Carboxy vinyl polymer, g | Ineff. ahs Inconclusive test due to pre- (Carbopol (941) cipitation upon dilution 2C, = concentration required (in weight parts per million) for 35% disk-torque reduction at 40 rev/sec with lake water as the solvent. M = approximate molecular weight of the polymer according to the literature. “The source of each polymer for this work is indicated by the letter after its name: b= Braun Div., Van Waters and Rogers, Inc.; d = Dow Chemical Co.; e = E. I. Dupont; f = General Aniline and Film Corp.; g = B. F. Goodrich Chem- ical Co.; h = Hercules Powder Co.; m= Meer Corp.; s = Stein, Hall and Go.; u = Union Carbide Chemicals Co.; w = Westco Research. CG. values in parenthesis are for solutions given heat treatment to increase polymer solubility. moment, the molecular chain flexibility which will produce a Gaussian-coil con- figuration for such long molecules. Thus the linearity of the molecule appears to play an important role in the drag-reducing effect. Molecular Weight Accompanying the linearity is a corresponding increase in molecular weight. However, from the experiments with Gum Karaya (Table 1) it appears that high molecular weight in itself is not as effective as the linearity. The poly(ethylene oxide) is some 65 times more effective than the heavier Gum Karaya molecule, on a weight basis. The effect of molecular weight (or linearity) can be demonstrated by replot- ting the disk data of Fig. 3 taken at a constant rotative speed of 40 rev/sec for poly(ethylene oxide) to give the logarithmic presentation of Fig. 5. In addition to showing the dependence of friction-reduction on molecular weight, Fig. 5 also indicates that substantial increases in molecular weight (degree of polymerization) 950 The Effect of Additives on Fluid Friction would be required to achieve better friction-reduction performance by, say, an order of magnitude, with this particular chemical. Such unusually large macro- molecules would suggest the possibility of finite particles also producing the friction reduction effect. Experiments with wood-pulp [7] show that this is in- deed the case, but friction reductions were much lower than those reported here. This is possibly because of the third requirement for maximum effec- tiveness, solubility. Solubility Referring again to Table 1, tests with Carrageenan indicate the greater the solubility the more the friction reduction effect. Further, molecules which otherwise would be expected to be effective, such as Amylose, do not show up well, probably because of poor solubility. FURTHER WORK WITH ROTATING DISKS Because the large-scale rotating disk apparatus described in the previous section required large amounts of experimental solutions, a smaller apparatus was developed consisting of a 7.6 cm diameter disk rotating in two liters of so- lution. Figure 6 shows experimental data obtained with this equipment using guar gum. The maximum torque reduction obtained was on the order of 40%. Similar data are shown in Fig. 7 for solutions of poly(ethylene oxide). The values of the torque reduction which were obtained on this apparatus as com- pared with the large-scale equipment, together with the variation of torque re- duction with rotative speed, suggest plotting these data as a function of Reynolds number. Such a comparison is shown in Fig. 8 where data from the 7.6 cm, the 45.7 cm, and also a 76.2 cm disk are shown. The resultant envelope of maximum torque reduction obtained in this way seems Surprisingly similar for many poly- mers, that is, the same maximum torque reduction at any given Reynolds num- ber can be obtained with any of the "effective'' polymers, with only the concen- tration required to obtain this reduction varying from polymer to polymer. The Reynolds number used in this plot is based on water viscosity without consider- ing any viscosity increase due to the polymer. As some typical data for the maximum torque reduction curve of Fig. 8, Table 2 gives concentrations of var- ious materials required to attain 70% reduction at a Reynolds number of 1.3 million with the 45.7 cm disk facility. Effect of Sea Water The work presented so far has been based entirely on tap water or water drawn directly from a fresh water lake. Additional tests were made with the 45.7 cm rotating disk to show the effect of sea water on the performance of polymer additives. As shown in Fig. 9, friction reduction data taken in simu- lated sea water agree closely with those obtained on fresh water for guar. The tests shown are at three different temperatures, ranging from 13°C to 27°C. Poly(ethylene oxide) is even less affected by presence of sea water salts. 951 Hoyt and Fabula Table 2 Concentrations (wppm) to Achieve 70% Torque Reduction at a Rotating Disk Re = 1,300,000 Guar gum (J-2F P) Locust bean gum Gum Karaya Polyhall-27 Polyox-WSR 205 Separan AP 30 (The source and molecular weight of the above materials is given in Table 1) Rheological Studies Since high concentrations (above 1000 wppm) of these polymers are known to be shear-thinning, early explanations of the friction reduction were based on the "non-Newtonian" (i.e., variable) viscosity with rate of shear. Considerable effort was thus placed upon the rheology of these substances and how their shear-thinning behavior could explain drag reduction. It was quickly realized, when rheograms were available, that at the concen- trations where maximum friction reductions were obtained, these solutions were not 'non-Newtonian,"' but of essentially constant viscosity, greater than that of the solvent. It was only at higher concentrations that departures from constant viscosity were evident. For example, Fig. 10 shows a rheogram for guar, and Fig. 11 for poly(ethylene oxide) of 4 million molecular weight.2 At the concen- trations of most interest (under 500 wppm for guar and under 100 wppm for poly(ethylene oxide) it is difficult, from these data, to ascribe a variable vis- cosity with shear to these solutions. The constant viscosity extends to very low shear as shown in Fig. 12.4 Thus the term "non-Newtonian" is inappropriate for these fluids, unless one allows the possibility that non-steady measurements will show that these solutions display shear rigidities at high frequencies which ideal ''Newtonian" fluids would not. J. L. Lumley [8] has recently argued that friction reductions shouldnot be expected from the purely viscous, non-Newtonian class of fluids. Since many of the effective additives produce highly viscoelastic solutions in higher concentrations, it is possible that the drag reduction phenom- enon is related to viscoelasticity. However, viscoelastic solutions are not nec- essarily effective drag reducers: e.g., Carbopol (Table 1). 3These data were obtained under U. S. Navy contract by the Western Company, Research Division, using Fann and Burrel-Severs viscometers. These data were obtained by J. M. Caraher of the Naval Ordnance Test Station, using a new type, helical-coil viscometer of his design. 952 The Effect of Additives on Fluid Friction Furthermore, additional experiments have shown that the effect is not en- hanced by increasing the viscosity of solution of guar by ''complexing" with so- dium borate [9]. Increasing the viscosity in this way resulted in lowering the drag-reduction effect based upon the weight of guar in solution. In a typical test the viscosity was increased by a factor of 22 over the guar solution alone, by addition of sodium borate, and the drag reduction than obtained was only 70% of that which would have occurred using guar only. However the friction reduction is produced, it seems clear that the action involved is suppression of turbulence intensity. Figure 13 shows test data from the 45.7 cm diameter disk for guar, correlated with disk Reynolds numbers based on water. At concentrations of guar up to 311 wppm, the slopes of the test curves are roughly parallel to, but lower than the turbulent flow water data. For 621 wppm and above, the slopes are roughly parallel to, but higher than, the ‘laminar water flow case. From Fig. 10 it can be seen that no significant changes in Fig. 13 wouJd result from use of Reynolds numbers based on the measured viscosities of the solutions for under 500 wppm. PIPE FLOW EXPERIMENTS The friction reducing effect of polymer solutions can be easily studied by measuring the pressure drop occurring in a given length of pipe in which the polymer solution flows. Many experimental facilities of this type have been constructed, and in general they are similar to that shown schematically in Fig. 14, except for the use of air-pressure pumping to minimize degradation of test solutions [10]. Pre-mixed polymer solution contained in tanks is forced through the pipe test section where the static-pressure gradient is measured. Flow rates can be determined by weighing the amount of polymer solution discharged in a given time. Discharged solution is discarded to minimize bias due to shear degradation which occurs very rapidly for many of the solutions. By compari- son of similar data taken using pure water as the flowing medium, drag reduc- tion may be calculated. Typical data using poly(ethylene oxide) of 4 million molecular weight are shown in Fig. 15. Drag reduction of well over 75% is easily obtained. Similar data using the same polymer in sea water, but in a different apparatus, > are given in Fig. 16. Another pipe flow apparatus, which is essentially a turbulent flow rheome- ter, has recently been constructed according to the sketch of Fig. 17. The pis- ton of the cylinder is moved upward at 1.245 cm per second, forcing fluidthrough the small diameter pipe. The entire apparatus is mounted vertically to allow entrapped air to escape. Some representative data from this instrument, taken at a constant flow ve- locity of 12.65 meters/sec (Reynolds number based upon water at 21.1°C of ap- proximately 14,000) are given in Fig. 18. >Data taken by the Western Co. under U. S. Navy contract. 953 Hoyt and Fabula Reynolds Number Correlation Data from poly(ethylene oxide) of 4 million molecular weight has been cor- related on a pipe flow Reynolds number basis using the viscosity of pure water. At a concentration of 100 wppm, drag reduction reaches a maximum of 78 - 79% at a Reynolds number of about 10.2 At a lower concentration (30 wppm) the ef- fect falls off at higher Reynolds numbers. A possible explanation for the fall-off is rapid shear degradation of the polymer at the higher flow velocities. The envelope of maximum drag reduction shown on Fig. 19 is the maximum effect obtained for any polymer in pipe flow as a function of Reynolds number. Thus it is an empirical relationship for pipe flow corresponding to that given for rotating disks in Fig. 8. These pipe flow data are consistent in general with those reported in Ref. 11. To further demonstrate the validity of Fig. 8, Table 3 gives some concentra- tions of materials required to attain the maximum drag reduction of 67% at a pipe flow Reynolds number of 14,000. Table 3 Concentrations (wppm) of Material Required to Achieve 67% Drag Reduction in Pipe Flow at Re = 14,000 Guar (J-2FP) Colloid HV-6* (refined Guar) Polyox WSR-301 Colloid HV-2* (refined Guar) *Source of polymer: Stein, Hall and Co. Source of other materials listed in Table 1. OTHER EXPERIMENTS The drag reduction phenomena has been suggested [12] as a possible expla- nation of certain erratic fluctuations of measured resistance in some towing tanks.© Frictional drag measurements on the same model in the same towing tank are known to be subject in some tanks to considerable variation, always down from the "'standard,'' and as much as 14%, with no other complete explana- tion than a ''change in resistance characteristics of the water.'’ Since it is known that many algae and marine organisms secrete mucous or slime, it is conceivable that these may act in the same manner as the compounds studied above. 5Data taken by the Western Co. under U. S. Navy contract. 6it appears that these fluctuations are reduced in tanks where the water is chemically purified. 954 The Effect of Additives on Fluid Friction The experiments shown in Table 4 were not intended to be rigorous, or even very quantitative, but simply tests to show the possibility that organic materials similar to those which might be present in towing tanks or other hydrodynamic facilities would affect the measured drag. Table 4 Drag Reduction of Living Materials ; Observed Algae from fresh water 7.6 cm disk aquarium (principally Ankistodesmus falcatus) Same with green cells 7.6 cm disk centrifuged out Green cells resuspended 7.6 cm disk in tap water Bacteria-free culture of .109 cm pipe sea diatom Chaetoceros Same concentrated to .109 cm pipe 1/6 volume Slime from sea snail in .109 cm pipe sea water Same concentrated to .109 cm pipe 1/3 volume Scraped slime from sea .109 cm pipe fish in sea water Same concentrated to .109 cm pipe 1/6 volume The experiments shown in Table 4 were not intended to be rigorous, or even very quantitative, but simply tests to show the possibility that organic materials similar to those which might be present in towing tanks or other hydrodynamic facilities would affect the measured drag. From Table 4 it is seen that sizeable reductions in drag can be obtained from a variety of natural substances. While concentrations required for signifi- cant effect were high enough that the contamination was apparent in these tests, it is conceivable that other, more effective natural contaminants may occur which approach the synthetic polymers in effectiveness at very low concentra- tions. The search for such contaminants in tank water at the time of sucha drag reduction excursion must be directed at concentrations of a few parts per 9595 Hoyt and Fabula million, since 40% friction reduction or more for 2 wppm of high molecular weight polymer is demonstrated in Fig. 15. It is interesting to speculate on the idea that some marine animals might have evolved the release of friction reducing agents into their boundary layer. This appears to be a possible area for further research. APPLICATIONS The only known present application of these materials as friction-reducing agents is in oil-field pumping operations. However, the attractive power reduc- tions which seem attainable should promote extensive interest in the further use of these polymers. In considering applications, however, careful thought must be given to prac- tical matters such as surface roughness, mechanical polymer degradation, and economic feasibility. Surface Roughness A preliminary check on the effect of roughness was made with the large- scale rotating disk facility. The data shown in Figs. 2, 3, and 5, and Table 1 were obtained with a smooth, polished disk. Another disk with about 100 micro- inch rms machine-turned roughness was also tested, but showed no change in torque required for either water or guar solutions. A rough surface was then produced by means of wrinkle-finish paint. In water tests, the torque for a given speed was increased about 35% due to the roughness. Figure 20 shows that two or three times the concentration of guar gum was required to achieve a given torque reduction with the rough disk. Also, effects of rotative speed ap- pear at low guar concentrations in contrast to the smooth disk data. Neverthe- less it seems clear that the additive can be effective on practical structures. Mechanical Degradation The polymer molecules are subject to mechanical degradation as the friction- reduction process continues. For example, concentrations of 15 wppm of poly (ethylene oxide) of 4 million molecular weight were repeatedly tested in the large disk apparatus, with the results shown on Fig. 21. Each test was about 15 sec- onds in duration, repeated at intervals of 3 minutes or 10 minutes. Each test run with this polymer evidently contributed to the mechanical degradation. A Similar test with guar gum did not show this effect, and this is the main reason for continued interest in this less effective, but apparently very sturdy polymer. Economic Feasibility The additive concentrations used in oil-field applications are about 1000 wppm and up [6]. Such concentrations, if assumed across the full turbulent boundary layer thickness are out of the question for boundary-layer applications. 956 The Effect of Additives on Fluid Friction This difference in feasible concentrations is simply because the unit of additive in pipe flow is used over and over again, until the end of the pipe, while in a ve- hicle boundary layer, the unit of additive is effective only on a certain wetted surface area for a certain time, before it is discarded into the wake. This key difference can be seen more clearly by comparing fully estab- lished pipe flow and high Reynolds number flat-plate boundary-layer flow. A useful measure of performance is Additive Effectiveness (A.E.) = —— = with units of hp - hr/kg. In the following it is assumed that speeds are kept fixed and that only pipe length and plate length are varied. For pipe flow, assume that an additive weight concentration per unit volume, C, produces a percent pressure-drop or friction reduction, R, for fully estab- lished turbulent flow in a pipe of diameter, D, for a throughput of Q with the mean flow velocity U = 4Q/7D?. The pumping power saved is RP, where P, is the power required for C = 0. Thus RL(P(/L) RL CQ E x const. where P|/L is the pumping power per unit length for C = 0. Thus if polymer degradation is negligible, A.E. will increase indefinitely with L. For boundary-layer flow, one can assume for a first approximation that the local percent skin friction reduction will require about the same mean concen- tration across the turbulent boundary layer thickness, 6, as in pipe flow for 6 = D/2 and freestream speed U, = U. The friction reduction factor, R, will be assumed to be determined by C as in the typical pipe flow results given earlier. Because the additive concentration in the turbulent boundary layer will be continually reduced by mixing as the boundary layer thickens, more additive will have to be injected at intervals along the plate length, or else the concentration will have to be very large near the leading edge. In either case, the total addi- tive supply rate per unit width will be C(5- 6*) U,, where &* is the boundary- layer displacement thickness. If 6, is the momentum thickness for C = 0, then the thrust power saved per unit width is ine) (ay) WEN Thus for a flat plate Re f2)U3 Additive Effectiveness = Reo Eee : CG com Us 957 Hoyt and Fabula For high Reynolds numbers, reasonable approximations are f= 8 = ikh = ik"@ where k and k’ are constants as L is varied. Thus A RO, ah = ce x const. and since 6/6 = 1-R is R A.E. = eGas x const. Thus in boundary-layer applications the additive effectiveness is helped by the reduced boundary-layer throughput as R is increased, but the increase with L is lost. Since a concentration ofa few wppm is 2 to 3 orders of magnitude smaller than used in the pipe-line applications, the newly discovered effectiveness at such concentration of the extremely high molecular weight, linear, soluble poly- mers now makes the situation more hopeful for boundary-layer applications. (Fortunately, for such applications, the extreme sensitivity of the same poly- mers to mechanical degradation may not be a major problem since the use-time of the polymer is short.) However, calculations indicate that even the increase in the factor 1/C by about 1000 still leaves the technique of reducing ship fric- tion by boundary-layer additives economically uncompetitive. Hence until additive costs can be brought considerably lower, this method of drag reduction appears to be reserved for applications where an emergency speed increase would be required. Of course, in an application where a large proportion of the total drag is frictional, such as a slow speed ship, the tech- nique may look economic. In any event, the applications of the rather basic experiments presented here are difficult to foresee. Certainly the possibilities of achieving substantial drag reductions with relatively small amounts of additive are attractive enough to warrant intensive further effort. REFERENCES 1. Toms, B. A., ''Some Observations on the Flow of Linear Polymer Solutions Through Straight Tubes at Large Reynolds Numbers,'' Proceedings of the International Rheological Congress, Scheveningen, Holland, 1948, pp. II- 135-41. 2. Mysels, K. J., "Flow of Thickened Fluids," U.S. Patent 2,492,173, Decem- ber 27, 1949. 3. Shaver, R. G., and E. W. Merrill, ''Turbulent Flow of Pseudoplastic Poly- mer Solutions in Straight Cylindrical Tubes,'’ AM INST CHEM ENGR, J, Wl. D3 NOs BIL), oe Weil. 958 10: las 12. The Effect of Additives on Fluid Friction Dodge, D. W., and A. B. Metzner, ''Turbulent Flow of Non-Newtonian Sys- tems,'’ AM INST CHEM ENGR, J, Vol. 5, No. 2 (1959), p. 189. Dever, C. D., R. J. Harbour, and W. F. Seifert, ''Method of Decreasing Friction Loss in Flowing Fluids,'' U.S. Patent 3,023,760, 6 March 1962. Melton, L.L., and W. T. Malone, ''Fluid Mechanics Research and Engineer- ing Application in Non-Newtonian Fluid Systems,'’ SOC PETR ENGR, J, March 1964, p. 56. Massachusetts Institute of Technology. The Effects of Fibers on Velocity Distribution, Turbulence and Flow Resistance of Dilute Suspensions, by J. W. Daily and G. Bugliarello. Cambridge, Mass., MIT, October 1958. (Hydrodynamics Laboratory Technical Report No. 30.) Lumley, J. L., ''Turbulence in Non-Newtonian Fluids,'' PHYSICS OF FLUIDS, Vol. 7, No. 3 (March 1964). Whistler, R. L., and J. N. BeMiller, Eds., 'Industrial Gums'’ Academic Press, New York, 1959. Fabula, A. G., ''The Toms Phenomenon in the Turbulent Flow of Very Dilute Polymer Solutions,'' Fourth International Congress on Rheology, Brown University, August 1963. Interscience Div., John Wiley (In press). Ripken, J. F., and M. Pilch, ''Studies of the Reduction of Pipe Friction with the Non-Newtonian Additive CMC," St. Anthony Falls Hydraulic Laboratory Technical Paper 42, Series B, April 1963. Hoyt, J. W., and A. G. Fabula, ''Frictional Resistance in Towing Tanks,"’ Tenth International Towing Tank Conference, London, September 1963. * * x DISCUSSION H. Schwanecke Hamburg Model Basin Hamburg, Germany At Hamburg Model Basin experiments are performed concerning the effect of polymer solutions on the viscous drag of the model of a surface ship. Addi- tives of several molecular weights are used. The main problem is to distribute the polymer solutions all over the wetted surface as a film of sufficient concen- tration. May I ask Dr. Hoyt, if he has done any experiments in that way or if he knows about such experiments having been performed elsewhere? May I ask 959 Hoyt and Fabula Dr. Hoyt further, if there is any upper limit with respect to the molecular weight of the additives beyond which a film of a polymer solution is no longer obtained ? * * * DISCUSSION Marshall P. Tulin Hydronautics, Incorporated Laurel, Maryland The authors are very much to be congratulated for their fine experimental studies. Their data should tend enormously toward a better understanding of the unexpected and puzzling effects which small concentrations of macro- molecules seem to have on turbulence. Our own experiments on a flat plate with leading edge injection confirm that a maximum drag reduction results when as little as 10 parts per million of Polyox WSR 301 is present in the boundary layer at the trailing edge. Unlike the flows in pipes and on rotating plates, however, a rather rapid decrease in effectiveness of the additive occurs when the concentration is increased only slightly beyond its optimum value. Perhaps this has to do with the special cir- cumstances which accompany injection of the fluid containing additive. We were curious whether macromolecules would affect ''free'' decaying turbulence as distinct from maintained turbulence in a shear flow in close prox- imity to a wall. Therefore we have studied the decay of a cylindrical cloud of turbulence. Rather, we have measured the diffusive spread of the cloud. These experiments show that additives do affect free turbulence and tend to increase the rate at which it decays. I have been doing some theory on the effect on turbulence of weak solutions of macromolecules. It seems to me that the shear stiffness of the resulting viscoelastic solution is the crucial characteristic and that the generation of elastic shear waves by turbulence offers a mechanism for significant 'damping" of turbulent motions. Figure 5 contains very clear evidence that the elastic shear stiffness controls the turbulence damping effect; it may be shown using certain results of the molecular theory for weak polymer solutions that this stiffness is virtually constant on the lines of constant torque ratio in this Figure. * * * 960 The Effects of Additives on Fluid Friction REPLY TO THE DISCUSSION J. W. Hoyt and A. G. Fabula U. S. Naval Ordnance Test Station Pasadena, California The authors would like to thank the several discussors for their comments regarding this interesting new field in fluid dynamics.* It is to be hoped that theoretical attacks on the mechanism of drag reduction through the use of high polymer solutions will soon give the firm foundation needed for further advances in the application of the method. Perhaps the approaches of Prof. Eringen and Mr. Tulin will provide the keys to this understanding. With regard to Dr. Schwa- necke's questions, the only published data now available on the ejection of ad- ditives over a body seems to be the Vogel and Patterson paper in this Sympo- Sium. Our experience with various molecular weight additives seems to indicate that the higher the better, if the molecule is also fairly linear. Fig. 1 - Large rotating-disk apparatus *See contribution by Eringen to the paper by Lumley. 221-249 O - 66 - 62 961 Hoyt and Fabula PERCENT TORQUE REV/SEC 20 fe) Bo “oF 40 TL 2) ee ee == “o as | 20+ | lob | ple l l l i l l 500 1000 1500 2,000 2,500 3,000 3,500 CONCENTRATION, WPPM Fig. 2 - Rotating-disk torque curves for guar additive 100¢ T T T T T T T ian T I T 1 30 4 SYMBOL MOLECULAR WEIGHT 80 A #% 2x10° || oO x 6x 10° ° ~ 4x 10° 707} © > 8 xio® 1 DISK TORQUE RATIO IN PERCENT < l tL ! L Nl 1 n O 1 1 1 1 1 50 100 150 200 250 300 350 400 450 500 550 600 650 CONCENTRATION, WPPM Fig. 3 - Rotating disk-torque curves for poly(ethylene oxide) additives 962 The Effect of Additives on Fluid Friction ee -CH.- OjCH:CH,-O}CH:CH.” Poly(Ethylene Oxide ) CheOH H 4 CHpOH ao) Qo 0 Hf Vy \ OH O# / \ wy ‘ Vi OW OH HW Ow OH Oo oO Oo HW H# : le HW H H OH O OW H H ro) ChQ OH f? Guar Fig. 4 - Chemical formulas of two effective additives 963 60 on fe) | 1S (eo) Ol fe) TORQUE REDUCTION, % pe) fe) Yo fe) C, CONCENTRATION, WPPM Hoyt and Fabula POWER-LAW FITTINGS C~m™* x=0.43, 0.47, 0.49 DISK TORQUE RATIO | IN PERCENT 35 100 10 ht 1 1 a a 1 n ES SS Se 10° 108 10" M, POLYMER MOLECULAR WEIGHT (APPROXIMATE ) Fig. 5 - Dependence of required concentrations for various disk torque ratios on molecular weight of poly(ethylene oxide) 4 R AN Rn 4000RPM — A a 0 3,000 ras a O | O 5 2,000 I O 100 200 300 400 CONCENTRATION, WP PM Fig. 6 - 7.6 cm disk torque reduction vs guar gum concentration 964 The Effect of Additives on Fluid Friction 50 4,000 RPM b ° 3,000 1% w {e) 2,000 TORQUE. REDUCTION jo) fo) io 20 30 40 ~ 100 CONCENTRATION, WPPM Fig. 7 - 7.6 cm disk torque reduction vs poly(ethylene oxide) concentration ‘SD eel eam ee nal ab J =] 60 J 3250 L =| 2 (e) . = 2 40 a Gi | jog 30 2 F DISK DIAM.,CM is ra O76 = 2 A 0 457 i A 762 4 iobk 5 SE MEE ee ete ee ae Tec os melee Mea a 10° 10° 10" 108 DISK REYNOLDS NO. Fig. 8 - Maximum torque reduction as a function of Reynolds number 965 WALL SHEAR STRESS,DYNES/CM= Hoyt and Fabula 40 aT T Se = i 30 fF GUAR IN SIMULATED SEAWATER 20 + GUAR IN LAKE WATER nm (0 3 9 5| 5 8 | i x 7 | o Oo 6 5 4 4t 4 3 = 2 1 1 1 1 4 1 —! 10 5 20 25 30 40 50 60 DISK SPEED, REV/SEC Fig. 9 - Effect of seawater on guar solution 4 10 [— ] 5 3 10 F + 5} CONCENTRATION ia “| WPPM:10 000 2 og ———— a [= | g000 3) ae lt ia 2,500 \ “$A HY 5 == 000 Ly =F ale ’ | 500 ‘° 250 TEMPERATURE 19°C : | | 10 L | shige =e aes ee _| oe ee) 190 5 0! 5 02 5 103 5 104 5 109 WALL SHEAR RATE, SEC"! Fig. 10 - Rheogram for guar additive in water at 19°C 966 5 106 The Effect of Additives on Fluid Friction 107 ,OISTILLED WATER WALL SHEAR STRESS, DYNES/CM“ TEMP=25.0£0.1°C io! flee ia 10° io* 10° ie WALL SHEAR RATE, SEC™! Fig. 11 - Rheogram of poly(ethylene oxide) of 4 million molecular weight in water 967 WALL SHEAR STRESS, DYNE/CM? Hoyt and Fabula O DEIONIZED WATER @ 100 WPPM © 200 WPPM & 400 WPPM 0 1,000 WPRPM TEMP= 225225 10! 10° 10' 10 WALL SHEAR RATE, SEC™! Fig. 12 - Rheogram for poly(ethylene oxide) of 4 million molecular weight at low shear rates 968 The Effect of Additives on Fluid Friction SYMBOL C,WPPM nm - oqdO°p+0 eume)) NN ——s GOLDSTEIN FORMULA FOR TURBULENT NEWTONIAN FLOW TORQUE /(4/2)W2R> ° © Sgog000 VVVVVVY TORQUE COEFFICIENT Cr 2 ly Sener 2 3 1o& fe) REYNOLDS NUMBER BASED ON WATER Fig. 13 - Torque coefficient as a function of Reynolds number HIGH PRES. AiR ——~> REGULATOR ot S ro) 5 o ce) 2 cv ro) 2 es =< a N m t | yee. | | I 2 3 6 4_1.02-cM-ID PIPE WITH SIX PRES. TAPS TUBE INLET CONTRACTION HOSE MIXING WEIGHING TANK BARREL Fig. 14 - Schematic diagram of blowdown pipe apparatus 969 8 (0) DRAG REDUCTION,% £ 100 DRAG REDUCTION, % a ~ .e) fo) Oo (o) Hoyt and Fabula a 'om/sec | 5 M/SEC (eo) i fe) ] seb |.02-CM ID PIPE al l | eee | 2 5 10 50 100 POLYMER CONCENTRATION,WPPM Fig. 15 - Drag reduction curves for poly(ethylene oxide in blowdown pipe apparatus a (os) W fe) fe) D ON fo) silted [pease 0.46 CM ID PIPE Roe | pa | ee oll Te) 10 100 000 10,000 POLYMER GONCENTRATION, WPPM Fig. 16 - Friction reduction curves for poly(ethylene oxide) in pipe-flow apparatus 970 The Effect of Additives on Fluid Friction PRESSURE TAPS AUXILIARY, LARGER DIAMETER FILLING PIPE NEST PIPE 0.1I09-CM INNER DIAM pe THREE-WAY VALVE 100-CC HYPODERMIC SYRINGE CONSTANT -SPEED |/2-HP MOTOR BRAKE Fig. 17 - Turbulent-flow rheometer oa DRAG REDUCTION ,% Hoyt and Fabula 0.109-CM. 1D PIPE 10 20 30 40 50 60 740) 80 90 POLYMER CONCENTRATION ,WPPM Fig. 18 - Turbulent-flow rheometer data for poly(ethylene oxide) 972 The Effect of Additives on Fluid Friction 100 @ (eo) ENVELOPE OF MAXIMUM DRAG REDUCTION 30WPPM IOOWPPM PIPE DIAM..CM 4 fay 0.109 fe) '@) 0.46 x x) 102 DRAG REDUCTION,% £ a (e) (e) 20 10° 1o* 10° \o8 PIPE FLOW REYNOLDS NO. Fig. 19 - Reynolds number correlation for pipe flow 90 - 80 - WRINKLE-FINISH DISK, + TA) = REV/SEC PERCENT TORQUE oD (eo) T 20 © | 40+ | SMOOTH DISK 30 : os aa fe) — 1 1 ssssesaa! sesh = 1 == Hl ) 50 100 150 200 250 300 CONCENTRATION, WPPM Fig. 20 - Effect of high surface roughness on the percent torque reduction with the 45.7 cm diameter disk 973 DISK TORQUE AT 40 REV/SEC,% Hoyt and Fabula (ee) oie) o> R ib & 60+ 0° A A 4 (2) A A ie) A fe) am & Oo 50+ 4 A (e) A 1 TYPICAL TIME SYMBOL BETWEEN TESTS 40r S| (e) 3 MIN A lO MIN 30 4 o) aire yucese SE 1 1 mei 1 | O 20 40 60 80 100 120 140 60 TIME AFTER START OF MIXING, MIN Fig. 21 - Rapid mechanical degradation of a dilute poly(ethylene oxide) solution of 15 wppm concen- tration, seen in disk tests x * * 974 AN EXPERIMENTAL INVESTIGATION OF THE EFFECT OF ADDITIVES INJECTED INTO THE BOUNDARY LAYER OF AN UNDERWATER BODY W.M. Vogel and A. M. Patterson Pacific Naval Laboratory Victoria, British Columbia, Canada ABSTRACT The effects of injecting solutions of three linear, high molecular weight polymers into the boundary layer of a three-dimensional streamlined model are being investigated. The following are the preliminary results of this experiment: (a) The drag of the body decreased with increasing molecular weight of the polymer. (b) The drag decreased as the concentration of the polymer solutions increased. At concentrations above 500 ppm for the highest molecular weight polymer used, the amount of drag reduction decreased. (c) For increased flow rates of the polymer solution, the drag reduc- tion increased. (d) The flow rate of the solution injected into the boundary layer, and not the injection flow velocity, was the controlling factor at the injec- tion velocities used. (e) Turbulence and average velocity measurements in the wake of the body indicated two effects when the polymer solution is injected: a change in the mean square of the turbulence velocities, and a change in the velocity profile. INTRODUCTION B. A. Toms (1949) pointed out that, in turbulent pipe flow, dilute solutions of linear polymers reduced the pressure drop along the pipe to a value below that of the solvent. Since then there has been a growing body of literature on the flow of polymer solutions which exhibit non-Newtonian characteristics. Experimen- tally, most of the studies have been concerned with the rheological characteris- tics of the fluid, or with pipe friction. The work by Shaver (1957) showed that in 975 Vogel and Patterson some cases the addition of a long molecule polymer to a liquid resulted in higher friction losses at low flow rates and lower friction losses at high flow rates. This is analogous to the behaviour observed by Daily and Bugliarello (1961) in wood fibre suspensions in smooth pipes. Shaver and Merrill (1959) observed the velocity profiles of dilute polymer solutions in circular pipes and found that at high flow rates the profiles were sharper than for a corresponding Newtonian flow. This did not check with Dodge and Metzner's (1958) prediction of profiles blunter-than-Newtonian. This disagreement was attributed by Dodge and Metz- ner to the possible presence of elastic effects in the fluids used by Shaver and Merrill. Recently Fabula, Hoyt and Crawford (1963) investigated about twenty-five water-soluble polymers. They discovered that whenever the polymer had both a high molecular weight and a linear molecule, significant reduction in friction occurred in the high Reynolds number flows (Re > 10°). .This phenomenon was first observed with a rotating-disc apparatus and later confirmed using a pipe flow apparatus. The very dilute solutions studied were often superficially indis- tinguishable from plain water, and their apparent viscosities for the high shear rates involved were nearly that of water. One of the polymers tested in the rotating disc apparatus, poly(ethylene oxide), gave about a 70% torque reduction for a .01% solution (Hoyt and Fabula 1964). In the pipe flow apparatus, cases of 50% pipe friction reduction were found for very dilute solutions of poly(ethylene oxide) in water (Fabula, 1963). Because of these large changes in the turbulent flow produced by very low con- centrations of poly(ethylene oxide) in water, it was proposed to study the effect of these polymer solutions when they were injected into the boundary layer of an underwater body. EXPERIMENTAL APPARATUS There are a number of variables which should be considered when a fluid is injected into the boundary layer of a body. These are: 1. Type of polymer solution 2. Concentration of the polymer solution 3. Velocity of injection of the solution 4. Position of the injection slot 5. Tunnel velocity A body of revolution (Fig. 1) was chosen as the most convenient to use for these exploratory experiments. Because our low-turbulence water tunnel has a working section 35cm by 35cm, the maximum diameter of the body was limited to about 5cm so that tunnel blockage would be minimized. The body as finally constructed was 41.3cm long and had a maximum diameter of 5.08cm. Five slots were made in the body; the positions of the slots are shown in Fig. 2. The section 976 Additives Injected Into the Boundary Layer of an Underwater Body Fig. 1 - Model used for drag reduction experiments 41.3cm. 22.7cm. 13.2cm. Tubes to slots support Nose Slot 2nd.slot 3rd.slot 4th.slot 5th. slot Fig. 2 - Schematic of model 221-249 O - 66 - 63 977 Vogel and Patterson up to the second slot is elliptical in shape, the section from the second to the fifth slot is cylindrical and the tail section is faired to a pointed trailing edge. Each slot can be individually adjusted, and the polymer solution can be injected through each slot independent of the flow through the other slots. Only the nose slot was used for this work. The wings are elliptical sections symmetrical about the line joining the lead- ing and trailing edges. They are mounted on the body so that the planes of sym- metry of each wing pass through the centre-line of the body. The wings serve two purposes: to support the body in the three-component force balance, and to act as a Shield for the lines running to the slots. Three lines are in one wing and two lines in the other. The pumping system and tunnel set-up are shown in Fig. 3. In order to min- imize the possibility of degrading the fluid before it is injected into the boundary layer it was decided to use air pressure as the pumping force. When the fluid is mixed it is drawn into the pump by reducing the air pressure in the pump. To inject the fluid into the boundary layer a known positive pressure is applied to the pump. The pressure is adjusted until the desired flow rate is achieved. The flow rate is determined by using a stopwatch to measure the time required for a known volume of the fluid to leave the pump. ‘luorometer | Vacuum a) | a | 4° Line | y ea a4 Turbulence Hlectronics Fa di z man Ba aa ; : lance = | 2 cd » ? ers i z a & ‘ < } K s f Bf Ys ‘ aN H a : a s i ¢ 2: RK ; x : | i + cm. | 2 CROSS-SECTION OF WAKE 7cm DOWNSTREAM OF BODY- WITH 100ppm. POLYOX WSR- 30! INJECTED FROM NOSF cint Figure 17 990 Additives Injected Into the Boundary Layer of an Underwater Body TURBULENCE STUDIES OF THE WAKE To obtain more detailed information on the effect of the injection of a poly- mer solution on the drag, a study of the turbulence in the wake was made using two concentrations of POLYOX WSR-301, 100 ppm, and 500 ppm. A tunnel ve- locity of 400 cm/sec and an injection flow rate of 30 ml/sec through the 0.25 mm nose slot were used for all the runs. A hot-film probe was used to measure the turbulence and the average ve- locity in the wake. The sensitive element is a thin platinum film, mounted near the tip of the conical nose of the probe, which is maintained at a constant tem- perature by appropriate electronics (Evans, 1963). This type of probe essen- tially responds to the turbulence fluctuations in the direction of the mean veloc- ity in the tunnel (Ling, 1955). The probe was mounted in the same holder as used for the dye work, with the platinum film 7 cm behind the tail of the model. This is the same position as used for the dye runs. The frequency response of the probe is reasonably flat to about 1000 cps, and the average velocity output of the equipment was adjusted so that it had a linear response over the velocity range encountered in the wake. The procedure in each set of runs was to record the turbulence signal for about a minute starting at the centre of the wake without a polymer solution in the boundary layer. The polymer solution was then injected, and the turbulence again recorded for about a minute. The average velocity at that position in the wake was read from a meter, and any changes in the velocity when the polymer solution was injected were recorded. After each run the probe was moved up 0.5 cm and the procedure repeated until the probe was out of the wake. The recorded turbulence signal was passed through a digitizer with a sam- pling rate of 2500 samples/sec, and the resulting digital tape was processed on our Packard-Bell PB-250 computer to obtain the mean square of the turbulence velocities, and the power spectrum of the turbulence at the probe positions in the wake. Figures 18 and 19 show the average velocity profiles and the mean square of the turbulence velocities for the wake with no fluid injection, and with 100 ppm, and 500 ppm solutions of POLYOX WSR-301 injected at 30 ml/sec. The tunnel velocity for these runs was about 400 cm/sec. Two effects of the poly- mer solution injection are shown. For the 100 ppm solution the mean velocity increased and the turbulence level decreased; for the 500 ppm solution, the mean velocity increased markedly, but the turbulence level also increased. There is a reading error of about plus or minus 2 cm/sec in the average veloc- ity curves. The significance of the differences shown in Fig. 19 for the probe positions between 2 cm and 3 cm above the wake centre-line is not known. Figure 20 shows a set of spectrum of the turbulence taken 0.5 cm above the wake centre-line for the 100 ppm solution. Figure 21 shows three sets of spec- tra for the 500 ppm solution taken at the centre-line of the wake, 1.5 cm above the centre-line, and 3 cm above the centre-line. These curves are plots of log ¢ vs log k where ¢ is the mean square of the turbulence velocity per unit wave number k, and k = 2, (frequency of the turbulence signal) divided by the average velocity passed the probe. 991 Vogel and Patterson PROBE POSITION 7em. DOWNSTREAM OF BODY TAIL. ol fe) ©- No polymer injected A - 10Oppm. WSR-30! injected at 30mI/sec. oO [e) ©- No polymer injected. A -l1O0Oppm. WSR- 301 injected at 30mlI/sec. © uo WS) a = Ly) u fe) = by a fe) fe) 9 a Vertical distance from centre- line of wake — cm. Vertical distance from centre -line of wake - cm. (@) A"——_O 20 300 400 ifo) 20 30 40 50 60 Velocity — cm/sec. Meon square turbulence velocity - (cm/sec.)@ (a) (b) Fig. 18 - (a) Average velocity, (b) mean square turbulence velocity through wake PROBE POSITION 7cm. DOWNSTREAM OF BODY TAIL. 3. ©-No polymer injected. 3.0104 ©- No polymer injection. & - 500ppm. WSR - 30! A- 500ppm. WSR- 30! injected at 30ml/sec. injected at 30mi/sec. iw i) S f oO uo S) a Vertical distance from centre-line of wake - cm. Vertical distance from centre -line of wake -cm. 200 300 400 20 40 60° 80 {00 120 Velocity - cm. Mean square turbulence velocity - (cm/sec.) (a) (b) Fig. 19 - (a) Average velocity, (b) mean square turbulence velocity through wake 992 Additives Injected Into the Boundary Layer of an Underwater Body 1.0 a i 0.5 _--No additive ; probe O.5 cm. above wake centre-line Additive —__ 2 O probe O.5 cm. above wake centre-line log © -1.0 Tunnel velocity - 400 cm./sec. 100 ppm. Polyox WSR-30I Flow rate — 30 ml./sec. -1.5 log k Fig. 20 - Turbulent velocity power spectra The curves in Fig. 20 show that the injection of the polymer solution re- duced the turbulence intensity over the frequency band analyzed. The highest frequencies used for this analysis are about 1200 cycles per second because tl frequency response of the hot-film probe falls off in this region. In Figure 21 the curves for the probe at the centre-line of the wake show that, for the smal wave numbers, the turbulence energy is increased when the fluid is injected, k as the wave number increases the curve for the additive crosses the other cur and the energy at the higher wave numbers is less for the polymer in the boun ary layer. For the 1.5 cm position, the additive curve is again higher, and the point at which the curves appear to cross over is at a higher wave number tha for the centre-line case. With the probe 3 cm above the centre-line of the wal the character of the signal, as observed on an oscilloscope, contains many lar spikes which indicate that the wake turbulence is intermittent in this position. The curve for the additive case are still higher than for the wake without the < ditive but the slope of the curve indicates that a cross-over might occur ata wave number larger than for the 1.5 cm case. DISCUSSION OF EXPERIMENTAL RESULTS This is essentially an exploratory experiment which attempts to add to th knowledge of the behavior of polymer solutions in reducing the friction of a fl along a solid surface. Previous work by Fabula et al (1963) had shown that th 221-249 O - 66 - 64 993 Vogel and Patterson +_,-Additive - probe centre-line of wake Additive - —_, wr | probe up |.5cm nN | SSO GB a No additive - m o probe up (sem. OSS | OQ i i i OF Additive -—_., probe up Scm. “~~ Ri Ne No additive - probe centre-line ot woke “4 * log : \ | Ne + q No additive —.+__ Sr \, \ probe up 3cm. Sn *, ~~ +. .045 BL Qear g, = local velocity head outside boundary layer. According to the reference for laminar stability, the Reynolds number based on u,, the local velocity outside the boundary layer, and the boundary layer thickness should not exceed approximately 3400. In this case, the boundary layer thickness was defined as the value of 5 for which u/u, = 0.707. A velocity profile through the boundary layer of the form ues 2 ay Dig) at] was assumed. In lieu of calculating the actual boundary layer profile, this seemed to be a reasonable choice since when based on the displacement thick- ness, this profile lies between that of Blasius and the asymptotic suction profile. These results were then substituted into the Karman momentum integral equation and the boundary layer thickness obtaining by numerically integrating along the length of the body for different values of AQ/Q,,. At each increment in length, x, along the body the value of the boundary layer Reynolds number was compared to the critical value quoted by Ref. 1. If R, equalled, or possibly slightly exceeded 3400, the value of x was printed out and Q through the slot at the x location calculated from A OS 2m ky ea Qpr- (3) BL 5 immediately after the slot was calculated from Eq. (1) and the numerical in- tegration continued to the next slot location where R, once again reached a value of 3400. In this manner the slots were located along the body and the total suction flow requirement determined as the sum of all the Q’s obtained from Fae): After a series of trial designs, a value of AQ/Q,, of 0.17 was selected. This resulted in slots 0.007 inches thick spaced every 2 inches starting 6 inches back from the nose. These continue back to within 7 inches from the tail at which point the axial flow pump is installed. On the recommendation of Ref. 1, the slot thickness was chosen equal tothe boundary layer thickness. The required suction 1003 McCormick flow quantity was calculated to be 1.37 cfs for a design velocity of 50 fps. The pump was estimated to require 8.34 hp. In order to integrate the Karman momentum equation it is necessary to know the body radius and static pressure distribution. The body shape of TRI-B is composed of three parts: (a) a modified ellipsoid nose, (b) a parallel mid- section and (c) the afterbody of the DTMB series 4166 body. The final shape is similar to a Reichardt constant pressure body. For this shape, the pressure is nearly constant over 80% of its length be- ginning 5% back from the nose. The measured pressure distribution obtained from wind tunnel tests is presented in Fig. 2. Included on the figure are empir- ical expressions which were used in the numerical integration. At 50 fps, the laminar skin-friction drag on the body was estimated to be 11.7 lbs. The drag of the ring tail was estimated at 29.4 lbs giving a total drag of 41.1 lbs. If laminar flow were not achieved, the body drag was estimated to be 132 lbs giving a total drag of 173.1 lbs. The pump was designed to eject the suction flow at 50 fps; hence in evaluating the drag from the terminal velocity, thrust (or drag) from the pump must be considered. In terms of equivalent flat plate area, f = D/q, f was predicted to be equal to 0.069 for a turbulent boundary layer and 0.0164 for a laminar one. The method by which the body was designed has been presented only briefly because of various shortcomings in the method which became obvious as the project progressed. These will be discussed later in the paper. TESTING OF TRI-B The first tests of TRI-B began October 1962 at the U.S. Naval Torpedo Sta- tion, Keyport, Washington. Over a period of two months, 12 runs were per- formed of which 7 yielded valid data. A photograph showing the body exiting from the water is presented in Fig. 3. For these runs, only the velocity as a function of time was measured. From the results, the disappointing conclusion was reached that the expected laminar flow had not been achieved. The fact that the boundary layer with the pump operating was turbulent was substantiated by running with a trip ring on the nose for which the body attained the same terminal velcoity as without the ring. There were several possible reasons at this time why laminar flow was not being achieved. First, a calibration of the suction pump showed that at the de- sign hydraulic pressure of 2000 psi, it was delivering only 0.85 cfs instead of the design value of 1.37 cfs. Secondly, the suction slots were not continuous around the circumference, instead they were interrupted by small, structural, carry- through bridges. Finally, the body exhibited a tendency to depart from the ver- tical in its travel to the surface. 1004 Drag Reduction by Suction G-IUL IF uotynqia3stp einsseaq - 7 ‘31q Ol 8:0 Z190 600- ,,,, 2690 = %) X*HLON37T AGOS 4O LN3D Y3d 90 fe) SS Ol vO 12O 1005 McCormick sso Fig. 3 - TRI-B body exiting from water In view of these questions, the program at NTS was temporarily suspended and the body returned to ORL for modification and laboratory tests. Tests were conducted in the low turbulence wind tunnel at the Garfield Thomas Water Tun- nel, a division of ORL, which showed that even at a length Reynolds number of approximately 4.5 x 10°, lower than the design value by a factor of 8, extensive laminar flow could not be achieved. In light of these results, the suction slots were modified to assure a continuous suction around the circumference. When this was done, laminar flow was achieved at the low Reynolds number over ap- proximately 90% of the body as determined by listening to the noise of the boundary layer with a total head tube connected to a stethescope. Hence it ap- peared that the interruptions to the slots were the cause of the difficulties. At this time, tests were run to determine the suction coefficient, C,, re- quired to maintain full-length laminar flow for different length Reynolds num- bers. Cy is defined as where S, is the wetted area. For uniform suction Co is simply the ratio of suction velocity to free-stream velocity. 1006 Drag Reduction by Suction Transition Reynolds number for different length Reynolds numbers are presented in Fig. 4 as determined experimentally in the wind tunnel. The de- sign value of C, at 50 fps is 12.3 x 1074. This value is appreciably higher than an extrapolation of Fig. 4 to the design length Reynolds number of 39 x 10° would indicate is necessary. This was the first indication that the design method was not sufficient. 6 Be a HO TRANSITION REYNOLDS NUMBER ~ RS 5 1x10 Oe 4 6 8 10) 12 46 SUCTION COEFFICIENT ~ Cgx 10° Fig. 4 - Transition Reynolds number vs suction coefficient Field testing of TRI-B was resumed at NTS in March 1964. In addition to having the slots modified, the body incorporated more refined instrumentation. 10 channels of information were recorded on a galvanometer; measured were the velocity, depth, pressure across the pump, time, pump rpm, velocities at the tail at 3 different radial locations, and the deviation from the vertical in two mutually perpendicular planes. The project was plagued with instrumentation difficulties throughout the second series of tests. This included the failure of pressure transducers, the sensitivity of differential transducers to change in temperature and absolute pressure and shifting in the zero settings of the bridge outputs, possibly the re- sult of a mechanical hysteresis in the transducers. 1007 McCormick Though somewhat inconsistent, the data: namely, the time history of the velocity, did point to the fact that laminar flow was still not being achieved with TRI-B even though the slots were now modified. However, the tilt traces and visual observations of its surfacing confirmed the fact that the body was under- going violent excursions during its rise to the surface. This had been experi- enced to a lesser degree during the first series of tests and had apparently been cured by adding 10 lbs of lead in the tail. For the second series of tests the CG was even Slightly behind that of the configuration with the lead. At this point, it was realized that the contribution of the hydrodynamic forces on the tail to the Slope of the pitching moment curve completely overshadows that due to the dis- placement between the CG and the center-of-buoyancy above about 10 fps. Hence on a buoyant, vertically-rising body, moving the CG aft improves the stability at low speeds but, due to the shortening of the tail moment arm, is detrimental at higher speeds. At this point in the program wind tunnel tests of a model of TRI-B showed it to be statically unstable, contrary to calculations of its dynamic stability made early in its design. These same tests indicated that an increase in the chord of the tail from 4 inches to 6 inches would provide static stability. Hence a new tail was made and shipped to the field. Successive runs with the new tail showed the stability problems to be solved. The body repeatably rose with no indication on the tilt traces of any deviations from the vertical. Unfortunately, solving the stability problem did not result in a reduction in the drag according to the terminal velocity. Thus in the latter part of May, the body was returned to ORL for additional laboratory studies. It is planned to test this body in the Garfield Thomas Water Tunnel at the design Reynolds number. However, these tests must await the installation of a honeycomb in the tunnel designed to reduce the turbulence in the test section to a level acceptable for such tests. ANALYSIS BASED ON KARMAN-POHLHAUSEN METHOD The analysis on which the design was based was felt to be inadequate for several reasons. The assumed velocity profile was too approximate. In addi- tion the stability limit having a fixed value did not consider the dependence of the stability of a laminar layer on the shape of velocity profile. Also there was no means to calculate the change in velocity profile across the slot. An exact prediction of the stability of laminar boundary layers involves the solution of the eigen-value problem defined by the Orr-Sommerfeld equation. Fortunately, enough cases, with and without suction have been investigated so that one is able to specify a stability limit, Reta as a function of some meas- ure of the shape of the velocity profile. Figure 5 taken from Ref. 2, presents Rsv , as a function of the shape parameter H, the ratio of displacement thick- ness to momentum thickness. More recently Tollmein in Ref. 3, presented the curve shown in Fig. 6. Here, the shape parameter used is related to the curva- ture at the wall measured in terms of displacement thickness. Qualitatively both criteria are in agreement. A profile having a relatively higher velocity at the wall will have a greater value of K and a smaller value of H. 1008 Drag Reduction by Suction crit 20 21 22 23 24 25 26 H= 8°70 Fig. 5 - Stability limit Rs* vs shape parameter H Hence the problem is reduced to calculating the boundary layer growth over the body and, at each location along the body, comparing Rs* to Rs*_,, obtained from Figs. 5 or 6. To do this, the Karman-Pohlhausen method modified to account for suction at the walls was used. A brief outline of the method follows. The velocity profile is represented by a fourth order polynomial (4) 1009 221-249 O - 66 - 65 McCormick ASYMPTOTIC 6 SUCTION O 0.1 0.2 2 can | J vo dy WALL Fig. 6 - Stability limit Rs* vs shape parameter K where er Sepa The boundary conditions to be satisfied for a suction velocity of v, are: ou 2S 1 dp iy, 37u (5) 1010 Drag Reduction by Suction At VS. 8 = eee eg ay ay? From the above it can be found that: 12 + A 66—3 A fl. = — - b = —— 6 + pb 6 + B (6) =12-86 +34 6+36 - A e264 2 6 + B 6 + B where Les du, ; v,° - yp dx Ean ty The displacement thickness 5* can be expressed as: b c d (7) while the momentum thickness is given by: 2 2 2 g= 8-9 |S + 2, Gactd®) , Gadtbe) , Greed a, a (8) The shape parameter K can be calculated from 2 La. (9) For an axi-symmetric body with suction, the Karman momentum integral equation is written as du 2 dé * 1 2 @ dr at fo) Cie (22 8) Wy a aas aul gee (10) to} r, is the body radius at any x. In the above all velocities are dimensionless with respect to the free-stream velocity U, and all distances with respect to a reference distance. The dimen- sionless shearing stress 7,/oU,” can be determined from 1 uy a 1011 McCormick The dimensionless velocity u, isfound fromthe static pressure distribution, uy = 1-Cp STR ead) Sr (12) dx De, Cb where C,, the pressure coefficient is defined by 1 = |p) Sa OL if 2 mets Cp = The numerical integration is started close to the nose by estimating 5* on the basis of the exact solution of viscous flow near a stagnation point. 6 is then assumed to be equal approximately to 3 5*. Actually the ensuing integration depends very little on the initial choice of 5. Knowing 6, C,(x) and specifying Vv.) One can now integrate Eq. (10) numerically along x with the aid of Eqs. (6), (7) and (8). This integration has been carried out for the two Reynolds numbers of 2.3x 10° and 39x10° and for Cg values from 0 to .0003. The lower Reynolds number is typical of the wind tunnel tests while 39<10° represents the design value. The results of these calculations are presented in Figs. 7 and 8. In each case the suction was assumed distributed uniformly over the body starting 6 inches back from the nose. Also included on each figure are the stability limits predicted from Figs. 5 and 6. These results are very interesting and in agreement with the experimental observations. From Fig. 7 for zero suction, the transition point is predicted to lie between 8 inches to 14 inches back from the nose. With a Cg of .0001, the shape parameter H predicts transition 33 inches from the nose while K predicts it at 9.5 inches. Finally for a C, of .0002 both criteria predict laminar flow over nearly the entire length of the body. Observe that the lines of critical R;* and actual R;*, aS Cg increases, become nearly parallel. Hence as Cg is in- creased slightly above some value close to .0002, the transition point shifts sud- denly from the nose rearward. This predicted behavior was observed experi- mentally. It should be noted also that the flow is stable at a Cg of .0002 not simply because the suction is inhibiting the growth of the boundary layer but, equally as important, because the suction is causing the profile to become more stable. Now consider the predictions of Fig. 8 made at the design Reynolds number. Both shape criteria predict transition before the first suction slot at 3 or 4 inches from the nose. Thus it appears that transition may be occurring before the suction can take effect. In fact it appears as if the velocity would have to be reduced to about 17 fps in water to move the transition point behind the first suction slot. However, calculations, not presented here, have shown that a Co starting 3 inches back from the nose would be sufficient to prevent transition. 1012 Drag Reduction by Suction 6 5 = 6 Re 2.3 x10 4 K LIMIT Cg=0.0002 R.¥x 10° H LIMIT Cg=0.0 H LIMIT Cq= 0.000) ka H LIMIT Cg= 0.0002 Rsx Cg=0.0001 (ae Rsx Cg=0.0002 O (@) 10 20 30 40 50 60 70 80 90 x (in) Fig. 7 - Calculated boundary--layer thickness and stability limits for TRI-B according to Karman-Pohlhausen method for low Reynolds number CONCLUSIONS It is concluded from the results of the Karman-Pohlhausen method that the probable reason why TRI-B has not yet achieved full-length laminar flow at 50 fps is due to transition occurring before the first suction slot. Wind tunnel experiments at low Reynolds number and predictions based on the Karman- Pohlhausen method were found to be in close agreement. This method assumes the suction to be continuously distributed over the surface. A method, based on calculating the decrease in 6 across a slot, did not prove fruitful. It was found experimentally that, even at low Reynolds number, continuous suction in the circumferential direction was necessary to the maintenance of a laminar layer. Interruption of the slots probably results in secondary flows or streamwise vortices which cause instabilities. This paper would not be complete without pointing out the experience which has been gained in handling a body of this type in the field. It is very important to provide for proper handling equipment in the planning of such a program. All dollies and packaging equipment must be lined with soft coverings. Field per- sonnel, particular ordinary seamen, must be impressed with the importance of not allowing the slightest scratch on the surface. This is not as easy as it may sound. A navy diver bobbing up and down with the body along side the ship is naturally more concerned with his own skin than with the skin of the body. It 1013 McCormick 6 r Rex C=0.0 Seach) = 6 4 2 Q al! 40 U2 dv If df(x' R = seus ee) | : a a cos [yA(x- x’)] dxdx’ (4) D A Neat eo “a where y = ¢/U?. The area of the water plane is given by the integral 0 e 2 | f(x) dx -0 0 a cltiGs) ae 7) dx > ll (5) Since sf( Ll) = “0. The determination of the function f(x) so as to minimize Michell's integral (4) for a fixed value of A, leads to the equation derived by the theory of calculus of variations, cos [yA(x-x’)] dx’ + kx =0. (6) df(x') x © f feet) te Gee ike, el ae ae an Sretenski [5| concluded that no solution could exist among square-integrable functions, but there is some doubt in his reasoning as was pointed out by Wehausen [6]. Karp, Kotik and Lurye [7] has proved explicitly that the integral equation has really no solution except a trivial case df(x)/dx = 0 when k=0. Being integrated by parts with respect to x and x‘ remembering that f+) = 0, Eq. (4) becomes 1021 Maruo 4p 2? © e e = dA 1 f d (7) Rats ——— f(x) f(x') cos [yA(x-x’)] dx dx’. 7? i a is From this equation, the condition of the minimum wave resistance for a fixed water plane area gives the integral equation | ee | fics cos) Ib\Gx— x. idx! ik = 0): (8) 1 = Because of the integral representation of the Bessel function of the second kind, the kernel is expressed by a known function. e 7 al f(x') Y, b(x-x')]dx' =k. (9) 0 This equation was dealt numerically by Pavlenko [8] without regard of the exist- ence of the solution. Wehausen pointed out that the solution of the integral equa- tion has a type of U(x) L? - x? where U(x) is bounded. Karp, Kotik and Lurye calculated the function U(x) nu- merically for several Froude numbers. It was found that U(x) did not vanish at x = +4, so that the solution becomes singular at both ends. If f(x) gives the ordinate of the surface, infinite horns appear and the condition f (+t) = 0 is violated. A similar situation appears in the case of finite draft because of the logarithmic singularity still existing in the kernel. As far as original Michell's assumption is employed, there is no admissible solution of the present problem. However the formula of the wave resistance may have a different interpre- tation from the definition of original Michell's integral. It can be shown that the Eq. (7) gives the wave resistance of a distribution of x-directed dipoles over the vertical plane y=0. Then f(x) does not mean the shape of the strut but gives the density of the dipoles. Karp and others calculated the boundary streamline when such a dipole distribution was placed in a uniform stream. The integral Eq. (9) belongs to the family of equations of the type £ ie f(x") Y, [(x-x')] dx’ = g(x) (10) which was solved by Dorr [9]. By the change of variables x=-£cos 0, x' = -4 cos 6 Eq. (10) is converted into 1022 Ship Form of Minimum Wave Resistance 7 | o( 8") Y, [y,(cos 8'- cos 8)] dO" = 9(6) (11) 0 where y, = ¢t/U’. Dorr has shown that Aa | ee-(0" aq) Y.. [2i/qu¢cose cos ji de = ce. 6, qa) (12) 0 where ce,(¢,q) is the even Mathieu function of the integral order. AS ce, is orthogonal in the interval (0,77), the functions o(6) and 9(@) can be expressed by Fourier series of ce,. If 9(@) is expressed by toe) W8) = D) agce,(6,4) 3) n=0 the solution of the integral equation becomes foo) aie) = oS a Gen(@a sc) ie n=0 (14) Then the optimum dipole distribution f(x) takes the form 2 -1/2 >< (4° - x?) o COST = 5 Bessho [10] calculated the eigenvalues \,, and showed numerical results for the solution of Eq. (9) at various Froude numbers. The function o(¢) does not van- ish at 6=0 and 7, so that the singularity in the dipole distribution always ap- pears, but becomes less remarkable at lower Froude numbers. The best form has blunt cylindrical nose and tail, but the radius of the cylinder decreases rap- idly according to the decreasing Froude number. Though the solutions at higher Froude numbers show so to speak dog-bone shapes and are hardly regarded as practical, the shape appears quite plausible at moderate and lower Froude num- bers. It can be noted that negative ordinates which have appeared often at ap- proximate solutions by Pavlenko and others never appear. Therefore the prob- lem to minimize the wave resistance of infinite struts under a single condition of constant sectional area always has a solution, if a slight deviation from Michell's original assumption is allowed. The similar situation holds in the case of elementary ships of finite draft. Though the kernel of the integral equa- tion cannot be expressed by known functions and eigenfunctions which are given in the case of infinite struts are not known, a numerical solution is possible. A few results at Froude number 0.4 were published by Kotik [11]. Weinblum's in- vestigation has assumed not only the condition of constant volume but also other side conditions such as the fixed beam. For elementary ships, the constant beam together with the constant volume means a constant block coefficient. To seek the best form among those of constant block coefficient seems to have greater importance from the practical side because the solution under a single condition of constant volume often presents a ship form of too small block 1023 Maruo coefficient. However Bessho has proved for the infinite strut that there is no solution under such dual condition. This situation is similar for the elementary ship of finite draft. The wave resistance of an elementary ship is expressed by a general form as e t 2 R = — | f(x) ax | f(x’) K(x- x’) dx’ (15) -f -0 where y = f(x) is the equation of the load water line and the kernel K(x - x’) depends upon the shape of the frame line. Change of the variables x = -¢ cos 6, x’ = -¢ cos G' and substitution of the expression for the solution, remembering that the opti- mum form is symmetric, #Gx) = oe As GOS BE + a, COS VE +p =>) (16) lead to the equation such as R = Qe bv y ys Fon FamMon om (17) n=0 m=0 where Me ons i vol cos 2n@ cos 2m@" K(4 cos 6’ - £ cos @)dé' (18) 0 0 b being the half breadth of the ship. The condition of the constant volume is a — constant =) C (19) while the half beam which is also assumed constant is b= b (ay rane (20) n=0 Now let us determine the coefficients a,, in such a way that the right hand side of Eq. (17) becomes minimum. Consider a function TiC@ieh laos Ee Dy apy aig Myo yeh ak She Oro . (21) n=0 1024 Ship Form of Minimum Wave Resistance In order to make the wave resistance minimum under the condition of Eqs. (19) and (20), the coefficients a, andk should satisfy the following equations. 2n oT oT oT =0, =~=0,...==0 (22) oa 2 - : da together with Eq. (19). These are equivalent to the simultaneous equation 2 oz Bom Mon, 2m = (=) Dyes Sepa (23) m=0 If the infinite series of Eq. (16) is truncated at nth term, Eq. (23) together with Eqs. (19) and (20) presents N+1 equations for N+1 unknowns. The coefficients can be determined provided the characteristic determinant is non-zero. Assume that the coefficients of the Fourier series, Eq. (16), satisfy the Eq. (23) and sub- stitute in the integral e | TOR” \ INGR = se") Cbs” - -0 Making use of the Eq. (18), it is easily found that 0 7 | fi Gxen K(x exe ichxen = pe] 0(@') K [4(cos 6’ - cos 6)] dg" -0 0 (24) where When one tends N toward infinity, Eq. (24) will give an integral equation which the minimal solution f(x) or o(@) should satisfy. However there is a relation N N 1 n me) cosvCNaaye Ht = (-) cos 2n@ = eT ON GP n=1 and the right hand side of Eq. (24) does not converge to a continuous function. Bessho has proved that the solution diverges in the case of infinite strut. For the infinite strut, the solution is expanded into a series of eigen-functions as mentioned before. The coefficients can be determined analytically. By virtue of the asymptotic behavior of Mathieu functions, a few terms at the beginning of the series becomes dominant when the speed parameter », increases. Though the minimal solution gives a diverging series, the latter may be regarded as an asymptotic expression for small Froude number. According to the numerical results, the asymptotic value obtained by taking first three terms gives a 221-249 O - 66 - 66 1025 Maruo reliable approximation of the present problem, if the speed parameter y, is greater than 5. Instead of the condition of constant beam, Bessho proposed an- other side condition as a substitute. That is a condition of constant moment of inertia of the water plane with respect to the transverse axis. g = 2| f(x) x?dx = constant . (25) = (? In this case, the integral equation satisfied by the minimal solution becomes e | iG yn Gx odd) dix an ke ee (26) ¢ 1 2 The solution exists and is unique. If the solution of the problem with a single condition of constant volume is designated by f,(x), the solution with dual con- dition is expressed as fGs) = Gs) # GF as) (27) where £ is the difference between the given midship beam and f,(0). Bessho published the function f,(x) for the dual condition involving the constant moment of inertia. Figure 1 shows a comparison between the asymptotic approximation for constant beam and Bessho's substitute. DOTTED LINE BY BESSHO Figure 1 1026 Ship Form of Minimum Wave Resistance It has been shown that the best form does not exist under a single side con- dition of constant volume unless the elementary ship with prescribed vertical distribution is assumed. However Krein pointed out that a solution could exist if another side condition such as a fixed area of the wetted surface would be added. From the mathematical point of view, the solution under this dual condi- tion is equivalent to the ship form for which the sum of the wave resistance and the skin friction becomes minimum. Lin, Webster and Wehausen [12] computed the ship form of minimum total resistance, which was assumed as the sum of Michell's integral and the frictional resistance according to Schoenherr's mean line. Their results are quite plausible except undulating lines which seem to be a consequence of an improper choice of the series used for the expansion of the solution. According to Froude's hypothesis, the frictional resistance of a ship is equivalent to the frictional resistance of a flat plate of same length and same area. However the frictional resistance of a curved surface is an integration of the longitudinal component of the tangential stress. If the local frictional coef- ficient C; at a point where the normal to the surface makes an angle a to the longitudinal axis, x say, the total friction is given by 1 ' Ree 4 wwf fet sin ads. (28) When the surface is expressed by an equation, y = f(x,z), one can put of Ox coS a = 1+ (Se) + (Ey Ox OZ . ae of \? ds = 2 fat (S$) + (S$) axaz Therefore the frictional resistance becomes ae ov? |fc, yi+ eS) dx dz. (29) Taking the mean value of the local friction, one may write Rp = ovrc, |{ fA iF 2 dx dz (30) where C, is regarded as the frictional resistance coefficient of the ship. The area S.=2 {fy 4 = dx dz (31) 1027 Maruo is called the effective area which is the product of the length and the mean girth. Let us consider a dual condition of constant displacement and constant effective area. Let us start with the formula for the wave resistance of a ship of length 2¢ and draft T as t ene aa; R= 20uy | { I f(x, Zz) fCx yze) K(e ay x=) dxdx Wdzidz (32) -f J-0 Jo Jo where od 5 P 4 K(ztz , x=x) = -| Ba mane [yA(x-x')] ds (33) 1 a Woe i which is obtained by the integration by parts of Michell's integral. According to the principle of the calculus of variations, the minimization of the wave resist- ance under the conditions of constant volume of displacement V= 2[[ecx2) dx dz = constant (34) and of the constant effective area DoS 2| J y1 tr (st) dx dz = constant (35) gives a non-linear integro-differential equation as @ of of ome ; : ere: 3 Oz i | f(s, 2) Kate", Ros) ck” dz” = ie NS) ae —_—_—_—_—_—— (36) -C Jo Or \F 1+(s] where k, and k, are constants. Integrating with respect to z, one obtains Coat of root (1) 1 1 p 0 Oz CO 5B) i | (ara ”, eas) abe" Gig” = kom + kk, SSS emmy) a ae) oz where eo) 2 ; 2 Kae AEZ a i x= X 7) = = SR AN ag lyA(x=x')] sy (38) MS V2 -1 and 9,(x) is an arbitrary function of x only. Since the Eq. (37) is non-linear, an iterative method is employed. In the first place, the vertical gradient of the surface of/dz is assumed as small. Then a linearization of the integro- differential equation is made by the exclusion of the non-linear term (0f/0z)?. 1028 Ship Form of Minimum Wave Resistance (eer ; 5f i | fixer) K' Ug ee Se Sel tae Ole Sk at k= + Q(x) - (39) -0Jo a Integrating again with respect to z, one may obtain a linear integral equation as Bin an ; i | fiCx ys ZO) POE erent meee NCbe ly = 3 Ez? t eAi(z) E ZOy (mer Cae) (40) - Lo where co P : 2 Boe Cz) Bee PNG dr (41) my? J, Ma il 2 K‘ deat, ase) = and »,(x) is another arbitrary function of x only. Since K‘*)(z+z', x-x’) is absolutely integrable in the domain - ¢ < x < 4, 0 < z < T, one can write ee (2) t i / / \ | |K ‘(zt+z', x-x') |dx’dz’ M. Therefore the linearized integral Eq. (40) has a solution, and the latter is unique except for the arbitrary constants k, and k, and the ar- bitrary functions 9,(x) and 9,(x). These unknowns are determined by side con- ditions. There are already two of them, the given volume and the given effective area. However the solution is still indeterminate owing to the functions 9,(x) and 9,(x). Two other conditions are necessary in order to determine the solu- tion. It is understood easily that the vanishing ordinate at the keel line, i.e., Gx Py S10 (42) © can become one of the required conditions. The other can be a condition im- posed on the shape at the water line z=0. As the integral on the left hand side of Eq. (39) is bounded in the domain - 7 < x < ¢, 0 < z < T, one may have Cover | i f(x,z’) Ke ae, eX) dada! =k, (+) + 0,(X) - (43) -lJo Therefore 9,(x) can be determined by giving the slope of the surface °of/ez at z=0. The vertical sides for instance corresponds to of/oz = 0 at z=0. This condition is equivalent to the implicit assumption employed by Lin, Webster and Wehausen in their calculation. As the non-linear factor has always a non-zero denominator \/1+ (of/dz)?, the integral equation at any stage of the iteration has 1029 Maruo a solution. It can be shown by the Eq. (40) that f(x,z) is finite (or zero) at both ends x = :¢. ASYMPTOTIC FORM OF THE OPTIMUM ELEMENTARY SHIP FOR VANISHING DRAFT It has been shown that the elementary ship of given vertical distribution has a minimal solution for modified Michell's integral under the single condition of constant displacement. The wave resistance is given by 0 Baie. Aa R= 2nur* | I l | X(G)Z(Z) XG Cz KG ez xx) dxidx: Sdizicizam (44) -€ J-0Jo Jo Letting eit | | Ui) TAB) WS tea! , Re") Cla Gls” = IK(sx=52") (45) 0 40 and writing f(x) in place of x(x), one obtains oe R= apu?y | | f(x) f(x’) K(x-x') dxdx’. (46) eee Therefore the optimum form is given by a solution of an integral equation such as 0 | f(x') K(x-x') dx’ =k. (47) -f As mentioned before, the above integral equation have a solution which can be determined only by a numerical way. Though there have been some examples, the solving procedure requires very tedious and extensive calculation any way. As the basic assumption of Michell's theory is that the beam of the ship is very small in comparison with the length, it applies to the thin ship. However actual ships have draft which is smaller than the beam. The slender ship stands on the idea that the draft length ratio is of the same order of amount as the order of the beam length ratio which is much smaller than unity. The lineari- zation is achieved by means of these parameters. Attempts have been made to find out a slender ship form of minimum wave resistance [13]. They seem not to be successful from the practical point of view. The reason is that the solu- tion involves only ship forms of a very restricted class and is by no means the - best among whole admissible ship forms. Results which will be reported here deviate from the original slender ship assumption. The basic idea is to return to Michell's integral and to look for an asymptotic form of the minimal solution when the draft becomes infinitesimal. 1030 Ship Form of Minimum Wave Resistance The solution to minimize Michell's integral for elementary ships under a single condition of constant volume exists as mentioned before. Assume an ele- mentary ship, f(x,z) = X(x) Z(z), and write Michell's integral t t Re= 2our* | i X(x) X(x') K(x- x’) dx dx’ (48) -@vs-f where 2 2 mq 4 K(x=x') = Al cos [y\(x-x’)] | mae tas] pil . (49) 1 0 The integral equation to determine the minimal solution is Y | K(x VKG ox dx = ke (50) i Since the kernel has a logarithmic singularity at x=x', the solution takes the form £2 — x2 If U(x) is finite at x = +4, the function X(x) becomes singular at both ends. Now let us consider the asymptotic behavior of the wave resistance when the draft T tends to zero. The simple slender ship theory expands the integral 1 2 | Wezwe fied 0 by an ascending power series of T and takes the first term that makes the kernel K(x-x’) have the order of T?. Though the kernel has a higher singularity at x =x’, the integral with respect to x and x’ is regarded as the finite part due to Hadamard. Then Eq. (48) becomes finite only when dx(x)/dx = 0, otherwise the integral diverges. Since the finite part of the integral is taken, the singularity of the kernel does not matter except at the end points x = +4. Therefore the in- finity appears from the behavior of the integrand at the ends. This phenomenon may be called the end effect. It has been shown that the end effect gives a term of the order of T? {nT when dx(x)/dx is finite there. The order of the end effect can be evaluated if z(z) is assumed as a simple function such as Z(z) = 1 and the behavior of the resulting integral is examined at the limit of zero draft. It can be proved that the end effect has the order of T'’? when the water line func- tion takes the form of Eq. (51). Since the volume is proportional to T, the re- sistance per unit volume increases infinitely when the draft decreases. Itseems to be natural that this case is excluded from the admissible solution. The case that X(x) is finite at both ends is also excluded by the same reason because the order of the end effect is TfnT. Therefore the only case that the width of the 1031 Maruo water plane vanishes at both ends is taken as the asymptotic form of the opti- mum ship. Then the left hand side of the integral Eq. (50) is integrated by parts Nie | tO Con") dx! = ke (52) where K°’ is an integral of Eq. (49) with respect to x. Integrating Eq. (52) three times with respect to x, and taking account of the fact that dx(x’)/dx’ is an odd function of x' and K(x- x’) is symmetric, one obtains Q | Ee) ) Ko? ¢x-x") dx! = Zig + k’x (53) " dx 6 where (oo) T 9 2 ses) - =| cos [y\(x-x’)] [| Way ev “a ain : (54) Oe 5 0 AZ -1 K‘*? has an asymptotic form when T tends to zero as follows: {oo} T 2 To ees) es eos [osxe 23] [| Zz) ee an WY Sah 0 Yr2- 1 ee (55) T 2 == = Me [y(x-x’)] | Z(z) el y 0 where Y, is the Bessel function of the second kind. By putting at ING) SDE) | Z(z) dz (56) 0 that means the area of the transverse section, Eq. (53) becomes ia ae 1 se ges Sats x Sane DR ge 3 1 57 ; = aya | ea iu lly Cxeoexe lclx B Kx + k'x. (57) This is an asymptotic form of the integral equation. If the condition dx at x = +¢ is employed, the end effect does not exist and the minimum wave re- sistance is given by 1032 Ship Form of Minimum Wave Resistance ( Reais 2p uty | X(x) dx . (59) =? min This is the case of a simple slender ship theory. The method of solution and numerical results are given in literature [13]. In order to solve the integral equation, let us employ the dimensionless coordinates SP cosmen x 0 = =4Gose 9 (60) The sectional area is non-dimensionalized and to facilitate the solution, one may put dAG) voce) dx A A? sin 6 — (61) Then Eq. (57) is equivalent to the following integral equation: | GE) NCO Cee ACOs ME Ile) = Ne COM! ae IRG EOS Sie! (62) 0 where y, = gt/U*. The displacement becomes j dA Wes | ce) xdx dx afl | a(@) cos 6 dé 0 so that (63) | a(@) cos @ d@ = 2. 0 The integral Eq. (62) can be solved by means of Fourier expansion by Mathieu functions. There is a relation for even Mathieu functions ce (¢@,q), that 7 Aa | cen(a a) Mie [2Va (cos @- cos @')] do Seen(Gr ayy 0 Since ce,(@,q) makes an orthogonal system such as iI ro) 5 + 3 7 2 7 || cen(C,q) ce7 (0, 1q)d7 0 1033 Maruo any function can be expanded into Fourier series of ce. As A(x) is an even function of x, the expansion for o(@) contains odd terms only. ae) = 2, Ca y(@ Gh) * A €e,(2G) 7 A, CE(2,G)) a 29° There is also a relation 2n+1) ces. (Die ap er = De oe CE 5441095) r=0 (2n+1) where A,.., is the Fourier coefficient of ce,,,,, such as (2n+1) CExony (CoG) = De Moonug GOS (Berri ye n=0 Substituting these relations in Eq. (62), the following equation is obtained: foo) {oo} (2n+1) (2n+1) DE Bant1 C2 ants©F>D/roned = 2, ia + kA A,; CC ony 1 (9,9) - n=0 n=0 Therefore the unknown coefficients are determined as (antl) (2n+1) 64 Aont1 = None1 ca + ky, A, He ee The condition (63) gives (2n+1) 4 2 Aone Ay ~ Fe ) n=0 and together with the condition o(@) = 0 at 6 = 0 and 7 the arbitrary constants k, and k, are determined. Since GVA ake | Z(z) dz (66) 4 0 the wave resistance is given by k, pU2 V2 2) ee ee (67) £ Necessary coefficients for the calculation of Mathieu functions have been given by Bessho and the optimum forms of simple slender ships are calculated. Fig- ure 2 gives the best curves of sectional area of the simple slender ship. It has been found that the minimum wave resistance of the simple slender ship given 1034 Ship Form of Minimum Wave Resistance by Eq. (67) is not the minimum of the asymptotic value of Michell's integral for vanishing draft, as a result of comparison of it and the wave resistance of a slender ship with vertical stem. In the latter case, dA/dx or o(@) does not van- ish at the ends. If the condition Eq. (58) is discarded, one of the two coefficients in Eq. (62) becomes undetermined unless another side condition is introduced. Then solutions of the integral Eq. (62) give a family of curve of sectional area by which the wave resistance excluding the end effect is minimum for a constant displacement. One may have a doubt since the above indeterminateness seems to contradict with the fact that the minimal solution for Michell's integral of finite draft is unique. By a proper choice of the midship ordinate, k, can be eliminated. Then the principal part of the wave resistance given by Eq. (67) vanishes. Though difficult is it to identify the true asymptote, the above solu- tion may be regarded as the asymptotic form of the minimal solution with the single condition of constant volume for vanishing draft. Figure 3 shows a com- parison between the curve of sectional area obtained from the above method and the dipole distribution for the optimum infinite strut. The difference between them is small especially at lower Froude numbers. Kotik calculated the opti- mum form of the elementary ship of finite draft at Froude number 0.4, one of which concerned a 4th power vertical section and the other concerned a wall- sided section. His results with respect to a draft-length ratio 0.05 are plotted in Fig. 3 for comparison. They fall between the result for a infinite strut at Froude number 0.397 and that of the aforementioned approximation. When finite value of k, is retained, a family of solutions with various midship section area is obtained. As mentioned before, there is no minimal solution for dual condi- tion of constant volume and constant midship section. Then the above results seem to correspond to the asymptotic solution for the condition of constant volume and constant moment of inertia. Figures 4-12 give the asymptotic opti- mum curve of sectional area at various speed coefficient y, = g?/U? with pris- matic coefficient 9 as a parameter. In some of the figures Weinblum's results [14] are given by dotted lines. Difference is not remarkable except the case of Y.= 2 where the polynomial representation employed by him seems to lose its accuracy. In Fig. 6, curves of forebody sectional area of the Taylor Standard Series (T.S.S.) are shown for comparison. There is a surprising agreement between the T.S.S. and the theoretically optimum form for medium prismatic coefficient at Froude number 0.25. On examining the chart of residuary resist- ance coefficient, one may find out that T.S.S. shows an excellent behavior at Froude number near 0.25, if the prismatic coefficient is around 0.60 where the best agreement is obtained. It is of some interest to observe that a hump ap- pears at Froude number 0.25, if the prismatic coefficient is reduced to 0.48 or raised to 0.68 where deviation from the optimum curve becomes remarkable. At Froude numbers other than 0.25, T.S.S. does not agree with the optimum curve. Therefore better results than those of T.S.S. can be expected by em- ploying the theoretical curve of sectional area. 1035 Maruo aesoseees INFINITE STRUT KOTIK 4T# POWER } EM SIGVA KOTIK WALL-SIDED Wee: 1036 Ship Form of Minimum Wave Resistance O 0.5 1.0 XTi Figure 5 1037 Maruo WEINBLUM SS = 2858: 1038 Ship Form of Minimum Wave Resistance 1039 Maruo I) t A(x) A(o) fo) a5 X/| — Figure 10 0.5 Xx/| — Figure 11 1040 Ship Form of Minimum Wave Resistance Qs X/i —- Figure 12 SOME CASES OF SMALL WAVE RESISTANCE As shown by Krein and Bessho, there is no definite solution of the problem to minimize Michell's integral for a given volume. This fact suggests that the theory of minimum wave resistance discussed so far is not the only way to ob- tain a ship form of small wave resistance. Inui [15] has shown that the wave resistance can be reduced to a great extent by addition of a bulb at the bow which enables the cancellation of the wave generated by the main hull. This method was refined by Yim [16]. He considered a combination of a source distribution representing the ship's hull and a distribution of dipoles along a vertical line of infinite length at the bow. According to him, the wave resistance can be elimi- nated when a suitable choice is made in the combination of sources and dipoles. The vertical dipole distribution of Yim's model shows a vertical cylinder of in- finite length at the bow. Instead of it, one may consider a source distribution on the vertical line. In fact, it is possible to make the wave resistance vanish by a suitable choice of source distribution along a horizontal line and those along vertical lines at both ends of the horizontal distribution. As the simplest exam- ple, let us consider a source distribution along a horizontal line of length L = 2? on the free surface. Choose the density of sources given by the following equation: eo = je sin(—),, Abe Sore Ss Abc (68) If a distribution of sources along an infinite vertical line at x= 4 and that of sinks along a vertical line at x = ~~ have density distribution given by WD = ; (0) < < © (69) 7,(2) m, exp ee ) Zz 221-249 O - 66 - 67 1041 Maruo the wave resistance becomes 1/2 Ry asaory— P? sec36 dé (70) “ty 2 where 0 bs b= | G(x) sim’ Gx see O)dx 2 sin (% SEC | o,(Z) exp (-yz sec 70) dz. (71) -2 0 } Substituting Eqs. (68) and (69) in (71) and carrying out the integration, one obtains mim, \~ (P= [Pasian (7. see @) 4 R= 647py~ (m,— : sua domne Shee sec30 dé (72) fo) 2 oY 2 2 2 fo) A Vege SES Q@- 7 if y,>7. If there is a relation m, = m™m,/Y (73) the wave resistance vanishes. Though Krein and Bessho have shown that wave- free distribution of sources gives zero linearized volume, the wave-free distri- bution without negative ordinates does exist if the draft is allowed to be infinite. The horizontal distribution corresponds to a half immersed body of revolution with cross sectional area given by the equation 8st m, A(x) = cos” ze (74) U yy The vertical distribution corresponds to a vertical strut of infinite depth, the horizontal section of which is the Rankine oval. The resultant shape is a com- bination of them and is so to speak a yacht shaped ship with infinite vertical Keel. As the infinite keel cannot be realized, it must be truncated at a finite depth. The truncation invalidates the perfect cancellation of the waves gener- ated by each system of sources. Figure 13 shows the results of calculation of wave resistance when the vertical keel is truncated at a depth 0.25L and 0.1L, when the designed Froude number at which the wave resistance vanishes for the infinite keel is 0.316 or y,=5. Though the truncation of the vertical keel does not matter much at lower Froude numbers, it weakens the cancellation of the wave at high Froude number especially when the vertical keel is truncated at smaller depth because of the practical requirement. In order to compensate the weakened effect of the vertical keel, the strength of the vertical distribution should be augmented considerably. An investigation has been made so as to find out the vertical distribution which makes the resultant wave resistance mini- mum. According to the result, a remarkable peak appears at the bottom of the vertical source distribution. This fact suggests that the best form has a con- centration of the source at the bottom. Instead of pursuing the best distribution along the vertical lines, a discrete source and a sink are assumed at the depth 1042 Ship Form of Minimum Wave Resistance R (72 pU*[A(0)/L]? 0.2 0.25 Q3 0.35 0.4 Figure 13 of the bottom. Then the system of sources is the combination of a discrete source at the forward end of the bottom and a discrete sink at the after end to- gether with the horizontal distribution. The source-and the sink form the so- called Rankine ovoid, and the resulting shape is the combination of a submerged ovoid and a surface piercing hull. At a lower Froude number, the surface piercing part is much greater than the submerged part, so that the former can be regarded as the main hull, while the latter forms bulbs on both sides. At higher Froude numbers on the other hand, the submerged part becomes the main hull, while the upper part is like the super-structure or bridge of a half submerged submarine. Such a type of ship as this may be called a semi- submerged ship which has been discussed from time to time [17]. If the strength of the submerged source at the point x=, z=f, is designated by m,? and an equal sink is placed at the point x = -t, z-=f, the wave resistance when com- bined with the horizontal source distribution given by Eq. (68) becomes 1/2 2 f ma E exp (-7, 7 sec?) + | sin? (ysee 2) sec 26 dé. R= 47py2 e he sec 26 0 (75) Write the area of the midsection of the Rankine ovoid as A, and that of the bridge by A,. Then there is approximate relations due to a linearized theory such as im = s and MGs = hewerae (76) Maruo Then the wave resistance can be written as 2 A 1/2 2 4 if R= pg == Ye | | exe —Y¥, — sec?6| + ae ae sin?(y_ sec 0) sec20d@ 4, ; 4, Tia yy see " (77) where m A sale Se el Ge ae (78) The ratio \ can be chosen in such a way that the resulting wave resistance be- comes minimum. If the volume of the submerged part is kept constant, it is merely given by the equation OR Oss =) . 19 — = 0 (79) Calculation has been carried out for cases of ¢/f = 4 and 5. Models for tank experiment were prepared as shown in Fig. 14. Figure 15 shows some of the results of the experiment together with the computed curves. The designed speed at which the relation Eq. (79) holds is indicated by the arrow. As the ex- perimental value is the residuary resistance coefficient, some difference exists between the experimental curves and the theoretical wave resistance coefficient. However, the general feature of the curves is similar. There are also shown theoretical curves of wave resistance coefficient when the submerged body, the Rankine ovoid, moves alone under the water surface, and one can observe how the wave resistance is reduced by the interference between two parts. L/¢f = 8 L/4=10 fe.) L=1.6m Figure 14 Kotik calculated the value of minimum wave resistance of elementary ships at Froude number 0.4. For a wall-sided ship of draft length ratio 0.1, the wave resistance coefficient defined by 1044 Ship Form of Minimum Wave Resistance Figure 15 a Wy 5) hy Cape RZ eu B where 2B is the mean breadth, becomes 0.32612 and for a ship with 4th power section, it is 0.35665. The corresponding value for a semi-submerged ship of minimum wave resistance at Froude number 0.4 was calculated. It was found to be 0.08837, and a considerable reduction of the wave resistance is achieved. The experiment of the semi-submerged ship was conducted under a finan- cial support by Uraga Heavy Industry Co. Ltd. The author wishes to express his gratitude. REFERENCES 1. Weinblum, G., "Schiffe geringsten Widerstandes,'' Proc. 3rd Internat. Congr. Appl. Mech. Stockholm (1930) 2. Krein, M. G., Doklady Akademi Nauk SSSR, Vol. 100, No. 3, 1955 Kostchukov, A. A., ''Theory of Ship Waves and Wave Resistance," Lenin- grad, 1959 oe Bessho, M., ''Wave-free Distributions and Their Applications," Inter- national Seminar on Theoretical Wave Resistance, University of Michigan (1963) 1045 Maruo . Bessho, M., "On the Problem of the Minimum Wave Making Resistance of Ships,'' Memoirs of the Defence Academy, Japan, Vol. II, No. 4 (1963) . Sretenskii, L. N., "Sur un Probleme de Minimum Dan la Theorie du Navire," C. R. (Dokl.) Acad. Sci. URSS(N.S.)3(1935) . Wehausen, J. V., ''Wave Resistance of Thin Ships,'' Proc. of the Symposium on Naval Hydrodynamics Publication 515 (1957) . Karp, S., Kotik, J., and Lurye, J., "On the Problem of Minimum Wave Re- sistance for Struts and Strut-Like Dipole Distributions,'"' Third Symposium on Naval Hydrodynamics, Scheveningen, September 1960 . Pavlenko, G. E., "Theoretical Ship Forms of Least Wave Resistance," Proc. 4th Internat. Congr. Appl. Mech. Cambridge (1934) . Dorr, J., "Zwei Integralgleichungen erster art, die sich mit Hilfe Mathieu- scher Funktionen Losen lassen,'' Z.A.M.P. 3 (1952) Bessho, M., ''On the Minimum Wave Resistance of Ships with Infinite Draft," International Seminar on Theoretical Wave Resistance, University of Mich- igan (1963) . Kotik, J., "Some Aspects of the Problem of Minimum Wave-Resistance," International Seminar on Theoretical Wave Resistance, University of Mich- igan (1963) Lin, W. C., Webster, W. C., and Wehausen, J. V., ''Ships of Minimum Total Resistance," International Seminar on Theoretical Wave Resistance, Uni- versity of Michigan (1963) Maruo, H., ''Calculation of the Wave Resistance of Ships, the Draught of Which is as Small as the Beam," Journal of Zosen Kiokai, 112 (1962) . Weinblum, G., Wustrau, D., and Vossers, G., ''Schiffe geringsten Wider- standes,"' J.S.T.G. 51 (1957) . Inui, T., ''Wave-Making Resistance of Ships," Trans. SNAME, 70 (1962) . Yim, B., "On Ships with Zero and Small Wave Resistance," Internat. Semi- nar on Theoretical Wave Resistance, University of Michigan (1963) Lewis, E. V. and Breslin, J. P., "Semi-Submerged Ships for High Speed Operation in Rough Seas," 3rd Symposium on Naval Hydrodynamics, Scheveningen, September 1960 1046 EXPERIMENT DATA FOR TWO SHIPS OF “MINIMUM” RESISTANCE Wen-Chin Lin, J. Randolph Paulling and J. V. Wehausen University of California Berkely, California ABSTRACT This report presents the results of towing-tank tests carried out on two of the models of minimum ''total'' resistance described in an earlier report by Lin, Webster and Wehausen (1963). One model was sym- metric fore and aft, the other asymmetric with a prescribed afterbody. Each was supposed to be optimum within its class for a Froude number 0.316 and a Reynolds number 1.18 x10°. Although the forms showed resistance qualities near the design speed as good as the equivalent forms for Taylor's Standard Series, they were not significantly better. The occurrence of separation behind the stern bulb of the symmetric model may have masked possible superior wave-making qualities as indicated by a rather small surface disturbance. INTRODUCTION In a paper presented at the International Symposium on Theoretical Wave Resistance in Ann Arbor in 1963 (Lin, Webster and Wehausen, 1964)* two dif- ferent minimization problems for ship resistance were considered. In each problem an estimated "total'' resistance consisting of the equivalent flat-plate frictional resistance plus the wave resistance as given by Michell's integral was minimized for selected values of the Froude number. In one of the problems, only the volumetric coefficient C, = v/L* and the ratio H/L were fixed. The re- sulting optimum hull-form was necessarily symmetric about the midship section. In the other problem, H/L and a particular afterbody were prescribed, and an op- timum forebody was found. In each case the class of hull shapes within which an optimum was sought was limited to a 6X6 double Fourier series: 6 6 f(x, Z) = De DD anp COs 5 (2m- 1) mx cos - (2p - 1)7z. m=1 p=l1 “References are identified by author(s) and date and collected at the end. 1047 Lin, Paulling, and Wehausen Here the variables have been made dimensionless by measuring distances in the x,y,z directions by “4L, 4B, and H, respectively, where B, is not the true beam but is fixed at 3H for this purpose. The optimum forms which were obtained for the symmetric ship for values of the parameter y, = gL/2v* varying from 5 to 10 were reasonably shiplike in appearance except for some waviness in the lines and two small areas at the waterline near the bow and stern where the ordinates became slightly negative. The waviness seems almost certainly to be a result of the limited number of trigonometric functions used to describe the hull. The amount of negativeness was so small that these regions could be deformed to zero without significantly altering the lines. The wave resistance at design speed for each of these forms, as predicted by Michell's integral, was very small compared with the frictional resistance, in fact, negligible for the forms corresponding to y, = 6 to 10. Fig- ure 1 shows the wave-resistance coefficient Ry,/egV for each of these optimum forms as predicted by Michell's integral for Froude numbers between 0.18 and 0.50. Ru/pg¥ x 10% 0.50 Fig. la - Michell resistance for optimum symmetric forms The results obtained in the problem with the fixed afterbody were not as sat- isfactory as those described above. With the exception of the forebody obtained for y, = 5 (Fr = 0.316) the forebodies were generally unacceptable as ships. Partly this was a result of the occurrence of negative offsets of substantially 1048 Data for Ships of Minimum Resistance 0.09 y ° 0.08 0.07 0.06 o 0.05 0.04 0.03 %o=7 0.02 %=g %=8 0.0! %=10 IN 2 — Wy, 0.20 0.25 030 0.35 0.40 % % Ry /pa¥ x 102 Fig. lb - Michell resistance for optimum symmetric forms larger amount than for the symmetric bodies, partly it was the result of exces- sive waviness in the waterlines. In the present context the latter was objection- able not because of the practical difficulty of fabricating such shapes but be- cause of the great liklihood of boundary-layer separation behind the bellies. As was stated in the cited paper, imposition of restraints like 0 < f(x,z) < M, -C < f,(x,g) < D, which would have prevented the excessive waviness, presents a much more difficult problem in computation. The wave resistance for these forms, again as predicted by Michell's integral, is no longer negligible compared with the frictional resistance for comparable forms. For example, at the design speed the coefficient R,/og¥Y for the form corresponding to y, =5 is approxi- mately half the coefficient R,/cg¥ for the equivalent ship from Taylor's Standard Series, and about one third the frictional resistance coefficient R,/pg¥. Following the obtaining of these results, several courses of action seemed open: there were mathematical questions to be resolved; the effect of increasing the number of Fourier components upon the waviness of the waterlines could be studied; a feasible method of incorporating inequalities among the constraints could be devised. However, more important than any of these seemed having some experimental evidence that the optimum forms derived from theory did, in fact, have good resistance characteristics. The main purpose of this paper is to report the results of testing two of the forms. 1049 Lin, Paulling, and Wehausen Some preliminary comments with regard to possible expectations seem in order. As noted above, for the symmetric forms the Michell wave resistance at, and for some interval below, the design speed is generally negligibly small compared with the frictional resistance. If the ''real'’ wave resistance does in some sense approximate this, any attempt to observe it experimentally will be plagued by the uncertainty in estimation of the "viscous part" of the total resist- ance. In particular, a region of boundary-layer separation or even an excessive form drag may have the effect of masking completely the quantity being meas- ured. In addition, one must bear in mind that Michell's integral is based upon linearization of the boundary conditions and represents the first term in a per- turbation series in B/L. However, the fact that this first term is very small for a particular form does not imply that the second-order term is also very small for this form. Under such circumstances it may, in fact, be considerably larger, although still of second order. Consequently, there may exist an appreciable wave drag in an inviscid fluid even though the linearized theory predicts prac- tically none. CHOICE AND CONSTRUCTION OF MODELS One hull form was selected from each of the two series. For each, the form optimum for y, = 5 (Fr = 0.316) was selected. As has been mentioned above, the choice y, = 5 for the hull with prescribed afterbody was hardly a free one. For the symmetric ship this form was chosen because 0.316 was the largest Froude number for which the corresponding optimum ship had waterlines of small enough slope so that boundary-layer separation did not seem likely to occur and thus render invalid the fundamental assumptions underlying the com- putation. Figure 2 shows the section curves, waterlines and area curve for the optimum symmetric ship for y, = 5. Figure 3 shows the prescribed afterbody, both as designed and as represented by the Fourier series. Figure 4 shows the optimum forebody for y, = 5. The models as actually constructed differed slightly from those designed by the computer. For the symmetric model the lines in the neighborhood of the regions of negative ordinates were modified slightly so that the ordinates were zero in these regions. In effect, this created a submerged protruding bulb, as in some of Inui's optimum forms, but not as deeply submerged. The optimum forebody as shown in Fig. 4 has rather noticeable wiggles in the midship section and in the section just ahead of it, a result of trying to fit a U-shaped section with only six terms of a Fourier series. In this case the afterbody was built as originally designed and not as approximated, and the forebody was modified slightly near the midsection to make it join smoothly to the afterbody. Figure 5 shows photographs of each model. CRITERIA AND STANDARDS OF COMPARISON One way to judge the performance of a proposed hull form is to compare it with others of acknowledgedly good performance. Of the usual measures of per- formance the dimensionless ratio R,/ogv¥ at and near the design Froude and Reynolds numbers seems most appropriate and has been used in this paper. The 1050 Data for Ships of Minimum Resistance 8] |/'10 AREA CURVE FOR OPTIMUM SYMMETRIC SHIP GAMMAO = 5.00 H/L = 0.0500 . B/H = 2.64 CSUBB = 0.455 CSUBF = 0.00190 CSUBV = 0.003 CR-MIC = 062654E - 03 CSUBX = 0.743 CR-TOT= O.I3080E-0! LINES DRAWING OF OPTIMUM SYMMETRIC SHIP Figure 2 coefficient R,/%Sv*, although convenient for working up model data, has several obvious disadvantages as a figure of merit for comparing different hull shapes. Of the available standards of comparison, the two which have been used here are Taylor's Standard Series and Series 60. The "equivalent" hull in each case has been taken as the one with the same prismatic and volumetric coefficients and the same ratio B/H. Other geometric parameters such as H/L and the block coefficient cannot be kept constant in this comparison. Furthermore, an equiv- alent hull for the ship with prescribed afterbody did not seem to be available in Series 60. Table 1 below gives various geometric parameters for the two opti- mum hulls and the equivalent ones. The sources of data for Taylor's Standard Series have been Gertler (1954) and for Series 60 have been Todd (1963). There is a second method by which a comparison can be made with Taylor's Standard Series. One can try to carry out within the series the same minimiza- tion problem as was formulated for the symmetric ships, i.e., with C, = 0.003 and L/H = 20 fixed, one can look for a Taylor-Standard-Series hull which 1051 Lin, Paulling, and Wehausen SNOILDAS aN | 31s0u8d Se oh Oe eel et oe ee 006'0 x9 ~9S'0 d9 80S°0 89 Sanne NI 29'2 isvua NI OG'2 wvaa 144 00G¢ “IMQ-HLON3T SOILSIWSLIVYVHD T30QOW 1052 28680 = Xo ogs9go = 9D sols = 99 S3IW3S YSIdNOS AG GSLNS3SSYd3Y ATIVNLOV AGOBYSLIV YOS Nvid AGOS GNV SSNIMYSLVM ‘3ANND vauv Data for Ships of Minimum Resistance sane rail ame a AREA CURVE FOR OPTIMUM FOREBODY GAMMAO = 5.00 H/L = 0.0437 B/H = 3.00 ee CSUBB =0.500 CSUBF = 0.00190 ZA CSUBV = 0.003 CR-MIC = 054367E-02 SE a a Se = CSUR © ORRE CR-VIS = OI5597E-O1 OPTIMUM FOREBODY CSUBX = 0.899 CR-TOT 0.21034E -O1 LINES DRAWING OF OPTIMUM FOREBODY Figure 4 Table 1 Geometric Parameters t Opt. - t ao S Series 60 Forebody diese = St. Series Ship St. Series ; .64 : , 2 23.0 3.00 x10~2 | 3.00 x 10-2 | 3.00 x10-°| 2.87x10~* | 2.87 x10 * .613 .613 .614 .096 .096 .067 60 ; 015 1053 Lin, Paulling, and Wehausen Fig. 5 - Photograph of the two models tested minimizes R,/og¥ for the design Froude and Reynolds numbers. The various steps required are incorporated in Table 2 and are explained below. The suc- cessive lines in the table are obtained as follows. Fixing L/H fixes L/B for each of the three available values of B/H. Then C, is fixed by the given value of C, i [see Gertler (1954), pp. 10-12]. Since there is no hull form with B/H = 3.75, L/H = 20, Cy = 0.003, this column now drops out. The associated values of C. = SL*C,"4 and of C, = R,/% Sv’ are read directly from Gertler [1954]. The value of C, = R,/4 Sv? is the Schoenherr coefficient for Re = 1.182 10°, cor- responding to a 400' ship in salt water at 63°F. with a ship's speed correspond- me tOy- —so. hen eC. esand Ry peyv The hull with B/H = 3 and C, = 0.536 is evidently the best within Taylor's Standard Series which meets the constraints L/H = 20 and Cy = 0.003. Although this is not an ''equivalent"' hull, it does seem to be also a legitimate one to use in a comparison with the optimum symmetric ship for y, = 5. = i 2 77 i - Pi 72 2 C.Fr Coy Pe C, Yo CoC, : TEST PROCEDURE The models were each tested in the Ship Towing Tank of the University of California. The models were attached to the dynamometer so that they were free to both heave and trim. Figure 6 shows the symmetric model being towed at a Froude number of 0.316. Each model was tested both with and without a tripwire. In the region for which data are presented there was a small constant difference in the resistance coefficients R,/% Sv’ with and without the tripwire. This was taken as evidence 1054 Data for Ships of Minimum Resistance Table 2 Optimization Within Taylor's Standard Series 3.00 6.67 0.580 0.536 2.007 2.024 124 lies 1.03 ~10; 4 3 3 1.90 x 10° 1.90 x 10 B14 x 10" 2:93 x 10%? 0.734. x 107 0.676 x 10°? Fig. 6 - Symmetric model being tested at FR = 0.316 that the region of laminar flow was confined to the region ahead of the tripwire. The data were appropriately corrected for the added resistance of the tripwire where necessary. In order to test for separation of the flow behind the bow and stern bulbs of the symmetric model, thread tufts were attached to the model and observed vis- ually. There was no evidence of separation behind the bow bulb. However, be- hind the stern bulb there appeared to be separation at all speeds tested. This 1055 Lin, Paulling, and Wehausen will, of course, cause an added resistance associated with viscosity which is not taken into account in the expression used for R,/eg¥. Furthermore, if contravenes to some extent the fundamental assumption of streamline flow upon which Michell's integral is based. TEST RESULTS Figure 7 shows R,/og¥ for the symmetric model extrapolated to a 400" ship by using Schoenherr's friction coefficients and a roughness allowance 0.0004. On the same figure is shown the same resistance coefficient for the equivalent ships in Taylor's Standard Series and Series 60 and for the ''optimum" ship in Taylor's Standard Series, as explained earlier. The results speak for themselves. The optimum symmetric ship is slightly but insignificantly better than either of the "equivalent" ships near the design speed but is not as good as the optimum Taylor's-Series ship. 3.0 © SS N fe) ; = ‘ > y/ o fe} a fi? we 7 = se Oo 1.0 y oe = Ps oe ee: ie ° ee O Ss Ce Z =a 0 ¢ ee — _ FROM EXPERIMENT @ ee an, —-— TSSequiv Be i at —--— TSSoprt siaiearaant ——-—-— SERIES 60 eho) 0.2 0.25 03 0.35 04 Fr Fig. 7 - Total resistance coefficients for symmetric model and equivalent models Figure 8 shows the residary-resistance coefficient R./egv for the symmetric ship and the equivalent Taylor's-Series ship, and also the Michell wave resist- ance R,/egv¥. It is evident that the residuary resistance of the model is much greater than its Michell wave resistance in the neighborhood of the design speed, but that the two become more nearly equal both below and above this speed. On 1056 Data for Ships of Minimum Resistance 3.0 FROM EXPERIMENT FROM MICHELL TSSEqQuiv TSSopr ins) fo) Re/ pg ¥ x 102 0.20 0.25 0.30 0.35 0.40 Fr Fig. 8 - Residuary and Michell resistances for symmetric model. Residuary resistance for equivalent models. the other hand, observation of the water surface during test runs near design speed shows remarkably little surface disturbance. This leads one to suspect that the residuary-resistance coefficient in this region may contain a significant amount of form resistance, a suspicion partly confirmed by the observed sepa- ration behind the stern bulb. Figure 9 shows R,/ogv for the optimum-forebody model, and for the equiv- alent Taylor's-Series ship, both extrapolated to a 400' ship. Over the range from Fr = 0.25 to 0.35 the two are practically indistinguishable. Figure 10 shows the residuary resistance R,/og¥ for this model together with the Michell wave resistance R,/og¥. It is evident that the agreement is much better here than it was for the symmetric ship. SOME CONCLUSIONS As is evident from the foregoing, the "optimum" computer-designed ships have not shown any dramatic improvements in resistance properties over the equivalent ones in Taylor's Standard Series. In fact, they are hardly distin- guishable. For the ship with prescribed afterbody this should cause no sur- prise, for the predicted improvement was a fairly modest part of the whole. The situation is somewhat different with the symmetric ship. Here the predicted 221-249 O - 66 - 68 1057 3.0 [o) Ry/pg¥ x 102 0.0 3.0 GS) (e) Re/pg x 102 0.0 Lin, Paulling, and Wehausen FROM EXPERIMENT —-— TSSequiv 0.2 0.25 0.3 Fr 0.35 0.4 Fig. 9 - Total resistance coefficient for optimum-forebody model and for equivalent model FROM EXPERIMENT FROM MICHELL ] 02 0.25 0.30 0.35 040 Fr Fig. 10 - Residuary and Michell resistance for optimum-forebody ship 1058 Data for Ships of Minimum Resistance improvement was substantial and has not been realized. Unfortunately, for the present purpose, the reasons are not clear-cut and one cannot ascribe the failure entirely to unreliability of the linearized theory in a situation where it predicts unusually small values of the wave resistance. As has already been mentioned, there was, in fact, remarkably little disturbance of the free surface at and near the design Froude number, so that a small value of the residuary resistance might have been expected. It seems possible that the contribution of the observed boundary-layer separation behind the stern bulb to the residuary resistance may have increased this so much that the favorable wave-resistance properties of the hull were lost. With the wisdom of hindsight it seems evident that for our first symmetric model we should have chosen the one designed to be optimum for y, = 6 (Fr = 0.289) or y, = 9 (Fr = 0.236) instead of y, = 5. Their Michell wave resistances are negligible in the Froude number range 0.2 to 0.3 (see Fig. 1) and their maximum waterline slopes at the stern are smaller, about 16° and 11°, respectively. Even though the two computer-designed ships have not shown any marked superiority in resistance qualities, there is another sense in which the attempt to let certain over-all requirements and the optimization procedure design the ship can be said to have been successful. All forms have been designed without aid of the naval architect's practiced and expert eye and yet the two tested ones have performed as well as the equivalent Taylor's-Series hulls. This in itself is encouraging and seems to indicate that it is worth the trouble to refine the method, in particular, to devise computational procedures for taking into account more complicated kinds of restraints. SYMBOLS R, Total resistance Re Frictional resistance R Residuary resistance Wave resistance according to Michell Prismatic coefficient Ry CG & Block coefficient Cy Volumetric coefficient = V/L° Es Area coefficient = SL™ = ats e R, /20S8v 2 2 & R_/4pSv Ec. R,/%eSv? 1059 Lin, Paulling, and Wehausen REFERENCES 1. Lin, Wen-Chin; Webster, W. C.; Wehausen, J. V. - Ships of minimum total resistance. Proceedings, International Symposium on Theoretical Wave Resistance, Ann Arbor, 1963. 2. Gerter, Morton - A reanalysis of the original test data for the Taylor Stand- ard Series. David Taylor Model Basin Report 806 (1954). 3. Todd, F. H. - Series 60. Methodical experiments with models of single- screw merchant ships. David Taylor Model Basin Report 1712 (1963). * * * DISCUSSION P. C. Pien David Taylor Model Basin Washington, D. C. This paper gives us the theoretical and experimental results of two mini- mum resistance models. It is quite clear as to how these results have been ob- tained. However, it is not easy to digest these results. Why the theoretically predicted low resistance has not been obtained ex- perimentally ? Why are the relative resistance qualities of these two models just opposite to the theoretical predictions? In explaining the results of the symmetrical model the authors suggested the possibility of the second term in the perturbation series being greater than the first term. If it is so then this second term would also be much larger than the first term of the asymmetrical model since the experimental results show that the symmetrical model has much greater resistance than the asymmetrical model. This is not likely to be the true situation. Based on Professor Inui's important research work, we know the linearized ship-surface condition is not accurate for the beam value used in the paper. Therefore the theoretical model of singularity distribution used in the wave- making resistance computation is not in correspondence with the physical model used in the experiment. In such situations, we should not be surprised to see that the agreement between the theoretical and experimental resistance values is not good. Based on the experience of Professor Inui as well as our own, I believe a much better agreement between theoretical and experimental wavemaking resist- ance results can be obtained, especially for the symmetrical case where the free-surface disturbance is small and the Froude number is not too low, if a higher order approximation is applied on the ship-surface. It would be interesting 1060 Data for Ships of Minimum Resistance to see the experimental results of a new model which is more exactly corre- sponding to the theoretical model used in the wavemaking resistance computation. Even though it means additional work it is a rather essential step. I would like to know the authors' views on doing this additional experiment. * * * DISCUSSION Lawrence W. Ward Webb Institute of Naval Architecture Glen Cove, Long Island, New York This is a very interesting paper following the Ann Arbor paper and signifi- cant in including model tests of the forms. I would like to quote a sentence from the paper and then make a comment on it. In the last part of the ''Abstract"’ we read: The occurrence of separation behind the stern bulb of the symmetric model may have masked possible superior wave- making qualities as indicated by a rather small surface dis- turbance. Thus, the authors clearly recognize that the residual resistance is not a good measurement of wave resistance in that it is not based on the waves in a direct way. I suggest at the least one should take a qualitative look at the wave pattern as is done by Inui, using stereo-camera pairs. Better yet, one should make quantitative measurements by means of a wave survey according to methods such as the ones which have been proposed by Dr. Eggers at Hamburg, Hogben and Gadd at NPL, or myself. Such methods are not as difficult to use as some might think, and while subject to certain approximations they should form a much more accurate means of obtaining the wave resistance in cases such as those in this paper. DISCUSSION G. P. Weinblum Institut flir Schiffbau der Universitat Hamburg, Germany The discusser has tried to popularize the application of polynomials for the determination of hull forms of low wave resistance. There were two reasons for 1061 Lin, Paulling, and Wehausen this preference: the theorem of Weyerstrass from a mathematical point of view and the attempt of using the wisdom of art embodied by "spline curves.'"' The "ugly'' wiggled forms obtained by the authors justify my dislike of trigonometrical series as long as a relatively small number of terms is admitted. Exact solu- tions (Karp, Maruo, Kotik, etc.) indicate that the wiggles are not significant re- sults but an outcome of the use of the functions mentioned. In the meanwhile it follows from a kind information by Prof. Wehausen that the wiggles are smoothed out when a larger number of terms dependent upon x is used. Notwithstanding the offense against beauty committed by the authors the theoretical investigation has furnished valuable information on the (a) low magnitude of resistance up to relatively high F when a large num- ber of terms is used, (b) influence of vertical displacement distribution on resistance, and (c) dependence of resistance upon number of form parameters. In agreement with my experience when testing forms of extremely low wave resistance (by calculation) the authors are disappointed by their experimental results. The high resistance measured may be due (a) principally to the fact that viscosity destroys the calculated favorable interference effects. This ap- plies even to fine streamlined forms in the afterbody (compare our tests with a model C, = ¢ = 0.52, communicated at Ann Arbor Proceedings 1963, (b) to sep- aration and excessive viscous form drag due to the stern bulb. The author's decision to test a more normal form is commended, further, forms with bow bulb alone could be tested. But in the light of our earlier experiments (Schiffbau (1936) point (a) may be more decisive than (b)), (c) the authors point out at the possibility that second order terms in the resistance integral may become im- portant when developing optimum forms based on first order theory. This is a new idea which may be checked by evaluating a second approximation using Sisov's formula. Such work is going on under the guidance of Dr. Eggers at the Institut fiir Schiffoau, University Hamburg. * * * REPLY TO THE DISCUSSION Wen-Chin Lin, J. Randolph Paulling and J. V. Wehausen University of California Berkeley, California First of all we should like to state that we are pleased that Dr. Pien has found it quite clear how our results were obtained, even though their significance may remain cloudy. It seemed particularly important for these tests that this should 1062 Data for Ships of Minimum Resistance be the case and that there be no question of finagling with data in order to "im- prove" it. The authors cannot agree with Dr. Pien's statement that the "relative re- sistance qualities are just opposite to the theoretical predictions."’ For both models the residuary resistance is greater than the Michell resistance at the design Froude number 0.316. This is, in fact the usual occurrence at this Froude number for hulls with these prismatics. The only thing which really seems out of the ordinary is the very low ratio of Michell to residuary resist- ance for the symmetric model, but here there are no similar experiments to compare with. However, it is because of the extremely low Michell resistance in this case that we called attention to the necessity of considering the possi- bility that the second-order term overpowers the first-order term. This neces- sity does not seem so pressing for the asymmetrical model. An accurate assessment of the effect of viscosity is also correspondingly more important for the symmetric model. Although Dr. Pien might appear to have deprecated the importance of the second-order approximation, he is, in fact, also proposing that we take it into account. He states, ''... we know the linearized ship-surface condition is not accurate for the beam value....'' Indeed, we know much more, for we know that the linearized approximation is not accurate for either the hull shape or the wave surface. In a paper by one of the authors presented at the Ann Arbor Symposium in 1963 it was shown that the more important error is associated with this phenomenon in a related situation. Dr. Pien's argument that, in cases where the first-order resistance is very low, it is legitimate to use the linearized free-surface condition together with the exact body boundary condition is tempt- ing, but assumes that low wave resistance is associated with small surface dis- turbance everywhere. However, it is still possible that the local disturbance is substantial and it is just in this locality where the inaccuracy is most important. With regard to further experiments, it is our own opinion that the influence of viscosity should be clarified before any attempts to improve the approxima- tion are made, and, in particular, that one should have a more reliable experi- mental determination of the wave resistance. This also appears to be the import of Prof. Ward's remarks. It is a pleasure to add that he has later volunteered to undertake an investigation of the wave resistance of the symmetric model ac- cording to his method. Prof. Winblum states that he has preferred polynomial representations for hulls partly on mathematical grounds because of Weierstrass's Theorem. We hope he will take it as good-natured malice if we point out that there are two Weierstrass approximation theorems. One states the uniform approximability of continuous functions on closed intervals by polynomials, the other by trigo- nometric sums. Thus there is no mathematical reason for preferring polyno- mials. The advantage of the trigonometric sums lies in the orthogonality of the expansion functions, which results in smaller coefficients for higher harmonics. The same can, of course, be achieved with Legendre or Chebyshev polynomials (Prof. Maruo worked out the details for the latter during a visit to Berkeley), but now the numerical computation becomes somewhat more complicated. 1063 Lin, Paulling, and Wehausen With regard to the unaesthetic aspect of our wavy forms we are pleased to report the following. Since the forms from which the tested models were made were computed, we have extended the computations from MXP = 6x6 to MXP = 10 x4, 12 x4 and 15 x4 for the symmetric model. The numerical evidence sug- gests that the 15 x4 form is already quite close to the limit with P = 4. The im- provement in resistance is negligible. However, there is a quite noticeable change in form as one proceeds from 6 <4 to 15 <4, and, although the latter form is no longer wavy, it does not ''smooth" the wavy lines of a 6x4 form, and it would have been misleading, if not even somewhat dishonest, to have drawn smooth curves through the wavy ones on this basis. It may be of interest to note the following. In going from 6 x4 to 10 x4 the beam decreases, the middle sec- tions become more U-shaped and the ends much bulbier. In going from 10 <4 to 15 x4 the middle sections remain practically the same, but the bulbiness con- tinues to increase, although the difference between 12 <4 and 154 is slight. If one keeps M fixed at 10 and lets P be successively 2, 3, and 4, the section-area curve hardly changes, the sections near the ends change little, but the middle sections become more U-shaped. The authors thank the discussers for their interest in and comments on their paper. 1064 SOME RECENT DEVELOPMENTS IN THEORY OF BULBOUS SHIPS Be Yim Hydyronautics, Incorporated Laurel, Maryland INTRODUCTION The history of the bulbous bow on ships may start in the early 19th century with submerged rams on combatant vessels projecting forward along the water- line at the stem, or with the projecting underwater hulls of many old French warships built about the same time. Later, the British armored cruiser Levia- than had such a projecting ram bow. D. W. Taylor suspected that this ram bow played a definite part in the ships superior performance, and he based the par- ent model for his famous Standard Series (D. W. Taylor 1911 or 1943) upon the lines of Leviathan. Systematic bulb bow experiments were made by E. F. Eggert in the early 1920's and the general data were reported upon by D. W. Taylor (1923). It had been generally understood that the decrease of resistance due to a bulbous bow is a wavemaking phenomenon, such as a decrease in bow wave height due to a bulb wave. This understanding was more strongly supported when Havelock (1928) calculated the surface wave due to a doublet immersed in a uniform stream. A deeply submerged sphere is equivalent to a doublet. Hence according to his calculation, a sphere moving through water at a constant speed causes the surface wave to start with the trough just aft of the sphere. It is natural to imagine that this trough has something to do with the bow wave crest which is seen to start just aft of the bow in ordinary ships. However there was also some other suspicion that the bulb effect is due to a change in the effective ship length owing to the alteration by the bulb of the position of the bow wave. This suspicion was removed by Wigley's mathematical and experi- mental investigation (1936). He used Havelock's formula for wave resistance (1934) in terms of the regular wave heights due to the ship hull and a point dou- blet. He separated the wave resistance into three parts: the hull wave resist- ance, the bulb wave resistance and the interference resistance of the hull and bulb. The most favorable case occurred when the negative interference resist- ance was largest. He derived the following six rules for the bulbous bow as the conclusion of his investigation (W. C. S. Wigley, 1936): (1) The useful speed range of a bulb is generally from V/VL = 0.8 to V/VL = 1.9 (or in Froude numbers based on ship length, from 0.238 to 0.563), V being the speed in knots and L the ship's length in feet. (2) The worse the wavemaking of the hull itself is, the more gain may be expected with the bulb and vice versa. 1065 Yim (3) Unless the lines are extremely hollow the best position of the bulb is with its center at the bow, that is, with its nose projecting forward of the hull. (4) The bulb should extend as low as possible consonant with fairness in the lines of the hull. (5) The bulb should be as short longitudinally and as wide laterally as pos- sible, again having regard to the fairness of the lines. (6) The top of the bulb should not approach too nearly to the water surface; as a working rule it is suggested that the immersion of the highest part of the bulb should not be less than its own total thickness."' G. Weinblum (1935) dealt with this same problem by expressing the form of a ship with a bulbous bow in terms of a polynomial according to Michell's thin ship approximation. His theory was also supplemented by model experiments. He expressed a different view from Wigley's, concerning the best vertical posi- tion of a bulb (Wigley's rule [4] and [6]). According to Weinblum's result for an extremely hollow form of ship, a uniformly distributed bulb along the stem line was superior (taking into account the wave resistance only without considering other effects like spray) to the bulb located near the keel, both having the same sectional area. However neither Weinblum or Wigley suggested any optimum variation of bulb size with the speed. Since then, some experimental investigations on bulbous bows were per- formed by Lindblad (1944) in calm water and by Dillon and Lewis (1955) in smooth water and in waves. However, after Wigley (1936) and Weinblum (1935), no significant theoretical development on bulbous ships seems to have been made, until Takao Inui and his colleagues made a great contribution on this sub- ject. This will be discussed ina later section in some detail. In this report, first the necessity of a bulb for minimizing wave resistance will be discussed, followed by a brief review on Inui's explanation of the bulb effect. Inui, using the concept of Havelock's elementary surface waves brought us a Clear understanding of the mechanism of bulbs and an easy approach to their design. Yim (1963) found the ideal bulb or the doublet distribution on a semi-infinite vertical stem line which completely cancels the sine regular waves starting from the stem of a given ship. For the cosine waves from the ship bow, a source line or a quadrupole line are considered. The separation of waves and the wave resistance into the components as in the diagram of Fig. 1, simplified the analysis of the bulb effect at the bow or the stern of a ship. The size and the form of the bulb, which are functions of ship shapes and Froude numbers, are supplied extensively. The location of the bulb is of course related to the ship shape and the type of bulb. However, the higher order effect is found to be non-negligible. These are discussed in the next sections. Throughout this report, inviscid, homogeneous, incompressible, and poten- tial flow around a fixed ship is considered. The origin of the right handed Car- tesian coordinate system is located on the bow of the ship and on the mean free 1066 » elopments in Theory of Bulbous Ships LOCAL DISTURBANCE HIGHER ORDER WAVES REGULAR WAVES SUPERPOSITION OF ELEMENTARY WAVES SINE WAVES COSINE WAVES FROM puLes eal SHOULDERS STERN INTERACTIONS WAVE RESISTANCE Fig. 1 - Diagram for the characteristics of ship waves surface. The intersection of the ship's center plane and the mean free surface is taken as the x axis, with the z axis perpendicular to the free surface, posi- tive upward. The flow at x = -~ is considered to be uniform with the velocity V parallel to the x axis in the positive x direction (see Fig. 2). SHIPS OF MINIMUM WAVE RESISTANCE AND BULBOUS SHIPS Since Michell's wave resistance formula (1898) was found, problems of finding the Michell's linearized ship which has the minimum wave resistance have been attacked by many hydrodynamists in various forms and ways. Sretenskii (1935), Pavlenko (1937), Karp, Votik and Lurye (1958) and Maruo and Bessho (1962) treated symmetric infinite vertical struts. Weinblum (1930, 1957), Krein (1955) and Martin (1961) dealt with three-dimensional symmetric ship 1067 Fig. 2 - Schematic diagram for a surface ship and the coordinate system with a given vertical distribution of volume. In their solution, they all found either some singularities in the functions representing hull shapes at the ends of ships, or bulblike forms around the bows and the sterns. Wehausen, Webster, and Lin (1962) treated the optimum forebodies of ships with a given afterbody as well as three-dimensional symmetric ships without any restriction on the verti- cal distribution of volume. However they took the ship surface area into account to minimize the wave and friction resistance, and they too found big bulblike forms near the bottom of bows for higher Froude numbers. Havelock's wave resistance formula (1934) from the regular waves due to the singularity distribution on the center plane of a ship is essentially the same as Michell's, as long as the linear relation of the ship hull form with the singu- larity distribution m( x, Z) = SF (x,2) (1) is used, where m(x,z) is the source strength and f(x, z) is the ship hull form. Inui (1957) calculated an exact hull form (body streamlines of a double model) from a given source distribution for zero Froude number (flat free sur- face), and he used this hull form for his model experiment to test waves and the wave resistance. He compared his experimental results with his calculated wave heights and the wave resistance due to the source distributions. He found that the calculation agrees better with his experiment on his model than the corresponding Michell's model satisfying (1). The way Karp, Lurye, and Kotik (1958) interpreted their result to a ship form of infinite draft is similar to the idea of Inui's which we have just described. The singular behavior of Michell's ship hull can be easily treated by reinterpreting Michell's ship hull as the dis- tribution of various singularities like sources or doublets either distributed or concentrated. ; 1068 Developments in Theory of Bulbous Ships Krein (1955) proved in a rigorous manner the existence of a lower bound for the Michell's resistance of ships with a given center plane, a given velocity, a given displacement, and a given vertical distribution of volume. However he concludes that the lower bound of the wave resistance due to a submerged ship is obtained only with generalized functions (i.e., linear combinations of Dirac delta functions) of a ship hull shape; and for floating bodies the wave resistance achieves a lower bound but only for functions of hull shapes having integrable Singularities at the ends of the ship. In the Michell's ship hull representation (1), it is easy to see that the hull shape f(x,z) is proportional to the doublet strength distributed on a given center plane of the ship. Therefore, if we consider the body streamlines due to the doublet distribution in the uniform stream instead of considering f(x,z) asa hull shape, we may be readily convinced that the ship form of minimum wave resistance has a bulbous bow. In addition, it is worthwhile to note here that, the Dirac delta function of the distributed doublet at the bow is the concentrated line doublet, and the integrable singularity of the doublet distribution at the bow may also be interpreted as a doublet concentrated around the bow. ELEMENTARY WAVES AND THE WAVE RESISTANCE FORMULA By Lord Kelvin (1887), it was found that the surface wave due to a point dis- turbance in a uniform stream consists of two parts: the local disturbance which is limited to the neighborhood of the point disturbance and the regular wave which propagates far aft of the point, mainly restricted to the sector of |6| < 19°30’. This is a mathematical solution of the equation for the potential ¢ perturbed by the disturbance, Vd = 0 (2) with linear boundary conditions at the mean free surface z=0, considering the wave height is small compared to the wave length, Sh BES, (3) where k, = g/V? (g = acceleration of gravity) and at x>-» and z+-%, V=0. (4) Now it is well known that a point source of strength m located at a point (x,,0,-z,), where z,>0, produces a regular wave height ¢ at a large x 1 1 1 iif?) Grr Alen mexp|(=k-z, see-@) see20 — Tie x cos [k, SEG Gr (Cv —aks NCOSGE y sin e}] dé (5) 1069 where _ ll _ Hg k = v2 5 k = y2 . (6) L is the ship length, H is the ship draft, m is nondimensionalized with respect to LHV, x,x,,y,¢ is nondimensionalized with respect to L, =z is nondimensionalized with respect to H. = Zig For a distribution of sources at a ship center plane S,(y=0, 0 : a,,a Bintan) = 10 n 2? (40) for b, when a, are given. Since the bow resistance due to sine waves and that due to cosine waves are additive as shown in (23), the concentrated singularities for each case can be dealt with separately. The optimum distribution of the concentrated singularities at the finite stern line for several given ship source distributions are calculated (Yim 1963) and shown in Figs. 3-7. These indicate that the strength of the singularities at the deepest point (the same level as the keel) is the largest. Especially for the higher Froude numbers, the optimum distributions appear to be almost concen- trated at the keel. This rather supports Wigley's fourth rule. However the op- timum size of the bulb is extremely sensitive to the Froude number. We notice in Figs. 3-7 almost a linear distribution of the doublet for the low Froude num- bers. If we were given the volume of the bulb, the optimum distribution would be also sensitive to the Froude number and the displacement of the bulb would gradually move from the keel closer to the surface as the Froude number in- creases, since the effect of a bulb is stronger at a smaller depth. This would clarify the difference in the opinions of Wigley and Weinblum mentioned before in our introduction. However, in actual ships, the wave resistance is not the only problem. 1077 NON -DIMENSIONAL DEPTH, -Z/H FOR OPTIMUM BULB 1.0 ers 1.4 1.6 1.8 =) | 0.4 = ~~ __ |OPTIMUM Fy, SS \ a a \\ Rie Pi NSs gS Sa \\ = aoe a ~ ~ \ SS a ae) ie 2 \ ~ 0.8 1.0 0 0.4 0.6 0.8 eye: NON- DIMENSIONAL DOUBLET STRENGTH 12”/(HBV) OR SECTIONAL AREA !2r/(BH) Fig. 3a - Doublet distribution for sine ship (L/H = 16) 1.2 Tee —— -—— WITHOUT BULB / Sut aes WITH OPTIMUM BULB | —~ ‘ & xl“ |.0 5 ih | Din o|> Ne y \ if 0.8 al a ee lela hi 0.6 F 0.4 NON- DIMENSIONAL BOW WAVE RESISTANCE OPTIMUM Fy= Was 0.75 10 1.25 1.50 1.75 FROUDE NUMBER Fy 0.1875 0.25 0.3125 0.375 0.4375 FROUDE NUMBER F, 1078 Fig. 3b - Bow wave resistance of sine ship (L/H = 16) 288R (Pv? Be 77) oO @ NON- DIMENSIONAL BOW WAVE RESISTANCE Developments in Theory of Bulbous Ships fo] ; (ee — —— — FOR OPTIMUM BULB 0.4 0.6 NON-DIMENSIONAL DEPTH, -Z/y 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 NON- DIMENSIONAL DOUBLET STRENGTH, !2”/(HBV),OR SECTIONAL AREA !2r7(BH) Fig. 4a - Doublet distribution for sine ship (L/H = 24) SINE SHIP WITHOUT BULB WITH OPTIMUM BULB 1.50 175 FROUDE NUMBER Fy, 0.153 0.204 0.255 0.306 0.357 0408 0.459 FROUDE NUMBER F,_ Fig.4b - Bow wave resistance of sine ship (L/H = 24) 1079 NONDIMENSIONAL DEPTH, - z 0.2 0.4 0.6 0.8 Yim 2 V F a oH | aia Fyzl.75 Fyzl25 8) GOs Gb sy Sim 2 At a point (x,y,o) which is not on the singularity plane, the last quadruple inte- gral J(x,y,o; €), say, can be written sec Bienea ets NGA) cos 6+ y sin@] 1 7 co J(*, y,0;1) - J(x,y,0;0) = >Re i ————SS Sooo EG Lae Kietesee te) — Vij Sec? T (ee) ts ; 1 seerO Mines ee Ces vesting) ae J fT, ae Less Se unos a did. (44) - 7 0 Kee sec*@ - ip sec 6 When we consider the limiting case of y>0 in J/x,y,o;1), this becomes zero for any k, since the integrand is antisymmetric in ¢. Now if we change the variable k >k Jk 1 [tin ey Ge) Bet ikk,(x cos 6 + y sin@) Iox,yroit) = ERe [ f See 4 sin’ ¢—___ cae. (48) k- sec?@ - ip sec @ -7 0 1087 Yim This is a function of only k,x and ky. The case when x->0, y-0 for a certain k, is exactly the same as the case when k,~0 for certain fixed values of x and te) y. For k,> 0, or the case of infinite Froude number, Therefore, for any k, d (46) we O. The above argument can also hold for a point (1+ x,y,o) as x>0, y>0. Although we considered points only on z=0, we notice from the potential theory that physical quantities change continuously into the potential flow field from the boundary. This indicates that every surface ship which is represented by a centerplane source distribution has as strong an influence of the free sur- face on the shape of the ship in a certain neighborhood of the free surface as in the case of infinite Froude number. The influence of the free surface can be explained much more eloquently by Green's formula for the velocity potential ¢ which satisfies the Laplace equa- tion (2) with the boundary conditions (3), (4) and Bite Samir (47) (n is the normal vector at the ship hull surface into the fluid.) Ona given ship hull, 1 ie - 7 = p — An [[eno CACSa Mh Gos re) =] Pal Ss; @) GCSEs Hs een 2)| ds (48) S) where S includes the free surface S, and the ship surface S. (see Fig. 2). G is the well known Green's function (see e.g., Stoker 1957) which is a harmonic function for ¢ <0 except at (x,y,z) where it has the singularity [(G=22 4 Genes Gene) and G satisfies the boundary conditions (3), (4) and The integral on the free surface S, in (48) can be written by using (3) 1088 Developments in Theory of Bulbous Ships I ia [46,, at b,G| dS = - ie [¢G, - Gd, | dé dr Sp z=0 1 fe) B) f a ko i Oe (PG -) = Be (p.G) dé dn z=0 = - = $ a6, - £-G) dy (50) ie} f where ¢ is the intersection of the ship surface and the z=o0 plane. Since the ship beam length ratio B/L = « is considered to be small, in general, (50) is omitted in the first order theory. Wehausen (1962) considered a systematic, formal, yet thorough estimation of the order of magnitude in the Green's formula with the exact boundary condi- tions of the potential. For a ship with the draft H as small as the beam B, he estimated I in (50) is 0(«3) while the main integral around the ship hull in (48) is O(«2). In fact it has been known that the effect of the draft behaves like exp (-CH) where C is a function of Froude number and even for the case P/B = 2, the wave heights was comparable to the case H>™ (Wigley, 1931). Therefore the above estimation may be true even for the case of an infinite draft ship, and the line integral I, in this case will be the most important contribution to the higher order terms which have been previously neglected. Indeed, in (50), I is the influence of the free surface on the potential. However, it is extremely difficult to understand the higher order effect just by the formal estimation of the magnitude and without actual evaluation, since the property of Green's function is very complicated particularly near the free surface. As a simplest case for the evaluation of the line integral, Yim (1964) considered a source distribution {I So = i < < =o) < (x-¢€) we can evaluate the line integral (50) at large x and y=0 neglecting higher or- der terms, d k,(x-a) x las Mai anit aes 2 tan a [ecegx-t) at vo] (o} =k_x (o} + 4 tan ak, | 4([s ¥,(0)| dé. 0 (al (CaaS ) If we take only the lower limit of the above equation, it can be considered from the equation for the surface wave to represent a regular wave starting from the bow due to the influence of the free surface. From here Yim (1964) calculated the amplitude and the phase of the regular bow wave /,. far behind the ship on y —0 due to the line integral, Ci ee sim (kx + eo p) : It is easy to see from Havelock's result that the regular bow wave ¢, from the first order theory is, In Fig. 10 are shown the phase difference £ and 1090 Developments in Theory of Bulbous Ships P Q tan a % Roa) which are functions of only k,a. The amplitude of the total wave °, een corer cn € : TT y = 0? # p? + 20p'cos 6 sin (kx +4 ) and the phase difference ¢ between the total wave /, and the first order wave ” are shown in Figs. 11 and12. 2 and ¢% are shown in radian, considering that one wave length (27/k,) is just 27. PHASE DIFFERENCE 8 FOR RADIAN Fig. 10 - Phase difference between the first and the higher order waves 5, and the amplitude ratio/half entrance angle, f£(k, a) These show that the total wave phase is indeed advanced considerably com- pared with the first order wave, while the amplitude of the total wave height does not differ too much from that of the first order wave. Namely the second order effect is quite large. It is proportional to the slope of the entrance on the free surface, for a givenrun, a. Therefore, the smaller the entrance slope near the free surface is, the less the second order effect to be expected. As we see in the integrand of the line integral (50), this effect mainly depends on the potential and the wave on the free surface waterline where the waterline slope is large. Since the local effect is usually big near the bow and the shoulder, the influence of the local effect on the second order wave may be quite important. 1091 1 .004 X/L=15 LENGTH OF ENTRANCE a/L =0.25 HALF ENTRANCE ANGLE&=0.12RAD. 003 je) (e} is) RADIAN WAVE AMPLITUDE INCLUDING HIGHER ORDER FIRST ORDER WAVE AMPLITUDE WAVE AMPLITUDE/LENGTH OF SHIP (e) {o) | PHASE DIFFERENCE BETWEEN THE FIRST ORDER AND TOTAL WAVES O- Os Z2 "20 (5 aL 0 5 223 25. «258 F 316 a4 Fig. 11 - Comparison between the first order wave and the total wave (a = 0.12 rad) 1092 Developments in Theory of Bulbous Ships ‘ < Booth og = X/L=15 WwW = a/L=0.25 a Ir a=0.1 4p) “Nn = 2 2 =) E =! a = aq = a > Si | | S | fs) PHASE DIFFERENCE ) 2 25 3 35 4 45 5 F 20 13 i0 9 7 6 5 4 k.L Fig. 12 - Comparison of the first order wave and the total wave (a= .071 rad) We notice that when we cancel the regular wave by the bulb, the line integral due to this wave will be also cancelled. This study of the line integral (50) has just started. However it seems to be quite promising for furtherance of a proper understanding of ship waves and of their reduction. CONCLUDING REMARKS Theory and experiment are always stimulating and helping each other. Al- though this report is on the theoretical side, it does not mean that the influence of experiments are underestimated. This report is merely intended to further appreciation of our great predecessors, Michell, Havelock, Wigley, Weinblum and Inui for the theories related to the bulbous bowed ship, and to add a slight theoretical illumination to them. The mechanism of the bulb at the ship bow (or stern) is completely clari- fied. The type of bulb for a given ship hull, and the size and the vertical area distribution of bulb for a given Froude number are derived. The higher order influence is known to be the major reason for the phase shift of the regular 1093 221-249 O - 66 - 71 Yim waves. Although the stern problem in the non-viscous fluid is exactly the same as the bow problem, it should be studied separately due to the large influence of viscosity, wakes, propellers, etc. Because of these influences, the bow waves are more important in practice than the stern waves. The humps and hollows of the curve of the wave resistance due to a ship without a bulb may be applied to that for the ship with the bulb without any considerable error. The bulb has an effect of smoothing out the humps and hollows of the resistance curve to a con- siderable extent (Yim 1962) in the vicinity of the designed speed or for larger speeds. Pien (1962) seems to have obtained this effect using the principle of wave cancellation by distributed singularities rather than concentrated ones. Naturally, a ship with a bulbous bow would have much the better performance if it has a better stern. At the present time, shapes like the transom stern seem to attract the interest of many naval architects for high speed ships. The higher order effect and the influence of viscosity are extremely difficult to analyze, yet they should and will be gradually exploited in the near future. The theoretical study on the seaworthiness of the bulbous ships remains to be done, although it is known from experiments that bulbous bows are still effective in waves. ACKNOWLEDGMENT This work was kindly sponsored by the Office of Naval Research, Depart- ment of the Navy under Contract No. Nonr-3349(00), NR 062-266. Thanks are due to Mr. M. P. Tulin for his helpful discussions. NOTATION a = Length of run of wedge strut a, = Coefficients of polynomial representing source distribution for a ship b, = Coefficients of polynomial representing concentrated singularity distributions for a bulb B = Beam f(x,z) = Ship hull form F,,F, = Froude numbers with respect to draft and length respectively g = Acceleration of gravity H = Draft of ship H. = Struve function Z = ‘x72 se = gH V 1094 co Developments in Theory of Bulbous Ships k, = el? L = Length of ship m = Nondimensional source strength R = Nondimensional wave resistance v = Uniform velocity at x = - x,y,z = Right handed rectangular coordinate system with z positive up- ward, x in the direction of the uniform velocity y, and the origin on the mean free surface Y_ = Bessel function of the second kind a = Half entrance angle ¢, = First order wave height C,- = Second order wave height E,n, © = Coordinate system equivalent to 0 - x,y,z -y = Nondimensional doublet strength \ = Nondimensional quadrupole strength (CE) S (C8 = 6) COS CO se s¥ Swi Ch REFERENCES Bessho, M., ''On the Wave Resistance Theory of a Submerged Body," 60th Anniversary Series, Vol. 2, SNAJ, 1957. Bessho, M., "On the Minimum Wave-Resistance of Ships with Infinite Drafts,'' Proceedings of International Seminar on Theoretical Wave- Resistance, University of Michigan, 1962. Dillon, E. S. and Lewis, E. V., "Ships with Bulbous Bows in Smooth Water and in Waves," Trans. SNAME, Vol. 63, 1955. Havelock, T. H., ''The Wave Pattern of a Doublet in a Stream,"' Proc. Roy. Soc., A 121, pp. 515-23, 1928. Havelock, T. H., "Ship Waves: the Calculation of Wave Profiles," Proc. Roy. Soc., A 135, pp. 1-13, 1932. 1095 iL), tie: 25 13. 14. 15). 16. Ws Se 18). 20. 21. Nauta Havelock, T. H., "Wave Patterns and Wave Resistance," TINA, Vol. 76, pp. 430-443, 1934a. Havelock, T. H., "The Calculation of Wave Resistance,'' Proc. Roy. Soc., A 144, pp. 514-21, 1934b. Havelock, T. H., ''The Forces on a Circular Cylinder Submerged in a Uni- form Stream,'' Proc. Roy. Soc., A 157, pp. 526-34, 1936. Inui, T., ''Wave Making Resistance of Ships,'' Trans. SNAME, Vol. 70, pp. 283-353, 1962. Inui, T., Takahei, T., and Kumano, M., "Wave Profile Measurements on the Wave- Making Characteristics of Bulbous Bow,'' SNAJ (Translation from University of Michigan), 1960. Karp, S., Kotik, J., and Lurye, ''On the Problem of Minimum Wave- Resistance for Struts and Strut- Like Dipole Distributions,'' Third Sympo- sium on Naval Hydrodynamics, ONR, Department of the Navy, 1960. Kostchukoy, A. A., "Theory of Ship Waves and the Wave-Resistance,"' Leningrad, 1959. Krein, M. G., Koklady Akademi Nauk, SSSR, Vol. 100, No. 3, 1955. Kelvin Lord (Sir W. Thomson), "On the Waves Produced by a Single Impulse in Water of any Depth, or in a Dispersive Medium," Proc. Roy. Soc. of London, Set A 42, pp. 80-85, 1887. Lindblad, A., "Experiments with Bulbous Bows,"' Publication No. 3 of the Swedish State Shipbuilding Experimental Tank, 1944. Lunde, J. K., ''"On the Linearized Theory of Wave Resistance for Displace- ment Ships in Steady and Accelerated Motion,"’ Trans. SNAME, 1951. Lunde, J. K., "On the Theory of Wave-Resistance and Wave Profile," Skipsmodelltankens, Meddeleke Nr. 10, 1952. Martin, M. and White, J., "Analysis of Ship Forms to Minimize Wavemaking Resistance,"’ Stevens Inst. of Tech. R-845, May 1961. Maruo, H., "Experiments on Theoretical Ship Forms of Least Wave- Resistance,'’ The International Seminar on Theoretical Wave Resistance, University of Michigan, 1963. Michell, J. H., ''The Wave Resistance of a Ship,'' London, Dublin and Edin- burgh, Philosophical Magazine, Ser 5, Vol. 45, 1898. Pavlenko, G. E., ''The Ship of Least Wave Resistance,'’ Trudy Vsesoyuz Nanch-Inzhen-Tekhn. Obshch. Sudostroen, 2, No. 3, 28-62, 1937. 1096 22. 23. 24, 25. 26. Zit. 28. 29. 30. 31. 32. 33. 34. 35. 36. Developments in Theory of Bulbous Ships Pien, P. C. and Moore, W. L., ''Theoretical and Experimental Study of Wave-Making Resistance of Ships,'' Proc. of the International Seminar on Theoretical Wave-Resistance, Michigan, 1963. Sisov, V., Isvestiya Akadem Nauk, SSSR, Mechanics and Engineering No. 1, USGL. Stoker, J. J., "Water Waves,"' Interscience Publishers, Inc., New York, NEY -5 L997.. Sretenskii, L. N., "Sub un Probleme de Minimum dans la Theorie du Navire,"’ C. R. (Dokl) Acad. Nauk, USSR 3, 247-248, 1935. Takahei, T., ''A Study on the Waveless Bow,'' SNAJ (Trans. from University of Michigan), 1960. Taylor, D. W., ''The Speed and Power of Ships," 3rd U.S. Gov't Printing Office, Washington, D.C., 1943. Taylor, D. W., Marine Engineering and Shipping Age, pp. 540-548, Sept. 1923. Wehausen, J. V., ''An Approach to Thin-Ship Theory," Proc. of the Interna- tional Seminar on Theoretical Wave Resistance, The University of Michigan, 1963. Wehausen, J. V., Webster, W. C., and Lin, W. C., "Ships of Minimum Total Resistance," Proc. of the International Seminar on Theoretical Wave Re- sistance, The University of Michigan, 1913. Weinblum, G., "'Schiffe geringsten Widerstands,"’ Proc. Third International Congr, Appl. Mech., Stockholm, pp. 449-458, 1930. Weinblum, G., "Die Theorie der Wulstschiffe,'’ Der Gesellschaft fur Angervandte, Mathematik, 1935. Weinblum, G., "'Schiffe geringsten Widerstandes,"’ Jahrbuch der Schiff- bautechnischen Gesellschaft, V. 51, 1957. Wigley, W. C. S., "Ship Wave Resistance,"' Trans. N.E.C.I.E.S., Vol. XLVI, pp. 153-196, 1931. Wigley, W. C. S., ''The Theory of the Bulbous Bow and its Practical Appli- cation,"’ Trans. N.E.C.I.E.S., Vol. LII, pp. 65-88, 1936. Yim, B., ''Analysis of the Bulbous Bow on Simple Ships,'’ HYDRONAUTICS, Incorporated Technical Report 117-1, 1962. 1097 NZ aligat 37. Yim, B., ''On Ships with Zero and Small Wave Resistance,"’ Proc. of the In- ternational Seminar on Theoretical Wave Resistance, The University of Michigan, 1963. 38. Yim, B., ''The Higher Order Effect on the Waves of Bulbous Ships,'' HYDRO- NAUTICS, Incorporated Technical Report 117-5, 1964. THE SHIP BULB Ata Nutku Technical University Istanbul, Turkey The merits of the ship bulb as a resistance reducing mean has first been de- tected by Admiral D. W. Taylor. However, a great amount of testing has since been carried out to utilize it as an improving medium of ship form. The testing has been confined to minor changes on its size and form, and no attempt has been made towards a scrutiny on its basic concept or characteristic function. As a matter of fact, the bulb today stands as we have inherited it from our forefathers who designed and used it for ramming the enemy ships during ac- tion. The original form of the bulb has been conservatively retained with only minor changes, which has satisfied its experimenters within the limits of 2 per- cent to 5 percent gain in total resistance of a ship. Some of the explanations for the action of the bulb may be summarized as: (a) lowering centre of pressure zone at bow, (b) displacing the bow wave to forward, consequently changing the phase of the wave system as to their order of synchronization, and (c) causing a suction on the surface wave phenomena. All the above will consequently cause change of flow pattern at bow. The section of the bulb has attracted my attention from the observations made on the behaviour of a submerged circular streamlined body towed near the surface at different depths, and from the analysis of the results of its resistance and trimming moments. The purpose of these tests, conducted in the years 1956-57 has been purely academic, parallel to Wigley's and Gawn's experiments with fish form bodies. I acknowledge the help and directives given by Prof. Dr. Gunther Kempf, who was then a visiting professor in I.T.U. 1098 Developments in Theory of Bulbous Ships A circular streamlined body of L/D = 4 has been used as a basic model which has later been utilized for different purposes as: the submerged body of a hydrofoil supported catamaran ship, as the ballast keel of sailboat tests and later as a bulb for Turkish fishing boat model tests. Bulbs as large as one third the length of the model were tried and interest- ing results were obtained, which however not published has served to inspire the visitors to Turkish Tank, to promote new strides in chapters of wave re- sistance of ships with bulbs. The action of the bulb as to its characteristic of producing suction can be visualized by the head-on trim it causes on the surface ship. This suction be- comes highly distinctive when it is towed under a flat bottomed pontoon, or near the water surface. The pictures of a fish form circular body of L/D = 3 taken at different speeds are shown in Fig. 1. Fig. 1 - Fish form circular body It is noted that, at speeds lower than (the critical Froude number for depth), a wave trough is produced immediately after the bow wave of the fish, which moves aft as the speed is increased. This trough, the focal point of suction when coincides with the bow wave of the ship is swallowed in it. The effect be- comes more pronounced as the bulge nears the surface. At greater speeds a sheet of water covers the top and the centre of the suction moves further aft over the tail. The ships which are sensible to trim, when fitted with bulbs, have some- times indicated increased resistances, at certain speeds, due to dive in of their bow, resulting from the suction produced by their bulbs, consequently increased bow waves, instead of reduced ones. This will mean a wrong shape, size and position of the bulb. This complex interaction of bulb and ship necessitated 1099 Yim systematic testing with bulbs fitted as separate appendages at the fore end of the ship. Unconventional means and methods were tried to assess the behaviour and interaction. For this purpose, geometrical bodies like spheres, cones, cylin- ders, etc., were included in the programme (Figs. 2a, 2b, 2c, 2d, and 2e). The science of hydrodynamics already reveals the individual resistances of geometrical bodies, also when they are towed in tandem formation at differ- ent spacings between them. In choosing the unusual devices, the aim has been to study their comparative interactions with the hull, rather than their direct adoption as a resistance reducing mean. The circular streamlined axisymmetric body has been selected as the near- est geometrical contemporary to the existing ship bulbs. Two ship models: one of a coastal tanker and the other of a motor launch were selected to be subjected to systematic testing. Some of the devices as fitted are shown in the accom- panying photographs. The devices as tried may be subdivided into the following categories according to their functions: (a) interference effect, (b) bow wave suction or flow deviation, (c) wave suppressors, and (d) wave scrapers or spears. Geametricol Lodies ieee Ipterterence eV GE Fig. 2a - Passive means — geometrical bodies interference 1100 Developments in Theory of Bulbous Ships Suction elements & Ceviotors Fish form bodies B Fig. 2b - Passive means — suction elements and deviators — fish form bodies in Aw) Tool wy Fig. 2c - Passive means — suction elements and deviators — fish form bodies 1101 3 Weave Suppressors Mezzle Segments AS Fig. 2d - Passive means — wave suppressors —nozzle segments ae Stee eeee ee Fig. 2e - Passive means — wave scrapers 1102 Developments in Theory of Bulbous Ships The axisymmetric fish form body has been split into two and has been fitted in different positions on the ship as shown in Fig. 3. The comparative curves of resistances of the original naked model and that of a composite configuration having a bulb fitted at stem on the designed water- line in combination with a circular segmental suppressor of hydrofoil section (curved on top, Fig. 4). This model with (WL bulb plus suppressor) has shown itself of having less resistance after a model speed of v = 1.60 m/sec approxi- mately equivalent to F, = 0.217, F/ = 0.292 and a V//L = 1.00. Fig. 4 - Naked bulb and bulb with circular segmented suppressor 1103 Wikaa It has shown a 17.5 percent gain in total resistance at maximum speed of v, = 1.75 m/sec and up to higher speeds (from V//L = 1.10 upwards). Compar- ative wave formations at certain speed ranges are shown in Fig. 5. It may be concluded that, the ordinary ship bulb as fitted near the keel does not perform as well as a bulb fitted at the designed waterline. The wave forma- tion being a surface phenomena, the surface bulb becomes more effective, in taking the core of the bow wave, transforming thus the original solid bow wave into a sheet wave. The water at the trailing edge is accelerated at its lower edge, trailing aft. Submerged bulbs of greater sizes may similarly influence the downwash, but the penalty paid for their extra resistances, due to their bulkiness thwarts off the advantage brought by their adoption. A badly designed bulb, is therefore, worse than having no bulb at all. The bulb is destined to kill the bow wave which is the father wave and once it is killed, next of kin will not be as predominant. However, the effect of shoul- der wave does still retain its place of importance and however the use of shoul- der bulbs were also resorted to, it still needs careful considerations, calcula- tions and a good programme of experimenting, to find its proper shape and place. It might be a denting instead of bulbing. The devices shown in Fig. 2 as wave suppressors, scrapers or spears are impressive and effective in quenching or suppressing the waves, which is dem- onstrated by smoothed surface around the hull, yet their resistances are so high that their use for calm water alone may not be justifiable. Therefore, the term (waveless form) should not essentially implicate a form of least resist- ance, in every Case. The type, form, size and placement of the bow devices have to be decided according to the designed speed/length ratio, angle of entrance and other form characteristics of the ship. Some of the tests carried out with the model of a motor launch and the placement of the bulb or spear and the resulting wave formations are shown in Figs. 6, 7 and 8. The spear, solely an experimental device, piercing the water with a finer angle of entrance is also seen at speed. The waterline bulb may invite suspicion of many of us as conservative naval architects, also due to its higher resistances up to the cruising speed range. Yet, apart from the fact that the part of the resistance curve we are most in- terested in,is inthe high speed ranges, we may well go to introduce inflated rubber bulbs or appendages to suit the different speed ranges of the ship. Nearly every modern vehicle, from cars to ground effect machines are benefitting from its advantages. We may thus inflate it only at the speed ranges we want. Naval architects of today trying to design sea kindly ships with solid walls of steel are preoccupied with problems of seakeeping and slamming. A bulb properly designed and fitted at design waterline may be a better antipitching device than its submerged contemporary, also insuring less loss of power ina seaway. 1104 Developments in Theory of Bulbous Ships % ag 9 / fsa, 413 f 4 (6 47 8 i9 ties —de— ts de dy — bey te sda sh is ga —$—— Fig. 5 - Comparative wave formations at certain speed ranges 1105 Yim mie . seit ae Fig. 6 - Spear placement 1106 Developments in Theory of Bulbous Ships Fig. 7 - Spear placement 1107 Yim Mounted bulb and ship igi, & 1108 THE APPLICATION OF WAVEMAKING RESISTANCE THEORY TO THE DESIGN OF SHIP HULLS WITH LOW TOTAL RESISTANCE Pao C. Pien David Taylor Model Basin Washington, D.C. ABSTRACT Despite its limitations, the existing wavemaking resistance theory can be applied effectively to the design of better hull forms with practical proportions. Proper application of the theory can produce not only the direct benefit of reducing wave drag but also anindirect gain in viscous drag. Most of the numerical work involved in such application has been programmed into the 7090 IBM high-speed computer. Some nu- merical results obtained by using computing programs are shown. A ship design example to show how we can reduce both wave and viscous drags is also included. INTRODUCTION The total resistance of a ship consists of two parts, wavemaking resistance and viscous resistance. If wavemaking resistance theory can be used to minimize the wave drag of a ship, we can not only have the direct benefit of low wave drag but also a great possibility of reducing viscous drag. It has often been said that the application of this theory to ships currently designed to operate at low Froude numbers holds little promise because wave drag is a very small portion of the total drag. It is true that we cannot reduce the total drag of a ship very much in such cases even if we can eliminate the wave drag entirely. However, if the length of a ship is reduced, the wetted sur- face will be reduced, and as a result, the viscous drag will be decreased. If ship length is decreased, and speed and displacement volume are kept constant, the operating Froude number will be increased. Any experienced ship designer will agree that the increase in wave drag will far exceed the decrease in viscous drag. If the wavemaking resistance can be kept low through the application of the wavemaking resistance theory, then reducing the ship length will achieve a great gain in total resistance as well as a reduction of construction costs. This 1109 221-249 O - 66 - 72 Pien concept of applying the wavemaking resistance theory to reduce the total resist- ance of ships can be applied advantageously in the design of "practical" ships, i.e., Ships with practical L’/B and B/H ratios. To date, numerous attempts to utilize this theory have had disappointing results. However, this lack of success is not necessarily due to the limitations of the theory. It is my belief that, despite its defects, the existing theory can be used in the design of practical ships with low resistance. The justification for this view is fully discussed in this paper. In the belief that much better forms can be obtained by using this theory, I have undertaken a hull form research project at the David Taylor Model Basin. The first part of this project has been to program for automatic computation all the numerical work involved in the application of the theory to ship design work. Once this has been done, the application of the theory becomes a fruitful, enjoy- able task rather than tasteless, tedious labor. The second part of this project is devoted to the actual application of the theory to the design of ships. Models will be designed according to the theory and then tested, and the model experi- ment results can be applied immediately to the shipping industry. After suffi- cient theoretical and experimental data have been gathered, further improve- ment in the present wavemaking resistance theory can be expected. The first part of this project has already been accomplished. Two comput- ing programs have been developed. The first is used either to compute the wavemaking resistance and free-wave amplitudes of a given singularity distri- bution or to optimize a singularity distribution to fulfill a ship design problem. The second is used to compute the hull geometry from a given singularity dis- tribution. With these two computing programs, the second part of this project becomes relatively simple and easy. One model has already been designed and is under construction. The theoretical results for this model are given. This paper is essentially a progress report of the present hull form re- search project. The second part of this project has just been started. Another paper will be published upon completion of this phase. JUSTIFICATION FOR APPLYING THE WAVEMAKING RESISTANCE THEORY TO THE DESIGN OF PRACTICAL SHIPS Two important assumptions are involved in the development of the existing wavemaking resistance theory; these must be carefully considered if the theory is applied to ships with practical L/B and B/H ratios: 1. The free-surface disturbances created by a moving ship are small, and So wave height will be small in comparison to wave length. This assumption justifies linearizing the free-surface condition. 1110 Application of Wavemaking Resistance Theory 2. The viscosity effect is negligible and the potential theory can be used in the study of ship-created waves. The first assumption is usually satisfied by selecting a beam that is very small in comparison with the length and the draft. Such ships are called thin ships. Since a thin ship has no practical value, the theory has also been applied to ships with practical beams in the hope that some good may result despite the limitations of the theory. Fortunately, while a small beam is a sufficient condition for a small free- surface disturbance, it is not a necessary one. If a practical (thick) ship can be designed which disturbs the free surface as little as a thin ship, the linearization of the free-surface condition should be applicable to this practical ship as well. Since the main portion of the free-surface disturbance is due to the free waves which cause the wavemaking resistance, the theory should be applicable to thick ships of low wave drag as well as to thin ships. Therefore, the pertinent ques- tion to be asked with regard to the linearization of the free-surface condition is whether or not the wavemaking resistance is small rather than whether or not the beam is small. If we limit our study only to hull forms with very small wavemaking resistance, the theory is valid so far as the assumption about the free surface is concerned. In later sections, a procedure will be given for obtaining low wave-drag ships under the restraint of practical design conditions. Let us first examine more carefully the argument for using the theory to design low wave-drag prac- tical ships. For this purpose, the comparisons made in the past between theo- retical and experimental results have been carefully re-examined. Unfortunately, most of these comparisons have severe defects except those of Inui. He has clearly shown that the linearized condition on the ship surface is not accurate enough to obtain the singularity distribution of a given hull geometry for thick ships, or vice versa. If this situation is not improved, the theoretical model (singularity distribution) and the experimental model are not equivalent. Inui has been criticized by many people for employing a higher than first-order ap- proximation on the ship surface while keeping the first-order approximation on the free-surface condition. His approach has been fully justified by the impor- tant results he has so obtained. Since at this point we are examining only the consequence of the linearized free-surface condition, our study is confined to the comparison in the Froude number range where the viscosity effect is relatively small. In many cases, due to the fact that the theoretical and experimental models are not equivalent, such comparisons are rather confusing. Generally speaking, the percentage differences between theoretical and experimental results are smaller when the level of wavemaking resistance is lower. Emerson's paper [1], based on Wig- ley's experimental work, definitely shows this tendency. Fortunately, we have the comparisons of the S-series models made by Inui [2]. In each of these cases, the theoretical model and the experimental model are equivalent. Table 1 gives the theoretical and the experimental wavemaking resistance coefficients and the corresponding Froude numbers taken from Inui's published curves. Some geo- metrical parameters of these models are also listed. Figure 1 shows a simple comparison of the theoretical and experimental wavemaking resistance 1111 Pien ‘JUSTOTIJIOO QOUCYSTSAI SULMEUIOAPM PaaNnseseW oy} ST OS) oR “JUSTOTJZEOO QoueISISeL SuTyewaAeM poynduiod ayj st “y ‘“¢ “SOUTT UOTJOIAT SoySN YIM posn 1O4DeF ULO;Z 9yy ST ig fe} . “SouUT[Ie}eM FO o[sue aDUeTIUS 9Y} 0} [eUCTIJZOdOAd st qJ *pue ay} ye AyISuap ada4n0s BdRFINS 9Yy1 ST W ‘I :S310N 9400 °0 80T0°O0 | ~SS00°0 96000 | 9S200°0 | 924S00°0 GOSTO | PSPTO S€c00 0 | ¢4c00°0 | 90c00°0 vc00 0 | 8I100°0 | 77000 9SETO | 6€80°0 v6c00'0 | 84€00°0 | €7200°0 9€00°0 | L7T00°0 €¢00°0 6460°0 | 6¢ccT 0 68000°0 | 76000°0 | 28000°0 | 68000°0 | €S000°0 | 8s000°0 9080°0 | 8740°0 Va 1112 Application of Wavemaking Resistance Theory Model S-101 Model S-201 = Theoretical Prediction Model S-102 Model S-202 Dae *—~~weasured F= 7 <——Measured F = Measured F = .40 Fig. 1 - Comparison of theoretical and experimental C, values of S-series models coefficients. All the curves show a rather well-defined trend. Near the origin, the linearized theory gives quite accurate results. As the wavemaking resist- ance increases, the experimental values deviate more and more from the line- arized theoretical results. At higher Froude numbers the experimental values are Closer to the theoretical predictions. The theory always overestimates the experimental values. This is a rather familiar experience when using linear- ized theory for nonlinear problems. In view of the fact that these four models vary greatly in beam, draft and angle of entrance, Fig. 1 is rather an interesting plot from which the following remarks can be made: 1. If the wavemaking resistance theory is applied to the forebody only, where the viscosity effect is small, the theoretical prediction will be an upper limit to the possible experimental wavemaking resistance values; and 1113 Pien 2. the theory based on the linearized free-surface condition is more accu- rate when the level of the wavemaking resistance is very low (disregarding the value of beam) and when a higher order of ship surface condition approximation has been made. In such a case, the linearized free-surface condition is suffi- cient even though the ship-surface condition must have a higher than first-order approximation for practical beam values. Therefore, Inui's approach is both logical and practical even though it may seem inconsistent. The second defect in the existing theory is that the viscosity effect has been neglected. At the present time there is no reliable method of estimating this, and the existing theory cannot predict the wavemaking resistance of a given hull form accurately. However, this fact should not prevent us from using the theory to search for forms with good wavemaking resistance qualities. The statements may seem to be self-contradictory, but it is hoped to show in what follows that they are quite consistent. Because, for practical purposes, the viscosity effects can be neglected on the forebody and because the linearized free-surface condition will always over- estimate the wavemaking resistance, we can use the theory to compute the upper limit of the wavemaking resistance of a forebody alone. This is equivalent to that of an infinitely long prismatic form fitted to the after end of the forebody. Since the forebody contributes most of the wavemaking resistance, the capability of the present theory to predict the upper limit of the forebody wavemaking re- sistance immediately gives the theory a very important role in the search for hull forms with low resistance. The most frequent use made of the theory in ship design problems is to op- timize the wavemaking resistance of a whole ship without checking the forebody free-surface disturbance alone. It is conceivable that the optimum value so ob- tained might be attributable not to the fact that both the bow and stern produce very small free waves but rather to the favorable theoretical interference effect of large bow and stern free-wave systems. Due to the viscosity effect, the ex- isting theory cannot accurately predict either the amplitude or the phase of the stern free waves, so that the favorable interference effect as predicted by the theory may not always be realized in practice, thus leading to a large wavemak- ing resistance. Therefore, it is rather important to minimize the forebody free- surface disturbance. It will be shown later that by proper application of the existing theory, we can obtain hull forms with theoretical wavemaking resistance values much less than those of existing designs. Due to these low levels of wavemaking resist- ance, such theoretical predictions will be quite accurate, any remaining errors no longer being of great practical significance. In concluding this section, I feel that the present wavemaking resistance theory can and should play an im- portant role in the design of future ships. 1114 Application of Wavemaking Resistance Theory NUMERICAL COMPUTATIONS Theoretical Representation of a Hull Form — Singularity Distribution For theoretical analysis of the wavemaking resistance of a given hull form, the latter is theoretically represented by a singularity distribution. Since our aim is to obtain a hull form with low wavemaking resistance rather than to pre- dict the wavemaking resistance of a given hull form, the singularity distribution has been chosen as the starting point. After a suitable distribution has been found, the hull form is then generated from it. A distribution of singularities in space is defined by their location as well as their density. Our ultimate objective is to find an optimum singularity dis- tribution which will generate a hull form with low resistance and practical pro- portions, and at the same time is convenient for theoretical analysis. It is obvious that a central plane distribution cannot yield practical hull proportions, and it must be discarded. On the other hand, if we choose the hull surface as the location (as has been done in Ref. 3), the density is automatically fixed. In such cases, even though we can always choose a satisfactory hull ge- ometry to start with, we have no room left for improvement of the wavemaking resistance. A logical choice of the location is somewhere between the central plane and the hull surface. The gross overall ship dimensions can be effectively controlled by the loca- tion of the singularity distribution. Our procedure is to select this location first and then to determine the density distribution on the chosen location such that the wavemaking resistance will be kept low. Let ¢, 7, and ? be the nondimen- sional coordinates normalized by one-half of the ship length. The origin is lo- cated at the midship section on the undisturbed free surface. The positive di- rections of €, 7, and ¢ are in the forward, port, and upward directions respectively. Equation (1) defines an 7-surface on which our singularity distribution is placed. = Bee a Ua) aa =) bs] (1) with x= Ss Borne ORG abe) ECS) and = P(5) Hor —b 4@)exe 10 where B(¢), P,(¢), and P,(¢) define the midship section, bow profile and stern profile of the 7-surface, respectively. Parameters a, b, and n are needed to 1115 Pien obtain a large family of 7 surfaces. At present, 8(¢), P,(¢), and P,(¢) are chosen to be constant. Later, if necessary, the general case will be examined. We choose Eq. (2) as the expression for the singularity density, which is defined as the singularity strength per unit velocity of a moving ship. Meer ha wank yas diane O (2) fej These surface singularities can be either source or doublet. For the purpose of generating a bulb, a line source and line doublet located at the end of the 7-surface are also included in our scheme. Equations (3) and (4) define the line source and line doublet strength, respectively. line source S(C) = ), st) (3) j line doublet D(Z) = )- dt (4) j To obtain a flat keel line or a flat bottom, an additional surface source and doublet are placed on the horizontal bottom of the 7-surface. Theoretical Analysis of Wavemaking Resistance We first assume that our hull form, theoretically represented by Eqs. (1) through (4), has a very low level of wavemaking resistance. Under this assump- tion, the theory can be used to analyze wavemaking resistance characteristics of the forebody of a hull form quite accurately. If, at the end, the theoretical wave-resistance level of the hull form under consideration is not low, we reject such singularity distributions. We are interested in two different kinds of theoretical analysis. First we must obtain the theoretical wavemaking resistance curve as well as the free- wave amplitudes of a given singularity distribution. Second we must find the optimum singularity distribution under a set of design conditions. A computing program has been developed to perform both kinds of theoretical analysis. The general scheme and procedure for performing the double integrations for free-wave amplitudes and triple integrations for wavemaking resistance numerically have been fully discussed in Ref. 4. Computing the free-wave am- plitudes and wavemaking resistance curve of a given singularity distribution is a relatively straightforward procedure. To find the optimum singularity distri- bution under a given set of design conditions is more complicated. Our aim in such theoretical analysis is to develop a hull form with both low wavemaking resistance and a satisfactory hull geometry. It should be emphasized here that the wavemaking resistance theory is used to obtain a hull form with low wave- making resistance rather than to predict the wavemaking resistance. The wave- making resistance of a final design is obtained by model experiments. It should also be mentioned that when we write down a set of design conditions, we have 1116 Application of Wavemaking Resistance Theory to forego the usual way of specifying a number of hull proportions and hull co- efficients intended for good resistance characteristics based on past experience. Basically, the chief objective of a design is to produce a ship which is safe to operate and economical to build and run, to carry a specified displacement at a specified operating speed. The conditions imposed in any design problem should not include any hull form coefficients related to the resistance. They are not to be spelled out as design conditions, but rather are to be determined in the proc- ess of design. In our design problem, the objective is not to obtain the optimum hull form among the family covered by our theoretical representation scheme, but rather to find one hull form in this family which satisfies the design requirements and has an acceptable low level of wavemaking resistance. From a practical point of view, further reduction in wavemaking resistance has no great significance after such a level has been reached. Our theoretical representation of hull forms is rather general, and the de- sign conditions to be specified vary from one problem to another. In order to have a computing program that will cover a large variety of design conditions and perform the optimization, we split the surface source distribution in Eq. (2) into four elements. Equation (2) can be viewed as a polynomial of ¢ with coeffi- cients as functions of €. Each of the ¢ terms is considered as a singularity distribution element. These elements are denoted by £,,E,,E,, and E, re- spectively, corresponding to the zero, first, second, and third power terms of ¢ in the case of surface source distribution. Similarly, E,, E,, E,, and E, rep- resent the four elements of surface doublet distribution. The line source dis- tribution is denoted by E, and the line doublet is denoted by E,,. Altogether, we have ten independent singularity distribution elements. Consider E, as an example. It is expressed as follows: ; : 2 aS 4 “Ss E,(S) = 49S + p96 + 8395 + AgoS + A505 (5) with E,(-¢) = -E,(¢) and define: 1 Th { E,(é)x€ dé (6) 1 B, | E,(é) dé (7) 0 mel) (8) Equations (6) to (8) define three possible restraints to be imposed on the ele- ment E,. They are grossly related to the displacement volume, the midship area, and the entrance angle of the waterline. Similar restraints are defined for the rest of the nine elements. In the case of the surface doublet distribution, the first restraint is related to the ICB position of a half body and the second one is related to the displacement volume. In the case of the line source or line 1117 Pien doublet, the first restraint is related to the VCB and the second one is related to the volume of a bulb. In any specific design problem, we can choose any number of the ten elements and impose any number of the three available restraints on each of the chosen elements. The computing program performs basically one operation. The free-wave amplitude is computed from all the elements specified in the input data. Then a chosen element which is not specified in the input is determined under the spec- ified restraints so that the resultant free-wave amplitudes of this particular element and that specified in the input will yield a minimum wavemaking resist- ance at the design Froude number. If no element is specified other than those in the input, the program simply computes the wavemaking resistance and the free-wave amplitudes of the singularity distribution given in the input at the specified Froude number range and interval. To obtain a singularity distribution with a low level of wavemaking resist- ance, only two or three elements are required for a main hull form. The re- maining surface singularity distribution elements are provided mainly for the purpose of meeting the hull geometrical requirements. The computing program is very flexible. We can start either with the de- sign of main hull form alone and later consider the size and shape of the bulb, or we may first specify a bulb and then design a main hull form in conjunction with this bulb. A number of interesting theoretical analyses have been performed by using the computer program. The results are given in later sections. Hull Form Tracing From a Given Singularity Distribution A second computer program has been developed which can be used to develop a set of hull lines from a given singularity distribution. This program is an im- portant link between a theoretical model and its corresponding experimental model. The basic assumption made here is that the free surface can be replaced by a rigid plane. Inui has shown in Ref. 2 that in the low Froude number range, ~ the error resulting from this assumption is not serious so far as developing hull lines is concerned. Therefore, at the same Froude number, the less the wavemaking resistance, the closer the free surface will approach the rigid plane assumption. That means if we limit ourselves only to hull forms of low wave- making resistance, the error involved in the rigid plane assumption will be even less serious. The input data for this program specify all the singularity distribution ele- ments involved in the theoretical representation of the hull form under consid- eration. The first item the program computes is the additional bottom surface singularity distribution required for obtaining flat keel line or flat bottom. The program will trace a specified number of streamlines generated by all the sin- gularity distributions involved. The output of this program consists of a table of offsets which define a hull geometry. The details of this computation are given in Refs. 2 and 4. 1118 Application of Wavemaking Resistance Theory If the hull geometry so obtained is not satisfactory, we may either introduce additional singularity distribution elements with the necessary restraints or modify the restraints on the original singularity distribution elements. Based on the gross effects of either modifying the restraints of a particular element or introducing a new element to the singularity distribution, we may decide what modifications should be made on the restraints or which additional elements should be introduced and then make corresponding changes in the input data for the first computer program. The output will give a new singularity distribution optimized with new elements or with new restraints. The iteration between these two programs is necessary in order to obtain a good compromise between hull resistance and hull geometry. NUMERICAL EXAMPLES IN WAVEMAKING RESISTANCE In the previous sections two computing programs have been described. The first one is used to obtain an optimum singularity distribution for a ship design problem or to compute wavemaking resistance curve and free-wave amplitudes of a given singularity over a specified range of Froude numbers. The second program is used to compute the hull geometry generated by a given singularity distribution. This section gives a few numerical results obtained from these programs. The first example is intended to show that a thick ship can produce less free-surface disturbance and wavemaking resistance than a thin ship. The ques- tion of whether a ship is thin or not is a relative matter, and so is not easy to define. It may be thin enough at high Froude numbers and yet not be considered thin at low Froude numbers. Model S-101 of Ref. 2 is arbitrarily considered to be thin for Froude numbers greater than 0.30, based on the fact that the theo- retical and experimental wavemaking resistance values are then in reasonably good agreement, as shown by the comparison of the computed and measured C , curves in Fig. 2. This model is generated by a surface source distribution on a central plane having the following density expression: M(é,0) = 0.4€ (9) with -1 < € < 1, and -0.10 < ¢ < 0. The body plan is shown in Fig. 4. The L/B ratio is 13.37. We can now show that a model can be found with much smaller L/B ratio and much greater displacement-length ratio, but with less wavemaking resistance —— - —— CALCULATE (UNCORRECTED) 0.006 S -—10! MEASURED Rw Fig. 2 - The computed and measured Cc, curves of Model S-101 1119 Pien than Model S-101 at F = 0.30. At first, as in the case of Model S-101, only one singularity distribution element E, over the same distribution area as in Model S-101 is used. Only the restraint of a certain displacement volume requirement is imposed on the optimization of E, at F = 0.30. The E, so obtained is shown below: 2 3 Di(Eo6) = So SIAIE = Sealse se WS .GASS = 24.4694e° + 13.6865. (10) Figure 3 gives the plot of the corresponding density distribution. eeueae 0.8 V | l yall Ale | WY ie SLs 04 NS ve 2 Lf SORTS s oe 4 eee cae ees - 1.0 = 08 = (O15) =O. -O0.2 O 0.2 0.4 0.6 fe) cS) Fig. 3 - Surface source density distribution of Model A The body plan of the model generated by E, is shown in Fig. 4. It is denoted as Model A. It has a L/B ratio of 6.06 which is less than half that of Model S-101. Figure 5 shows the comparison of Cy curves of Models A and S-101. Up to a Froude number of 0.31, Model A actually has less wavemaking resistance than Model S-101. This result proves the point that a thick ship can have less wave- making resistance than a much thinner ship. If a singularity distribution is uniform in the draft direction, the free waves produced by layers of singularities at various depths are all in phase even though the magnitude is reduced as the depth is increased. There is no cancelling ef- fect between them. To obtain favorable interference, the density distribution should vary with depth. To demonstrate this idea, a new singularity distribution element, say E,, is added to the singularity distribution of Model A. Let us 1120 -16 .20 24 Application of Wavemaking Resistance Theory + HHA ~12 20 A -08 SESE! SPs = Vy Wp FA i alia) ws XS SS ss ! Ht abe Vis 24 MODEL A —————— — _— COO R Fig. 5 - Comparison of theoretical C, curves of Models A and S-101 1121 Pien assume that the only restriction put on E, is that the displacement volume of Model A should be increased by one-third. The optimum E, so obtained is shown below. 2 3 E, = [15.977 - 137.8429 + 441.3259 - 575.0504 + 259.4315€5] 0° (11) This is derived in such a manner that the wavemaking resistance due to the combined singularity distributions of E, and E, is an optimum at F = 0.3. Let us denote the model, generated by E, and E,, as Model B. Figure 6 shows the comparison between the C, curves of Models A and B. Despite the fact that Model B has one-third more displacement volume than Model A, it has less wavemaking resistance at F = 0.3. To illustrate the importance of section shape upon the wavemaking resist- ance, let us consider a third case, Model C, which has the following singularity distribution: 2 2 MQRC SEG Es/3e (12) where E, and E, are defined in Eqs. (10) and (11), respectively. It is obvious that to the first order of approximation Models B and C have the same SeCtional area curve. Figure 7 shows the comparison of the C, curves of Models Band C. The differences between these curves are quite large. This Sei a eee AEE EE ae nese: 0020 Model A = = _hodel B apie (eaie SCE mn oe eee weal Se .OOI6 0012 .0004 eer = [cla ieawiees cre earl ef 20 2 a A Fig. 6 - Comparison of C, curves of Models A and B 1122 Application of Wavemaking Resistance Theory MODEL B ———— MODEE GS ~ ~ ~ Fal Fig. 7 - Comparison of theoretical C, curves of Models B and C figure also indicates that a good vertical displacement-volume distribution at low Froude numbers may not necessarily be good at higher Froude numbers. In Fig. 8, C, curves are given for three cases with M(¢,¢) = ¢, 4/3 Bas and 7/3€°, respectively. To a first approximation, all three cases have the same displacement volume. At low Froude numbers, the differences between the three curves are amazingly large, mainly due to changes in angle of entrance. It is also interesting to note that in the case of M(é,¢) = 7/3é°, the last hump is much less pronounced than the preceding ones. REDUCTION OF VISCOUS DRAG The viscous drag constitutes a major portion of the total resistance of a ship. A great potential, therefore, exists for reducing total resistance by de- creasing the viscous drag, which is mainly a function of wetted surface and Reynolds number. However, if not designed properly, the hull form can produce large eddies, resulting in a large form drag. Therefore, to reduce the viscous drag, we have to reduce both the wetted surface and the form factor. We know how to shape a hull to keep down viscous drag for a deeply sub- merged body, but such information cannot be directly applied to designing a ship hull subject to free-surface effects. In ship design, the principal dimensions 1123 Pien ake Sa ea ee a a Ws Pree Shee eee eee ES AONST i H q H \ ‘ } i ' \ * | \ 1 \ I } 1 T ' \ i ' ] i eal ae ' \ aaa i \ Css. XV \ Jee ! aaa eo alte Se Ay LY B2a205s De a No Mee ne mi 004|4 cee : ~~ MEE) = “73 : Poo of E Ce WE ali ty 4 002 at eu te Icy Am €ér€ | bs 2 A il V ii aoe ig = i ! W a ooo | A eG aia lala ri 10 20 30 40 50 60 big. 30) J.) Pheoreticalic curves) of WM(Ese y= hy WE 5 W/Z 7 EOS pSChivEly are chosen to give a proper balance between the viscous and wave drag, rather than for optimum viscous drag. Knowing how to keep the wave drag at a low level, as previously described, we can select principal dimensions without the danger of increasing wavemaking resistance materially. This fact immediately opens the way to reducing the wetted surface. As an example, let us consider Model 4210 of Series 60. It was designed for V//L = 0.9 and has the following characteristics: L/B = 7.5 B/H = 2.5 A/(L/100)* = 122. Assuming the length to be reduced by 20 percent, and the displacement and ship speed kept the same, V///L becomes about unity and A/(L/100)* becomes 190.6. The reduction in wetted surface is of the order of 16 percent. For this model, such a change in length will greatly penalize the performance, the increase of wave resistance being much greater than the gain in viscous drag. If, based on theory, we can design a hull form of these proportions and displacement, with 1124 Application of Wavemaking Resistance Theory very low wave drag at V/VL = 1.0, such a gain in viscous drag can be realized without the penalty of increased wave drag. This idea of reducing the wetted surface through the application of wave- making resistance theory is quite useful. However, this is not the only way the theory can be used to reduce viscous drag — the form drag can also be reduced, as described below. Ever since the comparison of the results of Models 4946 and 4953 (repro- duced here as Fig. 9) were published (Ref. 4), some uneasiness has been felt. The body plans are given in Fig. 10. Model 4953 has 28 percent less displace- ment and 7 percent less wetted surface than Model 4946, and yet has greater total resistance at the lower Froude numbers. It was thought that this was mainly due to the wavemaking resistance, but considering the fact that the dif- ference between these two models is confined to areas much below the free sur- face, it should not produce a big difference in wavemaking resistance, especially at lower Froude numbers. One possible explanation for the larger total resistance of Model 4953 is a greater form drag. Due to the flat bottom, large eddies may be created in the water which flows over the bilge to reach the flat bottom. Such eddies are not likely to be created in the case of Model 4946 because of its rounded bottom. However, the turn of the bilge in the case of Model 4953 is not particularly hard. If large eddies do exist under the flat bottom of this model, it is likely that a majority of flat bottom models have the same drawback. In searching for evidence of eddies beneath a flat bottom model, the wake survey results behind a smaller version of Model 4210, reported by Wu (Ref. 5), have been studied with great interest. Some of the figures of Ref. 5 have been reproduced here as Fig. 11. This model has a draft of only 0.53 ft and yet the wake is still quite strong at a depth of 0.7 ft. This cannot happen without the presence of large eddies underneath the flat bottom. It is quite possible that although Model 4953 has less displacement volume and wetted surface than Model 4946, it may have a stronger wake belt trailing behind it. It would be very desirable to conduct wake surveys behind these mod- els, but these are quite tedious and expensive, and it was thought that a flow visualization test might give a general picture of the flow near the bilge and bottom. Such tests have been conducted on both Models 4953 and 4946 in the circulating water channel. Figures 12 and 13 show the corresponding pictures of these two models. Ink was introduced at nearly the same longitudinal stations. All the photographs were taken at a speed of 3 knots. It is quite clear from these pictures that large eddies do exist in Model 4953. These may account for a large portion of the in- creased total resistance. The bottom picture in Fig. 13 shows the ink flow near the stern of Model 4946. The ink was introduced at the bow, and there was no noticeable change in the thickness of ink marks as viewed from the other side of the model. If there is strong eddying, the diffusion of ink is very great, as ob- served in a similar test on Model 4953. (Unfortunately, the corresponding pic- ture was not successful.) 1125 221-249 O - 66 - 73 Pien fi a Goel ios a iedelear P| ec, srl it mstapnce Dn ba ee LY 0.8 0.7 0.5 0.4 0.2 FOR MODELS 4946 & 4953 : | Pl ws 67 625 62 63 BULENCE INDUCED BY TRIP WIRE ail) Aa SIT aS al) ORO) ER ee IS nt} As) cs os sl) ote FROUDE NUMBER 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 565 MODEL SPEED IN KNOTS Fig. 9 - R, and C, comparisons for Models 4946 and 4953 1126 RESISTANCE IN POUNDS Application of Wavemaking Resistance Theory MODEL 4946 —————_-—— 4953 — — — —— Fig. 10 - Body plans of Models 4946 and 4953 It is obvious from the foregoing that if water is prevented from flowing across the bilge, as in the case of Model 4946, formation of eddies can be avoided. However, the round bottom of Model 4946 is not practical, and we have to search for other means. A sizable bulb can be used for such a purpose. Starting from the stagnation point, the water can be guided in all directions by a bulb, so that it is properly channeled toward the flat bottom from the very begin- ning rather than spilling over the bilge to reach the bottom at a later stage. A bulb can be used in this way to prevent the formation of eddies and thus reduce the form drag. However, the total resistance will not necessarily be re- duced. If not properly matched to the main hull form, a bulb will produce a large wave drag, and any gain in form drag may well be exceeded by the penalty of wave-drag increment. Again the wavemaking resistance theory can be used to great advantage in this situation. To start with, we may choose a proper sized bulb and place it at a correct location based on the consideration of 1127 Pien y (tl HERES Sea eee Ze Curves of Hp. — Fh in fe, at s = 8 ft y(t) Curves of H, — hilaft,azs = 4h Curves of H, — My in ft, at x = 2 fe 1T various distances behind Model 4210 in ft at = lel - Curves of H, Fig. 1128 Application of Wavemaking Resistance Theory Model 4953 Model 4946 Fig. 12 - Pictures of flow on the forward end of Models 4946 and 4953 1129 Pien MODEL 4953 MODEL 4946 —_ oe — Fig. 13 - Pictures of flow on the after end of Models 4946 and 4953 reducing form drag. Then the matching main hull is designed by using the wave- making resistance theory to ensure that the combination of the two will produce very low wavemaking resistance. Only in this manner is a bulb effective in re- ducing the total resistance of a ship by reducing the form drag. This may ex- plain the reason why placing a large bulb on tanker models can result in a great reduction in total resistance at relatively low Froude numbers. It is believed that the application of wavemaking resistance theory may have more practical value in the reduction of viscous drag than in the reduction of wavemaking resistance itself. From this point of view, the wavemaking resist- ance theory can always be used to reduce the total resistance of a ship regard- less of its design speed. i 1130 Application of Wavemaking Resistance Theory SHIP DESIGN AND MODEL EXPERIMENT Having finished the ground work in the first part, we now proceed to the second part of this research project. A number of ships will be designed and their models will be built and tested for resistance as well as self-propulsion. In each design problem, two models will be designed and built. The first one will have a simple stern profile. It will be tested for resistance only. The the- oretical wavemaking resistance curve will be computed for comparison with the experimental curve. The second model will be obtained from the first one by modifying the afterbody for the purpose of self-propulsion tests in such a way as to obtain better propulsive characteristics. However, the original afterbody sectional area curve will be kept intact as much as possible. One ship design has been started already, and the first model is now under construction. Perhaps it is worthwhile to discuss some of the thoughts incorpo- rated in the design of this model. The design conditions are very broad. It is required to develop a fast cargo ship with a displacement of 21,500 tons and a designed speed of 24 knots. A ship length of 550 ft will give a V/VL value of 0.98 anda A/(L/100)* value of 129. If normal practice is followed, a ship length of more than 550 ft would be chosen. Based on the idea of reducing viscous drag, we limit the ship length to 500 ft. This will increase the designed speed-length ratio from 0.98 to 1.07 and the displacement-length ratio from 129 to 172. A bulb of moderate size has been adopted for the purpose of reducing eddy- ing underneath the flat bottom. This bulb is placed above the base line so that the keel line is bent upward toward the bow. In doing so it is hoped that the favorable flow condition on the bottom of Model 4946 will also exist on this de- sign. We have thus shaped the bow first entirely from the consideration of re- ducing eddying. Then the main hull is optimized in conjunction with the chosen bulb such that the forebody free-surface disturbance is very small. To start with, only E, shown below is used to generate the bulb. Bae s0l 056? Esse (13) with -0.8< €< 0. E,, has not been included here in order to avoid excessive narrowing between the bulb and the main hull. The next item to be considered is the 7-surface. It has been found that sat- isfactory results can be obtained by approximating the 7-surface waterline to the sectional area curve of a Standard Series Model with about the same pris- matic coefficient as the model under consideration. The width and the depth of 7-surface as well as the singularity distribution placed on it determine the L/B and B/H ratios of the design. To start with, the width and depth of 7-surface are estimated. Satisfactory solution is obtained by trial and error. From the eight available singularity distribution elements, we arbitrarily chose E, and E, for the main hull. The only restraint imposed upon the opti- mization is the required displacement volume. However, if the midship section 1131 Pien area so obtained is too big, we can add one more restraint on the design condi- tion so that a desirable prismatic coefficient can be obtained. From the experience obtained from Model 4946 and many models tested by Inui at the University of Tokyo, we can anticipate a phase shift between theoreti- cal and measured wavemaking resistance curves. The experimental curve is always shifted to the right of the theoretical curve. Therefore, in this design the optimization is done at F = 0.28 rather than 0.32. At this point the computer programs are used to carry out the lengthy, tedi- ous numerical computations. After a few trials and errors, we obtain the fol- lowing singularity distribution for our final design. q@= &(oOlse? + Slee 2 ,1Giige4 = cil]. On the side of 7-surface we have 3, = (O55 = 29 ORES 6 67 “ORO? = 74 720SEY + 20. 26922?) ica} ul [MOR99342 = 6l.3783e2 4) 158. 84566" — 183. 88592" 4+ 75.0307 eer or MG) Eee On the end of 7-surface we have E, = .008 + .04¢7 + .04¢%. On the bottom of 7-surface we have surface source M,(€,7) = [-.1882¢ - 2.3841€? + 15.9590€° - 28.8343¢" + 13.6760€°] + [2.2616 - 5.37697 - 625.5772€2 + 163.9480é* - 82.5272€°17 + [-13.3743€ + 107.5863€? - 208.1482¢3 + 151.7266¢* - 88.7849€°] 7)? + [12.2606 - 121.1156? + 338.7764€3 - 419.1692¢4 + 232.28014°]7° surface doublet M4(€.7) = (.1131€ - .4594€? + 1.5838é3 - 2.7567E* + 1.6938€°] i (ED. SEODS & FOCANEZ LA IGCNES 4 SOTTE® = STEN + [-.3740€ + 2.4491€? - 8.803662 + 12.8106€* - 5.2930€°] 77? Pule4IS7e =) 32469262 4 15309222 =) 20ee145405— AR0316e>) ae 1132 Application of Wavemaking Resistance Theory with -0.08 < ¢ < 0 and 0 < € < 1, where 7 is the nondimensional distance from the central plane as normalized by the local offsets of 7-surface. The afterbody has been chosen as the mirror image of the forebody. This model is denoted as Model 4996. .0024 Rw .0OI16 | Fig. 14 - Theoretical C, curve of Model 4996 Ba 35 The theoretical C, curve for this design is given in Fig. 14 and the body plan and waterline endings are shown in Fig. 15. Due to certain limitations in the second computer program, we cannot obtain a true flat bottom. Some hand fairing is necessary at the present time. However, such fairing is limited to the bottom portion of a model only. In the case of Model 4996 as shown in Fig. 15, such fairing is done below the 0.3' W.L. An effort is being made to eliminate this shortcoming in the computer program. Model 4996 has a B/H ratio of 2.64, a L/B ratio of 5.82, A/(L/100)° ratio of 172.2. These proportions are very desirable, especially the large displacement- length ratio which has a large influence on the per ton construction cost. The experimental result of Model 4996 is anxiously awaited. 1133 Pien HE HH 1,650'WL fe cc 1.475'WL hd a a el 1,300'DWL th 200'WL 1.100'WL eel 1.000'WL .900'WL .800'WL 700'WL .600'WL .500'WL .400'WL .300'WL .200'WL . }00'WL BASE LINE | | TOP OF MODEL 5o99'we 1825'WL 1650'WL 1475 WL 1000'WL 900'WL 800'WL 700' WL 600'WL 500 WL 400' WL 300'w 200'WL 100'WL = BASE STA3 SsTA2 STAI STA.6 STA3 ) LINE (b) Fig. 15 - Body plan and waterline endings of Model 4996 1134 Application of Wavemaking Resistance Theory CONCLUDING REMARKS While fully aware of the limitations of the existing theory, we believe that useful results have been achieved without exceeding these limitations. By re- stricting the analysis to forms of absolute low-wavemaking we have not unduly violated the linearization at the free surface, and by recognizing the relative importance of forebody wavemaking we have avoided some of the problems of viscosity. The idea of reducing the viscous drag of a ship through the application of the wavemaking resistance theory is rather interesting. It may have an impor- tant influence in the design of future ships. However, even though the arguments used in this paper to support our views and ideas are quite logical and plausible, the final proof of the validity of these arguments rests on model experiments to be carried out in the very near future. ACKNOWLEDGMENT The author wishes to thank Mr. Louis F. Mueller of the Applied Mathemat- ics Laboratory for his help in the computer work. NOTATION a,b,n Three parameters defining 7-surface a. General coefficient in Eq. (2) B Ship beam B, Defined by Eq. (7) C Total resistance coefficient C, | Wavemaking resistance coefficient d. General coefficient in Eq. (4) D(C) Strength of a line doublet E,,E,,E,,E, Surface source distribution elements E,,E,,E,,E, Surface doublet distribution elements Ds Line source distribution element Eno Line doublet distribution element F Froude number 1135 Pien g Gravitational constant H Ship draft H Total head referring to the undisturbed condition H, Total head referring to the behind condition K Form factor used with Hughes friction line L Ship length M(é,¢) Density of surface singularity distribution M Surface source density at é = 1 R Total resistance R, |Wavemaking resistance s. General coefficient in Eq. (3) S(¢) Strength of a line source T Depth of singularity distribution , Defined by Eq. (8) Vv Ship speed , Defined by Eq. (6) —€,n,6 Coordinates A Displacement. REFERENCES Emerson, A., ''The Application of Wave Resistance Calculations to Ship Hull Design,"' INA (1954) pp. 268-275 Inui, Takao, ''Study on Wavemaking Resistance of Ships,'' 60th Anniversary Series of the Society of Naval Architects of Japan, Vol. 2, pp. 173-355 Breslin, J. P. and Eng, K., ''Calculation of the Wave Resistance of a Ship Represented by Sources Distributed over the Hull Surface,'' Davidson Lab- oratory Report No. 972 (July 1963) 1136 Application of Wavemaking Resistance Theory 4. Pien, P. C. and Moore, W. L., "Theoretical and Experimental Study of Wavemaking Resistance of Ships,"' Internation Seminar on Theoretical Wave Resistance, University of Michigan, Ann Arbor, Michigan (August 1963) 5. Wu, J., "The Separation of Viscous from Wavemaking Drag of Ship Forms,"’ Journal of Ship Research, Vol. 6, No. 1 (June 1962) DISCUSSION G. P. Weinblum Institut fiir Schiffbau University of Hamburg Hamburg, Germany Leaving aside basic theoretical considerations in the field of wave resist- ance, we consider Dr. Pien's recent proposal a valuable contribution following which bodies are generated in a uniform flow by distributing singularities over a suitably chosen skeleton surface instead of over the central plane. By these "Pienoids'' a serious difficulty has been mitigated when investigating hull forms of least or low wave resistance; the recent trend to study flow conditions by de- termining singularities over a prescribed body surface makes an optimisation of the latter obviously impossible. In the present paper an attempt has been made to apply theory to the solu- tion of a rather general engineering problem, the determination of hull forms of low total resistance (instead of low wave resistance, etc.). The exposition of this important task is in my opinion slightly impaired by some global and dep- recating statements made by the author. Some aspects of the problem have been clearly described by D. W. Taylor in his "Speed and Power of Ships"; cf., his famous sketch representing the total rest and frictional resistance R,, R_ and R, of a given dimensionless form and ¥ = const. as function of the length. The essential difficulty consists in finding the wave and viscous drag components leading to an optimum. It is typical and unavoidable that one has to face the viscous resistance problem when dealing with the wave resistance. The author asserts that we know how to shape a deeply submerged body of low viscous drag. This is cor- rect as long only as a qualitative reasoning is concerned. Reference is made to the pertaining formulas Ci = (1+ n)C,, with a =) DoD BVAL Ses 5e 1137 Pien for a cylinder; no=.0 76. D/IN2en. for a body of revolution; nel (CRB/N))- Granville's formula for shiplike bodies. The primitive character of these relations indicates that quite a bit of re- search work should be done before the author's optimistic statement can be ac- cepted, e.g., with regard to dependency of the drag of full forms upon propor- tions. Contrary to his optimism, Dr. Guilloton has recently expressed the opinion (Bull. Ass. Technique Maritime, 1964) that our knowledge of viscous drag as function of the hull form is almost nil. The difference in the total resistance R, of Model 4946 and Model 4953 can be explained by viscous as well as by wave effects. The former are estimated by ah at low si (as pointed out by the author), the latter by the intersection of resistance curves at F = 0.30. The difference in the prismatic coefficients is helpful for such a phenomenological discussion. The author emphasizes as a new result that the wave drag of a fat ship can be smaller than for a thin ship. In the light of Taylor's findings (and those de- duced from theory) this may be trivial in a range where R, depends strongly upon the prismatic coefficient. Examples based on theoretical calculations have been frequently given; some caution, however, in the quantitative applica- tion is advisable. The shift of the measured wave drag curve to higher F as compared with theory has been firmly established by Wigley and Havelock. The author asserts that in the field of comparison between theory and facts almost only the work done by Prof. Inui counts. Although I am an admirer of the valuable contributions made by our distinguished chairman the author's state- ment is in my opinion erroneous; the most valuable experiments are those by Mr. Wigley (TINA 1924) and the TMB Report where the so-called friction plate furnished by mistake the ideal thin ship model. It is erroneous to assume that wave resistance results by computation are always larger than those derived by experiment; this certainly does not apply to hull forms which by theory are extremely advantageous (due to strong interfer- ence effects which may be destroyed by viscosity). The hull form proposed by the author appears to be promising for medium Froude numbers es = d= 0.58, moderate bulb, gentle turns of bilge). The raised bulb, however, may be unfavorable in a seaway, especially under ballast condi- tions. The attempt of applying theoretical reasoning to actual design problems is highly appreciated. 1138 Application of Wavemaking Resistance Theory DISCUSSION K. Eggers Institut ftir Schiffbau University of Hamburg Hamburg, Germany I have to make a general remark concerned with the method by which Dr. Pien and other colleagues find hull forms for which certain singularity distribu- tions are considered representative for calculation of wave resistance. We know that by the Hess-Smith procedure we can determine source distri- butions on surface of these hull forms, and that wave resistance for such distri- butions then can be calculated along the lines developed in the paper of Breslin and Eng. I declare that there is definitely no convincing argument for the assumption that resistance calculations for these alternative singularity distributions should, precise numerical methods assumed, lead to identical values. Furthermore, we can create systems of arbitrary high wave resistance, which still generate the same flow around the’double body under infinite gravity, just by proper linear combination of both kinds of distributions! Which wave resistance then is to be considered the 'correct'’ one, assuming now the form to be given? We could select the lower limit from the class of all distributions representing the form under infinite gravity and constant speed at infinity. But probably this value is not attained by a single distribution over the whole range of Froude numbers. It is easily shown that for any form of nonzero volume there must exist more than one distribution to represent it in infinite fluid. We can, for instance, at any interior point add a source layer of constant strength on a surrounding sphere, compensated by a corresponding sink layer on an exterior concentric sphere such that there is no resulting flow outside. In case of a submerged body this will not change the wave resistance. In case of a floating body, however, only the part of the additional system below the undisturbed free surface will contribute within linear theory. The flow due to this lower part only will in general not vanish outside and will thus induce additional waves. Take the case of a semi-submerged spheroid. This can be represented by volumetric dipole distributions in any confocal spheroid, equivalent to source layers on the surfaces. As a limiting case we get a line dipole distribution be- tween the foci. This latter gives the largest, i.e., infinite resistance. For a singularity distribution found by analytical methods to be optimal within a certain class, we may determine some associated body form by tracing 1139 Pien stream lines. But if the body is piercing the undisturbed free surface, why should just this distribution be selected for calculation of wave resistance ? Intuitively, I would prefer the combined source-dipole layer on the surface used in Green's theorem, as this has minimal, i.e., zero-inner kinetic energy. In any case we have to formulate proper restrictions for the flow within a ship's waterplane area to keep variation of resistance calculated in reasonable limits. DISCUSSION J. N. Newman David Taylor Model Basin Washington, D.C. There has been considerable discussion this afternoon concerning the rela- tive importance of nonlinearities in the free surface condition, and now Dr. Pien has advanced the suggestion that the linear free surface condition is valid for "fat" ships if they are ships of low wave resistance, or that fat ships of mini- mum wave resistance are equivalent to thin ships, as far as the free surface condition is concerned. This may in fact be a valid analogy from the engineer- ing standpoint, but I hope that it will not be confused with a rigorous mathemati- cal development. A necessary condition for the linearized free surface assumption is that the elevation of the free surface is everywhere small, compared to the wave length of a characteristic wave. This is clearly true for a thin ship since (in an ideal fluid) the fluid disturbance and free surface elevation can be made arbitrarily small by making the ship sufficiently thin. The free waves or far-field disturb- ance associated with a ship of minimum resistance will also be small because wave resistance implies wave energy radiation, but the free surface disturbance near the ship will not necessarily be small since this is a local disturbance and is essentially independent of the wave resistance of the ship. An obvious exam- ple is the waveless (infinite draft) ship discussed by Dr. Yim; for this case there will be no free waves and the linearized free surface condition will clearly be justified in the far-field, but close to the ship there will be a local disturb- ance which can be made arbitrarily large simply by increasing the singularity strength. In other words, a ship of low wave resistance will satisfy the linear free surface condition over most of the free surface, but not necessarily close to the ship. Please let me emphasize that my objection is based upon the rationality of the theory, and not upon practical considerations. For practical purposes I 1140 Application of Wavemaking Resistance Theory would encourage the use of the linear theory, as long as it gives satisfactory re- sults. Intuitively I would agree with Dr. Pien that a ship of small wave resist- ance will probably have less associated nonlinear effects from the free surface than another ship of the same principal dimensions but larger wave resistance. DISCUSSION Lawrence W. Ward Webb Institute of Naval Architecture Glen Cove, Long Island, New York Dr. Pien has presented a very stimulating paper and one which I feel is es- sentially correct, but I would like to take this opportunity to discuss two points which I feel are of importance in connection with this work and with some of the other papers given this afternoon as well. The first point is that of the question of the definition of wave resistance which is essentially that of the definition of wave resistance in real fluid since in the case of an ideal fluid all definitions seem to lead to the same result. There are a number of definitions possible, depending on the use to which the definition is to be put; and I would like to review this matter with you at the risk of boring those who were at Ann Arbor with the help of a table similar to one shown at that time (Fig. 1). Since the theory of wave resistance in a real fluid ie 1 DIMENSIONALLY es VECTORIALLY ls, Brevomewox aGicaLly METHOD OF 2) FROUDE 4) MODERN FROUDE 66eRr) BREAKING DOWN A YPOTHES/S AY POTHESIS (Yeues) Vyerous G- SON FRICTION C044) TANGENT/AL Vscous-G TERMS AO € ARM SYMBOLS“ USED ese) eae ee C, PRESSURE - GC u4ve -G, Wave (Form Resistance) EXPERIMENTS F@ICTION GFOSIM PRESSURE SURVEY SURVEYS Ar CONTROL NEEDED 76 PLANKS TESTS Ay, rats LVALUATE Mute SuRrACE || 7ROM AMAL. J ke * TE VARIOUS KESISTANCES ARE EXHIBITED (hd COEFFICIENT FORM, C ~ SH, WHERE K 45 THe RESISTANCE W/ POUNDS, p THE DEnsiTY, S THe WerréD StrFace 1 FEETF And \, Wie Speco I FeeT PER SECOWO. Fig. 1 - Various breakdowns of ship resistance into components 221-249 O - 66 - 74 1141 Pien has not been developed to any useful extent, I have included only those definitions which can be related to experiments in some way. In Fig. 1 various breakdowns of ship resistance into components are shown in historical order from left to right. First we have (1-a) Froude's hypothesis and (1-b) the modernization thereof by Hughes. Here the goal is mainly that of model scaling, and the breakdown into frictional and residual components is done on the basis of dimensional analysis, that is, Buckingham's Pi Theorem. Froude's original hypothesis was that the total resistance could be separated into a "residual" part, C_, depending on the Froude number and a frictional part depending on the Reynolds number. In practice, the latter was estimated as being the skin friction, C,, of a plank of the same length and wetted area. This results in the residual resistance including some viscous effects due to separa- tion. These are sometimes termed "form" and ''eddy'" resistance. Hughes added the concept of a form effect (1+ 1) on C, derived from tests of geometrically similar models (Geosim tests), this being a practical improvement only if such factors do not depend strongly on the Froude number and can be estimated with- out recourse to such tests. By assuming no Froude number dependence, the form resistance can also be estimated on the basis of the resistance at low Froude numbers where the wave resistance is expected to be negligible. The corrected residual resistance, C’, includes the wave resistance but also an un- defined portion of the eddy and form resistance, probably that part which is Froude number dependent. The second listed breakdown is with respect to the vectorial nature of the local fluid stresses at the hull boundary, i.e., tangential shear and normal pres- sure. The latter are determined from a pressure survey over the entire hull and then are integrated in conjunction with the known hull surface slopes to give a resultant pressure drag component, Core This can then be subtracted from the measured total drag to deduce the integration of tangential viscous shear stresses. It should be pointed out that the major effects of viscous separation are not included in this force component but in the normal pressure drag com- ponent. The required experiments and analysis, originally done by Eggert and more recently by Townsin and Hogben are quire extensive. While historically interesting, this has not yet proved to be a practical means of meeting any im- portant goal. The third breakdown is with respect to the physical phenomena involved, i.e., the formation of waves and the development of a viscous shear wake. Here the ‘question of breakdown reduces to that of separating the total momentum survey around a closed control volume away from the hull into (a) that portion involving the viscous wake and (b) that due to wave orbital velocities, and then integrating these to obtain C, and C,, respectively. The experimental tech- niques available to measure the wave resistance, C,, are therefore either (a) a direct momentum survey of the waves making a proper correction in the wake region or (b) a valid viscous wake survey adjusted for the presence of waves which is then subtracted from the measured total resistance. The latter method, which has been employed for example by Landweber, is less direct and might suffer from inaccuracies due to the process of taking differences of large num- bers. In addition, one must assume that there is no third mechanism of energy dissipation present, which has-not been proven yet. It is the third breakdown of 1142 Application of Wavemaking Resistance Theory total resistance that I tend to favor. The wave resistance is in this way defined in terms of the energy actually going into the wave system in the real fluid (not for example what might be the energy going into the wave system of the same ship in an ideal fluid). It is evident that no direct relationship between the wave resistance, C, so defined, and the pressure resistance, C ay OF the residual resistance, Gu (or Ga); need necessarily exist, and this is the point I wish to make. Recognition of this can eliminate pointless arguments as to which method of measuring wave resistance is correct. My second point deals with the various statements by Dr. Pien on pages 1110: 1116 and the results given in Table 1 and Fig. 1 of Dr. Pien's paper. The state- ments which I refer to and which seem to be backed up by comparison of theory and experiment are those which infer that the theory approaches the experimen- tal values in some monotonic way (a) as the Froude number gets larger and (b) as the wave resistance gets smaller and furthermore that Michell's prediction forms an upper bound to the experimental wave resistance. I find it hard to be- lieve that the situation in general is that simple and would like to point out some evidence to the contrary. The first involves experimental results I obtained in the Webb tank from direct measurement of the wave pattern using the 'XY"' method of analysis. The first (Fig. 2) shows this result for the ATTC Standard Model which is also the parabolic form Wigley tested and reported in 1926-7 in the INA. It can be seen that there is in fact a region where Michell's estimate is less than the wave survey result, and it would also be less than the residual resistance with a suitable Hughes form factor. The second result (Fig. 3) is a series of tests using the same method on the Series 60, 0.60 block model car- ried out to very high Froude numbers and while there is some question of the circular cylinder used in the method being large enough at the high Froude num- ber end of the curve, it is obvious that a very major adjustment in the data would be required to bring theory and experiment together in this range. © [a C4 dER 12" OFF y 0/4" CrAmoeR 6” Ore £ soe) LS heal LK v | I 1 deseo | | SC. Never ¢ ee X¥ MeTHab (Warn, | | 20 - 50 40 Fig. 2 - Experimental wave resistance of the ATTC standard model (L= 5 ft 4 in.) 1143 Pien M/ i! (XY Meron) (Gu | (Weaszee, U peala i) ; | Yee Fig. 3 - Experimental wave resistance of the 5 ft 0 in. Series, 60 Model (0.60 Block) Finally, I should like to say in reference to the suggested improvement in agreement of theory and experiment as either value becomes small, Wigley him- self as most of us know tested two models of the parabolic form of 3/4 and 1/2 the beam of the original, which was already quite a thin ship, with no such im- provement in agreement between the experimental values obtained and the Michell's calculation which, of course, remained constant (on a coefficient basis). I do not mean to imply that Dr. Pien's results are not correct but that they should not be looked at as being general. DISCUSSION A. Silverleaf National Physical Laboratory Teddington, England Dr. Pien's work in applying wavemaking resistance theory to ship design is held in the highest regard at N.P.L., and this latest progress report contains 1144 Application of Wavemaking Resistance Theory many fruitful ideas. A general programme of research into low resistance hull forms is now being undertaken at N.P.L.; this includes experiments to examine Pien's suggestion that the double model approximation should give closer agree- ment between calculated and measured resistances for source distributions hav- ing low wave resistance than for resistful forms. Two bodies are being designed from wave source theory, each consisting of a bow shape followed by a long par- allel afterbody, and an attempt will be made to measure their head resistances alone. One of these forms does not have a particularly low calculated wave re- sistance, but the second has been optimised following the same general princi- ples as those adopted by Pien. If good agreement between theory and experi- ment is obtained for this second form, we believe that it will aid significantly in using wave source theory as a practical design tool in the way indicated in this paper. The assessment of the results of any such study of calm water resistance effects depends on the establishment of a recognised yardstick or resistance criterion. In assessing wave resistance alone, this criterion should preferably involve only the displacement, speed and length. Displacement and speed are the primary specified operational requirements, and length may be regarded as a primary limiting parameter. The hydrodynamic criterion of quality should be based on the maximum immersed length, thus imposing a penalty on a device, such as a projecting bulbous bow, which reduces resistance at the expense of increased underwater length. Not all comparisons have been made on this basis and it is more than possible that this has influenced the conclusions drawn from them. DISCUSSION S. W. W. Shor Bureau of Ships Washington, D.C. In commenting on Dr. Pien's paper I first wish to congratulate him on the persistence with which he has pursued his search for a practical solution to the problem of reducing the total resistance of a ship's hull. The fact that this search seems verging on a successful result with even more general applicabil- ity than we had dared hope is most gratifying. As to the details of his paper, I wish to invite attention particularly to two of his statements which are corroborated by my own work. First, Dr. Pien is quite correct that the best approach within the confines of existing ship wave theory is to optimize the shape of the forebody of the ship, and then to design the stern separately. This means that he does not count on using the stern waves to cancel the bow waves, but instead sees to it that the 1145 Pien forebody does not generate bow waves. This approach is suggested by Inui's work. We may recall that Inui found that bow wave amplitudes are close to those predicted by theory, particularly if they are small, but that stern wave amplitudes are significantly smaller than those predicted. The ratio of observed to predicted stern wave amplitude is Inui's parameter £, and it becomes quite small at low Froude numbers. At the Froude numbers around 0.3, for which Dr. Pien has designed his models, the value of £ for Inui's Model S-201 is under 0.7, and for S-202 is even smaller, just above 0.5. This means that the stern waves, which by theory for these double-ended hulls should be as big as the bow waves, are in reality only a little over half as big and so cannot do much to can- cel the bow waves. Worse, they are generated in the frictional wake which moves in the same direction as the ship, and so their transverse components must have a shorter wave length than the transverse components of the bow wave if the stern wave pattern is to move with the ship. Waves must both move in the same direction and have the same wave length if they are to cancel. It is therefore evident that in practice we cannot expect cancellation of the trans- verse portions of the bow wave by the transverse portions of the stern wave even to the extent suggested by the existence of non-zero values of 6. Itis possible to show, also, that transverse components of the bow wave should not penetrate into the wake at all, but should be reflected from its boundary so that cancellation becomes impossible. This, of course, can also be deduced from Inui's experimental observation of the wave-shadow effect. Because of this we should not expect complete cancellation of bow waves by stern waves even at Froude numbers much higher than 0.3 where the value of 6 approaches unity much more closely. Actually, it is possible not only to provide an explanation for Inui's semi- empirical parameter 6 from the fact that viscosity causes water to be dragged along with the ship, but an estimate of how much this reduces the velocity of water relative to the stern. As a result of viscosity the stern waves are gener- ated by water moving at a velocity relative to the hull which is somewhat less than that of the water which generates the bow waves. How much less can be deduced by working backwards from Inui's results. This has been done in Fig. 1, where the ratio c_/c is that ratio of relative speed of ship and water at the stern to the forward speed of the ship which is required to fit Inui's curves of 6 vs Froude number. As shown in Fig. 1, the ratio does not change much over a wide range of speeds. Although the bow, as Pien points out, should be optimized at a speed close to the speed of the ship (he optimized at F = 0.28 for a ship with actual speed F = 0.32), it follows from the data shown in Fig. 1 that the stern should be optimized for a much lower speed. For example, referring to the figure, if we were to optimize the stern of hull S-202 to operate at a ship speed of F = 0.32, we should optimize the stern at a Froude number (c,/c) (0.32) = (0.66) (0.32) = 0.21. A second point which Dr. Pien makes is that the conventional hull form pa- rameters must be disregarded when hull forms are optimimized. Certainly I find this to be true. I have just finished a calculation using the method of steep descent to decrease the wave resistance of a destroyer type ship intended to operate at 30 knots, and in the calculation I held constant the sectional area curve as well as the load waterline, the sound dome, and the draft. The calcu- lation, which started with a hull designed by a good naval architect, had to make 1146 Pien auxiliary functions arising from each term obtained by squaring the left-hand side of Eq. (2) are analogous to his functions tabulated in TMB Report 886. The main difference is that Eq. (2) is used to express the singularity distribution which will generate the hull form rather than to express the hull form directly. With this remark, I shall attempt to answer a number of points raised by him as follows: It would indeed be erroneous to make a general statement that wavemaking resistance values as computed are always larger than those obtained experimen- tally. Remark 1 in the justification of application section of my paper was based on the observation of Fig. 1 that if the wavemaking resistance theory is applied to the forebody only, where the viscosity effect is small, the theoretical predic- tion based on Professor Inui's method will be an upper limit to the possible ex- perimental wavemaking resistance values. I am fully aware of the fact that strong favorable interference effect may not be realized due to the viscosity effect. I mentioned this fact as a source of difficulty when the wavemaking re- sistance theory is applied to a whole ship. Hence the importance of minimizing the forebody free-surface disturbance has been advocated. In the past, many comparisons have been made between the theoretical and experimental wavemaking results. No consistent conclusion has been reached from these comparisons. One of the main causes is due to the fact that the theoretical model and the experimental model are not "exact" as explained clearly by Inui in Ref. 2. Therefore it is extremely important to know whether the theoretical and the experimental models are equivalent or not, when we study the comparisons. To illustrate this important point, let us study the com- parisons of three models made by Mr. Wigley in 1927. Figure A is a replot of a portion of Mr. Wigley's original comparisons. In this figure, C, versus Froude number (and V/VL) curves are plotted instead of y versus (@)as in the original figure. Models 825, 829 and 755 are identical except in beam scale. Model 825 has the smallest beam and its theoretical pre- diction at high Froude number, where the viscosity effect is small, should be closer to the experimental curve than in the cases of the other models. How- ever, this figure does not show this fact. This apparently puzzling situation had been cleared by Professor Inui thirty years later in Ref. 2. Figures B and C are taken from Ref. 2. Figure B shows how the singularity distribution per unit beam varies with beam instead of being constant as in the Michell's thin ship theory. Figure C shows how the wavemaking resistance coefficient R/% pv*B” varies with beam rather than independent of beam as Michell's theory asserts. Even though both of these figures are for the case of infinite draft, it is quite obvious that the theoretical wavemaking resistance curves computed according to Professor Inui's method would be higher than the experimental curves at high Froude number range for all of these three models. The overestimation by the theory varies depending upon the level of wavemaking resistance: the higher the wavemaking resistance level, the larger the overestimation. This agrees with remark 2 made in the justification of application section of the paper. 1147 Application of Wavemaking Resistance Theory Fig. 1 - The parameter 6 for Models S-201 and S-202, with calculated values for the ratio c,/e varying with speed. Curves are Inui's experimental values. Triangles are values calculated for Model S-201 and squares for Model S-202. a considerable change in hull shape within the given constraints to work a one percent decrease in the wave resistance. It was obvious from some of the in- termediate quantities computed that with less constraint much more improve- ment could have resulted from the same amount of departure from the given hull form. The calculation was therefore repeated with the constraint that in- creases but not decreases in hull volume could be accepted. Under this con- straint, which is much less rigid than one which holds the sectional area con- stant, the same amount of calculation as used before resulted in a forty percent decrease in the wavemaking resistance instead of the one percent achieved un- der the constraint of constant sectional area. REPLY TO THE DISCUSSION Pao C. Pien David Taylor Model Basin Washington, D.C. PROFESSOR WEINBLUM Professor Weinblum's comment has been studied with great admiration. His work has greatly influenced my thinking in carrying out the work reported in the paper. For instance, the surface singularity distribution expressed by Eq. (2) is quite similar to his polynomials representing ship surface. Likewise, 1148 Application of Wavemaking Resistance Theory DERIVED FROM EXPERIMENTS USING NORMAL WETTED SURFACE — --— DERIVED FROM EXPERIMENTS CORRECTED FOR TRUE WETTED SURFACE — — — DERIVED FROM MICHELL'S INTEGRAL MODEL L A p FT 755 16 1250 847.2 .636 829 16 5 .0938 6354 .636 825 16 10 0625 4236 636 on oO SCALE FOR MODEL 755 " mM 9 bs top) ah Sep ue nN = tae a va) -|ln O39 2 = is} ow 3p u2 | Oy o | 4 4 : aad ot : : ; : & ats ot a er es ~ tc : ater " acetate bg oat :