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2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER
I. REPORT NUMBER
5. TYPE OF REPORT 6 PERIOD COVERED
TITLE (and Sub title)
4.
Proceedings of the Thirteenth Symposium on
W
Proceedings
Naval Hydrodynamics (October--&41O,--1980) 7.
AUTHOR(a
,
EAD INSTR CTIONS
.
.EFORE
6. -7FORMING ORG. REPORT NUMBER
OR ORANT NUMBER(-)
TA
-,p(RF
Takao Inui, editor ,. 9. PERFORMING ORGANIZATION NAMe AND ADDRESS
-Shipbuilding -h%
-
(
Research Association of Japanr]
10. PROGRAM ELEMENT.PROJECT. TASK AREA & WORK UNIT NUMBERS
,
The Senpakv - Shinko Building 1-15-16 Toranomon, Minato-Kv Tokyo 105, Japan I.
RT DATE
Office of Naval Research
.41981 /
Code 432
-l.
Arlington, Virginia t4.
-
-
. R
CONTROLLING OFFICE NAME AND ADDRESS
NUMBER OF PAGES
905
22217
AND
15. SECURITY CLASS. (r' this report)
Naval Studies Board National Academy of Sciences Research Council Washington, D.C. 20418
unclassified -
National Isa.
DECLASSIFICATION/DOWNGRADING
16. DISTRIBUTION STATEMENT (of this Report)
Approved for Public Release; Distribution Unlimited 0, It different from Report)
17.
DISTRIBUTION STATEMENT (of the abstract entered In Block
Is.
SUPPLEMENTARY NOTES
19.
KEY WORDS (Continue on reverse side If neceseary and identify by block number)
&
design practice hydrodynamics
wave energy oscillation boundary layer hydrofoils
hull form viscous flow propulsion cavitation
ABSTRACT (Continue on reverse side If necessary and Identify by block number)
The technical program for the Symposium was constructed around one core theme of "the impact of hydrodynamics theory upon design practice with emphasis on high performance and/or energy saving ships," and consisted of three sessions on Hull Form, two on Viscous Flow and one each on Propulsion, Cavitation and Wave Energy. The authors of the forty-five papers which were presented were drawn from the international community of ship hydrodynamics research scientists with fourteen nationalities represented on the technical program. -~~
i
FORM _U7 73
-.,AN
01O
EDITIO~~VSI
S/N 01
-6601
65
OBSOLETE
UNCLASSIFIED
II~2'P
.,?
SECURITY CLASSIFICATION OF THIS ,rAGE (Whn Data Entered)
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K-5
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'LTERTEENTHSYMPOSIUM
ON NAVAL HYDRODYNAMICS, Impact of hydrodynamics theory upon design practice, with emphasis on high performance and/or energy saving ships
OCTOBER 6-10, 1980 SASAKAWA HALL, TOKYO Accession For NTIS GRA&I DTIC TAB Unannounced
PROCEEDINGS
Justification
Edited by
By --- -- -Distribution/
Takao Inui
Availability Codes jAvail and/or
Sponsoredjointly by
Special
The Office of Naval Research The Naval Studies Board of the National Academy of Sciences -
National Research Council Washington, D.C.
The Shipbuilding Research Association of Japan
Tokyo
Published by
THE SHIPBUILDING RESEARCH ASSOCIATION OF JAPAN 1981
I
I
Support for the publication of the Proceedings was jointly provided by the Oifice of Naval Research of the U.S. Department of the Navy and by the National Academy of Sciences. The content does not necessarily reflect the position or the policy of the Navy, the U.S. Government, or the National Academy of Sciences and no endorsement should be inferred. Partial financial support for the preparations for the Symposium was provided by the Japan Shipbuilding Industry Foundation. We express here our sincerest gratitude to them.
The
ceedings are availablefrom
The i buil i Resea ch ssociaion of Japan Addr s: Th Se aku- hinko Bld 1 -16 ran monkMina o-ku o 10S, ap n \ Tel: (03)
2-2059
PROGRAM COMMITTEE
I
Emeritus Prof. Takao Inui (Chairman) Prof. George F. Carrier Mr. Ralph D. Cooper Mr. Lee M. Hunt Dr. Yasufumi Yamanouchi Dr. Koichi Yokoo
The University of Tokyo Harvard University Office of Naval Research National Academy of Sciences Mitsui Engineering & Shipbuilding Co., Ltd. Shipbuilding Research Centre of Japan
JAPAN ORGANIZING COMMITEE Mr. Shigeichi Koga (Chairman) Mr. Muneharu Saeki Mr. Eiichi Tasaka Prof. Seizo Motora Dr. Yoshio Akita Mr. Takashi Nakaso Mr. Ryo Sanada Dr. Koichi Yokoo Mr. Toshimasa Mitsui Mr. Kotaro Nemoto Dr. Kazuo Hamano Mr. Toshikazu Yuguchi Mr. Akira Takeuchi Mr. Kenzo Nihei Dr. Yoshitomo Ohkawa Emeritus Prof. Takao Inui Dr. Noritaka Ando
President, the Shipbuilding Research Association of Japan Director-General, the Ship Research Institute Director-General, the Japan Shipbuilding Industry Foundation Vice-President, the Society of Naval Architects of Japan Vice-President, Nippon Kaiji Kyokai Executive Manager, the Shipbuilders' Association of Japan Executive Director, the Japanese Ship Owners' Association Executive Director, the Shipbuilding Research Centre of Japan Executive Vice-President, Mitsubishi Heavy Industries, Ltd. Managing Director, Ishikawajima-Harima Heavy Industries Co., Ltd. Executive Senior Managing Director, Mitsui Engineering & Shipbuilding Co., Ltd. Vice-President, Hitachi Shipbuilding & Engineering Co., Ltd. Senior Director, Nippon Kokan K.K. Executive Vice-President, Kawasaki Heavy Industries, Ltd. Executive Managing Director & General Manager, Sumitomo Heavy industries, Ltd. Chairman of Program Committee Managing Director, the Shipbuilding Research Association of Japan
-
iii
-
Preface The Office of Naval Research, the National Academy of Sciences and the Shipbuilding Research Association of Japan jointly sponsored the Thirteenth Symposium on Naval Hydrodynamics which was held in Tokyo, Japan during the period 6-10 October 1980. The Symposium was the culmination of several years of intense and careful preparation and organization, and its success from the technical and scientific point of view as well as from the cultural and social point of view is a reflection of these factors. The technical program for the Symposium was constructed around the core theme of "the impact of hydrodynamics theory upon design practice with emphasis on high performance and/or energy saving ships," and consisted of three sessions on Hull Form, two on Viscous Flow and one each on Propulsion, Cavitation and Wave Energy. The authors of the forty-five papers which were presented were drawn from the international community of ship hydrodynamics research scientists with fourteen nationalities represented on the technical program. As always in the case of a symposium of such magnitude, many people contributed in many ways to the success of the Thirteenth Symposium on Naval Hydrodynamics. First and foremost among these, however, is Professor Takao Inui of the University of Tokyo who served as chairman of the Program Committee and was the focal point in Japan for all activities involved in the organization and management of the Symposium. He was ably assisted by Dr. Yasufumi Yantanouchi of the Mitsui Engineering and Shipbulding Co., Ltd. and Dr. Koichi Yokoo of the Shipbuilding Research Centre of Japan, who served on the Program Committee. and by Professor Seizo Motora of the University of Tokyo and members of the Working Group of the Symposium, who assisted in the planning and execution of the many details associated with the technical program. Many thanks are also due to Mr. Shigeichi Koga, President of the Shipbuilding Research Association of Japan, for the invaluable assistance and support rendered by his organization and for his gracious words of welcome during the opening ceremonies of the Symposium. A similar expression of appreciation is extended to Dr. Saunders Mac Lane, Vice-President of the National Academy of Sciences, and Dr. Rudolph J. Marcus, Scientific Director of the Office of Naval Research Scientific Liaison Group in Tokye, who also gave introductory addresses in behalf of their respective organizations during the opening ceremonies. The National Academy of Sciences was further represented by Mr. Lee M. Hunt, Executive Director of the Academy's Naval Studies Board, and by Professor George F. Carrier of Harvard University and the Naval Studies Board, who participated on the Program Committee and provided valuable counsel and assistance throughout the entire planning period for the Symposium.
•~~
--
Ralph D. Cooper Program Director Fluid Dynamics Program Office of Naval Research
[
°i- i CONTENTS
PROGRAM COMMITTEE
iii
JAPAN ORGANIZING COMMITTEE
iii v
PREFACE INTRODUCTORY ADDRESSES Address by Mr. Shigeichi Koga Address by Dr. Saunders Mac Lane
2
Address by Dr. Rudolph J. Marcus
3
SPEECHES AT BUFFET PARTY Opening Speech by Emeritus Prof. Takao lnui Congratulatory Speeches 6
by Mr. Marshall P. Tulin by General Max Aucher Kagami-Biraki Ceremony
7
by Prof. John V. Wehausen Information about the 14th Symposium
8
by Prof. T. Francis Ogilvie 'Closing Speech
8
by Prof. Seizo Motora
-vi-
._,,
.
-
=.
r
-.
L---
~-- -----
- -
,=-
-
=
= _
_
=
=
,
-
_L_____
Session I: Propulsion On Application of the Lifting Surface Theory to Marine Propellers Koichi Koyama
I
13
Prediction of the Transient Cavitation on Marine Propellers by Numerical Lifting-Surface Theory Chang-Sup Lee
41
Practical Applications of the Discrete Vortex Element Method for Calculation of Propeller Induced Excitation Forces Hajime Yuasa, Norio Ishii,
65
Bror Persson, Oddvar Frydenlund and Kjell Holden Prediction of Propeller-Induced Fluctuating Pressures and Correlation with Full-Scale Data Noritane Chiba, Takao Sasajima and Tetsuji Hoshino
89
Session II: Cavitation
T
New Applications of Cavity Flow Theory Marshall P. Tulin and Chun Che Hsu
107
Off-Design Performance Prediction Method for Supercavitating Propellers Okitsugu Furuya
133
Cavitation on High Speed Propellers in Oblique Flow - Influence of Propeller Design and Interaction with Ship Iull Olle Rutgersson Recent Research Results on Cavity Flows about Hydrofoils Alain R. Rowe and jean-Louis Kuenv
181
Boundary Layer and Cavitation Studies of NACA 16-012 and NACA 4412 Hvdrofoils Jan H.J. van der Meulen
195
The Influence of Hydrofoil Oscillation on Boundary Layer Transition and Cavitation Noise Young T. Shen and Frank B. Peterson
221
~-
vii
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AM--~W~-
159
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Session IhI: Hull Form 1 Wave Making Problems Mathematical Notes on the Two-Dimensional Kelvin-Neumann Problem Fritz Ursell
245
Numerical Solution of Transient and Steady Free-Surface Flows about a Ship of General Hull Shape Robert K.-C. Chan and Frank W.-K. Chan
257
On the Time Dependent Potential and Its Application to Wave Problems Hiroyuki Adachi and Shigeo Ohmatsu
281
Second-Order Theory of Oscillating Cylinders in a Regular Steep Wave
303
Apostolos Papanikolaou and Horst Nowacki
Characteristics of Nonlinear Waves in the Near-Field of Ships and Thier Effects on Resistance Hideaki Miyata Flow Past Oscillating Bodies at Resonant Frequency Gedeon Dagan and Touvia Miloh
335
355
Session IV: Hull Form 2 Problems in Waves 373
The Unified Theory of Ship Motions J. Nicholas Newman and Paul Sclavounos Diffraction Problems of a Slender Ship with a Blunt Bow Advancing in Head Seas Shoichi Nakamura, Matao Takagi and Ryusuke Hosoda
399
Added Resistance in Waves in the Light of Unsteady Wave Pattern Analysis Makoto Ohkusu
413
Rolling and Steering Performance of High Speed Ships Simulation Studies of Yaw-Roll-Rudder Coupled Instability Eda -Haruzo
427
.°VIII-
Session V: Hull Form 3 Problems in Design
;
Vertical Impact of a Disk on Compressible Fluid Surface Chiu-Si Chen
443
Seakeeping and Resistance Trade-Offs in Frigate Hull Form Design R.T. Schmitke and D.C. Murdey
455
Optimizing the Seakeeping Performance of Destroyer-Type Hulls Nathan K. Bales
479
Prediction of Resistance and Propulsion of a Ship in a Seaway Odd M. Faltinsen. Knut Minsaas, Nicolas Liapis and Svein 0. Skjordal
505
The Seakeeping Characteristics of a Small Waterplane Area. Twin-Hull (SWATH) Ship James A. Fein, Margaret D. Oehi and Kathryn K. McCreight Hull Form Design of the Semi-Submerged Catamaran Vessel Yuzo Kusa'ka. Hiroshi Nakamura and Yoshikuni Kunitake Theoretical and Experimental Investigations of Non-Equilibrium Jet of Air Cushion Vehicles Tao Ma. Wei-Lin Zhou and Xiong Gu
531
=
555
569
Session VI: Viscous How 1 Boundary Layer
__
Calculation of Viscous-Inviscid Interacdon in the Flow Past a Ship Afterbody Martin Hoekstra and Hovte C. Raven Calculation of Thick Boundary Layer and Wake of Ships by a Partially Parabolic Method Kenji Muraoka
58.
601
Influence of Wall Curvature on Boundary Laver Development on Ship Hulls Jfirgen Kux
617
Prediction of Viscous Flow around a Fully Submerged Appended Body
631
Nicholas-Christos G. Markatos and Colin Brian Wil!s -ix71
Effective Wake: Theory, and Experiment Thomas T. Huang and Nancy C. Groves Experiments on the Reduction of Bilge Vortex Formation by Discharging Air into Boundary Layer Valter Kostilainen
651
675
Session VII: Viscous Flow 2 Interactions and Scale Effect
__
Viscous Effect on Waves of Thin Ship Takeshi Kinoshita
%93
Numerical Viscous and Wave Resistance Calculations Including Interaction Lars Larsson and Ming-Shun Chang
707
Calculation of Near Wake Flow and Resistance of Elliptic-Waterplane Ships Kazuhiro Mori
729
GEMAK-A Method for Calculating the Flow around Aft-End of Ships A. Yficel Odabasi and Oner Saylan
74)
An Investigation of Certain Scale Effects in Maneuvering Tests with Ship Models Peter Oltmann. Som D. Sharma and Karsten Wolff On the Nature of Scale Effect in Manoeuvring Tests with Full-Bodied Ship Models Evgeny Nikolaev and Marina Lebedeva
779
803
Session VIII: Wave Energy Wave Power. The Primary Interface Brian M. Count and Edward R. Jefferys
817
Extraction of Energy from Wind and Ocean Current Th eodore Y. Wu
829
Characteristics of New Wave-Energy Conversion Devices Masatoshi Bessho. Osamu Yamamoto. Ko-lan and Mikio Uematsu
841
INorihisa
______
Theoretical and Experimental Study on Wave Power Absorption Hisaaki Maeda. Hirohisa Tanaka and Takeshi Kinoshita
857
Rafts for Absorbing Wave Power Pierre Haren and Chiang C. Mei
877
The Sea Trials and Discussions on the Wave Power Generator Ship "KAIMEI" Yoshio Masuda. Gentaro Kai, Takeaki Miyazaki and Yoshivuki Inoue
LIST OF PARTICIPANTS
-PREVIOUS
BOOKS IN THE NAVAL HYDRODYNAMICS SERIES
887
899
905
Introductory Addresses Address by Mr. Shigeichi Koga President of the Shipbuilding Research Association of Japan
Ladies and Gentlemen, It is a great pleasure for me, on behalf of the Shipbuilding Research Association of Japan, to expre:'s my cordial welcome to all the participants at the Thirteenth Symposium on Naval Hydrodynamics. Taking this opportunity, I would like to express my sincere appreci.,,on to the co-sponsors, the Office of Naval Research and the National Academy of Scierces, for providing us with great assistance. Without their kind co-operation, we could not have organized the meeting here today. I would also like to thank the Program Committee for their utmost efforts toward the realization of this Symposium in 1980. It was the Committee that decided to hold the Thirteenth Symposium in Tokyo this fall. We will never forget many others for their kind support to the symposium. I have to express my warm appreciation to the Japan Organizing Committee, its Working Groups, the Shipbuilders' Association of Japan, Japan Shipbuildirg Industry Foundation, and Dr. Noritake Ando, and other staff members for their contributions, Since the first meet.ig in Washington, D.C. in 1956, the symposium has been held continuous) every other year. After the third meeting held in Scheveningen, the Netherlands in 1960, it has become customary that the even-numbered symposiums would be held in the United States and the odd-numbered ones be held outside the United States. Our Scheveningen symposium was followcd by ones held in Bergen, Rome, Paris, and London.
This is the symposium held in Asia for the first time. The core theme at each symposium has always reflected the most important issues in the field of naval hydrodynamics at the respective time. At the first symposium to be held in the 1980s, we have selected the subjects on the High Performance Ships and the Energy Sav;e,. Ships as the core .. greatest concerns theme because they hav since the outbreak of the world oil crisis. Forty-five reports will be presented in the symposium by famous authorities from fourteen different countries on the theme of the "Impact of Hydrodynamic Theory upon Design Practice with Emphasis on High Performance and/,r Energy Saving Ships." I am confident that the symposium will no doubt achieve excellent results through your active discussions. To our happy surprise, more than three hundreds have participated today, while at first we expected only more or less two hundreds. Participants are from twenty-one countries, that is, Brasil, Bulgaria, Canada, China, Denmark, Egypt, Finland, France, Federal Republic of Germany, Israel, Italy, Korea, the Netherlands, Norway, the Soviet Union, Spain, Sweden, the United Kingdom, the United States, Yugoslavia and Japan. We are especially delighted to welcome so many beautiful ladies here in Tokyo from abroad. October is one of the most beautiful months in Tokyo and I hope the climate will be very comfortable to you. I wish all of you will enjc, he autumn of
Japan.
you will bring our most "Friendship", to your home.
In closing my speech, I sincerely wish that the symposium will be very successful, and that all of
important
ship,
Thank you.
Address by Dr. Saunders Mac Lane Vice-President of the National Academy of Sciences
It is my particular pleasure and privilege to welcome you to this symposium on behalf of one of the sponsors, the National Academy of Sciences of the United States of America.
organizing these symposia. Happily, at the time of the Twelfth Symposium in 1978, the National Research Council again appeared as one of the sponsors, this time in terms of a new committee, the Naval Studies Board of our National Research Council. That particular symposium, as I know directly, was an exceedingly successful one. Many of us, George Carrier, I, and others profited from our participation. Now I am pleased that the same Naval Studies Board of the National Research Council is active in sponsoring this Thirteenth Symposium.
The National Academy in my country covers a great variety of science, running from the social sciences through the medical sciences, the biological sciences, physical sciences and mathematics, to engineering. Moreover, the National Academy of Sciences maintains and guides our National Research Council. This Council carries out studies on many varieties of science as they are applied to or needed for problems of national and international interest,
So far, I have spoken to the continuity and tradition of this symposium.
It so happens that this National Research Council has had a long and effective connection with the Symposium on Naval Hydrodynamics. I believe that the first symposium of this series was held in 1956. One of the sponsors at that time was a committee of the National Research Council, namely, the then-active Committee on Undersea Warfare. Therefore, right at the beginning, the National Academy was involved in this symposium. This was also the case in the second symposium in 1958, when the same NRC committee was engaged in planning for the second symposium.
Happily, there are also new and exciting additions to the content of the symposium. First, it is held in Japan. This we can especially weklome, because this ensures an active participation by our many scientific colleagues from Japan, with their wide-ranging and penetrating knowledge. It is right and proper that Japanese scien As should be interested in matters of naval hydrodynamics. Clearly an island country such as Japan must be vitally interested in connections with the sea, and indeed Japan has long been so interested. We are especially pleased, therefore, that this symposium is held here, in the lively city of Tokyo, with the sponsorship of the Shipbuilding Research Association of Japan.
There was then, as far as I can discover, a considerable hiatus. The biannual symposia ,ontinued, but the National Research Council, for one reason or another, was not directly connected, although our Office of Naval Research was active in
This time, also, the international composition of this symposium is especially impressive, as -2-
A
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indeed it should be, because of the wide in. ternalional interest in the problems with which we are dealing. At this Thirteenth Symposium there are, according to my count, papers presented by
hydrodynamics and also wih aerodynamict, during the many centuries over which Man has gone adventuring out to sea, propelled only by sails driven by the wind, either on old-type vessels or on
scientists from fourteen different nations. These
the most modern Catamarans or 12-meters as raced
nations are: Brasil, Bulgaria, Canada, China, Denmark, Egypt, Finland, France, Federal Republic of Germany, Israel, Italy, Japan, Korea, The Netherlands, Norway, Spain, Sweden, the United Kinghm, the United States of America, the Union of Soviet Socialist Republics and Yugoslavia. We are especially happy that this symposium is able to include such a wide range of international experts and interests, because we do know that scientific questions such as those that we meet in naval hydrodynamics are questions which have profited directly from the international exchange of ideas and the stimulation of contacts across national boundaries. The United States is pleased to be a part in this activity,
in the raies today. Finally, this symposium has the great attraction of dealing with a number of fundamental subjects about naval hydrodynamics. You can easily imagine that this is of particular interest to a person of my own background, since I have worked myself in mathematics and I am especially proud of the way basic mathematical ideas have come from, and have been fed into, the topics which we deal with here in this symposium, topics such as cavitation, wave-making, boundary layer problems, and wave energy questions. Not only because of the international and other interests, but also because of the highly fascinating technical problems, I will be interested, as all of us will be, in the ideas to be developed in this symposium.
The special topics of this symposium are also particularly welcome. It deals both with highperformance ships and with the very timely consideration of ships that so can be propelled as to save energy. For my own part, I am especially interested in this, being from a long time ago a sailor, and I think with pleasure of the great privilege of sailing where the only propulsion is that of the wind. I realize that this symposium will hardly get down to the consideration of windpropelled vessels, but I submit that for such vessels we have the longest tradition of connection with
On these traditional grounds, a longcontinuing symposium, and on these specially new grounds, the particular features to be examined in the present symposium, I am happy to be able to act here as Vice-President of the National Academy of Sciences, one of the sponsors of this symposium, in welcoming you to participation in the five exciting days that we have before us. Thank you.
-3-
Address by Dr. Rudolph J. Marcus Scientific Director of the Office of Naval Research Scientific Liaison Group in Tokyo
It is a pleasure for me to bring you the good wishes of the Office of Naval Research and of the American Embassy for a productive meeting.
T
first Director, Dr. Alan Waterman, had previously been Chief Scientist of ONR. ONR continues to be a major sponsor of fundamental research in the physical and social sciences in academic and industrial institutions in the United States and elsewhere. All ONR-sponsored work in universiti's in unclassified and ONR policy is to encourage publication of research results in the open technical literature. The breadth of ONR's interest is indicated by the names of its divisions: physical sciences, mathematical and information sciences, biological sciences, psychological sciences, arctic and earth sciences, material sciences, and ocean science and technology.
I know that I speak for Dr. Jerome Smith, who unfortunately could not be here today, when I express our appreciation to the two sponsoring organizations and their representatives here, Mr. Koga, President of the Shipbuilding Research Association of Japan, and Dr. Saunders Mac Lane, Vice-President of the U.S. National Academy of Sciences. I am sure that all of us also appreciate the excellent work done by Professor Inui of the University of Tokyo, and his many able and dedicated co-workers, in organizing this meeting.
-a
Visits by ONR staff scientists are made to numerous scientists, no matter who sponsors their work, in the United States, Europe, and the Far East. These visits have as their purpose to foster information exchange and to encourage cooperative research. The Office of Naval Research Scientific Liaison Group in Tokyo was established in 1975 to facilitate such visits in this part of the world. ONR staff scientists based in Tokyo visit laboratories and other facilities and discuss matters of common scientific interest with their academic and industrial colleagues in the Far East. The Group's liaison activities as well as its reports are wholly unclassified.
Th o is the thirteenth meeting in a series, each with s...olarly participants of many different countries. After each meeting a proceedings has been published. Collectively these are widely used reference volumes. Since my organization, the Office of Naval Research, strongly believes in wide dissemination and further use of research results, I feel that your activities at this and past meetings are very useful and constructive, Let me tell you a little bit about the Office of Naval Research. The Office of Naval Research (ONR) was the first, and trom 1946-52 the most important, federal government agency which funded a broad range of basic and applied research in university and industrial laboratories in the United States. When the National Science Foundation (NSF) began operations, its initial personnel was recruited largely from ONR; NSF's : °:
Speaking for the Office of Naval Research and for the American Embassy, where our office is located, it is my honor to wish your good luck and success in your work. Takyu Thank you.
-4-
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Speeches at Buffet Party Opening Speech By Emeritus Prof. Takao Inui Univ. of Tokyo
itLadies
was considered as a possible Japanese host organization. The year 1980 was selected as the first possible data. Otherwise we must wait till 1992 or 2004, i.e. twelve year interval. The symposia outside the United States are to be held at four-year intervals and in Japan we have three-year interval, i.e., off-season of !TTC and ISSC. This multiplied interval resulted in twelve years. Later, the Shipbuilding Research Association of Japan kindly accepted to be the host for the symposium in 1980, and the official announcement was made at the Twelfth Symposium held in June 1978 at Washington, D.C.
and Gentlemen, It is a great privilege for me this evening to be able to greet you as Chairman of the Program Committee. The Symposium on Naval Hydrodynamics has been blessed with great achievements during its twenty-four year history which began in 1956. And I have realized that we, the Japanese researchers, have benefited greatly by attending these Symposia. We had long been studying the possibilities of holding a symposium in Japan, in the hopes that we could repay all the good will which has been shown to us. Unfortunately, it was difficult for the Society of Naval Architects of Japan to host this symposium due to the special circumstances of our country after the war.
-
It is unfortunate, however, that we had to face a serious depression in the shipbuilding industry after that due to the second oil crisis. In fact, we had even the third "crisis" or "Koga" crisis in January 1979. I was shocked when we were ad :-.3d by President Koga to reconsider the plans for tne Tokyo Symposium.
In February 1976, the International Seminar on Wave Resistance was held in Tokyo and Osaka. On that occasion, the first preliminary meeting for feasibility study of the symposium in Japan was assembled. In addition to five Japanese members, Mr. Cooper, who was one of the Seminar Session Chairmen, Dr. Bertin and Prof. Kovaznay from the Tokyo Branch of the Office of Naval Research were present at the meeting.
Tonight, I am very pleased to see that the symposium was finally realized. I would like to take this opportunity to thank all of you for your participation which contributed a great deal to the symposium. At the same time, I would like to express our special thanks to the Office of Naval Research, the National Academy of Science, the Shipbuilding Research Association of Japan, the Shipbuilders' Association of Japan, and the Japan Shipbuilding Industry Foundation, for
Either the Shipbuilding Research Association of Japan or Sipbuilding Research Center of Japan -S-
session tomorrow, which, in Japan, is a national holiday, "Sports Day". I sincerely hope that you will have a fruitful and happy day. Have a good time here tonight.
extending to us their kind offices throughout the preparation stages. I am afraid that all of you are quite tired by now after going through the tight schedule since last Monday. You are also requested to attend the
Thank you very much.
Congratulatory Speech By Mr. Marshall P. Tulin Hydronautics, Inc.
Thank you Mr. Master of Ceremonies, Professor Inui, our Japanese host, and all of our Japanese friends. I think there are, by official count, a hundred and thirty-one of us who have traveled to this wonderful country from abroad to participate in this Symposium, and I don't think any of us would like to leave without expressing certain thoughts. I am honored to have the role to try to do that in my own words. I would like to address myself first to our Japanese host, Professor Inui, and all of our Japanese friends. You know it's not possible for us to visit this beautiful and fascinating country without being almost overwhelmed by your personal kindness and your graciousness. In addition, we experience every day at the Symposium your extraordinary efficiency and organization. This is of course a most appropriate locale for the Thirteenth Symposium because in addition to these attributes you're carrying out here throughout the country intensive and high-quality research in our field which is making its impact and has made its impact for many years throughout the world. And so, I think you are something like the New York Yankees of the Naval Hydrodynamics Symposium with all of these assets going for you. And in addition you have, of course, a secret weapon, the beautiful and charming Japanese women who have mace their presence known to us in subtle ways; I think the twenty-five foreign
women who are here would like especially to thank these Japanese women for their great kindness in showing them Tokyo, acquainting them with Japanese ways, and in otherwise charming us. I think, too, some note should be made on this occasion- mention should be made of Professor Inui's particular role, as of course he is responsible for the Symposium being located here in Tokyo, and we have to thank him for that. It happens to be a time when he has just retired from Tokyo University, and I think we would want to show our regard and appreciation for his contributions which have taken place over a period of at least thirty years. He is, of course, known to us as a man who eliminates waves, referring to development of his b-lbous bow which made such an impact in ship design. But I also think of him as a wave maker, and I think that his effect on Japanese naval architecture both in its practice and in research will be propagated far into the future through his own research, through his own students working in industry, and through his students who are now teaching. I think there are some ten of his past students who are teaching in Japanese universities, and only when I came to Japan four years ago was I able really to appreciate in detail the important role Professor Inui has played throughout Japan.
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J
I
I
responsible for keeping this Symposium in progress* since 1960-it was founded in '56-is Mr. Ralph Cooper of the Office of Naval Research, who has been responsible for its organization. This is his last year, as ,most of you know he is retiring, and I think note should be made, and appreciation should be paid to his very important and vital role.
This is as close as we are going to come to the twenty-fifth anniversary of this Symposium because it was founded in 1956 and next year is its twentyfifth anniversary: but the next time we meet will be twenty-six years; therefore on this twenty-fourth year perhaps some note should be made of this anniversary. I think it is especially fitting that in that time we've reached halfway around the world, and our numbers are growing. I make special note of the presence here of representatives from the People's Republic of China for the first time at this Symposium. And finally, I think most of you must know that the man who has been primarily
I hope-if I have left anything out, that you will please express it personally to our hosts. I thank you very much.
Congratulatory Speech By General Max Aucher Bassin d'Essais des Carenes European people this staying in Tokyo should be more pleasant if we could understand the Japanese language. How much it would be beneficial if we were able to appreciate your pretty city, to understand the information on signs in town, to know
Ladies and Gentlemen. Dr. Inui and Dr. Takahashi, on behalf of the ONR Organizing Committee, asked myself to make a short speech as a representative of the European participants at the Symposium.
the name of the street and to be understood easily by the taxi driver.
Dr. Takahashi justified his choice saying that the French language was the most harmonious of the world, and so my speech could be understood by Japanese ears perfectly. I do not know if the French is more harmonious than German, Swedish, or Dutch, but what I know is that my French is the baddest of my European colleagues. You can't appreciate it.
Many ITTC Technical Committees are profited by the occasion of ONR Symposium to be added to their own meeting in Tokyo. So, we have to thank for their hospitalities and their kindness, Dr. Sugai from the Ship Research Institute, Dr. Murakami from Meguro Basin, Dr. Yamanouchi from Mitsui Basin, Dr. Tamura from Mitsubishi Basin at Nagasaki, and Dr. Kato from Tokyo University.
For us, European peoples, Japan is a marvelous and exotic country, and everybody had dreamed in his youth or later of dong touristic cavort in Japan. For many European people who are present, the Eleventh ITTC Conference and the success in Tokyo some fourteen years ago had been such an occasion. Since our Symposia concerning hydrodynamics or naval science took place in Tokyo, I am sure that all the participants have got a good remembrance of the city. Nevertheless, for us,
Now, to conclude, I also add to express our grateful thanks to the member of the ONR Organizing Committee for the manner they are allotted the program doing a pleasant cocktail for us to ease this section and gastronomy such as this buffet party. V Thank you very much. .7.
Kagami-Biraki Ceremony By Prof. John V. Wehausen Univ. of California LThe
toast I would like to propose is to all the
this is a very impressive group of young people, intelligent, well-educated and enthusiastic. It speaks well for their teachers and for the future of Japan. I propose a toast to them.
young ship hydrodynamicists of Japan. I have the same feelings now as I had four years ago at the International Seminar on Wave Resistance, that
Information about the 14th Symposium By Prof. T. Francis Ogilvie Univ. of Michigan
r
be host at the University of Michigan for the next ONR Symposium. Obviously, this is going to be a difficult act to follow in every way. The second thing is, and I'm sorry Ralph Cooper is not here, but after talking to many many people about this, we have deci(! - d to dedicate the next ONR meeting, the Fourteenth Symposium to Ralph Cooper in thanks for his many many years of organizing these meetings as well as a lot of other things.
I understand the reason that Ralph Cooper is not here is rather a typical reason, namely that he is out working trying to arrange transportation for a lot of people. He is always on the job. I have two announcements to make about the next meeting-one I make with some trepidation, and the other with complete pleasure. The first one I make with trepidation is that we have offered to
Closing Speech By Prof. Seizo Motora Univ. of Tokyo for your generous efforts in making this ONR Symposium such a great success. I am very happy that we were able to hold the Symposium in Japan this time, and firmly believe that the sessions have proved to be very fruitful.
I am greatly honoured to have been asked to say a few words in closing this party, on behalf of the Society of Naval Architects of Japan. Let me first express my appreciation to all of you, especially to those of the Office of Naval Research, and the National Academy of Science,
I would like to thank you very much for giving -8-
young researchers in Japan this great opportunity to meet and speak with you all. I am sure that these young people will contribute gieatly to the field of Naval Hydrodynamics in the future. It is also my hope that the Buffet Party this evening has helped to further mutual understanding, both academically and socially, between all participants. I was particularly surprised to learn that we had so many musicians and other talented performers among us. The abhreviation "ONR" this evening could have stood for "Office of Nice Recreation." Let me take advantage of this opportunity to inform you of future international symposia now
being planned by the Society of Naval Architects of Japan. One is the 2nd International Conference on Stability of Ships to be held in the autumn of 1982; the other is a Symposium on Practic:-l Design in Shipbuilding (known as PRADS) scheduled fc7 the autumn of 1983. We hope that many of you will be able to participate in these meetings. This time of the year is considered the best season in Japan. I hope that you will be able to take back good memories of Japan, and that our country may be attractive enough for you to come back soon. Thank you very much, and gooo night.
7
7-9- 7
Session I
PROPULION Chairman George F. Carrier Harvard University Cambridge, Massachusetts U.S.A.
_1-9-
On Application of the Lifting Surface Theory to Marine Propellers Koichi Koyama Sni Researc' institute
I
Tokyo. Janan
ABSTRAC'"
bound vortex
Three methods of propeller lifting surface theory proposed by Hanaoka and their numerical investigations performed by the author are shown and discussed. Method I is devised to be able to calculate the unsteady propeller lifting surface with a rather small digital computer. Method 2 employs the doublet-lattice method from the view point of wide applications. Method 3 deals successfully with the difficulty in the singularity at the blade tip. The uantitative evaluation of the numerical lifting surface theory is performed by conparing the resUlts of calculations and exnuerperiments. ThL investigation into numerical results reveals the advantages and disadvantages of the three methods from the practical point of view. It is shown numerically and experimentally that Method 3 yields the accurate solution near the tip of blade.
Y
circulation density
ro
f1 =ciyoj
d
r
f 1 =cj Y
dE
w
=
woe
circulation of blade section upwash on lifting surface
ivt =
¢e
p =v/h
acceleration potential order of harmonic of inflow field
/3n,3/nnoml derivative on the helical surface at the control and loading s points induced velocity at a blade by wI steady free vortex a=- I wt= wIsin cI
NOMENCLATJRE
sin c1 lt~ 2 Cos w radial mean value of w a a l radial mean value of w /r t t V*=V+wa, t + h=V*/f,* r0 propeller radius 0 r boss radius o n Znumber of blades u=r /h, =r /h 0 0 b b s distance measured along the helicoid S1 T wition of leading edge position of trailing edge 2 , T 2 , -2)I2 -/ 5- * *
Propeller x,r,6 cylindrical coordinates .0 Ui, helical coordinates of control point O,a',u' helical coordinates of loading point u=r/h . c=6-x/h, t-&e+x/h, : . 2rh pitch of helicoid D mass density of fluid V advance velocity of propeller angular velocity of propoller angular frequency of oscillation t lift p 71=11e lif density, desiy or or pressure presur Y=Y e circulation density of
.0-
1;/ =(U0+1b)/2 ;=(t-t 0 )/to
'(T-5)/T',
pellers. At present we are going to confirm numerically the quantitative relation between upwash distribution and loading dis-
4*=(TA)Ita
/,tribution +r
'(-,,
2
c'=h/lT' c =h/l4u
71/2 7 /2
on blades of propellers by the
theory. High speed computers make it possible to achieve such a complex calculation. There are many studies in development of such calculations[4]h[81.
half chord length halt chord length
it iS significant that the form of vor-
c=-2c 1 (r0 -rb )=1+
tex sheet is refined to adapt the actual phenomenon or that the reasonable viscous coefficient is introduced. In analysis of lifting surface however much depends upon numerical calculation. For the discrepancy between theory and experiment, it is difficult to decide wnach is more dominant cause: analytical inaccuracies in the numerical method, or physical differences between the idealized model and the actual phenomenon. So it is an important subject that numerical results for hydrodynamic characteristics by the lilting surface theory are estimated with confidence. The methods are not satisfactory if their results show some differences with each other even for the cxistinc simple propeller liftinc surface model. Comnarative calculations on pressure distributions o; the propeller blade performed at 15th -i1 show that the results obtained by many methods are apart from each otherf9]. Although there is a great variety of the methods, the disagreement between the mathods raises a problem from the practical oix t of view. How is the degree of agreement of the results of the numerical liftinc surface theories ? On the other hand the lifting surface theory can be applied to various aspects such as prediction of performance, vibratory forces, cavitation, etc., so it is convenient that there exists suitable method for
2/(2j*)
Wing x ,y ,z
Cartesian coordinates of control point Cartesian coordinates of loading point
:y',z' w y b 11,12
upwash on liftino surface circulation density half span positions of leading and trailing edges of the blade element including the control point li,l positions of leading and tralling edges of the blade element including the loading point c =(12-11)/2, c'=(i -!)/2 half chord Lenqth x 0 =(I+1 2 )/2, xA=(li+l)/2 c=c'/b, ;.=b/c' .-y/b, r;'=v'/b /c,= dc'
;=(x-o
')4'
T
e,, 6
d
dy'
y
asi-ct.
_____each
this paper, three methods, named Method 1, Method 2, and Method 3, of lifting surface theory proposed b.. Hanaoka and their numerical investigations performed by the author, are shown and disMethod I was developed in an early of the study of liftinc surface theory 1013). Numerical prccedure in the mothod is devised to be able to calculate the rnsteady propeller lifting surface with a rather small digital computer. ith te current of monularxzation of larcer canacitied dicital computers, Method 2 was devel-
*-in -- --r
\
-
-here
. .
p
i/Z
N" \cussed. ~stage ./
I
/
%1
_____
Fio.l.1 Helical Cooroinates
oped with doublet-lattice method from the view point of wide applications[143,15]. These two methods don't give converged solution near the tip of blade, which becomes arn obstacle to "he calculation Of cavitation. Method 3 deals successfully with the difficultylil5]-l8J. In chapter 2, the degree of agreement between them is revealed, which is considered to show the situation of accuracy of the present lifting surface theory. And further, calculated results are compared with experi-ents. Although doublet-lattice methoc employed by Method 2 is easy to deform the wake vortex sheet or to co-bine duct system, the method is eployed fre the view point
INTRODUCTION 1.1 Main Theme and Background It goes without saving that th.e liftitg surface theory plays an important role in the hvdrodynamic analysis of marine propollers. The theory has been developed for i n fong ield of aeronautics. ile almost all of the applications of the theory are to wings of airplanes, there is an old application to screw propellers by Kondolli. We can see Sparenberg's12] 3,:d Hanaoka's[3) contributions as pioneers in application of the theory to marine pro
U)I
AVITY
RE-ATTACHMENT/
Sc
CAVITY DETACHMENT
TIME.t
TIME.t
a.
(SUPERCAVITATING
CAVITY
TRAILING EDGE
1.0 i, -FI
LE
tN
CA aS
XX
I
TIME
tdld
a.
~
R:E:DARMVIUITYj7
FI'-EGE STREAK 22 LINES ST
a
I
EAHIENTAGA
S
TIME, t
CVT
xiC
a condition
I
b.o
THNFI
,LMR b nTIthis casd
operating in wakes (42), proThis very important result ides the basis for an essentially simple unsteady theory asymptotic to the frequency limits: tas sa l cIn Ssibilit the qrc o eqb =
"BEKIG
the steaIins
a
e. teps
viously flowed over the foil, a condition lihely to be very disruptive of the cavity. Cavity "breaking" can occur only when the slopeof the streak lines varies from line to line, as would occur whenao = (t), in a ventilated flow undergoing psations. this case the streak lines have the posto interset. The condtion for this intersection ca n be shown to be, see Figure 23b:
fact, te second of these conditions
SIn -former
s a
b
(moderate reduced frequency) implies the
in the usual case of thin cavities
tn)
/at(o,t )
the usual cavt o ucavitating flows. o = In const., and) (42) implies everywhere parallel streak lines, as shown in Figure 23, and a boundary value problem, excluding the the closure ~~~~steady flow condtion, case where identical (42) also to applies. %
e
Th i dS se l i e t e dentinu s or, adosin h very (36), growing kinks in the flow, someho rei i c n f w v r a i g
As i- we shall see in the subsequent section, by be effected will closure condition the unsteadiness ~the to the o'-'ker of our approxi-
bc
Cavity Growh and the Closure Condition.
main.Te streak diagram immnediately allows
(the body ae shapefchne is assumed fcaiy not to change): oum sThe
the prediction of two pathological phenomena
~~2) £ vin cavity "breaking" or shocks." The formthe flow:ve is 1.)cavity n flSu "ploughing," and
t
er of these can arise during the expansion of a sheet cavity when the expansion rate ~~is excessive (I t > U ), as in the region between b and c in Figure 23a.
b d
There. the
-125-
__
Cs
__
,
S)
Using the definition of t e f i l s d ~ .
(46) and since the
4>0
ad
Ct
GROWIH
0c
d(S 0
S resents le n rep 0 ~ te d tIre cav1.t ergens'
Sine tr
the
and in ee ttl
(4d
l
v
I
u
i
-~
For all collapsing cavities (V < 0 there are, as in the steady case, two solu2 tions for each value of CL/ 7c less than a
~a
critical value (one long and one short cavity) and no solutions for larger values,
increases, the hysteresis lcop will become evident, see Figure 27. It arises since growing cavities must be shorter than stable cavity. while collapsing cavities must be longer.
In the case of growing cavities (V > G) there are either none, one (long), three (two short and one long), or one (short) solution. The situation is illustrated in Figure 26. During cyclic motion, solutions 4 must generally be found in the regions® and(®in the Figure 26.
-
p
tt
7k-
I ''
SNORT EIt1EIYAIC Fp OGE SoU-FO SOL-
e
udo
(70 S#ORT
annot occurutl
clo
-Q
FIGURE 27
'0 SOwia10.S
FIGURE 26
-
t
o
-
.0%E SHORT.
COt~LS
n
/e
SLUT;CMIUCAL
ThEE
:Z s-n
/n
Cn
THE SIX REGIONS FOR DYNAMIC CAVIIEScollapse
Stability of Acosta'a Solution, We note that any point on the upper e nseem branch (b-c) of the zero growth curve
r
CL
MOTIONS, HYSTERESIS (SCHEMATIC)
-CAVITY
Note the following effects which 1.) the m.aximum cavity aczompany hysterc,.sis: length is reached after C. has reached its mxiu n eu odcie oeta cannot occur until after C.,4-c 2.) the r-xideclines to a value 1) the mean wake is sensitive to speed and trim variations the flow is oblique relative to the propeller shaft
As the wake changes with trim variations at constant speed one could assume the wake to be rather much influenced by the potential flow. The total wake is then the sum of the effect of viscosity, wv, potential flow (without wave making), w ,and wave making, ww . S-161-
M
i
P-MFN 9 .JL..
Fraujd* isnbw 006
_____E
to
Z
0 1410sated Ce n t e r
U e ra
Co L clo t e l
rassarol pr e ssure s
Filled points:
t n
towing
tn
points: Cavitation tunntel
1Popen !
plating
Sottorn
Ling
-olT_
Fig 3
Influence of ship speed on the static pressure measured on a triple-screw model
v
Distance below bottom plating i
0
010
Wing prop measured
-_6- Wae colmn to
X
0.0
I
kig Coe
Iin
Fg C
Comparson ofMeasuredadlu t in r s u e o r d s a i
and ca ctlt
of me su e
om a i o
1,02"ticIl
triple-srewCmode
-~~~ --
~
t
-
~
'~ -'
-r
-
-
-
-
-
-
~
-
A J1=--
a+
apg
Sstatic
~The Svalues
p/2Vs2 ( C
-
w)
number of revolutions at different speeds. The purpose of the cavitation tests is for most merchant ships to check the erosion properties of the propeller and to measure the vibration excitation forces and the noise generated by the propeller. The characteristics of a high speed are, however, very much influenced by cavitation. Further, this influence is peeriwoknbhndteul the proin constant flow and when(aso different is working behind the hu2l (as shown later on). Thus both propulsion tests and cavitation tests are necessary for a relipower prediction. In the latter case tests have to be carried out in behind condition. The purposes of the different tests
Tand +
+
(4) (1- w) 2 Itapropeller the water column up to of using method the approximate the transom an ap te valunof the ater value mtansom as using ttcpesr.peller pressure. the to agreehigh speeds, FN is1.0, be shown between the methogs ment At nable e wi At lower speeds where the trim and good. d Athe sinkage at the stern are large the calculated static pressure is too high." For the wing propellers this difference is not negligible. theoretical calculatiops described in theaortincled catiosessur hed o open water tests in towing tank give the hutadavnec rlto oetween ewe thrust The in 2.1 also included and advance corelation lobe i pressure. aeistatic vausotie in Fig 5 also been efficients and form the basis of calcu obtained have wae b a i o n compared with measurea static pressures.efficonntf ffective fraction wake effective of lation basin towing the The measurements in and in the cavitation tunnel happened to give liost identical peller thrust and wake for different ship gieiits dnia results eut for o the h pror-speeds spedions peller centre. This is not usually the te n bhind tonue, caiation t case and correction for this difference is the relation between thrust, torque, made when calculatina the cavitation numefficiency and number of revs at cavi ber tating conditions. Input parameters are the cavitation number according to Eq C_5 T = V (5) and the propeller loading according +
M
+
OT
OVA
(I
_
W) 2
to Eq (6) 3. SYSTEMATIC PIOPELLER TESTS
In Fig 5 the theoretical calculations also agree very well with the measurements at the propeller centre in the tunnel. The differences for lccations closer to the hull are probably due to disturbances from shaft and struts, which were not included in the calculations. The calculations far the towing basin show also here that useful results cannot be obtained unless the free surface is taken into account.
P
2.3 Propeller Leading The propeller loading is also a fundamental parameter for the cavitation test. The estimation of the propeller loading includes all the traditional towing tank problems: o o o
*
-propeller
towing tests for measurements of resistance of hull and appendages [5] self propulsion tests for determination of the thrust deduction factor and wake fraction [3, 5] calculation of propeller loadings by the use of scale factors and correlation factors, empirically estimated on the basis of earlier experience ST 2 (6) V 2 KT/J2 pD
For most merchant ships the cavitation does not develop so far as to influence the characteristics. The self propulsion tests in the towing tank can therefore form the basis of the prediction of power -163-
A number of systematic propeller series for propellers specially designed for high speed ships have been presented in the literature, for example [6, 7, 8, 9]. Tnese propellers have, however, usually been tested in uniform flow. Very little is known of the influence of propeller geometry on for example erosion, interaction with hull and pressure fluctuitions when the propellers are working behind a hull. It is the purpose of the present investigation to improve the knowledge in this field somewhat.
M 2
3.1 Propeller Geometry
a
Six 3-bladed high speed propeller models with the diameter 250 mm were chosen for the investigation. The main propeller parameters are shown in Table 1. The three first propellers (Conv 1.05, Conv 0.75 and Conv 0.50) represent a blade area variation of propellers of rather conventional design, with symmetrical blade shape and NACA sections. Propgller Warp 0.75 is a blade shape (warp 120 ) variation of the conventional propeller with blade area ratio 0.75. The two last propellers have two different supercavitating sections. Propeller S.C. 0.50 has a face shape according to tne 3-term distribution and a modified 2-term thickness distribution (9]. The last propeller model was designed to have improved cavitation oroperties in the partially
A
z
Table 1
Main propeller parameters
Propeller No
J
Design point K a T
P1391 P1477 P1514 P1714 P1439 P1790
1.16 1.16 1.16 1.16 1.15 1.15
0.19 0.19 0.19 0.19 0.146 0.146
0.58 0.58
AD/A 0
Profiles
Pitch distr
Blade shape
Designation
1.05 0.75 0.50 0.75 0.50 0.50
NACA 16 NACA 16 NACA 16 NACA 16 wedge spec
unload unload unload unload opt opt
sym sym sym 120 sym sym
Cony Conv Cony warp SC PC
cavitating (PC) region. The profile is a combination of 5-term face shape and an empirically derived back shape. Further information about the propellers, such as blade shape, pitch distri-
warp
1.05 0.75 0.50 0.75 0.50 0.50
Notation in figures O A
-I
.....
---.
1.25 m absorbing about 1.5 Mw at about 700 r/m. Table 2
Propeller loadings and cavitation numbers tested
VS
bution, camber distribution and design
method is given in the Appendix. KT/J2
1 -w
aT
The ship model used for the investigation is a twin-screw patrol craft in scale 1:5.0 with a propeller arrangement
(knots) 20 30 35
0.248 0.156 0.134
1.05 1.02 1.0
2.07 0.941 0.70
according to Fig 6.
35
0.134
3.2 Ship Model
1.0
0.70
3.3 Test Facilities The cavitation tests were carried our in SSPA large cavitation tunnel. The tunnel
c o
one circular, high-speed test section one rectangular, low speed test section, large enough for tests with combinations of propellers and complete ship models
A sketch of the tunnel with the large test section in place is given in Fig 7. The most important data of the test sec2.0-0
tions are given in Table 3.
I ,I
Table 3
Main data of test sections of cavitation tunnel 2
High speed Low speed section section original with insert Fig 6
Length (m) 2.5 Area B x H (m2 ) diam Max speed (m/s) 23
Propeller arrangement on ship model The main interest
in
this
investiga-
Mn cay number
9.6 2.6x1.15 8.8
1.45
0.50
The low speed test section is covered by a recess, in which the ship model was placed. The model was the one used for tests in the towing basin and was at this investigation made of fibre-glass. The model was placed in the tunnel with the correct draught at the stem and with the same trim .164-
-_
9.6 2.6x 1.5 6.9
*Epty tunnel. At propeller tests cavitation on right-nne arpr o eer ets cvit a s tn right-angle gear dyamoter set o = 0.15 as te lower limit.
tion is devoted to the speed range 30-35 knots. In order to have a typical low speed point 20 knots was, however, also included in the test program. Thp propeller loadings used for this study were chosen to suit the pitch and diameter of the propellers. The full scale ship correlating to these loadings could be a twin-screw patrol craft with a displacement of about 100 m 3 operating at speeds of about 30 knots with propellers with a diameter of
-
0.06*
im
Rii 15S3
I
Q*5
-- --
---
I
I
1
/ !
-
L~IT
Fig 7
The SSPA large cavitation tunnel with low-speed test section in place. Dimensions in metres
as at the tests in the basin. Individually cut wooden plates were then fitted to simulate a flat free surface. At the tests in this study also a dummy simulating the wave behind the transom was mounted under the plates behind the model. With this movnting the static pressure at the propeller centre was measured to be 0.16-p/2V2 lower than the free stream pressure, as is also shown in Fig 5. The test section and the recess were completely filled with water at the tests. To drive the propeller models one AC electric motor was used for each propeller. Strain gauge dynamometers for measuring thrust and torque were placed in the shafts close to the propellers. The operating range of the test section with insert covers ship speeds up to about 45 knots at the water speed 8 m/s. At this investigation the water speed 7 m/s was used. A more thorough description of the tunnel and its background is given in [10]. 3.4 Primary Results The primary results from the cavitation tests are:
t
--
r--
.5-
o o o o
cavitation patterns thrust, torque and number of revs risk of erosion pressure fluctuations induced on the hull
In Figs 8 and 9 photographs of the cavitation patterns with the blade in top position are compared for the six propellers. Due to the oblique flow the back cavitaton has its maximum at blade position +90 and its minimum at -90 , the variations being more pronoinced for the inner radii than for the oute. ones. Thus the cavitation patterns in Figs 8 and 9 represent a kind of mean cavitation extension. One of the most serious problems with high speed propellers in oblique flow is root erosion. At SSPA a paint test technique has been developed, which shows the risk of erosion after only 30 minutes test in the cavitation tunnel [11]. Pig 10 shows the results of testing the propellers with this technique at the 30 knots propeller loading. The most severe erosion is found on the conventional propellers with wider blades. The propellers of the supercavitating type show, however, very little erosion. Especially propeller P.C. 0.50 seems to be very successful from this point of view. The propellers Cony 0.50 and Warp 0.75
-
--
Vs= 20
Prop e Le r
V3
knots
s3
nt
nt
Cony.__
1.051
Cony.
Cony.
i 1-1q
8
Cav2Aation
~
ttSS
t 11
coieICte
I166-
Sill)
Vs=30kotsVS3klt
Vs= 20 knots
Pro-
peLie r
-arp fZ
S0,75
A
o.Al
PQCAll
ra-A
9)
si 1. tes~s with a con:pIetc Ca-.itaLiofl :)dttens at
MdeIC
Leading edge
BACK
FACE
Leading edge
PROPELLER Root erosion on face and back.
Conv. 1.05
C
Root erosion on face. Erosion on back.
. 0.75
Conv. 0.50
No erosion.
Warp 0.75
No erosion.
S.C 0.50
5No --- 0.50
R
C
erosion on face. Root erosion on back.
0.50No
erosion on
P C 0.50
Fig j0
face. Slight root erosion on back.
Cavitation erosion obtained with the paint test technique after 30 minutes test at propeller loading corresponding to 30 knots -168i
a
-
2.5
-
-
1____
-i
C 0000.-
Xg60 -
' sc oso
) ._____
PCo 0.5
1 -
___Iov
0
.
IE
../420253035.. i
4"
I
"hip
!
'
Speed Vs (knots)
l
i
_
_
I 500
_______I
i |
12I When comparing the propellers in FigL 12tretig ar noiebe
_I
700
,
Comparison of blade frequency amplitudes induced on the hull of the prototype
Fig 12
__t1
930
1100
Number 0f revs. Np (rlm) Fig 11
0
Power and number of revs for the fro cavta-type prottypepredcte frmcvt-than ptotype peted
o
propeller Cony 0.50 induces very large amplitudes
o
propeller Warp 0.75 induces rather low
o
the propellers of the supercavitating do not induce larger amplitudes the conventional propellers.
1
werecomletly fee romerosondepnd3.5 Influence of Cavitation on Pressure Apiue ing on almost fully cavitating conditions i 2hv mltdsi Tepesr at this propeller loading. o tan When he cnvetioal wit bWhen compit ngssh thpr atpeller in Fonthe ime,2ha inpligt13es of alo e plottue tests the number running the popeler i 3a h iei as enpotdi : ~revs is adjusted so that the propeller sionless amplitude coefficient K used for loading according to Eq 6 is obtained. At the scaling of pressure fluctuations. Using loading the torque coefficient and !this nt helvsar sow tob advance ratio are then measured, formingthscefc eraigcvtto fil osata basis for calculations of power demand Sthe numbers in spite of the fact that the cavi" tation extensions are increasing. A probandnumer pofereas for therootye.en explanation of this, also proposed in ifeetable ih oe dmnsfo-h ntth rsueapiue (3.cudb shown in Fig 11 are fairly clost S -propellers are-increased at lower cavitation numbers for 30 propellers Cony 0.50 and arp 0.75ts knots). Note especially that the de e d aat ecauiseon rducedon the e ll propellers of the supercavitating type oa do re Shave
only slightly higher power demands onventioal propller wit bae differences oscoefficient between the propellers are,
hw ta pfrann oain(1i.a cavitating propeller divided by the thrust K Using is almost of the advance ratio. this independent parameter the in-
considerable, however, The pressure fluctuations induced onthe the hull are important results from theae forming advanc ratio.. are the measured, s xcied ibatinsand cavtatontess, tion tne, ase te -arl noise emanate from them. Fig 12 shows the evalues just above the propellers. The values given are the mean blade frequency amplitudes of analysed evs by the digital method described tpresent
fluence of clearance ratio on the pressure
tanea
=
-from theoreproduced [2]. The in Fig 14, retical curve wasat- calculated bycavitation a method derasn out l contan ne of Cavtate ex Prental emanate from different investigations
Thpre s amelthe s in he nw lagsareth prpelerai the investigation.
po
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Fig 14
-
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exrsso
3.6- Influence +... -'+Charac- +on :Propeller :q Hull - of teristics
. I
0OC Oh 0.2 a iD Vertical learance clearance on the blade Influence of frequency pressure amplitudes of a
In [2, 9] the influence of hull on the c.aracteristics of a cavitating propeller was reported to be a kind of wall effect thrust and torque up to 10%. In in this 17 the hullis influence Fig on the influence given as obtained investigation thrust and efficiency. The thrust coeffi-+ at cavitation measured n ) were tion(w in behind cients accordmodelcondicomplete ship tests and efficiencies
non -cavitating propeller
ing to Fig 6 with the cavitation niuber
-::;Iireducing 0 -_______ -
-£
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Fge 14. bycor
16. The majority of the propellers end up values around 2.0 at o = 0.7. For the propellers with complete thrust breakdown, however, the amplifications are about 3.5.
•at
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vste cavitation
the cavvitationis ..Theoretical cuv €
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in Fig ,-&. ,".,. given pellerWarp to be is 0.75 shownisof influence the very hull different shown in Fig the influence
:
are17. The thrust coefficients and efficiencies
I
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deinclination influence of the sh-ft grees on the tested procara except pro-
-
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Z_./
,
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been shown to be very small
ohas Cii
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-
The influence shown in Fig 17 thus includes not only the influence of the hull but also the influence of rudders and the shaft inclination. The influence of rudders
_______
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cavitation nu,'er at the propeller centre the ship model is n-cunted in the tun-
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tin nmber
Iwhen
07
Fi
order to com.pare identical cavitation nunmeasured at the caviber KT0and 10were =
!
'-
0.8____
, ?,-Cavitation _
cx-w-aso _____
~s" -
r
-
:o : -
in the tunnel. These values have been cornoared with thrust coefficients and efficiencies (KT0, nO measured with the same propeller mocel -,orking in homogeneous flow at zero shaft inclination in the high speed test section of the cavitation tunnel. In.!
2.0
a'
i
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.7
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Cvittion number
£
_
_
ol 0-
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-
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.the
Hull influence on thrust and efficiency
-. s
-
are generally increased a few per cent when the influence the Thus shaft,if While inclining of the obliqueness coefficients.
-
flow had been excluded the tendencies have bcen would probably given in Fig 17 emphasized. in even further When studying the influence of cavita- - tio 171
I
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Y-"
tfolt trust brftdown f SSPA
Leit tr upeft
Is
s e t i ..
tnl
22..
jo
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a !~
~~v
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(
°
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A __ 0.7
0.1
Fig 19
s p ee d t e s t
i.
with tip ctera bm
Call-
e l i m bM I
tclet*d beh,.i Ship mel
0.2
___
-2--
0.3
Influence o hull on the limit for incipient thrust breakdown TfcosQ0 7 - -.. G nV 7 eLc AD/A ttheo 0 - 112 1 /= / 0.7
__
ence on the advance ratio thrust breakdown, as shown in Fig 17. In Fig 19 some exerirental values obtained propeller similar to Cony 1.05 are also given. At these latter tests also a plate with clearance ratio 0.22 was mounted in tunnel. The agreement between these Splate tests and the tests with the complete ship model seems to be very good. The li for thrust reakdo n obtained in unbounded
Is
flow apparently give too small blade area ratios. This is also in accordance with the full scale experience given in 117]. The very small hull influence shown in cavitation conditions at Warp the extreme Fig 1? = 0.7 0.75, Con 0.50) is in agree-
tion number on the curves for propellers arp G.75 n Fig Conv 0 .7s c -: C..... ~(c 17 it is somewhat surprising to see the relations between cavitation numbers 0.94 met with studies of wall effects in 118] seems to be and [191. When discussing wail effects on larger at moderate cavitation extensions supercavitating and ventilated propellers c =0.94) than at the extreme cvitation mrgan in the suzmary report [20] states conditions at o = 0.7. This means that the that according to the theories of Tuln limits for thrust breakdown should be very [18] there should be no influence of wall much influenced by the hull. In Fig 19 the geosi3 of geosim -ov07 ieasuens on n ov05 aebe Cony sul of omeasurements e Results lte1.05, eff effects. loading conditions for propellers propellers in a cavitation tunnel given in Cony 0.75 and Cony 0.50 have been plotted [19] lead to the same conclusion. HowEver, in a graph showing the limit for thrust if the influence of the hull is considered breakdown for these propellers when tested to be a blockage effect then correction of in the high speed test section [161. Accordthe advance ratios (which means correction inq to this limit the loadings 30 Vnots for of the wake) should be more appropriate ConY 0.75 and 35 knots for Cony 1.05 should than correction of the thrust coefficient. not have thrust breakdown. However, when In Fig 20 this treatment has been used on working behind the ship model they did have -77222
0
the results of Fig 17. The hull influence given in Fig 20 is shown to increase continuously with decreasing cavitation number. The points for u = 0.7 and propellers Cony 0.50 and Warp 0.75 have, however, been left out of Fig 20. The reason for this is that they are situated in tha supercavitating region of the characteristics where the thrust coefficients are almost independent of the advance ratio. The ratio J 'J could therefore be anything from 1.0 0 .8. Treated in this herefore very difficult to tell way wh,
.
er.
.
.
10
.(.n rt
06
/ 1
-.
/06
o.9.2
is a large influence of wall percavitating conditions
Sueovtting proptUr AD/A 0 "0G
-
Z
i
r. he fact that the thrust coeffi* ;1ost indpendent of advance
h
Vaticat cwaranc, a/D 0.2 0
Z
--- onvnntio0ot propester AD/A o105 0
a o negligible influence of wall
effects. 4. INTERACTION ON A TRIPLE-SCREW SHIP
0.2
t
In this chapter some results from cavitation tests of a triple-screw :hip according to Fig 21 will be given. First the hull influence at single-screw cpera-
.
0.
06
I.a.
.
tion will be discussed and finally the
_
__,
propeller-propeller influence is shown.
Fiq 22
cavitating propeller at incieasing clearances. These results have, however, now been confirmed at the clearance ratio 0.2 At by tests with a complete ship model. the tests with the triple-screw ship also single-screw operation was tested as a comparison with the earlier .ested plate arrangement. These results are given in Fig 23. The agreement betwe- the test arrangements is shown to be very good at small and moderate advance ratios. At J 1.1 the tests with the complete ship model, however, give lower thrust and torque than
(2UD 12
t.2SxD
Influence of clearance on propeller thrust and efficiency obtained by tests with a plate arrangement
-
1280
the tests with tne place arrangement. The
results from measurements on the prototype, also given in Fig 23, are shown to give the same tendency as the tests with the complete ship model. 2 In order to confirm the results of Fig 22 also at large propeller-hull clearances some further tests with complete ship models are needed. Fig 21
Propeller arrangement on triplescrew model
4.2 Propeller-Propeller Interaction At tests with the ship model shown in Fig 2i identical wake fractions at atmospheric tests were measured at triple-screw operation and with each of the propellers alone on the ship model. With the same number of revs on all three propellers there was obviously no interaction between the propellers at non-cavitating conditions. For the cavitatinq propellers the interaction shown in Fig 24 was, however, measured. These results were obtained by
4.1 Influence of Propeller-Hull Clearance The influence of clearance on the propeller characteristics has been investigated at tests in the high speed test section with a plate located above the propeller [2, 9]. Results frot, these tests are shown in Fig 22. One notable observation at these tests was the very slow decrease of the hull influence for the super-173
:1
Model tests Plate orrangemint with rudders -- o-- Complete shipmolet with rudders -- a--1-only center proelter -
Fultscate '
-
Port propeller
tests
Measurements
,,,
on centerpropetter
Center propeller
-
Starboard propeller
"-'-_
_
c_ .2
-
6 -0.65
2N
0.9
-
_
_
_
__
_
_
_
_'
-- o-- Cay number iz0.8
KT 101K 0.6 0
--
-I
0-
-
-
=0.6
Filled points advance coeft J=0.8
1.0
0-
J=0.95
Open points:
br
CR
0.2I
.-
0
5. CO,9LUION
0.9
0.7Fig 0.9 Advance ratio
24 1.3 Ja
Interaction between propellers on triple-screw ship
5. CONCLUSIONS Fig 23
Compariqon of propeller characteristics obtained at tests with plate arrangement and at tests with a complete ship model
From the r'esults presented in the present paper the following conclusions may be drawn: 5.1 Working Conditions
*
comparing thrust and efficiency in triplescrew operation (KTB , nB ) with the thrust and efficiency measu ed wAen each of the prope.lers was working alone oa the ship model (KTB, n ) .•The propeller-propeller interaction in Fig 24 is shown to imply a further reduction of thruist and efficiency. The influence is larger for the wing propellers than for the centre propeller. It is also very much dependent of the advance ratio. In fact the advance ratio seems to be a more important parameter than the cavitation number. J = 0.8 with almost supercavitating conditions gives less influence than J = 0.95. This is a siwilar tendency to that shown concerning propeller -hull influence in Fig 17, where the heavily cavitating propellers were less influenced by the hull than the partly cavitating propellers.
o
o
Wake and static pressure at the propeller _.7ane are sensitive to ship speed and trim. Accurate loading conditions should be based on measurements of these. Useful results of theoretical calculations of wake and static pressure cannot be obtained unless the free surface is taken into account.
5.2 Systematic Propeller Tests o
o
o
Propellers of the supercavitating type give less risk of erosion, do not induce larger pressure amplitudes on the hull and demand only slightly more power than propellers of the conventional type. Propellers with highly warped blade shape induce considerably lower pressure amplitudes on the hull than propellers with symmetrical blade shape. The cavitation amplification of the pressure amplitudes induced on the hull was about 2 for "moderate" cavitation
-174-
. .. .. .....
g3
o
o
o
conditions and about 3.5 when complete thrust breakdown occurred. The hull influence on thrust and efficiency implies a reduction of up to 15% of the thrust coefficient and a reduction of up to 10% of the efficiency. The hull influence on the limit for thrust breakdown implies that too snill blade area ratios are chosen when using limits obtained in unbounded flow. The hull influence on the propeller characteristics may he a blockage effect. The influence at supercavitating conditions can then be considerably larger than it appears to be when thrust correction is used.
The hull influence on propeller characteristics seems to be remarkably constant at increasing clearances. Good agreement between measurements of tu.rust and torque with plate arrangement and with complete ship model. Propeller-propeller interaction implies further reduction of thrust and efficiency. Propeller-propeller influence is larger on wing propellers than on centre prooeller and is larger at "moderate" advance ratios than at small advance
5. blount, D L, Fox, D L, "Small Craft Power Pr.diction", Marine Technology, Vol 13, Nc 1, Jan 1976 6. Gawn, R W L, Burril, L C, "The Effect of Cavitation on the Performance of a Series of 16-Inch Model Propellers", Trans INA, Vol 99, 1957 7. Newton, R N, Rader, H P, "Performance Data of Propellers for High-Speed Craft", Trans RINA, Vol 103, 1961 8. Venning, E, Haberman, W, "Supercavitating Propeller Performance", Trans SNAME, 1962 9. Rutgersson, 0, "Supercavitating Propeller Performance. Influence of Propeller Geometry and Interaction between Propeller, Rudder and Hull", Proceedings of the Joint Symposium on Design and operation of Fluid Machinery, Fort Collins, USA, 1.978. See also SSPA Publ No 82, 1979 10. Johnsson, C-A, "Some Experiences from Excitation Tests in the SSPA Large Cavitation Tunnel", Trans RINA, 1979 11. Lindgren, H, Bjdrne, E, 'Studies of Propeller Cavitation Erosion", Proceedings of Conference on Cavitation, I Mech E, Edinburgh, 1974 12. Johnsson, C-A, Rutgersson, 0, et al, "Vibration Excitation Forces from a Cavitating Propeller. Model and Full Scale Tests on F High Speed Container Ship", Proceedings of the llth ONR Symposium on Naval Hydrodynamics, London, 1976. See also
ratios.
SSPA Publ No 78, 1976
5.3 Interaction on a Triple-Screw Ship o o o o
13. Weitendorf, E-A, "KavitationseinflUsse auf die vom Propeller induzierten Druchschwankungen", Institut fUr Schiffbau der Universitat Hamburg, Bericht Nr 338, 1976 14. Johnsson, C-A, "Pressure Fluctuations around a Marine Propeller. Results of Calculations and Comparison with Experiment", SSPA Publ No 69, 1971 15. Suhrbier, K, "An Experimental Investigation on the Propulsive Effect of a Rudder in the Propeller Slipstream", International Shipbuilding Progress, Feb, 1974 16. Rutgersson, 0, "Propellers SSPAFMV Series K131. Summary of Cavitation Properties for Propellers 3.105, 3.075 and 3.050", SSPA Report K131-24, 1974 (in
ACKNOWLEDGEMENTS The author wishes to express his gratitude to the Naval Material Department of the Defence Material Administration of Sweden for sponsoring parts of the present investigation and to Dr Hans Edstrand, Director General of SSPA, for the opportunity to carry out the study. Thanks are also due to those members of the staff of SSPA who took part in the investigation. Without their urgent work this paper would not have been possible. REFERENCES -1
. Reed, A, Day, W, "Wake Scale Effects
Swedish)
on a Twin-Screw DiLplacement Ship", Proceedings of the 12th ONR Symposium on Naval Hydrodynamics, 1978 2. Rutgersson, 0, "On the Importance of Rudder and Hull Influence at Cavitation Tests of High Speed Propellers", Proceedings of the High-Speed Surface Craft Conference, Brighton, UK, 1980 3. Taniguchi, K, Chiba, N, "Investigation into the Propeller Cavitation in Oblique Flow", Experimental Tank Laboratory, Mitsubishi Shipbuilding & Engineering Co Ltd, Report No 1800, 1964 4. Hess, J L, Smith. A M 0, "Calculation of Non-Lifing Potential Flow about Arbitrary Three-Dimensional Bodies", Douglas Aircraft Company Report No E.S.40622, 1962
-V
17. Blount, D L, Fox, D L, "Design Considerations for Propellers in a Cavitating Environment", Marine Technology, Vol 15, No 2, Apr 1978 18. Tulin, M P, "Suoercavitating Propeller Momentum Theory", Hydronautics Technical Report 121-4, 1964 19. van de Voorde, C B, Esveldt, J, Tunnel Tests on Supercavitating Propellers", Proceedings of the 4th ONR Symposium on Naval Hydrodynamics", 1962 20. Morgan, W B, "The Testing of Hydrofoils and Propellers for Fully-Cavitating or Ventilated Operation", Proceedings of the 11th International Towing Tank Conference, Tokyo, 1966
21. Johnsson, C-A, "Comparison of Propeller Design Techniquies", Proceedings of the 4thONR Syposium on Naval Hydrodya mics, 1962. SE also SSPA PubI No 52, 1963 22. Pien, P C, "The Calculations of Marine Propellers Ba".ed on Lifting-Surface Theory", Journal of Ship Research, Vol 5, 1961. 23. Nelka, J, "Experimental Evaluation of a Series of Skewed Propellers with Forward Rake", NSRDC report 4113 APPENDIX Geometry of Tested Propellers
and0.50 Cony
0.4
0.5 4
26
-IFig
03
02
01
01
0
02
03
04
05 15
1.5
1.6 2.0
Blade shape and ditributions of pitch and camber for propeller Warp and Co-376.5
I. xi=0
0.6___________
(010.0
.0, 0.75 o
0.2 Ig/0 Fig 27
11
0 ~ O1
I'A .1.5 1.6 0
0.2 If /D
P/D
.01 0.02 YMO~/O
Blade shape and distributions of pitch and face camber for propellers S.C. 0.50 and P.C. 0.50
The propellers with NACA 16 profile by the same been designed all with Shape have and the same rather unloadprocedure ed circulation distribution. The lifting line calculations were carried out according to a modified procedure for calculation of induction factors described in (211. The lifting surface corrections were then calculated according to Pien's method (221 programmed at SSPA. For propellers with symmetrical blades and no rake this design procedure gives very good estimates of pitch and camber. The propeller Warp 0.75 has the same chord lengths and circulation distribution as propeller Cony 0.75, the blades, however, being skewed 120 degrees. Further the skew-induced rake has been compensated by raking the blades forward about 60 degrees. In this way the clearance curve for propellers Cony 0.75 and Warp 0.75 are almost identical. When propeller Warp 0.75 was designed, however, very little was known about the effect of extreme forward rake on the propeller characteristics. The design was therefore carried out by the as for unraked propellers. slame procedure Later it has been shown that the effect of rake is to increase thrust and torque (231, which is not predicted by the design method. The propeller Warp 0.75 is consequently considerably overpitched at the inner radii. The propellers S.C. 0.50 and P.C. 0.50 have been deigned according to the procedure given in(8) with empirically derived pitch corrections according to (95. The radial circulation distribution is optimum. Propeller S.C. 0.50 has wedge-shaped sections according to (9) with a 3-term face combined with a 2-term modified thickness distribution. Propeller P.C. 0.50 represents a first attempt to improve the performance of the supercavitating sections .177.
at partially cavitating conditions. in this case a 5-term face is combined with an empirically derived shape of the back. In Fig 28 the free stream propeller characteristics are plotted in the following way. Propeller thrust and efficiency at cavitating conditions at the advance ratio J used in the behind condition have been related to thrust and efficiency at noncavitating conditions at the same advance ratio.
Covitation number a
102
0.7
Con 075/i
E 09
1/:5
S C 0.501
u
-
T_
I /,'i
co-
II /-.:
O'
'
0.7
"
I
I
ICfVOi
01~
/
I 105
•
nConv.
I
-:
.. ",,. .-
:2
0-9 - 0
-
0.7 I ,'Ai
'
Fi/8Efeto catatio Fig28ffeto
ontefe
onthefre (cton
stream propeller characteristics
A -178
L-
-
T-
found in the wake of merchant ships and secondly a decrease of the blade frequency pressures due to the thrust break down caused by the cavity. But in Fig.16 of the paper no value of less than 1.0 can be detected, at least for the conventional propellers. Can the author give an explanation for the cavitation amplification factor of this type ? Thank you.
Discussion K R Suhrbier(Vosef,Tornycof
.
I thank the author for his stimulating paper. Regarding Fig.17 I would like to say that comparisons of conventionally defined relative rotative efficiencies 11R can be misleading for inclined shaft arrangements extended shown to cavitating con~ditions The if Ychanges
A to'
should not
Au
be regarded as an 'exact' measure of propulsive efficiency changes, simply because the 'behind' thrust used is not the effective or net thrust; the effect of cavitation on the propeller normal force (due to If it not considered. inflow) oblique were included in is a more complete analysis it would either show up, as a somewhat compensating effect, in the hull efficiency '1 or in a different IR or Ky ratio, depending on the approach used. The data on the influence of shaft inclination (Fig. 18) is affected in a similar way. I have some difficulty with the author's findings on the propeller-hull or clearance influence. The effects shown in Fig.22 and in the author's earlier papers do not seem to agree with our experimental data. So far, we did not find (in several similar tests for different purposes) effects of this kind or magnitude. Further experiments will be carried out by us in the near future and they might help to clear up this discrepancy. A 20% reduction in efficiency, as mentioned in the paper, is not in line with our experience with high-speed craft. With reference to Fig. 12 I would like to add that it is not sufficient in our (and others') experience to compare just the pressure amplitudes of the tip position, because the peak pressures ma be further downstream depending on the extent of cavitation.
el
0 Rutgersson(SSPA) I thank Mr. Suhrbier and Dr. Weitendorf for their attention to the paper. I agree with the discussers that it is generally insufficient to compare pressure fluctuations measured in only one transducer. In this investigation actually 8 transducers were used. However since all transducers showed the same relation between the propellers the results from only the tip transducer were shown in the paper. Dr.Weitendorf seems to have misunderstood the purpose of Fig.14-16. The reason for extracting the amplification factor in Fig.16 is to give the magnitude of the effect the discusser is mentioning. The decrease in amplitude due to thrust reduction is shown in F'g.15 and the increase due to cavitation is shown in Fig.16. Mr.Suhrbier is correctly saying that the definition of q given in the paper only takes account for the forces in the shaft direction. An "exact" definition however means that forces must be measured both in struts and rudders and have not yet been considered. Finally i am very sorry that Mr.Suhrbier has not been able to find the propeller-hull influence in his investigations because I am quite convinced it should be there. The magnitude mentioned in the conclusions of the paper is perhaps somewhat provocative (15% reduction of thrust and 10% * reduction of efficiency). A closer look at Fig.17 will hc.ever show that these figures are valid for a 50% blade area propeller of conventional design on a 30 knots vessel. This is of course not a realistic propeller for this speed and I doubt if such a case exists in the discussers investigations. More realistic propellers of 100% blade area of conventional design on supercavitating propellers give a reduction of 0-8% in thrust and a reduction of 2-4% in efficiency. This effect is thus rather small but very significant and important according to the results of a number of projects tested at SSPA. A well defined testtechnique and a very good dynamometer is however necessary in order to measure this effect.
E.A Weitendorf (HS7A) In Fig.14 of the paper the influence of the tip clearance of the blade frequency pressures in the non-cavitating condition is given. By the way, these values can easily be calculated, In Fig.16 of the paper an amplification factor for cavitation is displayed. The first pressure impulse measurements in 1967 by Denny published and our measurements published in 1973 showed a decrease of the blade frequency pressures due to the thrust breakdown was found by using five pressure pick-ups in longitudinal direction parallel to the propeller shaft - a point already mentioned by the fore-going discusser. It is well known that there are mainly at least two physical phenomena; An increase of blade frequency pressures due to the displacement effect of the cavity as
*
In the preprints it was misleadingly stated 20% redsction of efficiency.
-'79-
-
- -~
--
---
Elam
Recent Research Results on Cavity Flows about Hydrofoils Alain Rowe Centre National de la Recherche Scontitique Institut de Mecanique de Grenoble Jean-Louis Kueny Unjversite Scientitique et Medicale de Grenoble Instlitut de MWcanique de Grenoble France
ABSTRACT P;SThe behaviour of a base-vented hydro-
tack the speed effects.
foil in a free surface channel is studied. The flow is simulated by a displacement surface outside which eddy effects, the influenc( of viscosity and air comoressibility are neglected. The model used to characterize the displacement surface is derived from Michel's model and takes gravity effects into account. The theory is developed according to a perturbation method which uses two small parameters : E to which foil thickness and incidence are proportional, and 8 the reciprocal of the square of Froude number. Unknown variables are expressed in the form of asymptotic expansions 0 + 8Xi)+E 0 +X 4E (X 0I IPc where coefficients such as X depend on the shape of the free boundaries. The classical resolution method consists in transforming the field of flow into a half plane, but consideration of the bottom and free surface leads to considerable distorsions which render numerical calculations delicate and requires a powerful computer (C.D.C. 6600, double precision). T'e contribution of the free surface is evaluated after stud:'ing its asymptotic bt:iaviour at infinity. The wake contribution "'hich is significant in the Oth-order proolem with respect to 13 , vanishes in the 1st-order problem. The simDlifying hypothesis consisting of replacing the w:ake by a displacement surface is thus justified. Comparison of theoretical with experimental results shows fairly good agreement :ith regard to - the effect of the ventilation number K on forces and on cavity length j - the variation in free boundary geometry with respect to angle of at-
iOMENCLATURE = length of chord measured on lower side (C=1) F = function characterizing the shape of the foil Fr = Froude number = Uao/Vg = gravitational acceleration coistant h. = submergence depth n = nominal water depth in the test section 2 h = h - h K2 = ventiiation number = (Pm -Pc)/pUw k = o / = cavity length P0 pressure on the free surface P, + p g hl Pc = pressure cavity Uc = free stream velocity U,V components of the speed ul,v 1 = components of the perturbation speed at the first order b,c,d,f,g = abscissae of the points B,C,D, F,G in the auxiliary t plane y = ordinate of the free surface Vc = ordinate of the cavity y = ordinate of the wake displacement y surface O = attack angle 2 1/Fr I C = small parameter characterising the thickness of the foil p = density of water. C
INTRODUCTION While a great deal of research has been carried out over the past years on supercavitating hydrofoils with unwetted upper side, the same cannot be said for
-*181 -
I
base vented hydrofoils which have received fairly little attention. This lack of interest would seem to stem from the fact that the first test made with these foils [1] to (3] indicated problems of flow stability and cavitation around the leading edge. In :act, these tests, which were carried out under unsatisfactory conditions, are not very significant and can hardly be used as a basis for understanding how a base-vented foil operates under real-life conditions. More recent tests, the results of which were published in 1979 [5] and 1980 [61, have shown that by choosing a suitable pressure distribution, and on condition that a wallmounted support system is used, thereby avoiding the presence of a vertical strut which disturbs the flow above the foil, a stability domain at high speed is obtained, always including the cavitation-free area. It is the development of cavitation on the upper or lower side of the foil that causes stalling. Thus, by using a wali-mounred foil of aspect ratio 2, Rowe obtained a stability domain of about 5 degrees for a cavitation number cr equal to 0.16. As far as the cavitation-free area is concerned, its extent is not negligible since, for a
vented hydrofoils. The model must be capable of restoring a geometry which is as close as possible to reality in order to take into account as accurately as possible the effect of the boundaries which limit the field of flow. This condition is particularly crucial when the foil is placed in a liquid with free surface since, in such a case the deformation of this surface, directly related to the cavity deformation, may induce a considerable angle of attack. With a supercavitating hydrofoil, the limitation is not as severe since the pressure acting on the upper side of the foil is known and, in addition, the slope dCL/d0 is half that of a base vented hydrofoil. Thus, the same error made on the value of induced angle of incidence AcK halves the error in lift in the case of a supercavitatinq foil. In the following paragraphs, the behaviour of a bse vented foil placed in a free-surface canal is examined in detail. From ti.e practical point of view, ths is perhaps the most interesting case since it allows consideration to be given to both tests a towinq tank Three and tes's a freesurfacein water tunnel. pointsin are exa-
cavitation number of 0.19, an angular deviation of the order of 1.7" is possible undew two-dimensional flow conditions. This value is slightly greater than the theoretical values calculated (-1.3^ - See ref.[43,[6] and [7)). Moreover, in view of the fact that the middle of the non-cavitating range coincides almost exactly with the ideal angle of attack, and that at this angle of incidence the lift of the foil remains approximately constant regardless of the operation copditions (variation in ventilation number, proximity of a free surface, wall effect), it is to be expected that basevented hydrofoils could in practice be used in certain cases. The detailed study of operation of such a foil is not therefore without interest. However, this study raises a certain number of specific problems which are not encountered in the study of supercavitating hydrofoils with unwetted upper side. For example, in the case of tests conducted in a tunnel, the correct measurement of the cavitation-free area requires special precautions in order to be sure of constant air content in the upstream flow. Moreover.
mined the effect of ventilation number on forces anv on the shape of free boundaries - the effect of a change in the angle of attack - the effect of speed. This study is based on the use of Michel's model [9]. previously presented by Rowe and Kueny [6), and modified to make allowance to gravity effects. A number of details on the calculation method are g;ven below, then the theoretical results are compared with the experimental results pu blished in [5] and [6]. -
1. CALCULATION METHOD The foil, with lower chord equal to unity, is placed in a channel at an immersion depth hl (Figure 1). The depth of water in the channel at infinity upstream is Y
-
i
Y
I(F)
.
the fact that the foil is held by a strut/ promotes separation of the boundary layer over the foil by causing the air in the cavity to rise, thereby modifying the value of the pressure coefficient. With a supcrcavitating hydrofoil, this defect is no longer apparent since the lower part of the strut that penetrates the cavity, and the upper side of the foil, are at a pressure known in advance. Other difficulties are encountered from the theoretical standpoint: the calculation must be non-linear, at least in the region of the leading edge. iow the non-linear theory proposed by FURUYA (8] for supercavitating hydrofoils with unwetted upper side falls into the case of base
-
I .'1
////////
/
//////////
I C ,OIJ oim ftc. h +h The foil is followed by a cavity wfosg length measured from the rear of the lower side of the foil has a value of J, and a wake assumed to extend to infinity. The shape of the foil, previously described
-182-
L3 -
-7
*i
by Rowe [5) is defined with regard to the axis Ux, by the relationship y'=EF(x ) where e is a small parameter characterising the thickness of the foil and F a function
K,
7,u, i Ryi,
~~--~
of order unity, except near the leading edge, wnich characterizes the shape of
I.&
8,
.Y
the
UM
of the Froude number, defined by choosing
c
X) -r
P.
The angle of attack is characterized foil. by number k such that o =(Ur,6-r')= 6k. f is taken to be the reciprocal of the square
E
F
Y.
u*" ,.
F -
-.
,1•
as characteristic length the length of the lower chord of the foil
.
:__ Geotntlryt
r!GU;ZE2 a
.U
m
A -
FIGuPE 2o
+ 7
qyf
=
-
hl
K
*u,(E)
= EK
-
.*c1s,
Y ys
=
Y y
cr (E
the displacement surface is defined by adopting the benaviour law :
r C
2E
e-
x -(( +1
6
)
.
where and z/ are two free parameters, and 9, is a parameter which is calculated in view of the conditions adopted to close the problem - zero perturbation at infinity
1I
c] (E)
- edge integrable singularity at the leading pressure continuity at tre trailing
(-)
edges of the foil - zero perturbation at infinity downstream - zero thickness of the displacement surface at infinity downstream. Outside the surface corresponding to the foil. its cavity and wake, perturbation speed u - iv is a harmonic function because it stisfils the hypothesis of irrotationality and incompressibility of a perfect fluid. It also satisfies the boundary conditions represented on figure 2a. These conditions bring into play the Froude number and the shape of the cavity and the free surface. Since y. and y are unknown, the variables are alculaid firstof all by assuming u -O on the free surface, and u40= KI/2 ln the cavity, thei the perturbat on di - to gravity is calculated by assuming that the previously calculated variables yj. and v are correct values. Thus successive so 1ions have to be found to two mi=ed boundary condition Hilbert pro-
(.5)
the problem can be considered as solved if coefficients a, v y K can be calculated. The fi 's)NrdeyctlluCi on of the problem in 5 is obtained by applying the first-order boundary ciinditions t, the flow boundaries corresponding to zero order (Figure 2a). The conditions of steady flow, incompressibility. 4rrotationality and inviscidity are not satisfied in the wake. As the perturbation produced by this wake cannot be neglected, the difficulty is overcome by assuming that there are two completely distinct areas behind the cavity, marked by a line representing a surface close to infinity, outside which the preceeding conditions are satisfied. This surface is identified with the displacement surface of the wake whose unknown shape is
~-
eol¢o 0-a
uostrea
W(
()
Yc
, z,
expressed in the form of the expansion
constant cavity length considered, ventilation number K is a function of E . The flow is then oisturbed by the presence of the foil followed by its cavity and wake. For a previously imposed geometry. i.e.. for fixed values of h , , and Y, and a given function F, the othir values characterizing the flow are expressed in terms of the parameter C . It can easily be verified that these values allow the following asymptotic expansions : = I -
'
3
When E is not zero, for the assuried|
U/U
f-
4 4i
length J being maintained constant wnatever the value ol small m os sE , the condi= Pparameter tion : limit Pc P mposes: 0 0),
o Int Zeto Ott-
(2)
On the cut representing the limit position of the cavity, when E-0. the cvity
0
-.
is
nfiity. The
the flow speed at upstream
ventilation number is designated by K K (P1,) pU
limi K fixed
wroes;-n
183-
_it
[
blems which are dealt with by the classical method. In the following, b, c, d, f, g
-.
IA-. - 1, 9
,(.
designate the absrissae of the points B,C,
u .2(i1, ,f.1) 4 D,F,G in the auxiliary plane t=r+is (Figure [(h,n2) 2b). Since bottom and free surface effects E()(u)I 2 are taken into account, considerable distorsions are introduced into the calculation plane t (for example, length of the upper wake image CB in the calculation plas',.].,,, ne makes the humerica1 for h, - calcOlation 1, h :IO) bothwhichA long A.(O+[ 0 ° "WI
= 10-14
f s-
-4,
C
and difficult, thereby requiring the use
of a high-capacity computer (CDC 6600 in double prcision).It is for this reason
r
'
-,
(pc__(j
.
-
0
-.
•.
,.
2B. ' K
.
.
e,
s" ,:5
D
,*. '.
.
where coefficients () correspond gravity.
~The
2
w
(e
.
T
...
0'1.0 -
" - 7
-
.
( I
)
(7)
el=
"°"
+O
(e
K : qKo
O
)a
y
I l~cyc +kK) -Ik
(yl
1
)0 -k(yj I
!}oy k(yc)
where coefficients -A and B depend on the wake boundary behaviour law [6]. Of course, a singularity appears at the leading edge since the tangent to the infinite slope, foil at this point has solution is then regularised according
(12) Quantities DR,R',T',S-,S',(II),(i2),(SI), (TI) have been defined in (61. Quantities (P3),(P4),(I5),(16) are defined in appendix.
to the previously described technique [7]. Taking into account the conventional symbolt as defined in the appendix as well as in [6), the un-formly valid solution of the problem with gravity is :
2. EFFECT OF THE VENTILATION NUMBER
.aft_
Altmann and Elata [l1] showed for the first time in 1907 that the lift of a basevented hydrofoil placed neir a free surface depends or the ventilation number. These autnors showed that when K- 0, lift increases. An explanation of this argument was given by Rewe ([41 and [7)) who also found that, before increasing, lift starts by -
.____
ob
t e
.
at infirmity is of solution The the asymptotic following form : h
+
:6
,
101o .
tion in a suitably chosen auxiliary plane.
eX#
.
--
6
The calculations were made for the case of p=h Irl < 1. The shape of the free surface is Obtained for the entire domain by matchinc the numerical solution calculated at finite distance, and the asymptotic solu-
B
;
,--
the position of the foil in the channel.
t=ex/2(h
o,) -( T'f
.
I
4
l
wake boundary is representedf by a displacement surface is thus justified. The calculation method, whicn consists in calculating the gravity effect on the basis of a gravity free solution, is justified by the fact that, for this problem, the behaviour of the free -urface is realistic for the hypothetical case of supercritical flow. Thus, the free surface is calculated as a jet line and cannot oscillate downstream of the obstacle. The value of the depth of water at downstream which is infinity is taken equal to h +h justifiable in view of the alovi-mentioned restriction. The effect of the free surface is calculated by making an analytical study of its asymptotic behaviour at infinity. This study is important as the convergence of the integrals in the second problem de0. The pends on the manner in which yJ expansion of yll to infinity de~ends on
at upstream infinity y at downstream infinity y
-I 1
Li
whe-eby the
~~-
AI-kB.
.1
that no attempt was made to determine the exact shape of the free boundaries taking gravity into account. The contribution of vity free problem, does not appear in the second problem. The simplifying hypothesis
"
]
21 f',=, .2
.
.
I
I
V
for this reasons decreasing. decrease are The still not very clear initial since,
when the length of the wetted part of the
0020
upper side of the foil is equal to the length of the lower side, the theory no longer agrees with experimental results. In this case, the effect of oressure acting on the rear part of the foil becomes difficult to analyse, and the loss in lift that is observed [4] is not a trivial result.
The theoretical results presented in
1
-
xi '
0
. 007
00150
// ,
-0_
_ ,
0
R 2
O0
$ [00100
o
o
h c'
E
ooo.
0000
U-
I/c z '
.
lU= . - 9 91 . s
It,
99, I.r '1119
0 175
+
a
--
.o ¢
.
. . 1u.- 9 9tM/" IF- - 11 9 iso-..So -0sh-
Z
two
subrgfence depts
-
obvious. Figure shows that the cavity length foreseen by the model SI is slightly too short in the case of small values of K. Figures 6 and 7 represent the free boundawith this model. rties that are obtained Figire 6 also shows that when cavityto length the increases, its thickness increases
om
ISo
0-"
0150 -
-0125 U
FIGURE 4. DOag coeffcrent versus vensioton number
'Is. 0
-Is,
M (fo
i
lm Cf 000392 z/0* O0 cs2 U 29c$3 1 A * . 99M/S 19 1f 119i h
h/'4
0025 0050number0075 0100 ventilation -. ( Poo P)
"=
0200
o0
Is
cient much closer to reality than that obtained with S0. However, in the case of
"
I
, h2h/C;2 I hi/cz2.t2/c 31
(13)
3, of it the is clear that this model From givesfigure a value lift coeffi-
i
I
hi
a
th- present paragraph have been obtained with gravity effects neqlected. The results obtained in high speed experiments (Fr 11.19) are compared with these theoretical results. The curves marked S- correspond to the double spiral vortex model proposed by Tulin. S corresponds to the model presented here, in which the following values were given to /a and z'
(S1 )
Ne
rear in a seemingly exaggerated manner. By changing the values of ,± and z/ , very different downstream behaviour is obtained (Figure 8). The correspording model is designated by S,
..
_0_4~
0-
0"0 000
0075 0050n.rbr o0025 vent,on K1 ( 0100 -15
3.Lacocflm-er v-vs -"F'GURE
(S2)
-
012 ....
eni,,cn.
012
u
.
7
;
6
(14)
lift with S 2 , a By calculating foil value nearly identical with that corresmodel is obtained (See ponding to th3 . This is because the Fioure 13. / : surface in the proximity slope of the f.is .ractically the same in both of the foil models (K fixed).
0150
015
3. EFFECT OF A CHANGE IN ANGLE OF ATTACK
er
The behaviour of
for wosubmetence etIhs
-
--
s.all values of K, the lift coefficient will be seen overvalued Asswllb is isoevle enblwbelow, and n despite the fact that the Froude number is high, the difference can be partly explained by a gravity effect. Figure 4 shows a very good agreement between experimental and theoretical results for the drag. The reason is that near the ideal angle of attack, the drag var.es very little. Thencirefore, 4t is not sensible to the d dence effect e induced by i surface b the free f upressure 3nd the models deficiencies are not -185.-
the foil ano the ca-
vity when the angle of incidence increases is studied. By reason of the procedure used the cavity (See Ref f4l to inject and [6]), air this into behaviour can oe studied a with a fixed air flow rate. When the cavity is long and the angle of attack not toolarge. a small variation in this angle has practically no effect on the pressure in the cavity. The cavity shortens and the same result as tnat shown on Fioures 9 and !I0 is found. When the cavity is short, the within varies more rapidly, and it is difficult to keep it constant by
4
141
cca
0
C-O08m
ty
Ex*IF 110 % S0 \A
4.31. cr' 2
h,/cv!
\ -
h2 /cz !Fr -11 19
001
0 03 0
002
I00 7
1
Ver~t~ots2K
P
P1~U
1Te
FIGURE 5. Cavity teng-. versus tntaatocn nureogor for tw S;;bmefgtnce depth%
0
m~-.-eI
Ccv.?) qeo're-r.
FIGURE 7
ccn
o
t*4*!.
5
A0
t
tih1 4
_2
0
FIGURE 6
2
Wiec
6
of
te
£
2
-
onc~~ 05 -.%t ?C
05"?!
0. 24
6
0.0223
8
0.0166
10
0.0134
14
16X~
varying the airflow because of the pulsing effect oftecvt.Any comparison with
A'
sment of the ventilation problem would necessitate a special study, and this goes teyond th, limits of this study. From the b tests carried out, some general ideas can be Oien the variation in the cavity length affected by modification in angle of attack is largely due to the change in the kinetic
___0-
I__s
in,
0.
0
~-
I
K 0
-
by the calculations although the conditions are not exactly the same. -this variation is also due to the fact that the charge in angle of attack modifies the conditions in which the air is evacuated with the flow. Tnus. for a fiair flow. if the angle of attack becomes large, the air in the cavity evacuates much more easily ;in this way, the pressure in the cavity decreases and
~xed ~ceo?~
V:GuE ce S.nt
F/
so does its lenotn. It seems that a rapid decrease in the cavity pressure is always b'5rCfui,~013appears at leadin ede when cavitation observed
101 4
-
~o
This is explairned by the fact that, after the onset of cavitation at the leading edge. the interface of the cavity becomes very turbulent and permeable, which confavours air entrainment. 1ccording to figure 10, a point wouid seem to exist on the free surface such that its position is independent of the angle of incidence. The result observed numerically has not been demonstrated analytically. it may perhaps be explained by linearisation problem.
5 5. o50sequently
A-093
.8
-3
-7-6.5-4
-1
4
ID .1.-2.3
of the flow. This is confirined
054-conditions
5.ofthe
EFFE:T Or SPEED &1GURE9. influence of t"e onqle of attock on
the cc"r
lent
it has generally beer. assumed that this 2ffect was due to gravity effects. Parkin ulin u'1 1 [1? and Street [13j have
K ie
__
T
_
_
T_
_
_
4
3 -3.
2
FIGIME 10
0
Infenct cl
-~-77
4
35*
6 7
2*30'
8
0261
10
0.-25s -1* 14*
1'24"
z :9
2
-
Me* ce.
of otc
S
a
on Mhe;ec.ne1r! ft,
-187-
-
_
$0
12
vten.8o1on lwf bet
16
14
K
*0
0158
SI
indeed shown that gravity can introduce significant variations in forces, while at the same time modifying the shape of the -free boundaries. From the theoretical noint of view, however, up till now no study has been carried out taking into account both the free surface and the gravity.effects, In the experimental field, Altmann and Elata have made a number of systematic tes.ts with a superdavitating wedge section, and they have obtained significant force variations. Previous tests made with basevented sections do not allow any concrete conclusions to be drawn. Figure 11 shows the variation in cavity length plotted against ventilation number for different values of the Froude number. It can be seen that for the tests corresponding to )9= 0.008, excellent agreement with the S2 model is obtained and this cannot be considered as pure luck since the coefficients /. and zv were adjusted to this effect. Even more interesting is the change in results when Je increases. For example, when 13 = 0.05, there is no longer agreement with experimental results. On tHe other hand, at this speed, if the S model is used to calculate the effect of gravity, excellent agreement between theoretical and experimental results is once more obtained. On the basis of these results, it can be assumed
_
to
Exp.
h 1 /C,1
(Se)
= 0.04/3
v= 0.04/)3 -I (15)
thus giving the model S 1+ 1 -x (16)
=2e
eys
From a practical point of view, the neces-sity of using a type (16) law, dependent on speed, to start off the calculation ofI free lines, means that the slower the speed, the more the model must give a thick cavity at infinite Froude number. This is quite coherent with the fact that the effect of gravity is to close the cavity. Figure 12 shows what is obtained with 13 = 0.089 using this new model. It can be seen that the results are in very good agreement with experimental results. Thus, the model S provides a good description of the flow for a Froude number range of
j
A $*0050
( =0' -
-
I
-
Exp
--=4
3,o08j 0O-0089
h2 /C4
hI /C
15
that speed has an effect on the flow, not only through direct gravity effects, but also through a wake effect which is itself influenced by gravity as well as by unidentified dimensionless parameters dependent upon speed. In order to harmonise the re= 0.008 and at 13 = 0.05, sults obtained at the value of coefficients / and v can be varied in relation to 13 according to the following relations
6
0002
o
Ep0
4
15-
0.050
5
=x.
h/3
S2 0 0.0506 9 (o
7
-5
N
S,,
.-
-
1 :
N
20
-
0.01
002
5',/3- 0050
--
*
t
J
0.
i5 to3
9
_
22'3-
,..4
-
003 004 0.05
0089
9
=00508
0.2
0.01
FIGURE 11. Influence of the speed onthe cavity length.
/
0.02
F:GURE 12..
0.1
0.03 004 0.05 0
0.2
Influence of the speed onthe cavity lengjth. 111-
__
il
__
-1880.0-
00 00
00
00
.102001
00
.0
0
.0
I-
P-'0050
-
-1
T -
.
.
hi~ I' 0.150
N
h,
4
F
UO
c O.08m
S S,
0.-J0.
U
0125 0.2
0.100
0.050
0.025
0.10
Ventilation number K-(P-P FIG 13.
Effect
Si and
S2
three to infinity. It would be tempting to use this model now to study the effect of gravity on forces. However, such a study nmly seem a little artificial as the model S is derived from a parametric adjustment. If the gravity effects on the lift that are obtained with the two models Si and S2 are then compared (see figure 13), it is found that these models give almost the same results. This is mainly because of the fact that: the contribution of that part of the free surface downstream of the cavity is almost negligible compared with the contribution of that part located above the foil the and two cavity. models Thus, $I and despite S 2 the the contribution fact that for
an angular shift of less than six minutes. This shift can be explained by experimental error. For example, the three-dimensional effects caused by the side walls may induce an angle of attack. CONCLUSIJNS In order be ableacting to make on a a correct baseforecast of theto forces
he ontibuionuse the S anwo S2 odes
of the downstream part of the free surface is very different, in neither case does this contribution count for very much. To calculate the effect of gravity on lift, only the model S2 wis used, in the knowledge that the model S would give very simiN lar results. Figure 14 shows that the calculated reduction in lift due to gravity effects is very close to the reduction in lift observed experimentally, at least for small ventilation numbers. When the ventin ihtetereia NEELnmbrilationflow experimental results do not agree, but it has already been indicated that when the cavity is small, the theory seems to have a number of failings. In addition, it seems ME that, in this case, the upper boundary of 97: the cavity tends to wet the rear part of the foil. After having taken into account the gravity effect, there is still a very slight difference between theoretical and experimental results, even for low values of ventilttion number. This difference, exprebsed as an angle of attack, represents
vented hydrofoil, the following experiment t a) From the experimental standpoint, foil to avoid the presence of a
-n
a wall-mounting system for the
support strut which disturbes flow srt tru wich ditresfo theo b) The theory must take into account
bottom surface, the test free section, the the cavity and of the the
AM
cavity wake. For long cavities, it is it also necessary to make allowance
P
fre g Independently of the gravity effect, the Independen wof tem gavty effect t probably linked law t(K). This effect on to an effect seem toishave rate would
R
mechanisms which do not come within the scope of the present study.
ACKNOWLEDGMENTS This research was supported by the Direction des Recherches, Etudes et Techniques, Group VI, division Hydrodynamique, under contracts n0 77/352 and 78/490.
-189-
7-
PU 2
of transverse gravity field on relation between lift coefficient and ventilation number
Comparison of the laws
-
/
4
L__
at
0
Fr
Fr 0.1t50
11 .1 (/3' 0.0O0 8. 95 /.012)
-
2
h.I at
02-
.-
_j
0
Ex
6.72 (-.022) 47C/0O5)a
s, P-.008
0.125
c0.08M
U
1..5
2
00.10F,008
0.125
.
S
2.3 0 0008
S2
13'.
0 012
S.30.022
0.100
A
S
6s
so00
0 075
L.2
0
0
0075
0050.050 Ventilation
5
number
K z
Pix-
0.100 1p
PC
I
FG1feto trnvregaiyfield on reainbetween lift coefficient and ventilation number Comoorisoi of theoretical and experimental results
REFERENCES
I-
fl) Christoffer, K.W., 1961, "Experimental investigation of a high speed hydrofoil with parabolic thickness distribution and an aspect ratio" , Langley Research Center, NASA Tech. Note D728.
[3] Lang, T.G. and DAYBELL, D.A., 1962, "Free 6urface water-tunnel tests of an uncambered base vented parabolic hydrofoil of aspect ratio one" China Lake, California Narweps Report 7920.
RASNICK, Johnson, V.E. ofanda high 1959, 14 'Investigation speedT.A., hydrofoil with parabolic thickness distribution" Langley Research Center, NASA Tech. Note D 119.
A.R., 1973, "Profils hydrody-L [4] Rowe, namiques bidimensionnels A base ventille en presence d'une surface libre", These doct. ing. 20 Partie. Institut de M~canique de Grenoble. -190-
=
-
~-
~~
=
--
7
[5] Rowe, A.R., 1979, "Evaluation study of a three-speed hydrofoil with wetted upper-sides" , Journal of Ship Research, Vol.23, N01, pp.55-65.
[9] Michel, J.M., 1977, "Wakes of developped cavities", Journal of Ship Research, vol.21, n04, pp.225-238.
[10] Altmann and Elata, 1967, "Effects of ambient conditions, the gravity field and struts, on flow over ventilated hydrofoils" Hyronatics, Tech. Rept. 605-1, Laurel tic.~
[6] Rowe, A.R. and Kueny, J.L., 1980,
"Supercavitating hydrofoils with wetted upper sides", Journal de Mfcanique. vol .19 n*Z Juin 1980.
[11] Parkin B.R., 1957, "A note on the [71 Rowe, A.R. and Michel, J.M., "Two cavity flow past a hydrofoil in a liquid dimensional base-vented hydrofoils near a with gravity', C.I.T. Pasadena California free surface :influence of the ventilation Report N*47-9. number", Journal of Fluids Engineering, haeo 95 "h Special section on cavity fows, Dec. 1[2]TlnHP, "Tesaeo pp. 4 65- 4 74.[1]TlnM..195 cavities in supercavitating flows" Ilyjronautics, Tech. Rept. 121-5. [8] Furuya, 0., 1975, "Non-linear calculation of arbitrarily shaped supercavita[13] Street, 11.1., 1965, "A note on grating hydrofoils near a40free surface". J. vity effects in supercavitating flow", J. Fluid Mlech. 68, pp.21- . Ship Research, March ,pp.39-45. -
4
APPENDIX 0.5
-1
1
C~[f(Ycj)O =f32
S22
=5C
1
1dr
r'dr'
)
1S24
0.5
0~
3 31
=5C
E(c)
_S
__
-'rdr'
0.
C!f 0
0 0.5
dr
~
33
0
C
J
CI
+
(
1
T
c
P
CIdr
r'2dr'
1
dr
,)b)
(b-rr)b
]' dr
co
+
([
Vib)0 T'
Jj
r'dr'] b-rC b
dr 1r-r
L23
-JfC
f [S
f3 P
T' b
L33
'r1d
b rr,
1-r
,f
0.5
(b-rr')(b-r')J
00,) T'
]r
r'%-r* (b - rr') ( b-ri)
b
0. [+_C
r.2 dr'
,+
T'
d0
0 oz
x--
c
d0.5
0.5
0
1d
(b-rW)(br4)J
+I
r'dr'l dr S2 CS34T' b-rr'J 1-r
L31
T'
d
~
______
L21
r' 2 dr'
-
C
dr rdr T'J b-rr'J 1-r
05 [(Yc
'(c)[Jf = [5 5 1
C
[
'
dr '2 r (b-rr') (b-r')jd
7
P3~ P4:
S2i--S23) -(S22-S24)-(M21 -L23) S-31- S33)-(S32 -S34) -(L3 - L33-)
(y
J T
r'dr'
C
b -r'
fJT'
Jf f
r'dr'
____
f
b -r'
d
fa
16:
cio
y
-1
c
T'
-
r'dr
-
f
b -r'
d
T'
__)o
___
I
b -r'
b
(yc1)o(
rd'i~ rdr
TJ
bVT
Y
-
-r
___
r b
I
-192-
Discussion _that
0. Furuya (Tet,a Tech,) According to the lift data obtained from the e':periment shown in Fig.3, the lift-to-drag (L/D) ratio of the base-ventilated hydrofoil is only in a range of 6 to 10. On the other hand, this value can easily reach 20 for the supercavitating hydrofoil having the sharp leading edge and 10 even for that with leading edge radius of 0.5% chord 1). It is understood that many hydrodynamic and structural limitations exist for the former type of foil in order to keep the upper foil portion wetted, whereas for the latter no such limitations are designers' concern. I would like to know, therefore, if the authors have tested such a foil that will provide a higher L/D value and can still maintain the upper portion of the foil fully wetted. If this is not possible, what is the reason for using such a lifting device which is more vulnerable to the flow environment than supercavitating foils ? The second question goes to the comparison of lift data between the experiment and theory. When the authors used the doublespiral vortex model, the apparent discrepancy existed over a complete range of cavitation number (see Fig.3). The authors modified the flow model by introducing a continuously changing pressure distribution from the cavity end point to the downstream infinity. The amplitude and slope of pressure change were arbitrarily determined by two free parameters v and v. They demonstrated in Fig. 3 that, with v = 1 and v = 0 chosen, a better agreement between the theory and experiment was obtained for large ventilation numbers K but not for smaller K's. This fact is quite easily understood if one pays attention to the cavity length. By changing the velocity distribution on the wake, it is possible to influence the flow around the foil if the cavity length is short (or for larger ventilation numbers) but such effect diminishes as the cavity length becomes longer (or for smaller K's). A question now arises as to the current theory's inability in accurately predicting the lift force for the smaller K's cases. It is a well-known fact that the analytical result for the cavitating flow is insensitive to the selection of type of wake model if the cavity length is large (or smaller K's). If there is a theory which is unable to predict the force coefficient accurately for smaller K's, one should realize that such a theory must have an inherent deficiency in its force prediction capability. It seems to me that the modification of the wake velocity with arbitrary amplitude ,and decay factors is just an ad-hoc remedylbut not an essential one. 1* s found in our computations 2) an the nonlinear theory with double-
coefficients for supercavitating hydrofoils under a free surface over a wide range of cavitation parameters. It is my suspicion there might exist a limitation of the linearized theory for this type of baseventilated flow near a free surface. I would like to ask the authors if they intend to apply the nonlinear theory to the present title problem in the near future. REFERENCES 1) Furuya, 0., 1975, "Nonlinear calculation of arbitrarily shaped supercavitating of hydrofoils near a free surface", Journal Fluid Mech., Vol. 68, Part I, pp. 2 1-4 0 . 2) Furuya, 0., 1975, "Numerical procedures for the solution of two-dimensional supercavitating flows near a free surface", Proceedings, First International Conference on Numerical Ship Hydrodynamics, October, at the National Bureau of Standards, Md. 3) Furuya, 0., and Acosta, A.J., 1976, "Experimental Study of superventilated finite aspect ratio hydrofoils near a free surface", lth Symposium of Naval. Hydrodynamics, sponsored by the Office of Naval Research, London, March.
W
Auhor's Reply A.R. Rowe,, eroe The first question of Dr.Furuya has three reply elements. First element When we have begun to study base vented foils with wetted upper-sides, many studies had been developed concerning supercavitating foils with unwetted uppersides. On the other hand, only few information was existing concerning base vented foils. It looked interesting to have more information. Second element If a flap is added on the rear part of a base-vented foil, it is easy to obtain a good subcavitating foil. With a true supercavitating foil this is more difficult. Third element : The results presented -on Fig.3 correspond to an angle of attack of zero degree. In those conditions, the flow is not well adopted. At the ideal- angle of attack, the lift coefficient reaches the 0.19 value and the lift to drag ratio is of the order of 17 for the ventilation number 0.025 (see Ref.[61). The second question is very interesting; in fact, it is the reason why we have developed this study. As at low ventilation number theoretical model gives all the same results, we have thought that he
-
4 L
gravity effects could explain the gap ob" served. This point of view seemed justified because when the ventilation number is low,
-1933-
_
the characteristic length to take into account to calculate the Froude number is the chord plus the cavity length. So at 10 m/s, for K - 0.025, -the cavity length is 7 and the characteristic Froude number is only 3.9. In fact, the calculation shows that the gap observed becomes ACL = 0.008 when we take into account the gravity effect. This ACL = 0.008 corresponds to AacS5, which is very small. I think that this Aa - 5' is not due to the linearized theory.
In fact, this theory is well adapted to base vented foils which always are used with very little angle of attack. On the contrary, it is quite possible that the experimental conditions give a discrepancy of
influence of the side walls this order giving three-uimensional effects is not negligible as well as influence of the level of turbulence of the tunnel, measurement of the true geometrical angle of attack. It is much easier to test true supercavitating hydrofoils : as the flow is Very-perturbed the causes of error are minimized. Moreover for the same Aa, the- CL is twice more little with a true supercavitating hydrofoil.
t
Concerning the last point, a numeri-
cal non linear method has been developed, the method is very efficient,-but it is difficult to compare because the closure conditions are not the same. Wc will tryin the next future.
LI -41
iI !A
.194-
I --
.#
oF
Boundary Layer and Cavitation Studies of
NACA 16-012 and NACA 4412 Hydrofoils Jan H.J. van der Meulen Netherlands Snip Model Basin Wageningen. The Netherlands
ABSTRACT Boundary layer and cavitation phenomena on two hydrofoils are studiea in a high speed water tunnel provided with a rectangular test gectiop. The boundary layer is visualized with a tracer, using an in-line holographic technique. Depending on the Reynolds number and angle of attack, typical boundary layer phenomena are observed, such as laminar separation at the leading edge or downstream, transition to turbulence of the-separated shear layer with or without reattachment, and transition to turbulence of the laminar boundary layer. Cavitation phenomena are studied visually, photographically and by means of in-line holography. It is found that the appearance of cavitation on both hydrofoils, such as sheet, bubble and cloud cavitation, is related to the viscous flow behaviour.
C Bc 0 v p a o ad
1. INTRODUCTION During the past decade, sea-ral studies have been devoted to the relationship between the boundary layer flow about axisymmetric bodies and ca.itation. Using the Schlieren method for boundary layer flow visualization, Arakeri and Acosta [I] were able to relate cavitation inception on a hemispherical nose with the occurrence of a laminar separation bubble. Further studies by Van der Meulen [, 31, Gates (4] and Gates and Acosta [r revealed the influence of polymer additivies on laminar flow separation, and hence on cavitation. Arakeri and Acosta (6] classified axisyinetric-bodias into four groups, depending on the viscous flow behaviour. In each group characteristic types of cavitalion are found. The authors indicated similarities with practical flow situations. Most applications, however, refer to the flow about foil sections. It may be inferred that the viscous fiow about foil sections is related viith the section shape, the angle of attack and the Reynolds number. It seems questionable whether all viscous flow regimes found on hydrofoils are found also on axisymmetric bodies. Therefore, a classification into groups can better be based on hydrofoils. The aerodynamics of wing sections hasbeen studied at great length. Studies of the boundary layer flow are mainly concerned with the influence on lift and drag forces.
NOMENCLATURE C, C, mi CPT T P P __~
Angle of attack Zero lift angle of attack Kinematic viscosity Liquid density Cavitation number Incipient cavitation number Desinent cavitation number
Pressure coefficient minimum pressure coefficient Pressure coefficient at transition
Static pressure Undisturbed static pressure in test sction Pmin Minimum static pressure P Vapour pressure Re Reynolds number V Undisturbed free stream velocity in test section c Chord length of hydrofffil x Distance along chord xC Position of leading edge of cavity x Position of laminar separation XT Po sition of transition -.
-195-
E
1
In general, these studies do not provide adequate data that can be readily used for corresponding cavitition studies. This implies that the relationship between viscous effects and cavitation on hydrofoils can only be assessed by performing both flow visualizatii and cavitation experiments. A first and iwpurtant example of this approach is due to Casey f7]. He measured the position of a laminar separation bubble on a NACA 0015 hydrofoil in a wide range of 1ngles of attack by means of an oil-film technique. Thir position was compared with the position of the leading edge of attached cavities at inception and an excellent agreement was found. Similar experiments were made by Blake, Wolpert and Geib 8' with a Liebeck N112 h.drofoil. For the untripped hycrofoil the position of (attachex) bubble cavitation was found to coincide with the position of a laminar separation bubble, whereas for the tripped hydrofoil travelling bubble cavitation occurred. Van der Meulen -9. rela'ed the appearance of cavitation on a NACA 16-012 hydrofoil at angles of attack of 7° and 10 with the boundary layer flow. A holooraphic technique was used to visualize the boundlayer. The present stud. is anxtension of this work and is aimed at classifying cypes of cavitation on hydrofoils and
the hydrofoils, in-line holography has been used. Visualization of the boundary layer was effected by injecting a sodium chloride solution from a small hole (dia. 0.21 mm) located at the leading edge of the hydrofoil. This boundary layer flow visualization technique, in which a liquid is injected with an index of refraction sightly different from the surrouiding liquid, and in which the three-dimensional flow pattern of the injected fluid is recorded in a hologram, has been used earlier by Van der Meulen 2, 3 to visualize the boundary layer flow on axisymmetric bodies. However, in the case of axisymmetric bodies, the image of the body on the hologram is a single line giving the contour of the body as illuminated by the laser beam, whereas in the case of twodimensional hrdrofoils, the image of the hydrofoil on the hologram is a single line representing the whole surface of the hvdrofoil as touched by the laser beam. Hence, small deviations in the direction of the laser beam relative to the surface of the hydrofoil will mask the surface and thus mask the boundary laver flow. Such deviations are difficult to eliminate. Thi len was overcome by slightly cur-ina backward the nose line of the hydrofoils and slightly decreasing the section thicknesz further downstream toward the side walls.
relating them with the viscous flow behaviour.
Thus the chord lenath at the side walls was only 68 -m. In this way a center section was -itained with a chord length of 70 m suitable for flow visualization. initially, the chloride concentration was 2%, cf but sodium in subsequent te:ting a Concentration
2.MCEINENIL ETOS :LDPROCEDURE 2.1 Test Facility And Hydrofoils The experiments were made in the high speed water tunnel of the Netherlands Ship Model~~~~~~ A ealddsrjino Bain Model Basin. A detailed description of this tunnel with its air content control system is given in [10i.For -he present study a new test section was used. It has a 40 mm x 80 mm rectangular cross section and the maximum water speed is 40 m/s. TWO hydrofoils were tested. The first one is the NACA 16-012 hydrofoil, which has a symmetrical profile shape and a leading edge radius of 0.703% of the chord length. The second one is the NACA 4412 hydrofoil, which has a 4% ca r and a lading edge radius of 1.58% of thn chord length. Both hydrofoils had a chord length of 70 mm and a span of 40 mm. They were both made of brass.
A -
One of the reasons for selectinq the NACA 16-012 hydrofoil is that the same section is used in a comparative cavitation test program initiated by the International Tow.ing Tank Conference Cavitation Co=mittee. The NACA 4412 hydrofoil was selected because this hydrofoil had been subjected to an extensive cavitation test program at the California Institute of Technology, as reported by Kermeen NI"2.2_ffolographic Method To obtain detailed information on the cavity type and the boundary layer flw on -196-
5
I -
51 was used. The ratio of the injection to tst t velocity was usually around 0. 1. In-ection was made a pune neto a aeb ~ner moving witl, an adjustable speed in a cylinder. A schematic diagram of the in-line holographic set-up is shown in Ficu-e IA raby laser (A = 694 nm) with a 30mJ single mode pulse duration of ]C ns was used as a liaht source, serving a plane parallel beam of 60 mm diameter with the aid of a telescopic system. Agfa-Gevaert 8E75 Holotest plates (5000 lines/m-m) were used as a re cording medium. -econstruction from the holograms was made with a 2 mW HoNe laser ( 633 nmi. The reconstructed image wa studicd with a microscope: the applied magnification was usually 40 x. Further details on the holographic procedure and sresolution are Viven by Van Renesse and Van der Meulen 112 . 2.3 Procedure The tests performed in the high speed tunnel comprised holographic recording£ o the boundary layer flow and-of cavitation, cavitation inception measurements and photographs of cavitation. Theze photographs were made with a camera mounted vertically above the hydrofoil so that its field of view through a plexiglass-window in the top of the test section covered a length in chordwise direction of 40 =m and a width in spanwise direction of 27 mm. Prior to each
1
LASER
PULSO
~IBY ~Table
2 Test angles of attack for Holographic recordings
meas. and photographs
IMIRRORio
0 20
FL40 IJft.CIONY
GASWINDOW
60
60
0
80
00
160
160
0
to the geometric ancle of- attack.
-refers
-ale of attack
1 S-'iematic diagran of in-li,.e holograp hic system for making holograris of cav itation and 'fowm'encmena on hvdrofoils.
series of tests (at a certain angle of attack), the hydrofoil was cleaned and the tnel refilled. To adjust the air content, wat-r was vassed through the deaerat4 te circuit for a period of 1.5 h at a constant Dressure in the do'acration tan,.. All tests were made at-a constant air content of 3 o air per liter of20 about 4.6 cm~/ 1 water at S'IP correspo*nds to 1.3--5 puma bv 1.eight). The watex tenerature-was usually 0 around 20 OC. However, to extend nheRenolds number range for ,o neto esrmns at the highest test spxerd the temperature was allowed to rise to a m~aximum value of about 3 0 A survey of all angles of attack at which tests were made is given in Table 1 ad 2. 7he angle of attack in these tables
P'--
:.n.r
31 'NACA 16-02- Hy.drofoi The 1bounar- layer flow on the s:Ctin side of --he AC-A 16-012 hymvdrofoil has been studied for thse following an-eles of at-tack ('
,
.10ndI.Snc
3 anzeaed to be a critical ancle-. same additional observations were =7de for 2 2. observations were ftade I 1 - 20 m/s, where V_ is the the r an ge t el4t i tra undi t~ec 0e est sectic- At low speeds, a lam-inar boUnd-
Table 1 Test angles of at-tack for NACA 16-012 hydrofoil. Inception mecas. and photographs
Holographic recordings 0
0 40
30
3
60 0~) 10
.0
0_
__
______
Theflw
10
-
_
_
_197-
_
right. x 1c= 0 _70 .
~to _
_
The
and the pressure
distribution about the !-vdrofoils are affected bv the restricted4 !'Aght of the tet section. N~o attenot is made to make an!corrections- The ceometrc comi~f5o the flow Visualization and ca-vitation tests were tne same, and thus those results Can be comnared with~out anv restriction.
CAER.OOGRAPNICPLATE
r
40
8 100
______________effetv
F..
40
_
__
s fomle
where c is the chord length of the hydrofoil (c = 70 z=), and . the kinematic viscosity. Lamrnar sevazarion occurs at /c = 0.70, where x is the d-stance along te chord. Further downstream, transition to turbulence occurs, as shown ,±n Figure 3. The postion of transition is at x /c = 0.96. The positions of separation or transition for all - 2 are plotted in holograms taken at Figure 4. Below Re = 2 x 19, the mean value of the pcsit-on of senaraton is
-II
(Xs/C) 5
= 0.70.
Cp till Re =
x 10
lai-
•ar separation is still present, but above Re - 5 x 10 a laminar boundarv layer is abserved with normal transition to urbulence. In this and all subsequent figures, the position of transition to turbulence refers to tne .nosition of the first eddy and should thus be d's-ingu-shed from the . prositi -on" of c lnstabilztt which refers to r.e Fig.
:
1Tran.sit-on to -rbulence of !aminar eshear 3aer on N.CA 16--12
p>osition of the first wave.
=Vo='G -:t. XT =096. oundwi ' lan~nr separatzo ='" 'a"-- : rd::nt o a followed zby tran-sitzon to turbuience V the sepearaed shear :a.er o0o raP of *he reconstruCted imace shy.4:nl !am-nat sepiration is presented allC fot0m- U;--aa rom left -: rignt. T-V Reyrnolds n'er is u.72 -1
as
_
=rVRe7--
.e
F. -
Laminar separation on NACA 16-012
hydrofoil for a=30 and Re=0.Sx10' t, 97. "XS/ (Vo
0
°.
*t"~z
'
Fig. 6 Transition to turbulence of laminar
senarated shear layer on NACA 16-012 hydrofoil for a=30 and Re=0.8xl05 {Vo=1.15 m/s). xT/c=0.843.
-ig. 4 Positions of separation and transitio- versus Reynolds number for NACA 16-012 hydrofoil at 1=2 °. -198-=
-
_
-_
H
For a 30, a laminar boundary layer is found at Re =0.8 x le~ (V =1.15 M/s), followed by laminar separation, and transition to turbulence far downstream. Photographs of Jaminar separation and transj~tion are shown in Figures 5 and 6, with x /c =0.697 an~d x /c = 0.843. Observations mAde at Re = 1.47 x 105 (V = 2-0 mis) showed the presence -ifa snort lamainar separation bubble, as shown in Figure 'A.Data
_____
. .. .. .VnFigo -
____Re _____Figure
oni the positions of separation and transition are plotted in Figure 8. The mean value of the position of lan-inar separation for the short bubble is (7 /cj = 0.0061. Some observat-ions of ~he goundary layer fow were made at =2.90. The quality: of' the holograms in this series was rather poor and accurate data were hard to obtain. However, the main features of the boundary layer could still be distinguished and it was fotind that aboDve V0 = 4 =n/s a laminar boundary layer eitdwith nomltransiti-on to turbulenz-e. The absence of laadina edge separation (short bpble ana 0dt have a arge influence on cavilaina or~ 0 ala~arseaation,babl will be discussed later. is found up till Re z5 x 10D. Pn ex:anleA at Re = 0.731 x W0 (V,= 1.0 m's) 1s shownre 9. Ab-ove Re v 5 Ialmnar baparation wa~s no logrpresent, sInce turbulence began-earlier. An example a = 7.6 V 10~ 'V = 12.0 --1s' is shown in 1.Data on the uositxons of senara-
and transition are plotted in 7F.gure I1I.
_______tion
Fig. 7 Lam iar seoaration bubble on NACA
16-0 12 hivcroloil fo
A=30 an'd
Fi.
0
0
Fig. 8
Lam.i-ar senara-4o- bubbl- on ktACA 16-^012 nVdrofoil for a=5 and ..
01 2
0
2
3
5
4
6
Posi tions of separation and transition versus Reynolds number for NACA 16-012 hydrofoil at oL30 .-
Fig. 10 Transition to turbulence of laminar boundary layer on NACA 16-012 hydrofoil for *=S and Re=7.6x1 0 b (V,=12 .0 m/s).
.199.
_
_
__
_
_
___I
0 1PARATON 0 TRANITION
3. 0,
1
1.
I
-
0
Fig. 12B Continued trom Fig. 12A. PhOtOgraphs show overlap. 0
S 1 a
S
S S 1I-A
*
40
5 REYNOLOS NUMB.ER,.Re -
Fig.
11 Positions of separation and transition versus Reynolds number for NACA 16-012 hydrofoil at a=5 0 .
In some holograms the precise position of separation was difficult to establish, but separation did occur. These cases are marked with '"S". For a = 70, laminar separation occurs at the leading edge, followed by transition to turbulence of the separated shear layer and, although hardly visible and apparently being a limiting case, regttachment. An -xample at Re = 0.76 x 10 (VO = 1.15 m/s) is shown in Figures 12A and 12B. These
tions of separation and transition are plotted in Figure 13. The mean value of the position of laminar separation is (3z/C)n= 0.0035, which is very close to the va ue mentioned in [9]. For a = 100, laminar separation occurs at the leading edge, followed by transition to turbulence of the separated shear layer without reattachment. Data on the positions of separation and transition are plotted in Figure 14. At the highest speed tested (V = 8.0 m/s; Re = 5.73 x 105) transition to turbulence occurred on the nose of the hydrofoil, prior to the position of separation. The mean value of the position of laminar separation is (7 /c) 1 9 = 0.0030, which is close to the value mentioned in [9] i
'
photographs were made from the same hologram and have an overlap. Data on the posi-
,
I
0
SEPARAION
*
TRANSITION
36
4-
0'
mI.
00 0
1
2 REYNOLDS
Fig. 12A Laminar separation and transition
0
0
3 NUMBER,
4 .10-
Fig. 13 Positions of separation and transi-
to turbulence on NACA 16-012 hydrofoil for a-70 and Re-0.76x10 5 (Vo=1.15 m/s).
tion versus Reynolds number for NACA 16-012 hydrofoil at aw7 0 .
-200-
6-A
0 SEPARAION 0
TRANSITION
0 ------00
i~~~i
~RENO "
=
-
0 -5 S NUN4ER Re 1,,
-"-- "
Fig. 14 Positions of separation and transi~~tion versus Reynolds number for
so
NACA 16-012 hydrofoil at a=100
Fig. 15B Continued from Fig. 15A. Photographs show overlap.
The behaviour of the boundary layer for a = 120 is very similar to that for a = 100. Laminar separation is observed, followed by an unattached free shear layer. A clear example of this phenomenon is shown
in
Figures 15A and 15B (V
= 0.73 m/s;
Re = 0.50 x 105). These pRotographs were made from the same hologram and have an overlap. Data on the positions of separa-
0
4
tion and transition are plotted in Figure
SEPARA!,ON
.s,,o
16. The mean value of the position of laminar separation is (RS/c)12 = 0.0030. 3.2 NACA 4412 Hydrofoil
3 -
The boundary layer flow on the suction 0
?0
011 02
A
6
RYNOLDS FUMBER. qe x 10'
Fig. 15A Laminar separation and unattached free shear layer on NACA 16-012 hydrofoil or a=120 and Re=0.50xl0g (Vo=0.73 m/s).
Fig. 16 Positions of separation and transition versus Reynolds number for NACA 16-012 hydrofoil at a=120.
-201-
80
I
0 SEPARATION
* TRANSITION
"0
0 401.
Fig. 17 Laminar separation on NACA 4412 5 0 hydrofoil for a=2 and Re=0.72x10 (Vo=1.0 m/s). Xs/c=0.506. side of the NACA 4412 hydrofoil has been studied for the following angles of attack o: 20, 40, 60, 80, 100 and 160. For a = 20, observations were made in the range V0 =1 - 20 rn/s. Up till V0 = 4 m/s, a laminar boundary layer is found with laminar separation far downstream followed by transition. An example of laminar separation at Re = 0.72 x 10D tVo = 1.0 m/s) is shown in Figure 17. The position of separation is at xs/c = 0.506. At higher speeds, a laminar boundary layer is observed with normal transition to turbulence. An example of transition at Re =14.1 x 105 (VO 19.9 m/s) is shown in Figure 18. The position of transition is at 45 xT/c = 0. 7. Data on the positions of separation and transition arg plotted in Figure 19. Below Re = 2 x 10 the mean value of the position of laminar separation is (Rs/c)4 = 0.509. The behaviour of the boundary layer
Fig. 18 Transition to turbulence of laminar boundary layer on NACA 4412 hydrofoil for a=20 and Re=14.1x105 (Vo=19.9 ms). X/=0.457.
0
2
6
S
10
REYNOL.DS NUMSER.
12
14
Re10-
16
18
5
Fig. 19 Positions of separation and transition versus Reynolds number for NACA 4412 hydrofoil at a=20. for a = 40 is very similar to that for = 20. An exaMple of laminar separation at Re = 0.63 x 10 (V = 0.89 m/s) is shown in Figure 20. The position of separation is at Xs/c = 0.427. An example of transition to turbulence of he laminar boundary layer at Re = 4.25 x 10 (V = 6.0 m/s) is shown in Figure 21. The position of transition is at XT/c = 0.480. Data on the positions of separation and transition are plotted in Figure 22. Below Re = 2 x 105, the mean value of
_
Fig. 20 Laminar separation on NACA 4412 hy0 drofoil for =o4 and Re=0.63x0 5 (vo=0.89 ms). X /c=0.427. -
-202-
__
-
_
-
A'
__
~ ~ J-
'~"
-
~ --
---
--
.-
~
-.
.
-=--____
___
___
___
JA
MOW
Fig. 23 Laminar separation on NACA 4412.hydrofoil for a=60 and Re=0.79x10 (Vo=1.0 m/s). xs/c=0.37 5 . shear layer. The positioz. of transition is
Fig. 21 Transition to turbulence of laminar boundary layer on NACA 4412 ydrofoil for a=40 and Re=4.25x10o m/s). XT/C=0.480. ~(Vo=6.0
at XT/c = 0.558. Data on the positions of separation and transition are plotted in Figure 25. Below Re = 1.75 x 10', the mean value of the position of laminar separation is (x /c)4 = 0.386. Por a = 80, observations were made in the range V o = 1 - 9 m/s. Up till V = 2 m/s, a laminar boundary layer is found with laminar separation downstream, followed by transition. An example of laminar separation at Re = 0.63 x 105 (V = 0.89 m/s) is shown of separation TheThe position in photograph in Figure is at Figure Xs/c = 26. 0.333.
the position of laminar separation is s = 0.426. The behaviour of the boundary layer for a = 60 is also very similar to that for E = 2. An example of laminar separation at Re = 0.79 x 105 ," = 1.0 m/s) is shown in Figure 23. The position of separation is at x /c = 0.375. The photograph in Figure 24 wAs taken from the same hologram and shows transition to turbulence of the laminar free -r
80
27 was taken from the same hologram and shows transition to turbulence of the laminar free shear layer. The position of transi tion is at x /c = 0.523. At V = 2 m/s, the position of laminar separation has moved to the leading edge, whereas transition to turbulence is still located far downstream. Hence, in this case a long separation bubble
o SEPRAYON
STRANSITION
*
0*S
60- -
40
20-
Z0
20
0 =
=
-:a-
2
4
~0
12 8 8 Re 10 REYNOLDS NIUMBER.
14
18
le
~
MIT
inn~ 429
Fig. Fig. 22 Positions of separation and transition versus Reynolds number for MACA 4412 hydrofpoil at a=40
-203-203
24 Transition to turbulence of laminar separated shear layer on MACA 44125 hydrofoil for a=60 and Re=0.79x10 (V =1.0 mis). x Tic=O. 5 5 8 .
2N X
MI
I
o
SEPARAION
6 TRANSITION
0
28. The mean value of the position of the leading edge separation is /0ic 8 = 0.0233. For a~ = 100, a long bubb e is found at low speeds. An example of transition to
-1
I
27 Transition to turbulence of laminar separated shear layer on NACA 4412 5 hydrofoil for ax=80 and Re=0.63xl0 (VO0.89 mis). xTi/c=0. 5 2 3.
Fig.
20
turbulence of the laminar free shear layer = 0.89 mis) is shown
-~_________________________
2 RENOD
7
. UU B.
a
at Re =0.64 x 105 (V
I
in Figure 29. The posiltion of transition is at xT/c =0.405. When the speed is in-
Q
Fig. 25 Positions of separation and transition versus Reynolds number for Ni\CA 4412 hydrofoil at a~=6 0 . %01_
is found. In the hologram, the position of separation was difficult to locate since the bubble was very thin. At increasing* speed, the position of transition moves further upstream, so that at the highest speed tested (V0 = 8.8 mis) a short bubble is found. Data on the positions of separation and transition are plotted in Figure
0
SEPAATIO
*
#A$1 C
0 AM
V~
0
0
~0
Fig. 26 Laminar separation on NACA 4412 hydofol or 8 0 ndReO63x0 (V10 =0.89 mis). xsicO0.333.
5
0
0
Fig. 28 Positions of separation and transition versus Reynolds number for NACA 4412 hydrofoil at a=-80 . .204-
aA -
-
~
-~~------
_
-.
-
T
~
--
~=-
-.
-
-~-=-
=-
~ ~
M
*
0
SEPARAIMON TRArSMO1
Fig. 29 Transition to turbulence of iam:nar separated shear layer for long bubble NACA 4412 (V,=0.89 hydrofoilm/s).--:for x=100 and on Re=0.64x105
'
XT/c=0.405.f creased, the position of transition moves suddenly upstrea, close to the l; ading edge, thus creating ashort bubble. An example at Re = 2.0 x 10' (V = 3.0 m/s) is shown in Figure 30. The positions of separation and transition are at Xs/C = 0.0169
0 *
95
and XT/C = 0.0'47. Data on the positions of
9
9
separation and transition are plotted in rigure 31. The mean value of the position of the leading edge separation is (Rs/C)1 /C.). 4= 0.0164. For a = 160, laminar separation is found at the leading edge, followed by transitin to turbulence of the laminar free shear layer, and reattachment. Apparently, this angle is a limiting case since reattachment could not be distinguished in some of the holograms. An example of laminar separation at Re = 0.62 x 105 (V = 0.89m/s) 0 is shown in Figure 32. Anot,,er example at Re = 3.47 x 10t (V0 = 4.9 m/s) showing the whole bubble is presented in Figure 33.
Fig. 31 Positions of separation and -ransition versus Reynolds number for NAC,, 4412 hvdrofoil at A=JO.
Fig. 30 Laminar separation bubble on NACA 4412 hydrofoil for a=10 ° and Re=2.0x]0 5 (Vo=3.0 m/s).
Fig. 32 Laminar separation on NACA 4412 hydrofoil for a=16 0 and Re=0.62x10 5 (Vo=0.89 mIs).
-20i.
.
reference to the relevant hydrofoils and angles of attack, are listed below. A. Laminar boundary layer with, far downstream, transition to turbulence. NACA 16-012 hydrofoil, a = 20 NACA 4412 hydrofoil, a = 20, 40 and 60. 3. Laminar separation at leading edge with subsequent transition to turbulence and reattachment (separation bubble). O NACA 16-012 hydrofoil, a = 30 end 5 NACA 4412 hydrofoil, ai 80 and 100. - -4 -
C. Laminar separation at leading edge with subsequent transition to turbulence with or without reattachment (transitional case) NACA 16-012 hydrofoil, a =7 NACA 4412 hydrofoil, I = 160.
~ Fig. 33 Laminar separation bubble on NACA 4412 hydrofoil for a=160 and Re=3.47x10 5 (Vo=4.9 m/s).
D. Laminar separation at leading edge with subsequent transition to turbulence without reattachment (unattached free shear layer)0 NACA 16-012 hydrofoil, a = 100 and 12
Data on the positions of separation and transition are plotted in Figure 34. The mean value of the position of separation is (xS/c)1 4 = 0.0060. 3.3 Analysis In analyzing the results of the boundary layer flow visualization experiments for both hydrofoils, some typical similarities are found. When the boundary layer flow phenomena at very low Reynolds numbers are disregarded, the boundary layer flow behaviour may be classified into four groups. Descriptions of the observed behaviour, with
*
5
-
0
0 S
0
0
The zero lift angle of attack a for the NACA 16-012 hydrofoil is 00, whereas for the cambered NACA 4412 hydrofoil it is -4'. Hence, a comparison based on a - a instead of a would have shown more distinctly that the behaviour of the boundary layer for the NACA 16-012 nydrofoil is much more sensitive to an angle of attack change than for the NACA 4412 hydrofoil. This is mainly caused by the large difference in leading edge thickness. The distinction between the observed long and short laminar separation bubbles is based on the general description given by Gaster '13]. Long bubbles were only observed for the NACA 4412 hydrofoil at a = 80 and 100. The free shear layer of the long bubbles was located at a very short dist3nce from the body surface. 4
a4.1
2
Cavitation Appearance
1
*
3-
I
* 0-
3*
*
$
!
b
8a 0
101_
1
9
CAVITATION STUDIES
8
o E
8
The appearance of cavitation on the hydrofoils has been derived from different sources. First, visual observations were made under stroboscopic lighting during the cavitation inception measurements. Secondly, each series of cavitation inception measurements was followed by a series of tests during which photographs were made of limited, moderately developed and developed cavtaken at specific test section 8itation, speeds. Thirdly, holograms were made for specific conditions. These hclograms provid-
# O
2 3 REYNOLDS NUMR.R.10 S
sed
detailed information on the cavity shape (bubble, sheet) and cavity location (attached. unattached). In many cases, however, the smalland gaps the endsin ofthe thetest hyglass windows the between drofoil section walls caused premature cavitation in the corners. In those cases, cavitation on the centre section of the hydrofoil was
6
Fig. 34 Positions of separation and transition versus Reynolds number for NACA 4412 hydrofoil at a=16 0 .
'1
-206-
----
-
-
-
i=
eg
Fi. 5 hoorahssh-un
apernc
o
cvtaio
o
AC
1-02hyrooi
t
O=0b,
0[ and for angles of attack of (a) 20 (b) 3', (.) 4', (d) 5', and (h) 12 'The flow direction is from left to right.
(C) 60,
-207 -
A3
(f)
70
(g) 100
th led~in :uasedexcet fr p f tgr sk e 4.1.1 NACA 16-012 hvorofoil
ed3e.
A phojtoraphic record of the cavitation -0 OfO7 appearance on the0O. Ni,A 16-012 hvdrofol! for 20, 3, 40, 50, 60, 70, 100'and 120 is shown in Figure 35. These photographs of moderately developed cavitation were all taken at V0 10 mls. The flow direction on
emanedabseLt. Final tests were made at 0.10) and an air content of 5.4 cm'/1. After some experxmenting, bubble cavitation was observed at lcw speeds. For 40 the apearance of cavitation
-
is sheet cavitation (Figure 35c). A sequence of photographs taken at V = 10 m/s and at
-
different cavitat-on numbers is presented in Figure 37. The cavitation numbe; is
these sand all following) photographs is from left to right. The nhjtographs cover about 50% of the it-ngth of the hv'drofoils 0btarting from the leading edge (x/c =
=2 is typically bubble cavitation ( Figure 35a). Most cf the bubbles are attached to the surface and travel wltn the main flow. Tnis type uf cavitation is also calle,. travelllnq bubble cavitation. The bubbles appear in a wide area (Wei 0.2 - 0.8): the largest ones are observed at x/c 0.6. At higher speeds (up to 30 m,s) some streak cavitation was obscrved. T;e streak cxtv:i-es were caused by small particles, trapped on the hydrofoil nose. Tnaearance of cavitation at 30, is very much dependent on t.u, precise angle of attack. initial observations at 0 = 2.19 ;.. 0.10) and an air content of 8.4 cm3 1 showed bubble cavitation at low speeds (Vo = 10 - 15 mis) and streak cavitation at high speeds. Subsequent observations at i = 3 090 ( 0.10) and an air content of 5.2 cml/1 showed streak cavitation at low speeds (Figure 35u' and sneet cavntation
(with some streaks) at hig
-
I
sneeds.
An example, taken at V = 20 m/s, is shown in Figure 36. Bubble cgvitation did not: occur. Initiall;, it was assumed that this was caused Dw the lower air -ontent. Therefore, additional tests were made at
=
.09°
during which the air,content ,.as gradually increased to 13.7 cm'/i by bleeding air into the tunnel, but still bubble cavitation
[",I
Fig. 37 Sequence of photographs showing development of sheet cavitation on NACA 16-012 hydrofoil for a=40, V =10 m/s and '-values of (a) 1.22, (5) 1.05 and (c) 0.98.
Fig. 36 Photograph showing sheet cavitation (with some streaks) on NACA 16-012 hydrofoil for i=3.09 0 and Vo=20 m/s.
I •
I
--
. .
.
-0:
Fig. 38 Attached sheet cavity on nose of NACA 16-012 hydrofoil for n=5, Vo= .9 m/s and =1.77. defined as p-p 0
~
where P0 and V denote the undisturbed static pressure ang flow velocity in the test section, Pv is the vapour pressure of tie its density. liquid and Also for a = 50, the type of cavitatior is sheet cavitation (Figure 35d). A photograph taken from a hologrim., showing the attachment of a sheet cavity to the hydrofoil nose at V = 4.9 m/s and c = 1.77 is presented in 79gure 38. For a = 60 at inception, cavitation appeared as a symmetrical band of small :ttached bubbles and sheets. When the pressure was lowered, an irregular sheet was created near the nose. Further dowistream, the sheet transformed into cavitation with a cloudy appearance (Figure 35e;. Foi a =
70
A
-
Fig. 39 Photographs showing cloud cavitation on NACA 16-012 hydrofoil for a=100 and different speeds: (a) V=5.1 MIIs (c=3.07) and (b) Vo=15.1 m/s (o=3.41) "
at inception, cavitation
appeared as a symmetrical band of small attached transient bubbles near the nose. A detail of at attached cavity is shown in '9]. When the pressure was lowered, the cavitation obtained a cloudy appearance, but remained attached to the nose, occasionally showing areas with a transient sheet (Figure 35f). For a = 100, the cavities were no longer attached to the hydrofoil surface 191. The type of cavitation observed is cloud cavitation (Figure 35g). At lowspeeds the clouds appeared to have a finer structure (consisting of many isolated bubbles) than at high speeds. This phenomenon is illustrated by the photographs presented in Figure 39. which were taken at (a) V0 = 5.1 m/s and (b) V = 15.1 mn/s. for is e cloud 120 , cavitation the type of (Figure cavitation Also observed tioi obervd cludcavtaton Figre 35h). Holographic observations clearly show that the cavities are not attached to the hydrofoil surface but are located at some distance. Examples, taken at V o = 4.9 m/s
rn-
-
Fig. 40 Unattached cavities (cloud cavitation) near nose of NACA 16-012 hydrofoil for a=12 0 , Vo=4.9 m/s and o=3.13.
-209-
-
.
-
Fig. 43 Photograph showing irregular, threadlike appearance of cavitation on NACA 16-012 hydrofoil for u=16 ° VO=I0 mis and =.2 of cavitation is almost identical. At low
Fig. 410Unattached cavities (cloud cavita*ion) near nose 8f NACA 16-012 hyrofoil for a=12 ,V =7.0 m! 6 and
i !_o=.21 i
and 7.0 m/s, are presented in Figures 40
Scated
and 41. For a = 16, the cavities are irregularly shaped and often have a threadlike appearance. Most of'the cavities are loat some distance behind the leading edge. Examples, taken at V O = 10 m/s, are presented in Yigures 42 an8 43. Similar patterns are reported by Kermeen and Parkin T141 behind shazp-edged disks, and by Keller i-151 oT a blutnt axi'sym.etric body. 4.1.2 NACA 4412 hydrofoil A photographic reLcord of the cavitation on the NACA 4412 hydrofoil for appearance 00, 1-0, 4° , 60, 80, 100 , 120 and 160 a i s presented in Figure 44. Tne photographs moderately developed cavitation at Vshow O = 10 m/s. 0° , 2 and 4° the appearance For a
i
,_
.
: :: PFor
speeds (till about 14 m./s) bubble cavitatio is found (Figure 44a, b and c). At high Is) transient speeds (above about 18 -. attached cavities are observed and, occasionally, travelling bubbles. The attached type of cavitation is named transient spot carlration. At moderate speeds (around 16 m/s) both bubble and transient soot cavitation occur. Three examples of transxent spot cavitation, taken at about 20 m/,are presented in Figures 45. 46 and 47. The cavities are located at a large distance fro the leading edge. They seem to originate from weak spots on the surface of the hydrofoil but. because of -thetransient character, these soots do not have fixed positions. Quite frequently, bubbles are are-shown observed with phenomenon has been in Figure 44c and "tails". d. This Examples described before by Hol-I and Carroll ['16'. Examples of holographic recordings showing bubble cavitation for a = 20, are presented in Figures 48 and 49. In Figure 48, an attached and unattached bubble are observed forV =14. m/sand =0.85. The position of thoe attached bubble is at x /c = 0.171. Figure 49 shows an attached bubIble for V = 19.8 mis and =0.80. The position of a = 6° , thS appearance of cavitation
"
[ at low speeds (till about 14 m/s) is bubble cavitation (Figure 44d). At high speeds streak cavitation is observed. and 10° is almost identical. The type of cavitation is essentially sheet cavitation. Occasionally, however, some bubbles were observed at low speeds (till about 10 m/s} Zfor
:V
Fig. 42 Photograph showing irregular, threadlike appearance of cavitation =6, o yrfi -o AA1-1 and o=3.22. 10 m/s ° " -_T
a = 8.The sheet has a rather irregular appearance, as shown iri Figure 44e and, f Apparently the sheet is very sensitive to small imperfections -at. the leading edge of the hydrofoil- A detail of the leading edge of the sheet cavity is presented inFigure 80, 50. This photograph refers to a
-
-a -c
b*f
Fig. 44 Photographs showing appearance of cavitation on ?NACA 4412 hydrofoil a- Ve,=10 m/S and for angles of attack, of (a) 00, (b ~ c 0 d 0 e 0 5 00, (g) 120 and (h) 160.
Fig. 45 Photograph showing transient spot cavitation on MACA 4412 hydrofoil for a=00 , V,=19.7 mns andc=0.60.
Fi.46 Photograph showing transient soot (and bubble" cavitation on NACA 4412 hydrofoil for a=2 0 , vJ =19., .m/so=.0 and c=0-55.
Fig. 47 Photograph showing transient spot cavitation on NACA 4412 hydroil for a=4 0 , VG=19.8 rn/s and ::1.06.
Fig. 4Z Attached and unattached bubble on NACA 4412 hydrofoil for a=20, v0 =14.9 rn/s and -=0.85. xC/0l.
Fig. 49 Attached bubble ~nNACA 4412 hydrafoil for a=0 1:0=1 9 .8n/ and /C .28 .. xccO28
Fig. 50 Attached sheet cavity on nose of NACA 44!2 hydrofoil for a=0 v 0 =9.7 =-Is and c=2-06. xe/c=0.0186.
-
-
~~Fig.51
of rata ig. 53 Attached bubble (sheetj on nose hydrofoil for i=16 ° , NACA 4412 Inception 4.2 Cavitation Vo=7.1 n/s and -=5.34. Xc/C=0.00189.
he Photograph showing cavitation on NACA 4412 hydrofoil
The and , Vo 2.06. 9.7 mis a=120 and position -=3.a0. of =5 m/s,
Sfor
Sthe
leading edge of the sheet cavity is at = 0.0186. 20. the appearance of caviFo tation depends rather strongly upon the 5 m/s (Figure 51), the first At Vo m",s AtV=10 is a the sheet. the cavity song shtstate part of patia 51gPhot Fig lead etcviyisa ing edge tac small and the downstream part of he Fvery
To examine the beginning of cavitation, cavitation inception measurements were carried out for both hydrofoils. Cavitation. inception is the limiting state between a di erent This cavity flo4. was: flow andi a two nOn-cavitating can be attained starting from a non-cavitating condition and
Xc
__ _ =speed.
cavit9 has a cloudy appearance
decreasing
Finaliy, at
the ,ields ate of cavitation incipient and starting numer 7,and incipieontx cavitation trom a cavitating condition a-d increasing leads to the desi-ent which eressure the desinent theyields state of cavitation and cavitation number c.. The Present study
refers to V0 = 7.1 m/s and = 5.34. The Position of the leading edge of the cavity is at xe/c = 0.0089. When the pressure is lowered, the leading edge of the cavity remains tltached to the hydrofoil nose, but further downstream the cavity has a cloudy appearance (Figure 44h).
com-zrised both desience and incenti-r measurements. For each test velocity, desinence was masured prior to incention. The precise definition of inception is important in obtaining consistent results. Two asnects have to be considered in performing inception measurements on hydrofois Fiist, thej cavity ray start on the leading edge in a small part along the span and spread out gradually when the pressuie is lowered. Secondly, intermittent cavitation may occur, such as bubble or cloud cavitation. The problems associated with both aspectz were solved by using the well-known 50% cyiterion of inception. The 50% my refer to the extent of cavitation along the span or to the time during which cavitation is present. The inception measurements were made under stroboscopic lighting. When possible, inception data are compared with the pressure coefficient C defined as P Po 0
P %~0o2I where P denotes static cofficient pressure on is the If thethe pressure hydrofoil.
drofoil for a=12 0 , V =13.8 m/s and 0 c=4.10.
related to the minimum static pressure Pmin on the hydrofoil, the minimum pressure -213-
_
__
__ _ _
A
:he pressure which leads to the
2 =Fiattached to "he nose, most ofis the but c006.T s9ill caviy has a cloudy apparance of atand = 16 at inception, a For A deobserved. i2 smll attached bubbles tail is shown in Figure 53. This Photograph
Fig. 52 Photograph showingon cavitation NCA 4412 wit hycloudy apparance
-
M
M
K!
-4
0I
so
A
70
'&T
I~
000
* *A
43
06
A IA lb
o6
I
I A
04
AD
A
*
I
,
-
A0
11
AA A
A 00 AA
2-
,, A
A
A
AfV A
A
0
03
01
R2NOD
z
NUMBER
-
2exI
V
-
°
Fig. 54 Cavitation inception and desinence number versus Reynolds number for-NACA 16-012 hydrofoil.
i
coeffici ent CPmin is given by i0
_
CPmi n
p
o
it has been shown that for this angJle lami-
_nar Pmin
W 0 2this Z 7",
:
flow separation occurs near the leading edge without reattachment. Aliparently, above the lift coefficient andstall -CPmin decrease.angle leading edge occursand A
The results of the cavitation inception measurements are plotted in Figures 54 and 55. In general, it is found that the ai-: and ad-values are almost indentical, Cavitation hysteresis effects F171 are observed only at low Reynolds numbers for the The inception values NAA1-1 yrfi(or desinence) ta= 60ad7.separation for the NACA 16-012 hydrofoil- increase gradI ually with an increase in^a, but reach ilimiting values at a 10 .In section 3.1,
This explains the lower inception values for a - 120 and 160. =For -the NACA 4412 hydrofoil, a -gradual increase in inception values fs fotand with an increase in ° a up to the highest angle tested (a = 16 ). In section 3.2, it has been shown that for this afigle laminar flow with reattachment. A direct comparison occurs near the leading edge between the present inception values and those measured by Kermeen [11] is dlifficult to make because the present results are not
-214 -
i
0
*0
0
6 -
0(
dJ
W16 0 12'
&
A
Wf
A
31 A
A
AaI
b
IOCP
(3
J"A
)
I'
ii. 3a
*
00 0 0*
0
A
0, 2
V
6
~~'±g.~~ -AAA ja D
2
A RANLS'~M~ RAAat o . i d
aj
55dOnnenkbrvruieyod
ubr arAesrdvle
WL
~~~
~
o
f~pjacrigt
---
-
-
acodn
to-
AA4l ikro lJ
Piketo
corrected for blockage. The most important difference, however, is that Kermeen's data for a = 00, 20, 40 and 60 show a Reynolds number dependency, whereas the present data for these angles are independent of Re. Xermeen [iI] compared his inception data with accurate wind tunnel measurements of 6 -CPmin made by Pinkerton [18] at Re= 3 x10 , and found a reasonable to good agreement. In order to compare the present inception values with Pinkerton's values of -Cpm, a correction method is used, which simply consists of multiplying the measured -Cp . -values with a constant factor of 1.2 This factor represents the square of the racio between the height of the test section and the height of the test section minus the hydrofoil thickness. These corrected -Cp . -values are plotted in Figure 55. The nvalues for a = 60 and 10 have not been measured and were obtained by interpolation. The agreement between measured inception values and corrected -Cp values for a = 00, 20 and 40 is excellen~n -- but for larger angles the agreement is rather poor. The inception values appear to be independent of the Reynolds number for all cases that bubble cavitat-on is found, at low or moderate speeds (NACA 16-012 hydrofoil, a = 20 and NACA 4412 hydrofoil, = 0, 20, 40 and 60). Since the position of transition to turbulence of the laminar boundary layer is a function of Re, as shown in Figures 4, 19, 22 and 25,.the pressure coefficient at transition, CpT, is also a function of Re, and thus it may be concluded that a i d Coes not correlate .ith CpT. Such a correlation has previously been reported by Arakeri and Acosta [j on a 1.5 calibre ogive body. In our opinion, a correlation between Gid and -CP seems more appropriate, provided the waer contains a sufficient number of nuclei, 5. DISCUSSION AND CONCLUSION
=
The appearance of cavitation on both hydrofoils at low to moderate speeds can be classified into four main types. The first type is bubble cavitation. It has been observed on the NAC 16-012 hydrofoil at a = 20 and on the NACA 4412 hydrofoil at a = 00, 20, 40 and 60. In most of these cases the boundary layer behaviour was studied and a laminaz boundary layer with, far downstream, transition to turbulence was found to occur. Bubble cavitation is the classical type of cavitation and has been extensively analysed by Knapp and Hollander [19] and Plesset 120] . It is merely dependent on the pressure distribution and nuclei content. At high speeds, streak cavitation or transient spot cavitation is observed. Streak cavitation is related with small surface irregularities or dirt particles on the leading edge of the hydrofoil. Transient spot cavitation is probably related with weak spots on the hydrofoil surface or in the turbulent boundary layer (bursts).
The second type observed is sheet cavtation. This type has been noticed on the NACA 16-012 hydrofoil at a = 30, 40 and 50
and on the NACA 4412 hydrofoil at a - 80 and 100. it is closely related with the occurrence of laminar separation at the leading edge of the hydrofoil, with subsequent transition t) turbulence and reattachment. The mechanism responsible for this relatiozship was clearly established by Arakeri and Acosta [i] and Araker-i [4l]. Van der Meulen [3] compared the length of the separation bubble on a hemispherical nose with the length of the sheet cavity at equal Reynolds numbers and found corresponding values. The length of the sheet cavity on the hydrofoils, however, is much larger than the length of the separation bubble. In rigure 37, for instance, it is shown that the length of the smooth part of the cavity (the sheet) increases without bounds when j is decreased. Whether this is due to a more favourable pressure distribution caused by the cavity itself or to other factors remains, as yet, unknown. The dominating role of the separation bubble in the cavitation inception process is clearly demonstrated by the flow and cavitation observations made with the NACA 16-012 hydrofoil at angles of attack around a = 30 . At a slightly smaller angle (a = 2.90), a laminar boundary layer is found with normal transition to turbulence and the type of cavitation is bubble cavitation. At u = 30, however, a separation bubble occurs and the type of cavitation observed is sheet (or streak) cavitation. This type of cavitation appeared to be dominant, since it could not be transformed into bubble cavitation by a large increase in dissolued or free air content. The third type observed is bubbleband cavitation. This type was observed on the NACA 16-012 hydrofoil at a = 70 and on the NACA 4412 hydrofoil at a = 160. These cases refer to leading edge separation with subsequ2nt transition to turbulence and, hardly present, reattachment. This is typically a transitional case between the separation bubble and the unattached free shear layer. When the cavity becomes slightly more developed, the downsLream part of the cavity hows a clouiy appearance. The fourth type observed is cloud cavitation. Th"s type was observed on the NACA 16-012 hydrofoil at i = 100and 120. The boundary layer observations for these angles showed the presence of an unattached free shear layer. Transition to turbulence is accompanied with the formation of vortices. These vortices appear in the bulk of the fluid flow, away from the hydrofoil surface. At inception, the vortices provide the locations for isolated cavities to be created.' The continuous formation of vortices results in a formation o., a large number of small cavities resembling a cloud. A type of cavitation usually related with the flew around sharp edges and not with hydrofoils, is the type observed with the NACA 16-012 hydrofoil at a = 160. The irregular cavities of threadlike appearance -216-A
ifzM
R~--
i
72 A -
m4
in the flow downstream from the hydrofoil nose resemble the patterns observed behind disks [14] and blunt bodies [15]. In conclusion, it is stated that cavitation on-hydrofoils at low to moderate speeds can be classified into four main types: bubble, sheet, bubble-band and cloud cavitation. Each type is uniquely related to a specific type of boundary layer flow behaviour. At high speeds, the bubble type of cavitation transforms into streak or translent spot cavitation, whereas the other types remain essentially unchanged. REFERENCES
2
IS
WGeib, Ba
FEFlows",
1. Arakeri, V.H. and Acosta, A.J., "Viscous Effects in the Inception of Cavitation on Axisymmetric Bodies", Journal of Fluids Engineering, Transactions ASME, Vol. 95, Dec. 1973, pp. 519-527. 2. Van der Meulen, J.H.J., "A Holographic Study of Cavitation on Axisymmetric Bodies and the influence of Polymer Additives", Ph.D. Thesis, Enschede, 1976. 3. Van der Meulen, J.H.J., "A HOlographic Study of the Influence of Boundary Layer and Surface Characteristics on Inci-lent and Developed Cavitation on Axisymmetric Bodies", Twelfth Symposium on Naval Hydrodynamics, Washington D C., June 1978. 4. Gates, E.M., "The Influence of Preestrea Turbulence Freestream Nuclei Populations and a Drag-Reducing Polymer on Cavitation Inception on Two Axisymmetric Bodies', California Institute of Technology Report No. 183-2, April 1977. 5. Gates, E.M. and Acosta, A.J., "Some Effects of Several Freestream Factors on Cavitation Inception on Axisymmetric Bodies", Twelfth Symposium on Naval Hydrodynamics, Washington D.C., June 1978. 6. Arakeri, V.H. and Acosta, A.J., "Viscous Effects in the Inception of Cavitation", International Symposium on Cavitation Inception, ASME Winter Annual Meeting, New York, Dec. 1979, pp. 1-9. 7. Casey, M.V., "The Inception of from Laminar Separation Cavitation Attached Bubbles on Hydrofoils", Conference on Cavitation, Edinburgh Scotland, Sept. 1974, pp. 9-16. 8. Blake, W.K., Wolp.rt, M.J. and F.E., "Cavitation Noise and Inception as Influenced by Boundary-Layer Development on a Hydrofoil",oJotirnal of Fluid Mechanics, Vol. 80, Part 4, May 1977, pp. 617-640. 9. Van der Meulen, J.H.J., "Viscous Effects in the Appearance of Cavitation on a Hydrofoil", ASME 1978 Cavitation anA Polyphase Flow Forum, Fort.Collins, June 1978, pp. 7-9. 10. Van der Meulen, J.H.J., "Cavitation on Hemisphericdl Nosed Teflon Bodies", IUTAM Symposium, Leningrad, June 1971. 1j. Ketmeen, R.W., "Water-Tunnel Tests of NACA 4412 and Walchner Profile 7 Hydrofoils in Noncavitating and Cavitatifig Caljifornia Institute of Technology Report No. 47-5, Feb. 1956.
12. Van Renesse, R.L. and Van der Meulen, J.H.J., "In-Line Holography for Flow and Cavitation Visualization on Hydrofoils and for Nuclei Measurements", International Symposium on Flow Visualization,!Bochum Sept. 1980. 13. Gaster, M., '!The Structure-and. Behaviour of Laminar Separation Bubbles', AGARD Conference Proceedings No., Separated Flows, May 1966, -pp. 813-854. 14. Kermeen, R.W. andParkin, B.R., "Incipient Cavitation and Wake Flow Behind Sharp-Edged Disks", California Institute of Technology Report No. 85-4, August 1957. 15. Keller, A.P., "Cavitation Incepti0 Measurement and Flow Visualisation on Axisymmetric Bodies at Two Different Free-Stream Turbulence Levels and Test Procedures" International-Symposium-on Cavitation Inception, ASME Winter Annual Meeting, -New York, Dec. 1979, pp. 63-74. 16. Holl, J.W. and Carroll, J.A., "Observations of the Various Types of Limited Cavitation on Axisymmetric Bodies', International Symposium on Cavitation Inception, ASME Winter Annual Meeting, New York, Dec. 1979, pp. 87-99. 17. Holl, J.W; and Treaster, A.L., "Cavitation Hysteresis", Journal-of Basic Engineering, Transitions ASME, Vol. 88, March 1966, pp. 199-212. 18. Pinkerton, R.M., "Calculated and Measured Pressure Distributions over the Midspan Section of the N.A.C.A. 4412 Airfoil", NACA Report No. 563, 1936. 19. Knapp, R.T. and Hollander, A., "Laboratory Investigations of the Mecnanism of Cavitation", Transactions ASME, Vol. 70, July 1948, pp. 419 435. 20. Plesset, M.S., "The Dynamics of Cavitation Bubbles",, Journal of Applied Mechanics, Transactions ASME, Vol. 16, Sept. 1949, pp. 277-282. 21. Arakeri, V.H., "A Note on the Transition Observations on an Axisymmetric Body and Some Related Fluctuating Wal-lPressure Measurements", Journal of Fluids Engineering, Transactions ASME, Vol. 9.7, March 1975, pp. 82-86.
IN
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3
T
DiscussionT .a
i
e
As shown in Figs.36 and 37 the flow is not quite two-dimensional in- your experiment. two-dimensional flow condition will be much easier and in fact is of fundamental important for the future analyses of your valuable data. Is it possible to achieve a two-dimensional flow condition and to eliminate tunnel blockage in your experimental set-up ?
aaeA
-The author has presented valuable new information On the relationship between boundary layer flows and cavitation forms on two hydrofoils. These data-will proyide us with most useful guidance for our own work in years to come. In this discussion I wish to limit my questions to only one aspect of the V.H. Araketi (lndlan Inst. ofScience) experimental technique related to the author's ingenuous use of slight taper of his in order to mitigate the laserT a beamfoils alignment author contribution. is to be congratulated for twgnn problem associated with this The excellent It is unforpurely two dimensional experiments. Thetuaehttetsswrecnced ih tunate that the tests were conducted with cavitation patterns shown in Figs.35 and hydrofoil sections having aspect ratio less elsewhere in thd paper seem to suggest some than one. This could lead to unknown strong departure from purely two-dimensional flcw in these experiments. For the research three dimensional effects. It is also carried out by the author this might not be curious that the author has not compared his incipient cavitation index measuiements with a serious matter. However, in other applithose of Kermeen (11) on the NACA 4412 hydrocations of tapered foils departure from two s angle For example, foil. dimensionality needs to-be quantified.- Can at a Reynolds number at of 160 about 5 x of10attack the the author give us further information on author finds n value of close to 6.5. this aspect of his experiments as far as ahrfs e in value of .5. 3.2. is finds a ?Whereas, isconernd flw is noncavitatingnoncvittin flow concerned ? there anyKermeln explanation forvalue this offantastic difference ! In addition, Kermeen finds a Reynolds number dependence on his ci meaH.Murai and A.Ihara(rookuUn.) surements at all the angles of attak. It should further be noted that a comparison We would like to comment about the of author, a. or ca values with -C . meainception and desinence of cavitation as surements of Pinke~ton (18) may noml well as the cavity length at the attack appropriate due to possible three dimenangles of 8* and 100 on NACA 4412 hydrofoil sional effects in the authors tests wit~h a which appear in Fig.55 and section 5 of this low aspect ratio hydrofoil. Finally, it paper. would be interesting to know what new flow According to the diagram, the incipient regime has been found by the author on hydroand disinent cavitation numbers show larger foil sections studied by him that is not values than -Cpmin- This tendency is difpossible or has not been found on axisymferent from those in case of a hemispherical metric bodies. nosed body, on which ci and Od are smaller than -Cpm according to the reference (1] of this paper. It can be deemed that this was caused by a following reason. Since the effective attack angle is increased owing to the effect of flow shear near the wall, -Cpmin in the neighborhood of corner is considered to be larger than that at the core region. And the boundary layer separation near the corner may take place at a J.H.J. van der Meulen (NSMB) smaller att3ck angle than that at the core region. According to our observation in The author would like to thank the the high speed motion pictures, which have discussers for their valuable remarks. All discussers raise questions about the been presented at the twelfth symposium on Naval Hydrodynamics, the sheet cavitation two-dimensionality of the flow in the exof this kind has taken place at first in perimental set-up applied. It is true that the neighborhood of the corner, and has deviations from a two-dimensional flow:V propagated toward the spanwise direction, occur due to the presence of a boundary to cover throughout the span accompanied by layer on the side walls and the slightly a small decrease of static pressure. In tapered hydrofoils. However, the boundary the present case, as the aspect ratio of layer flow was visualized in the central the hydrofoil is fairly small, the corner plane of symmetry for which the above inflow can affect considerably on the values fluences are believed to be small. Cavitation inception measurements and observaof ci and od as well as the cavity length compared with the length of separation tions of limited or moderately developed bubble. cavitation were related to the centre sec a-
Autor's Reply
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7'_
_
_
_
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_
_
A M A
15 I
i
tions of the hydrofoils in order to minimize three-dimensional effects and to obtain direct comparisons with the boundary layer flow ir the central plane. The author agrees with the comments by Prof. Murai and Dr. Ihara that cavitation inception on the NACA 4412 hydrofoil at a = 80 and 100 takes place near the corners, but in the present experiments inception was related to the centre part of the hydrofoils. the finding in Figure 55 that the ci and od values for a = 80 and 100 are -somewhat larger than the corresponding -Cpmin values may be attributable to fluc-
of about 5 x 105, Kermeen (11) found f r o i a value of 3.2, but for ad a value of 11.7 Lonverted to the present test secC tion, these values become 4.0-and r4.6 " respectively. At a Reynolds number of about 7.5 x 165i Kermeen found'i ; 5 6and od =- 7.65. C6nvfed -t thepresent teSt section, these values becmi 7 0 and 9.55 respectively. Hence, the "fantastic differences mentioned by br. -Arakeri are to be found within Kermeen's own-measurements; Apparently they are due to nuclei effects associated of the resorber in thewith CIT the high presence speed water
tUting pressures related with transition tO turbulence of the separated free shear layer followed by reattachment. Referring to Dr. Huang's question about tunnel blockage, in the-author's opinion there are two possible methods of reducing blockage effects for the present test section configuration. One-method is to decrease the chord length,-the other method is to inVestigate thinner hydrofoils. Both methods, however, have -the disadvantage of increasing the experimental difficulties involved in visualizing the boundary layer flow.
tunnel. In the present experiments,no Reynolds number dependency is found-for In these casesa 00, 2, 40 and 60. a laminar-boundary layer occurs followed by natural transition to turbulence. The fact that Kermeen (11) did find a Reynolds number dependency for these angles may have been caused by a lack of nuclei in-the oncoming flow so that the pressure fluctuations related with transition became the predominant mechanism in the -inceptironvprocess. In the author's opinion, flow regimes to be found on hydrofoil sections that have not (yet) been found on axisymmetri.c dies are : (1) laminar separation for which the distance to tne stagnation point iS either small, moderate or large, whereas for axisymmetric bodies this distance is moderate, and (2) the unattached free shear layer leading to cloud cavitation.
Dr. Arakeri raises some interesting questions on the comparison between the present Oi,d measurements and those by Kermeen (11) for the NACA 4412 hydrofoil. For o = 160, the present Oi,d values are still smaller than Cpmin , and hysteresis effects do not occur. At a Reynolds number
AM
V
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-219-
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The Influence of Hiydrofoil Oscillation on Boundary Layer Transition and Cavitation Noise Young T. Shen and Frank B. Peterson Davic W.Taylor Naval Ship Research and Oevelooment Center Bethesda, Maryland, U.S A.
ABSTRACT
--
faces of hydrofoils and marine propellers. For many years, scientists and engineers have dealt with the problams of cavitation-induced noise, structural vibration and erosion associated with the operation of marine vehicles and hydraulic equipment. All three problems are related to the inception, growth, and-collapse of vapor cavities.' Inception of cavitation in a fluid is the condition under which cavitation is first detected, either visually or acoustically. It had often been assumed that when making analytical predictions, cavitation inception occurs immediately after the static pressure on the body surface becomes equal to or less than the vapor pressure of the fluid. A research model, the International Towing Tank Conference (ITTC) headform, tested in different cavitation facilities demonstrates that cavitation-inception on a given model can have many different physical forms and cavitation inception indices, de-3 pending on the environment and body surface condition. Departures of cavitation inception from the traditional rule are attributed to so-called "scale effects." The influence of boundary layer pressure fluctuations on cavitation inception is known to be one of the mnjor sources of scale effect. 2.4 Rouse5 showed that the high pressure fluctuations generated in the turbulent shear layer of a free-jet can produce cavitation at static pressures which are higher thani vapor pressure. Daily and Johnson6 showed that pressure fluctuations in the middle of a fully established turbulent boundary layer can initiate inception. Levkovsldi and Chalov7 reported that turbulence in flow past a wing causes cavitation f ror numbers Reynolds to inceptindicates earlier. Recent by itical Arakeri and AcostaU and sere that forwork Huang which bodies do not have laminar boundary layer separation, cavitation inception take place in the region of bersition
Significant effects of boundary layer characteristics on cavitation and the effect of unsteady cavitation on noise have been widely observed experimentally. In order to better understand these effects, an experiment with a two dimensional hydrofoil, under sinusoidal pitching oscillation, was conducted in the 36-inch water tunel at DTNSRDC. Three pinhole-type pressure transducers were installed on the foil surface to measure pressure fluctuations and a hydrophone was used to measure the radiated cavitation noise. Two subjects are discussed in this paper: 1) relationship of boundary layer transition and pressure fluctuations with the hydrofoil in oscillation, and 2) noise generated by unsteady cavitation, The magnitudes of pressure fluctuations in transition and turbulent regions are found to be independent of oscillation amplitude and frequency. However, the development of boundary layer and occurrence of transition are delayed with an increase in oscillation frequency. The influence of hydrofoil oscillation on cavitatiun inception is also discussed. With the occurrence of leading edge sheet cavitation, a significant effeet of foil oscillation on cavitation noise is measured. At high reduced frequencies, extensive cloud cavitation is formed during th final phase of sheet cavity collapse and the noise level is significandy increaned, PREFACE An earlier r by the present authors at the 12th two subjects, the inception of s.,Vo u deat -with ONR cavitation on a two-dimensional foil and the physics of leading dge sheet cavity stability and subsequent formation of cloud cavitation. The method developed to predict inception was based on Giesins unsteady airfoil theory and assumed that viscous effects were of secondary importance. The present paper will explore the importuace of the boundary layer development. In addition, as a continuation of the early study of cavitation instability, the importance of cloud cavitation in producing acoustic noise will be presented.
from laminar to turbulent flow. For subcritical Reynolds numbers, cavitation inception is found to occur in the reattachment region following laminar boundary layer separation. Significant boundary layer pressure fluetuations on body surfaces have been measured by Huang and HannanO0 and by ArakeriI in the reattachment region following laminar separation. In addition, Huang and Hannan reported that measured pressure fluctuations in a natural transition region are higher than fluctuations in a fully established turbulent boundary layer flow by a factor ranging from 2 to 3. Huang and Petersont- reported that a significant scale effect
1. INTRODUCTION Cavitation frequently is unavoidable on the lifting sur-
-221-
5549
7
s----m-
-~~~
w
__
on cavitation inception induced by boundary-layer pressure fluctuations exists between full-scale and model propellers due to large differences in Reynolds numbers. The properties of the wall fluctuating pressure field associated with the growth of turbulent spots in a transition boundary layer on a flat plate have been measured-by DeMetz et al.,13 DeMetz and Casarella, 14 and Gedney.15 Each reported that pressure fluctuations in the transition region are smaller than those in the fully developed turbulent iegion. This result is not in agreement with the measurements of
length c of 24.1 cm, a span of 77.5 cm and a relatively laige leading-edge radius. The maximum thickness to chord ratio is 10.5 percent. The foil surface was hand finished within 0.38 ym rms surface smoothness. Pressure transducers were installed at a distance of 7.96, 24.1 and 60.3 mm from the leading edge. These locations correspond to 3.3, 10, and 25 percent of chord length from the leading edge. Kulite semiconductor pressure gages Of the diaphram type (Model number LQM-10-250-305) were mounted within aHelmholtz chamber connected-to the foil-
Huang and Hanna.'
surface by- at pinhole. This arrangement permiAtted measure-
0
A basic question has been raised as to whether the boundary layer pressure fluctuations in a natural transition region can be greater than those in a fully established turbulent boundary layer region. The significance of this point is that if the pressure fluctuations at transition are greater than in the fully developed turbulent boundary layer, then one can expect cavitation to preferentially occur at the transition location. In view of the significance of this question in cavitation scaling, fluctuating pressure measurements for a twodimensional hydrofoil are given in this paper. When a hydrofoil operates in waves or a propeller operates behind a ship hull, the lifting surfaces are subjected to temporally and/or spatially varying velocity and pressure fields The time varying pressure field can be expected to have a significant effect on the characteristics of cavitation inception, growth, and collapse. As a first approximation, the unsteady effect on cavitation has been studied with oscilla-
ment of unsteady surface pressures due to foil oscillation and high frequency pressure fluctuations inside the boundary layer over a pressure range of j207 KPa (*30 PSI). In order to increase the spatial resolution in measuring the local pressure fluctuations inside the boundary layer, the diameter of the pinholes installed on the foil surface was kept at 0.31 mm (0.012 inches). This arrangement also reduces the danger of cavitation damage to the pressure transducers. Extreme care was taken to fill the Helmholtz-type chamber through the pinhole under vacuum with deaerated water to minimize the possible occurrence of an air bubble trapped inside the chamber. A flat pressure response was observed up to 2,000 Hz in dynamic calibration tests. The calibration procedure used here was developed by the National Bureau of Standards, 2 ' modified to the extent that water, rather than silicone oil, was the fluid medium. Since it was very important to determine the relative phase difference between the foil angle
ting hydrofoils by Mfiyata et al., 1619 Radhi, Bark and20 van 5
ad hhe ressure gage signals, all amplification and recording an
Berlekom,' Shen and Peterson, and van Houten. The hydrofoil approach is especially attractive for simulating a propeller blade operating behind an inclined shaft with the effective aigle of attack oscillating periodically during each cycle of rotation. Available data demonstrate that the cayltation-inception-angle-of-attack at the leading edge is different from inception angles determined from quasi-steady nal yis bmed on uniform flow experiments. Shen and Petersonr 9 showed that the computed potential-flow pressure distributions are subject to a significant phase shift with respect to the foil oscillation angle, which in turn influences the occurrence of cavitation inception. As previously mentioned, significant physical effects of boundary layer characteristics on cavitation have been well documented. So far most transition and boundary layer pressure fluctuation studies have been limited to flat plates and headfornis. In the first part of this paper the influence of oscillation on transition boundary-layer pressure S-uctuations, and the possible effect of oscillation on cavitation inception for a two-dimensional hydrofoil, will be presented. As pointed out previously, cavitation frequently is unavoidable on the lifting surfaces of hydrofoils and marine propellers In the second parL of tUi paper the influcnce of osscillation nn foil
equipment was selected to minimize the introduction of unwanted phase shifts. The cavitation noise was sensed by a flush mounted hydrophone 117 cm downstream from the foil axis, on the top of the water tunnel's clos6jet test section. The hydrophone was uncalibrated, so all noise measaremerts are relativdto an arbitrary level.
The closed-jet test section of the 36-inch water tunnel was modified by the insertion of sidewall liners to provide two flat sides. On each end of the foil a disc was attached. This disc rotated in a sidewall recess; thus the foil could be rotated without gap cavitation occurring between the end of the foil and the sidewall of the tunid. One sidewall assemblywas fitted with clear plastic windows to permit side view photography. PTIAIR MBGS -- ,
SLID
OCLAO
2. EXPERIMENTAL APPARATUS AND TEST
FI
PROCEDURE 2.1 Foil and Instrumentation
W
SLD
A foil was machined from 17-4 PH stainless steel in the form of a rectangular wing of Joukowski section with the trailing edge modified to eliminate the cusp. The offsets are
CONCTN
(t c
given by the following equation:
ROD
D-RIE BYVARIARE
um
SPIEED D.C. NOTOR
=
y/c = 0.04077 sin 0 - 0.02039 sin (26) Cos-1 (1.88 Y/c - 1)05x 0-
o
ytc w 0.08590 (1 - xlc)
0.7945sxYcs 10
-
cs-Figure (
S.74
c-
Conceptual Design of Foil Oscillation Mechanism The foil was oscillated in p itching motion arouid the
To simtdate prototype viscous effects at the leading edge as
quarter chord by a mechanism whose conceptal design is
closely as possible, the model was designed with a chord
shown in Figure 1. With this type of desig, the foil mean -22
-A
_____ -
--
.-
~
-
2.2 Test Section
instability and noise will be presented.
-cavitation
I
-~- -
__3W
2
p2 as shown in Figure 2. This sound power level is also digi,ized along with the signals corresponding to the foil angle-ofattack and the mean foil angle. In order to obtain the average relative sound power, the digitized signal is averaged over the whole run. In order to determine the average waveform of the hydrophone signal during one cycle, a cycle is defined-as being initiated and terminated when the increasing foil anigle passes through its mean position. Each cycle is divided into (1600/f) time increments, i.e. at a f 4 Hz oscillation frequency, 400 time increments are used. The digitized hydrophone signal in each time increment is averaged over all of the cycles in a run. The averages in each of the increments then collectively represent the averge waveform occurring for one cycle.
angle (a0) can be adjusted statically and the amplitude of foil oscillation (a) can be continuously adjusted between 00to 40 while in operation. The oscillation frequencies cover the range between 4 Hz to 25 Hz. Air bags were installed to reduce the fluctuating torque requirements on the motor drive system. 2.3 Data Reduction Due to installation of two sidewall liners in the test section, the tunnel speed was corrected according to the arearatio rule. The tape-recorded time histories of foil angle and pressures were digitized by using a Hewlett Packard 2100 minicomputer and reduced by using algorithms implemented on the DTNSRDC CDC-000 digital computer. The time
zhistories
were recorded on one inch magnetic tape at 15 inches per second (38 cm/s) using IRIG standard intermediate band frequency modulation techniques. During digitization, pressure fluctuation data were filtered using four-pole Butterworth bandpass filters that have a -3 dB signal attentuation at 80 and 2,000 Hz for boundary layer pressure fluctuations. The run lengths used in the data reduction were nominally 40 seconds. For the oscillating foil data, the computer output consisted of values of mean and standard deviations of pres-
3. BOUNDARY LAYER PRESSURE FLUCTUATIONS ON A STATIONARY FOIL
1 -
Consider a steady uniform flow past a two-dimensional hydrofoil. Let the local pressure p on the foil surface be expressed in the following way: p
=
pp + Apt
(2)
Sure fluctuations. O
GI
05 ®
3
where pp is the mean pressure ard Apt is the pressure associ; ated with the boundary layer pressure fluctuations. Let Cp denote local pressure coefficient: p Poop
N
aft,) ; CAVITATION
4
NOISE
(3)
p-,t
.....
-_
C-- + ACt
where
-
ht-T-V
PpP - oo
.RSPL
(4)
ISOUND POWER)
and
-CUT' AND ORDER oft
EACH CYCLE
Q V--
---
(5)
here Q,p,, and V.. are fluid der.ity, reference free-stream
RFLt T
static prc.uie fnd reffcence free-stream velocity, respec-
0015 sec,
tively. aftI =
.
T
3.1 Laminar Boundary Layer Stability Calculations (NLet
NN
R-P (V) =
ORSPL((k- VT+
RSPL(a1)-
=
-
denote the reduced frequency where c is
limiting case k - 0 corresponds to a stationary condition. A uniform flow past a stationary foil at an angle-of-attack a of 3.25 degrees will be considered first The computed steady potential flow pressure distribution for this foil angle is shown in Figure 3. The flow field to be measured by the pressure gages at 3.3, 10 and 25 percent.chord length is seen to be in a strong adverse pressure gradienL This fact will be shown to have a significant effect on the magnitude of boundary layer pressure fluctuations. The prediction of laminar boundary layer separation is based on the criterion of zero shearstress. The Smith disturbance amplification method2 is u'ed to correlate and predict flow transition. Let A/Ao denote the Smith cumuladve spatial amplifi cation ratio A to its amplitude Ao at the point of neutral stability. Let R. denote the Reynolds number based on chord length. As a numerical example, the computed amplification ratio versus chordwise location x/e is given in Figure 4 at
CJISPL
IS
k
2V, the circularfrequency associated with foil oscillation. The
k=1
AVERAGED
I--
Figure 2 - Sequence of Ensemble Averaging 0
_-a
(From an unpublished report by R. Pierce, DTNSRDC) The hydrophone signal was first bandpass filtered between 10 kHz and 40 kHz to minimize signal contamination from noncavitation related sources. This signal was then squared and passed through an averaging filter to obtain a "running average," i.e. quasi-stationary sound power levels, -223-
16
-15
1
THEORY
-
0
MEASUREMENT
UPPER SURFACE (THEORY)
12/
2 1
V
0
U:
~~ I
~
D
4-1
LOWER SURFACE ITHEORY)
0S5
/
4
2 5x
j
~25 x104
10) LAMINAR SEPARATION PftEDICTE
0.
0.2
0.
0.6
0
0
1.0
-00
.0
01
Atxic
0.23 e
Figure 4a - Computed Amplifi tortLAatos 6at Foil Angle 1.6 x 0 a =3.25 deg and R.
Figure 3 - Measured and Predicted Pressure Distributions R = 1.6 x W0.The Smidth cumulative spatial amplification ratio was computed for a rang of nondimensional boundary layer disturbance frequencies Q, V2
(6)
where Qis the dimensional disturbance frequency. Figure 4a shows that disturbances will he significantly amplified in the
frequency range of %pond
1.7 x 10-' to 32 x 104 which corres.
to dimensional frequencies of 1,25 Hz to-2350 Ky~. Figure 4a also indicates that the frequency m35 x 104has the largest amplification ratio up to xlc = 0.11. The freestr~eam velocty used in these computations is V.,, - 6.71 nis. The same method was used for stability calculations at other
I
C 2.0-
\
Reynolds nunbers. The computed results arm summarized in Figure 4b.
\
3.2 Experimental Results The experiments -with the fogl at a 3.25 degreesI were performed in the DTNSRDC 36-ich water tunnel. As seen in Figure 3, the measuremenits of statir. pressure coef fi-I cieats are in good agreemnent with prediction. At speeds of 4.88, 6.71, 9.75; 13;11 aid 14.94 meterlsec, the presure responses measured at three foi locations are given mn Figure 5a to 5e. The pressure gages located at 3.3, 10 and 25percent chord lenths on the tipper surface -aredenoted by PP0 aiid P2s, respectively. The purpose of steady runs was:RtO --224-
a
______________ .
Fig'Jt 4b - ffect of Ren t.
02
03
Xflflibers on Anplificatiou 32 e -
iII RUN NO. 8044
to provide reference data to be used for comparison with dynamic runs. Consequently, the test runs given in tigures 5a to 5e were conducted with the oscillating mechanism in motion and the oscillating amplitude a, set to zero to incorporatephenomena. the possible effect of noise and vibration on the meas~~ured
FOIL ANGLE
t
33 PERCENT CHORD LENGTH P3
RUN NO. 7001 -___
____________.
.10
______-
i
FOX ANGLE
-. I
PERCENT CHORD LENGWH PI0
'.
.
....-
J
74;-" -,
-.
P.RCENT CHORD LLEGNTG PP
2.
10 PERCENT CHORD LENGTH PI
Figure 5d - Measured Pressure Fluctuations at R, =3.2 x 10G in a Steady Flow at a 325 25PECENTCHOPMLENr. P2,
,-
.
NO. SGEB
•RUN
Figure 5a - Measured Pressure Fluctuations at R, =1.2 x 106 in a Steady Flow at a = 3.25
FLtA 33 PERCENT CHORD LENGTH ?3
-
PUN NO. 80 FOIL -UIGLE 10 PERCENT CHORD LENGTH P19
WF11 3.PEOR 3
P
________!
PRETWMLENGTH % S
~
r
'~ As 0
10 PERCENT CHORD LENGTH PICR~3.z
Figure 5e - Measured Pesr lcutostR in a Steady flow at a 3.25
5PERCENTCHORDLENG1H P2
-.
-rv
---
RUN NO.804 -
-- v-
-.-
crv rcbo nWENGTH Pe 3 -
.,
,P,
~ ~
~
~
-in
2rower
3PERCEMI
Figu
W mEa
~-
~-~'~ ~S
(~%~ .-
5cM Measur Pressure Fluctuations at R in a Ste:dy Flow at -325
W
bsed on narrow band frequency analysts vere codctdbr several test Runs. Figure 6 shows a sharp peak at 4.7 Wcz at, pressure gage Pi 0. The same kind of sharp peak at 4.7 kHz was obser.,ed in all the zun analyzed- A imlrpbediomenon3 was observed at 5.0 kHz at r gage Pr. It is sected ~~~~~that this nuv be reson00ancephnm onascte pinhole-typ pressure gages. The prese gages had been dynamically calibrated up to 2,000 Hz with flat reos.consequently, t.e data given in Figures Sa to 5e were bendpass filtered with 4 pole Butteworib ftes fiam 8D to2,000 Hz. Burtoum3 has shown that the enegy spectrum of a flow fiedd an adverse prer re~afient is cornpmded in a noch narand lwer fouei -Tranges than the ener, spectrum associaed with a fho feai in a zero pressure graient Acon parlsou Of res ponse measurements and sAdbility calculation suggests that the boundary layer at Pms (25 percent cthcrd Iength~fbr all the runs giem in Figur 5&a
-
-
-
E.,enTcEMoALEnum
0
The energy.spectrn based on narrow hand frequency anabsisfron 80 to 5040 Hz of Run 8028 at pressre gage P10 is given in Figure 6. The ma-ium fluctuating signal at P1o corresponds to 2650"Hz. This value is in close a t with the computed frequency of the most unstable ances; see F gure 4a.-Tl e W Voectra of pressure -----es
Figure 5b - Measured Presure Fluctuations at R- = 1.6 x 106 in a Steady Flow at a = 325
-FOX.AhIGLE
.
=
24 x 106
to 5e is fully rubukLt. The measure roo mea quare (rn) pressre uctuations nomalize by the fteestream ymM
A-
I:
predicted to occur at Wec -- 0.23. Theoretical computaticns suggest that natural tranution from lamnr t&turbulent
W 0-
2
boundary 'ayers is to beexpected to occur upstrearn of the
4M~z 2660 Oz Ilaminar
10-3-
~0-
separation point at this Reynolds number. Thus, natural transition precludes the possibility of laminar separa iha n codlnt.aelmnr nd 0 At a speed of 4.8 m/s, Figure 5a shows that the flow
I
inspeed to6.71 m/s, cuwrespr.lding to R. = 1.6 x 101% a strong pressure fluctuating signal -Aith intermidttency is noticod ai. 10 percent chord; zee Figure 5b. The bursting signal observed in Figure 5b, is related to naturlal transition.
0~crease
g
o i0j A~a _It
~. ~
%rasfound that the computed amplification ratio of e6 corwell --ith the mea-sure location of natural transition
_ -related .
U MOREN"dC
-
MO Hi
Figure 6 - Energy Spectrum of Run 802 at R,,
1.6 x '106
head are given in Table 1. The values of AC14 range between 0.0065 to 0.011. Burton reported that the measured rins pressue fluctuation coefficient normalized by the free-streamn dynamic head is measurably smaller in an adverse pressure gradient (around 0.0078) than in a fa,.orable pressure gradient (around 0.0101. Huang and Hannani n-easured 0-015 on an axisymmetric bodiy. The present data were digitized through a bandpass filter fromn 80 to 2.000 Hz. The loss of high frequency range may influence, the resolutions of pressure gage measurements and underestimate the magnitude of pressure fluctuations,
~
picaal
pressure repor field ofthtt" individual spots are approximately equzl to those in uefully turbiulent boundarl ayer regimn. As the internttencv factor Tincreases. the power spectral densities increase systematically to a maximum value in the fully tur-
I
4"
'12
anm 621
is
111 2
36i -
--
so
OS:1
*0"0 O D*S W
w
0117
Intl
elm
6M
69Wr
cam
*a
Gas
0
:: ::W
for the preent experimental setup at R,, 1.6 x 10s. For other Reynolds numnbers the comp~uted locations of the amplification ratios of e6 were used to indicate the start of natural transition. At a speed of 9.75 m/s, corresponding to R~=2.4 x 106, the stabilty calculations predict an amplification ratioof es' at Plo-teasured pressure responses suggest that the flow at 10 percent chord is in the final phase of transition with an intermnittency factor close to unity; see Figure 5c. The rms pressure fluctuations normalized by the free-streamn dynamic head, are given in Table 1L The measxred rms pres!Pre fluctuation at 10 percent chord in the transition region are seen to b,- greater han those measured at 2-5 percent chord in the turbulent region. This observation is based on data analysis u,-th a bandpass filter of 80 to 2.000 Hz. In the few cases when the bandpass; filter upper limit. increased to 5,040 Hz. the same conclusion held. The properties of the wall fluctuating pressure field associated with the growth of Utulent spots in a natural transition boundary laye on a flat plate have been measured byDeMetz et ai,'3 DeMetz and Carela.14and Gednev?5
bulei t
.
p?( y)i? ( y)n
(7
Thee rsuls sgget tatthe pressure fluctaions measwed
in anatural trnsition region ame smaller than those ina Uinbuent region, wiiis notminagreement wih the p-esent
SKNUSRTRACIEDmeasurements.
Pressure fluctuations in regions of natural flow trasilioni on an axisvmmetrical body were also measured byv Hag TABL 1- oryrMEANSQURE BUNDAY LAER aPRESSURE FLUCTUATIONS AT 25 PERCENT CHORD2nHanL'Co ardtthprsuefcttos ured in a fully established turbulent region, the intensit o (P~j AND 10 PERCENT CHORD (Pig) in the r-aa;Swth i present measuremnents were made in I ....sqwith adverse prer-
Additionll, 1Tle 1 shows that at a given location Ciute preMefctuto coefficients decrease with the an inres aspee. - sme tred wa also reported by Burtmby Ldwagand Tllma-- sm~edzero
ainEquana ebosp sr mins hzate tion (7) is dtrived fronm measurements on a fla Plate with pressr gradient. It is suspected that the evuse o ff
tha th msic fft~reun eadshid e sedto hathboundcollpsethedoULAD fas ft~gs uggst fixuatm mffiien tobe ed or dl"kMxieaog s o aunvrr-couisbtnt htea i afuctonofReynolds as se i R.t Stalihty11.. anyctio.ratios Of es, et mid ell we reacied at zlc reqfetidy.-Lmiac sepoaim is 0-10. 0.13 and 0.lf%
m~s, c0rtoponding 2 xz106, wegivem n Figae 5d. 1is figure sugests that the flow is turbalent at 10perent chord; this eoerelates with the staliliy caklaiis reached about e' at; wigch indicated that the arlii 5 b eattsi.Fiu thatthe loraion i to mve aea o Pe.The measured pressr Asueda at this speed is sii, hnteoem ~coefficienttrasiimha las fact sees to suport the aun speed of 9.75 m&~s that the peeuu-e fluctuation in a transidan reion is greater1
__________
-~ -
-
-~=.-~-,
-%
a _
7
-~~~~~5
-
_
_r=--=-
than
in a turbulent region with an adverse pressure gradient. A further increase in speed to 14.94 rots reduces the pressure fluctuation further as shown in Figure5e. As a final remark, Figure 5d shows that the frequently-occurrng large negative pressure fluctuations are approximately-2.5 times higher than-their rnms value in the transition region at Pio and ?_0 tie ihrta hi i au nthe fully establihe tubletreinatP. hs rslt r in close agreement with the Huang and Hannans mea.,rnients.10 Huang and Peterson' 2 suggested that the frequent-
periodic part has been bubtracted. To accommodate an oscillation cycle of the foil on a reasonably siicd sheet, the trace .--s processed at 2 inches per second paper speed. (Figures 5a to-5e were processed at 10 inches per second paper speed.) The first line from the top gives the instantsneous foil angle. In this example, the frequency of foil oscillation is 4 Hz. The instantaneous foil angle is given by a -3.250 -l 2.l1 sin wT_ The speed is 9.75 ails, corresponding to R, = 2A4 x 10' . Recall that in the steady run (u - 3250) at R., 2A4 x 106 (Figure 5b), the flow was in the final stage cd tranisition at
ly-occurring large negative pressure fluctuations in transition may be responsible for cavitation incepton events,
the 10 percent chord length. Thie low amplituide, unsteady signal appearing at 3.3 percent chord length is dte to mechanical noise caused by the oscillaitor motion and the elec l ihntercri telnn
4.BONARY LAYER PRESSURE FLUCTUATIONS
WITH FOIL IN OSCILLATION
RNN.O
4.1 The Oscillating Foil
5JJC~
ms
osIl il oscillate ion euocd escvly pichn .moio Let tedn he instantaneous e anglfoil tgvnoai dnotef hy l P~t)=P, where O q r
+IP sin q (wt)P
ozetema.Mage osc~~Sample andlogap
denoe th phs anl the
(8)~.
mltd
ffi
t&hfi
Recordio at~ecy Resl2.4 location pressure response age at onie h
foil
ofsiio prcatio
we.owese£p sin
*
+fe P,AP.T +sue
106. 2
tnstaulentu thati-A th locatio ofao iseafedofg. fhe angmebeginsen toca decreaseethe of tansiion e tooor tharalngeg. A.t*en irmeiae ol nge the fo+ P-t to P.1 trasiio
the heewdt ~ ~~~ ~ eemn ~ ~ h ~nluneo ~ woder ~ eldecroe ~ ~~~I fo eoesclyribentsre agazi = in. rw onse 25cn oation wit et consid the pressure resp 0at osIll VItraoiio I anangigesole.drylvr peren chord lrgh rIFigure u-ggests that flothe mnra w is P1 . whenthei ang at he location of transiti aedo P., te ndrvdengthAstheigl ageMgs te when the fOpJi AC." Fratpclpoee atnbehd an-inane nle egnsto moeese fraithe in oftanslitio ovesin shaf, te litin suracerqwrentngaroplle lae wll tar ce rafil ange.e- A urithermeicrae fi ngefoi em~t a7AeIoiand j r eld.Tenomlte ea pesur art.gte sioguaes ofullesturelenctuation areItsuld e th fluctuaotio pes con t sl sp e ntel flo presur rasitid isatsmthe occure at P. Aftethe folil eachby a teioari. andte ustady vartio offi .aegssocat nto fa thefa to~ angl a nto ireace ind gte memnte if th 0.7ndenper ressare fluctedations r ebe tiv ane of tadirect cri i f pressowrte ladtig etwelially and pt wtrk a es.e Ther wasecto deted rmietiluentce offol lo ae to bemadiesonlltheouirbul pomt Th flo re- in vriation on trntin and bmagnud-ae o boundar laers ext. *b Wentthe olrngesrehes. at 10~ fent prenssur flctigate bychar h iwimydscie d ynth a wi Intesatas e ,flo naglenA. tatonl is tw2~om hydife saaidfy agan 1-1s at sugAests decha enfoi roueat M7pa A. Desction epmswt Pl~ir~ aina lwhe the fo l Tge, i a-: Thes at Pof tdn i -
t i a oc en prss ure The of prur-e Ia I' m fraon Sto20 .The W~therhe lowpfrpeedcin
cb.d
Aif--f-ftcl o 05
o
-0deresatfa vrabri
eproiuesare ulgien of rUFre O ain the
0,115toacuut
igure . &gre8t shols thte h M xnwnM05 at r thebfute
in
o
wrece
i
gge-
i
-i
-
angle a 2.6 degres PFlgure-7 gives ameastured large regaive prsuefluctuatin okSf A = 0.042. which is up-proximately 2.7 ties hier than the-peak value (mis) shown in Figuire 8-This result is in agreemient -with-the -previous$-ationary runs
[
_____
___ ___
___
__oscillation.
z7
iraw AT tirnVsKcoo W
m AT
w
J
-~~~~~
merrda nressaanA oilage- of abiout 3.5 egrees-(wt-i180 degriees)the locatio-nof tanD.!toni apro imately coincides with Pie. A further decreas in_foil angle mio-es the transition location toward the-trailing idgeanlhe ~pressure fuctuations mesue at 10 biento dease Tstion is seen to pass th,,ow,4 t -cein every cycle6 This results in -theapparnc of-two-peaks-4 in presre fluctuafions. The .vol sew~enceofaraonn presmr, iAutuations wihLtnaeu olangl cmn he-
best illistratedwith largeoslltoaptds -oc0aihafp
AmGtE
-
Figre8 )earre
R
S
Figuree 8-esue -M
JFO
on5Run 8043 r onith-S
Pressure Flctaton arssr Phlu
4.3 Unsteady Effects
bfistream
A series of dvnaxnie runs was conducted to exainir uRnstead effects on boundar layer pressure fluctuations. A
peed of V.-
an3
9.75 rms was used for this series
to
t
=
of experiments- The isantaneous foli angle a is given by aI range fr0to21 egeeThe requenc, f cv~x, covered therange fro4 to 15Hz. Due*.theincreased background noise. the presr fluctution arsociated with the 2 Hz filu oscillation are not pented here. Recalthe case a, -0. orrespondingto Run W40 as given in Figuze 5c. The-flow at P3 (3.3 permet, chord length) i . to be laidnr. 77eflowat P 10 isseento be inthe final_~as of tiansition The flow at Pr-,is seen to be fully
I I 0'---. Si-.eu, -----
4-3.1 Pressure Fluictuations at 10 Percent Clard length (PO) Mrat case with.or - 0-5 degrets is considered
.
125
4=
am S
lust. It
is remarked tha the flow at 3.3 percent chor-d is laminar
iroioiiut; the wiole series of ns. Because the pressure fludatii~i naric ms ae nn-s onay.t'sda were diiie to .,oa TinS ralue (see Figue 2).Thi cmreqiobili rms pressure fluctuaton at Plo are shown in
~
-
2
3m 21
ft .iq
~ rsueFutain tPnwt MesrdRSPsue ueainatiewh Amplitud of 0-f7 Degeesand R., -4 x 106-
1 Wm1 Ostn
..
veric;Al ais gives rms pressue ffictuatim normalized by tie free~stiem dynAii head lie hoizontal axis giWs e
=
magnitdes of the prmme fctuations are seen to vary wit): the intantnecis a-IsFgrscn~ngsta the lciA of biafion is shly aft of Pie itw -125 eee.As the foi anglesam- i~creasKd tmnsition, Nov65s foruaItoward Pin. The tnre rzrs presure fluctuatons at Pie are -4ste matiody lnter f~ As t1-_ Jlell angleapOwrdies 3.6 dene- the loctio of Utrasition oodn at Pie. ai h agnitu-de of ressrefluctuations attais a nuximumA ~forietbaa n fil q --1 the -- w ion aheil of P -and the mantudeW Of pR I e floduaios -wed atP,.is --ind. Miner the foigle r---I us Uta fd angle he i~orease *=dsiton inisdown-
I tumd Pie ani the intensiy dfpreswure flatmtioiis
*taw.
i
I
kg 0
.
-
-
s
Fgre1-
3
.
a
5
'
-'
asedISPes
a'
s
u
i
.ue
Fluctatinsat P,, with
Oscilaing AmqWitde Of 1.V Degems and I.~2.4x10'
7 J " .. =
whether the magnitudes of adverse pressure grain these two locations control that difference. 5. Consider-the flow characteristics in the region
__.____dients
4t ,'z srn
Soaround 0of
o.
-c ^
270 0
0.
a
.
0-
- 000,0
0
.moel.'
mo
5.1 Cavitation Inception at. Cm,,
00R
~
3.35~ 4.O35t US~~~~~~ 53
0
46
oo
225
will first consider the case when cavitation incep-
.We
tion on the model occurs at the location obf C.. Since the-
35.(o
Ii115IM
ISO
3S
o
31S
3o
t WOW'
Figure 21 - Measured RMS Pressure Fluctuations at P25with Oscillating Amplitude of 2.1 Degrees and R,, 1.2 x 106 RK
~t
R_,,S'MBoLf 4-
004-
0
6I
-for
~~~~12X 10'
0
-
~reduced
S/ 0.033
frequencies investigated (K = 0.23 to 2.30) tion angles, respectively. Led bK e t4 iodynam k angijr prdssure gradient (dad)a versus sta&i ang~ular =74tr gvadienz (dCrJa) at a given location on the foil,
W
0 0
0~ xl
_
rimn4y
IrWcANGLE
45
90
135 I
15~_
10 225
Wj0
te Based i~a Yqaton (),Shen and Peterson showed that teunsteady izrp;flion cz,. for b -z,-- reduced frequency K can be obtained from:
5. INFLUENCE OF HYDROFOIL OSCILLATION ON CAVITATION INCEPI'ON
-* si + I
(
)
~ V~2shift
11 2
at~cavitation inception in pure fluids In practice significant been widely observed in te deviation from this equation has model cavitation tests. Arndt 26 stated that the so-called scale eets aie due to-deviations in two basic assumptions in-
-
-232-
W
-_
_3
AO)
The Shen and Petersonl 9 experimental results showed that cavitation incepition alway,,s.fnitiated aroun xlc -a 0.02; This result is in good~agreement with the Assumption that cavitaDu~ tzWhs tion inception occurs at the locationof~, in the pressure distition, a significant delay in iiception of leadingredge cavitation was observed *ith the foil inoscillation. It was concluded,9 that the influence of hydrofoil oscillation on inception at Cp,, of leadi edge sheet cavitation can be reasonably prediced by.Equation(1) It is remarked that full-sctale Propellers are-gemmrally associated with high Reynolds numbers.-The locations of boundary layer transition and C~may be coincided.2Inthscase, thle pressure f~ltation~ term ACp. must be added in Equation (12).
where P,, is the vapor pressure. If thermodynamic equilibrium prevails, then C
% 5101
I
-(14)
12
~~ -.
Letoa denote the vapor cavitation number.
a
13
/d%
z *9
Figure 22 - Measured RMS Pressure Fluctuatior- -1225 with Thre eynld Nmbes ndOscillating Amplituft of 07Dere(f=1 )angle
Po- P , 0 -(10)
-
where IACpuI and 4are the amplitude and phase ange. An earlier series of experiments conducted by Shenand Peterson19 reported misre values of tie dynamic pressure coefficient C,(t). Fully wetted, timie depej3ee t, ex. perimenital pressure distribtoswr om Ared withrsults Giesing's meathodz, for cilctlatin~g unsteady potentialperimental measurements was obtained for both dynamicA pressure amplitude and phase angle 'within the range of
0from
ci
magnitude of Oressure fluctuations associated with a laminar flow is essentilly zero, the present experiments show that the flow at 3.3 percent chord-length is laminar for all of the dynamic runs so that AC,,t = 0. Theoretical computations a 3.250 show that C,. ocursaround x~c - 0.018; see Fgure 3. Thus, Equation (9)reduced to: (12) ~ Ct +A.6snct4
0
iVW'V
-
_
U 4
1;
5.2 Cavitation Inception at transition
[
~
We now consider-the second case, when cavitation in. ception occurs-in the transition. region -of the model. Aside from the-pressure field, cavitation also requires a time scale in order for nuclei to grow. Experiments on axisymmetri cal bodies by-Arakeri rAnd Acosta 9 and Huang 4 showed thatcavitation inception can occur in the boundai -layer transi-
OVNAMIC PRESSURE DIST-RIBUTION
\5
0
C-CIHuang and-Peterson tion assumed-that:
.4
3.660 3.250
1%\50 0.
tion region wheb the value-of (- CP, - ACt)is smallern 12
4.3
450
V.
.
in dealing-with steady cavita-
where q~and Ct, are model cavitation inception ubr L and potential -wpressurecoefficients, respectively. On the other hand cavitation inception at full scale was assumed to occur-when owp -CN.. The difference in-these twof asswiffptionis is due to time scale associated with bubble 0 growth and flow regime asoitdwith boundary-layer wU STATIC PRESSURE DISTRIBUTION pressureflcuios Consider a previously discussed dynamic run-with an oscilltion amplitude-of-1.57- deg, frequency of 5.5 Hz andoU .5 freei.~feinspeed -of 9.75 iIis,correspondineto R,, 2.4 x -106 and-k oo-0.432. This isa typical value of X for a surface ship propeller if the chord length at 0.7 radius is uised as the characteristic length.-The computed unsteady-potential flow pressure coeffifeenti-Cp, -Cgt) are shwn in Fig-re 23 for sevralvalesof wt; vsee :Tabre 2.- The computations are-based on Giesig's nonr-linear unsteady potential flow thry. Recall that a, =825 + 1.57 sin cot; For purposes of comnparison, the dynhamick and static pressure distributions are I II I 0.15 0.20 sonin Figure 24.-This figure -gives a i6 4:0 degrees at cot 0 0.06 0.10 30 -degrees. At th6-saffn& foil-giemetric-apgle, due to the x~c ~phaO lag,- the magnitue fthe Viynnicpressures-i ~ie abl te sati ~mlleprssues thn earthel~aing~.Figure 23 - Dynamic Pressure Distributions with a This result suppot the experimental finding 1' that leading edge shecavitation isdelayed with the foilin oscllation; ee+1;snt Eualloi114). The saetbiic has -ams been reportedfo aerdynamic stall.-. Figiire 5esuggests that the flowis inthe final gage ofCP1 ~ transition at 10 percent chord, P10 for R, ;;-2A x-106. -
__
__ --
ing transition attain a maximuni value-at Pl 0 around wt -30 dijrees. h measured rms pressure fuctuations giv
__Reference ___
7Fuirther -
-tion __
__moves
__
0.0145. FjgurEd? shows that the measured frequentlyoccuring large negativh rue fluctuations are approx-_ imnately 2.7 titmlarger thnterms value in the transition region; if cavitaton occurs at-the natural transition-point of the ffodel, we have ow- 0.98 +-0.04= 1.02. A much higher value of ACpt = 0.257may be uhed if transitio n -iscaused by 4lmuwa separation. Tis value of am is smaller than'the value of -C .In the previous cavitation tests reported in 19 with leading edge shetcavitation, inception ~always odcund near the lodation of Cp. However, cavita-D3 tion inception on axisynimetric bodies by Arakeri and Acosta,and H~g didoccur in the transition region where-the quanstudies required notably in three areas to determine: (1) why the magnitude of pressure fluctuation terms associated with natural transition and laminar separaare smaller in the present measurements than the values measued by Hun and Hannan'O -with different-instruments; (2) theretically. how the Iocation of tranisition periodically with the foil in oscillation: and (3) the validity of the assumnption used above that the location of transition measured at a steady mean foil ani~ge is-the critical location to brigger cavitation inception-when the fell is in oscillation. This assumption requires further verification. Two different types of cavitation inception phenomena have been considered in this study. One type-isassociated -
-Sa-.
-
-
--
-
M3
-10
0=7
93
1
-
00
152
-jM
-
00
-
z
-9
15
OW
-&3
13J
-. 801
-
0 S
.2
.
09*
0,1
0200
011
GO
05
GAO3 0132
24
0M
0213
0280
51
-0an21
Z6
S2
-0246
09M1
27
7
D
M
-52
ME TO
04
..
W.Ow
3.25
.
00a OM
0.25
0040
CO"04
.1
02m
095
0.120
07
057
01
TI!
M042
cIi
*StL=55H.,3
-
11iLj
-
TABLE 2 - DYNAMIC PRESSURE COEFFICIENT Cpu4t)H with cavitation inception at the location of Cm. The other type is assoeiated with cavitation inception in the transition region.4 Headformi experiments by Arakeri and Acosta, and Huang were correlated with the second type of cavitation inception. On the other hand, the leading-edgpsheet cavitation on a hydrofoil observed by the present authors corresponds to the first type of inception. It is possible that a cambered -233-
-H
1
6. CAVITATION INSTABILITY AND NOISE
hydrofoil with a smaller angle-of-attack, nmnely with a less severe suction pressure peak, could encounter cavitation inception at transition as observed by Kuiper 29 in his model propeller experiments. Due to the existence of these two different types of cavitation inception, Huang and Peterson2 computed a significant scale effect. They provided a method for correcting modellfull-scale propeller cavitation scaling in a steady flow. The present work is intended to provide needed information to compute cavitation scaling corrections in unsteady flow. It is shown that a diagram of mean pressure coefficient, Cp, and low frequency dynamic pressure coefficient, Ct) versus chordwise location must be constructed first. Secondly, the location of boundary layer transition with the foil in osclltin ustbedeerined either theoretically or experimentally. Thirdly, the magnitude of boundary layer pressure fluctuations associated with frequently occurring negative pressure is determined. The selection of this magnitude depends on the Reynolds number and amplitude of oscillation angles u%shown in Figures 8 to 24.
6.1 Foil Oscillation and :a1oud Cavitation Unsteady sheet cavitation has been recently reviewed by the 15th ITTC Cavitation Committee. The emphasis of this portion of the paper is on the salient features of sheet catty instability. Tanibayashi$0 provided an insight into the -ubject with his description of cavitation on a propeller in both uniform and nonuriform flow. He concluded that the presence of sheet and bubble cavitation in nonuniform flow can be predicted by quasi.steady methods, but that the collapse process cannnot be predicted. Unfortunately, the details of the collapse process are the controlling factor in erosion, noise, an nuedsrcual vibration. When the sheet cavity produces "cloud" cavitation, erosion, noise and vibration are observed to significantly increase in magnitude. For example, Chiba and Hoshimo3' found that the induced hull pressure had superimposed upon it pressure impulses produced by the cloud cavitation formed from the breakup-of the propeller sheet cavity. A physical description of cloud cavitation and its formation has been given as follows: 19 1. A large portion of the sheet cavity becomes highly distorted and undergoes a significant increas of overall cavity height in the distorted region. 2. Once this distorted region begins to separate from the main part of the sheet cavity, the upstream portion of the sheet cavity develops a smooth surface and reduced thickness. 3. The separated portion of the sheet develops the appearance of a cloud and moves downstreamand expands away from-the foil surface. The trailing WVg of the smooth-surfaced region then moves down, stream, becomes unstable at itstrailing edge, and quickly develops the characteristic appearance-of the leading edge sheet cavity elsewhere along the span. Alternately, the trailing edge of the Smooth portion of the sheet cavity moves upstream to the fol leading g as the eavity collapses and disappears. Photographs depicting this proces can be found in reference 19. -
2.0
STATIC PRESSURE DISTRIBUTION AT =4 0 DEGREES
-e SAT
U
~
DYNAMIC PRESSURE DISTRIBUTION =30 ANDu =4 DEGREES
0I
Ito,32 reporting one of the first detailed experiments on the subject, compared unsteady cavitation on propellers in a wake field with pitching three limensional hydrofoils He concluded that the reduced frequency for the blade element of a propeller in a wake field has an important influence on the formation of cloud cavitation. The imp "cation of Ito's Work is that the wake field and the propeller should be considered together to minimize the adverse effects due to the formation of cloud cavitation. by Miyata, et al.S with oscillatin wo-
W
-o
dimensional hydrofoils instrumented with surface mounted 325
157 sin
pressure gages showed that unsteady wing theory was ueful in explaining the relationship between the time-dependent pressure distibution and cavitation. They concluded that the cavity collapse proess is strongly influenced by the unsteadiness of the pressure field and the reduced frequency associated with foil oscillation. The present authorsS provided further details on the
t
instability of sheet cavitation and the formation of cloud
0
cavitation.lTheir experiments were done with an oscillating two dimesioral hydrofoil over a Reynolds number range from 1.2x 106 to 3.6 x 106 and reduced frequencies up to 2.3. The results indicated that the principal controlling parameters were reduced frequency, K, cavitation number, a, and foil oscillation amplitude, .The maximum cavity length, is a function of these three parameters and cannot be predicted on the basis of K - 0 conditions. The role of reduc-
Weo
Figure 24 Comparison of Dynamic and Stai Prssr Distributions at a = 4.0 D Stgrees rlac,
ed frequency can be demonstrated in the foowing exame -234-
ET
IF 160 For constant o, it is possible to have marginal or.no cloud cavitation at some finite K, even though it was present at the maximum unsteady angle when K = 0; -6131.-1316 The importance of reduced frequency has also been 1/0 shown by Matveyev and Goishkoff.33 They reported that for propellers in a uniform flow field, sheet cavitation was less i1o noisy than bubble cavitation. However, when the propellers
-
V_ (MIS)
RUN
0 01411-1416 - 43
ic-iaan 3
-
11
1.13
16A
121
16.4 wt 5+°Asin
1.16
11431
were in a nonuniform flow field, the sheet cavities become
more unsteady and sheet cavitation then was noisier than bubble cavitatiott. As the work of Matveyev and Gorshkoff points out, it is of crucial importance that a similarity in cavitation time history exist between model and full scale. The noise scaling relations assume this similarity as a founda tion and thus the importance of work such as reported by Bark and van Berlekom' s is reenforced.
Bark and van Berlekomn tried to assess the cavitation
noise produced by a propeller in nonuniform flow by studying the cavitation noise generated by a pitching hydrofoil. Based on a correlation of photos of cavity life cycles and their associated rad'ated noise, they concluded that good simulation of specific events is important and that these important events are not generally described by simple parameters (e.g. cullapse time, T,; variation of cavity area with time, A(t); and maximm cavity volume, V, ). One of the most important aspects of-the process is the separation of the cavity (i.e. cloud cavitation formation) which must be correctly scaled. They found that cloud cavitation formation can occur at an
in early stag~bf the sheet cavity collapse. Furthermore, agreenentith tOW results of the present authorsl 9 a com-
1zr
-
0
% e Ma W s 3 C CLOUD
HA M
VTATION
&*/UGHT
CLOUD CATAiom
40
ie) city length and high reduced bination of lng (e d frequency causes extew.re cud cavitation. As can be concluded from the data presented in their paper, large noise transients were associated with the cloud cavitation.
i
6.2 Cloud Cavitation and Noise 43 " The observed phenomena to be discussed in this paper 0.4 0 can best be describeWby reference to examples shown in ofREDUCDFRUENCY.K Figure 25. Table 3 summarizes all of the parameters
L
fi
_16.4
__
___ _ _
-
_
_ _ly
__
_-235.
cd wi,
th tests repo t
here.r
t
0.
12
1
f
Of
the test program, air content of the water was not varied. The air content was measred with 70 percent saturation in reference to atmospheric ;,resure at a water temperature of 22.20 C and tunnel pressure of 103.6 KFa. For a velocity of ms, - 3.25 + 0.95 sin wt. o = 1-21, a plot of relative sound power P2 versus K shows the existence of a "noise bucket." When a - 4.30 and K = 0, extensive cloud cavitat-tion is developing from the sheet cavity and the noise level is high. As the foil is oscilated, a leading edge sheet cavity experiences.an inception, growth, and collapse cycle related to the impressed pressure distrun of the foi. At low reduced frequencies no cloud cavitation is produced, the sheet cavi-
strong influence of velocity is again apparent. The influence of velocity on the radiated sound power is ju as dramatic when the amplitude of oscillation is increased from 0.5 to 1.55 degrees, keeping o constant, as shown in Figure 28. There it is seen that with the larger loud cavitation is present over the amplitude of osclation full range of reduced frequences. Although the violence
ty collapses toward the foil leading edge, and there is F
associated with the cloud cavtaton at lrge reduced
significant reduction in the noise relative to the conditton of a 4.3 and K = 0. As the reduced frequency is increased further, a cloud cavity is produced during the collapse of the sheet cavity and the noise level again Lncreases due to cloud cavitation collapses.This variatinin2as aftio foi offfi ciicsfunction angle is shown in osculograph records, Figure 26, for 2 foil oscillation periods and in Figure 27 as a mean noise variation based on the average of the cycles occurring in a 40 second period. Figure 25 also presents data that demonstrate the strong dependence of 2 on the water velocity. For t = 4.3, K =0, and o = 1.13, heavy cloud cavitation is present and the noise level increases by a nominal factor of 30 when the velocity is increased from 11.5 ms to 16.4 mis. Once foil oscillation starts the amount of cloud cavitation is significantreduced and the velocity difference appears to have far less impact on the radiated noise. At large K when the clo d
cavitation is produced upon sheet cavity deinence, the
Figure 25 - Influence of V and Reduced Frequency on Relative Sound Power fort = 3.25 + 0.95 sin wt
4 -a
-
des limited the ability to collect data, it is apparent that has a strong dependence on K when the cloud cavitation is present. Reference to Figure 25 shows that this latter effect is present. is not present when little or no cloud cavitation o perod is shown p2 during the The h vaion aito in n2drn h foil oloclainpr hw by oscillograph records in Figure 29 and as a mean variation based on the average of the cycles occurring in a 40 second period in Figure 30. Based on the limited data presented in Figures 25 to 30, it appears that for cloud cavitation originating from an unstable leading-edge sheet cavity V, VL when o, K, o, ai are constant The maximum cavity volume, area coverage, etc., of a leading edge sheet cavity have in the past been used as parameters associated with the magnitudes of the cavitation noise. Figure 31 presents the maximum length achieved by the leading edge sheet cavity as a function of the reduced frequency. From these data and the noise data of Figure 25, it is
clear that the maxinmum cavity length has essentially no cor-
1 _
-
~~ ~
relation with 2. This basic onuion wso deduced by Bark andBErlekom. 18 Az has already been shown, the princia sourceof noise is th colla e of the cloud cavitation gnratdbteshecaty As shown in Figure 32. forK smiall reduced-fte-uendes, hd sheet cavity-disappears after12 the rided noise level hs peiked. However, is the reduced n of cloud cavitation for frei~ncs icres p ressivly delayed until it occurs- at the sheet cavity d6sinent ciitioin, the peak amplitude of the noise occurs afterthe sheet ca~vity has disappeared. In fact, for the educ. ed frequency of 1.65, maximum radiated noise does not occur until the foil reaches its mininam angle of attack. Before the doud cavitation comletely disars the inception of a new lf-&-ing edge sheet cavity has occurred.
RUN-NO. 1411 026...
l 018V,1MV
-
I , -FOLAJt
CAVITATtCN NOlSE
T
-: UN1vz
-
Mo
14W
1_
14-"1
1307 1
S-
28
143 147
11.145 3,3141
"
2.8
11 "41
1642
143 7, 143 10
1~ 2 16 2.
-
-
j5
Zn
55 ,
~
UN so1 '151 151
015
K
-- 5
826
1.14
115
022
1.14
13 ,,5
GM
121
&M3
625
75
6345
4-M00.
1=
Is :
is
GAP 113
1W42
4.
1642 W142 16.42 1W42 16M 1642
43o 4.3
0
40 U*
55 S
8 63-1.15 6
4A 49
4 is9
am14 886
152 *642 W642
4* 44
7. W6
am5 6182
4*
15: 161
.
26
RU
4
'" -
--
43
14 &1142
42 -- 141
6OM
FIL
/ /
K
O.on VIf
1642
=1
IsU0/ 15is ga1l 78p 1.1531
0 a--
+= M an _
VI
16 an
960 an3 M 11" 63 so3
328'1
81)
0432:
iiI
63 3.6' an on. 02 an
4
U
27U- Relamv Sound Po*&i(.j as s
~Wa
.
X14
UBESADTABLES-qiz TASSEO3 -TET
"
-Ii
9M5
0-
*
-
Paws1411andI1416Gaw.25+O.95 sid ct_
iss/ 1 -
121 121
SS
~.
26- Cavitation Noise Signals-/P2 and Foil Ahgks-
--
114 121
M
-
~ -..
-:
E
1
s 1
for
1/./
-513 0.15
C
..
-
~ ~SFigure ~'for
13
,
V1
1.16
A-\
Is$ 1.123 11CAVITATION ,S,-4, .4 135 113
az 9
401
-
1-,
1.1
o --8 -
0.4
is 4
40
1.13
11 3 ,1
0.46
28
75
23 3A
UM1
no0
:1.5
,842
"a
-
1.13
IM.4 1414W 1642
-- -13M
_CM
6am6
55
Il142
03.5
a6 51
4 1. 5.
2_1 -
USK
ta1t
1.13 113
13
1474.34
MR 14
038 01o62
033
7S
3
~11,45
141-
4
S,
11i 13W
10S 5.5
is
116 ,
Imaje€ -0oas 9
f 25 HZ
ORis
NO. 1416
cRUN
l.a
DEG
-A
H&
t0
"
to.
~
i'.
-33
oll 6 Artie for R=_ 1411-to 1416 loith a. & a for Sheea Caviiu
ADASSOCUTEPARAX, -236-
V.Wms)
RUN
,RUN 1.3No.
Alot14 5 A140141" 143 olm--14w 14*
2 40
1.13 1.5.nII
a m325 15si114
x
V.tows) 1143
f(izi 4~ 5
lIMA 1141
10
-~1401 U--146 30
75
K 0264 I 0212 ' 0.4 CASSu
j-
ItA
i
0
*
o
HEAVY CLOUD
100
43-
0
0*
A
12
1A
25
Figur 28 - Influence of V.* and Reduced Frequency on Relatie Sound Power forea
_
3.25
+
24
3
0 I
ogs 4ANIAU"-A
-S4.4
1-55 sin cot F~m30 Reaiv SondPower P(a)asa uboofFi
RUN NO. 1401 f - G4 NZ
frR s10t1406-
An
-0 1W W-
02
.
lV
-Z
m
I
IM
IW TO2*
0g2
0
1.12
S
sm
1.12
0*
0
6.02 4
-W
TOI
es
U1
14
is
s
READUEDOOMYC.K
-
CAVITIOff NOISE
ene
a~u
af
e~f
-~ue3
RUN NO. 140 1
L-MI an -Kn'
IA
2A FOU.ANGES AT~
-
__
I. ANGE
A
___
~Fiur 29 - Cavittion Noise Signals for Runs 1401 and 1406, a
_____-
0 FOIL AMEUS AT C,
0 1
CAVITTO 'fP2 and Fbo
3.255+s155 +
~~~~ FOIL. ANGLES AT
L$______ 62 *A4
OS
F~gur 32
Aqgles a
92
1 is
uEouc:EoF&oUUACV.
12
IA
Is
12
MeX&aad Foil Angles at Maiufm MftMa~dzw Nonse Peak and Loftiu Edge Slee CaVit Desinerace f&r Rus 1401 to 1406
Cavty Lentb.
237-
-6E CAITY e~E~
so
In many-instances, partially cavitating hydrofoils are
subjected to conditions that can be effectively simulated by small amplitude-osiilation during which the leading edge sheet cavitation is continuously present. Based oi results for the intermittent sheet cavity, it is-known that sheet cavity stability, and hence propensity to produce cloud cavitation, is -dependent on the reduced frequency. The noise level variation associated with this type of cavitation is found in Figure 33. As with intermittent sheet cavitation, the noise level is low when cloud cavitation is not present Once cloud cavitation
32 + 0 "2i ut/ V.= 164 ms O_ (RUS 150 TO 119
"
X
cc .
a
formns, then there is a dramatic increase in the mean sound power level and in the time variation of sound power level, as shown in Figures 33 and 34 respectively. It is apparent from these results that the desinent condition for the sheet cavity is not required for high radiated noise levels, but rather the existence of conditions that promote the formation of cloud cavitation. If a leading edge sheet cavity is considered as similar to separated flow at a foil's leading edge, then some parallels can be drawn with the vast body of recently published data on dynamic stall. For example, the following conclusion from McAlisier and Carr34 closely parallels the description of cavity stability given by Shen and Peterson.19 "The free-shear-layer that was created betv:een the region of reversed flow and-the inviscid stream was not
4o
to
!;i- 3
"
()
0-
ot*
G
stable. This instability resulted in a transformation of the
REDUCED KAEGUENCY. K
free-shear layer into a multitude of discrete clockwise vortices, 'out of which emerged a dominant "shear-layer vortex'." McAlister and Carr go on to further describe the upstream movement of a thin layer of reversed flow along the foil surface. When this reversed flow reached the leading
edge,"
IS
1-
Figure 33 - Relative Sound Pressure P2 Over One Cycle for a Continuously Cavitafing Hydrofoil
a protuberance appeared over the first 6 percent
of the surface in response to the sudden influx of Did. This protuberance grew and eventually developed into the 'dynamicstall vortex' that has been observed in high Reynolds number
RUN NO. 1510 t- 4H K- W85
experiments." This dynamic stall vortex moves downstream
and away from the airfoil surface just as cloud cavitation does on a cavitating hydrofoil.Furthermnore, if the location of flow oscilatin K - against 0.25) will suppress movement reversal isji.e. plotted airfoil angle,3the 4 itforward is seen that foil
-
-----
knw~c - 047 V. -
s
_
a03
--
of the reversed flow region. This is again similar to the cavity stabilization at low reduced frequencies relative to the sta-
=
- FOILANGLE . -
__
tionary hydrofoil.
These types of analogies must be used with great care. For example, one of the conclusions of Telionis and Korom-lasTfrom their study of unsteady laminar separation is that separation isnot affected by the amplitude of oscillation. The parallel with cavitating flows mayconvidefations break down here due to, among other reasons, the inertial of growing
-
-
-
RUN NO.1515 -=
K-a 1.1
-
.
-In
7. CONCLDIING REMARKS
-
bnwdcio.S V.
G.W-S--
.
Depending on the cavitation resistance of the liquid, cavitation inception on a model may occur either (1) ;t the location of Cp-, or (2) in the transition region of te model. the present experiments for a hydrofoil with a large suction peak, !eading edge sheet cavitation was observed to take place at Cp.. In this case, the boundary layer was laminar at
the location of Cp,
-
5
cavities. Telionis and Koromilas also have concluded that for. finite oscillaton frequencies, the point of reverse flow is shifted downstream from the quasi-steady location. -
I CAVITATnONOI
.
FOIL AGLE -
VA
:z -.
and an unsteady potential flow theory
-
was shown to provide a good correlation between prediction and experimental measurenrnts of cavitation inception. For the case of cavitation occurring in a transition region, it is shown that a diagram of mean pressure coefficiA ent and low frequency dynamic pressure coeffient versus chordwise location must be constructed first. Secondly, the location of boundary layer transition with the foil in oscila-
lik.o
lion must be determined either theoretically or experimental-
Fi
ly. Thirdly, the magnitude of boundary layer pressure fluctua-
-238-
e3--Caton Nose Sial for Runs 1510 and 1515, a
fpa and Fol Angles
3.25 + 0.32 sin *t
L
--
cloud cavitatioa is present. From the limited dataobtained from these experments, the sound power ars to ypry as: cavitation a associated with cl
tios associated with frequently occurring negative pressure is determined. A general-theory on boundary layer properties ii is not yetavallable:The Present witha hydrofoil in o
wbk is tended to-provide some needed iformation this subjet the location of trnsition and the mnitides of bcuzndary layer pressure fluctuations. Experimental results -show:
5.T!he npltudeof 6@cl~atona 1 ,lms an ifuence,on
1.The movementof boundary layer transition can behe detected by the earement of pressure ft-availabe of~. dtnote foil f
ca
-- on the foil ,ions t i e. 2. The development of the foil's boundary layer is -
ti-S point 6. The aittion DbOise geneWa1 by a Sationmi-foi-
delayed with awincrease in oscillation frequency.
3. IThe
nise-in tha lage s mo data u However, the fii t do not Permit more detailed dscud&i of
s
indtiveofJhoise p 'dued
t
foil
-is oscillated.&
p r~ alcs associated with advancing bran-
siti6n are independent of oscillation amplitde and feecy and we identicl with the values
7 The redu
fe ncy p. does not inThus cavity dyn rp t o it isD_-sufficient to p i -twi:>ueme of f e-
ured in the transitio regions on a stationam. t
ifeam velocity on the groas stability of the 16dn
4. In a ful established turbulent region, the mag
niades of pressure fluctuations are ilmdependent of oscillatio amplitude and ftequency and identical
edge shet cavit. bs influence ofvelocit should be In.es1ited furtm-r In order testabls the
with the values measured in the t sainrfol.
criticl Kat whh th filscalespeedsM
t region on
in a 5 .Toift f pressur or.the reent foilreoges higher MV ty in a r v tan i a .tc r etem followidg wer deftieil&hyrpoe esr sihdrooiwaterrpehe -n
stswere derie fomeddur teil STme
mi
J
a-
noie
reait
e
.The unsteady -caviAion is a toi justi its y nic state of understanding. Recent res c - ina i
t
requires i a _iW4i
uniform flo
ydaeofeet ay
d
cannot a
6f the Rfo
t
eld i which thi operate
-
te som of the riticaldetais of
the i p ,growth and collapse process of leading edge sheet cavities ,
ii stbt-of a leading edge sheet cavity deterthe xent to which cdoud cavitation is pro-
I2une formation of loud cavitation during the life of
AO231
DGM
a sheed ca-it is suppressed at srja3 reduced fieTe authors would like to thanik Ir&Robert Pieie for quencs and corrspninl the noise level is low3.At high reduced frequencies, extensive cloud cavita~ asalloed _ t h a rceair.Hi evot nt f e tion is formed during the final phase of sheet cavity chiiealwdamn orqatttieeamtonhn W~ ises'"e been hle Tore ams istn g - y is anmi&ani icras collapse~~1 stabilit cmklaflo oflair m "nis ~ ~ ~ plotted Thus agaonst ~ K shos ~ ouda i geeatly appreciated. cavity ~ ~ iscninoswihlyer The orkrepotedherein was fwnde by the Naval ~ e tem Ccimd, Code 035. unde the Gemieral Hydrotime ~ ~~ uce ~ saprn.s~ ~ ~ ~ aSiiayos dynic Research Program, Element 61153N, Task Area SR 4- Whn a. , f and al are kept constant, the in0230101.j fiunce&of velocity on PI is found to be very lag if
_______ _
° __
-239-
;
__ ___
--
gz
-
-=
n~FERENCES 1. Peterson, F.B. mid IE. Amdt, "Unsteady Cavita tion," 19th American Towing Tank Conference, University of Michigan, Ann Arbor, July 1980
16. Miyata,,H et aL, -Pressure ( ractaisltic and Cavitation on an Oscillating Hyarofoil," Journal of the Sockty of Naval Architects of Japan. Vol. 132, No. 10, 197, pp. 107-115
2. Aeosta, AJ. and B.R. Parlin "Report of the ATTC Cavitation Inception Conitte," 19th American Tow-
17. Radhi MH. "Theoretische und Experimentlle
Untersurhung liber den Kavitaiionseinsatz an Sclhwingne Tr," PhD Thesis. Techniscben Universtit
Tank Conference, University of Michigan, Ann Arbor, Julyin1980
Ber n D63,.1975
3. Acosta. A.J. and B.R. Parkdn, -CAvitation Inceplion - A Selective Review.' Journal of Ship Researd, Vol. 19, No. 4, Dec. 1975, pp. 193-205
i.Huang, T.T., "Cavitation Inception Observaions S
I& Bark, G. and W. van Berlekorn, "Expermental Investigations of Cavitatio oise," 12th Smposium on ,Nm Hydrodynamics, Washington. D.C-, June 1978, pp. 470-493
o.
Axisymmetric Headforms," paper presented at the
19. Shen. Y.T. and F.. Peterson. "Unsta" Caita-
ASNE Iiternafmio., Symposhum on Cavitatka Inception,
tion on an Oscillating Hydrofol," 1Mt Soiu!
New York, Dec. 2-7, 1979
Hydrodywunis Washington, D.C, June 197 pp. 362-38,
5. Rouse, H., Caiation in the Mixing Zone of a Submerged Jet," La Houile Blanche Jan.-Feb. 1973. pp. 9-19
20. Van Houten R.J., The Transient Cavitation on A 1wo-Diensional Hydrofol - Comparison of T "ery and Experiment," MJ.T. Department of Oca Engineering, OSF
6. Dily. J.W. and V.E.Joson'd
2,268, Aug. 19,9
Boundary Layer Effects on Cavitation Inception from Gas
Nuclei," Trans. ASM , Vol.
, 1956, pp. 1695-1706
8. Arakeri, V.H. and AJ. Acosta, -it-cous
21. Hilte J.S. et a.. "ASmple Sinusoidal Hidi-li
22. Smith, AMo
- -Transition. Pessure Gradien and
Stabiit Theory." Proceeding, of the inth International Congress of Applied Mechanics. Brussels. Belium Vol. . IW1O57
Efects in
pp. 234-243
the Inception of Cavitation on Axisymmetric Bodies," Tans& ASM, Journalof Fluids Engineering, VoL 95. Series 1,N-
=4,
7
Pressure Calibration," National Bureau of Standards, Technical Note 720, 972
7. Levkov" Y..L and A.V.Cha.o ,"-Influenaof Flow Turbulence on the In q and ,Growth of CaVoalio&'n March-April 1978 (English TrAnslay Soy Phy AcousL 24 tion) -
on Naval
23.I urto.. TE.. -Wall Pressure Flucumations a-
Dec-.191% pp. 519-Z27
Snmooth and Rough Surfacs Under TW tlent Boundary
Laes With Favorable and Adverse Pressure Gradien-.-MT. Acmi c and V 1ionLaboatory. pcrt No-
9. Arakeri V..andAJ Acosta, "CavitabonIncep-
lionObmrraion onAxiimmtricBodes t Sperritcal70208-9 L. Jurnal of Fluid EngrReynolds Nu.mers," Ta AM, i 1. 1975, pp. 82-87 Deering, VOL 97, 10. Huang, T T arid De
Hanmn, .2-Pressure
24. Ludwieg. H. ad W. T1lknan
tcen, Ing-Ard 171949, pp728299. E gnish tansation) i:-
ctua-
-ACA TM 1285,1950
tion in the Regionts of Flow Transition. David W. Taimo Nav'al Ship Research and D)eveloprnent Centar Report 472.11.yArakeri. V _
Unter1sudm nge
,BodrvLM-
Pressure Ftictirations on Sanootl and Rou6 A WaII of Fluid uechanics. Vol 44, O197 pp. 6r-660
-A Note on the Transition ObsvA.
tio*as on an Axiymnmetric Body and Sonve Rebae Fluctuating Wall Presue 1kasurments."Journal of Fluids
alJournal
26.. Arndt, ILE and W.K. George. "Presure Fiels
Engineering, Trns. ASE, VoL. 97, Series 1, Noe1, March 197pp. 82-86 1975,
and Cavitation in Turbule Shear Flows 12th 1968aval 2F5.F.zbe Blake.J. BonayLvrWl -on HydrodynamP.cs. Washington. D-C. June I 98 pp-
12. Huang. T.T. and FAB Peterson, "Ifluence of Viseam Effects an MaddI"tl Scale Cavitation Scalin," Jornial of Ship Research, Vol. 2D. Dec. U076, pp. 215-223
27. Ging. JT... '-oD *nnsionalPotentiial Flow Theory for Multiple Bodies in Snal.Anmgtwde NIoma.n Douglis Aircraft Comny. Report No. DJLC-67, April 1968
13- De~etz, F..G., Farahee, T.M. and M.J Casare&4a "Statistical Features of the Intermittent Surface Pressure Field in a Transition Bon yLayer," Trams ASMdE, Nonateadly Flud Dynamis, Edtdby D-E. Crow and L-A. Miller, 1978, p.3M-9tu,
28. McCroskey. WL. -Some Comaen Researh in UnVe- Flidynarsics - the 1976 Freeman Scolar Lie.Trans. ASE, Journalof Fluid Engineems Vol 99. series 1, Marc lsui pp. 8&3
14. DeMeta, P.C. and N.J. Casarella. -An Expeitenta Study -ofthe Itermittent Prqprte of the Boway Layer Ps emne Field Duriiig Transition wna Flat Plate," NSRDC, Roport No- 4140. Now.. 1973
29. Kuipe G., -Scale Effects aa Propeller Cavitation Inception," 12hSyrmi-no ava Hvdovaris Wairion D., June 1978 30. Tardmyasta. H.. 'PacuWa Appuach to Unsteady Prc, of Propellers." Pro. Second Lip Proneler Symn. posiu, May 10-11, 1973. p 65-728
I1.. Gedmey. C.J., "Well Preware Fluctuatimas During Tmauton an a Fla Plat, MIT., Acusc and Viktion lbauulcy Report No. 806184, April 1979 -240..
-reuet
31. a-u-b. N. and T. Hoshino. "Effect of Unsteady Cavity on Propdlw-r Indkiced Hydrodynamie Pressure.- Jour. nil oi the Society -ofNaval Arcets of Japan. 139. 1976 32. Ito, T.. **An Expmrmental Investigation into theUraeteady Cavitation of Marine~ Propelles Prooedigi of LAHRI - Syrnt~dm on Cavitation and Hfydrauli Macliner. Sendui. Japan, 1962 3.Maneyev. GAL arnd AS. Gorshoff, -Caitation -Noise Modelling A Ship H)*odanie L-Akatories," 12th Synmpommw on Naval Hydrodynmics Wagangtoli, D C.
[
June 1978
&S.Telloris and Kon=las. -1ow Vilation o Tamt and Omsilat Sepradwing iau - T w ram ASUE. Nonstendy Fluid Dmarnics, Efdtd b~rD.. C-rm and 7 J.A&M3lier. 1978. pp. 21-32 = =
Sesikm Ill
HULL FORM 1 -WAVE
-~
MAKING PROBLEMS-
Cha~aira
john V. Wcbausen Univeisty of Cailornia Berkele. Calfornia, U.S.A.
F-
Mathematical Notes on the Two-Dimensional Kelvin-Neumann Problem Fdtz Ursell University of Manchester Manchester. United Kingdom
is often neglected; then the fluid motion is irrotational and can be described-by -a velocity~potential, the stress in the fluid is a pressure-which can reasonably be assumed to be constant at the free surface, hull to the ship'-s velocity normal and the -1ormato the free :surface and the Velocity both vanish. Even this simplified prcblem cannot be solve.d. One-difficulty is that is not the determined free surface hmust th prescribed-but lto o be duling
ABSTRACT A horizontal cylindrical body moves with constant velocity in the horizontal direction normal to its axis, near the free fluid under a frictionless surface ofThe calculation of the resulting gravity. fluid motion is an important problem of ship hydrodynamics. For a-deeply submerged body pressure of be constant the condition fb but at in the tome linearized, free surface can
problem. Thus additional simplifying assumptions are usually made. Many calculations are concerned with thin-ship theory the ship is assumed to be so thin that the fluid motion relative to the ship is ne rly a uniform stream. A perturbation procedure can then be set up in terms of a small thickness-parameter. The first approxim ation is -linear and can be treated mathematically, but its range of validity is not adequate for many applications, and-various non-linear approximations have therefore been studied. For instance, in some work the full non-linear condition on the- shi 's hull has been used together with thr linearized free-surface condition on the mean free surface. The resulting problem is the so-called Kelvin-Neumann problem with which the present work is concerned. This
recent calculations the same linearized condition has been used also for partially immersed bodiesi The resulting linear boundary-value problen for the velocity potential is the Kelvin-Neumann problem. In the present paper the two-dimensional KelvinNeumann problam is studied for the halfimmersed circular cylinder. There are arguments which suggest that in the corners the velocity potential must be strongly singular but it is shown here that a unique velocity potential exists which has velocities bounded in the corners and at infinity. Similar results hold for other cross-sections intersecting the horizontal at -right angles. It is hoped that the experience gained in this work may be of use in treating the full inviscid three-dimensional problem in the _future.
approximation can be justified when the
translating body is deeply submerged, but in recent work it has also been used for bodies intersecting the free surface. It is well understood that the approximation is then inconsistent even for second-order thin-ship theory since only- sme of-the second-order terms are retained while others are omitted. The corrections that are obtained are therefore of doubtful practical value but may perhaps turn out to be in the right direction. It is hoped also that- the experience gained in this way may be-of-use
1. INTRODUCTION
___ -
Consider a ship in steady uniform motion on the free surface of a fluid. It is well known that the ship is accompanied by a wave pattern which is steady relative to the ship; the calcu lation of this wave pattern and of the associated wave resistance is a cantral problem of ship hydro dynamics which however remains largely unsolved even although drastic simplifying assumptions have been made. Thus viscosity -245.
kF
_---
-___
-
----
j7
future. in treating the full inviseid problem in the In the present work, however, we shall not be concerned with the practical applicability or logical consistency of the Kelvin-Neumann formulation for surfacepiercing bodies but merely with certain mathematical aspects of the two-dimensional Kelvin-Neumann problem. The free surface of the fluid-is then represented by a horizontal straight line, and the body is represented by a curve intersecting this line. It will also be supposed that the curve intersects the line at right angles. (According to the full theory, the c.rners would be stagnation points where tne perturbation velocity is equal to the forward speed and where any perturbation scheme must fail, but we are ignoring this difficulty.) It has been suggested that any solution of the problem must hive-singularities in the-corners. that such We need shallnotsee, singularities however, occur if at infinity the velocity potential is allowed to tend to infinity logarithmically with distance. The corresponding velocity components remain finite in the corners and at infinity. 2. STATEMENT OFTHE MATHEMATICAL-PPOBLEM
Y = 0 at Neaicthe corners, riatigh angles. Near rn rs linearization cannot be justified. :Let it nevertheless be Applied along the whole of the free surface. Then the Kelvin-Neumannfree-surface condition is +
= 0 on v = 0 outside -C,(2.4) 2
where /U . -(See e.g.- Wehausen and Laitone, 1960, eqn. 10.15.) 3. THE SOURCE POTENTIAL In subsequent work we shall need the potential of a source at ( n). This will 0, be denoted by G(x, ; , n); it also depends on K. It is known (Wehausen and Laitone, 1960, eqn. 13.44) that, except for ano arbitrary additive constant, G(x,y;EJ,n)=log{K(x-)
2
+(y-_)? +.giKv((X-. ..
-k ) + 2eek(y+n) k-K
-
2 re-K(Y+
2
(y~n)
-
sk (-0
)sinK(x-0)
(3.1)
where the symbol indicates that Principal Value 6f the integral isthetoCauchy be taken. Let us write k-K coskXdk= C(KX, KY).
(.
+-)O(x,y) =0
Properties of this function are given in the appendix at the end of this paper. in particular, (K(X-O) K(y+n) ) - ne-K y s 0K) 0
A cylinder, w' its generators horizontal, is moving alon. the free surface of- the fluid with a constant velocity -U which is horizontal and normal to the generators. Take coordinate axes moving with the cylin- der, the x-axis horizontal and the y-a:is vertical, y increasing with depth. We shall be concerned with motions which are steady relative to the cylinder. The velocity potentia is then written in the form potential is thwill Lix + 4(x'y). since the density is assumed uniform we have 92 a2 = 2 (x +-7)(x,y) =0 in the fluid. (2.1) Since there is no we flow across the boundary of the cylinder have Ccwdipole)
and it follows that there are no wave term Thu G(w e in (3.1) when x - F - -=. satisfies the radiation condition. Evidently G(x, y; , i) is not symmetric in (x, y) and ( n). k It be noted that the source potential is should logarithmically infinite-at iotentai th a iufinite at infinity, this Indsuggests that he solution proole- (see 92 above) of maythe-boundary-value have the same property.This in fact be shown in g4 below. it is
CCC
when
'< I SC
=
i
x Ul-U-
on C
(2.2)
x-
--
=,
important to note that G( ) is not an even function of x- i It is also important to note that the surface singularity is replaced by a weaker singularity when q 0. The potential G(x, y; r, 0) wili bedescribed as a weak surface singularity. as y &, surac (appartya oiztal -G(x, wa y; )/ax (apparently a similarly horizontal is a vortex. These results follow from the expansions in the appendix.
4. THE LEAST SINGULAR-SOLUTION FOR THE HALFto C.IMMERSED SEMI-CIRCLE Since there are no waves formed upstream of the cylinder we have fomdCosdrthe Consi de t e potential p en i l flx, ( ,y) y) == (rs r-in6in r cos ), given by the multipole expansion ax
when x
-P.(2.3)
curve C is assumed to intersect the line
Ua
A AKx, Ky) +BQ(Kx, Ky)
_
f
-r
~~
s(2m1) +
ad bounded in the.corners. Il ~be~ seen, that am 0l/~
.
costl~~~~ mlcs2m1O
~8'
0,Proof-
oi-Theorem 1 fsc ail converging solution exists then we may-put
2M+1
0
~r~~
r r~. P(Kc,Ky) l.ogKr + -e
co
xt~kJ+
--- " snMJ
4
IB//
-\/-p
0
v.I (4s6)
say, Similarly,. on Putting e find that
whers the Cauchy Principal Value is to be taken at = K - (An arbitrary cofistant may be added to ,(x, y) .) it is readily verif ied that each term of the right-hand side of (4.1) satisfies Laplace's equation (2.1) and the free-surface condition (2.4) and the radiation condition (2.3). The boundary condition
/U \i
~gnO, - .
\ /~ --
+B +~ a Q. A/a P \r2 O obnn hswt
~
P \ sn 6~ /
a
in (4.4) We
=
-Ka~Bgsin(mA~) - -1. 47 44 eseta
r~
P o\r2/
+
A
-sn
0
(4.2) (m2
i
1n2
I/ (
/&
sn
on the~ semi-circle C is also satisfied if tne coefficients A, B, a., , are chosen ,so tat in the interval -2-v ~ - he equations A/a&
(4.3) and obtain
kx-@,B -4XySi . I +12~C -e =IQ (Kx,Ky) I + U02(xy)~
£
!in
~msm? = 0-(4.5) On conbining this with (4.3) we see that
1KxKy))1i, say, and Q(Kx,Ky)
(Actually it and
a.-,
(4B
,,Q(i 2+)_ -i~.nmh-
P p
BB
_-Q0(cosA-Ka) -
(2n$+.l)n-a
We shall now verify that the Ltolutions of 14.6) and (4.8) satisfy am = O(l/m1) and am = 0(1/mir. by transforming (4.6) and (4.81 into systems of simultanaous equations. e.g. (4.8). M&ultiply in turn by the complete set sin(2t+2), U~ 0,1,2,., and integrate over (- -. 1T). An infinite set oi equations of the form
Q (4.3
r s2mI t cs 2r rn4 n:2Yf+1Consider
=0
and Si 'aP\ B-a-Q ~\r2/ \r /(4.4) (2nf2)8n(~m(2 (2+2 sin (Z22 sin (2m+l);-
-
2+2)B.4a Z )a
=~
0.
0
(--r+2) i
c_, z. 012..
.fl(49
obtained where cz is the Pourier coefficient of the left-hand side of (4.8) and where the coefficients am can be found explicitly bx- elemenfary integrations-. Write (4.9) In the form
ITOis
(Here and elsewhere brackets are satisfied. < 's are used to indicate that r is put squal to a- inside the brackets.) The equations (4.3) and (4.4] are series expansions resembling Fourier series. When A these and B have been chosen arbitrarily equations can be shown to have unique solutions am, Sm (m 0,1,2, ...) exceptm posibl dscrte ata et f ireglarBy values of Ka, and it can also be shown that 2 am= 0(1/n 2 ) and.fim =0(1/n ) by arguments (The used in the proof of Theorem I below. details are omitted.) The series for the two Velocity components are thus seen to converge only slowly; at 0 6 ±3," theth horizontal velocity is in general d~is-th velcit isthe an verica th t cntiuou ancontnuos th veticl vlocty s logaithicaly ifintepossibly We now obtain~ our principal result.thr
(2+)8~ 9.d
~2
)a(+20
2
2c
4-O integration 3by parts it can be shown thtcZ-(/ ),-tu X2+)4~C2 is convegent and1ithuaso be sowntha hw ta lob a cnegn;Adi ( 2 -1-2 is c2vret h hoyo 2m2 t sy em(;0 i hrfoea lgus o syem(.0 is hrfoealgusequations o of theory of Fredholm integral he second kind. it follows that (except at a set of irregular values of Ka) isauqesoton()sch ht
Theorem 1. The coefficients A and B in (4_.1) can be chosen uniquely so that 0(1/n ). The corresand 0(1/n am ponding solution (the least-singulara solution) has velocitie which are continuous
To Show that 1(2-X+2)58 2 is convergent. rm(.1)ta O(l/Z41 oe ZN 8t
-247-
£
i(2£)3+(2)+2)2c2(a)
iz(K)
2
IV(a2I (m) ( 2
immersed dei-circle; for other cross-
3
(t 2m
t n ia(2) tcan be gec neralized by conformal mapping (see e.g. Ursell 1949 for the corresponding construction 6 wave-freet potentials at-zero mean speed)-. -An--alter-
IN
6
m)
m 1= by cauchy's inequality,
'(n)
S2z+2 mt. +)"m" ( It can be shown that
(2a+21a =O/ 2 mEmL
and it follows that s = o(l/t"). from (4.8), 8£ is of the form , 6z =AB (P) + BB(
2
, )
Evidently,
0
2
-+B -Ka
B are uniquely determined except when the (4.14) the determinant dete n of ohene a, system and4-5. e (4.12), determinnull 2 4re4 vanisheS; fo ( hence c£ and are detemined f ( 3 dh scan determined from (4.1) except for an arbitrary additive constant. It is obvious that the potential 4(x, y) does indeed satisfy the boundary condition (4.2). it remains to be verified that the determinant of the system (4.12), (4.14) does not Thus Ka. by the system considering thisfor cansmall be done vanish identically; the procedure described above is effective except possibly at a discrete set of values of Ka and provides a construction for the least singular potential *(x, y). It can be shown that for general values of Ka the logarithmic terms in (4.1) do not cancel, thus this potential is logarithmically infinite at infinity. This concludes the proof of Theorem 1.
___248-
(5.1)
G(xy;aO) G(x,y;-a,O) ,
-
(5.3)
K
denoted by a~./n. Then, by Green's " t theorem, sources of density (2fl-!Aint/an and normal dipoles of density -(201 cin int distributed over C, generate a -(f20) field exterior to C. By subtraction it follows that the exterior solution ¢(x, y) be represented by a source distribution of density (2r)-13(-4int)/an over C. Similar ideas will now be applied to the Kelvin-Neumann problem. Proof of Theorem 2. (Not all the details will be giveni) Apply Green's theorem to G(x,y;rjn) function ¢(x, source solution the reversed y) and to the least singular = G(r,n;x,y) in the region bounded by the
contour C, by a large semi-circle S(R), and by the two segments of the x-axis between C and S(R). (Note that € and E invo]-ve arbitrary additive constants.) It is assumed that , i lies in this region. Then we find that =
a2n)
5. R.EPRESENTATION BY SOURCES-(xyn__The
,aos )
where the coefficients (tt) in (5.2) And r) in (5.3) are the limits of the source-density function u(S) when 0 : =. A corresponding result holds for the least singular solution exterior to an arbitrary boundary Curve C intersecting the horitontal at right ahgles. To motivate the following proof let us the corresponding results inpotential theory and in acoustics (Lamb 1932, §58 and These are obtained from Green's theorem which states that the exterior solution A(x, y) can be represented by = o(1/£), sources of density (2m -06/an and normal d over C. Consider now the interior potential such that tint on C: let the (4.14) corresponding normal velocity on C be
m(0 ) -os O. From (4.12) and (4.14) the values of A and I a
u(MG(x,yasi
(xy)=U
(4.M1)
in an obvious notation, where 6(P 2 ) and 8.(Q) ate independent of A and B, and where = 0(1/t4) and B(Q) = 0(1/0,). Thus, from (4.7), ' ,u(A a-T_ Ka (P)sin(m+)z r 2 4 (4.12) u +B ' sa=-l. -mQis f (recall S a Similarly, from (4.6), A290). z( I t where c(Pi) = O(1i) and a£(Q2) thus, from (4.5), p\ -)cosm, >a 0/int
nat-Sve distrib;utions -of -~asources approach-uses over the boundary curve C but additional terms are needed at the ends. Thus for the half-immersed semi-dircle e find Theorem 2. The least singular solution (obtained in Theorem 1 above) can be represented in the form
method of multipoles, described in 54 above, is appropriate for the half-
(xdcsxy an
I
0M
-
-
SR ai
S(R1 1,
+ dx+
(R,0) dningAS8
Write
E
4;-d_--4)-=
where the normal gradients are- to- be evaluated at -(x,y) mi(asin-o, acos-v) -also note that G.(x, y; ~ G t= if;- xi-y) it is- seen that -(5.7) =Is eqiiiva-ent--to ~ R(a2)Then I)----~dxfomth feesufaeTheorem 2,_ except fo-r th-e two terms Arising = ~ from the large semi-circle. I-It- can _-be shown a a yconition that these contribute at mnost An-additiVe This concludes tepofo 1_ _3G_:-iconstan~t. Term2 - &- dx from Laplace's equation, Note that for reasons of brevit. -we a have-omitted proofs that the least singular (R,O) inter' r potentiaL exists,_ and that thei(4 l'G-E arge semi-circle contributes at most a K ~ XiaO~' ax axi(a,01 const)ant t- the notential. We can noW see how~the least singular can be constructed by-meanis-of-an and-simiarlypotential integral equation- -Wr-ite (-a,O) a-
WE [!pr a'
-&-S
_3dx
-2~(Zzr
n
__1SR)K
let
weethe last two- terms (apparently -waivesources) Are actually wealk -surface if-The fundtion u(6) and te ariti s. constants p and q are to be detrmied. On fidnha applying the boundary condition (4.-2) we
a G--
ds1' ds-
-LB
=U!U( )~xy~sinea-asO)
(RO)-(x'
CR
R I (ROer (RO
(5.5
in(x, y) denote the least singular on C. potential satisyfing -in
e~r
+_
(r sin amcos ;a sin eja Cos-)a
inero
(The construction of this potential involves the superposition of regular wave potentials resembles the construction of the least exterior potential. The details Apply Green's theorem to tare omnitted.i) and to a in the interior region bounded At\~ Then by C and by the x-axis.
__and __singular
=r/ .(rsina,rcbsu; "
/a-! G(rsina, rcosa;-,
y !5v
-a kn
-
in
~(a,0)Oq()
-xit
___
-_
or
y int;
___ ___I
-from
56
-(-a,0)
in (5.5) above. On subtracting (5.6) (3.5) we find that ___
(5.9)
is a Fredholm equation of the -seconid kind, and (except possibly at a discrete set f ireqlar-Values of 1(a) there i~ unique solution which is evidently of the
nThis
if
___x
__as
)
-sina.
__
__
a, O)
-;~(,~
(5.10) 0) + where the functions u+(S), u_(0-), ~()are the solutions-corresponding to thepthree known functions on-the right-haud side of fS. 9). From Theorem 2-,we have- pu(= u /Ka and q = ut- ie)IKa;- thus, on putting 0
~and
=-
0-= Kap
___
i(-a,01
Kc
it
and
(a,o)
Rag
~iin- (5.10) =pu~(~
we -see that
+ qui(QD0
+ -0-)
pu,-(-h:)- + qU (-L7) +
.-
)7
From these equations the two constants P and q can be found, and the source-density is-_ .249-
then given uniquely by (5.10). if p and q are instead given arbitrary values, thenthe solution (5.10) can be shown to have uribounded vertical velocities in the cornets. This corresponds to the solution -(4.1) above when A and B are given abritrary values-
-
6. DISCUSSION It has been shown that for the halfimersed semi-circle the Kelvin-Neumann problem has a two-parameter set of solutiom if the singularities in the two corners are at most weak surface sihgularities. There is just one solution, the least singular solution, forwhich the velocity is bounded in-both corners- It has been shown how either by can be constructed, solution this method or by adistrithe of multi-les
=
bution of wave sourcesover the boundary. (The source denity-satisfieS an integral equation (5.9) involving additional endcontributions.) There is however no obvious physical reason why the condition of boundedness should be imposed in the corners. In-the physical problem the velocity is in fact not small -prturbation in the corners, and- the -inearization is not valid there. It would thus be ecually reasonable tolook for solutions of the nerturbation equations Which have weak or strong singularities in-the corners but there is then no obvious way of deciding -what singularities would be appropriate. in an earlier unpublished version (1978) of this work the representation (5-i0) was used with p = q = 0, and it was shown that the Vertical velocity in the corners in then-unbounded. This choice is arbitrary, as was pointed out to me b- Mr. Katsuo Suzuki, whose criticism led me to a .-more thorough study of the problem. Suuki's own choice of-p and q is based on adtionai physical conditions and does not lead to the least singular solution; Although -here is ib obvious physical reason why the least singular solution should be preferred in two dimensions there may well be-physical reasons why a corresponding -boundedness condition should be applied in three dimensions, and this is one of the motivations of the present study. The-perturbation potential (x, y) • becones logarithmically unbounded at infinity, the velocities due to the logarithmic terms tend to 0 at infinity. There appears to be no physical reason for excluding such a solution; it will be recalled-that the total potential is linearly unbounded at infinity. It is sometimes stated that the perturbation potential must have strong singularIt iin-the corners; it has been seen in the present work that this conclusion is incorrect. The usual argument is based on (5.5) above which from the corner at (a, 0) contributes terms (2K) *(a,O)-G(,-a,o)+(2nK)F'
to them=potertial 4(,in). Since X, Y) represents a source at (-, Y) G(C, r 0 this is often-interpreted at the when Y sum-of a dipole term -and a isource term. This interpretation is in any case-inconn h (at01 sistent since-thedipoleStreng! and- the source strength a(a, 0/d would not be finite if there were strong singularities in the corners. Actual!y G( , n; a, 0) is a weak surface sirgularity, and G(, '; a, 0)/3T is a vortex as c-an bWe seen from the expansions in the appenaix but even these weaker singularities are nOt present in the least=singular solution. The correct interpretation can be inferred from the representation (5.1-5.3) of the least singular solution, whichconsists of a continuous distribution of sources together in weak surface with a discreteNear each-ornersingularity the end-effect each corner. of the source distribution is like a weak surface singularity and is cancelled by the corresponding discrete Weak surface singularity term. Similariy near a corner the=end effect of the dipole distribution is like a vortex and is cancelled by the corresp-onina discrete vortex teim 7.
I am grateful to Mr. Katsuo Suzuki for his valuable comments on an earlier version of this paper. APPENDIX. Power series expansioa associated with the source potentia-. we consider the functions C(Kx, K
C
=
e
c
S(Kxi 1Kv)
e
sir -x
-
l
..
.
and
2)
where the Cauchv Principal Vale is to be -
taken at k
K.
Wri
(
wT-en )r pe-P'F(
= -
dki (k-K) (k+--) the order of Integraticm,
1|n&-l pn L'Ik.
1
..
k.
m =
i
l
(A 3)
m=O p Also, from Euler's definition of the gamma function, (K¢ K w e whence by e F(v+l) P differentiation with respect to v, dtto (A.-4) F)g -p; j = e (d
(a,O)G(r,n;aO)
0-20
ACKNOWLEDGEMENT
=
On comparing--this-with-(A.3), we find that
~~~
-~
~m
.(~kK
r
-)q
whence
We also need theexaso
lrKF0
i!
t.)
~+ix
Now put C-e
d--e
'inc
,then
re'
frth-orbtl
dipole-
(,cosk)xc- i sin kx)~
-T
e -*
~
(lh*A4 ~
-KS
on taking -real dnd imaginary narts we find that
~
-.
lW
~
(;j rjosi~ 1
"nl)
1 lgKr
~
'
-s*
ml
=-
=ccs Kx
K(xX) tf.U)
m
-F "
cosr
(A.12)
=
IL
-Y inv e-' sinlx in_
Ky)
-
1*.6) and-that
e *
eM' snkk-~
and the well-known expansion§
Zy7kK
C(K'cKy) -~ -
sinkxcdk--
+ a.sinxn9
co
I (M+1r-r 'r~
=
(A.9)=
m+lfl
'
mMK)
a
~
I
-_KY;_
6-kMdk
~
-snxx---r-o t8VX-
C(A'xKy)z,.*-
t(nr-(rifl)
=
X
) is -an even f unction-of -Kx and Since C StU is an- odd -function of _xx we= infer that
Vft
n(C) .?=I~v{~
hn
csK
snxde
-
si rw
(A-13
dk
siin-
LM
It
Ul) -,and ()cylinders + whiere
is known that 1 +'mil
flinMF 1 2
in'
I=0-5-172 x
-- is -uler's constant. To fine the behaviour of F-(y4=ix) when Fi=e
I
10-e
-k; dk k-K kk-dk
=ae
-K
~
adding ano subtracting thd integral along a smnall indentation below k =K, =exp(- iw)
C
-!L'
S
Iwhere k-axis
t
-Ebounded.
has been deformed into the path k on which the integrand is
SKy~x ae
(A.8)
-
The integral in (A.8) is easily
seen to tend to 0 when Xx find that
-
ri-iC,
k-KX the indented path along the real
arg k=-
-Ftc)
_
+-, consider
k
Sby
.HH dodynmcs 6t' EdCambridge University Press. 1932. 2. fursell, F..i On thi rolling motion of in te surface of a fluid. Quart. J. Miech. AppI. Math., 1949, 2, 335*353.= 3. Wehau~en,,j.V. and atnL. Surface Waves, Handbuch der Physik, Vol. 9, 1960, 446-713 Berlin: Springer.-
=C(KM,Kyl
-
-4,and
we thus
iS(rGXjy)
-
snw
_CS
D iscussion
()G(Xy;a _si
y)
coSoad
-U/Klt.i(vA/2)G(x, y;8 -a0J) (--/2) G(xyVa,0)] 1
K. Suzuki (Ja--a;efflsACI) -The author has settled the problem of the uniqueness of the 2--D Kelvin-Neumann solution. The solution can be represented in various manners, for example, -Green's representation, mutipole expansion and so an. Furthermiore, the solution can be determined arbitrarily, as shown by Bessho-izuno (Sci.Eng.Rep,Defense Academy. 1-1,1963) and Eggers (Disc.to Bessho,ISWR,1976). Prof. Ursell has made clear that the least singular solution can be determined uniquely if one assumes continuity and boundedness of the velocities at the corners. He has also indicated that there are infinitely many solutions according to the properties prescribed at the corners, I had earlier calculated the 2-D Kelvin-Neumann solutions of the following -form (1978,unpublished);
+U/KC IG(x,y;a,0l-G(r,y;-a,0)] S l) P When C =0, this repre6sentation gives -the least gingular solution. The streamlines are shown in Fig.S-l(i-d) for various singular solutions. the I have a question concninp multipole expression (4.1), whether thelogarithmic terms do not cancel ? Once~-we assume that both the velocity potential and the flow are continuous (least sincular) even in the corners, we had better replace Id the logarithmic singularity-by a dinie singularity. This results from two reasons. To show one of them we shall first deal with the lihearized free-sLrface cdition (Wehausen-Laitone,1960,p-471) as
$R
IL
-
_
jt
b) Zero-vertical flux flow (Cp=2)
a) Least singular solution (C=0)
440
Z
3.
0-
I30
2-r
d) Wave-free flow (C -1.568)
c) Approximate flow for slightly sunken 'idcle (C =2.1)
Fig. S1I Comparison of streamlines around a semi-submerged circle for various singular solutions ( Ka=0.4
-252-
_
_+qconst
~form:
(-2)Kt+t-0tI)
for y=O
where V is a°:conjugate (stream) function t-=-s 1 for V-. We can ptit the constant in-the I tst right-hand side zero upstream from the _U_ body without losing validity, on the assum-ion that 0 as x-= (1) (see the t=yt==K (5) number in Fig.S-2). As x tends to the forward corner point Pr along the line y=O, F. 2 rig we take *=I-/K (*x=-l=incident velocity) from the continuity of the stream velocity at the function,• PF (2). Since the conr Pp()*ic corner h temfnto, g(z) instead of g(z), we can show that the =y+0, is constant on the body surface, corresponding function- (z) is Writtih inthe form-of eq.(4.l). But such a solitc we have -2=-I/K there from the continuity cnntstsye.(-) of the stream function at the corner PF (3). -The ne t reason is that both the least singular solution (5.1-3) and a sinBy using the continuity of the stream function and the velocity at the corner PA, we gular solution (S-i) do not become logarith, can also derive the fact that 4 --le W-/Kiclyubnddaifnt.Tes solulThes infini aton unbounded mically v=O tions can be--looked surface as-a-point soudrce free the PA in at the corner (4). Finally we obtain the zero value for ti a looke ash int or the constant in the right-hand side of eq. (S-2) downstream from the body (5). Thus the linearized free-surface con.,= /2( 1 dition can be written in the following y d 2 x log(KR), (S-9)
-strebgth
where R=V(x 2 +vy) We can show that -the of the point source is equal to the difference of tthe x-velocities at Pwhere L=d/dz-iK (differential operator) and PA,#Z--XQ , both of which ar to-b& taken as the limiting values to PA ahd PFand f(z) is the perturbation complex potential. We introduce a new function, on the line y=O. This relation was shown analytic for I:!zIl(assume that the body for the 3-D Neumann-Kelvin solution -by (1976, unpublished) and is to-be -Tsutsumi is: a semi-circle) as at continued Washington Workshop ~z)L~fz)]reviewed Re Llf(z)]=0 for y=0, outside C,
K __
(S-3I
For the least singular solution- the
wscbecause Re g(z)=O for y=O.
I; U
t
I
difference of the x-velocities vanishes they both become -1. As fot the singular solution (S-3I), we have Cx= =A_ -1+1/2 Cp. Accordingly, the strength of
(S-5)
the point source in eq4(S-9) becomes zero. This tells us the above mentioned. From the above two reasons, must we say that the logarithmic terms in eq.(4.1) cancel or that the multipole exnansiin for the least singular solutionzis to be expressed by eq.(S-8) instead of eq.(4.1) ?
Such a function can be written as follows: g(z)=i S am/tm. (S-6) m=l It should be noted that the function g*(z) =ia~in(z) does not satisfy the condition (S-5)_ Substitution of (S-6) into eq. (S4) gies the function f(z) as, f(z)=1/Ltg(z)] iKz - -K; =se
e-K
.
.ts
am - dc.
(S-7)
--=
The above solution is rewritten as a combination of the two terms of wave doublets and the sequence of wave-free potentials as shown in the following: f(Z)=flWr(t
0
2w
-
(z;O)+t cn+2h (z), (S-8) nnl -
where hn(Z)=i/zn(l-i!KZ), n are the wave doublets in the x- and x-directions respectively. if we use g(z)+
14ro s
I believe that oreat progress has been made by Prof. Ursell from a theoretical view point, especially in its mathematical aspects. I want to make some comments in the following. As the author says in his discussion in the paper concerning a rather popular misunderstanding that the perturbation potential must have strong singularities in tle corners. This misunderstanding was also mine before I read this. I think this misunderstanding originates from two reasons. As the author points out, one 6riginates from the ontribution from the ornor (See Eqs.(5.2) & (5.3)), or the so-called -253 -
-_ _-
-
L
}
line-integral term in the three-dimensional broblem. -And-the other may originate from a confusion with a stagnation point -singularity in- slender-body approximation. The -corner-contribution is derived from a free-surface singularity distribution by processes such as shown by Eq.(5.4). Hence, in the original form; the potential is expressed by the sum of a singularity distribution on the cylinder surface and on the free surface. From this expression, -we may have a feeling that the velocity at the corner may not be strongly singular.a But this-is nothing more than-a feeling. We should pay enough attention-to Prof. Ursell's finding that the-source distribution on the cylinder surface given by Eq. (5.1) shows a -weak singularity at each corner and this singularity cancels -the similar singularity due to the corner contribution given -by Eqs.(5.2) or J5.3. Concerning t-he-misunderstanding, if I'm not wrongi originating from a confusion with a stagnation point singularity-in the slender-body approximation, my comment is as folows. in the- case of a-slender body, the -potetial makes a change of 0(1) in the neighbourhood of the stagnatich-point, and -the size of the- neighbourhood is 0(c). This may :be the reason whv the singularity appears there. On the contrary, in -the present problem, the potential makes a change of 0(I) in the neighbourhood of each corner but the size is also -(I). Therefore, the potential may be expressed without introducing a-strong sicularity at each corner, and this may be the reason why the least singular solution in the form of Eq. (4.1) is possible, Finally, I would like to express again my surprise and admiration for the really great originality of this paper.
a -(K
0 0I.e.
)
+
K
_
+
0,
)
iie. K0 BY free surface xOr on the Thu for any -pint tiv X-ais outside C we have, K4(X,0) + "!(X,0) -Ka,0) +
osi-
(a;0), -
C)
a K (rX,0) + "y
I -X,0) = K-(-a,0) +
(-a,O)
+
Since the total mass flux out of -C vanishes we -have= V(a, 0) (-a,0), and from-4(2 Z) we have 'a 0) =-(-a,0) -U. By TyIt follows that (KO +
aX
-) (X,0) = {X
+
Y)
C-x,0)
-
(l)Consider this equation when X From the expansion (4.1) we have 4(k. Ua
-Ae-K)san Kx + 2,Be-KYcWS-Kx +(A
Kau)log Kr,
when x is large and positive, ana is near 2 -(Y),(a - Kaoa)log Kr, U=:
when x is large and negative, and 0is near
s Reply
it folluws that the conjugate stream func-
tion V( ,y) is given by Kx + 2zBe-KYsin KX, Kcot Xx+ U--y) 14 22Ae Ky x *Ax, 0
Ua
F.Ursel (U.P,oft ar---ste
+
-
Kaao)0,
when x is large and positive, and 6 is near 1 2 ; (x (A - Ka f ,-, Ua
In my paper I showed that there is a unique least singular solution which has finite velocities M the-corners, and I suggested that the velocity potential would in general be logarithmically infinite at infinity. In his discussion Mr. Suzuki has given a convincing argument (based on equations thepotentibl5.;3), representation, ) to show that the (5.1integral
=
(a
large and negative, and 6 is when xl near edd cdb a r ~ Exressions~f differentiation. On substituting in (1) above and noting that 6 = :4f at (X,0) we obtain A - Kao0 = 0, i.e. the terms A r -
is in fact bounded at infinity. I have now established the samie conclusion for the potential in the alternative multipole form (4-1), using a simplifisi Version of Suzuki's argument. Let W(x,y) denote the stream function conjugate to A (x,f). On each segnent of the -fred surface we have
combine to form a dipole singularity which is bounded at infinityi This is -equi__lent -254-
V1________
____
_
to Suzuki' s result.
I wish once again to express my gratitude to Mr. Suzuki for his contributions to my workI also thank Dr. Isshiki for his- valuable comments which help to elucidate the -orn of the least singular solution in the
corners. No physical arGument tad iet :been given for preferring the least :singular solution to other poSsible solutions.>Aard other -Solutions may turn--out to -be nhy ically nore appropriate. -This is a prob1em -which requirs further study.
°-l
N
-Z
I
Numerical Solution of Transient and Steady
Free-Surface Flows about a Ship of General Hull Shape Robet K.-C. Chan and Frank W.-K. Chan Del Mr. GQF&
S
ABSTRACT O A single-valued function -which This paper describes a finite-ie measures the distance of the hull ence numerical technique for simulating surface from- the shipts -center transient and steady-state three-dimension-plane in the rectangular c-rtesian al potential flow about a ship with blunt system ithYrz)i n bow. The primary features of the method in-Y diude the use of special coordinate transG An arbitrary scalar function formations to allow rigorous application of hull surface and free surface boundary cong Gravitational acceleration ditions, and application of Orlanski's numerical open boundary condition to prevent R The distance between the origin of nonphysical wave reflections from boundthe coordinate system and the cenaries of the computational region. The ter of the semi-submerged sphere method has been applied to wave resistance in Fig. 7. of a tanker with and without protruded bow. It is found that the breaking of bow wave T Spatial 7oordinate along one of occurs at a higher- Froude number for the the mesh line directions (Fig. 6). protruded--bow, thus reducing- the wave drag considerably at higher Froude numbers. IMAX Maximum value of the subscript i (Fi. 6).4 NOMENCLATURE
a coefficients defined in Eq. (19) 1i,B2
Coefficients defined in Eq. (19)
S ,C2
Coefficients defined in Eq. (19)
14 fl1 .D2 /flDT
Resistance coefficient defined Eq. (31)
in
X, Y.
F
An arbitrary cralar function
r
Froude number;
f
A single-valued function which describes the hull surface (Eq. (6)1 in the spherical polar coordinate system C:r, 6i 6,)4
Fr
coordinate along
one of
the mesh line directions (Fig. 6).
Coefficients defined in Eq. (lte) Particle derivative in tha Z, T) system
spatial
J3AX
Maximum value of the subscript
K
Spatial coordinate along_ one of the mesh line directions (Fig. 6).
KNAX
Maxinum value (Fig. 6).
L
Characteristic length of the problem
N
Time as measured at 4 given spatial point in the [1 3, K) system.
A
-Unit normal vector at the bull surface. Components of At in the (x; y. z) system te
U/ JgL 4
nlin 2 n3
-
57 -
of the subscdript o
-
k
One of the angular coordinates me
Fluid pressure
One of the angular coordinates i Fig. 4.
exper-lenctd
force
No. 'rtorizonta'11 by the shi.IP Radi1al
inE.
tCordinatp defined
In.
Fluid-densit
p
hull surfaceoe-r
aala
the as measured ordinate s'5-v~r .r, t Tim
frrene
.
I.wt' Gra
'-iyt-- in
rre-~ren.c
h~1;
co-
p011100
oliae
a
1ned in Aalt=o'v-
f
±i-ne as ne-Asured in the absolute Inrilfrane of reference.
the
ano--1 ar c 0oonae5de (
of trtinco in
COne
ii-
-
tera
eoct
*
RadiAal cor-ponent of the spherAcal polar coordinatle svstnson nrea of wette
M7
fined in Eq.
sphere.
Radusofa seisb-rqed
enl
the shin. f imal Sneed co-st8.At s-mn i n Its f-rwar- notion.
a
in
Define=
u0
-
__
uesr~ vol i-n u-II w'rrsoect totne
-e,
in
noints a11r--=a
__A-
v
0 --
X,Y,7
=';ned irf
31
Re-rnuar
carte-san
In an 'neriua!
coordlinates ofth of rotation in the ten_=vr
aoeficcients -72)
Coefficients
sipscne ix. Y. ZI sts-
defined
tor.
defined
tn
_' -
9122)
Q*7 Coefficients defined in Sq. '77) Agle between the forward d irecship- and tht X-axis lion of th(Fig. 1). Yt Co efficients defined inEq. Ttine increnent oration. Affscalar n
In nis-erical
2) into-
"'octn definca in Las. (5) nd ~1.puttional
iqeiatidfnto and t, whic psitin the [Eq. (101.
..*
r-e r-.
COOM rates frane Of referene Ti
coordinate5 te-sian ecta nular _cz R~~ reence fixed4 in of '-a'-a-
Xc#Y_
n e
4. fr. odsribe i=ue ofthe free surface
!-e
s aur
-x
"ob4ct'1
e~ o-
a ,unr~'ca_
Pl-
j
as-m
the
aIona
__
nt#b
~
u
for
= r a surface shi n 0M a 'a -_ Z-. an 5=onte=mrate=n,,7--rIA h w- a1 mw' si _'In wiha -!owisr assun-tion of potenti o s te o rltis stud-y. However. ,.or-1 ,e-Ir irn-"'lz boundary conditions. a-aat Oka core~ temaeapid 1 -nf--a a ecss approach tions. Thi _.ee _cn= a etotIninertn ne atI, ~ dicensiona- equations inar fitted connu-tational znest'. Za aw nsleI odcan e usedi to obtain 'b" h tadyftaeslto. .hero are three naj-or coo- erns when calcul~ation_ three-dinensiona' flos abu a teczoe shpby fiiedference byv many other nunerical metIhods1=- = ca.
coof thk size finite renide due to can re flections reg-ion -0sn totally n-eanimless. 2. 1he fi.teOcffr-
so=ranv ence schemes outet .Te-=t acuacy renrmn" r. onaC joesh must confontth surfaces) hull surface atfree
anonal o
6-
tne
problem, so that boundary conditions can be applied ri gotusly. To utilize the UIited computer storage efficiently, it is preferable to select a- frame of reference fixed in the ship so that the sesh resolution is concentrated -ear the ship. in this reference frape, the ship appears to be stationary in a running strea-. Conputationally, with arrangement creates probiens this boundary conditions at the "open-boundary," ihere the flow and waves are supposed to
axis of the ship, pointing to the front. Usin formal coordinate transformation 14, we obtain the- following governing-ecuations for the velocity potential 2n the Ix, z, t) system: Laplace Cquaion
the way
through
--x
*
MI) lV, with the appearance of Orlanski's excellent technicue for numerically proparegion, this natina waves out of a finite difficulty is now removed. With implicit treatment of those terns that covern wave pro-ations, the finite-difference method gtven in this pacer prodUces soluti¢ns that
all
in
ae
he
-
SHE P
Potentia
~ Study. u.
v
a
v
-
3 3
-
cos g +
andt
(x 0 J
-
sin
z-3)
scin 4
ia lt cos
YE .:n§Y:iai: a
rota io- in the (X, Y,r T fraz- of r-eference. The forward motion o= a s-ip-es speo-ia! case in which = Y = constant, o cneneceC and, i1 we ~'ce -= te that X c v - and ar e manermllv functons te fore, arbitrary planar motions of t
tsereship
Z# Hull
$
rac
Ro....
a ref ren-_ w-t he their conep movimo frae wnich -s .ix& In the ship. We assume furth'er that the ship under oes an
v Condition
,,n) a, re the cartesxa nents cC hull s-ae svste
_abitrary Planar nation suc that nts t (i.e., the vertical axis of the sin- a!ways nintS vertically uwA e postPve x-direction is the same as 'thelo-itudinal
h_ a
-itnapo-
normal vector At at the eas-e in the (x, y, z, t
eni
inre-tic Free Surface condition -'-:t -0
i
forms for Vq .
Tl Exlii
C } !I i
(5)
Where C is a scalar function defin ui sU-a that ¢ a C at the free surface is the Particle deivimes-. D0 for al ative of in the inertial f,rr, C,
>
&y
(2)
= 0
In the eq-quations ab-ve -subscripts s, . z, and t denote partial dwfhereneiation. wief VxC)poIino -are the coordinates of the instanta4ou hpsc-.e o-
he tb- cartesian coord nates referred to a. absolute inertial frame. and ne , v, . t'
-
-
+ my
-
=
=
.... +-d
As so~.,in shown n Fi Q . 1, let MX, Y,
z +
-
gien.
s-LE
V)2
0
Somme sacDle caiculations o t e flow ahgut <eci-submrged sphere, an- comparison etween a HSVA tanker an a noc- tied tanker wih*rotruded -boware Ascussed4
Fo. vtrIA OMow
m2
where
=
Coorcinate and e-quations relted transformations ot thesstn gi- e-rnin are give.,
2
[
-
the tran-
the su6ions that follkw, a dedescripti¢n or the derivation of the
:ot
Vtv+
2
sient neriod to the steady state. As to the problems of ceonetry, a body-fitted coordinate Svstem nas been developed by deforming the sphericall "W cordinate syste= in such a way that _ =ew coordinate surfaces conf r the free surface and the halt -alles
_
Bernoulli eouaxon
region. Improper the computation leave trea t -ent of this suit type in nonphysical can re"reflectof boundary ions- Fortunate-
are stable
)xx
zz
(5)
can not
-be
-iv the final body-fitted curvilinear c-ordinate syste -is selected. We wiil return to this equation later-
.
Te urinary
I
o=:.,,
*oal of our effort is
to
develop- a conoutational procedure utiliti.g
a mes= sytste +i ch conforms to the fre surface and the surface of the boady. such a
z esh
-
system
would
-3lk-w rigorous
apulica-
~
tion of the complete sets ot free surface ontions at an ... bcll sunface boundary
Pi . I
Framne
Vl--r exact positi-ons. Reference coiides c u to s hipmadea i a sh te it ata-
of Reference
El~ed s __
:59
,tr r
t -r- ship hlabl
.ia
Ic
be described by a- single-valued function y = fo(x,z)i, where y is the distance of the hull surface measured- from t,e ship's center plane (i.e., the x-z plane). The class of body shapes was therefore limited to those representable by f,(x,z), with a sharp-edged keel (Pig. 2). To extend the method to hulls without the restriction of sharp-edged keel, a different choice of body-fitted coordinate system must he made. With applications to tankers and other wide-team blunt-bow ships in mind, we choose a mesh which is obtained by deforming the usual spherical polar coordinate system (i.e., stretching) in the radial direction in such a way that the new coordinate surfaces conform to the free surface and the hull. At the time this work was started (1977), many researchers had explored the use of very general, bodyfitted meshes to solve problems in-computational fluid dynamics (3] with considerable success. This type of approach has been applied to two-dimensional, time-dependent problems with free surface and body interactions. For three-dimensional problems, however, this general approach is not practical because of its demand on computer time and storage. As th6 free surface moves, the new curvilinear mesh must be regenerated at each new time step and the coordinates (x, y, z) of each node point have to- be stored. It would also be very expensive to solve the Laplace equation V 2 ¢ = 0 in such a system, unless all the coefficients of transformation are stored in the
computeri which again consumes a -large fraction of the precious storage. Fortunately, -for the ship wav obem under consideration, a general -curvli r mesh is not needed. The computation canmbe efficiently performed in-a -compositemesh syftem as shown in Fig. 3. The bulk of the flow field lies in the "lower region'whi°ch can be covered by a fixed mesh conforming to the body surface. -In the "upper region," a more general, time-dependent mesh is used tv conform to the hul:l surface, to the interface between the two regions, ar to-the moving free surface. The upper r~gi6n is relatively Small, the only requirement being that its lower boundary (iie., the "interface") be placed low enough so that it is not crossed over by the free surface at any time. Since the more general mesh syste sue ny in h ml pe ein the expensive part of -he computation is minimized. We now describe the coordinate transformations necessary to obtain the desired mesh system. Let (r', 0', *') be the spherical polar coordinates with origin at pont 0 in Figs. 3 and 4. We use t' to- designate the angle as shown in Fig. 4 to aoid confusion with the velocity potential . The origin is suitably located inside the body and placed at the same level as the Interface between the upper -and lower regions. For most hull shapes, the hull can be -described by the single-valued function
Total Length of Sh. ' Nominal Length of Ship
To generate a body-fitted coordinate system (r, 8, qY), we make the following change of variables
G FE
-08
D
-04
00
r'
f(0,(,)
.
(6)
BA-
04
08
Longitudinal Distance, x/L
-
(a) Top View Stern
Free Surface
Bow
Rer
00
. -1
,
-o
i
ier
-0-3 -0A
-05
02
01
00
0.'
A
0.2
(b) Cross Sections Fig. 2 -Shiplike Body with Sharp Keel
Fig. 3 -260-
Composite Mesh System-
F
Ox
sin 0-cos
rr
/~G
tr04~ in e- sin
c'
/r
+
~~cos
0- sin 4' V
- I
do
0 ___7_-in
e.
0y
Cz
Cos0' G
If F is have
GO,.
sin
another
r
scalar
function,
2
(r
r'
(9) we
also
91
x +
+
Fig. A
Spherical Coordinate System(r
r
r'
0
2
-
C (7)
=
The surface r =0 coincides with the hull surface, and -the -plane 0 = irf/2separates the upper region from the lower one. As shown in Fig. 3, the mesh points on and below the plane 0 = n/2 are fixed in space relative to the ship. The mesh pitsat the free surface, however, move with the free surface, and the region in hetween is divided intc subregions with equal angular intervals, e.g., between points 1 and 2 in Fig. 3 (point 2 is fixed hur point 1 is alloved :o slide up and down along the mesh line r = constant). This ariangement allows effective resolution of the flow field ne the intersection of the free surface ano0r the hull.-
([r+
)
+ sino *0
-
y
=r'
sin 0' sin
z~~~
(7),
2f
Go,.
C8
G'
G 4'
-
fer(1 f~_*
f)]sine +() 04
0
+
sino0
2f 4 4 4
-sine0 cos e f0 sin 0 cos 0-
sin e
-feel
(12)
Bernoulli Equation +
4'
u 1I
.(i 1 +v
A)4r + 2 )4 4
e ef ]
r1o(r+[
For any scalar function G, defined in the space unde~r consideration, we can write -261
we
=
f
The relations between the Cartesian coordinates (x, y, z) and the spherical polar coordinates (r a', 4')are sin 0' cos
P(0 4' ' of Equations
Laplace Equation
+ J[2(r + f)
=r'
G
Using Eqs. (8), (9), and (11), Eqs. (I)-and (2) are transformed into
Governing Equations in the (r, 0,4) -Coordinate System
x
2
si By the transformation obtain
+
+ 11uB 2
+
+ g(r + f) Cos 8 + P -
f) sin 0
e -(13)
;Hull-Surface-Boundary- Condition The hull- surface condition can now he derived. -First, -we observe that the- function _r, where r = r' - f(Oj*', has the Value zero at the hull- surface by definition, and that its gradient Vr is normal- to vr/I7rI is the hull surface. Therefore, the unit -vector whose components are (n1 , -nl~ n3 -) in -the (x, y,- z) system, -as mentione& in connection- with Eq. (4). Thus Eq. (4) can be written as v
*
Using -Eqs. (9), r , Eq. (1.A)
-
vr
=-ur
-tial
r
-
x
- y~y +
4T + IX;X
y
~
l)
ZtZ
or +
~r rmEs Frm-s
V
~
7, 1) 7,(1)
=0(7
1) 1)
n n
+u~+
which relates time derivatives in the inersystem to those in the moving system, Eq. (17) becomes
and applied at
(0), (11), becomes
0
~~r) 0 2
N
~
0
Or +
+
2
u
+v
1
+
2
+f
(r +f
+
(108) where
f
-
f 2
f
+ UA
Di
+ vA
-O
uA 1 +v
2
()
sing_01
k'T~k+f
Kinematic-Free Surface Condition
sinr
+
+
We defin- a scalar fucto (r 4-f)
(r +f) The free surface-corresponds to the surface =0, or 8 = n(r,4~,t)4 Instead of the usual definition as vertical displacement -of
0.15 in the experimental data is in fact due to the formation of breaking waves. Thus, by comparing our computed value of Cfor the modified HSVA and the measured C. we can see that a 50% reduction of the wave drag is possible at Fr 0.17, and even more at Fr = 0.18. r The development of free surface wave pattern near the hull of the modified HSVA tanker is shown in Figs. 15(a)-15(h). The motion of the ship is from left to right. The steady state is reached after t > 3.70, as in Figs. 15(g) and 15(h). The computational mesh on the center plane is shown in Fig. 16 for the HSVA and in Fig. 17 for the modified HSVA, with the bow of both pointing to the right. Close-up of the mesh near the bow for the HSVA is shown in Fig. 18(a). velocity vectors associated with each mesh point are shown in Fig. 18(b) in an inertial frame of reference, while the same velocity field with frame of reference fixed in the ship is shown in Fig. 18(c). Similar plots are shown in Figs. 19(a)19(c) for the stern region. Figures 20(a)20(c) and Fins. 21(a)-21(c) show the same type of information for the modified HSVA tanker.
where R is the net horizontal force experienced ty the sphere. As another test case, the flow about an HBVA tanker j6] ad the a~sociated wave drag were simulated. The hull geometry of an HSVA tanker is shown in Fig. 11. Again, the tanker is initially at rest, and then accelerated to a constant speed. The computation is term-nated when the steady state is reached. The computed values of wave drag for several Froude numbers Fr are compared with experimental measurements in Fig. 12. The agreement is judged to be reasonably good. Steady states were obtained for F 4 0.15. For Fr = 0.161, however, the flow fieli does not settle down to steady state. For Fr 0.17 the bow wave steepens sharply and its amplitude grows rapidly. This behavior seems to be consistent with some experimental evidenci that breaking of bow waves may occur at these values of Froude number. An interesting application of the present method is to study the effect of change in hull geometry on the flow pattern and wave drag. The HSVA tanker in Fig. 11 is modified to add a protruded nose at the lower portion of the bow, as shown in Figs. 13 and 14. With the modified bow, we found the instability of the bow wave occurring at 0.18 < F < 0.19, higher than that for the originaf HSVA tanker. Thus, the protruded bow seems to delay the threshold of wave breaking to a higher Froude number. The wave drag coefficients for this calculation are also shown in Fig. 12. Although the coefficient for the modified HSVA is greater than that for the original HSVA (at F. = 0.15), it must be noted that for tankers most of the resistance comes from wave on wave breaking, can and significantly therefore the delay breaking contribute to
ACkNOWLED-EMENT This work was sponsored by the Office of Naval Resarch under the Fluid Dynamics Program. REFERENCES 1. Orlanski, I., "A Simple Boundary Condition for Unbounded Hyperbolic Flows,* Journal of Computational Physics, Vol. 21, 1976. 2. Chan, R. K.-C., "Finite Difference Simulation of the Planar Motion of a Ship," Proceedings of the Second International Conference on Numerical Ship Hydrodynamics, Berkeley, California, 1977, pp.39-52. 3. Thompson, J. F., Thames, F. C. and Mastin, C. W., "Automatic Numerical Generation of Body-Fitted Curvilinear Coordinate System for Field Containing Any Number of Arbitrary Two-Dimensional Bodies," Journal of Computational Physics, Vol. 15, 1974, pp. 299-319. 4. Harlow, F. H. and Amsden, A. A., *A Numerical Fluid Dynamics Calculation Method for All Flow Speeds,' Journal of Computational Physics, Vol. 8, 1971, pp.197-213. 5. Baba, E. and Hara, M., "Numerical Evaluation of a Wave-Resistance Theory for Slow Ships," Proceedings of the Second International Conference on Numerical Ship Hydrodynamics, Berkeley, California, 1977, pp.17-29. 6. Collatz, G., "Mass-stabsuntersuchungen fur ein Modelr grosser Vo11igkeit, des28, Deutschen Schiffbaus, Forschungszentrum Hamburg, Bericht Nr. 1972.
269-
______
=S
-
:
----------
--)-----
(a) T
(d) T-3.1000
=0.2000
N
IL
(b) T
(c)
(e) T
=0.SODO
T'
Fig. 9
()T
1.3000
3.4000
=347000
Close-Up of Free surface DiSplacement Contours for a Semi-Submerged Sphere (Fr 0.10)
_270-
A:A
-(h) T
(i)
1j)
4.4000
g)T
850
()T
=5.1000
T!
7.2000
T
(1)
5.7000 Fig. 9
(Continued)
T!
31.3000
A= -
-
4-0
V
~.7
I
xpirimental 'Wave Reiisistance Data, 6dllez 161 25.66 ft Model 2217,--. 9nf4del 2153, L. n31l11 ft -
CLa
* Present Results
0.6-
f
4474 ft
AModeL 21-54, L
1. -
B
*H1SVA 4corputedj7 Aki Moified HS5W. (co-mputed)
3-A
0.4
J:
~.Linearized
-
a
-f
Theory-1 ha and liar
2
on 0a
0.5-
00
0
0.n 0.10 0.08
0.12
.
I
0.12
0.14
0.14
1
- 1
-
MO 16
Froud4 number,
0.18 Fr
Fr Fig. 12 Fig. 10 Comparison of Wave Drag Coefficient C for a Seni-_SubetgeiF Sphere
I
Comparison of wave Re-sistance Data for the BISVA Tanker
Fig. 13S Body Plan of ModiOfied HSVAI with Protruded Bow
Stern
Pig. 11
BOW
-
Body Plan of IISVA Tanker (from Collatz [61)
I-
=
H
HVTakr(rmCollatz_16)
A--
The
(b
.
(C)
T
OD
f
T =1.6000
045000
-9)
T =3.7200
()
T
1=
-273
(d)
Fig. 15
T
0.6400
=
8.4400
Contours of Free-Sur~face Displacement fot the Modiffied HSVA tanker -
l
Fig. 16.
Computational mesh on the Center Plane of an HSVA Tanker
4
Fig. 17
Computational Mesh for the modified HSVA Tanker
-274-
t
(a)
Fig. 18 __the
(a) Mesh
Mesh
(b) Velocity Vectors in the (X,Y,7.,T) System
(b) Velocity Vectors in the (XY,Z,T) System
(c) Velocity Vectors in the (x,y,z,t) System
(c) Velocity Vectors in the (x,y,z,t) System
Mesh and Velocity Vector Field Near Bow of HSVA Tanker (Fr 0.'15)
-275 -
Fig. 19 Mesh and Velocity Field Near the Stern of HSVA Tanker (Fr 0.15)
(a)
Mesh
(a)
(b) Velocity Vectors in-the
(b) Velocity Vecto3rs in the
(X,Y,2,T) System
(X,Y,Z,T) System
(c) Velocity Vectors in the (x,y,z,t) System Fig. 20 %of
Mesh
(c)
Mesh and Velocity Field Near-the Bow 0.15) Motlified HSVA Tanker (Fr
-276-
Veloci-ty Vectors in the (x,ytz,t) System
Fig. 21 Mesh and-Velocity Field Near the 0.15) Stern of Modified USVA Tanker (Fr
_
Msides,
to the discusser, the present results seem not to be free from the above errors; in spherical polar coordinates, the hullsurface panel sizes around bow and stern or free surface and the grid spacing near centerplane or free surface get unevenly coarse which are the most important domain in ship wave problems. The discusser is of convince that it is primarily important the numerical scheme should be always checked for simple cases (the present computations for the sphere do not play that role). Only from the wave resistance results we can not evaluate the method properly.
Discussion
The authors' numerical method seems not to be effective for the estimation of the linear dispersive wave system but is effective for the estimation of local nonlinear waves in the nearfield of ships, such as free-surface shock waves, One of the authors developed a modified RAC method (SUMMAC) in 1970*. I think MAC method is preferable to the present method for the calculation of nonlinear flows around the bow because it is a direct calculation of the Navier-Stokes equation. The authors' opin%,n will be greatly appreciated. *
REFERENCES 1) Newman, J.N., Discussions:Session I papers, "The proceedings of the Second International Conference on Numerical Ship Hydrodynamics", pp.53 (1977). 2) Hess, J.L., "Progress in the Calculation of Noninear Free-Surface Problems by Surface-Singularity Techniques", ibid, pp.278.
Journal of Computational Physics 6, 68-94 (1970)
Kop. 2% K. Mori (Hiroshima
Uni)
Three Te=
The discusser appreciates the paper presented by Drs. Chan and Chan. The method is quite general and promising. The generality brings forth, on the
I
rIx.
.
*
4
o
f/
*
0
computing time and huge memory storage. Then computations are forced to be carried out by use of coarse mesh sizes and domains of computation to be restricted insufficiently for our demands. According to the discusser's primitive experiences, the mesh size is crucial for the finnl results. Fig.Al shows the wave profiles due to a 2-D submerged point source compared with
:o=
-
-
I
A =
.
(Const) A*xo0.125 (Const)
o =\ °/ - , o
/
&KoXOZ5
oo
(fig*,Approx.) o& For Terms Dutwtou e hjrdTermAwrox.
analytic solutions, where Kof=l.0; Ko: wave
number, S: depth of submergence- The method is the Rankine-source method different from that of Drs. Chan and Chan. The mesh size of-0.25 may be the minimum in 3D cases for usual computers. The results are not satisfying; the wave length is short-
o fl. 2 I 0
0 0
-o 5
er and the amplitudc is a little large (of
,.
course this depends on differencing schemes). The discusser recalls that the authors' last results were noted by Professor Newman for
2 |°\
7 i o.
x
/
7
*
,
the same tendencies in wave length 1). Dr. Hess has also examined the errors through his pilot computations 2). He proposes, in order to be free from such errors, the use of the higher approximations for freesurface figures and related quantities. The introduction of them improves the errors (+ in Fig.Al). Though it is not clear how finite the mesh sizes of AI, AJ, AK=l are compared with the wave length, they are suspected to be rather coarse because the wave length of the present cases is very short. Be-
.
-
\f \
* '
Fig. Al Wave profiles of a 2-D submerged point source obtained by Rankine Source Method for different numerical schemes (K0f=l.0, oc2U!K , c0 :source strength, U:uniform velocity)
2'77 -
movement is seen 3ust below the disturbed T'his free surface where the wave breaks. fluid region, which may be considered as spread widely in turbulent flow, does -;ot The thckness (f this turbulent dep'th. flow is nearly the sa~me as the wrve hei-nt itself. From this observatlon 'he bo,.-wave breakinc seems to resemble the phenomen of the decelerated stagnanion c !1n. with separation as shown, in P-.2 - . known even in the case of free-surface flows that an upstrca- vortex is formed. The free surface is then considered to behave like a membrane as pointed out by in an experimentai study around HonI.I Re l10 5 . The very thin layer of the free surface moves with the body while the fliuid below the layer does not move so fast. Therefore a thin layer such as a laminar boundary layer is develouod beneath the free surface. The present discasser cmnsa-ders that the shear flow,. at the free surface is the cause of the bow wave oreasino or a rull fCrm. To clarify- the mentioned instabi 1!-
Y Kayo,.V'M
I
I have enjoyed reading this paper and am. imoressed with the authors' effort to solve nuerically the exact boundary-value problems within the framewore of r otential theory. The authors found that the numerical simulation of the flow field around a full hull form did nor settle down to the steady state at higher Froude numbers. The authors explained that the waves might break beyond these speeds when the bow wave steepened sharply and became unstable. To get a better understanding of the wave-breaking phenomenon, the present discusser made an exper mental study of a bow wave of a full ship model by a flow visualization technique. A 6r model in length was used to observe the flow field around her bow in the Nagasaki Experimental Tank of itsubishi Heavy Industries. Fig.! is the photograpn, of the fiow field observed at the central plane in front of the model. A vortical fluid
Fi.
1
Decelerated staqnat - on fIow with separ ation .as -notc-r-ph J by and printed in Ref.i)
Flow around the bow Fn=0.167 Vm=I.2R4m11 4Foetxincer V
ziG.
Fi.
1ave breaking aroun m m- .2-04
s
,
itn Wave breaking around the ucw -h thin frcdecelerated- fl of surfac layer Vi.284n's rn= .
t-he bw 6 .28
of the two methods ?
ty of the free surface in the case of a flow of large Reynolds number, around e fRe10 7 , the present discusser made some observations. A sheet of vinyl film was made afloat
E
K.Nakatake (Kyushu Univ.)
I appreciate Drs. Chan and Chan for and was towed with the same speed as the their pioneering work in this field. ship model in front of her to retard thwo i nsf t h a e ilow of a thin free surface layer. When c I have two discbssionsd the sheet was present the wave breaking In the Orlanski's boundary condition, was developed more heavily and the area of expanded widely.(Fig.3 and 4) breaking It was alsoItfound that the bow-wave In their muststate solve,? for steady the finalthey C at method, fundthatthebow-avevelocity ws aso mens a by breaking could be controlled a huge number of unknowns. Mosm of them usually used to prevent boundary-layer have no relation with wave motion directly. separation. Namely, when accelerating the Is it possible to reduce them by using the thin surface layer by using a water spray solution of the double-body flow ? or a blower, the wave breaking decreased considerably. Judging from the above mentioned behavior of the bow-wave breaking, the present discusser thinks that it may be considered that the breaking of bow wave is
Author's Reply
due to flow separation at the free surface.
=
I would like to suggest that such flow characteristics of the free surface should be taken into account in the next stage of the development of numerical work.
R.K.C. Chan (JAYCOR) Reply to Prof. Miyata The present method is indeed designed for studying waves in the near field of ships. Both linear and nonlinear waves can be effectively con.puted if the mesh resolution is fine enough. As it stands now, of course, it cannot handle the formation of free surface shock waves, which involves rotation of fluid elements. The "upper region" of our model can be easily
REFERENCES 1) Schlichting,H., "Boundary Layer Theory",Sixth Edition,McGraw-Hill,Now York, (1968) 2) Honji,H., "Observation of a Vortex in Front of a Half-Submerged Circular Cylinder",Journal of the Physical Society cf Japan,Vol.40,No.5, (1976)
extended to use Navier-Stokes equation.
H.MaruoYooama Unv.) H. NThe The authors have developed a purely numerical zethod to calculate the free-surface flcw around a ship whose shape can be quite general. This paper is very interesting because the computation procedure is des--ribed in considerable detail. The method seems very versatile and is promising in the sense that the exact non-linear boundary-value problem can be handled directl-_ This is a kind of space-discretization metaod, and its shortcoming is the requirement of large computer capacity. On observing the computation results, I feel that the mesh division is not fine enough to obtain wave patterns which are comparable with measurement. I wish to hear the authors opinion about the feasibility of this method by the use of existing large computers. There is another method of numerical computation of the free-surface flo,4, that is the method of boundary discretization. This method assumes source distribution over the free surface as well as hull surface. Starting from the initial rest, the solution is calculated in succession by a finite-difference method. Thus the mesh division is two-dimensional in contrast to the three-dimensional diseretization of the present method. What is the authors' opinion about the reldtive merit
SUMMAC Method (Stanford-University Modified-Marker-and-Cell Method) was developed by myself about 10 years ago. The problem with SUM4AC when applied to ship problem is that boundary conditions at the hull surface are difficult to apply for general hull geometry, because SUMMAC uses a fixed rectangular mesh. However, for simple geometries, such as a wedge-like prism with a truncated flat bottom, SUMMAC may be quite useful as a tool for nonpotential flow simulations. Reply to Prof. Mori Professor Mori is quite correct in pointing out the fact that the resolution of the computational mesh can affect the solution to a large extent. And, in fact, we have not used the kind of fine resolution which we'd like to use in the sample problems. In the sample calculations the mesh size was largely dictated by the small core memory of the CDC7600 system. We could have used the large core memory to increase resolution in our calculations. It is also quite true that generality of the method requires much higher cost of computation. A typical computation to reach steady state from the initial rest position costs about (Which may not be too bad 100 Us Dollars. for our Japanese colleagues if the US dollar
.2'9-__
-
keeps devaluating with respect to Yen) But seriously, our general approach is indeed much more expensive than other schemes that take advantage of the properties of potential flow. However, our model can be easily extended to include no.ipotential flow features, such as the shear layer near the free surface as suggested by Dr. Kayo. For problems like that, I don't see how the Rankine source method would work. I think there simply isn't such thing as something for nothing. The objective of our work was to experiment with a fairly general numerical method, making use of fewer assumptions. Therefore, our present method must be considered as a research tool with which various hypotheses and mechanisms can be tested, but not as a tool for day-to-day calculation in ship design.
crete distribution of velocity potential s below the free surface is the major reason for requiring large amount of computer time and storage. Nowadays, there exist several very powerful methods for solving discretized elliptic equations which may be used to drastically reduce the computer time. In addition, the new vector processor in some computers can reduce the computer time even further. To the second question, I would agree that the method of boundary source distribution is superior to our method, if one is interested in steadystate potential flows only. But to include nonpotential-flow features, such as the free surface shock waves, it is not clear to me how the source distribution method can be extended, while our approach seems to have a reasonable chance of including additional physics which may not be representable by potential flow.
Reply to Dr. Kayo From his experimental observations, Dr. Kayo offers an explanation for the bow wave breaking of a full form. He considers the shear flow at the free surface as the major cause for breaking. Since our present model assumes a potential flow, we are not able to compare our results with his observation. However, the "uprregion"~in our model can be geraliz"upper segiont nopotential, shear flows, which, together with better to a beter moreexprimetalwork lead toa may lea work, ma more experimental understanding of the breaking mechanism and perhaps a more realistic mathematical model to handle the complicated interaction between the shear layer and the free surface.
Reply to Prof. Maruo Professor Maruo has asked two questions. The first is about the feasibility of our method using existing large computers, and the second is the relative merit of our approach compared to the method of source distribution. To the first question, I would say that it is quite feasible to use our method with good spatial resolution in large computers, such as the CDC7600. As stated in my reply to Prof. Mori, our sampie calculations reported in the paper were dictated by the size of the small core memory of the CDC7600. And the mesh used in these samples was rather coarse. By using the large core memory, we would be able to increase the resolution quite a bit. I should also point out that the dis-
Reply to Prof. Nakatake Professor Nakatake's first question refers to our outflow boundary condition which is described in an earlier paper by wi idescried an eaer paerby me and presented at the Second International Conference on Numerical Ship Hydrodynamics in Berkeley three years ago. As I wave is localpoint) of C (the value recall, the velocity at the outflow boundary veociatthe outlo bo ua ntis C is used only in d .riving the ultimate finite ied on he uie fiie difference expression which gives a prediction for the flow variables at the boundary point as a function of the time step and the information inside zhe flow domain and backward in time. When steady state is reached, there is no more wave propagation through the boundary, and, therefore, the value of C (if computed) would approach zero. The second question is whether it's possible to perform calculations in a more limited flow domain, using the double-body flow as the base flow, and just focus on the perturbation. The answer is certainly yes. That, I think, would cut down the computer time and storage requirement considerably. In fact, by comparing the solutions produced by the more complete model and the doublebody linearization, one can determine whether the simpler, double-body approach is sufficient for all practical purposes. I appreciate this suggestion from Professor Nakatake.
-
_
LU i-
g
I
2801-
- i
/ i
On the Time Dependent Potential and
Its Application to Wave Problems Hiroyuki Adachi and Shigeo Ohmatsu Ship Reseamcn Institute Tokyo.japan
ABSTRACT
E(x,y)
Source distribution on free surface Fe, Fi Free surface outside and inside of c F.(t) External force in i th mode F(x,y), F.(x,y) Source distribution on free surface G(y,z,n,4,t-) Green's function G*(y,z,n,,A Green's function (x,y,z, ,n,;) Green's function
The 2-Dimensional time dependent potential, governed by the Laplace equation and satisfying the linearized free surface condition and the initial condition, has many applications to ship wave problems, especially to the problem of bodies making ariitrary motion starting from t=O. The corresponding boundary value problem to the potential has some interesting features which are considered to be important from the point of the numerical analysis of the problem. We discuss this problem and propose the method of numerical treatment of the boundary value problem. Three applications of the 2-D time dependent potential will be discussed. First the response of bodies to the given incident waves and the radiation wave problem of the motion of bodies are discussed. Secondly the so-called "high speed slender ship theory", that is a kind of "strip theory" with forward speed effect, is discussed. Thirdly the flow field around the bow near field in the steady ship motion problem is analysed by making use of the slender ship theory under the low speed assumption.
1)
De, Di D(Xy) E(x)
Gravitational accelertion
K, Kj
main dochin domain wave making resistance /g!gveju/b, =
ij) in.caso M.4i N3-D ni(Q) P, P 0 SB
NOMENCLATURE A a B(y,z) c(t), c(x) C..
g
v(Qt) (x,y,z) y(X), y*(%)
Constant in (56) ilf breadth of cylinder Normal velocity on hull Hull contour at t or x Restoring force or moment coefficient and interior reExterior Source distribution on fySore sribu o(x,) fresrfceVrequency Memory effect function
6
-281-
fnon en cei n n inof inertia ertia Mass or moment c i coefficient normal directed into cB i th component of generalized vector Field point in De and Di Point on hull Ship hull surface Normal velocity on hull Cartesian coordinates Line source distribution and its Fourier transform Small parameter representing slowness Small parameter representing slenderness Coordinates on hull Free surface elevation of motion, j th irregular frequency
iIZ.
Added mass or added moment of inertia
1)
=g/U 2 , wave number P Density of fluid o(Q,t), o*(QX) Source distribution and its Fourier transform 4(Pt), P(P,X) Velocity potential and its Fourier transform (P,t) Temporal velocity potential Motion frequency V
1. INTRODUCTION In ship hydrodynamic problems, especially for motions of a floating body on the free surface, the full 3-D problem may be reduced to the 2-D problems. Notably if the ship were slender in some sense, the well known technique of the slender body theory will be applied to get the 2-D problem which is more easily analysed and calculated than the original 3-D problem. In many cases such simplified treatment gives us satisfactory results. For an extraordinary example, we can refer to the ship motion problem in regular sea waves by the strip theory. [I] Now we confine ourselves to the potential flow problem of slender ships which make motions on the water surface either steadily or unsteadily. In the field near to the ship body, the slender ship theory suggests that the 3-D field equation, in the concerning case the 3-D Laplace equation for the disturbance potential, can be reduced to the 2-D field equation in the lateral (y,z) plane that is either the 2-D Laplace equation or the 2-D Helmholtz equation according to the mode of ship motion. The former equation appears in such problem as that for a ship running through the calm water making forced oscillations 12) or that for a ship running steadily on the calm water surface. 13] Tle latter equatonfixedappearSunning a ship regularWith its motiof relf nolowhich atively short wave length. [43 Therefore, we can consider the following degenerated
[~X 'il [Ii
y
qutos
VY
++
= 0 on z=0 0on¢zzxx
where v=g/Uz. And when a ship is running with uniform speed making forced oscillations, the strip theory whose basic boundary problem is the combination (l)-(3) may become inadequate with increasing forward speed. It is believed that the forward speed effect has to be introduced in the free surface condition. For the attempt of the introduction of the forward speed effect in the free surface condition, the oscillation problem [71 leads to [F]
I iI +
€ + V;~
0
on z='
(6) Ti the body boundar condition and the suite!e causality condition gives us a new basi boundary value problem in ship hydrodynans..s is different from those mentioned previously. The combination (l)-(6) seems to be much more complicated than that of (l)-(5)
problem of the combination of (l)-(5).
and
2 y zz " k d 0 (2) yvariable The 3-D effect enters only indirectly into t when the boundary condition does not include the 3-D effect. In ship motion problem [21 and [4] the free surface conditions tary [LI 2 D
K¢
0 on z=0
-
(3)
where K=W-/g 2her and in the steady wave making pwith p" 31ent
0 0
n
on
0)
-
M
(5) (
mental key problem can be reduced to the
=
-yy *zz
[ IF)
IF]
however as will be shown in §3.2 the found-
ix
[F]
These conditions apparently lack the 3-D effect which may be represented by the differentiation with respect to x, so that the problem in the near field must be solved as the genuine 2-D boundary value problems. From the combination of the 2-0 field eauation and the free surface condition we can consider the three types of the basic 2-D boundary value problem which have been fully studied so far. They are composed of the combination-of ()-12) 21(3) and (l)-(4) with suitable body boundary conditions and radiation conditions or matching conditions. In the process of the development of the slender ship theory, another type of the basic boundary value problem has come out. When a slender ship runs with uniform speed, it is well recognized that some irregularity near to the bow appears in the slender ship theory. [5] To avoid this irregularity, the bow near field has been introduced. The bow near field concept for the steady motion problem leads to the free surface condition, [6Y'
The new basic boundary value problem_ which was first proposed by Ogilvie [6] for a simple wedge shaped bow, has other side of meaning. To understand this conceptually, let's change the space variable x into time t by t=x/u, getting [F] + = 0 on z0 i7) gz tt I) The combination (1}-(7) with the body boundcondition which may change with time and the initial condition defines a well established 2-D water wave problem. Since the free surface condition (71 depends on time, wtcall h the nta we odto value h basic boundary tm eed problem the initial condition the time depend boundary value problem", even if the
condition 5% were adopted. (4)There may be many applications in the ship hydrodynamics whose basic problem coin-
I
-E
cides with the time dependent boundary value problem. First of all the analysis of t.,_ the motion of a floating body which makes arbitrary motion from time t=o is a direct application of the problem. If the motion amplitude of the floating body is so small for the body boundary condition to be translated onto its equilibrium position, the time dependent boundary value problem becomes the most basic problem. It has applications in the 2-D ship motion problem in the time domain. [8],19],[10] Secondly within the framework of the slender ship theory, the time dependent boundary value problem has possible application to the 3-D ship motion problem which will be reduced to the strip theory with forward speed correction. Thirdly the application to the steady wave making resistance theory can be expected. This paper will deal with the above mentioned three applications of the time dependent boundary value problem. However, the general treatment for the problem has not been investigated so far, while the counterpart problem of the combination (l) (3) has been fully studied both theoretically and numerically. Especially the integral equation approach in the (l)-(3) problem clarified the existence of the irregular frequency 1111 and the methods of elimination of it from the solution. 1121 Therefore we will mention in brief the analytical aspect of the time dependent boundary value problem when it is treated by the integral equation approach. Then the theoretical backgrounds for these applications of the problem will be discussed. As numerical example the 2-D ship motion in the time domain will be demonstrated with comparison to the experiment.
For simplicity the body shape c(t) is as sumed not to change with time. This ma may -not lose the generality of the problem. Making use of the Green's theorem the potential is written as, t dTj dX{ I 0
+ A -+~
IF]
ao
[H]I
N
0
=
on c(t) oWe
AIt
t 2.2. However the wave excited motion in Fio. 16 is not larger than the calculated in the same region. Then the discrepancies in the motion are not necessarily attributed to the wave excited force FDI. As for the roll motion of model B, judging from the experimental result, we should conclude that Fw3 is more dominant than FD3 as Dr. Lee said. Only the roll motion of model B shows the discrepancy in the free motion. Since we did not perform the forced motion tests, we can not say anything about other modes of motion.
-301
-
4) The basic problem (62) with the inhomogeneous terms D and E in the free surface condition is a direct derivation from the full nonlinear problem under the low speed assumption. (62) can include all the nonlinear influences condensed in D and E terms. Then it can include the local nonlinear effect that is defined as the inter-
action between ship waves and the flow around the ship zody. In this slender low speed probliam, E x'inction contains some of ruch interaction ezfect, though the slender hnip assumption makes thd problem very simplified one. So we do not entirely neglect the effect n the local nonlinearity in the analysis in §3.3.
-
II
MI
M in
f
-302-
jL
Second-Order Theory of Oscillating Cylinders in a Regular Steep Wave A. Papanikolaou and H. Nowacki Technical University of Berlin, Berlin Federal Republic of Germany
ABSTRACT The nonlinear two-dimensional hydrodynamic problem of a cylinder with arbitrary cross-section shapd performing finite amplitude oscillations in the free surface of a regular steep wave is treated on the basis of dynamics, potential and of nonlinear system theory. A complete second orperturbation der transfer model is developed from the assumptions of a second-order incident wave, a quadratic dynamic model of the system, and a second-order perturbation expansic. of the nonlinearities in potential hydrodynamic flow.
-
f(1)
j k
.F
The given nonlinear wave-body flow system is decomposed into six second-order subsystems to which a perturbation expansion in several small parameters is applied. This results in a set of linear boundary value subproblems of uniform type, which can be solved by close-fitting methods. On this basis second-order expressions for the hydrodynamic pressures, forces, and moments are obtained and introduced in the of the body motions to obtain the equations motion response. motin rspone.1 Numerical examples illustrate the solution procedure and several physical aspects of the second-order model.
_a
____
w
'jik
(n) (t)
g (n)function
i _r
144
j junit
B
waterline beam, Fig. 3
acceleration of gravity for order n,
orio n hi of order n imaginary unit (space), e.g. eq. (54) unit vector in positive x-irecton x-direction moment of inertia (roll), eq. (99) imaginary unit (time), e.g. eq. (55) vector in positive y-direction
wave amplitude
area of cylinder cross section maximum half-beam reference quantity for motion
hydrodynamic force or moment of order n in direction j, eq. 8 F (nF(n(n), (n) (n) =(n M(n) _M 3 =V n , 4 F
(n)
Description
A b bi
amplitude of order n in direction j caused by problem k or applicable, i and k, as lensinteraction between probby eqs. (90) to (92), n = i or 2
eq. (53) water depth
h
NOMENCLATURE Symbol
generalized restoring force coefficient of order n for (i,k), eq. (106) (2) hydrodynamic force or moment
(
k k 1(2)
i, eq. (2)
-303-
wave number, eq. (58) unit vector in positive roll axis direction
free surface inhomog. for sec.-
Symbol n)
-
i
Description
angle between
generalized bodyn mass coefficient of order for (i,k),
x-direction, Fig. 3
eq. (106) cylinder mass per unit length, eq. (99) hydrodynamic roll moment of order n
M M(n) M (n) R
6(n)
phase angle of order n, general, eq. (93) w incident wave phase angle, eq. regular sine-=Ofor (57), wave (t=O) in positive x-direction perturbation parameter for wave, eq. (2) 7perturbation parameter for diffraction, eq. (2)
moment of inertia of order n, eincident outar, unit normal ig vector, 3 positive unrdi)3 directional cosine components of ~order n for direction j,~perturbation c i nfordirctin ofordr eqs. (21) to (24)
n nik
Xk
inertial right-handed Cartesian coordinate system, Fig.3 body-fixed right-handed Carteo-x-y sian coordinate system, Fig.3 P~n (n)(R, ,t) hydrodynamic pressure of o-x-y
11 (-, ), pk (x~y) (2) ik
-
-)
r r
rk
S(t)
t T
draft, Fig.
V Vn
velocity vector of body motion normal component of body velocity, eq. (9) body motion in direction i,i eq. (14), for sway (i=2),
S
xi W
X(n) R
(n)caused
problem k or by interaction of problems (i,k), as applicable, eqs. (67) to (79) position vector in o-x-y-system, eq. (81)
wetted cylinder contour at time t, Fig. 3 wetted cylinder contour at time t=0, Fig. 3 time
.(n)
k4 oi (n)
'i
body boundary for first (k) inhomogeneities or second-order (i,k) problems, eqs. (30),(44), (47), arc length on cylinder contour unit tangent vector, Fig. 3
s s
5(n) Oxn)
n
order n hydrodynamic pressure amplitude of order n caused by
,
and pos.
parameter for motion in direction i, eq.(2) perturbation parameter in general complex variable, source point, eq. (54) incident wave length, ea. (58) dimensionless hydrodynamic damping coefficient of order n, by (i,k, eq. (108)
-ik
dimensionless hydrodynamic mass coefficient of order n, caused by (i,k), eq. (108) frequency parameter, eq. (58) fluid density
0 I(x,y;t) ((x,y) 9-j)(xy! (Xy)
body mass density, eq. (99) velocity potential, eq. (13) space potential functions of orders (1) or (2), eq. (13) decomposed space potentials
t'2).x,y)
B( Z
of orders (1) or (2), eqs.(16), (17) frequency of incident wave and of first-order motion response body bo'"uary differential operator, eq
(19)
Laplace operator free surface differential ope2 at re (8 bdmoinidrcinradiation differential opera-(20 tor, eq. (20) F(n-V){
heave (i=3), and roll (i=4) body motion amplitudes of order n, eq. (14)
V{ I Hamilton's nabla operator Subscript Conventions:
coordinate of body vetia center of gravity norizontal inertia force of
The paper contains several multiply subscripted quantities of the general form in symbol expressions
order n, eq. (100)
mj
(
oik
Y(x,t)
free surface elevation profile
where some or all of the subscripts may be
y
vertical inertia force of order n, eq. (100) complex variable, field point, eq. (54)
present. These symbols denote:
Z
a) Superscript n in parentheses refers to the order n of the quantity in the per-304-
two-dimensional hydrodyna The current expansion. turbation is approached by means of nonproblem ~mic fnn en prahdb perturbation mcpolmi dependence denotes the m b) Left subscript theory linear system dynamics, (d=e)pnde-e otionseoffirsth b the m and potential flow methods. This implies ant (mlw) upon ohe m=2) of nonlineof the all, but not that several, cond order (m=2) of the following quantitles: potentials, eq. (17), t associated arities present in the physical problem can be taken into account and that the results disturbances, e'g. eqs. (44) to may be valid for moderate wave heights and pressures, eq. (65), and forces, moderate motion amplitudes, in other words, eq. (87). for nonlinearities of such degree that the c) Left subscript j (j = 2, 3, 4) is used second-order perturbation approximation with direction cosines, eqs. (21) to will remain sufficiently realistic. (24), or with forces, eqs. (90) to (92). Specifically the approach will allow It denotes the reference direction j. for nonlinearities in the For the forces both subscripts m and j - Free surface boundary condition for inmay occur, eq. (87). cident and motion generated waves (by saincluding second-order effects and d) Right subscripts i,k (i,k=0, 2, 3, tisfying this condition in the true free relate to the cause of the subscripted - in a perturbation sense), qouantity. First-order quantities have asurface - Body boundary condition single subscript k, second-order quanti(by satisfying it over the really wetted ties require at least two subscripts regime, approximating the shape of the quantisteady-state k; carry second-order i, o. subscript the additional ties by a Taylor expansion around the tlatter rest position of the body in the spirit of perturbation theory). Other Conventions: All derived responses such as pressures, forces, motions are also evaluated Lengths are made dimensionless by b or ja!, anglesby maximum wave slope or with to the second order. However, nonlinearities of viscous origin (like separation at -, and maspect to 1 radian, time 2by sharp corners) are disregarded, although mass per unit length by b . subsequent, empirically based corrections of the system model remain feasible. .INTRODUCTION Much of the previous work on the subThe past decade has witnessed a rapid ject of this paper has been devoted to the arowth of interest in the nonlinear aspects incident wave and forced oscillation subproblems. Numerous higher-order wave theoof ship motions. It is well established that linear theory succeeds extremely well ries have been developed by Stokes 12) , in predicting many important phenomena in Levi-Civiti, Sjelbreia and others. Papanishin motions within the accuracy required kolaou has reviewed several of solution these theoof order a third ries, including for design purposes; but it has also been purpose prescnt the [3j For in his own, . of varange recognized all along that the a Stokes second-order wave will be suffilidity of the linear model is restricted cient to represent the incident wave flow. vby its fundamental assumptions of small In regard to the nonlinear forced mowave heights and small motion amplitudes tion problem Ogilvie [4] derived the second[1] . There has been growing experimental order steady force on a submerged circular evidence in recent years, accompanied by cylinder, which could be achieved without basic theoretical results, suggesting that explicitly solving the second-order boundaty nonlinearities in waves and ship motions value ?roblem. This was done later .y Parisare also a matter of important practical for the circular cylinder heaing s consequence. This holds for the prediction in the free surface and by C.M. Lee 16] for of any large amplitude motion effects, parLewis forms. Lee's solution to the secondticularly extreme values, and has received large mamin connection with ai much attention order set of boundary value problems is ol(asznghdo pltd based on multipole expansions and conformal plitude roll (capsizing), hydro jnamic immapping. Potash [7] , working with close-fit pacts (slamming, section flare effects), techniques instead, extended the problem to Swave-induced bending moments and other nonlinear phenomena p sway, heave, and roll, including their coupshapes. Papaling, and to arbitrary section nof i nres of no e preicset on of a similar one specific aspect of nonlinear shpbasis that is, the problem of an infinitetions, reexamined the foregoing results ldgapproach, ly long cylinder oscillating in the free for the heaving cylinder and removed some inconsistencies in the analytical expressurface in sway, heave and roll in response of of finite beam to a regular limiorwave water deep water and numerical results. Finally, Masu(onincident amplitude asions
17:
v5
r :
to the corn moto [10] developed an approach problem using plete second-order oscillation multipole expansions in analogy to C.M.Lee's treatment of the forced oscillation case. Several approximate solutions to the forced motion problem have been derived without solving the corresponding boundary value problem completely and rigo-
ted depth). This case is an important refesolving the much more generence point for mof ship motions in an irregular ral problem nonlinear a comprehensive seaway, for although to be purpose remains the latter ~theory ttpasecond-order developed. inn *305-
*
[
rously. C.H. Kim [1I dealt with forced heaving of triangular cylinders by an iteration which departed from a zero-frequency solution. Grim [12] derived approximate solutions based on low-frequency assumptions for large amplitudes of roll. Salvesen's in 13 to nonlinear heave (and aprch pitch) is also based on low frequency approximations. S6ding [14] used Green's theorem to derive the second-order force on an oscillating cylinder from first-order potentials of the problem. Yamashita L15] developed an approximate solution for "thin" oscillating cylinders with results up to the third order. Several authors, e.g. (16-18] , in contrast to the perturbation school of thought, have pursued direct time-domain solutions of the complete nonlinear problem, using initial value formulations and numerical integration schemes. These methods do not depend on livearizing assumptions and are attractive, in fact, sometimes perhaps the on-
ment of all these subproblems by close-fitting methods. The standardized hydrodynamic coefficients obtained from the submodels are assembled into the complete transfer model via the equations of motion to solve for the motion response and other responses. The paper describes the solution procedure in detail. 3ome numerical examples are given to illustrate the physical properties of the transfer model and its components.
ly available recourse, for dealing with very
the latter consists of the hydrodynamic cha-
large motion amplitudes and waves. However, aside from some unresolved questions regarding the treatment of the radiation condition in the complete nonlinear case there are also some practical limitations: Computer time requirements tend to be heavy, and validation and generalization of results are difficult to perform for lack of frequencydependent submodels. Experimental results for several aspects of the nonlinear problem have been presented by Vugts 119] , Tasai and Koterayama [201 , Yamashita 1151 . The agreement between individual test results and secondorder theories is encouraging, although the available evidence is far from systematic and complete. The research reported in this paper has the aim of defining a complete second-order model for a cylinder of arbitrary shape oscillating in the free surface in three degrees of freedom. This model combines a quadratic dynamic system model with a second-
racteristics assumed for the flow system. The two models must, of course, be chosen in accordance with each other.
2. FORMULATION 2.1 Dynamic System Model The nonlinear system of an incident wave and a body oscillating in response to this wave on the free surface of a fluid domain may be described by two sets of assumptions, called the dynamic system model and the hydrodynamic model. The former is related to the general dynamic behavior of the system in terms of its input and output,
Motions Wave Parameters it)
WAVE/SHIP SYSTEM o;(t)
Fig. 1: Dynamic System Model The dynamic system model of the nonlinear wave-ship system (Fig. 1) is characterized by the relation between its input signal i(t) and its output signals o.(t),j=1,
order hydrodynamic model. The incident beam
...
wave is thus a second-order regular wave, that is, a "steep" wave in the sense of Stokes. The system model is subdivided into several separate, but interacting flow systems:Incident wave, forced motion, and diffraction are three primary subsystems familiar from linear theory. In second-order theory it is not sufficient to model these individual flows to the second order, but it is also necessary to account for the mutual second-order interactions in these flows. This requires introducing three further second-order subsystems. A perturbation expansion in five small parameters is performed on the nonlinear system in orcVer to recognize the "smallness parameters" characteristic of each subproblem. The expansion yields a corresponding set of 20 linear boundary value problems. The great majority of these can be regarded as radiation problems and represented by Fredholm integral equations of the second kind. This permits uniform numerical treat-
in the steady-state behavior of the system, that is, in the time-periodic response of the ship to excitation by a regular steep wave. Initial transient phenomena of the response are disregarded. For this purpose we assume a nonlinear relationship between input and output of the form;
,J.
In the following we are inierested
N 0(t)
n=0
a
"
and in particular a quadratic time response model (N=2).,This model is introduced bere (as in earlier work by Lee 1211 and others) as an a priori working hypothesis, chiefly because it can be demonstrated to be compatible with the second-order hydrodynamic model to be developed by perturbation methods.
-306-
=
It is a particular property of this quadratic model that the response oj(t) to a monochromatic harmonic input signal i(t) of frequency w will contain harmonic terms of frequencies w and 2w , as well as a zonstant "d.c." shift term. In view of the complexity of the total flow process it is now of practical advantage to decompose the system model into a set of separate, but interacting nonlinear flow subsystems, each individually better amenable to hydrodynamic analysis. This will be done much in analogy to the concepts of incident flow, forced motion and diffraction flow, familiar from linear theory. At the same time it is convenient to introduce a system of several small dimensionless parameters ck suitable for defining characteristic "smallness ratios" in the perturbation expansions of the subproblems. We introduce: k -
= 2-t ia
w
II 1
0
7 =XM') I
.
=
of the motion amplitudes in each degree of freedom. Thus we haVe available a set of physically relevant smallness parameters for the major subsystems of the flow model. The use of five small parameters does not imply that they are meant to be independent of each other, but only that their physical interdependence need not be considered until after the subsystem flow problems are solved. In fact, co and c7 are both dependent on wave height, and £7(£0) and £i are physically linked by body dimensions. However, at least two physically independent smallness assumptions can be made, for example, the traditional "small wave steepness" and "small motion amplitude" assumptions. To this extent our approach parallels Newman's in (22 who used three perturbation parameters in a thin ship oscillation problem to characterize the orders of beamlength ratio (thinness), wave steepness, and motion amplitudes. The decomposition of the dynamic system model S into a se. of second-order subsystem models jS, assuming for the time being that all k are of equal order of magnitude, can now be expressed in terms of the characteristic small parameters present in each subsystem: S(x,y;t;£ 0 ,c7 c
b.
)
1 S(x,y;t;c 0
(aw in general complex) wave length
w
=22
=k
=
)
s(x'y;t;
amplitude of regular beam wave
with a.
s(x(3) 6 S(x'y;t;c
7 £.)
c(£)
wave number .. .....
:k b=dimensionless frequency parameter
b
+
+ 2 S(x,y;t;c)
C7 ) +
5s(xzy;t;c
&
and physically for a second-order wave on water depth h ( 31 kb~(v){(1
2C
kb-(v0C
C
= s/c
S
,C
= sinh
1
))
-1
=1(SC-8C
(kh) ,
+
0(c)3
-----
9)/8s
c = cosh (kh)
i
I b2
b4
o direction = reference quantity for motion i = b3 b =maximum half-beam = 1 rd unit reference angle
{
,
iJ
ULU
maximum half-beam of section = 2, 3, 4 for sway, heave, roll, respectively
i
D/
_-_-
Further: b
11%
2
Fig. 2 illustrates the mutual interactions of the six basic nonlinear subsystems: S: Nonlinear incident wave 2 S: Nonlinear diffraction Nonlinear forced motion
The pareter is a measure of waver steepness in the incident flow, 7Nonlinear racteristic of the magnitude of the diffract'on flow, and the ci define the smallness -307-
4S: Nonlinear interaction of tion interaction of
and 3S
S: Nonlinear interaction of
iS and 2S
-
and -
The second-order model of (3) differs from linear theory in two ways: The familjar basic flow systems IS, 2 S, 3S have to be extended to second-order level, and their second-order interactions have to be taken into account, which is done by the subsystem 4 S, 5S, 6S. Each subsystem jS corresponds to a nonlinear boundary value problem of potential flow, which by perturbation methods can be reduced to classes of linear boundary value subproblems _S(n). Some of these classes are further subdivided into elements of types jSP x n *s) that is, linear subproblems to be derived in detail from the hydrodynamic model. Fig. 2 shows the complete scheme of nonlinear flow subsystems jS and their subproblem classes and elements, connected by solid lines, whereas the dashed lines with arrows indicate how the nonlinear subsys;~ li ea S interact va u each other via their with tens linear boundary value subproblems-
snx
---
2 +
~+
3
X
O
~
*-
o
Y
(4
4
4
x4
x"
The notion is assumed to have existed long enough for all transient effects to have decayed. Further, we assume inviscid, irrotational flow, which ensures the existence of a velocity Potential -(x,y,t) satisfying Laplace's equation for an incompressible flid Combining the kinematic and dynamic boundary conditiops on the free surface y=Y(x,t), extending to infinity on both sides of the body, one obtains 81 A
tt
2(1
-
t
(XY(X;tht) + 9 t×.X
X2
-
Y 2 X €y xy
-
If the fluid has a horizontal bcttom at y--h, then
2.2 Boundary Conditions A cylinder of arbitrary cross section shape is oscillating in or just below the free liquid surface in response tc a beam, regular, steep wave of amplitude a and . frequency
(x,-h;t)
!n
with w
(6)
a,
The kinematic body boundary condition implies that the normal velocity of the fluid n equals that of the body Vn(Fig. 3):
n
n v(xy:t)
v
S(nV,)-xyc1 4{sina,-cosal
,
x x
n
:2
x =
Cos9
Nx!3 x 4 (Y cos x4
Y
+ x sin
x4 ) (10)
-
2Yi.
h 0b
Y
Y'
3+
sin tilt) f/
,
//
7
8
z oscz
Sn
ffff)/j
17)
s =Cosa~. ,ina't
sina
-
tsin
/
{) y yy
--
7
Cos Qct c77
)73/%-s =
Y"(
cos x,'
Y-cos X+
X'sin X4
4
_;-COs x 4
(I
-
-
ysin
--4
where the prime denotes the derivative with respect to arc length s, the dot indicates a time derivative, a(t) is the angle between unit tangent vector s and the nositive xdirectinn, and n the unit normal vector which is positive outward. At large distance from the body a suitable Sommerfeld radiation condition is imposed. Physically this corresponds to the fact that the incident wave and the motion generated by the ship are the only disturbances present. Mathematically this condition ensures uniqueness of the solutio.. potentials. In dete=ining the potentials the motions of the body xi t) will be assumed to and corpotentials normalized forces be known. The determined on hydrodynamic responding
Fig. 3: Coordinate Systems Two Cartesian-coordinate systems are employed (Fig. 3): The right-handed system o-x-v is fixed in space so that o-x corresponds to the undisturbed fluid surface and y is positive upward; the right-handed sysis fixed in the moving body and tem 1-coincides with o-x-y when the body is in its equilibrium position. The displacements of sway, heave and roll are denoted by x 2 , x 3 and x4 , respectively. Due to these motions a point (Ry} moving with the body has the following coordinates measured in the stationary system:
308-
A
this assumption wili be substituted into the equations of motion later to derive the actual body motions in a given wave.
placements x.-t) may be correspondingly expanded into a perturbation series in terms of the small parameters C i , characterizing the magnitude of the motions in each direction i, eq. (2):
2.3 Perturbation Expansions In order to reduce the nonlinear boundary value problem defined in the preceding section to a set of linear subproblems, we assume that the potential ' can be expanded into a power series in terms of five perturbation parameters t k (k=0, 2, 3, 4, 7) up to the second order in accordance with (3):
t(xY;t;ck) =--
C (Xy:t) k(12) 3 (2) Z i~ (xy;t) + O() k( Si kk i = 0, 2, 3, 4, 7 with 0(0 = 0 , ik Questions concerning the convergence of the foregoing expansion, or particularly its uniform convergence, must be left open for the moment. Current calculation results 'k gest that the second-order potentials are of equivalent order of magnitude to the first-order potentials ki)so that the magEk will govern the connitude of the ci, vergence of the expansion. The series should converge for sufficiently small values of these parameters, but how this limits the physical range of validity of this secondorder theory has yet to be found out. Special caution is in place for non-vertical section shapes in the waterline with regard to existence and uniqueness of solutions* although in our exjerience to date no serious practical difficulties have arisen yet. According to the quadratic response model (1) a regular steep wave of frequency will produce physical output effects with . Allowing for this and 2 frequencies fact in the separation of variables for the potentials in (12) by including terms of corresponding frequencies, expanding the potentials for small perturbations about the positions at rest, and treating the boundarv conditions accordingly, one can show [8] that the relevant potential terms up to the second order are included in the simplified expression: *(xy;t;c S C
=
S kZ k
(xy) e-t' 4 )13) Ikt(2) (xO
e-
2 x 1 (t)
=
2 n X
e- j k
k0 n0 Neglecting trivial terms: 2 x(t) = £ = -
t
+
(14)
x 2
(14a)
where we have introduced the zbbreviated notation to be used from now on: (2) E_n Xki nI - X.(n), kJ0, Ci2 x~i
x (2) .
((14b)
This change of notation is equivalent to saying that we will initially consider all hydrodynamic problems and their responses in a normalized way, namely for ci=l. However, ¢i will be reintroduced later in solving the equations of motions and determining the actual response. By substituting these perturbation expansions (13) and (14), and corresponding ones for the wave profile y=Y(x,t) and its derivatives, into the Laplace equation and the boundary conditions formulated in section 2.2, it is possible [81 to reduce the given nonlinear time-dependent boundary value problem for 4 (x,y,t) to a set of only space-dependent linear boundary value problems for (n)(x,y). The nonlinear problem contains boundary conditions on free and moving boundaries, the linear subproblems involve only fixed boundaries, namely the undisturbed positions of the liquid and body surfaces. This is in the spirit of the perturbation method where the conditions at the true positions of these surfaces are approximated by Taylor 'eries expansions about the positions at rest. The linear boundary value problems resulting from this perturbation development are described in the following section. We restrict ourselves to the deep water case from here on. Details of the derivation and regarding the consequences of limited water depth can be found in 123] 2.4 Boundary Value Problems In order to obtain a formulation for
(13)
O(2j )
the boundary value problems for the potenwhich is independent of the unknown motion amplitudes it is convenient to normalize the potentials in terms of the displacement~velocity and angular velocity components of the body or the exciting wave, where applicable. These components are for the first-order potentials, from (14) and the incident wave velocity:
)tials
This expression omits some trivial potentials as well as some time-independent second-order terms whose hydrodynamic effects are of fourth order. By analogous reasoning one obtains expressions of equivalent form for the wave profile y=Y(x,t) and for all physical quantities to be derived from the potential, that is, for example the pressures, forces and moment on the body. The unknown dis-309.
II
(n) *1
t
--
-
V
_
- _Qj.j n xs
n\
-
1
The phase angle 6w measures the distance of
the incident wave crest from the origin at time zero. In terms of these velocity components the first order potentials may be xpressed v 'ntroducing qnk sfolrs the normalized potentials
P.xvl
Fig
4: Gcometiy£ of the Boundar-yaValue Problems
6{I kkK'I "
E
CK
X,} )= {-,) 3 1
kv k
V..
k
I •
16
{16)
The tiane-dependent-direction cosines of the unit nora! vector n, defined in(S) and " (11' , may be approximated by means ofi a perturbation e"pansion about the body m-s tion at rest and thus exnressed in terms of body-fixed coordinates: (21= = n e-n.t + = t 3,4 -n2) I
for
k=--, 2, 1, 4. may be split The up second-order into two ses potentials of normlized ter: s, d s those due to firstorder velocities (left sunscrint eu=2al tc one), and those resultin fro second-order velocities (left subscript two):n (2) ({2) 2) A
(2)
I;ik
{ =
0
n--0
12=
ik
k
for ik= 0, 2, 3. 4
*
7
31 ,
U)
({
2.ii
i
for i=k=, 2
(
M
!2
For the sake of brevitv we introduce
(1' _
( ('I
i.22 _( X {1 2) M N4
(1) 2
4
2
l
=4
02.
1 T.
01
-.
the following differential oprators ( 16] IS}(8):
,o
[811
(23"
Free-Surface Differential Operator °
SF~v){
?(.y) }
F2
{F
({2}l -
...:r 2)
S_
hF} (x#.C5
I2i n
1i
2'y, {1)
21
M
rin
i
Bthe
~RN_}
-- x-
4
factor is Coilectina terms whose con m n-i e-jnwE (n=!, 2; i4,k0O, 2. 3,'s7) we obEain the following well-posed linear Radiation Condition Differential OeRaiaton ConditionDiferenboundary value probi=s, generally of mixed rator geealyo 'h for rc 'o- (third kind, Robin problem), n I potential functions unknorm sR Ii~) R.e {. }VI~}% (X= l g, --
)
The boundaries S trated in Fig. 4.
a
'3-0
-
vector s.
I
First-Order Boundary Value Problems
:L
j
n21
(224'
1
'-0, A similar expansion is performed for direction cosines of the unit tangent
Body-Surface Differential Operator
t
i=4
SO, S. and S. R
are illus-
-310-
-
~SE.F
-
nr n
F0.(x.y1
M
-
k.-dA-ar
eo-
-
--
r
w
-
~e
*
c
2.
k=Z. w. Secoilt-ndrnsid
au
k
rbtm
k
I..
i
.
-o--r--iase isiflfl)en
2.
eia
a
t-t
0'
r2
=
B~x~~cS k 1
I
-
~ (40
r~a-K =o
k
0SsxyaS4
-311-
~=-'--
-2s
-
''
=..a
r~;s
fa ce noe-genonU tens Conr ccs. Af45) and 14)1 wit
crresnondIng to (both coPlex), are
T.he only essential difference -between the f irst and t11e secondM order lies in the ir ~~~~" o rh~a 4nree s--acm- bo -ndary comfl;ion-
C ' r~~ement t-or Lt-) =o-nn= -rFowCr mor der-alc sfie !a-
LJ oc
1
~ee'
la--r
Bwar Pr~re~ Vaalue_ T-
~e
~(
0
tJd
Ce
-_
tnet
uniformrn .-a- of the w'u-nary Val ue nrcbt n des.i- big the linear su os - -e ncnna - -r O 5se. _-as 5eVe-a= n-"-c---a' adtra Qu..s est'o-s of e tee c irenets of solunis may be {iscussed in a er ee' way onCci a 4 aoea±, hower ii L~ Is 17. bnefit' th - a si -'Cl nmer.cai sol-utio = a -aeo to all suburoblena a-r-oa' t34s- -s' U-er 17Le
-eced-'ng
sec-4on
se
of linte-
-robs cs was xn-trcnc=ed DO) -'- the -.=-rs-.order noand -t a-s (32) to (50) fo
arobr'natM1 h ea-i..a~s
-Il secom-o er pote-'a-ls ~i)~ ctetUnrtiomS a-c '-a generC loRYplex 1with respec to timc, according to ecs- (131 to 'o or- -(17). ReferrInc b ack t*o F'-
tion, One can- now dcis'incr incide-t
-First-order
vt
-
~
z te tial
forced os-i'llatlon notentlils Rkt2. 34,4;n=-). three j k~r Cctions.
~
diftra~tion,no-en--al I-=-n=i), symnetri and antsmmti z5~ s, -two u Ikeown poten-~1 funel es= -irst-order
-ainMt
G-ee-"s
-First-order
l Scn-ori t-
~
the fsir
Bounary vale nreiAe.~ of type5 =a-be addresc as "ra_ 4 arion, nr~ber U'.e .n su arz S--toc '-'nerfeld, ac. ion tis name alltse nprcblem wtac may be
rTc'
SV=
fco
the= sezcnd orde.
-~hirA
o-
theorem of potential theL-eA .b~ona value
---=an ter-lncii o" neous !ntegra s--- tion ofr Fredboinm type of tes-cono kiz (-aelmi- 1 'integral eaua6
. 4:- m1k:-21,-
k=2.
nrheepo-ntials are *nimln cos're 01~1
this~eric-rs
an at=-sb
omiat
ats.-
a--iv--.i
rc
-3speve.
z
ar
.e,=
o fa-e
~a1o r~
wave a-ir
deen free
h~ Cfttions ~
t
ae
o6 h
re'
=ei
~~~~ 9sa
ofthe po-tent ie i ar
01-.
u o
looiz--io
ww
Asion fro
ala.e
Souiost -hd
VleT~bems-mla nde
at2
the incident wzve nrcnb ee. is rme e50=-
ti.
'
is:c.(13
rA aT1-
aa
~
jie
S s-
;ne
~
th ~ =.x
val
h
h
tin-denen-%Tla ordr ilde
tz)-obUnd o rC
-oudr
11
~ ~~~proper treatment o Iqz~~t C0G"
z;In)t
=ReG
tsefctievnmre im-ortant than in 1-near theory. Proposed cures to thiis probiem.=may.b analytical (Ursell 127- , OCllvie-Shin !78j Panankolao i81 semi-nalvtical and -nkical (Paullina an- Vou ! 291 Ohmatsu -301),
5
M55)
Teintegral etuation (5)rpresents actually a pair of Coupled interal eauations because the unknwn functions Z(n) are coplex. A physca 1 interretation this latin 's a ol s e =o -In, (.y n a ft t
or ourely n=erical (Faltinsen 011context , Papantoau S 3 nte present In 22 -1ro aou nuerical methods, based on interpolation f regular frequency results, were preferred to others up to moderately high frequencies because th.ey were convenient to use and gave results, though not always without substantial of computer time analv25'4 t Bu -Us- expense be for mentionec ti it mnthoAs this urthat ose purely have not ye
cow-oseo or contriutios-eliable aver of Iunr on) intensity e An) ( =--tential ) rodsce produ along d ae centra of =
knoW. source itnsiy ,) oni.n, a.. _. o heeo ... .. r~- oe...a~ and in the event of second-order potentials a soz.ce further single lpotential of known. IntensitV Z'' , 0) on Sv The Heslt 4ntegral e ouation lation (52), which was aparently first introduced into a-lvic ship motion. theory by Pf-tash 1 we;rp 7 is analvticaliv = re cenee-al t.-in Fran1's we!-kno-w conventiona! source-s!k
ben extended to second-order situations wn inhozuogeneous free surface b.,unda--, conwt nooeeu resraeb'~ayo-
iz
-nteral
::-iS:tion
o.
gration along S. h ch should be- extended to onfinitv -" *= analytical criterion for the truncaticon limit is required.- Our anroach to these ques-
I -tiouis
whez-_ 11b ==
-
)
a =
-C,
with the dimensionless frequency parameter
is discussed in section 3.3.3. 261 that inteura±
It is well-known equation foi-lations like (52), or as used by Frank, fail to provide finite, unique, phys-callv meaningful sclutions at or ar Ser- i irregTula-" Ire ncies c=rrespondin- to eigenvaies otte anjoirt interior potential boundar- value -roblem. 1is phenmenon is k-w as irre-alarit nroble" .n se-en-order treo=v ttis te of effect beans at much lower frequencies, the first secnd-order irro- ularity occu-rim at about one the freque-n-y' of th rresondin-o aarter first-order value. Tis suggests that
in fis
-r
.333
(-b) =
b591
.
Note that the wave corresponding to , exact to the third order although here nonlinear effects are visible only in the relation between wave frequency and w-ave length, eq. (58). Th4--e phase angle dw measures the position of the wave -ret relative to the origin at tine zero.
this ptential i. in fa
f U
potentials
3nietWv Wave Problem(S rbe 3.. ;2cident es;a~ deivation1e of secon-order potentase a a regular, steep (Stokes) wave ethod 1241 b including bounda- valuea so, - inh.geneous h = maybeaseo-e to be e0. 3) normalized special in aknou~n, wi= e presened o a e of problem addition,, eMen when applied su-riculy to a e ,w as n form in th present context. 4ea. r0bl'm xit~se~hnas aeof4in ra~e -umrshown 1 1 aI cerain~ no acvr'-Fo in-water of= infinite deot- (2 tans_ nuricail-yi, espeCialvr o ore coi al and need not be considered.For plcated section shanes, and with regard to -as s , .t . o e c-=' ' ow-a second-orcer regular, steep waA-e propagaitssen-st i tonthew-11 , : : rre5 g uar ring in -hepositive Y-direction on deep waZ->C 2S enmnon -rei ss the first-o-rer notential r-r ( r o e), dis:cretization :he of the intea f o-1r atio syIe 12inoaaebraicwith anlitude aw and lnth 1we must satisfy es. (25) 126)wave anof uations is conventiona' se2se Sn)q.e w)2 o. close-fit not be cussed here. methods Details and can wii be 4r-und in dis 8 , .1"231. n contrast to Frank's cl .s- i 1 ~ b _M ek -~- jx -% method -the ncral de~rivatives of! G reen s56 unctio in (52) are taken with respet soure Poi. co rdintes ich s-mlifes some integral expressions to be evaluated deiff'ities This may be normalized according to (15), Some parti-zlar analytiCal ex --s ev:la -it=,ie -e=n Axpr-s'o-s (6 in (52)_ One of these pertains to th.-e line ovewich contains s s ga -1 1 :ky -1I (k) riywhere SiV S-. intersects with S 7hni ca--._0 _ 3 + 5)ses nt rical In eva uatng tepotential an difficul-es its derivatives - the vicini"ty -f-e sin.gular point. -Further the i'nto--
I
3.3.2 First-Order Radiation Problems All remaining potential flow problems are rajfation problems of the form ("', with L =0 for the first order. These drob-lems can all be solved by a uniform procedure: The integral equation formulation (52) is discretized by introducing N straightline element panels of constant potential value on the body contour So and, in the spirit of the close-fit method, setting up an algebraic system of equations based on the boundary conditions at the midpoints of the discrete panels. Details of the procedure and the system of influence coefficients and right-hand side terms ia the algebraic equation system are giver in 123] . The forced motion first-order potentials (k~z, 3, 4) are determined in a single calculation usin symmetry or antisy ,atetry, whereas the diffraction problem (k=7) must be solved twice for the symmetrical and antisymmetrical parts of r('1 in (30). Once the potentials € ) are known on So , eq. (52) may be used again to calculate this potential at any desired point (x,y) in the flutd domain D. In particular the first-order potentials can be evaluated on the boundary SF (free surface) which is required to obtain the inhomogeneous terms 1I1W, eqs. (45) and (49), for the secondorcer problems. 3.3.3 Second-Order Radiation Problems The general procedure in solving the second-order radiation problems is the same as for the first order, that is, eq. (52) is again diseretized and applied to points on the body contour (x,y) c SO . In fact, the
Ssome
(44) to
(49).
'i
0
(60)
Nevertheless an independent criterion must be used to measure th! truncation error. in the presunt context the following indirect procedure was used: The first-order damping coefficient was first calculated from near fieid quantities (by pressure integration) and then compared to results derived froPm far field potentials (via radiated wavei amplitudes), extending tre range of invegration on Sp step by step until sufficient agreement was reacned asymptotically. This defined the truncation point x. . In practice, x. was found to depend or. frequency (v b) and body shape. A oarticLlar difficulty exists at the internection between S and SF where 1r2 and 11,1) are singular. Assuming the sifgularity to be integrable, which cannot be taken for granted for any section shape, we treat this problem numerically by closely approaching, but still exempting the pole in the integrations. However, the fundamental aualytical problem, particularly vertical sections, remains unresolved for des-nonpite John's valuable basic work F26] Forces. Moment 7
3.4.1 Pressures The hydrodynamic pressure P (x,- t), measured relative to atmospheric pressure level, according to Bernoulli's equation is 2 2 p(x,y;t) y (x,y;t) (61)
3,
Using the abbreviations from eqs. (13) to (17) (1) (1) (2)' 2) (62) ( 2k , k i,k 2¢ 2(i2i 2, , k 3, 2, 3, 4,7 i Z we obtain the hydrodynamic pressure on S(t) up to the second order:
This
P(X( ),y(S);t)=
hFs helped, in p ticular, to obtain stable on So. rY results for Regardi e numerical evaluation of t:.e term 1(27 on Sp some further problems arise. The improper integral over SF in
0.25y x
(52) involving this term in the integrand
(XX'v(')X
requires integration to infinity, but in practice integration must be truncated at
(
a "sufficient" distance "x_
1 (2)(x) I ik
3.4 Pressure,
problems ssociated with secona-order onset flows v()(k=2, 3, 4), eqs. (32) to (37), are completely analogous in form to the first-order problems and c.n be treated accordingly without difficulty. However, the remaining second-order radiation problems, hich are caused by first-order disturbances proportional to vk)(i,k=0O 2, 3, 4, 7), do introduce x i special questions regarding the evaluation of the right-hand sides, eqs. (44) to (49). These expressions involve first and second partial derivatives of the potential which are to be approximated numerically. The accuracy of these approximations must be examined carefully. In our compuzing experience for a variety of different section shapes it has been found advantageous t- transform some of the second derivatives with respect to x and y, where required, to expressions with derivatives in the tangential direction s, eqs.
Tis problem is also familiar from time-domain and finite element formulations of the present problem 1171 , 1321 , 133] , where in addition it is fundamentally difficult to iret the radiatioa condition at infinity. It can be shown by theory that dp 4 o the harmonic asymptotic behavior of
-
g
-
+ X x(
{0g(x
(1j ( x))et . {2g(x12]
,(1)2
+
n
.314-
)
2
) - 2jpw( 2 + ()) + 0.25 p( (1)2 1. -x - (1) s 2 3I (1)
-
0.jp
n
4
W,
from the body.
;
(1) e-2j{t 4
s
(-'x(1)
-x
2 3 _ .g(x (2), o3
1
X(2)
25-
-
o4 0( 1
(1)12 )+O
"
4
O.S
n
piY
0.SjPw(-xX
)+ yx.
3
+(xy-yx)
2+
(63)
0
These expressions were derived after 4(2) and M(1) expanding the potentials into Taylor series about the equilibrium position of the body contour S so that all potentials and their derivatives in (63)
(2) !Pik (1) is
Eq. (63) represents a complete hydrodynamic transfer model for the pressures on the wetted body contour S(t). This transfer model corresponds to the dynamic system model (3) and can be decomposed into terms of different orders (and
k
i 2, 3,
+
i=
1 ik (2)
-
IPoik 4, k- 0, 2, 3,
(1) in
-1
(2) e-2jVt ((y) +)(e))_____ Z
1 2 4, 7
i,k=0,
(64)
e)
i=k=0,
=
,
+
L-0 ,
4
k
3,
)
,
-
OW 5 0..
,
__
+
0.oj
___
pw(.n(0)
11i (2) (1) +
Sin
i 1t
'is'
) x.i
; Is
+
;xl l
i=4 (74) (
( 2 )
3
p g Xo3 -g -X
k=4 k=4 (67)
0,
J
1=3 (2) , o4
(75)
i=4
3, 4
b) i,k=2, 3, 4; i< k
4
P( Sin 0.2t5l (1)2 + 24 (0)A
I is
i
x -
(2)
P. (1) -(1)
=
in -0-.
s
-
14
i
j
K
.25 :g
a) i=k=2, 3, 4
1 p.
i
(72)
4s
(1)
i.nS
in
101ik
(2)
(7)x
+
3. Second Order (n=2, 1=1, m=1; hydrodynamics to the second order)
;iI __
*(s
a) i=k=2, 3, 4 2 ( n i -0 ()" Mis (2) =(n), I 02 in 1
4'
-,
ks )Xi
(2) - 0.25 o( (1)2 (1,2 1Pkk kn + ks (73) 4. Second Order (n=0, 1=1, m=1; quasi-hydrostatics of second order)
PLJ g X3(1)(2) '2Poiil pg k=3
-
+
n
7
(66 Pg (66) P9 y 2. First Order (t=1, 1=0, m=l; :: k=0, 2, 3, 4, 7; hydrodynamics to the first order)
j:I
(
)
n41) + n
p p(0) =P-
() ((1)
n
* is
kn
7; 1 < k ,(..) ______
0.5 p(
(2)
1. Zeroth Crder (n=0, 1=0, m=0; hydrostatics)
-
s
c) i=2, 3, 4; k=0, 7 (2)) (2) 2j pw (2) - 0.5 p( (1)
d)
Comparing (64) with ~~~~()(2) (63) yields the dropping individual pressure terms k e j n wt the factors m -In
""
I
(0) (1) (1) Is )x
(71)
where the left subscript is defined by the following convention: P(2) = () (2) (65) (2) f2 (2) X (2)
=
=
()(1) kn + (0) ()
x~~
+(2) 22
(2)
k
() 4 in (!) kn
(0) +
p(2)
(k
1 2
-
0 0
kIiin €i ks ) + 0.51 p.(.n
£k Pk
k
i
!
3, 4
,
ks )
)x
frequencies) in analoEk
^,.(2) 1 w lik
(1)
gy to (13):I (Y) +
)
b) i,k=2, 3, 4; i< k
pertain to this position.
()
(68)
(2) 3 2,
g
-(xYY')
)-(1)
P(x,y;t;Ck) =
i=4
,i4 2(
x
2
g
S0.25
+
(1)-
) +
x (I)
12
¢(t(1)
V n# +(n(0)ti) In k
+ i(1)2 S
(I)
-315-
4
+
i
)
-
4 kn is ks 1){(in(05-(1)
kn s s ) k (0)-(1) is
+ k
(1)++ i
ks
k(1)} k
(76)
c) i=2,
3, 4; k=0,
(2)
1
=oik 0 -0.5j 0W( n
7
T(1) + i)--1)) kn is ks +'' S. s ) x
0 in
jF(t;ck)
-
EC C
(77)
-0.
P(
i
) -ks1
(78)
oii (((2) @1)T(I) ks ks
jloii if(2) + j2 f2) oii
= V
==3,
i=2
(88)
H Collecting terms whose common factor is C, 1 e-jnL (1, m, n=0,1,2; i=2,3,4; 1 m -
(80)
Xy-yx),
(87)
The left subscripts denote force and moment components, respectively:
In these expressions in and -s denote the direction cosines of n and s at time t=0 with in(O) from eq. (23) and
i=3
j92 fi 2)i
ii (2)
(79)
(°)
x
f (2)
"k
jii (2 ) =j1
s(0)
(
where, in iccordance with (65):
(n ) kn ( )
-M p ( T2 1 () ( kn kn -0.25
-jwt
(86)
jCIk fi(2e e
e) i=k=O, 7
(2) I okk
e
k f(I)
k
d) i,k=0, 7; i< k
~~~~(2) 1Poik =
z
*F(0)+
=
i=4
k0, k3, 4, 7) the individual components are defined by the following expressions:
3.4.2 Forces, Moment I. Hydrostatics (n=0, !=0, m=O) 1=
Integration of the pressures P in (61) over the wetted body contour S(t) leads to the hydrodynamic forces acting on the body and the moment about the origin
F(0)
I p(0) n(0)ds
-
I
(89)
s 0
=-f P n ds,
(t)
(81)
2. First-Order Hydrodynamics
s(t)
M(t) =
f P( r x n ) ds,
-
(n=1, 1=0, m=1)
(x-x2 ,Y-X3 )
=
S(t)
By means of Leibniz'integration rule the integration over S(t) may be reduced to an integral over the wetted contour at rest, S , and a few additional terms of second order [7] , [23] ;so °I
I(S(t) ;Ck)
=
R
R+
f i(s;t;c k ) ds 4 I n(0) (0) - R_ n R+ n
-g~ 0 .2
3 +
- _ 0.2 -1e 2j(t whr - te -j g
4(1) M(
g-
pgj (1)+ =1
k
(82) (83)
(
2
ik f(2) " . -1 .
,nP _ "1ik
(84)
0)
o
i+05fP
n
i or k=4
:
--
X
So
:
(0)
ds deno -jv + 0.25.
d~
2
(2) (2)
(n-x(2) (1)2
n,
x4
)ds (91)
i=k=4 +
fr(2)
y;)= (-0.SB,0'
The transfer model for the hydrodynamic forces and moment to the second order is derived by substituting the pressures (63) and normal vectors from (21) to (24) into (81). The resulting expression is of ana-
logous form to (64):
__
(2) n (0)dsPi.
o
(1) (1)
1(85)
L4
(0)
()n
(1)
where the subscripts denote ()+ () ,)(0.5B,O) -
(90)
(n=2, 1=I, m=1)
2 (1) +
(1)
fp ( ds k n 4(1)d n' JPp(0) .n) 2 4 k4
3. Second-Order Hydrodynamics
(1)
2
_
4. Second-Order Quasi-Hydrostatics (n=0, 1=1, m=1)
f( 2 ) = -
2 oik
-316-
p( so
,n (0)ds i koi
(2
-
0
2
oiij
n(0) ds
I) (1)-Cl) cS 3
(0) Ss
+ 0.25 n
or k_4
12)
(2) (2). ( n0o 04
(1),2 x4 1 ds +
i
HR =
4 144
M{x 2 GCos X4
-
+(9+x)
Gsin x4 )
(92)
,0-with j oR f(2) represent
xi(t) from (14) and
+
fR2) and
The expressions
A
the contributions from the additional integrals (83) to (85) which approximate the effects of the actual wetted contour S(t) deviating from S. The time-complex expressions for pressu ep (n), eqs. (67) to (73), and forces f n, eqs. (90), (91), can readily be converted to real notation for any quantity a(n), whose amplitude and phase are AIn , and A A
a (n)
We2
y
(n
)
Re jna
= arc tgf
99
kM(9 4 0 G
The expressions in (98) can be developed into a perturbation expansion to the second order with respect to ci , using xi(t) from (14) and a power series expansion of the trigonometric functions. The resulting expressions are of the form 2
5
0
2
. dm
o - . A A
=
X(
(n)
(n) A
2
-y
-
sxn ( n.t + 6
A( n)
=_. -2 -2 I fIX +y
44
m (a
Y
E
Xn
2
y n) e J
2
en=l
t(0
)
R93 (
(100)
n=0 M
3.5 Body Motions
R
To determine the unknown motions x. (t), eq. (14), of the body in a given incident wave the equations of motion must be taken into consideration. These equations serve to determine, in a second-order sense, the actual motion amplitudes, hence the parameters ci (i=2, 3, 4), and further any other explicit hydrodynamic quantities of inter-
2
=z
n)
e
R
n=1
-jnw.t
+
(2) Ho
Substituting (100) and (86) to (88) into (94) and separating terms of different order (and frequency) the following sets of equations of motion are obtained for motion components of matching order (and frequency): 1. Hydrostatics (n=0)
-est.
H(0) =0
Force components relate to the inertial coordinate system o-x-y, the moment is taken with respect to the origin Z of the coordinates fixed in the body. Equating hydrodynamic pressure forces (moments), subscript P , to inertia forces (moments), subscript
= YR v M(0) =0
2. Hydrodynamics of First Order (n=l)
R , in the equations of motion, we obtain: F R=, R
FP
Ip n~ ds S(t)
,M
~
7
k=0
1)R
f
r x n Ids
-Ip(
(95
v~ 1
E
=
S(t)
FRm
A
fjg
f fr'g
+
A
(9)3.
fine& Ly. = -R M {x
-+
00
Y=
M{q
H((2)
H
(2) v
X- X4 YG x3
-
M 0--
((2) =
(97)
x4 YGsin x 4
o*._ X4 yGsin x4 -
7 , r i,0
2
H
x4YGCoS x4 } (98)
-317-
1
0
XR
7
(2)
E
=
M
ik=0
02-
=
k=0 Hydrodynamics of Second Order (n=2)
Af;Xg )dmAR
with F, M,, r from (81), v from (10) and A body cross Sectional area. The hydrodynamic pressure forces Fp (M,) were derived in 3.4 except for the factors .zior xn), respectively. The inertia forces (moment) are de-
(102)
=
H(1)
.)=
X()
=
k=0 7
.0
(101)
2 YR
(2) i(13
2
Al.Quasi-Hydrostatics of Second Order(n=2) Exciting forces and moment are made dimen(2) H°
7
sionless by i(n (n)
(2), , =0i~k ilk 0 oi.k 7(2) (2)
0 =
0~
7 o
V°
E
Xo
) =
The first set of equations, (101), concerns hydrostatic effects due to zeroth order pressures, corresponding to the law of Archimedes. The set of eqs. (102) consists of the first-order differential equations for sway and roll (coupled), and for heave. The unknown parameters ei(i=2, 3, 4), defined in eq. (2), can be derived from the solution of (102). Eqs. (103), with ei substituted, comprise second-order equations of motion for coupled sway-roll and for heave, and are used to find second-order motion amplitudes ,(2) (frequency 2-- ). in (104), finally, time-independent second-order effects, called quasi-hydrostatic, are present and can be determined using the Ci from the solution of (102). This yields the so-called drift components x(?), where the sway drift o2is equated toozero, by virtue of an assumed external force balancing the drift force. The dynamic equations (102) and (103) can be written in dimensionless form as follows:
4
(W n
=
=
22 33 in2 4
=
=-0.5 1 '-
G
4 2
cik'
=
4 a (2) ik
33 c
ik 0.5GM A,m
44
ik
W
ik
1
fl)
ik
_-0.5 f S0
df
-(2)
4=2.3
2 pgb' with
(i
1 kgb
ilk=AA 0
l
(2) (2)
1k
(2)
7 1
o
0.Sga2
0
4
-(2) o
-2
7
The nondimensional vertical drift is
2)
2
~
o3 w
X oth 3
-1
Xo3 b (2) o o
-(2) 7 E CiC I oik i=2 k=0 1-1 -(2) 1) (i6(2) =
w"
v3
C7
and, finally, the roll drift ("heel")
(107) o X
(n) n (0) i
4
)1
o4
ka
=
4(2) 7 7(2)ka w)
wka
i=2 k=0
2ds Ci
,n n
(2)
-(2)
b2,i=4
~F =
(2) )
H( 2
2 o
20d~kw
(n)
¢7
a momentum theory. From (86) and (87):*
(106)
2od dknw
(nI +
i=4 i
(2 ) Sr (2) The drift components (of second order) o result from the algebraic set of equations (104). Te horzontal drift force is derived from H(e (xo remains indeterminate) and may be compared to ?aruo's results[34i from
l = 1 (n ,i=2,3 i(n)
7 7
)s2
(14
o.44
1
In)}n pd dk
(112)
±-4 )
ping in (105) are defined as
2
(r.)
=(kb)
S0
Pik
n-1
(n) . (w (kiaJ
and
ele k 1lC k 0 ii The dimensionless hydrodynamic mass and dam-
'(n)
(n) -I ikX
= i=2
X( xi
I)
ik
i
k
with
n2(vb),((n)) ()a (n ) _Jk+ ik ik i&. (~ '(n) ~n) Cik =S+ i' ,i= 2, 3, 4 (105) --2-with: A A b -2 = A (
,a k
are obtained by matrix inversion
(n)-
ik' Cik
(
k=2
k
(1)_ ik -
(110)
The solutions to (105), which is of the form
4
(2)_ mik -
(lk I from (86))
Bk k=0,7 -(2) 7 E I (from (86) {87)) (111) i=2 k=0 ilik ]
2 R "Ro
(2) = i,k 0Soik
gbd= Z 1
(109)
=
,
(
,
7
(2)
(104)
()
'0
= vi
_ti
1, i=2,3
E
(.LO8)(0.5
M',
=(10)
- 0.25(0b) 7 1G x 3 - -1 (05
GMA
)
4
)(117)
2
-318-
_
_ _ _
_---
___ -
=
_
L
-
S
4. NUMERICAL RESULTS
are not in agreement with Yamashita's measurements regarding the absolute level al-
A computer program [35] has been developed to numecically evaluate the boundary value problems described in the preceding sections and to calculate the physical quantities derived from the resulting integral equations. The results presented in the following were generally obtained with N725 discrete pinels on the body surface SO and about 50 discrete elements in the free surface SF (second-order). The free surface disturbances for the second order on SF were evaluated up to where the potentials reached an asymptotic limit, usually no more than 9 half-beams away from the body. The frequency range was 10 5 (vb)S 2.5 in steps of A(vb)=0.05. The size of the program is 140 K words on a CYBER 175 computer. The program calzulates in one pass all pertinent hydrodynamic quantities (potentials, pressures, forces, moments, and motions, where applicable) for any standard problem case of either first order (k=0, 2, 3, 4, 7) or second order (i=2, 3, 4, k=O, 2, 3, 4, 7'. Compilation time is around 10 seconds per standard problem case. A complete evaluation up to the second order, comprising 13 standard problem cases after suitable rearrangements, takes about I minute of execution time per frequency. The results presented in Figs. 5 to 36 (in back .f the paper) with few Exceptions pertain only to second-order quantities.
though both show a similar flat tendency; we are uncertain of his definition of this quantity. In Fig. 6a fine Lewis form for B/T=0.8 and 8=0.5 (section coefficient) is investigated, Fig. 7 shows the ellipse for the same B/T. Hydrodynamic and steady state secondorder forces show excellent agreement with measurements. The phase angles agree better for the ellipse; for the Lewis form the agreement improves for decreasing motion amplitudes, proportional to E, as one must expect. In Fig. 8 for the ellipse at B/T=1.4 all results are in very good agreement. Figs. 9 to 11 are related to wider secticns (B/T=2.0) of different fullness. The triangle shows the strongest, the U-shape the weakest nonlinear effects. This appears reasonable because the nonlinearities should be responsive to how rapidly the section shape changes near the waterline. For the circle in Fig. 10 the overall agreement in all results with data from experiments appears extraordinary. The minor deviations that do exist increase with c, but remain acceptable even at c=0.6. The U-shape (Fig. 11) shows some, but not much grea'r scatter. It is of interest to compare the steadystate forces in Figs. 9 to 12 in the limit of (b)_0. The triangle with large positive flare in the waterline produces a net positi4e steady lift force, circle and U-shape
They are based on first-ordcr results, which
have zero flare angles and a vanishing zoro-
cannot be included in the prescnt paper. Nor does space permit a discussion of the "irregidar frequency" phenomenon. Earlier Dublications by Papanikolaou 181 , [23] , j25] may be consulted for details on these issues, including numerous first-order results for different section shapes over a large frequency range [25] Figs. 5 to 12 compare our numerical resuits for calm water forced heave mtjQons up to the second order (problem S )in Fig. 2 or eqs. (38) to (46) with i=3,3k=3) to available experimental data from Yamashita [15: and Tasai-Koterayama [120] . This comparison is intended to demonstrate the physical relevance and numerical stability of these results. All forces are made dimensionless as indicated in the figures. The assumed heave amplitudes have been standardized to correspond to E3=1 or xjl)=b. Negative phase anglesare always plotted with an increment of 360 Fig. 5 for the triangle of B/T=0.8 and /2 (finite flare) demonstrates encouL ragingly good agreement between theory experiment for the hydrodynamic force V 3 3 and, some allowance, for the phase angle The appreciable phase shift, whic], is more abrupt in the measurements, occurs at frequencies near where the force has a minimum, an observation made here for the triangle, although in the linear case similar effects are familiar from several other shapes. The calculated results Wgr the steady-state second-order force VO3 in Fig. 5
frequency steady lift force, and the bulbous form with negative flare causes a small negative steady lift. At finite frequencies the steady-state vertical force may become positive or negative. The results for the bulb (Fig. 12) should be viewed with caution due to the very large c-values in the experiments. The magnitude of tho nonlinear effects is rather small, the scatter in the measurements Zonsiderable, and comparisons with the theory for c up to 1.917 are of questionable value. In Figs. 13 to 16, for a Lewis form (B/T=2.C, B=6.94), further comparisons of forced motion results for pure sway, heave, and roll as well as coupled sway-roll motions (with reference to standardized parameters , Ck=1) are presented relative to data from Potash's second-order theory 17] (Similar comparisons with theories by other authors are also found in , [R 2] . Fig. 13 relatq to the pure sway forced motion problem (3S1) in Fig. 2). The agreement with Potash in the vertical forces, hydrodynamic and steady-state parts, is excellent. These can be calculated from first-order potentials exclusively. The second-order horizontal forces H ),which involve secondorder potentials, differ appreciably from Potash. This seems due to a deviation in his second-order problem formulation 181 and the presence of irregular frequency effects in his results. '2) For heaving (problem.3S 3 i Fig. 14 the hydrodynamic force V(3 depends on
-319-
A2
second-order potentials, but the agreement remains reasonable, in part because the heaving irregularities ar 2 Tilder. In rolling (Fig. 15, problem 3 S 4| the hydrodynamic nonlinear effects are relatively weak for this section shape. Fig. 16 illustrates the second-order effects in coupled forced sway and roll motions (problem 3 S ))upon the vertical force component. This encompasses the second-order effects due to first-order disturbances resulting from coupled swa, and roll motions. Comparisons with Potash's results are problematic because it is not clear whether he was dealing with the same flow subsy m. (fie have included the problems 3 S2 j and simultaneously), Figs. 17 to 19 present the steady-state forces and moments acting on a fixed body in a wave. These results st)from the steady-state part of problem 6S . These i estigations, and the ones that follow, were performed systematically for three section shapes, the triangle (8=0.5), the circle (S=n/4), and the rectangle (8=1.o). Forces, moments etc. are nondimensionalized with respect to wave amplitude law! , as customary in the literature. Fig. 170c 9 erns the horizontal steadystate force Ho7 . In the limit of (vb)- our results for all section shapes approach unity in agreement with Maruo's [34] analytically derived result from ATmentum considerations. The quantity Ho 7, plotted in dashed lines, expresses the considerable contribution by the wetted part above water (rest integrals in eqs. (83), (84)) upon the total steady-state force. For (vb). this term tends to the limit of 2 sin a in agreement with [36j . The contribution made by the underwater part is smaller and negative, the sum of the cwo yields the net force. The corresponding vertical steady-state forces (Fig. 18) are negative for circle and rectangle (sinkage force), and positive for the triangle (lift). The zero frequency limit of this force is zero for the wall-sided shapes(- 21 /2) and equal to I for the 1 (V Voitriangle 7) -ctg aL= ctg r/4=1). 1he steady-state moments, Fig. 19, are negative, that is, heeling in the direction of the incident wave. In order to determine the influence of the motions upon the hydrodynamic secondorder contributions, it is necessary to solve the equations of motion to the first order (eq. (105), n=1) to start with. The results are required to solve for the small parameters ci (i=2, 3, 4) in terms of the initially assumed E7 (co), which measures the relative size of the incident wave, using eq. (114). The first-order excitation forces for a circle in a sine wave (6w=0) are shown in Fig. 20. Figs. 21 to 23 present the corr sonding first-order notion amolitudes x :i=2, 3, 4), where the results 1 for sway and roll stem, of course, from their c-upled equations of motion. Near resonance all amplitudes are considerable because hydrodynamic damping is small, above
_
all in roll, unless empirical corrections are made to allow for viscous damping. On the basis of the amplitude results we may now assign some bounds to the ej and limit the wave heights accordingly via -7 or we may assume some wave height, hence C7, determine the corresponding values of €., and avoid frequencies where £ i exceeds a specified limit, especially near resonance. In Figs. 24 to 28 some first results are presented describing the second-order forces and moments which result from the second-order interactions between motion potentials and the combined incident w and diffnis 4 ( 21 and 5S subproblems combined). These results take into account the small parameters ci=f(vb ) of the motions, deduced from the linear response analysis (eqs. (105), (114)). The curves exhibit some more or less pronounced resonant peaks. These stem mainly from the behavior of the ci near resonance, modified in part according to the basic frequency dependence of the second-order hydrodynamic forces. The results presented here are still a function of the initially assumed parameter c7 . By solving all standardized subproblems of the system and associating them with their pertinent small parameters i-ek, it is possible to assemble all contributions of second-order for the body freely oscillating in the wave. The steady-state part of this summation corresponds to the so-called drifting forces. Fig. 29 compares horizontal drifting forces for three basic section shapese. The results are credible, except near resonance. Comparisons based on Maruo's familiar formulas 134] give similar answers. The asymptotic limit of this force for (vb)- - should be one, as in Fig- 17 for the fixed body, because the motion amplitudes go to zero at high frequencies. Vertical drifting forces, eq. (116), are shown in Fig. 30, drifting moments, eq. (117), in Fig. 31. Rectangle and triangle have positive vertical drift (lift) for most frequencies. The drifting moments are negative, they tend to heel howard the wave. Masumoto's results [101 show similar tendencies. It remains to solve the equations of motion of the second order, eq. (105), n=2. The excitation forces of second order ate obtained from eq. (111) by su-unation of all second-order hydrodynamic terms, which are caused by the motions and velocities of first order. The vertical second-order excitation forces for a semi-circle are plotted ir Fig. 32. According to egs. (107), (108), n=2, the second-order hydrodynamic mass and damping coefficie::ts for heaving have been made dimensionless with the displaced fluid mass of the semi-circle, as usual. The curves in Fig. 32 are obtained by contracting the fretuency scale cf the corresponding first-order curves, which are familiar, to one quarter so that the first-order results for (vb)=4 are shifted to (v b)=l. The motion responses of the heaving circular cylinder to the first and second order are presented in Fig. 33. It is interesting 320-
7__
_
_
-
to note that two resonant peaks are present in the second-order heave amplitudes, one at the resonant frequency of the first-order system which tends to excite the second-order system, and one at the second-order system's own resonant frequency which lies at about one quarter of the first-order resonance. The second-order amplitudes are relatively small compared to the first-order values in this instance. This need not to he so for other degrees of freedom. To obtain an idea of the relative i.-.ortance of second-order effects in forced heaving motions we may compare the curves in Fig. 34. They represent the ratio of secondto first-order force amplitudes for a heaving circular cylinder as a function of the amplitude parameter E3. The agreement beteencalculated results and experimental data from Tasai and Koterayama 1201 is very good. Only at higher frequencies and amplitudes (s) do some differences develop. Viscous effects as a possible reason for part of These differences at higher frequencies are discussed in [201 . A comparabls diagram for heave excitation forces, second- to first-order ratio, for a semi-circular cylinder oscillating in a wave is given in Fig. 35. This ratio depends directly on £7, i.e., the relative size of the incident wave to the body dimensions. The nonlinear influence on this quantity has a peak at some intermediate frequency where larger motions are present due to resonance. The time-dependence of the heaving motion of a circular cylinder in a "standard" wave (r7=1), approximated to the first and second order, is shown in Fig. 36 , together with the second-order steady-state term. The frequency (v b) of 0.25 corresponds to a peak in the second-order effects (Fig. 33). The nonlinear effects are not dramatic, but noticeable. Amplitudes are about 15% greater than in the linear analysis and a steady lift effect is present. The result confirms why for a shape like the circle linear theory haF been so successful in practice whenever heaving motions and waves are reasonably small. That nonlinear effects upon vertical loads (and, of course, local pressures) can be much wore substantial in certain frequency ranges will be appreciated from Figs. 34 and 35. Although we have not yet examined motions in other degrees of freedom by the current method, we would also expect significant hydrodynamic nonlinearities in nearresonant roll, coupled with sway., in view of the associated large roll amplitudes. 5. CONCLUSIONS By means of an approach based on nonlinear system dynamics and nonlinear hydrodynamic theory it has been possible to develop a complete second-order theoretical model for the motions and hydrodynamics of a cylindrical body of arbitrary cross section in a regular, second-order incident wave. A crucial first ster is the decompo-
sition of the total second-order flow system into a set of nonlinear subsystens comprising the second-order equivalents of the familiar forced Potion, incident wave, and diffraction flow systems,but also their mutual second-order interaction. Perturbation theory, using several small parameters, has then been applied systematically to derive a complete set of first- and second-order linear subsystems of the flow. These systems together form the basis for developing a second-order transfer model of the dynamic problem. The equations of motion have been derived to the second order with all hydrodynamic couplings present. They include firstorder terms of incident wave frequency i, and second-order terms of frequency 2w as well as a time-independent expression. All system responses are of the same basic form. On the basis of this theoretical model a numerical solution method has been developed using an integral equation formulation for a Robin type problem and a close-fit discretization. The fact that all problems, with one trivial exception, were of the same boundary value type, that is, radiation problems, paved the way for a unified calculation procedure for all subproblems. Numerical calculations were performed for a variety of section shapes over a wide frequency range. These calculations included some samples of each of the major subproblem types. No insurmountable, fundamental obstacles were found in the path of the numerical calculations. In those relatively scarce cases (forced motions) where comparison with experimental results was possible the agreement in second-order effects was generally between good and excellent. For the heaving semi-circle, for which most experimental data are available, the agreement is outstanding, for sections with flare it is slightly worse. The evaluation of drift forces as a second-order phenomenon is possible from subproblems of the system. The resilts obtained show good agreement with other theories. The response of the system can be evaluated by assembling the results of all firse- and second-order subproblems into the equations of motion with small parameter values assigned to each case on a physical basis. The overall transfer model is based on frequency domain techniques because the linear subproblems, of which the system consists, all have frequency domain transfer functions. This possibility evidently has significant practical advantages over timedomain solutions. investigations of the type reported are, of course, only a prelude to developing a systematic understanding of nonlinear ship motions in a nonlinear irregular seaway. We feel, however, that the remaining open questions, despite their great fundamental complexity, show a certain promise today of gradually being amenable to higher order analysis via the frequency domain by
-.321.
-
-.
j
suitable extension of bi-spectral analysis and generalized three-dimensional flow analysis methods.
11. Kim, C.H., "On the Influence of Nonlinear Effects upon Hydrodynamic Forces in Forced Heaving Oscillations of Cylindcrs", in German, Schiffstechnik, vol. 14, 1967,
ACKNOWLEDGMENT The work reported in thisa the of the Deutsche supported by a--grant=plitude", Forschungs-gemeinschaft[
pp. 79-91 0., " 2 namic Forces Caused by Roll oscillations with Large AmSchiffstechnik,vol. German, in ciftcnkVl piuei emn 24, 1977, pp- 143-160. 13. Salvesen, N., "Ship Motions in Large Waves", R. Timman Memorial, Delft, 1978. 14. S6ding, H., "Second-Order Forces on Oscillating Cylinders in waves", Schiffstech-. nik, vol. 23, 1976, pp. 205-209.
REFERENCES 1. St. Denis, M., "On te Spectral Technicue for Describing the Seaway-Induced 'lotions of Ships
15. Yamashita, S.
A Review of Develop-
2. Stokes, G.G., "On the Theory of Osciliatory Waves", Trans. Cant. Soc., vol. 18'9, , pp. 441-455 3. Papanikolaou, A., "On the Solution of the Nonlinear Problem of Waves of Finite Amplitude on Water of Limiited Depth by the vl. p. 6384.outside te: 2, ,196, Method of Perturbation", in German, SchiffstIntl. 4. Ogilvie, T.F., "First- and SecondOrder Forces on a Cylinder Submerged under a Free Surface", Journ. of Fluid Mech., vol. 16, 1963, pp. 451-472. 5. Parissis, G., "Second-Order Potentials and Forces for Oscillating Cylinders on a Free Surface", MIT-Rept. No. 66--0, Engineering, 1966. Dent. of Ocean"cien 6. Lee, C.M., "The Second-Order Theory of Heaving Cylinders in a Free Surface", Journ. of Ship Res., vol. 12, 1968, 313--327..
-
-
.
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"Calculations of the
Hydrodynamic Forces Acting upon Thin Cylinders Oscillating Vertically with Large Amplitude", in-Japanese, Journ. Soc. Naval Arch. of Japan, vol. 141, June 1977, pp. 67-76.
ments over the Past Two Decades and an Outline of Problems Now in Hand", SNANE T & R Symposium S-3 on Seakeeping 1953 - 1973, held at Webb Institute of aval Architec-ture, Glen Cove, N.Y., June 1974.
16. Paulling, J.R., Wood, P.D., "Numerical Simulation of Large-Amplitude Ship Mo0tions in Astern Seas", SNAME T & R Symposium S-3 on Seakeepina 1953-173, held at Webb In-t. of Naval Arch., June 1974. F . eo 17. Faltinsen, 0., "Numerical Solutions of Transient Nonlinear Free-Surface Motion or inside otieo nieMvnMoving Bodies", oisPo.ScProc. Sec.Conf. on Num. SSPHydrodyn., Univ. of Calif., Berkeley, 1977. pp-. 347-357. 1 N 18. Nichols, B.D.. Hir, C.W., "Noniinear Hydxodynamic Sores cn Floa~ing Bodies", Proc. Sec. Intl. Canf. on Nun. Ship Hydrodyn. Univ. of Calif.. Berkeley, f57, pp. 382394.
7. Potash, R.L., "Second--Drder Theory of Oscillating Cylinders", Jou.:n. of Shin Res., vol. 15, 1971, pp. 295- 3:24.
CoeffiaveHyadrodlynamic 19. Vugts, and Rolling Cy .. rot S,:av J., ng, Heaving linders in a Free Surface", TH Delft Rept. No. 194, 1968. 2n0. 20. Tasai, F., Koterayama, W., "Nonlinear Hydrodvnamic Forces Acting on Cylinders Heaving on the Surface of a Fluid", Rept. No. 77, Res. Inst. of Appl. Mech., Kyushu Univ., 1976.
8. Pananikolaou, A., "On the onlinear Problem of a Vertically Oscillating Cylin-der of Arbitrary Shape", in German, Diss., Techn. Univ. of Berlin, D 83, 1)77.
21. Lee, C., "Second-Order Theory for onsinusoidal Oscillations of a Cylinder in a Free Surface", Proc. of the 8th ONR Syp., Pasadena, Calif., 1970, pp. 905-950.
9- Papanikolaou, A., "Potential Theory of Second Order for Cylinders Oscillating Vertically", in German, Schiffstechnik, vol. 25, 1978, pp. 53-80.
22. J.N., "A Linearized Theory for the Motion of a Thin Ship in Regular Waves", Journ. of Ship Res., vol. 5, June 1961, pp. 34-55.
Masumoto, A., "On the Nonlinear Hydrodynamic Forces for Oscillating Bodies in Regular Waves", in Japanese, Journ. of Kansai Soc. of Naval Arch., vl.l12,l9 7 9 . pp17-31
23. Papanikolaou, A., "Calculation of Nonlinear Hydrodynamic Effects in Ship Mo-ticns by Means of Integral Equation Methods (Close-Fit)", in German, Rept. No. 79/1, Inst. f. Schiffstechnik, Techn. Univ. of Berlin, March 1979.
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i
C
26. John, F., "On the Motion of Floating Bodies", parts I and II, Comm. Pure and App!. Math., vol. 2, 1949, pp. 13-57, and vol. 3, 1950, pp. 45-101. 27. Sayer, P., Ursell, F., "IntegralEquation Methods for Calculating the Virtual Mass in Water of Finite Depth", Proc. Sec. Int. Conf. on Num. Ship Hydrod-yn., Univ. of Calif., Berkeley, 1977, pp. 176-184.
28. Ogilvie, T.F., Shin, Y.S., "Integral Zquation Solutions for Time-Dependent Free Surface Problems", Journ. Soc. N3v. Arch. Japan, 1978, pp. 43-53.
-
_
29. Paulling, T.R., "Stability and Ship Motion in a Seaway", Summary Rept. for U.S. Coast Guard, June 1970. 30. Ohmatsu, S., "On the irregular Frequencies in the Theory of Oscillating Bodies in a Free Surface", Papers Ship Res. Inst., No. 48, Tokyo, 1975. 31. Faltinsen, 0., "A Study of the TwoDimensional Added-Mass and Damping Coefficients by the Frank Close-Fit Method", Norske Veritas Rept. No. 69-10-S, 1969. 32. Bai, K.,Yeung, R., "Numerical Solutions to Free-Surface Flow Problems", Proz. 10th ONR Symp., Cambridge Mass., 1974, pp. 609-641. 33. Smith, D.A., "Finite Element Analysis of the Forced Oscillation of Ship Hull Forms", Nav. Postgr. Sch, Monterey, Calif., 1974. 34. Maruo, H., "The Drift of a Body Floating on Waves", Journ. of Ship Res., vol. 4, 1960, pp. 1-10. 35. Pananikolaou, A., "Computer Program NONLINEAR, Version June 1983", Techn. Univ. of Berlin, June 1980. 36. Kim, C.H., Dalzell, J.F., "Analytical Investigation of the Quadratic Frequency Response for Lateral Drifting Force and Moment", Rept. No. SIT-DL-79-9-2061, Stevens Inst. of Techn., Davidson Lab., Day 1979.
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As a previous pursuer of the same kind of
Discussion
problem, I can fully appreciate the pains-
F.Tasai and W. Koterayama (Kyushu UnivI The Writers would like to congratulate the Authors on this very interesting and valuable paper and would be most grateful if they be kind enough to comment on a few points. 1. Which boundary condition has the larger effect on the second-order force, the free-surface condition or the body-surface one ? forced heaving 2. The-~results aend; focetestsIf ~ ~~hof the fistfre indicate that the first-order forces and the amplitudes of progressive waves A 3 depend on the amplitude parameter c-,. with For example, A 3 tends to decrease an increase in £3. Can these phenomena be explained with this theory 3. In Fig.17 and 29, the horizontal drifting forc:, fio are greater than unity at a certain frequency. By Maruo's theory, the nondimensionalized drifting forces don't become larger than unity. Did you compare your calculated results with those by Maruo's theory ?
H. Maruo (Yokohama N Univ.) SIn this paper, the auth-3rs have develIn tis te ape, auhrshav deeloped a general method of computing secondorder hydrodynamic forces on a cylindrical body oscillating with three degrees of freedom on waves of large amplitude. The computation program seems to be very useful for the assessment of the lateral oscillation of a ship in beam seas where the motion amplitude is liable to be large so that the non-linearity in the motion is no longer negligible. One problem which I w'sh to point out is the fact that the theory upon which the computation method is based is a pezturbation analysis in any case. The boundary-value problem is formulated on the body surface at its averaqe position. It is suspected that the perturbation theorynor's may present some difficulty in order to take
account of the change of shape of the wetted portion with time which is liable to become large in the case of rolling. A question is, to what extent, is the perturbation analysis applicable at large motion amplitude ?te
taking patience the authors must have required during the course of this work, particularly, in keeping the orders of magnitude consistent. I congratulate the authors for this excellent milestone work. I have to confess that I did not have the patience to check every equation in this paper; however, with regard to the passage concerning the irregular frequencips, I have the following suggestion. This suggestion may only apply if the authors' statement* at the end of the Section 3.2 has been the main reason that they have not used the technique for removing the irregular frequencies described in [28] through (30]. we let fn e the solution for the h etesouinfo boundary-value problem given by Equation (51) except for the body-boundary condition, then let h(n) (n) (n) fy such that F(n
ih "n' B{h~n)} - R(n) Bn
f(n
(X,)E-SI , )E S 0
and the rest of the boundary conditions remain the same as in Equation (51). Since the new boundary-value problem for h(n) is identical with the first-order problem, one of the techniques in (281 [30] should resolve the problem of indefinite solution at the irregular frequencies. I would think such an approach WOLld save w the computer time more significantly than the interpolation method used in this paper. As to the solution for f(n), one can find the expression in Equations (21,22) of "Surface Waves" by Wehausen and Laitone in which the expression should be for _f(n but behave as eJfl I, the solution is given in my earlier work (21].
Auth
Reply
A.PapanikolaouardH. Nowacki ech Up-v Sefin We would like to thank all discussers for their valuable comments. Professors Tasai and Koterayama have
C.M. Lee (D'SRDC_
* Quote: But it must be mentioned that purely analytical methods for this purpose have not yet been extended to second-order situations with inhomogeneous free surface boundary conditions.
The authors have completed the secondorder solution for freely floating twodimensional bodies subject to beam waves, -332-
of our second-order wave also includes second-order contributions, which must be taken into account in any direct comp arisons with results derived from Maruo's theory. It may be noted that some experimental results also indicate the possibility of horizontal drifting forces exceeding unity. Professor Maruo's remarks remind us of the basic limitations of the perturbation method. It lies in the nature of this method that the boundary conditions on any free or moving boundaries are met approximately by Taylor series expansions about their positions at rest. This is why we have stated in the paper that a secondorder theory is apt to be valid for mouerate motion amplitudes and moderate wave heights. When these limits are exceeded depends on section shape and frequency parameter. The present second-order extends these limits relative to linear theory by allowing for second-order hydrodynamic effects and, in particular, for the existing flare angle in the waterline.
raised three interesting questions. In response to the first one, it is difficult to give a general answer because the relative importance of the body and free surface boundary conditions dependc on section shape, frequency parameter, and upon which second-order subproLlem is discussed. In many situations the body boundary condition and the free-surface condition must be regarded as being of equal importance. Second, our perturbation expansion does not show any effect of the small parameter E3 on the linear heave damping coefficient A 3 , which is consistent with the assumptions of linear theory. We appreciate, however, that a different situation may exist in analysing test results, where the subdivision of measured forces into contributions from different orders and frequencies does depend on the assumed orders in the evaluation model. For example, if a second-oider evaluation model is assumed, but is not adequate to represent the measured effects, this may show up in a "false" dependence of A 3 on E3. However, such dependence may also have other reaso,.i not accounted for in secondorder potential theory, notably viscous effects. Third, our drifting forces have been calculated from near-field potentials by integration of second-order pressures over the actually wetted body contour, whereas Maruo's theory [34] is based on far-field linear incident wave considerations. Our Fig.29 pertains to the horizontal drifting force on an unconstrained body, Fig.17 holds for a fixed body. In the former case the nondimensional drift force values for the rectangle far exceed unity in a certain frequency range, which is due to a resonant motion situation. As stated in the paper, the results near resonance must be viewed with caution because of the considerable magnitude of the small motion parameters. It also remains to be examined in which way Maruo's theory must be applied to this situation. For tho fixed bodies investigated, only the rectangle has a nondimensional drifting force slightly greater than one. It must be noted that the reference
We are grateful for Dr.Lee's comments based on his long experience with the present problem. His suggestion to transform the second-order subproblems to a homogeneous form equivalent to that oF the firstorder problems, in order to be able to rely on the well-known analytical methods for alleviating the irregularity problem is well conceived and should certainly be feasLble. However, this would be at the expense of having to solve two boundary value problems instead of one for every secondorder subproblem under discussion. In our experience, purely numerical schemes for the same purpose have worked very reliably for the first as well as for the second order. It may be added that we have also investigated analytical and semi-analytical methods for dealing with the irregularity broblem[8], [231, which have worked quite adequately, too, although they only shift the irregularity to some other frequency, which may still lie in the range of interest, particularly for the second-order problem. Finally it is important to note that the severity of the irregularity phenomenon is much reduced by using the Helmholtz integral equation formulation with the potential as unknown function.
quantity used for plotting the drifting forces is 0.5fg-aw, corresponding to the kinetic energy of a linear wave of ampli-tude aw l In reality- the kinetic energy
U.:
_
333.
Characteristics of Nonlinear Waves in the Near-Field of Ships and Their Effects on Resistance Hideaki Miyata The Univers't of T yo 0;-o Jc-an
ABSTRACT
The characteristics of nonlinear waves around ships are experimentally studied, These waves generate lines of discontinuity, and the abrupt velocity change across the wave front satisfies the shock relation. A kind of shock wave occurs around ships in translational motion in deep water in the same manner as it does in compressible fluid and shallow water flow. The characteristics can be analogically explained by the nonlinear shallot' water theory, introducing equivalent shallow water depth. The existence of this nonlinear waves produces a deal of influence on the flow around ships and on ship resistance. The role of the nonlinear waves in the inviscid resistance component of ships and their effect on linear dispersive waves are discussed. I. INTRODUCTION Linear wave resistance theories based on the works of Kelvin, Michell and Havelock have contributed to the improvement of hull form design, and as a consequence, to energy saving for more than twenty years. However, there exist significant discrepancies between experimental facts and explanations by these theories or improved versions 12). In case the discrepancies remain quantitative and one can obtain sound qualitative information from the linear theories the theories can continue to contribute to industrial purposes. In many cases, however, linear wave resistance theories turn out to be powerless neither quantitatively nor qualitatively depending on the hull particulars. it is a popular opinion that the defect of wave resistance theories developed so far is originated from the linearization of the -335-
problem and nonlinearity should be taken into account. The nonlinearity can be classIidit w id.Oei s osek nonlinear effect on linear waves, in which governing equation is Laplace equation. The other is nonlinearity in characteristics of waves, which is governed by a nonlinear partial differential equation. L. ear and nonlinear waves both have opportunity to occur. The former nonlinearity is treated in the framework of the linear wave theory, while the problem concerned with the latter nonlinearity is to treat nonlinear waves themselves. Most of the improved wave resistance theories recently developed take the former nonlinearity into account to some extent. For example, one of the most general method is one by Gadd[8] who has proposed a Rankine source iethod in which Laplace equation is solved under the nonlinear free surface condition. As is commonly recognized in hich-speed aerodynamics and shallow water flow, the nonlinearity often plays a decisive role inthe zharacteristics of waves and in fact nonlinear waves called shock waves occur. The appearance of nonlinear waves around ships in deep water was first pointed out by Baba[ll], which have been called wave breaking. However, breaking of waves will not be the essence of nonlinear waves in the near-field of ships. The nonlinearity will consists in much more comprehensive characteristics of waves. Occurrence of shock waves will not be a marvel for ships in translational -otion. The author has experimentally clarified the nonlinearity of waves in the near-field and found out that a kind of shock wave occurs, which is named as free surface shock wave in 1977. Consequently,the basical nonlinearity must be e~:amineI in detail for the development of resistance theories for
chapter 8. Characteristics of nonlinear waves are re-examined in wide range of velocity in chapter 9.
ships. In this paper, clarification of the characteristics of nonlinear waves and their effects on linear disiersive waves, wave resistance and hull form improvement are discussed. In Fig. 1, a wave pattern picture of a full-scale passenger boat is shown, in which wave systems which are entirely different from Kelvin pattern are obvious. Following the discussion on the limitation of linear theories in chapter 3, the nonlinearity of the waves is clarified with a deal cf experimental results with a simple :mdel ship in chapter 4. The nonlinear waves have characteristics that allow us to call them "free surface shock waves". Therefore, as one imagines, the zharacteristics are very similar to nonlinear shallcw water waves. in chapter 5, the analogy to nonlinear shallow water waves and application of the nonlinear shallow wave theory are discussed through experiments with wedge models. In chapter 6 and 7, effects of the nonlinear wave (free surface shock wave) on resistance components and on linear dispersive waves are discussed. The nonlinear waves directly render resistance to ships as momentum loss and, at the same time, they indirectly influence resistance by deformation of the dispersive waves. The nonlinear waves at the stern and a new practical method to reduce waves at the stern are described in
2. NOMENCLATURE Al(a) B c Cw Cwp d Fn Fh F: g h
4
H Pb L p a P R-
weighted amplitude function beam length of ship critical speed wave resistance coefficient derived from towing test do derived from wave analysis draft cf shin Froude number based on L do based on hI do based on n acceleration of gravity shallow water depth equ±valent shallow water depth total head behind ship do ahead of ship length between nernendiculars nr~ssure velocity in x-y plane loss resistance measured as nomentu viscous resistance = Rf(+k) wave resistance derived from towina test do derived from wave analysis
Rw" w u w disturbance velocity in x,y,zdirecticn, respectively; u incues un-;or velocity in charter 5 U, V
velocity of uniform stream
x
axis narallel to ship centerline, aftward positive axis parallel to ship beam axis vertical, upward positive entrance angle shock angle
y z Ot 8
wave heicht
__
asubscrints
....
1 2 n ±
ahead of shock front
behind shock front normal to shock front tangential to shock front
3. LIITATION OF LINEAR THEORIES 3.1 Wave Pattern Pictures of Model Ships Fig.l
Wave pattern of a full-scale passenger boat (from Ships of the World)
Table 1
The legitimacy of theories must be guaranteed by real physical phenomena. The
Principal oarticulars of model ships
remarks Cb Did c LIB B Lu 2.400 1.240 var. 10.00 var. 1 .67 parabolic water line, wall-sided, sharp-bow b , blunt D9 .68 2400 1240 var. 10.00 var. , shamp-bowp .67 12 5.00 var.18O var. i72.400 , sharp-bow D9 .67 5-00 var. 1.200 .240 var. W'14-A '134 2 -01 .30 -.111 6 .50 2.77 I .39iie shin M40 2.000 .308 105 6.50 2.94 .54 si-olified -rdificationof Sf138 series ship 0 347alenth 200 .364 .105 D_ .54 4.50 4.25 .05 444 2.000 M42 M43 364 105 ° 50 3.47 I .54 simnlified modification of 41 6.06 4.52 1 .75 bulk carrier on ballast cond. BULK,2 --2.000 .330 .073 6.49- 2.85 j .57 container carrier, full load cond. SR138(F) 2.500 .385 .135 , trial condition D_ 4.38 I 6.49 1-08 i.088 .385 1 2.500 _SR138(T) 2-0! 11.142 1M6 2000 2 .i3 T ' 14 13.7l08 Inuid m) WMl-C WMI-B
117
2 000
.
.255
-312 I
.177 = 7.841 'M04 6.471
1.44
_
1.53 1
jInuic Inuid
-336-
ZU
w
wave height is steep rather than wave height
______
-
itself. Three examples are shown in Figs.2,3 and 4. The principal particulars of model ships are shown in Table 1. One can clearly notice that the wave systems in the neighborhood of the models are not similar to Kelvin pattern, or rather, they are similar to shock waves in compressible fluid. The lines of wave crest originated from the hull surface are usually convex and their angles
to the center line of the ship varies depending on the hull particulars, advance
__
velocity and so forth.
___
Fig.2
Wlave pattern picture of a simple imodel WD42 (d= 6cm, Fn =0.22)
-
Wave pattern pictures at the University of Tokyo are present in references[l], [2], (31 and (5], which lead us to the same intuitive understanding of real wave making as mentioned above. 3.2 Discrenancies between Theory and Experiment Discrepancies are outstanding between what wave resistance theories so far developed indicate and what experimental results show in the form of resistance force, wave disturbance velocity and so on. The discrepancies are both qualitative and quantitative, which have seldom been experienced in other fields of hydrodynamics. So.me comparisons between calzulated wave resistance, measured wave resistance by towing test and by wave analysis are shown in Figs.5, 6 and 7. Wave resistance is calculated by 'ichell's approximation in Fig.5 and by the low speed theory in Fig.7. General tendency of discrepancies are as f llows.
=height,
Fig.3
Wave pattern oicture of M34 (Fn= 0.333)
I
02-
2k
;p5.
_________________20
X~~z!5~
Fic- 4Wave plattern picture of M41 (F-n0.267)
t.tfA4
development of the bulbous bow by Inui would not have been successful without the aid of experimental analyses. At the experimental tank of The University of Tokyo, measurement of wave height for two-diensional wave contours and photographing wave pattern have
been routine techniques of experiment for long years. The former presents uantitative infor.ation of wave height and the latter qualitative one. Wave pattern pietures by the latter technique have property to intensify the lines on which gradient of
.
0..
r
.- _'
N-
-
Ot
-
•
t_2z 00
Fig.6 -337-
_____-
Wave resistance curves of hMI-B
.--
Xcs,---w,--Towing
K
-test
draft:OLSA7m -
I
dralt:0fl647l
S-ee w
Wave naySiS
Ant -
Theory
i
.
.....
_
I_,__
0.183
t,00--n -- 088 -
!/1dr 4,.
30.1.7r.,tz-
-
"
I1
.7
,7:
draft
a'
--
Art Fig.7
- c(u )), respectively, respectively,
dy _-7 X--
4,71
du=
1 +"
6. DPSITPMC' COMPOMP.ENT nUE -0 FREE SURFACE SOCK WAVES 6.1 Momentum Loss Measured far behind Ships 'leasurement of .omentum loss far behind ships has been carried out for various ship .odels in 1978 and 1979 [2]1(5]. Two examples of contours of momentum )oss are present in rig.23 for WM2 at the draft of 200mm and 60mm. The area where momentJm loss appears stretches widely in the vicinity of the free surface. This kind of momentum loss will be attributed to the velocity drop by FSSW through the relation described in equation (2). FSSW produce lines of discontinuity and, consequently momentum loss behind them. To the first order approx-
c_____ (14)
u2/c-
/CU - V2/c
-1 (15)
=
on hodograph plane, shock polar can be drawn according to (15), and the obtained line of characteristic, settle that on physical plane because ies represented by (14) and (15) are normal each other. This method of characteristics is applied to an usual hull fo'm whose hull for- , is assumed to be represented by the wa erline. And besides, the water line is mdified into succession of straight short lines for simplicity. An example for a fine hutl form M34 is shown in Fig.22 at varions velocity of advance, Although the calculated results do not gantitatively agree with the experimental curves, the qua!itative agreement is good. These results im.ply that this method can be useful for hull form design, because qualitatively exact estimption of resistance can inprove shock an:le and strength of discontinuity is rather 5imple as is shown in Fig.19. The reduction of shock angle will naturally lead to reduction of resistance due to FSSW. The explanation described in this chapter is based upon irrotationality, which seems to be contradictious with the physical fact of the existence of discontinuity, because the flow with discontinuity is rotational. However, the rotational flow is simply a consequence of the occurrence isof tiona23.Cooweverotheorotationalsflow
-
imation the resistance due to FSSW is co'nsiered to be measured in this form instead of wave pattern resistance. ? ' .. 0 , -- 'N\ ,/
.,d/,
W'-2 d-O.Cds .--.2 %
f 50
_0____0
______
i/7
GOO0
K
-
L
100
ISO H,-H mmAq
200 250
(
t
1 20 apart fromAP M m Fn-022 d=0.20 di
300 (Mm) zig.23 Contours of momentum loss of WM2
344 -
Rf.408
Rw byonT&
1440 fn.022 I'02673
Apby NSMtlhcd MOMENTUM LOSS byWok. Sudwynoo.
..
*M- 4.0a8d2
. to
M41 Fn-O 22
31
f c0.2673
__
_
_
_
_
_
_
_
_t,
1
z02887 1442~~~lt
___________-
0i-?
___________
-Z
1
M43VU 0-00647_________
M-32
dz0'
A30___________
2 03 0
__3E_
_
6.2 uPrtclran
_
_
_
wtl Pathecuafr
pTte
3
Fe.salecmoet 1 _
_
reSufc
_
hk
C..0
fwaerssac
A
2
_
Mesrmn of0momntu loss is carried (g out aize Reintnc compon Tetsn ~re ety0 a6ree
75Cmoet
2
[..6
------_
8d'
8-800
30
;2
.,
otu Free and Surfac os
Cwpxi0
i
f resistanc6sae cu beeppoxmatly_ resistand evaluated byEthee Rw oav Ileasurresstanc moeryu
I:
asnd.2 B inercton due to~* resstnc twogato yhes Thisurs seaaintfs tis conenen for theidou onmode 11 -sand the recsgitio whsc copoetufmaeresigniican amon f-rvisclyd tohen is marizanc Surpose wth the su fmomentum lossutand by rsstneave xmpe r soni the reistace iscos i theresitanc ue Fig.26 fore s igie walsieB odl adi Pacerand weiaternthetnei h ist -R.(odmninaie sC rolen tefrspayasigncntecin to SS. due resistance +Ru arxmto of the The~~~~~~~~~~~n amoun pi ofw-~o cmol aerssane ofshps. Thse engrti-beam aret n Fg.4,hogh nehastotak ito ratioisacgreatery thani5.e ffecttoe upaternsctdb los wav Supose.that the vameu :esit~nen masued mdel whse engh ise 2isor s reistuulanc i tteslethan tt onI larer dfrnebt nwrels.t aneawave ntr resistaneotin fro tw i comonets.The ar liearwav reistnce (Rw~aproiato and th~ sprtino due to FSSWp.r ictly~ ~ resistance ~~~~~~~h spaigti rssaneTes ie nouxat because of% comp interctimons; freampe longi.4,thui nwehaeou faeo SWaeu soetime includhe stlept wave heigt b conee ol bethe linear -oevernc meisue wav 2ressac which hsusave a tled be h)oen
e urve is vsaizedbyth areifan eina treidtaed curvXDes atre soel sin whoseanc g 5'riulr pa wae ~i th mode0 ind in 7ig. 26 Legt-ba p ratsip od and eam-draf rtorsonsmgnstdtrie h Rw - RwD tends o ensall e a certawin Fouenmradhsolyiglhmp figurs c iiian bete t oudlies hull frmdsincers. Ships whose Cgt-Cb ratio is reater than onutb 6the. cccur eindwt ect anf mntion speaking Gurenerviale stentofarSes prcialtihulafrs whose w/is less tha
texperiencbued tffect ose in t fipeld nds Theefore liea wav a-x resstnc S~ det of resistance seato sum teig icatly
t~a65raeceove remarkablerssta n bys ases Hu R5 fomds'ihu entb mll Osierationtaon reibefoahsnsis wil n obe FSS -345.
is
forex
nt e
mp
at_
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e
log
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sg
W-
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W
i
.-
-m
Ch.= C..jon~
~
CAK~jirgtest)I GNcWaean~ss)I-0
~
Cw byfree surface shock wave (no snockar--edegree)
0.5:
3
--
x13'2
0302
015 -
Fi.
_E
020
05F
lg27Wave resistance curves of bulk carries on ballast condition
av
esisnce2
6.3Resstace
umpbVFree
livsfac ltir _
Surface Shock
I'ave inFoud lw nuberrecion. An example of two Tdels of bulk carrier on ballast condi tion is shown in Fig.27. A slight modification of the hull form at the bow give rise to great difference in the resistance humps around Fn = 0.15, while the measured wav.e pattern resistance do not have any difrere-ce; or rather, Cwv itself is negligible__ in the velocity rarnge ot Fn 0.20. P'case of a large container model on trial condi-_ tion is shown in Fig.28. Cwp is very small around Fn = 0.15 whereas Cw. has a hump there. These humns at low Froude number are commironly recognized for shallow-drafted ships, especially those with bulbous and they mr.ake the definition of form bee resoabl eplane. 'hewave at hebowofth cotanermoelin
Iu __
bow, factor
-__
_
_ _
Fig.29 W~ave Pattern pictures of SR138 on
trial condition
pattern~ Fig.28 of obiosl bsrvd.Te hokangle is mato small un wit Fe the5inr a of vltyn ~t esmal of shoc t e nlereazs greatness of RV Geatnss f shc.,* anle eansgretnes of reisanedue to FSSWI as has already been r4escitne which can qualitatively ex-n1pir the hump at Fr- =0.15. The theory for shallow water shock waves .,hicl-has been utilzedin5by captr itroucin eqivalent shallow water depth can roughly es-346-
culated resistance due to the second free surface shock wave is shown by dotted curves fo twobulk carriers in Fig.27. The qualiaieagreement of the theory with the
experimental results is, again, excellent. One
can conclude that the hump that is often observed in low Froude number is due to FSSW. With the decrease of draft the critical sDeed V?.r becomes small, which is shown in Fi.1as the dependence of h, on d or in Fig.20 as relations between draft and equivalent shallo-w water depth, and, in con-
the hump in Cw curve where shock
sequence,
..
Fn.0.2236
.. 02673
angle is great is shifted to the low speed
range where Cwp is negligibly small, because z. hump by FSSW is usuall-y present arcund the critical speed. For shallow drafted lowand middle-speed ships, consideration on FSSW is decisively important. 7.
. ,0S
j
,
-
V:-
.I
EFFECT OF FREE SURFACE SHOCK WAVES ON LINEAR DISPERSIVE WAVES
7.1 Phase Shift
.
FSSIs coexist with linear dispersive waves around ships in translational motion. The problem of wave resistance comes to be complex, and it seems to be impossible to develone a comprehensive theory that explains the two wave systems simultaneously. There
the near-field of ships.
D source)
.
r. ' : \ . "-I .. ,,_- , " "\.-Ni
In case both ways
05-
With the increase of beam-length
-
M
W91 \-C\ ofaV
-1n.
-0
Fig.32 l-ave pattern resistance curves of -Ml-C Wave anahysis approx .D (odfe ouc): modified source)
L-L 7 r.. 0DQ
I 5
-----Michell
Fn-.24-
I
4
~
iIprox M h---
-
the amount of forward 3hase shift becomes laree. FSqs, esneciarly those at the bow, become intense with th. enlargement of beamlength; the shock angle increases and discontinuity in wave heicht and disturbance velocity is strenqthenea. On sectional area series ship odels whose particulars are the same with M40, the intensity of FSSWs at the bow and forward ohese shift have intimate connection, though the result is abbreviated. FSSI's accomnany lateral displacement velocity as is shown inon Figs.9, 11, 12 and It has wave13.system influence the dispersive to shift forward and sidemwards.
at
4
waveanalys/
10-.
are successfully achieved, wave resistance can be estimated by summing the two results. The phase of linear dispersive wave system is commonly shifted forward. An examole on beam-length series ships is present in ig,30.
442
Cw,-
The second is to estimate the resistance due to FSSWs themselves. '.hat is described in chapter 5 belongs to the latter way, and problems concerned wit!i the former way are described in this chapter.
'i -"., wI
15-
are two possible ways to be urgently pursued.
caused by FSSI4s in
X-
Fi... Longitudinal wave profiles of (Y/L 0.2)
The first is to evaluate the wave resistance die to linear disr-.sive wave system which has been influenced by the nonlinear flow'..
~an
.03015
AP
-W~o
fav~e4
4
k !
Fig.30 Comparison of longitudinal wave profile of beam-length series ships
7.2 Attenuation Effect Longitudinal wave profiles of the widebeamed fine ship M42 are shown in Fig.31 at various velocity of advance. One will notice that the rear parts of the wave records are extremely low, which implies attenuation of transverse waves. attenuation cannot of cancellation effect as an This understood be
-4vi..4 0 0 10 20 30 40 50 60 70 0(deg) Fig.33 Comparison of wave soectra of W 4M-C at the bow and the stern, because it does not depend on velocity of advance. This phenomena is also considered as an effect of the
between the two linear wave systems generated -347
.......
-
__
rudder without bulb rudder with bulb
existence of FSSIJ by the two reasons. They are that the attenuation becomes conspicuous in case FSSW is intense and that FSSW accompanies a kind of wide wake region as a result of the presence of discontinuity in velocity. One of the great discrepancies between linear wave resistance theory and experiment consists in the humps and hollows in resistance curve. Linear theories show rerarkable humps and hollows as an effect of cancella-
.
0 -10j
10, X 500 0
they can be scarcely observed in measured resistance curves in low and middle ranae of advance velocity except for very fine ship models. A typical example is shown in Fig.32. This qualitative discrepancy should have been very sianificant for the assessment of the validity of theories. A simple calculation is carried out for W41 to demonstrate that PFSW attenuate linear waves with the w.ake region behind the line of discontinuity. -he wake recion is represented by a rigid body whose depth is
_________-___
i0 Xz00 0 2013OX 30
0-0
20mm which stretches laterally alone th0
.
-....
.
20
line of discontinuity to a certain point and continues to infinity extending oarallel to the centerline of the ship. Calculation is made by Michell's approximation for The results are shown in
R-N ..
X 600
tion and amplification of waves, whereas
simplicity.
-R-A
.
..
(mm)
-
.
ni.32
......
200
100
for wave resistance cuve and in Fig.33 for wave spectrum. Wave resistance hv incar waves should accord with measured wave pattern resistance and this is approximately accomplished by this calculation. Fio.33 shows that the gap between theory and experiment is remarkable in the transverse wave system and that it is considerably remedied by taking the effect of FS'.' into
.
300
tm) Fig.34 Transverse wave orofile at the stern of S9138 on full condition
consideration.
8.
FREE SURFACE SHOCK
WAVES ?T THE STER%
8.1 Occurrence of Free Surface Shock 'aves at the Stern It is naturally understood that FSS-appear at any points in
the near-field in
_____
*-
..
various fashion. It is not limited in the bow-near-field. Especially F5S's are clearly observed at the stern on high-speed fine
I
X -i.
kind of waves have the same characteristics
___..
Tansas those of FSSWs at the bow [3]. verse wave profiles are shown in Fig.34 for line of disa container shin model. continuity can be recognized.
..
.i
8.2 Develonment of Stern-End-Bulb
Sand
/
X __
ships. The passenger boat in Fic.l generates intense FSSI s at the stern being accompanied with breaking of waves. ?bruot change in velocity and wave height have been measured and it has been verified that this
A
FSSWs at the stern look dominant in the wave pattern pictures of full-scale high-speed passenger boats, destroyers and container carriers. To reduce FSSWs at the stern and, simultaneously, linear waves from the stern, a new type of bulb is developed by the author colleagues through extensive ewaterline
Fig.35 Configuration of stern-end-bulb for SR138
Z
f/
investigations [6], [7]. This bulb -is named stern-end-bulb (abbreviated as SEB). This bulb is rather small in comparison with the bow-bulbs and equipped at the rear end of the of ships. Stern-end-bulb is care-
-348-
=
A
X10
wH withtiit Sx
1u0O
CwW /1/2Pv 2 s Cw 09
I
test
SR138 1
-aralys
i
o7
--. -°Z
SRI3x2t"'
A 'Q375i
""-
V 101901
0.4
"(
03
4
OIL
-~
t
ff/
". ""!_
07
__
-
.0175
--o--
SRX_130
-7
/
K
0
RWWM
--
0015 0005F 3 Fic. 37 Resistance reduction by stern-end-bulb Fig.36 Comparison of stern wave pictures of SR138 and S 3BX (i-n model) f lly desi-ned so tha
it reduces wave
resistnce and ini"mizes increent or viscous reistarce. One o. te conf'igrations of SEE zor a container carrier is present in Fig. 33. -he origina for-- of S wBs a modified rudder called R-- whi -wa a rudder with a bulb siilar to SEB-v in Fig.35. SEr-S was connected to the hung ng ruder R-A instead Xditfied rudder is called o- the and -ull the P-N.
SFi.35
h
ffect of R--' o-
is
-aves
showr in .eight is
of bulb on wave resistance can-. be_ recognized o nI in cw -(e, resistance due to FSS , in the -ow speed range. -hefed-tion of wave resistance by the stern-en--bulb (SEE) reaches uP to 21 percent 'hin leds to saving ol propelling power of 6 percent for containar carriers. The effectiveness of SB ha-e been verifled on several high-speship rodels [7), and it is coins to !e u.t to practical use at Japanese shinyards.
-.---. 34. The uiscontinuitv 4n wave weakened by .R-N, which means that it reduces FSSI is a- te stern. "'-e ef-ect of S_-_x in is visualized in Fic.36. Wave resir'ance curves are shown in Fi.37,:hich are results of volume series experinents for the -nvestiaation on the ontimun size of SEB. Th decrease of Cc qualitativeiv accords with that of Cwn- Hoever, the rate of decrease of C-wcannot be- f9uly explained by that: of Cwn. SEE reduced Cwn and C. - Cwn si.multarneouslv, in other w.ords, SF3 is very effectire for the reduction of both linear wave
Examination of wave making properties in -ide range of advance velocity will be useful for comprehensive understanding. *Wave pattern pictures of two models are shown in 'Fig.38 and 39. The entrance angle or 19M4-A is about 20 degree and the fore-end of the rudder is round, and therefore. FSSlis around the models are evidently observed.
resistance and resistnce
The free surface
ue to PSSI at the
stern. -his interpretation is also true for bulbous bow. Bulbs generally contribute to reuction of two components of wave resi a ce Fig.27 shows an effect of bulbous bow of bulk carrier as an extreme case, in which the effect of the variation of the form
9.
NONLINEAR WAVES
VELOC-TY
I'J
WIDE RANGE
ADVANCE
9.1 Wnave Pattern Picture
enoena of a floating bodyf
in translational mtion seems ultimately complex. Linear and nonlinear waves coexist and the nonlinear waves generate unsteady turb.ilenc flow on the free surface. 9.2 Lonitudinal Wave Profile
-349-
-
~----
-
-
--
W
-
t
a
.7|_
Fn=.,
o
7. S
= ?ni 0.22
a-- comnaredi longitud-Ia wave z~rofiles th~eory - ..... -i.0 he discrepancy inf uei depends on ernude an exper al wavhe roile . b coa. att!rsi jO' re ! by ane calcated-r- rs irly well in -o aors n at apro except 4oFr he -irs: wave cres t -0.5o
S
S.46e.
__ __--
:---n
S-
_-----n
lie disaree-ent low .roude number recion.
o .de s -- tn, the discrentenden This antics in wave resistance curves, which is The effect of FSSWs on co-ni recognized. w and ilolinear waves is very sicnlfica.. r l e range of velocity, whereas it is not hi 10in ranue of velocity. The discontinuity and its influence on free surface flow th;e seeM dominant in the pictures, however, aver t"ithe in limited is flow nonlinear whose thickness is independent of advance linvelocity, while the wave length of the r-tion ear wave system is elongated and its scales up with -he increase of velocity. can ost part of wave h In consequence, be explained by a linear theory in the high Froude number region-
C --
FS n is .remrkabe
.
Fig.38 Wave pattern pictures of WM14-A -350-
__n(1980)
'
.~--...
3. Miyata, H., Inui, T., Kajitana, H.: Free Surface Shock waves around Ships and Their Effects on Shin Resistance, J. of the Soc. of ,laval Arch. of Japan, Vol 147,
Q MdWII approx
-. ...
(l84. Takahashi, M., Miyata, H., Kajitani, H., Kanai, M. : Characteristics : of Free Surface Shock Waves around Wedge Models, to be presented to the autumn meet02O ing of the Soc. of Naval Arch. of Japan, (1980) -5. Kawamura, N., Kajitani, u., Miyata, H., Tsuchiya, T. : Experimental InvestigaWtion on the Resistance Component Due to Free Surface Shock Waves on Series Ships, to be - nOpresented to the autumn meeting of the .20 Kansai Soc. of naval Arch., Japan, (1980) 6. Inui, T., Niyata, H. : On the
, 'I
-~-0Optimization
..-
j 1
;8
Fn=050
:
of Overall Performance of Rud-
ders (Second Report), J. of the Soc. of !val rch. of Japan, Vol. 145, (19791 7. '!iyata, H., Tsuchiya, Y., Inui, T., Kajitani, H., : Resistance Reduction by Stern-rnd-nulb, to be presented to the autumn et-ng of the Soc. of "laval Prch. of: Japan
"
80 ,0 -. _0
8. Gadd, G. : Wave Theory znplied to Practical Hul! rorms, Proc. of the interna°-0 tional Seminor on Wave Resistance, Japan, (1976) !8 9. Preiswerk, F. : A.wendung Gasdynamischer -ethoden auf Wasserstr~mung Fh O6t 40 mit Freier Oberflche, Mitt. Inst. Aerodnamik, Ede.Techn. Hochsch , Zujrich, 10 9 8 7 66 5 3 .A 17EP -, 0 10. Courant, R., Shock Friedrichs, O Supersonic Flow and Waves, K. Interscience " : Publications, (194 8) 80Baba, . : A New Component of Viscous Fi-.40 Longitudinal wave profile of wm4-A Resistance of Ships, J. of the Soc. of Naval Arch. of Japan, Vol.125, (1969) 12. Takekuma, K. : Study on the NonLinear Free Surface Problem Around Bow, J. of the Soc. of Naval Arch. of Japan, Vol-132, (1972) A.CK.N -DGE.M.NT 13. Wiehausen, J. V., and Laitone, E. V.: Surface Waves, Handbuch der Physik, Vol.9, The author owes much gratitude to Springer-Verlag, Berlin, (1960i P-ofessors T. Inui and H. Kajitani aweno dispensed generous]y advise and encouracement. This research has been supported by able techni cal staffs and araduare students at The Experimental mank of The University of Tokyo. The author would express a lot of thanks to Mr. Y. Tsuchiya, Mr H. Kanai, Mr. M. Takahashi, Mr.. Kawamura, Mr. 4. Tsuruoka and Mr. P. Suzuki. REFERENCES
1. Inui, T., Kajitani, H., Miyata, H.:
~2.
Experimental Investigations Making in the Near-Field of Kansai Soc. of Naval Arch., (3979) Inui, T., 7.ajitani, Tsuruoka, M., Suzuki, A., linear Properties of Wave of Wide-beam Ships, J. of Arch. of Japan, Vol. 146,
on the Wave Ships, J. of the Japan, Vol.173, it.,Mivata, Ii.,
Ushio, T. : NonMaking Resistance the Soc. of Naval (1979)
-351-
-T
54 -
Figs.A and B are prepared from Fig.13 and 14 of the author's paper. It is observed that the change of u/U and v/U in the direction of water depth is quite large in the vicinity of the free-surface. E Baba !.tw In 1972 Takekuma (Reference 12 of the present paper) already observed this phenomenon. and these observations of the flow this paper ~From I have enjoyed reading the fow Feom tesa an impressed with the vast extent of the inone wonphenomenon near the free-surface, vestigation of the free-surface iDnenomenon ders whether the partial differential equaaround shin models. such as (13) can be exploited as a ahi or introduetion arohnd governing equation for such flow phenomeThe author introduced a concept of equivalent shallow-water depth and tried to no. The author's view on this matter is n~terp.ret the free-surface phenomenon ob-Thauorsveontimters twoa of use by models ship served around very much appreciated. dimensional partial differential equation (13). one equationcomof this the derivation H. Mauo "YokamaN Uriv) of velocity that change shouldIn assume
io
ussin
Di
ponents u and v in the direction of water depth is small 2nough to be negligible.
in this paper, the author has analysed the phenomenon of breaking of the free surface near the ship hullassuming an analogy with the shock wave in compressible fluid or hydraulic jump in shallow water-flow. I cannot help regarding that this treatis due to nothing but the author's misunderstanding of the true phenomena. The shock wave is an attribute of the mothe media, in non-dispersive tion a phenomenon is while wave in deep water surface
s.s.
-I
-
"
,,ment 0.1 u1U.V/Uw/U
fee- |surface
-0.1
i 7
75/12 =240m
4
/u
vU
in a dispersive media. Therefore no analogy exists between the shock wave and the suraround a ship hull, when the water w s o restricted.
CMfr= sur ace
R.Ko-C. Che (JAYCORj
Change of u/U and v/U in the direction of water depth at y = 240 mm
Fig. A
, S.S. -s511. 4
.2
0,I
ace
-
0.-
| ".. W C,
.
I
wU t
-
The various properties of free-surface shock waves (FSSW) as described by Professor Miyata are very interesting, especially in consideration of their impact on future directions of research of wave resistance. The analcgl of FSSW to shallow water waves suggests strongly that conditions satisfied theory must be preby shallow-water-wave sent in the near field of the ship-wave system also. One fundamental feature of shallow-water waves is that the pressure is essentially hydrostatic, distribution i.e.,
n is free-surface elevation and z = 0 defines the undisturbed water surface. Much insight would be gained if one could measure the pressure distribution in the
Lwhere I faq iTT
free-surface layer and determine in which
region and to what extent Eq. (1) is satisfied by the flow.. One suggestion is to
jFig. B
define the parameter Eas
Change of u/U and v/U in the direction of water depth at v
-70m
~-
ghn
-352-
Z)
(2)
-
and plot the distribution of E on crosssectional planes along the length of ship. If the flow field is not of the shallowwater type, r would be quite different from zero. For example, in two-dimensional deepwater waves, k(zd1 =p-cgtn-z)= ga COS(kx-¢t)cosh cosh kd (3) which shows strong variation of with depth, unless kd 0. By plotting Z, Professor Miyata may be able to actually measure the "equivalent shallow-water depth" h: to further validate his theory. The instrumentation for meas-
The concept of equivalent shallow water depth was introduced mainly for the purpose of practical use. This concept can be helpful for the improvement of hull form as is exemplified with the method of characteristics. Notwithstanding the bold assumption, the application of nonlinear shallow water theory is successful to a certain degree. -3) is not a Therefore, eauati general governing eqt .n Lut a simplified equation which can be utilized for practical purposes. The general governing equation will be Navier-Stokes equation, which can be numerically solved with boundary conditions. Further continuous efforts should be devoted.
ring p and - simultaneously, however, may
be quite involved. I would like Professor Miyata to think about this idea and comment on the feasibility of such measurements.
To Prof. Maruo
Y. ftimomuram-romoHeazv-vin) The author describes the effectiveness of a stern-end bulb to reduce the freesurface shock waves (FSSWs) which have entirely different characteristics from those of linear dispersive waves. The theoretical studies about attenuation of linear waves by a bulb have been studied by Prof. Inui and other researchers. However, I doubt that the interference of waves, which is the fundamental physics of reducing linear waves by a bulb, can directly lead us to a sound explanation of the effectiveness of a bulb in reducing the FSSWs. The theoretical background for a bulb's reducing the FSSWs is not clear in this paper.
I think the analogy is evident in the experimental results. On the problem of introducing nonlinear shallow water theory, please see my response to Dr. Baba's discussion. I hope that Professor Maruo will change his opinion by reading our succeeding papers.
To Dr. Chan Thank you for your kind suggestion. The characteristics of free surface shock waves in deep water are analogous to those of shallow water shock waves to a cercain degree. However, there may be differences between the two non-dispersive nonlinear waves, which will be clarified by further detailed studies including Dr. R. Chan's proposal.
I would like tr a-'- the au-
thor's views concerning tL physil-al phenomena behind the fact that a bulb can attenuate not only linear waves but also the FSSWs. And I also would like to receive some of the author's suggestions about the design of a bulb which would reduce the FSsfacts
To Mr. Shimomura Bulbs have two aspects of wave resistance reduction. Bulbs at the bow and stern-end reduce both linear wave resistance and resistance due to FSSWs. These are clarified in two papers in the reference ([51 and [7]). About half of the effect of SEB can be measured by usual method of w~ave analysis, and at the same time, the nonlinear wave system is reduced by SEB as is shown with pictures. Bow bulbs usually reduce shock angle and this leads to reduction of resistance due to FSSWs. Further detailed studies on the design method and its basis are now under
Author's Reply H.Miyata turiv of 1 yo)
The design of SEB is now carried out making use of linear wave making theory ba-sed on the formulation of' Neumann-Relvin problem. However, the effectiveness of the application of this tbeory is limited. Towing tests are necessary. By the SEB, which was first imagined by Professor inui and developed by the author and his colleagues, it is shown that theory is not almighty.
To Dr. Baba Nonlinearity plays a decisive role in the thin layer near the free surface and it does not in the deeper region. The motion of water is quite different between the two region. This will give rise to a steep change of velocity in the depthwise direction, because the water flow must be continuous, -353-
Flow Past Oscillating Bodies at Resonant Frequency Gedeon Davan and Touvia Miloh
:ABSTRACT Ilis
system, g is acceleration of gravity anda water density. FT and - stand as symbThis paper analyses the flow field about an oscillating body which moves with a constant velocity U near and parallel to a free surface. Special attention is paid to frequencies of oscillation in the neighbourhood of the critical frequency uc = 0.25U/g. It is well known that the classical linearized solution for this problem yields infinitely large wave amplitudes at resonant frequency. The singularity in the velocity potential, and hence also in wave amplitude, lift and drag at resonant conditions is found to be the inverse of the square root of ±=wc-w for two-dimensional flow and a somewhat weaker singularity of a logarithmic type in _ for a three-dimensional flow. It is demonstrated in the text how this resonance may be removed by considering a relatively simple non-linear case in which the oscillating flow-field is taken to be a small disturbance about the steady non-uniform flow field caused by the body. Even though the analysis presented is general, two representative cases are discussed in greater details: a concentrated two-dimensional vortex and a three-dimensional lifting surface. For both cases, a uniformly valid solution for the velocity potential, the amplitude and phase of the resulting free-waves in the neighbourhood of the resonant frequency is presented. Also computed are finite D.C. c3mponents of the drag and lift acting on the hydrofoil.
ols of Fourier transform. A
,
B c
-
)
(0
C(o),C(l) D h
-
-
kc =1/4 L Lo,L1
-
MA ,Me
-
P¢, * F tx
pressure terms strength of pressure point time - horizontal coordinate on the unperturbed free-surface in the direction of translatory motion. - vertical coordinate positive upward - horizontal coordinate normal to x - variables in the Fourier space in threee dimensions - common notation zor i$, M 0 , PA
y z a,
f,
All variables are dimensionless with respect to U2/g, U/g and - as length, time and density scales, respectively. U is the translatory velocity of singularities
-
ik
C
NOMENCLATURE
-
coefficients in the integral equation of the FT of the potential. nominator in the decomposition of nonnomogeneous terms in the free surface condition FT of c(O), c(l) drag depth beneath unperturbed freesurface imaginary unit variable in Fourier transform plane in two dimensions critical k value lift free-surface differential opeations strength doubet (steady and respectively) unsteady, of
r,
-
-
AT
circulation (steady and unsteady
-respectively)
Al v a
total circulation for a lifting -
line steady-state perturbation, potential
-355-
Un
velo-citv D-otential (total) s~ngu'a1 an regular -entials
-
of5 --
ree-s-.r*-
-
-rec-u=.r,
(0o
_z-
t
_Furie
in
-.-
i
a...e
~osa
---. =rm
=A--!e
z. tv
Dee
'10D
Geenerat -ea
-frcenvrow
. th -hus, in '4'
or, _
.or
n;
ON
*s
r.
40
The~m tansina--o
par '
-
c to n
i's~
0tead
4 otafvroo: a
a
-
-
-red is
0-'-=c-0
n-,
yt
We slnauiar_4mv system.
ov
cs
-'e -Z--
t-.
a
--
Casr ct0'Z a:n=a:e s1-!U .. r'
t
In
-
o e~__% of
Fourie
-ne
.
t-e res..na~~e
_h 'e-ina
lnat
'
'nI .e.f d e-es
trc..
s -
O-t
_1 and
e o
oude tendse to-r In is reCOvei c
gem'r- :ized ioexturbat"!e:
two-o
c=
zml
- prts
i
work
wel-known resut
Poenta
-n singu
of teresem
soc==
l25
ranstC
sr Ci 0SS' z tyte -i-e'at'o-
1
I e ai -*~r.
me a
sm±_-o
0
ow=--~
-e
cr"---
may rep=sent a submeruea z cve
he n4rble =-tvdthe
untdy. o
lip
-O--
-
~ ana re xe- a aro= f--n---nU oenlr a t ngpese . -h.c=-
near...
andi -
ss
-- ac
Order'
--
~
es
r
~
~
ht
is freoenca atfec-14 e boundien tomoe- way rom tle Ast'la -/ coutin altoah t wsra" hv hees e peents -i' ev in theor form nolie IoI=tr..aU-s a_-. thmsUrr= resonan o tem lt ufar tards ths airt~-aV esidesar te ante alus "or iecal pobldem, thei b resonant efect a bei - s
~as
I
the~~ -- 0 '
t
-S--
rnm-ati
thaIa
v-=
ase t-W
-lnpttrib-.ate~eh 'ot etd aad h~~zo u1
o-nZ
=esm-i
ml UCC bels-m ofecder
Doi 5V-nn
r:n
ao-s
--
aa
-a
shz -~
'-'=
v T
dto~r-~ j~-I -smd '-rs e -ue-nres tenath h andw im escareso
as"ed t-h-e reative flenendin on in sea waves f ncuner. al veoct and -fo the anl motionsM In numerghic ,a simulr"'s-eomt aMcre,~ee~~s~ eit'ar% the resonantcfe, nsuna fe boeeen mada so the autaor iesols ha~errica W trgn crtialsitatin iott meod doe neotant efecs mayoe nava±h nvaodvraviciss nuei prevent theiaton wrfa t_ e mtte on to se ites joyfullyn I
l
"
oe~eno
uhZb
ME
~
n sNtoaou,
--
0-= en erh'n o- 3Procedur_ ntreadv a, at-less ~ cu- a I LI , n dI tw:73. Inboh =-sia asse rcra s.e.c a~~~~-e traelin sintl a e s tady-J - eze wa-na of fur to ave ofl difn -±o--su=n . fr__nAe For hedimensies me4mt thod casete donh e br e. trewvsronal tw ntreamlses wav- exren orn [5) ude. infiity of twoofth -es i eeenfud ~a Asin a fre-a rl y~~v on me sxnulait-sese
m.ephs
a---c-.
s h_ nomo C- LC -- -e cIc sv~ -at
tion for en sle--------=st n--aheoroi-~ r5 a~5 o-r -at The ftwodesoa igaedthoouhl 3ra 1n
o------
-S
-
-cttle -s~dt
of to be -e
n
-
respeti-v
ate-rn
we x staynr *ont arerd.AUvrib -1149 eoettoU
sc m~s, '4s 0ech for vim c-_ ct eo
r
s
tg~
a~w
_
(2)
-ef
gig!
+
r'tt+(V4.V(V4t
functions 6. 0 and o. for cl(x,z). L_ o are linear in- -and zero-order in
-""and
The equation L(tr)
the steady flow.
+p, hp, +z,
4
=
Z..(
The problem is now defined in mathe-atsatisfying ical terms: the solution for v (10) and (11) and appropriate rdiation conThe problem is recast ditions is sought. in terms of Fourier transforms in the next section. 2.3
An ordering is introduced by assuming is a small perturbation of the uniform flow and 4 is small comared to o. -rthermore, in the case of p=O, and- are written as follows +r;
-r
5
-
_(xOz)ep
(,,y)
22 )
r 1 i(a"z).(
arts and where the singugiven and they obek the condition t
*s(xOz) =0;
in 3D and .
(6)
are regular for y.O and satisfy * and de following well known equations, derived from (1)-(6) _2 (7) (for VW 2 hv obedltdfr~c if ro.=OA75) and (76) represent the resuits based on the usual linearizcd theory. Inthe neighbourhood of Luc the dray
this is the case for -;wc and -hord Froude numnber around 3 or larger). We have then (x~y)
Ikexp(-2k h)keph] 2h] 1 )+ 2 ex( 1
for
Yy
a 1) eemnsCO (Y=O). Ti which has to be substituted in (54) and across ~ ~~ ~ ~ ~ ~
-
/4-
-.
(75) reaches a maximum at
I
(72)
A /
wc- W=0/8 (~8
4-291
(147)
(~O)and (o)__
then drops quickly to the value oi the k~ k-Ikh
s
(2~)2The (73) Limiting the computations only to tile DC terms, i.e., to the drag and li4ft avferaged over a period 2-/w, we generally have by using Blasius theorem DRe
_'r
2 t
L
-'
'presented
(0,-h)
Re-
(74)
With~j' (4) an ae algbric e rrve algbricma~ipuligions wearv final expressions T2 1I DD/. 8
(Wc_0) _W +
+
/2 L
2 eh/
-_
jup
(78)
to
hevle
fte
sidered next is that of a lifting line withK
afw t hethat ttethe
2
r /64
[k~ep(-k~h+k~ep(-k~hJ
LDMA =+O
usual linearized solution, i.e.. the term related to k3 and k4 in (76)._ To illustrate the results we have reD and ILDCas function of w for r-=0.1 angh=O.l Oin Fig. 2. Th he-dimensional case to be con-
__
3) y
lift, which i.snegative, reaches its extremal value at w=.jc 0 and the contribution of the leading term is then
At
(0,-h) DC
last 'term of (77).
time varying vorticity r--,(z)cosu.t distributed along the spanlzlCs. Again we assume the frequency is low enough to warrant ngeto fse otct aito ngeto fse otct aito in the wake. Thus, the case is similar to the steady one and in order to obtain C(0) (,e) we recall the expression for ;s corresponding to a single doublet of
4
strength r.,(z)/4-' directed in the negative axis lyifig at (-,-h,4),
+y
.Ki 3 texp(-2k~h)-Ik 4 lexp(-
YSxyz,,h~
4-L
4(1+4wa) -y+h 2
+
(X-02
y-h +(y-h) +(Z-;)
32
I
-365-
ZD
(65) we obtain
ti'xce by (14),(62) and
s0 (80) in (7)yelscs
susttuin
~rg
(xiy-
cosQ~siO)
=
1d
(80
I
,
(r,;~)
-
sO)
1
Jd/1(-4 0
.>()cos(plsinO) +
in
cos(
))
and)
0
2
(os) Tsn~
o~~ e
co
OP
0'
I'cseo (zsin)-id 1yh-xcs
0e(d +_cosi_! 2 :i)
0-es''~~
0
f cos(Csini) +81
'h
siiar-
prbena
e l
0
6.
COSO zsn:V1d4- ( th +o hyorofoi
1 -( 32 -2 WO L'o
4
32-..) -1
d
td
2I -2
2
ld
sIn orer to
(t)
set O=O
0ecand
~
1 D2
d;;- 1 in thesinle
-/
(8811)ed
stit~os~tiinn
2hnreeta
lys
Foriahe imiarIydrfoi prble,
d-3;66d2-ec
otx
manner~~_
preen
onl
th
DC
tem -
o
helftad
e-1
-
_
Jo
a matter as flow (21) and has terms of steady of the that part reveals only (24) fact
and 2 16/2' eh/ (in )= IDociI.T V2 T ( 327 r in6&
NT -2+
to be kept in (13) and the basic free-surface condition which renders a uniformly valid solution is
--- )2
5tel
2 . (:)d 8
L--
I ~r
(8
DCr
I
[-E_
+
2(-l+u) z-
+
(-+u)
-+
)x
t
its maximum Avainrs =exp(-16&¢2/PT) cat cver coseto and alu tsis atleaingterm 2
(90)
(for y=O)
replacing (ii) where u=c/8x is the horizon component of the steady velocity on y=o. Eq.(90) may be in turn traced back to ( C(7ze) t (( e t 2 2
where .. T =
1 CoX,Z) y]'r=c
+
Zj€ln6w
-h/2 It is seen, therefore, that essentially
(88)T
ID
we have at whereas for the lift e-shows TLnC /AX'
=uc
-
eeplacin -l+u which modulates the resonant waves. thatFurther Uso,te onlyresuits t inspection of the a first-order at velocity induced by the vortex and its image
0 (89)
- 1 in results IZ~ inare are represented compared against 3(b:, andalso 3(a) and Figs. These
across y=o corresponding to zero Froude-numterm in at the leading contributing bar,ii t associated the far free-waves (90). Thus,
the linearized solution only for the drag. the given by force zero. in the increment The Conparidentical theory is lift linearized ing the three-dimensional results for a liftthe two-dimension(3))(2))withshos surface (Fig. ing that the peak (Fig. al equivalent
with the solution of (90) should be viewed waves modulated by the uniform flow plus as the current associated with the steady perturbation horizontal velocity u5 0 . Finally, method the Fourier the solution 4.2 indicates Sects. 3.3 and Transform in by developed
for the Izft and in the generalized solution DC.) irTa singularity in the reguForce as well as the lar (linearized) solution is much more localized in the 3-D case and resemble the befunction.. of a delta haviour were most pre- of whichHence solutions, the numerical formed for frequencies in the neighboI:rhood do not predict the occurrence of of this concentrated singularity with the exw ception of 7]. l
is sufficiensiy strong to that -rent the cur remove r s ythe o coresonance r s o d if n lim to zeu 5 (k)# o r u0 einn two m s K o diin three u (a,o)#O lim or if dimensions ...current5 removes the resonenin. Hence, the ant conditions if ic,
SUMMARY, and CONCLUSIONS "' and areND FOR FUU)
(x,o)dx o
;
dxdz u
(x,o,z) (92)
.-......
6.
(9
RECOMMENDATIONS
in two- and three- dimensions, respectively. Since a lifting surface, in contrast with a n conditions eigbo:rwedan n thefrequency fequnces frflow 2 frme The body, leads or a non lifting patch ifoiy pressure past near resonaf to a constant value of the integrals in (92), a singularities system has been examined it was possible to determine the finite ampfor the case in which the time-dependent litudeeofthe fe waves at =T in this term is a small perturbation of the steady, i case. non-uniform, flow term which in turn is a In other words, since the resonance in flow. for sma l disturbance of the the usual li earized approximation is interIt has been found that in the case of preted as the result of the inability of free a lifting surface the regular perturbation __ .. .. from -waves_ to propagate their energy away usual whose leading term is the expansion, the singularity because of the equality betfor uniform flow, linearized approximation thcurrent yet ad w en the re is not uniform near resonance and the steady locity, it is seen that the removal of resoflow term has to be retained in the zeroorder approximation. A closer srutiny of -367-
-can
Land
nance by a steady current satisfying (92) be as well interpreted as the ability of eslet wave energy such a current to cape to infinity. Another point of interest is the change of phase related to the non-uniform current, which results in a cancellation of the resonant drag and a substantial change cf the time averaged lift in comparison with the linearized soluticn. We pursue at present the removal of resonance for non-lifting bodies by incorporating higher order effects of the steady component. Another topic of theoretical interest is the nonlinear interaction of the time-depending terms and the solution of the initiai value problem. We hope that these related problems can be attacked along the lines of the present study.
p4
p33
.,
.
p (a)
k4
k3
k2 kl W FIG. I
REFERENCES
INVERSION PATH IN THE FT PIANE (a) 3D FL% :W
1. Haskind, M.D., "On the Motion with Waves of Heavy Fluids" (in Russian), Prinkl. Mat. Mekh., Vol.XVIII, No.1, pp.15-26, 1954. 2. Debnath, L. and Rosenbiot, S., "The Ultimate Approach to the Steady State in the Generation of Waves on a Running Stream", Quarterly Journ. of Mech. and Appl. Math., Vol.22, Part.2, pp.221-233, 1969. 3. Kaplan, P., "The Waves Generated by the Forward Motion of Oscillating Pressure Distributions", Proc. Fifth Midwestern Conf. on Fluid Mech., Ann Arbor Mich., pp.316-329, 1957. 4. Wu, Y.Y., "Water Waves Generated by
FOR
O 0. The analysis in this Section is abbreviated from Newman (1978), where more details are provided. The principal task is to solve for the complex velocity potentials *, due to heave (j=3) and pitch (j=S) mo ions of unit amplitude. with the assumptions stated in the Introduction, these potentials are governed by the three-dimensional Laplace equation
Sf
+ Un
Here the subscript n denotes normal differentiation, with the unit normal vector pointing out of the fluid do'ain, nj is the component of this vector parallel to the x- axis, and m3 is an auxiliary function efined in terms of the steadystate perturbation potential UT by the relation*
THEORETICAL DERIVATION
jxx +(f Jjy/ + jzz
Uxm
=,8
Here z = 0 is the plane of the free surface where the latter potentials satisfy (6), and z is positive upwards. Far from the (7), and, on tha hull profile, in planes ship the potentials .j must satisfy a x = constant, suitable radiation coneition of outgoing waves and, for large depths, the condition it icin 3 (9) of vanishing motion ds The potentials qj are distinguished = m by their respective boundary conditions on 3n 3' the wetted surface of the ship hull. With the instantaneous position of this surface 5= -x 3 , (11) replaced by its steady-state mean S, the appropriate boundary conditions are x (i/) 3 . (12) 3n in + U 3, ()ubscripts j=1,2,3 correspond respectively to (x, y, z).
-375-
form
The potentials in (8) also satisfy the extraneous two-dimensional radiation condition. Thus we add to (8) a homogeneous solution of (6), (7), and of the boundary condition on the hull. This homogeneous solution can be obtained simply in the form (;3 + T3) ahere the overbar denotes the conjugate of the complex potential €3" This homogeneous sclution behaves like a two-dimensional standing wave at large distance from the hull, and can be regarded physically as the superposition of two diffraction solutions with symmetric incident waves acting upon the fixed hull profile. In summary, the general solution of the inner problem takes the form (i)
+ Cj(x)
(t3 +
3)
1 -2-
*
G (y,z;k)
*
G* (v,z;0) is the two-dimenwhere G2D sional source potential, which satifies (6) and (7), and .kcos , (K/ 1 f= n (2K/Jk i ) + i COS (k /K -1)
2 2 (k2K > 1),
-
Cj(x) is an arbitrary "constant, in the inner solu-2 tion to be determined from matching. The outer solution follows by considering the complete Laplace equation (1) and free-surface condition (2), but ignoring the hull boundary conditions. Assuming symmetry about the plane y=0, an approngitudi-al priate solution follows from a distribution of sources along the ship's length, I (14) = qj(I) G(x-E, y, z)d . rL x*
(18)
+i sn k'1k) 2) 2 l 1)
cosh
i
(
(13)
')
- k2/.
S1(i S
(17)
(]+ Kz)f (k),
where the interacrion fu_, icn
(k2/ where K
(y, z; k)
4aj
(19)
< 1),
2 2/g.
=
It remains to match the inner and outer solutions (13) and (14). This mat be carrie out in the Fourier domain, using the convolution ther-em to transform (14), and the .,atching condition takes the appropriate form C
+
G
+
+
Here qt(x) is the source 3trength, and G denote the potential of a "translatingpulsating" source situated on the x-axis at the point x = 4. This potential is expressed generally in the form of a double Fourier integral over the free surface. Of particular utility in our analysis is the Fourier transform of G, with respect to x, which can be expressed as G
2
*
(20) (20)
Far from the hull in the inner domain, the two-dimensional potentials on the left side of (20) can be expressed in terms of the effective source strengths, in the form j =
G(x, y. z) eikx dx
G2D (22)
=
(G2 D) (8), (17), and the fact that T,,i Kz e cos Ky, and c'xating separately the factors of G2 ) in t20), it tollows That
21,Using exp[z(k 2 + u2) I /2 + iyu] i 1 idu 2 2 2 (k + u )i/2 (k + U
I
(15)
• *
•
+
.
3J
JJ
*
= q
*
(23)
where and
(16)
( + Uk)2/g.
When ikI < K, there are two symmetric real poles in (15), and the appropriate contour of integration is leformed in thpir vicinity such that Im[u (w + Uk)] > 0. In order to match the inner approximation of (14) in the overlap region, an asymptotic approximation of
(24
-i( =
-
The errgr 2 in thc last equation is a iactor 1 + O(K~r ).
(15) is re-
*Equation (16% corrects a sign error in equation (4.9), and in the denominators of (4.6) and (4.8) of Newman (1978).
The quired for small values of (ky, kz). desired result can be expressed in the
-M
-376-
~
__
--
-!
-7___
-.
Z7__A-
The inverse Fourier transforms of the last equations provide the relations C to.
G. + UO.-
j
2-ri C. a i
j
+
q.,
(25)
f(x-E) d ,
(26)
.)
3.
liie kernel (27) in the integral equation (28) is defined by the inverse Fourier transform of the function f* given by (18) and (19). This kernel can be interpreted as the value of the source potential on the x-axis, after subtraction of the two-dirnensional oscillatory source potential G2D. Singularities can be expected, especially at x0O, and a careful analysis is required. The singular behavior at x=0 can be mitigated by considering the integral of f~x), or the inverse transform of f*/(-ik). Lthis modification is offset by multiplying teghq y(i) tetransformed suc in (24), (291, Is replaced by
)
where f()
77
dk e k
f(k).i
25
n
(27)
Aftr limnaionofC. ro (2) nd(26), isdetermined ute the sorcestrnat th
from~~~-
2qi
iq(4
j
inerleuto
f(x-&) dC
o.(x)~Her a.x. +
30
28
denotes the derivative of the source-strength, apd the new kernel is
Assuming a numerical solution fox the two-dimensional potentials in (8), and the corresponding source-strengths _-,o, the integral equation (28) may be solved for the unknown outer source strength q.(x). The complete inner solution follows from (13) and (240, in the form
~ 2i. (s) +j (2, 1
1
(i
r.
REDUCTION OF THE KERNEL
j
FWx
'~
ikx f~ C(kjk/k.
(31)
Snef=~)a ,ceitad 3)i Since f*enOk asr-x) vaihes inas (31)'is There is a logarithitic infiiiy In r~k as 'k- , and hence in F(x) as x -0, huit
(r)f(x-:,)d F.
this singularity can be integrated in (30)0 without difficulty. The integral in (31) - an be zirmrdified by- consideriing the funcien
129) The first termn on the right Side of (29) iJs the strip-theory potential (Including the contribution fro;m ;.which usually is2 ignored). The remaining contribution to-(-/c) (29) represents the thr- ~dimei~sional interaction between adjacept sc~ioins. In the high-frequency domain the in-:egral in (29) tends to zero, and the Conversely strip-theory solution remains. in the low-frequency regimne the two-diw.ensional potentials in (129) simplify and the esult is recovered ordinary slender-bod. -h 16) eia as dervdb unified potential ( valid more(18,
I
RE
4(
i(2K/K)
2-/ 2 nVk(.
kl)
1
(32)
~ (ik i where -(k) is defined by (16). analytic throughout the finn-te k-plane, except for 3 branch cut on the negative real axis. With appropriate values determined on e-ch siae of this bra.icb cut, in the manner described by Sclav.,unoe tflosht
generally, for all w, jnumbers betwean these two limiting regimes. f (kt) I(k~iO) + "j± i H(-k) in the special case of zerc= forward -/ velocity (U=0), the unified solation (29) reduces to a foirn closely related to the 2/,1,37 (3311%K "interpolation solution' derived by lqjk) (197G). Maruo's apprcach is rather different, h-ut the only char~ge in l-le final reHerai Z(-k) is the Hleaviside unit function, sult is that the homogeneous solution equal to one for kc0 and zero otherwise, and ( j+T_.) is replaced by (1+1(z), and the amp] Aiude of the two-dimensional stripS+(k) =2-'i (-k~k) N±a' theory potential is modified accordingly to satisfy the boundary condition oai the(k=0(kkk34b body. /Kk 1 3b 4 () 2)
-377.
-A1
g(k)
,
=-2i
n
(k z7>--).
(8)
constants o3 are determined so as to satisfy
1R.
--
(7) + Ty.
f(-
LUnknown
oteb n sno
(y-y) 2
R - 1(ix x
(3)
Substituting eqs. (12) and (13) into eq. (15). we get 1j
X
where H(1)0 the first kind of Hankel Function,
W E exis(XK1 -~,
5
)cs;a
N
(~~ -
-
(16)
E x(K.jW'(~.j -401-
_
_
________
____
_
In eq. (16) a03 must be multiplied by 8 considering-the above mentioned assumption. And in the same way, if s determined, considering the finite draft of the ship, Oj becomes tlsi. 01u
didk,
R3
R, =
/(x+B) 2 +y2 (xL)
for 01,
(18)
for 0,.
(is
where
j o*
f
o(x)e-lkxdx.
(24)
Under the assumption of the slender-ship, the near field expansion of eq. (23) becornrws( 6 1 q-
K
2.2 Forward-speed Problem
-2wo(x) exp(Kz-2)+4vKlyletzo(x)
In this paper, since n- w U/g 0(1) is assumed, there is a region before ship to which the wave does not propagate. Therefore, it seems that there are qualitative
-4u(x)
differences between the flow field of zero-speed problem and
+.(Kvz)tdv
that of forward-speed problem. But, because we assume that the surfaces of the ship in bow and stern- pub are blunt and U = 0 (e"), the strength of the singularities in forward-speed problem may be determined by the same way in zero-speed problem. Instead of eq. (6), we assume )(K
,
i
expfl(wt-Kx)
02
is
(26)
++_.-.. 1 + I a4 r Br 0a 302
a
ff 42
(27)
and the free-surface condition is )
x
.
.
K#2
.
0 n n
(28)
.
(21)2 where
n-1 Oj
(25)
(20)aonand 02 is slowly varying function with x, then Laplace
JrK.
and
V(v +l>/-T'vcos(Kvz )
If the diffraction potential due to parallel part by =
then, through eq. (9) we get _-to30a
,exPl(K.+K)Zjn-ltxj.IcDSOiu'(tt)CRt
(;31
1(K)W-Kl)
*o-
(17)
izeYZ(l1Kf)0,l(KRj.).
RV,
*(k)el'-
-
ecfotefrilprlmte
Essentially. the sources should be distributed on the surface of the ship. Hence for the far field problem the above mentioned centralization of the sources is reasonable but not for the inner field problem such as the calculation Of effect of 4). In the latter case, we change the position of the centralization and assume R,
4.
Gojie¢K'(0-e-24)H , (K.Rj),
(22)
y
3. DIFFRACTION POTENTIAL DUE TO PARALLEL SLENDER PART
=
r sinO,
z
-r 6osO.
(29)
The symmetrical wave-free potential, which satisfies eqs. (27) and (28) and is zero at infinity, is given by
The disturbed velocity due to parallel part occurs from satisfying the boundary condition about the vertical velocity. Therefore it can be obtainied by the use of the ordinary slender-ship theory, but the effect of the fore and aft blunt parts must be considered. As slender-ship theory we use the method which was created by Faltinseni s l and improved by Maruo et al. 161 .
4'
= K,,-(Kr) cos (2n-2)01
where K. is the second kind of modified Bessel function.
3.1
combinations of eqs. (25) and (30), putting
+ 2K2.- (Kr) cosl(2n-l)81 + K2 n(Kr) cos (2n0),
(30)
Because we can represent the-solution by the linear Zero-speed Problem
In order to obtain the far field's flow, we put a singularity distribution on the centerline of the ship. Then the velocity potential cI becomesov
vef-o v2l
'+T. Ivcos(kvz)
+sin(Kvz)ldv+4KlyleK. , -402-
(31)
we write
e( cx) Kro
~
=
x(z ~
-- )
M4W i~d
+O(x)[1R +2jr'eiz+ 2:i
ro (32)
,
where pm isto be determined, On the other hand, the boundary condition on the surface is
-ship
r(0),
M-1
r,
r(-)
If Pm and 0(x) are obtained bY the method-already mentio"-.1 the velocity potential due to the parallel part is given L. he following equation. 02
an
•an 0.)
(0,%+
f'K-+
' (g1 +K*K 03 ,
-
npm 0ByVm(
V
,0
cos(()+,)- {(0B,-0Be
cK1r
I
* -' )
3.2
aydI Vj
K
(43)
e)K
M
In the same way as in zero-speed problem,-the far field solution of the slender-ship is obtained by the singularity distribution on the centerline of the ship -and s given by
0.---- u*(k),
-x
N -.
I x V(.,()l
- exp i~o+
=
Co sa) cos(O+-,)-VR (0) 1
(0)eK(.
((X)KE
Forward-speed Problem
V X cos(O-~+7)+Ysin(O+y)
-
Kr. iKx
o
(3)a+oZ)
where n is normal to the ship surface. But, assuming the slender-ship and following Maruo's processl 6 l, we obtain the equations for pm and o(x) as follows:
P~~m'
GW(X){I I~ gr
-
(42
-
dk
-
(44)
-, v cos(0+-y) + 0,Ysin(0+,y)1 X VR(--),
(34)
SOF Ix)
(3(z)-
-
The inner expansion of eq. (44) is given by Faltinsen l s l as follows:
and
~x +
es__ e ,.-,i
"V t+KU
,Kr+iKx
K e
--
=i reKZ(!-e
0
-" Oy
=
-2
r , eyt-''
r --rtEq. wbut = 2
(37)
, zaz _O
z =-Y
-
and r
--
8-z=
-
]
o •in (38)
r"(0)' is the equation of the ship's hull• Furtheris the angle between r and n, and we put
=Pi
VR(t) r0 To
a 8 (co"7'
-b-caOs
r
z r 89
"
)%
'(0
'
()
___
eX
- 4o(x)li
wheeformula, -aterm's -|
4*a(x)
Xi(
(35)-
KT)
-
J.
Vo (v 5Kiz+n(vl +l)2 =
'/
WUU
d
(45) is almost the same as Faitinsen's formula has a little difference in the term with respect to advance speed. The third term does not appear in Faltinsen's but the difference of the order of it fomother is O(e) and so must be considered. Now, because the equation of the near field solution case of forward-speed problem is all the same as in zerospeed problem, is given by the linear combinations of ,eqs. (45) and (30). Namely, putting(4 as t fmore, mu t t verK+of t. oi s
39
_
_
-f
asinnd ms be cn- e +
welht Now
__
sal hsni ' nUeo incseo polm orad-pe 4zKIyle',
the e
(46)
i
t-r
_
;:
M
07 43 e
+ Z P. ' 2(2 ---
(7 ')of
g
+
~(O)}
(B)-
-.
a
1
p
(2
sgvnb =-p-,(53)
0 +(0
P"41
X e-~
I
*sin(8~y)
+0.
(l(54)
If Kochin function is known, the resistance increase
Vf (O)eK(t.ro)c5O7"y
=~VR
"d~~'(2 . ,)eixe
'~
rsueP
1P.1 *3 cos(647)
7
tffe external pressure acting on the hull-is obtained as
[V(eK(r-xwi)CSO+Y ~j P [I~n cos(4j)follows: ()e.
-
+ 0
P.
The equation for pm is
co§(O+)
ln-o
and stern is shown.- In this- paragraph, external esir-ad resistance increase in head seas are treated. as. the application it. The total velocity potential (P is given by
The method of the determination of Pm and O~x) is all the same as in case of zero-speed prbebut th fet101+0 (iofthe disturbed velocity due to the fore and aft parts is taken into consideration by the eqs. (21) and (22). The Sneteetra results are as follows:
-;
the method how to calculate
Before this paragraph, thete difracionpnilsu
is given by Marwo' formula. VR + (-)I~z -
That is
Kr-cscs8-)
(48)
,si y OBCG C7+RAW
-
4
-K. +-
'pl-f-
--
f-
(m+K~s2) 2 (rn-tI) 4(m+Kcf-on
m5
The integral equation for a(x) is
eOO 4 f
w+KI
K'(49)
where
d
o Vixit
202 I + a
e~-ixK
where
+4S
(56)
l~=gL,(?
Kr0
R
Xenp (K1 z + imx)ds,
0)x
4w
(Cm + K0 11) 2 /K0 .
5)K,
+
in eqs, (49) and (50) Kzz
0.
I__2j
uoW'(KRj') uO
+ oesH('(KR'4,
(59)
For the calculation of resistance increase, the/ singularities in bow and stern parts are assumed to be located at x =0 and x = L-B respectively. Hence Kochin function due to the singularity at bow part is
WV
x
(58)
H
Ke
(51)
and the other notations are the same as in zer&Vseed problem but eq. (46) must be used for 's Finally 02 in forward-speed problem is given by the
I atI
-404.
h
1
aemneKci
-
exp(-K 1
-
K,);TI.
ucindet
(60) tr
-
~H3
___
faoexp1K 0
-*K
+ K1 )z + i(L-B)mldzwhr
e-(KesKi)TI e(L,.'R)M
(61)
Rw
rff
1mKf)-on
+ 2
-(69)
x IH1 (m)1 8w, function due to parallel part is given by
__Kochin
f
H2
-a
G(x).etmxdx.
(62)
RAW3
________
--K 2Wff
(70)
x IH3 (m)1dm
Because v(x) is zero at the fore and aft ends, we put a~x)= w(x)and o(x)
+f
91
using Maruo's approximation
-
A si(x+a)
8zpKf S
RAw2
(71)
__iox~idx.
Furthermore for the interference terms 2; L-(63) 12j wher
J.1
~~~2 L-a
§
a=*
LL
JXI
L-aj~~)-
f
+f
-
mKf).mK
%(m+K~flY-Ko m2
K
(tHIH2-+H2H,-)dm,
-(12)
K1mKfY~o, (H 1 H5 +HH 1 )dm,
1
-
eq(62),
-,x~i
c(-xadX
RA
=4p(-f
3+
fI -Ki*m
i Ieos
-osL)-j
+i)
X /(m2+Dj
,(6 L
w
(75)
I0
K
(U' cosnas(mL) (1 IUEIA
sin(ma~s-(-
1-
co~L)I 6) abv q (5,w
R~ + RAWIZ
abv405-fntosinoe.(5,w
EXAPLES-K
m
whrerKsasa.a",wihhdeecosnsasuet car(6)
L)
Substitutin ohnfntosit
Susttuin
-0 m,
ti~asn(L he fre
L
-juL +(I
(73)
-ii,(5
where
_
(73
I(M+o)l
(m+KiW*(nK) H21
_
+f(m+KOS)2(TnK) :zp J(~
e have(65)
L
L~s-a
4.pK[-f -
RAWU2
jxa a(x)sin-dx,
~
Vcj
sn
ofsuis ntewaeidue rssr cin nhllVr ae h rnia atiuaso h hpaesoni Tabl I.~a
paer uhweer, isa fora tcsonationswrcaieou was fu~l oe of shuies onthedive indoutent rasre ctionsan ae Tabe
e
rnia
atclr
o
h
hpaesoni
I7
j
Fn =j.1I...
Length between perpendiculars (Lpp) 247.00 m Centherahgraifom
si
(LCB
7.037 fore
%
AA-
0 A.P.
'F.P.
the beam, draft and sectional area of each section were usedI aparameters for the transformation. Fig. 4 Longitudinal source distribution There are two integral equations, (35) and (49), to be along parallel part(UL =0.8. Fn = 0,-0.1, 0.75) solved. Though both equations have the same sinjular kernel, it is shown that the integral equation of this type can be solved by means of numerical integration. In order to keep fluctuation in the parallel part is supposed to be due to the the accuracy of numerical integration, some devices weresiglrteatbw ndtrnptsndoentaparn (55 and intgrte (75) byetheaparin Eqs.lriie were numrictl applied. the solutions of strip theory Or slender-body theory. method of Gauss. For the trigonometric series of Kochin function (65), ten terms were applied to the numerical. calculation program. The factor of the decrease of the incident wave was assumed to be 0.5 in case of F0 =O and 0.6 in case of F"*
Ca.bOSA__
Fi- 0
-1
a- 0
AIL a 0
P
1
FnL= 0.
1.0-
Fni 0.1 Fna .15 ... at Bottom Center line 7
MA A..F.P. Inuced o bo andtrip Fig.3 velcitydue stern singularities (VL 0.8, Fn 0, 0.1, 0.15)
Fig. 5 Transes dsrbution of hydrodynamic pressure acting on hull surface (VLI 1.0, Fni 0)
the caesi of WAI--l.0, F,=0O and *WI"'l.C. Fn=0 . The values of the pressure arm plotted on the normals of the hull, In the figurei the numerical results by means of ordinary method and the measured results(") 1 are shown too. It is obvious that the'results of the present calculation agree very well with the measured results. They also agree with those of the strip method in case of Fn=0. But, in ease of FO=0.lS. the strip method gives a little bigger value.
I.Numerical examples of the induced velocity of the
diffracted waves due to the bow and stern singularities are shown in Fig. 3. In the figure the ratio of 8#918z to a~w/az2 along the bottom center line' is shown. Fig. 4 shows thle longitudinal source distribution along ch center line of the ship. It is obvious that there exists a significant fluctuato. of the souice intensity along the ship. Fluctuations are seen near the fore ind aft end parta ofithe ship. This is considered to arise from that !he sectional configuration varies rapidly in these parts. The
Me longitudinal distributions of the hydrody nic
p~eaesoni i. n .Fg6sostecsso W-0.r8; 'F.-=, 0.1. 0.15 and the values of the pressure on__ the centerline of the bottom The measured vahies in case of X/Li*.75; F=4015112 1 are also plotted Fig.7 Shows the caeo-.LL;F..5adtevalues of the pressure o the bottom centerline (8=0'). bilge (50)and side WW=8) repciey The measuredi values in the raM- Cndte_ t as in the calculationnarceplotted inthis fiume As is obvious fronm the figures. the agreement betweeni the calculation
m
I X/L
and the experument is Eenerally good and theithoeical calculation shows the-flUtuation. Thstnec fIuctuation is seen -obviously-tin-the measured -value' in -FiS.6-and
=0.8
15 Fn - 0 Fn -0.1 Fn x 0.15
supposed to be due to-the effect of -the sinpilanties at bow and stern parts,
-Is ---
Experiment [12)1 (Fn-.0lS Ak/L=0.75)
of
FO
-)
RAVI
"1 RM13
RA.W?
j 0.5-
RAW
..-
taculdtion by
'
-
.t
Gerritsma method (Series 60, Crj0.8)
JI
''I
0
Fig. 6 Longitudinal distribution of hydrodynamic pressure acting on hull surface0 (A/L 0.8. Fn 0, 0.1, Ji.15)
0
-
0
0.5
XIL
1.0-
(a) Fn 0 3.Or I
Fn
0.15, A/L
I I fBilge(0=50-) Cal.
-7
d
1.0
Center"~
Exp. [12]
A.1
-n
--
Wae
A
.IL
I
-c
0.
:Jt*
.. . . . ..
0=.
I
.. .
1.0
1A:A
A
A. IA
01;
--- - - -- --
A..F.P.
0
(c) R.
Fig. 7 Longitudinal distribution of WyrOdYnamic Pressure acting
if0
0.5
0.15
Fig. 8 Added resistance due to
on hull surface (A/L 1.0, Fn 0.15)
diffraction in regularI
head waves (FnO, 0.1, 0.15)-_
-407-
-The
-are
calculated results of the added resistance due to diffraction in regular head waves are shown in Fig.8. Since there is no experimental result directly compared with the present calculation, the results for Series 60, CB=0.8ship formlt 31 are plotted in the figure. The calculated results by Gerritsma's Method 1 141 are also Shown. For the pent calculation, the components of the added resistance are shown. But the interference terms are so small that RAwt2 an R neglected. The figures show that the Present method explains the qualitative characteristics of added resistance in short wave region, where full ships generally have large added resistance.
Al.0.4 Present Cal. Fujii's Cal.[2] Fujii's Exp.[21,
0.5
0.8
*
A
The authors deeply appreciate the valuable discussions by Professor INV. Wehausen. Thanks are also due to Dr. S. Naito for'zhis kind help in this study. For the numerical calculation, 'ACOS600 of Computer Center, -University of -Osaka Prefecture was used.
_
REFERENCES L Maruo, H. "Resistance in Waves", The Society ofI Naval Architects of Japan 60th Anniversary Series, Vol.8, Chap.5, 1963, pp.67-97._ 1. Fujii. H. and Takahashi, T.. -Experimental Study on the Resistanice Increase of-a Large Full Ship in Regular Oblique Waves7, Journal of the Society of Naval Architects of Japan, Vol-140, Dec 1976. pp.13 2-1 3 ?.
.... a
ACKNOWLEDGEMENTS
3. Havelock, T.H., "The Pressure of Water Wihles-upon7 a Fixed Obstac!e", Proc. -Roy, Soc. Series A No.963. VFol 175, July -1940, pp. 4 09 - 42 1.
*
4.Higo, Y.. Nakanua S. and Takagi, . -A Study on the Resistance Incrase of a Towed Pontoon Among ihe Waves?. Journal ot the Kansai Society of Naval Architects.._ Tapan, No.174. Sept. 1979, pp.4S-56.
-C
_
S. Faltinsen. 0.. -Wae Forces on a Restrained Ship in Head.sea WavesP, 9th Symposim on Naval Hydrodynamics.Paris, Aug. 1972, pp.1763-184. 0
0.05
0.1
Fn
0.15
6. Maruo, HL and Sasaki, N., "On the Wave Pressure Acting on the Surface of ank Elozvated -Body Fixed in Head Ses, Journal of the Society of Naval Architects of Japan, VoLM16 Dec. 1974. pp.10 7 -114.
Fig. 9 Effect of advance speed on added resistance-in regular head waves LkFig.9
. BesshoALM, -Ont the *awe Pressure Acting on a Fixed Cylindrical Body". Journal of the Societ of NavalArchitects of Japan. VoL 101, Aug. 1957. pp.1-.O
shows the effct of advance speed upon added restance in short wave region. In the figure Fujil's curve and his expervuental dara12l &-C also shown Though Fuilis curve shows that the added resistance inereases with speed. the present curve has a maximum point beyond which the curvn has the tendency of d-crease. But, roughly speaking,
&.Ogilv*e T.R_ and Tuck E.G., "A Rational Strip Theory of Ship Motions: Part V" Department of N=Val Architecture and Marine Eiigitteerip&g College of Engineering.
the agreement of the both curves is good.
The University of Michisan Ain Arbor. Michigan, Report N19
6- CONCLUDING REMARKS -~
The authors show an analytical method wbich approximately solves the diffraction problem of a slender ship with blunt bow advancing in head sea In order to satisfyr the longitudinal boundary condition at blunt parts vertical line sigularities are introduced. These cause fluctuation of hydrodynamic pressure and large added ristance in short wave region. Some numerical examples wem to explain the charpeteiistics of external pressure and added resistance. But, s-Ice the determinaition of vertical line singularities is-veri rough, the ouintitaive agreement between calculation and experiment is not so good. This is especially true in case ot= forward~speed piroblem. Further invstigation would be necessary.-
-408-
9. Maruo, H. and IshiL, T., -Calculation of Added Resistance mnHead Sea Wave by Means of a Simplified Formula", *Journal of the Society of Naval Architects of Japan, Vol.40, Dec. 1976, pp-136-l 4 1.
_
10. Naito, S.. "Disturbance of incident wave.. 1975. Private commurunication.11. Report of Ship Research Association of Japan. SR131. Report No.163. 1972. 12. Report of Ship Research Association of Japan. SRI3I. Report No.176. 1973.
-
IKamsa
13. Shintanu,-A,. -Influence of-Ship Form Particulars on Resistance Increase in Regular WaveC, Jourral of-the
=
Societyof Nav~J Archiitects.-4n No. 139, Mari
1971. pp.35-4 1 . 14. -Geritrina. L. and--Beukebnan, W.,-jAalysis of -the Resistance in-avs Wnrae of a Fast Cargo ShWp", International Shipbuilding Progress. Vol.19, No.217, Sept.
I972 pp.285-293.
FI-
IA
DiscusionThe T. Tkahas i
psDiscussion to apply it isto that present basically think the authors method
-I-
It has been considered that a consistent treatment is necessary for the estimation of the added resistance of large full in short wavelangth region. Now, ships the authors show us an useful theoretical method as well as interesting calculation resultsI appreciate this paper very much. Would you please let me ask some questions If~nt wea th want ffc (1)hi to evlut d iff ) e ant sh ad lape eitc ofof different bow shape on added eresistance in short wavelength, is it possible by authors' method using the plane distribution of vertical singularity ? (2) According to the calculation redeadded resistance in Fig.B, sults spe.aed ressahereswdvnc crass shown creases as advance speed increases were 0.2. This tendency seems to be reverse to our experimental results for full ship models in short wavelength. I suppose thaenrgydissipation due that energy un to ti reflected re se waves on be surfae per upit time increases when a ship is running, compared with the case of zero speed, owing to increase of How can be explained from encounter waves. the theory that added resistance due to diffraction waves on bow Of runninq ship is smaller than the zero speed case ? I am much obliged to you if a plain explanation is given,
more general and complicated problems, by replacing the vertical line singularities with a surface or plane distribution of sinquirity. But, some difficulties are expected in satisfying the body boundary conditions at the ship bow and stern. It may be especially true when the ship has a big bulbous bow. There will be much more difficulties in evaluating the diffraction effects from bow and stern on the fluid motions in the middle parallel body doain. Also, a lot of computer time wil be necessary for the numerical calculation. As for his second question, in 4ase of, of, ae zeo-forward speod qestinri his coments hat the eergy dissiation
(1) You mentioned that in Pig.4 the fluctuation of the source distribution in the midship region is due to the bow and stern singularities, and that it does not appear in the solution of strip method. If you have the results of strip method. would you show us the ccmparion of the singularity distribution between your method and the strip method (2) In Fig.8(b), how does your result behave wen the wave length becomes very
will increase with the iftenty of ernccuOf ter. when the ship has a spe-d of advance, however, it should ;e noticed that there exists a region in front of the shin bow where are no waves reflected from ship bow. In such a case, the rise of the advanceuhacsters Ispeed mean the increase does not necessarily fteavne of energy dissipation. However, the similar method as that in zero-speed problem was used for the determination of bow and stern singularities. Asatepeas we can easily understand from Eq. And, (21), the strength of vertical singularities at ow and stern parts decreases while the This is frequency of encounter increases, another reason why the rise of the frequency of encounter or the advance speed gives the small value of the added resistance. But, as is mentioned in the concluding remarks, the procedure employed here is simplified one, therefore the authors think that the results may not necessarily explain the real phenomena. Professor Himeno asked that how is the between the longitudinal source distribution obtained by meanus of the present method and of the strip theory Fig-A-I replies to his question. In the figure, solid line indicates the result the slender-body theory, namely the strip teory. It is obtained byv the saae way as and Sasaki's method As is mentioned jaruo in the paper, hump and hollow near the fore and aft ends are due to the rapid change of the sectional shape of the ship. At the middle parallel body part, the curve by strip theory does not have any hmp or
large ?
hollow.
y.H Y-
Mr.-W. of OUA-Opa-f) rdifference
He also questioned the behavior of the
added resistance when the incident wave becomes very lcng. According to the calcu-
lated results presented here, the added
resistance due to ditiraction seems to increase with the incident wave length. from This increase is considered to ce that in determining the source st-renth of forward speed problem. we used the same method as that in zero-speed problem. 'In the fundamental assumptions, we assumed
Authors Reply R. HAoda (L-
aom-P-
that the incident wave length is small cr-
pared with the ship length, namely f(=,eU/g) Therefore, not-hing should Le mention11/4.
mr.Takahashi asked the applicabi. ty
ad about the added resistance in the range
of the present method to more general case. .4O
-
of very lcng range i -e. in Q accordingly waves do not exist, except the incident waves, far ahead of the model. When 9 is smaller than 0.25, waves propagate forward from the model and their reflections on the tank wall will disturb the wave patof water. tern at almost all the surface So the idea of the analysis of the unsteady model in wave pattern generated by a ship this case. incident waves can not apply to It is evident that the accuracy of the Kochin functions derived with the formulae and (5) from the measured unsteady waves is dependent on the y coordinate of a line along which the waves are measured. But some results of the wave analysis done with values of y show that the valuesdifferent of 12, at least, are not so much influenced by the magnitude of y if it is larger 2 than one thirds of the model length(Ref. ).
)
2w- [C
II(-)
where Ai(i=l, 2) are complex constants and
IN
..
.
-
AM
Svantage __acquire
S
From the time history thus obtained we can get Co on the right hand side of equation (1) as the averaged bias of it from zero level and, Cc and , as the cosine and the sine components of the sinusoidal variation of it in that time.it This makes measurement it possible has for anus adto
wave records obtained by the repeated expe riments are not regarded to be those in the same incident waves but with different phases of the encounter. A r.-w system, even for the measureir._,nts of the effective diffraction waves, des-
reliable data on three terms on the right hand side of the equation (1) with other noise components completely removed. Measurement continued during one run of the model gives, however, no more than the values of them at only one location in the reference frame moving with the carr':ge. Accordingly if we want to know them at so many locations along a line of y=constant enough to carry out the Fourier integral of them on the right hand sides of the equations (4) and (5), we have to repeat the experiments many times with different locations of the wave probe, or a tremendously large num.ber of wave probes have to be set. This way of measurement is, of course, unpractical in spite data. of its advantage in providing reliable
cribed below was devised as a substitute for those two systems. We set several number of wave prebes 4i, W2, .'' and WN in the water tank'-as shown in Fig. i with an equal spacing x and on the line of y=constant along which we need to obtain Cc and Cs. When a ship model runs in the regular head waves, each wave probe will record wave motion at its location which, of course, includes that due to the incident wave. The expression, given by the equation (1), of the wave elevation at a point in the reference frame attached to the moving average position of the ship model gives us the expression for the wave record Cj(y, t) taken at probe Wj(J=l, 2, ., N) That is
A measuring system was derived for realizing the wave measurement mentioned above with only one wave probe placed at a location in the water tank but with repeated experiments for the same frequency and the same forward speed of the model. This measurement which was successfully used for the measurement of the radiation waves, is as follows. If the model is forced to undergo a prescribed oscillation cos(wt + ci) with the time t counted from a moment when the model's midhsip passed the spot of the wave probe set in the water tank, then the record of the. wave motions there (no incident waves exist for the radiation wave test) takes the form i(-v , Y, ) = C0 (-Vt, Y)
(-Vt,
y)sin(wt +
o(-Vt, y)
+ C0 (- Vt,
Y)
x cos[wt + c +(J - i).Ax/v.] + C5 (-vt, Y) x s
+
£
+(J
-
l).x/v.w]
+ the incident wave
(8)
of the probeWj and e is the phase of the
i) (7)
Repeated runs of the model at the same frequency of the oscillatory motion and the same forwara velocity of the model will provide as many wave records with different Ei as the number of the repeated measurements, since it is impossible to adjust Ci to be identical for all the runs. This fact is rather convenient for us. For a fixed y and t, the unknowns Co, Cc and in the equation (7) are determined with several different £i and also with several different data on the left hand side. Thus we can get the amplitude and the phase of the sinusoidal wave motions for every t, in other words, for every x=-Vt along the line with the coordinate y, with a finite number of repeated measurements by one wave probe. However this system can apply neither to the diffraction waves nor to the waves generated by a freely floating ship model in incident waves. it is because we can never repeat to generate completely the same regular in_ _-417-
j(Y, t) =
where the time t is measured from the instant when the model's midship passes the position
+ tC(-Vt, y)cos(st + £i) +
cident waves in the water tank and all the
encounter of the incident waves with the midship at the instant when it passes the position of the probe WI. If we can exclude the incident waves from the right hand side of the equation (8) and if we replace -Vt with x in every record, then Cj(y, t), even though really they are the records at different spots, are equivalent to the wave elevations which might be measured at a spot (xi y) in the reference frame moving with the modelbut at different time instants t-(J-l)Ax/V-w. So we can determine ;o(x, y), Cc(x, y) and ;s 's(x, y) from those N records at every x along the line parallel to the course of the model. The way of exclucing the effect of the incident waves from the record obtained by WJ is as follows. When the running model locates some distance behind the probe Wj, the wave record at this .)robe does not contain yet the disturbance by the model but the incident waves (the unsteady wave measurement is possible and done only for R>0.25). This record taken during several periods of the incident waves before the model comes close to the probe is
I -
_
expressed in a Fourier series with respect to t. Assuming this Fourier series expression to give the estimates of the incident wave motions at the probe Wj even when the model proceeds and the record there includes the disturbances by the model's diffraction and radiation, we can subtract it from j. The remainder gives the unsteady wave motions with the incident wave motions not included. In order to confirm the accuracy of such extraporated incident waves, we measured the motions of regular waves generated in our tank without putting any ship model disturbing them and compared them with the estimates of the identical regular waves extrapolated by the procedure described earlier on the assumption that we could not measure the wave motions after an assumed moment, The results shows that the regular waves in our tank seem to gradually change their shape. Their amplitude, for instance, decreases or increases by a few millimeters in about 10 seconds. Then the extrapolated waves may have probably the error of several percents. But this rather large error has the same frequency as a fundamental one, that is, the wave lengtn of the error is just the length of the incident waves. The elementary wave among the unsteady wave disturbances by a ship model which is influenced the most by this error is the one with the wave number k 2 which propagates right backward from the model and it hardly has the effect on the added resistance. 4. THEORETICAL PREDICTION OF UNSTEADY WAVE AROUND A SHIP MOVING IN WAVES
I
of heaving mode or pitching mode with frequency w are given in the far-field by the pulsating sources distributed on the longitudinal axis, coinciding with the x-axis of the reference frame employed here. The source strength G(x)el1 t is expressed by i c (x )
(x) 4iiw
(9) where X(x) and c(x) are the amplitude and the phase of the out going wave motions determined by solving the 2-dimensional problem of forced unit-amplitude heaving motion of each transverse section at x. We have already had quite a few techniques(Ref.8, 9) for computing them of any arbitrary shape section. 6 denotes the hea'ring amplitude of each transverse section and is not independent of x when we are concerned with the pitching motion of the ship. The Nochin functions of the radiation waves induced by the heaving or the pitching mode of oscillations is expressed using the source strength determined by the equation (9), in the form
2
[' Hj(O)
()xp[ k i e
cos
]dx
-/2 fL/
2
9
The time-dependent waves in the farfield of a ship advancing into incident waves are regarded as a superposition of the radiation waves and the diffraction waves. The former are the time-dependent waves caused around the ship made to advance with steady forward velocity in otherwise calm water and forced to undergo just the same oscillatory motions as induced in the incident waves. The latter are the scattering of the incident waves on ship running with its oscillatory motions suppressed. If we linearize our time-dependent problems, there is no loss of generalit, in so dividing the wave field around the ship between the radiation and the diffraction waves, In order to evaluate theoretically the Kochin functions of the radiation and the diffraction waves and compare them with those obtained in the analysis of the measured waves around a ship model in regular head waves, we adopt Ogilvie-Tuck's theory (Ref.6) and Adachi's theory(Ref.7) respectively. Knowing the amplitudes and the phases of the ship's oscillatory motions resulting from the wave exciting forces, we can evalua .= Zhe actual waves around the ship as a linear superposition of both the waves predicted by those two theories. Following the Ogilvie-Tuck's slender body theory based upon the assumption of the motions of high frequency, the radiation waves of a ship making an oscillatory motion
-418-
(10) Adachi showed a solution of the diffraction problem for a ship running into head waves which was obtained with the method of matched asymptotic expansions. He assumed both the slow forward velocity and the short incident waves, and gave the solution almost identical with Maruo's sulution(Ref. 10) at zero forward velocity. He also claimes that his solution is valid even for much larger . Anyhow we are concerned with the far-field velocity potential of the diffraction waves. The Aachi's theory gives it as follows. Regular head waves of unit amplitude defined with respect to the reference frame moving with a ship are expi
(wt + kx)
Then the diffraction waves are expressed by the waves of the singularity distribution o(x)exp i(wt +
2X)
on the x-axis just as for the radiation waves. This singularity strength is the solution of an integral equation
I TM
oPEP
_____
L2 -i0()section
(1
2
waves with the procedure proposed in the
The wavenumbers %L--11.7 and 17.42 (L2. is the model length) are those of the forced heaving oscillation. They correspond to the encounter frequency when the model is assumed to run at a forward speed Fn=0.15 in the head waves of X/L=1.0 and 0.75 respectively where X is the length of the incident waves. This example shows that the Ogilvie-Tuck's slender body theory can surprisingly well predict the Kochin function H2(0) even for a not-slender hull form such
VX r-
1
1
(x)
+ S-
22
4us)
ig -
= 0
-
4wpredicted,
(11)
as a tanker hull form.
(12)
although it is done so qualitatively. HI(6) very small compared with H2(e) is, and however, hardly has importance from the practical point of view. Hereafter the for Hi(e) are not illustrated in this report. Since we could get the results of thu same good correlation for a more slender hull form(Ref.2) when forced to make heaving and pitching motions, we may conclude that the practically predominant part of the radiation waves for those modes of motions can be predictea by the slender body theory. The good agreement shown in Fig. 2 seems to reveal another factandthat the accuracy of both the measurement the analysis of the
where =
E ]ds E = [f kzY(S)exp[ii2z(s) kYsexresults C
y(S') is the distribution of Helmholtz wave sourse on the contour c of each transverse section at x. It is determined as the solution of a 2-dimensional boundary value problem for Helmholz equation by solving numer ically the equation aG .2Y
W
i 2 y(s)
ads
kz
lmodel
where G is Green's function satisfying Helmholtz equation, the linearized free surface condition and a radiation condition. N4denotes outvard normal, Z(s) the z-roordinate of a point and s the distance measured on the contour. Substituting the source density thus obtained into the equation (10) gives the Kochin functions of the diffraction waves,
..
5. RESULTS OF WAVE ANALYSIS -
-
radiation waves is fairly high. Considering the very small amplitude of elementary waves propagating toward the direction of around 0=1800 and the large wave length, measured parallel to the x-axis, around 8=90
-
e P[2Z (S)(13) ex
H1(6) is not so well
Measurements and analysis of waves were done for a tanker model (L=2.0m, B(breadth) =0.312m, T(draft)=0.119m, CB=0. 8 1 7 and CM =0.996). Those experiments were carried out in the large tank of Research Institute of Applied Mechanics, Kyushu University (80mL x 8mB x 3.5mD). Eight wave probes were set on the line parallel to the tank wall and at a distance 50cm from the model center line. Spacings between the neibouring probes were selected according to the period of the model's 05cillatory motions and the forward speed such that we could get the best X/v.w for carrying out the first step oZ analysis to derive C and ;s from the wave records at all the probes. In the first we measured the radiation waves induced by the forced heaving and pitching motions of the model for various osciating periods and forward speeds, Fig. 2 is one example of the Kochin ftrnction H2(0) obtained analyzing those radiation -419.
(the
data Cc and Cs can be collected over three length, backward from F.P. of the model and they have to be estimated after that as described in the section 2), the accuracy is so good and we are convinced that the is correctly for almost allanalysis the directions of thedone elementary waves. As evident from both the theoretical and the experimental H2(e) shown in Fig. 2, it has a big peak around 8=900 and is very small in the vicinity of 0=1806. Moreover the integrand of added resistance integral is-much weighted for 0 close to or smaller than 900. It means that even the simpler 2-D theory can predict well the added resistance if X/L is comparatively large and the radiation waves are dominant in the added resistance. The results of the diffraction wave analysis are shown in Figs.3 and 4. The steepness of regular incident waves used in the experiments is about 1/50. They show also good agreements between the measured Kochin functions and the theoretical ones computed using the singularity distributions given by the Adachi theory expecially when the length of the incident waves are not so short. Considering the model hull form is not slender, this agreement is again surprising. We can conclude from those results that we can predict fairly well the diffraction wave fields around a ship by the slender body theory when we want to predict the added resistance in waves with taking into account the diffraction waves. However the tendency is found in those figures that the agreement is the worse when the incident waves are shorter and the forward speed is
-
I
H2 4wkg
~go
Fn =0. 15, 0, 0 Experiment
Ogilvie . Tuck Theory
75)
IZ42 (Y1LO.
... KL
.
*
KL=11.7 (A4L=iO)
4
0
U
5
-
0
40
60
80
Fig.2
H.-4xvKz
140
160
-0.20
K =
Ht4
F, ' . I Measured
Theoretical
1-0 0.6
20-
120
180 &(degree)
Kochin function of radiation waves
FaFn
.S XIL
-
100
Theoretcl hM -
tMeasured
0.6
20
o
A/L .0
00 000 G0
1000
60
80
100
120
0
00
140
160
60
180
100
120
140
160
180
(er1
a(degree)
Fig.3
80
Fig.4
Kochin function of diffraction waves
-420-
lochin function of diffraction waves
faster. Since the theory used here is based -upon the assumption of very low forward speed, such a disagreement does not seem to be strange. The disagreements found for shorter incident waves are remarkable around 03=1200%140c and their extent is the larger for the shorter incident waves (the peak around 0=1300 is much higher in the results not illustrated here for much shorter incident waves). The length of elementary waves propagating to the direction of 0=1300% if n.aasured not along the direction of propagating but parallel to the x-axis, is about one and half times as large as that of the incident waves. Such long waves are not found in the error of the extrapolated incident waves which are to -be removed from the measured wave records as described in the section 3. Consequently it may be true the
diffraction waves of a full hull form are not so completely for short incident waves as the radiation waves by the present slender body theory. The disagreement is thought to be attributed to the bluntness of the bow of the hull form and the better theory should be able to explain it. Fig. 5 and 6 are the comparisons of the theoretical Rochin functions of the radiation and the diffaction vaves of the model. We multiplied the theoretical Kochin functions of the heaving and pitching motions of unit amplitude by the amplitudes of those motions measured in the experiments. The diffraction waves are dominant in the elementary waves propagating backward but the elementary waves propagating there are not so important in the added resistance integral. From these figures we may understand the relative importance of the diffraction and the raidation waves in the total wave Diffraction field around the model. Finally the results are described of ------ Radiation (Hearing: 21; =0.35) R i t.the wave analysis when the model runs with --Rdtio (Pitching: s/k; 0.45) a constant forward speed in incident head 10 waves without any restrictions on its motions. In Figs. 7 and 8 are shown the records of unsteady waves along the line of y=5Ocm. The distance along the line is mea*. sured from F.P. to backward on a unit of the model length L. We must be careful that 5 they are not real records of wave motions taken with a wave probe, but the distributions of C and 4s along the line which are * -; !iobtained --through the processing of wave re%i\\'t ;/ -. cords taken with all the wave probes ,, "---. cight nrobes set in the tank as explained in the section 3. 180 IF1 120 140 100 80 60 8 0 1 0The vertical coordinate shows -c and t(deqree) ;s divided by the amplitude of the incident Fig.5 Kochin function of each waves A. Fig. 7 is the reults for Fn=0.2 , wave component '/L=0.6 and Fig. 8 for Fn=0.2 , X/L=!.0. (F.=0.20, l/L-l.0) The data untill 3L to 3.5L from F.P. of the model do not include the tank wall reflection and can be used as the integrands of the integrals (4) and (5). At and behind this distance they were found to be almost con.U2924X41 Diffraction pletely fitte with the expression (6) given ...... Radiati(Heaving:zl.O) in the section 2. In Figs. 9 to 11 the spectrums of elePa aioan (Pitching:GlK%-0.08) mentary waves are illustrated of the measured 10 unsteady wave field around the model runnin_ in head waves. Those spectrums are defined Vt in the form 1
'1J
/1
% S\
-
1
-and
/ !
0
80 Fig.6
100
k 2 /'k2 )
2(o
120
140
1
Kochin function of eacF wave component (Fn=0.2 0, A/L 1.25)
180 (degree)
- 4n cos0
(4
therefore integrating them from aoto gives almost the added resistance of the mod small, the contibution of HIl(r-) is very small. The theoretical spectruats shown in those figures were computed using the diffraction waves and the radiation waves predicted by Ogilvie-Tuck theory and Adachi theory respectively as well as the amplitudes and the phases of the oscillatory motions obtained experimentally. The spectrum of the added resistance is predicted fairly well with the 421 -
U
I
_
!
WAVE PATTERN
s
--
1.S
COS.
COMPONENT
SIN.
COMPONENT
H/P 3L 33
OL N60'2M SHM. . MW033 I
n_..n20 WLL=0.60 Q-. 20CM 11-0.877SEC. Fig.7
50C
Wave record
WPVE PATTERN
-OS.
COMPONENT
QI N. COMPONENT
HSA
-1.0
v"
I
LL
2L
'0IL N60/2M SHM.( MW0,17
J
FN=0.20 WL/L=..00 Fig.8
to the direction of 6=1300. where tr- discrepancy is found, does not have so much the weight in the spectrum of the form(14) and consequently the difference with such a magnitude as found for /L:0.6 does not have so much influence on the spectum of the added resistance. we may conclude that we can predict with the rational basis even the spectrum of the added resistance as well as the resulting added resistance of a blunt bow hull form, taking into account rationally the effect of the diffraction waves, unless the incident waves length VL is smaller than 0.6. For much shorter incident
-
_-;m.
-. 132SEC. Y= 50CM
Wave record
slender body theories as the superposition of the radiation and the diffraction waves even for such abort incident waves as 1= 0.6, since the diffraction waves propagating
__
-. 2OCM '
I
wavs, t he discrepancy between the theoredeal predictions and the reality in the diffraction waves is much more and the
present theory is not enough to predict accurately the added resistance, let alone the spectrum of it. Integrating the measured spectrts illustrated in Figs. 9 to 11, if a small contribution from H!(6) components added, gives theadded resistance derived from the measured wave field. They are compared with those obtained in the resistance tests in waves in Fig. 12. The added resistances measured with those two methods are correlated with each other, but the ones derived directly from the wave pattern are a little smaller than the ones measured in the resistance tests. This difference is tnterpreted
I
to be originated f Lom the errors in the reEistance t6sts or the effect of water visor th.e errors in the numerical intecoity gration of the measured spectrum especially around small 8 where we have to analyze the wave pattern with mudh more fine mesh of 0. To get so.e conclusion 6 n those problems, Here we we need further investigations. are satisfied -with concluding that we can derive the added resistance from the wave pattern, The theoretical added resistances shown 12 are obtained by integrating the in -ig. theoretically predicted spectrums illustrated in Figs. 9 to 11. it is remarkable that both the added resistances at A/L=0.6 obtained by the wave analysis and by the theory are smaller than the one in the re-
-
FnzO.20. AlL=0.6
IJH.....t' f l ffi m l, 0
0
sistance tests. It means there is- th-epossibility that the comparatively lare adde resistance measured in the resistance tets of a blunt bow hull form for shor-t-waves is not attributed to.-only the diffractionm waves. . 6. CONCLUDING REMARKS -For the objective of studying-added resistance of a ship, measurement and analy= sis of the unsteady wave pattern formedaround the ship running and moving in regular head waves were proposed and carried out for a tanker hull form. It can be concluded from the results that the slender body theory developed on the assumption of short waves predicts sux-
IH H11
..... Theoretical Measured
£0
Fn:O.20
=
12
[
|
-
__
-
A1Ls 1.25 " heo e ti cda l A- Meosed =
04
0
-
A0
20 20
-
0 00 00a
0
60
80
Fig-9
S
100
120
140
150 180 *(deree)
so
80
i:.1i
Spectrum of wave pattern in the far-field
0 O
16., 1 &(degree)
Fn:O0.20. 4/Lrl.0 With Force MeaSureJment.
Meaured+
f-W gg' ..
th I.t.- A..al.sis Ttworelir
0.
40-
0.1
___20-
___
0
spectrum of wave pattern in t-e far-field
Trort0a
__60
120
8-0
ig.10
100
120
KG0
16
0
8
Fig.12
Spectrum of wave pattern in the far-field_
-423 -
05
-t0
1.
Added resistance coefficient
-
9
prisingly well the detailed structure of the radiation-waves generated in the far field by the motion of heaving and pitching modes even if hull form is not slender. The prediction of the diffraction waves in the far field is accurate except for very short incident waves and for faster forward velocity. It means that we can get the precise added resistance of a full hull form except for Very short incident waves with the diffraction waves taken into account rationally with making use of the slender body theory. The total wave field of the ship freely floating in waves is supposed to be a super-
6. Ogilvie, T. F. andTuck E. _* "A Rational Strip Theory of Ship-Motions, Part I", Report No.013, Dept. of -Naval Architecture and Marine Engineering, tje University of Michigan, 1969. 7. Adachi, H, "On the Calcul*aon_-f wave Exciting Forces on Ship Translatin_ in Head Sea Waves", J.S.N.A.Japan. "ol.143,1978, pp.34-40. 8. Tasai, F., "On the Damping Force and Added Mass of Ships Heaving and Pitching", Report of Research Institute for Applied Mechanics, Kyushu University, Vol.26, 1959, pp.131-152.
position of tne radiation and the diffraction waves. The theoretical prediction based on the assumption of such a linear superposition does give so good results as of the total wave of the expected diffraction from spectrum the troublesome field apart
9. Maeda, H., "Wave Excitation Forces on Two Dimensional Ships of Arbitrary Sections", J;S.N.A.Japan, Vol.126, 1969, pp.55-83. N., "On 10. Maruo, H. and of the an on Sasaki, the Surface Wave Pressure Acting
waves for much shorter incident waves. The added resistance computed from the total flux of the theoretical wave field is in good agreement with that derived from the spectrum of the measured wave field. The agreement reveals that the added resistance as a wave pattern resistance is predicted well theoretically. Investigation should be done further -on the difference between the added resistance derived from the wave pattern and the one from the resistance test.
Elongated Body Fixed in Head Seas", J.S.N.A. japan, Vol.136, 1974, pp.107-11 4 .
ACKNOWLEDGE-ENT The author acknowledges his appreciation to Professor F. Tasai. Research institute for Applied Mechanics, Kyushu University for his encouragement and support in the course of this study. His thanks also go to Mr. M. Yasunaga for his cooperation in carrying out experiments and his efforts in developing the data processing systems. Further he is gratefui to Mr. H. Adachi, Ship Research Institute, for tendering a part of the computing program. §
1
REFERENCES 1. Maruo, H., "Resistance in Waves", 60th Anniversary series of JSN;ME, Vol.8, 1963, pp.67-100. 2. Ohkusu, M., "Analysis of Waves Generated by a Ship Oscillating and Running on a Calm Water with Forward Velocity", 1977, pp.36-44. J.S.N.A.Japan, Vol.142, "The Determination 3. Newman, J. N., of wave resistance from Wave Measurements along a Parallel Cut", International Seminar on Theoretical Wave Resistance. Ann Arbor, Michigan. 1963, pp. 353-376. 4. Hanaoka, T., "On the Velocity Potential in Michell's System and the Configuration of the Waveridges due to a Moving Ship (Non-Uniform Theory of Wave Resistance-4)", J.S.N.A.Japan, Vol.93, 1953, pp.1-10. 5. Newman, J.N., "The Damping and Wave Resistance of a Pitching and Heaving Ship", Jounal of Ship Research, Vol.3, No.l. 1959 pp--. :
-424-
be oter
.1i1
...
actor not cotained in Wave
n at f r field, PatterSI~
l
as s ug etia I* the
author, and it will g vel snaller added
resiszance compared wit the one- obtained resistance-test. i-f We would like to ask the author, in the case of not full ship as shown in the reference 2, whether same degree of differ!wave-analyences were found or not between es ~s~s reutonfrssac resistance test results. results-and
T. Takahast and E Baba (MH At first, we would like to pay our fiel author who developed a new respect to the field of unsteady wave pattern analysis". As very interesting results are shown in we expect further fruits of this new method. this paper,
sis
Our di scussion is related to the dis-
rtepy
crepancies of added resistance between analyssreslAuthior's and resistance test wave analysis results ones, shown in Fig.12. Author interprets that difference is originated from the errors in the resistance M.Ohkusu Kr-v=n tests or the effect of water viscosity or the errors in the numerical integration of the measured spectrum especially around
I appreciate your valuable discussions and instructive suggestions. As for the way to confirm-the accuracy
small 0. Furthermore, author suggests the existence of other factor not attributed to only diffraction wave in short wave
of the wave analysis l completely agree
length.
t
-
-
Abow
F
with you. However, the good ageemtbetween the theoretical predictions at the results of the analysis for the Kochin function might be interpreted to show the reliability of the wave analysis. I believe the wave breaking is possible near the blunt bow. Our knowledse on the detail structures of the unsteady waves around ships oscillating and running in waves is too limited to draw any definite conclusion on this possibility. -So I think we should continue to study those unsteady waves along-the line I proposed here. To my regret I do not have data on the added resistance derived fro1 the wave pattern analysis of not full ship.
As for the-errors in the resistance test, it will be settled by making use of a large size ship model. And, as for the error of wave analysis, the accuracy can be evaluated by generating numerically theoretical unsteady wave patterns and analysing them through the same process as in measured data, and checking the results how coincide with the original Kochin function of the generated wave pattern, when a full ship with Cb = 0.817 as tested by the author runs at relatively high speed Fn = 0.2, it can be supposed that wave breaking will occur near the blunt and incident waves superposed on1 it also will break. Sot we supposed that thex_
-2
-
Rolling and Steering Performance High SpeedShips
_of
SimulationStudies of Yaw-Roll-Rudder Coupled Instability -
U
Si (3) Relatively large rudder.
SASSIMT
+
Equations of yaaswayroll aid rudder notions ring are formlated to represent realistic a behavior of high-speed ships su h as containr ships. I mportant coupling terms betwceen yWm,
ctive
sway,
These particular ctaracteristics introduce the possibility of fairly sigeificant yaw-swa-rollrudder coupling effects during high-speed operations.
rol I
_
The omjor objective of this study is to examine
model test results of a high-speed ship.
f
-
s
(e-g.. bull form similar to
Roll-induced yawmoent can be explained by the
destroyers) through digital simulation studies.
concept of Ill-for-camberlint, which is equivalent to that of thewing Section. Aseries of coader rusas a&& by using the eqnctios of sway, rolla rudder nti0ns. Results idicate sustential coupling effects between yaw, roll, and rudder, which introduce ci-hanges in mamvering characteristics and reduce course stability in high-speed operation. These effects together with relatively small GH (which is typical for certain high-speed ships) produce large rolling motions in a seaway as frequently observed in actual oererations. Results of dinital simulations and captive model tests clearly indicate the major contributing factors to such excessive rolling notions at sea.
Due to the lack of available hydrodynamic date, no extensive digital simulation effort has previously b m in the area of maneuvering performancewith incluion lus. of roll motion effect which should have an iortamt iopact during high-speed operations. Recently, hi ed ships were extensively tested in the rotating-am facility with inclusion of roll notion effect. Test results clearly indicated fairly significant couplings between yaw-sway-roll-rudder ntions. One of the most important coupling term. i.e., the -oll-induced yaw-nt, cAm be explained by the concept of bill-form-cerline as described in the paper. Accordingly, a math-ntical node! was formiated on the basis of these exqerimental resuts co--,with analytical estimations, for a S00-ft-long hul form which is similar to that of high-speed comtamer sips or naval ships. A series of comuter rus wre made by using equations of yaw, sway. roll and rudder notion on a digital computer. Results indicated substantial coupling effects between yaw, sway, roll and rudder, which introdJce changes ha. in maneuvering and rolling behavior. For exle, coupling term introduce destabilizing effects on course stability and increase turning performance at high speeds. This tendency agrees with actual observation madea during ship trials and free-running model tests of a high-speed container ship. These coupling effects together with relatively small CGand cutopilot feedak rode Ersa rolling notions in operations in seaways. Effects of
NWIMOIUTIOU
shi is prceding at a high-speed in 2a Whenashpsprceigaa a aay, serious rolling notions are frequently observed in actual ship operations Aid in model behavior of Anmalous no,j. testing in waves rolling and steering was Clearly evident, for exaple. in full-scale tests of a high-speed cootaimer ship during cross-Atlantic and cross-Pacific operations Mist of the hig-peed container ships and naval shps have the following hull fom characterasticS which h-ave major impacts on ship performance. in carticular. maneuvering znd rolling behavior: (i) High speeds witn large VS5 ratio and relatively small (2) Fore-and-aft as metry (e.g.. with a tot bulb at the bow, see figure 1 EMO. 42-
_
_
_
I _
i= 1 -k -
+" +'I
-i
_____%
z_
+_ P -=
-=
:!______-______-__IF-ra., N
FIG-l.Bod
-
i
.N
.
Plans of Rerese tativet
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0
Mag Ship opeed
/I-.L~tdniAy-OtyDet-fl I
'
j
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ca - i 0-_vkum t
i~
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'
"
Container i
sSpeed
II
Lbr
x-.-
-+ =I +
+
lii ilt
=-o -
-wow
-
B "
I .I. o -i -ir+l
l
.
t. +fillFI
kU-
L- Og - i tia l A si oefCo
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FILL2 Lonitudinal Argutrt-y Oe to NaIl (Destroyer) Jfll4@t~~erlne]Fixed
Fig.
Crient;, t 4ofCoriaeAe oriaeAe in :,Nip
+-
yaw-sway-roll- rudder coupling on the possibility of yaw-roil instability were clearly demonstrated in simulation results. Such a possibility of rollcoupled yaw instzb;iity was observed during autopiloted-model tests of a high-speed ship under obliqbie-sea conditions.
(yaw)
K
1;,f
(Roll)
K
m(44ur) = V
(Sway)
ta(6-vr)
(Surge)
=X
I
HLL COIFIWRATIENS where M. K, Y, and X represent total hydrodynamic terms generated by ship motions. rudder and propeller. Hydrodynamic forces are expressed in terms of dimnsionless quantities, M'* KI. y9. and X! basedon non-dinensionalizing parameters (water density). u 'resultant ship velocity relative to the ater). and A. ie..-
A high-speed hull form to be -onsidered in this study includes the following c-racteristics as shown in the table belw: (i) High length-wbea ratio and relatively small &M fur high-speed operation. (2) Fore-and-aft asymmetry, which z mre po i for hig--sp ship 1th bow bulb.
(3) Relatively large rudder. Lenth.
500.00
.t ,
op gem at
,. 8
Draft, n
*
*
ft
60.0
ft
17.0
Rudder Area Ratio, Ar/ H
1/40
Block Coefficient, Cb
0.
Y
IaAL
22
,t.A
nydrodynamic coefficients vary with position. attitude., rudder angle, propeller revolution, and velocity of the ship; fr example, in the case of hydrojynamic yaw momt coefficient.
N'
$i(v
,6.y',.'J',n, nW(3) u,,&.'
we
The above-mentioned hull-form characteristics in-
y
traduce fairly substantial hydrodynamic coupling
vim V
effects bete ya-sway-roll-rudder motions. Figure 2 shows two curves which indicate the distance of C;C of the iocal sectional area f., the lo--t.dinal centerline at roll aniwle and 15 degrees. Te curves can -be re-&Idcred to -be hullerline, which is Cquivalent to caerlne for-c of the wino section. Fiqure 3 shows the other example of the bullform-camberline shown in the top of the figure. -b
#=r,,
nu&
y,
etc.
Finally. the following polynocials were obtaimed for predictions of ship dynic rat;ons: 2 r'+a rv 2 N'=a tarvt..r'+a. ;+aa.yrv,r 7 7v 3 1 yni 'nQ'ea,.a ta r03 4a !. 3a 13j. 12 0 9
her roll "gle is not zero, the caerline is not a straight ine, as Shown in these figures introducing hydrodynamic yaw met and side force. This trend is more prono-nced -because of the fore-
and-aft asm-mtry of hull form
,')
Y''bbvb r4b6i'4' +b 50 34 1 -4b r e4b L341) 1 3-+* 9
x'=cr
in. particitar,
Ic1
3 °
I
4
OI5Y
3
r'b v 'r'bsv,' b 7 1* 2 P-' 13 - 13 4"b
Ir9.cv"4tLcS',X1
dur 'ig high-speed operation.
Figure 4 shoes, for example, captive model
K'=dtd.vl.d r'a,6n.od,tie1
tdi
3=
test results of yar-roll coupling effect, indicati'j hydrodynmic yaw -oment to port introducted by roll angle to starboard. Similar yaw-roll coupling was evident in captive model test results obtained by L. lotter and also It. ianaoto L7.
(* 4)
ROLL-U CO- -ED INMASILITY Figure 6 shows roll extinction curves obtained in sinilati- runs on a straight course at 30 knots
BAIC EUATlmlS FOR YAV-SIAY-MOL-RfI-A)0ER 1-ilon
having G' v, is
On the basis of captive model test results together with analytical est;itions, an effort was node to formulate the equations of yaw-swayroll-rdder motions to represent realistic flnet=vering and rolling behavior of a higi&-speed ship. Figure 5 slwa the coordinate system used to define ship motions with major symbols which follow the -omenclatureused in previous papers. Longitudinal and transverse horizontal axes of the ship are represented by the x- and -t with origin fixed at the center of gravity. By refere'ce to t-se body axes, the equations of motion of a ship iv the ho zontal plane can be written i the form:
ular result was obtaied in the roll eqution oncoupled from yaw and sway eiquations. Tt roll rspo*nse Sho.n in the figure can be considered to he realistic on the basis of comparison with results obtained from model tests of a similar hi--speed ship shnin in the same figure. Wnmw roll extin-ction curves were obt -d in sinlation runts in equations of roll-yaw-sway coupled notions, an i;wpor.ant change in rolling and yawing behavior was taken place. Roll-yaw zouled instability was clearly indicated in test runs. Figure 7 shows time history of roll and yaw notions Starting on a straight curse at 30 krots with aft initial roll angle of 10 deg s. The roll extinction curve is approximately the sane as that sho n in It
-429-
f3
et
n
2 feet.Ti
paric
-
To-otboacd
To .1 ftord fA,-~
30d GM -3 it nd 2 1
0U
r
lGM-Si
0
VA
3
H0
50..50.
Time,s
Iii!Roll
ponse to nmol angle at 0 lOfj 11,g g~e
ic t5 IS
anwo~ot fu
onstroghtto-;-$j
LFIG.7. Roll-Yaw Instability (With 100 initial Disturbance)
I
S S
5
10
0's~
(0 2 ets
1
F
0
-
o
S,ne,
20
5304
20 55
20
Charactcistic RRllln
Ft
_.i
5
FIG.6.
-43-
1
Curve (Wih
~t'e~
previous figure at the initial portion of the run. However, subsequent roll and yaw motions are divergent, indicating rolI-yaw coupled instability, When an autopilot Is adequately Included in these .-sway-roll coupled motions, stability characteristics of the ship system are Improved as shown in Figure 8 where the above-mentioned roll-yaw instability is eilminated.
generated to deviate the ship heading to the port. Subsequently, the rudder is activated by the autopilot to the starboard to correct heading angle deviation. This starboard rudder angle produces the roll angle further to the starboard. Under this con dition, the possibility of instability exists in the ship systems. Accordingly, simulations were carried out under conditions: the following 500-ft-long ship was proceeding on a
~The PREDICTIONS OF RESPONSE TO TURNING AND Z-MANEUVERS 0 0 Figures 9 and 10 show response to 20 -20 Zmaneuver having GM of 3.0 and 25.0 feet. The approach speed Is 30 knots in the tests. A comparIson of heading angle response is shown in Figure 9 which clearly indicates a greater overshoot angle with GM at 3.0 feet relative to that with GM of 25.0 feet. It is clearly evident in this figure that course stability characteristics are deteriorated with reduction in GM. This significant effect of GM on ship response during Z-maneuver was fairly well confirmed by free-running model tests and actual ship trials In the case of a high-speed container ship. Figure 10 shows a substantial difference in rolling behavior with GM of 3 and 25 feet. it should be noted in this figure that the largest roll angle Is generated for the case of GM of 3.0 feet when the rudder angle is shifted to the other direction.This clearly Indicates that the rudder angle has a .unteractlng effect to outward heel angle during -teady turning.
straight course at an approach speed of 30 knots. A stepwise roll moment (e.g., due to beam wind from the port) was given to the ship. The magnitude o' the moment is equivalent to a statically generatea roll angle of 5 degrees. The subsequent dynamic response of the ship was computed with inclusion of the autopilot system, which can be represented as. 8d
a0 -d)
+ b'*1
6-
desired rudder angle
4
desired heading angle
where
d
a bi
yaw gain m
yaw-rate gain
Figures 13 and 14 show oscillatory mctions for the case where GM = 2 feet, yaw gain = 3, and yawrate gain 0. instability of the ship systEms is 0 clearly evident in the figures.
When GM is Increased to 3 feet, the stability
Figures It end 12 show computer-plotted turning and rolling characteristics in deep water. The major parameter changes in computer runs were as follows: 1. Rudder angle = 350 2. GM = 2.0', 3.01, 25.0'
characteristics are improved as shown in Figures 15 and 16. When the autopilot is refineC with addition of yaw-rate gain of 0.5, further improvement in the stability characteristics is shown in Figures 17 and 18. It should be noted here that the autopilot refinement substantially improved the rolling behavior as shown in these figures. The results mentioned in the above clearly indicate the possibility of instability due to a stepwlse disturbance. During actual operations In seaways, continuous disturbances are given to the ship due to wind and waves. Accordingly, even marginal yaw-roll-rudder instability can introduce serious rolling problems in seaways. Such difficulties have frequently been indicated in fullscale observations and model tests [IE2). Figure 19 shows, for example, the possibility of yaw instability obtained by J.F. Dalzell during model tests of a high-speed ship in waves 2].
Roll angle during enter-a-turn Is shown, for example, in Figure 12, which confirms very well previous full-scale observations. Figures 11 and 12 clearly show the effect of GM on turning and rolling characteristics. Substantial changes in maneuvering characteristics (i.e., reduction in course-keeping and increase in turning performance) are clearly evident in these figures with a decrease in GM. Recently, Hirano repo-ted similar changes in turning trajectory due to roll motion For a carcarrier [I11. YAW-SWAY-ROLL-RUDDER COUPLED MOTIONS WITH AUTOPILOT Roll-yaw coupled instability was clearly indicated in yaw-sway-roll coupled motions in the previous test runs. In actuae ship operations,the rudder Is actively used, introducing important effects on yaw-tway-roll motions. Let us consider the ship dynamic behavior under the follos:ing conditions: When a ship is proceeding on a straight course, a certain external disturbance (e.g., the roll moment due to beam wind) is given stepwise to the ship. When the ship is rolled to starboard, for example, due to beam wind from the port, an asymmetry is formed in the underwater portion of the hull as shown in the previous figure (i.e., Figure 2). As a result, hydrodynamic yaw moment is
_.
CONCLUDING REMARKS The purpose of this study was to develop mathematical equations of yaw, sway, roll and rudder to represent realistic maneuvering behavior of high-speed nave. 5hips, and subsequently to examine yawing an6 -olling motions during high-speed operations through a series of simulation runs. Based on recent captive-model test results of a high-speed ship configuration, important coupling effects between yaw, sway, roll and rudder motions were included in the mathematical model. Certain terms such as yaw moment due to roll angle were not adequately considered in previous studies in the area of maneuvering and seakeeping. it was
431-1 .-
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40
4
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,
31. /
Rtader angle - 35*
-
25t
3
**
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-
2.
FIH FI.9
ae
e
t0 k
FIG.~
~~Rol
Turin Trjctr
Repos .
W
O
21
3
I
0k
Soo
angluvr epos
GM 3I
angleRl
IJ~~RdL
10
20
Trnn
20
S..
ouisrotceoiI tpwate Ioafln.1.RBearirn 2A.I,of . aqom of 3 3 h.p SpeedU 30 or. t..5C0 t 4 S5M-21t
Tostothoril --
c~tonhofl.I. Bean
otl angle -
20
.o
To starboard .
*
. .4.
20 to
toa
!E
Z.
0
*
Ruol angle
*
2w
too
To port
t
I
olYwRudrCul I.3 To port
FIG.1. Roll-Yaw-Rudder Coupled Motion
I
1To
To-larboard
storbocrd
~
Rodder orte (-S)
lo
15
iZ7 -I.6
R5r0~I
Hem
20-vl
To
-.
__
T.rre,
.4
0
it0
10
5
0 0*
i
ItOO sttre rffw000 o O l~ta yw~t g*;r. of 3 2. Autopmroi 3. Slopspd U-30St. -5,f.S~ CM-3ff
poot
tvIt)
ToEar
rolYwRd r=G14
-433.~ddrCupe
CuldMto
Mto
I .0
2 Autovilot Iwah yawgoo of 3 ard4 I# gar. Of0.5
Iya5-'
I I
I~
Stoooed V- 30 k%1 500 ft
'3
[
-
4 Gm-3f1
Roll Ooql.
A
Rudder angle
50
To
100
15
20 3
50
300
350
fI
FIG.l7.
Roll-Yaw-Rudder Coupled Motion
To sta(Waord
10
10
Rudder angle 150000 IS
to 0
200
250
0C
300
30
Roll angle
To-,at
[ RuntNo.285
N
FIG.18. Roll-Yaw-Rudder Coupled Motion W5 * Wh/shils le'0l1. 0.75
WO--1-"Gedtnq Origle*90- (beam ms4)
Rudder
Tam. s
FIG.19. Test Record of Yaw,Roll~and Rudder of a Container Ship Model (6.29 ft long) in a Beam Seaindicating__ Yaw Instability and Coupling Between Yaw, Roll and Rudder
-434-
REFERENCES
found in this study that these terms have important impact on maneuvering and rolling behavior, introdicing the possibilities of instability and serious rolling problems during high-speed operations in seaways. The major findings obtained in this study are summarized as follows.
1. Taggart, R., "Anomalous Behavior of Merchant Ship Steering Systems," Marine Technology, Vol.7, No.2, 1970. 2.
(1) Roll angle introduces asymmetry of underwater portion of hull form relative to the longitudinal centerline, which generates yaw moment due to roll (i.e., N p).
3. Baitis, A.E., Meyers, W.G., and Applebee, T.R., "A Non-Aviation Data Base for Naval Ships," NSRDC-SPD-738-Ol, 1976.
(2) This roll-introduced yaw-moment can be explained by the hull-form-camberline,which is equivalent to that of the wing section. When roll angle is not zero, the camberline is not a straight line and introduces yawmoment and -ide force, in particular, at a high speed. (3) The roll-introduced yaw-moment generates a tendency to turn to one side when the ship is heeled (e.g., to turn to port when the ship is heeled to starboard). This moment contributes to possible inherent yaw iastability along with autopilot feedback and other coupling te;s such as K1 and K1 (i.e., roll-moment due to sideslip and rudder angle, respectly).
=has =
=
-
Eda, H., "Directional Stability and Control 8 of Ships in Restricted Channels," TSNAME, Vol. 7, 1971.
5.
Eda, H. and Crane, C.L., Jr., "Steering Characteristics of Ships in Calm Water and Waves," TSNAME, Vol.74, 1965.
6.
Eda, H., "Steering Control of Ships ir Waves," Davidson Laboratory Report 1205, June 1967. (Presented at the International Theoretical and Applied Mechanics Symposium in London, April 1972.) Hamamoto, M., 1lydrodynamic Derivatives for 0 rectional Stability of Ships in Follow ng Sees." J. of N.A. of Japan, Vol.133, 1973.
8.
Eda, Haruzo, "Low-Speed Controllability of Ships in Winds," J.of Ship Research, Vol.12, No.3, 1968.
(5) The possibility of yaw-roll instability exists for the ship system with autopilot during high-speed operations with small GM.
9.
Eda, H., "Course Stability, Turning Pe-rformance and Connect on Force of Barge Systems in Coastal Seaways," TSNAME, Vol.80, 1972.
(6) Refinement of the autopilot characteristics important effects on yawing and rolling behavior of the ship.
10.
Eda, H., "Ship Maneuvering Safety Studies," 8 TSNAME, Vol. 7, 1979.
11.
Hirano, M., 'i0n the Calculation Method of Sh.p Maneuvering Motion at Initial Design Stage," J. of S.N. of Japan, Vol.147, 1980.
(7) =
4.
7. (4) When GM is relatively small (which is the case for most high-speed ships), the above mentioned coupling terms can introduce severe rolling motions in a seaway. This in substantial rollwas clearly indicated ing motions during turning and Z-maneuvers,
=
Daizell, J.F. and Chiocco, g.J., 'Wave Loads in model of a Container Ship Running at Oblique Heading in Regular Waves," Technical Report SSC239, prepared for the Ship Structural Committee, 1973.
Serious rolling problems frequently observed during high-speed operation in waves can be due partly to inherent yawroll instability (or marginal stability).
NOMENCLATURE
it should be stated here that a possibility of the above-mentioned roll-rudder coupled yaw instability is only pronounced during high-speed operations with a small GM.
ACKNOWLEDGMENTS The author wishes to acknowledge the Office of Naval Research and the U.S, Maritime Administration for the research program that made the preparation of this paper possible. The author also wishes to thank Mr. J. F. Dalzell and Dr. A. Strumpf for their valuable discussions during various stages of this study.
A a,b B Fr Iz
= ) reference area (A yaw and yaw-rate gain constants ship beam Froude number (U/g7) moment of inertia referred to z-axis
K
hydrodynamic roll moment
1,m N n r tr U u v X
ship length and ship mass hydrodynamic yaw moment propeller revolutions per second yaw rate time constant of rudder in control system = ) ship speed (U Ju+v component of ship speed along x-axis component of ship speed along y-axis hydrodynamic force in x-axis direction propeller force along x-axis hydrodynamic force along y-axis rudder angle roll angle heading angle of ship
&Xp Y 6 p
_
4-4_-
Q
roll and heading angle in the case of with-
Discussion
out spray strips, and Fig.2 shows the same
N.Toki (MH It is my pleasure to have a chance to discuss this interesting topic. My discussion consists of two parts. At first, I would like to mention our experience similar to the one pointed out by the author. In Nagasaki Experimental Tank, we have noticed roll-instability of a semidisplazement type high speed boat, during its esistance and self-propulsion test in calm water. To be more in detail, sway-yaw restricted model with small GM value unexpectedly heeled without any additional heel moment, when its advance speed was higher than a certain-value. And, for further continuation of the experiments, we had to lower the center of gravity of the model. We tried to find out the paper which dealt with the instability phenomena of high speed crafts, and found several ones 1), 2) 3 , For example, Suhrbier2 ) reported that a free running model with small GM and high advance speed suddenly heeled and began turning without any steering actions. He called this phenomenon "Broaching". Another information given by these papers is that attaching a pair of spray strips on the fore hull is sometimes effective to imprcve the stability of rolling. However, we could find out no paper which dealt with this phenomenon as sway-roll-yaw coupled motion, Referring to these papers, we measured heel moment, sway force and yaw moment acting on the model, varying heel angle and advance speed. The same series of measurement were carried out in both cases; modeX with spray strips and without them, to investigate the effect of spray strips, Thus, we obtained the values of hydrodynamic coefficient of roll moment and roll-yaw, roll-3way coupling coefficients. Making use of these coefficients and estimating the others, we composed the linear equation of manoeuvring motion and made an attempt to simulate sway-roll-yaw coupled motion. Two examples of the results are presented in Figs.l and 2. Their principal items are shown below. Dimension of ship 51.5m Length of water line Breadth = 7.2m, Draft = 1.8m Displacement - 274.5ton, GM = 0.53m Advance speed = 44 knots Initial heel angle = 2* Initial heading angle = 0* Rudders are fixed at 0* Fig.l shows simulated time histories of
-436-
in the case of with spray strips. From comparison of these two results, it is found that attaching spray strips can be an effective countermeasure to stabilize sway-roll-yaw coupled motion. In connection with our experience, I would like to ask two questions about the author's paper. 1. The latter part of the computer run shown in the Fig.? reminds me of time histories obtained from pull-out manoeuvre tests of an unstable ship. So, I presume this ship has an unstable loop with certain height in the rate-of-turn versus helm angle characteristics as a part of the roll-yaw coupling effect. In this case the r' curves in both cases, with and without the consideration of roll-yaw coupling, might show the effect more clearly. If the author could kindly present r' - 6 curves or time history of yaw-rate in Fig.7, I would appreciate it very much. 2. As to Figs.13 - 16, my impressions are as follows. Generally speaking, swayroll-yaw-rudder system with an autopilot has two free oscillation modes. In the one mode, rolling would play the main part, and it would have shorter period and larger damping, as shown in Fig.8. In the other mode, sway-yaw-rudder system would play the main part, and it would have longer period and smaller damping. Bee~'ue the oscillations shown in Figs.13 - lb have longer period than that in Fig.8 and converge very slowly, they must be the latter free oscillation. Therefore, in these computer runs, rolling caused by stepwise disturbance activates yawing, at first. After the first stage, however, sway-yaw-rudder system plays the main part and begins free oscillation. And, as the reaction rolling is activated again. In this case, I can easily guess that adding the damping effect by way of positive yaw-rate gain in the autopilot improves the convergence of Free oscillation, May I understand your result in this way? REFERENCES 1) Ferguson, A.M. and Conn, J.F.C. The Effect of Forward Motion on the Transverse Stability of a Displacement Vessel Trans, IESS. ol,113, 1970 2) Suhrbier, K.R. : An Experimental Investigation on the Roll Stability of a Semi-Displacement Craft at Forward Speed RLI:A Symp. on Small Fast Warships and Security Vessels, 1978 3) Marwood, W.J. and Bailey, D. Transverse Stability of Round-Bottomed High-Speed Craft Underway NFL, Ship Rep. 98. 1968
0.0'
30[Without
4
Spray Strips
020
-
0
A
1simulation result in the case of Fig. I high speed craft without spray strips
With
Spray
Strips
(
10.2
_
0
-0.3 1
Fig - 2
Simulation result in the case of high speed craft wIt spray strips
.437-
_
_
_
__
-
=
_SA=
_
1_
_
Y.Takaishi SRI; I appreciate this author who has presented an interesting paper relating to both seakeeping and maneuvrability of ship. The rudder movement of auto-pilot system should be included to estimate lateral motions as rolling, yawing and swaying motions in oblique waves. Today we can estimate these responses having the same period as the encounter wave period by using the strip theory inclusive the rudder etfects(Takaishi 1976). The yawing motion has a natural frequency when the auto-pilot system is used. The coupled oscillations between roll-yawrudder illustrated in Figs. 13 and 14 show this natural frequency which is lower than that of rolling motion. This motion cokld not be estimated by the strip theory. Since this coupled motion will become significant for the transverse stability of such a high-speed vessel running in quartering or following seas where the encounter frequency of waves could approach to the natural period of yaw, the phenomena pointed out by this author should be attracted attention not only by the ship operator but
-
0.6
i
- i
'i
2"
A0%£
3S
______
- t o ' |
L4~02\ LA7coupled ,,be
I i,._ n- i
0 .0.8
A
had similar experience with roll-coupled yaw instability for a semaidisplacementyehi> speed boat, although there exists substantial difference in configurations between a displacement-type ship and a high-speed boat. His simulation results clearly indicate typical roll and yaw instability, which is Introduced by a small initial roll angle. It should noted here that this instability of the high-speed boat was due to roll-yaw coupling
I
J[a
Ir
--.
-
en
The author was encouraged to learn Mr. Toki of Mitsubishi Towing Tank
jthat
.
0.2 SIP) 2&S
1) Takaishi, Y. and Saruta, T., J. of Kansai SNA, No. 161, (1976), po. 25-32 2) Mori, N. and Mori, M., Report of SRI, Vol. 16, No. 2, (1979), pp. 1-12
H. H Eda d St,-, eesls
0.. .
3"
REFERENCES
-
3.33
cI 3 ca 0.5 O.-0Th
tr
-
Author's Reply
0 3/4
also by t 3 ship hydrodynamicists. This yawing moment due to heeling of ship will become large on the ship having large asymmetry of fore and aft body. Mori (1979) has shown the effect of heel angle upon the course stability of a high-speed container ship with the quadruple propellers having an remarkably asynt.etric ship body between fore and aft, i.e. the fore body having an extraordinary large bulbous bow with a fine water plane configuration and the aft body having rather flat and wide stern shape. The result shows that the turning radius to starboard due to heeling angle of 6.7 degrees to port is equivalent to that due to about 2 degrees rudder angle to the starboard, as is shown in Fig. A-1. The seakeeping tests of this model have been carried out at Ship Research Institute. The re-examination of the test results from the viewpoint proposed by the author would be interesting.
effect without feedback control of rudder. On the other hand, yaw instability of a high-speed ship mentioned in the paper was introduced by a combination of yawo-swayroll coupling effects along with feedback controlled rudder. His first question is whether the ship is dynamically stable or not on course, with and without consideration of roll-yaw coupling effect. It is clearly evident. for example, Ln the overshoot obtained in (Accordingly, it is unnecessary to go out for spiral maneuvers to obtain r'-8 curves.) A large overshoot shown in a simulation run with an actual
S_VW nvwp~
=
t0 r FnZ-maneuvers. Fig.- A r '-d Curve, Single rudder Fn=0.283
.438-
U
-GM value (i.e., 200 overshoot in 200-200
studies of high-speed container ships. As he points out, oscillations due to roll yaw-rudder coupling cannot be adequately treated by currently available stripwise theories. Accordingly roll-yaw-rudder coupled instability problems mentioned in the paper should be examined from the viewpoint of both seak"eping and maneuvering with inclusion of feedback controlled rudder, where yaw moment due to hull-form-camber3ine plays an important role. The author is encouraged once again to learn that similar effects of heel angle on turning and course-keeping characteristica were observed in recent free-running model tests of a multi-propeller high-speed ship, at The Ship Research Institute, which has a significant fore-and-aft asymmetry with relatively small GM. The author is looking forward to reviewing these model test results from the viewpoint of roll-yaw-rudder coupling effects. Finally, the author wishes to thank the discussers of his paper for their valuable contributions and references.
Z-maneuver with GM of 3 ft) indicates that the ship is inherently unstable. With a hypothetically large GM of 25 ft, the magnitude of overshoot is reduced, indicating the improved course stability characteristics. The author fully agrees with Mr. Toki's interpretation of the results shown in Figure 13 through 16, which illustrate the improvement of inherent yaw instability by adding damping effect. Whereas the refined auto-pilot is effective to improve overall yaw and roll stability characteristics, the author believes that the best solution under this condition is application of an antirolling fin stabilizer together with a refined auto-pilot, which should introduce an excellent overall system performance with less rolling, less steering, and less increased resistance, The author wishes to acknowledge the useful comments given by Dr. Takaishi, who has been actively engaged in seakeeping
--
I
_
_
-
49
I
I>
_,
II
_
_
I
U
=
IN _
_
_
.439
_
-4 _
_
_
-
~
= =
_
_
_
-21
_
_
Session V_
HULL FORM 3 -PROBLEMS
IN DIESIGN
-
Chairman Odo Krappinger Hamburg Ship Mode Basin Hamburg, Federal Republic of Gemny
Vertical Impact of a Disk on Compressible Fluid Surface
STA -h The
problem of the vertical impact of a disc on the surface a compressible fluid has been dealt with inofthis paper. On the basis of linearized assumption. the equation fulfilled by disturbance velocity fluid has been redud -of to a wavepotential equation, The wave equation has been solved with the aid of spheroidal wave functions. The hydrodynraic pressure, force and pressure impulse, force impulse, and their changes with time have been given analytically. The dependence of impact force on the mass and = radius of tisc, and on the content ofloityof gs -iboles contained in the liquid have been given as well--=
-
speed of sound in undisturbed water speed of sound in water-gas mixture d= e-ution (11), @#I-. is the tconstant. coefficient c , = decay coefficient ofand iact (40-)force in formulas (40)
(c)
function defined by equation (43), or 44) expansion coefficients, eqs. (27) =) l and (281 impact force acting force on disc F, =impulse of impact ipl o a fThe =maest od isc, o hertio asof inwater density of liquid to that of gas in the M. Mxt number
and degree 2n+l of the first, third kind respectively
r. (,
t
=
second, and
radius of disc =cl riacirlina is cyindric oor nondiaensional = angle spheroiI wave functions of:te first kind of order zero a d ge te 2n+o 2nime
veoct non=imtnssnal osicr/r* veiocity or disc v i" j= volum of ionid and gas in the nixtare ofofofliquid ",%= velocitv disc at insant isc and atafirst fsgas respecivel insetnt oof imac mass number ratio tsheroidal co-ordinates, fi .2 ratio of oflon~ the mass te of the gas of inthemixt-e to water the mass of the li d in t n oasfwe and a-mean density of the -- -- o -o gas . == density water of water P, density of of undisturbed air i the mixture ti
air 1 = and density of liquid in the water - aIr
_ixture
___
-
f
problem treated here is related to entry of flat-or blunt-nos underwater wea-ons (e.g. aerial torpedo, project depth charges. antisubmarine rissile. may alorlaeo ae Pac Mach nu--e be n = outward normal to the disc plane also relate .e to water air = impct pressure acting ese iwct pactsnaso ~sonars er' pen-t. Tfhesla-n.e i=oc pressure isp=themnat fluid on dis, of discoro.other water sensor surface a reasonable ptopei ressure in thee fluidc ofre approximation the initial stageVs of impact atmospeic pessure se s rfaeouof a flat-nosedtocylinder and flat-or bluntimpulsive pressure nosed body of revolution on water surface. t r ,, .ulsiveprsure s o wIt may be noted that ship slammi-u, *radial spheroidal wave seaplane landing, and splash landing of space functions of order It
=
_-
___
-
443
_
_
_
_
_
_
1
f
_
capsules usually have water contact configurations apnroximated by a disc better than by a 2-dimensional vlane. In all cases where the impact velocity of flat-or blunt-nosed bodies is not exceptionally sm--all- and indeed, it is generally never very small, it is necessary to take the fluid comressibility into account, or"zrwase, unrealistically large calculated values of i=act force and body acceleration may be obtained. However, to take the fluid cpressibility into account, -eans to add comlexity to an already complicated problem of unsteady flow with free surface. So far, m st water entr-y problems were treated without taking fluid co=ressibilitv into c onsideration. Egorov !I] studied the 2-dimensional -roblem of vertical copressible imtact. He is the first author to take .the change of velocity of the falling plate into account, and approached the solution of a wave eration by the method of separation uever, there see- to be some of variablesmistakes in his calculation which influenced the correctness of his conclusions. Ogilvie (21 studied the compressibility effects on and gave very interesting ship slaing. rasults for the case of 2-dimensional flat plate. Poouchkwv [3] studied the problem of the i'moact of a ladisc sr _xis* o a coxaressible fluid and ragid d ottn/4. solved it for the timec interval on the .or-chkov's impact force edepe-nds the velocity of disc during ipcting; latter, however, is left undetermined in his solution. The werk presented here by the archor was essentially completed in 1364, conseqzu--y he did not enjoy lthe rivilege of referring to Dr. Ogilvie's and except that due to.-S. pprs, T_1 5c_-ch an ify the proble andorder to S or-o v. to study the effects of body sass, body dimensions and degree of cowressibility on it t-e transient process of water i=act, The S-necessary to work with simple for. author therefore attempts to treat the basic nr--= -= vertical impact of a disc on a ressible fluid surface. The phenmena of arbeing trapped betwe the t allincplate a.. the water surface was studied eal y be some investigators. The author did not take tis int_-o acc-cnt however in his work in 1964, therefore, the vnrclem treated in the paper is equivalent to that of a disc ia ating on the water surface, and then suddenly rushed down with an initial .It is interesti-n to note that velocitV many authors discovered by em-xriment the waerin the form of t .s.aniq f a bles. no
I
, ass m. Let a ri-id disc of radius ioct eticaily on the water surface, impact, is meant: The velocity yertical _f te disc noants -er3_-ally ~'now rds, the lane of th dise is always horizontal, and the t -rt cWletelv fills water -nitially velocity of iial r haif-snaco -e Obviously the v-ecity of W disc is dhe
___... _________-_,_______-7 .
.
i
disc will attenuate with time. we shall see , later that tl:. rate of this attenuat mn depend on the fluid density, the acoustic iscrduanth h infud velocit disc mass. At the initial stage, the motions are of an impulse nature. As compared with inertial force, the gravity force may be neglected. Because of the impulse nature of the fluid motion, the effect of fluid viscosity on flow is of no significance in most part of florw field excent where the velocity gradient is large. Thus, the fluid may be regarded as nonviscous and the motion of fluid set up bly the body is consequently irrotational. For a liquid containing small gas bubbles, it is natural that its compressibility should be greater than that of the "solid' liquid, the amount of increase of compressibility may he characterized by the decrease in sound speed in the fluid. As a matter of fact, water irmediately beneath the free surface owing to its disturbance by wave motion and the motions of the ship, always contain more or less small amounts of gas babbles. velocity of We assume that the initial impact of the disc is not very small, but that it is smaller than the sound velocity in the cndisturbed fluid. In practice, this is 'ly too true. Therefore, the subject may be treated under linearized considerations.
zzr
F-'
Sketch of the problem and coordinate system intrical cy
Let a cylindrical coordinate systema(rt, 7) be fixed with respect to the undisturbed fluid, its origin being in the plane of the undisturbed free surface with a-axis 7he pointing vertically upwards (Fig.1). linearized velocity potential asy -be as a wave expressed in this coordinate sysr equain
ev
In-A
wi hPurz te t sundzeed3mundisturbed fl d-. Or. try introf-cing non-dimensional variables .,,, l) becomes fl. a-
a
,a
t .
= -'
_
(2)
-
Thtere is a relationship between tin- instantaneous imulsive pressure (i.e. pressure inpulse) r,and the velocity potential
According to (7), rewritten as
condition (5) may be
Besides these boundary conditions, potential On the free surface, pressure is finite and constant, i.e. fl=B, and the boundary condition on the free surface is thus =0
'P>=
function m-t also satisfy the initial conditions; =
(4)
~(9)
O,,£
z=O. V=O
n the body surface Un, .,ZzAf the normal =0, r1 Cj, r =t0 ; j;=" velocity of the fluid must be equal to that of bodysatsfythi itself. It is difficult exactly bonday cndiioe,to sncetheTherefore, the problem is reduced to one satisfy this boundary condition since the o idn a solution ou o wave aeeuto 2 equation (2) for finding of moment. the for unknown is mion of body However, making use of the property of the under conditions (8) and (9). When V is known, the hydrodynamic presiiacting process, i.e. the velocity change sure is found by is finite, while the displacement is infinitesimal, the boundary condition on the body surface may be considered to be satisfied on plane z.-i
Hence,
th. boundary
'.
condition on the body surface may be written as
..I,
=-
I
--
-force.
Since the ye scity is actually verticalvy downwards, the values of v and v are considered as negative in the following calculations. In addition to above conditions, ,here
SOLVIN
THE WAVE EQUATION FOR
Considering the impulsive property of the motion, we look for a solution of 4 in the form of an exponential time-decaying function. We set
is the boundary condition at infinity to be
satisfied. The boundary condition on free surface, permits analytical continuation of the potential function across the free surface into the entire upper half-space. Let
9(rf)-$(,z)--.$(,,-a)=--l(,r)
(1z)
where c is a constant to be decided. Substituting (11) into (2), one obtains a Helholtzis ecration:
-6)
(12)I
then we have
Conditions (4) and (8) become respec-
:,} _ (,L-'.%)tively =- f(r..--, .
-.>!. -=0 t=0
It is easily seen that if P and P' are points of symmetry with respect to the free sUr--,ze and lying, respectively, on the disc surface Z and its mirror image Z' then
(13U
Apparently, conditions (13) and (14) are also consistent with the initial conditions. $ must also satisfy the condition of Heflaolta's equatin at infinity 141;
Taking into account thepeometric prop-
F
Therefore, the Jetermination of the velocity potential in a en wh exterior to the surface £1E ,- is a Neumann Problem. The normal derivative M on t is given by boundary condition (5) Nwhie that on X' is defined by relation (7). Here X ls the lower wailof of thbody disc, and n is the exterior normal the surface.
Z th _
u
Integrating p-p over the entire surface of the disc, we obtain the total hydrodynamic 3.
a (..-
I
'Sthe1'flr dis,
~nIi
wll
erty of disc, I shall next anopt an oblate spheroidal coordinate system (ye.tJ)ich iscrelated to the cylindrical coordinate syston by the transforiation (Fig.2) ,.ir,=wsi(p+fga.P
o
=MS44--
uI!)
3
!+ i(2
.h
2)
and making use of the expression for Laplace operator 72 in orthogonal curvilinear coordinates, equation (12) becomes [ L (I -q) J -_+ -.-- w +1)
.-
-
(, v1+ r)] 4-0
(23)
r/4- M4k
Finally, the problem is reduced to that of solving eq,(23) under conditions (19)-
system s(21). where
,
E.. =snh'
.
.- li~i,
Hence
0 i~ -
.2
Fig. 24 Models 316, 6 and 12 Pressures
55K
I
3.
I
X/L
Fig. 23 and 24 present comparisons of and impact pressures between accelerations ! The preModel 316 and designs 6 and 12. dictions are for 4300 tons displacement and in each case are based on model test results. Since the model test program did not include the measurement of impact pressure, these data were calculated from the measured motions using the method outlined in Appendix A. The data are non-dimensionalized using values predicted for design 6 at 22 knots in seas of 12 ft significant wave height.
-
6
.0 .
Model 316 fall roughly where they were expected, close to design 12. Design 6 has a lower resistance £'i'r the whole speed rnge under consideration due to its higher Mi, lower CB and narrow transom. On the other hand, design 12, with the extreme 'V' bow and wide transom has geee bow idetrasomhasgenerally nd higher resistance than Model 316, except at speeds over 24 knots, where the wide transom adds to the beneficial effects of higher i)and
Z.5 2
I
''i
I
20
12
6
316
16
(3) The NRC Hull Form Series for Fast Surface Ships provides a valuable data base for making seakeeping and resistance trade-offs in frigate hull form (4)
Accelerations
-471-.
A method of hill design combining studies, analysis of the results of methodical series for both
seakeeping and resistance, and interpretation of test results with models of specific designs enables evaluation of designs to be made more quickly ane with Bore confidence than by use of any of these approaches alone.
7. Bailey, D., "The NPL High Speed Reund Bilge Displacement Hull Series", Royal Institution of Naval Architects, Marine Technology Memorandum No. 4, 1976. 8. Lindgren, A. and Williams, A., "Systematic Tests with Small Fast Displacement Vessels Including a Study of the influence of Spray Strips", SSPA Report No. 65, 1969. 9. Comstock, J.(Ed), Principles of Naval Architecture. published by the Society of Naval Architects and Marine Engineers, 1967. 10. Moor, D.I., "Effects on Performance in Still Water and Waves of Some Geometric Changes to the Form of a Large Twin-Screw Ship", Trans. SNAME, Vol. 78, 1970. 11. Moor, D.I. and Murdey, D.C., "Motions and Propulsion of Single Screw Models in Head Seas", Trans. RINA, Vol. 1010,
1968.
12. Bowden, B.S. and Neville, E.J., "Pitch and Heave in Irregular Head Seas. A Comparison of Experimental and Theoretical Value for 50 Hull Forms", NPL Ship Division Report TM401, 1974. 13. Loukakis, T.A. and Chryssostomidis, C, "Seakeeping Standard Series for Cruiser Stern Ships", Trans. SNAME 1975. 14.Aspects Sospodnetic, D. and Miles, M.,Wave ::Some of the Average Shape of
__
0 , Spectra at Station 'India' (590 19 W)",
International Symposium on the Dynamics of Marine Vehicles and Structures in Waves, London, 1974. 15. April Mac~ay, M. and Schmitke, R.T.,
~~"PHHS
- A FORTRAN Program for Ship Pitch.
'
Heave and Seakeeping Prediction", DREA
Technical Memorandum 78/B, April 1978. REFERENCES
i
Displc Sehmntke, R.T., "The Influencs ea Displacement, Bull Form, Appendages, Metacentric Height and Stabilization on Frigate Rolling in reuaSesNMEhp Rolingin -regular Seas", SHAME
Ship
Technical and Research Symposium, June 1980. 2. Whitten, .T. and 2.DT. Witte, ad Schmitke, Scmite, R.T., RT., "SHIPMO - A FORTRAN Program for Prediction of Ship Motions in Waves", DREA Technical osAPower Memorandum in review. 3. Gertler, N., "A Re-analysis of the ~19. Original Test Data for Taylor Standard Series", DTMB Report No. 806, 1952. 4. Grim, 0. and Kracht, A., "Winderstand, Propulsion, Bewegund and Beanspruchung Schneller Verdringungsfahrzeuge in Glattem, Wasser und in Regelmissigem Seegang", Institute Schiffbau der July Universitat Hamburg, fur Bericht NR.167, 1966.
16. Murdey, D.C., "Seakeeping of the NRC Hull Form Series: A Comparison Between Experiment and Theory", 19th American Towing Tank Conference, Ann Arbor, Michigan, July 1980. 17. Drummond, T.G., "A Program for the Comparative Estimation of Displacement Ship Resistance and Poer", DREA orkng Paper Rita ad Pr", D e beri19Ppe Schmitke, R.T., "A Computer Prowith 18. Limited Circulation, September 1976.3 gram for Fast Surface Ship Resistance and Estimation". DREA Technical Memorandum Ochi, M.K. and Hotter, L.E., P ci M and MotterL.E., Hull Responses for Ship Design", Trans. SNAME, Vol. 81, 1973. 20. Schmitke, R.T., "Improved Slamnfor Program", DREA TechnicalhMemorandum 79/A, Fbur 99
"Prediction of Slamming Characteristics and
21.
5. Yeh, H.Y.H., "Series 64 Resistance Experiments on High Speed Displacement Forms", Marine Technology, Vol. II, No. 3, 1965.in 1965.
Conolly, J.E., "Standards of Good
Seakeeping for Destroyers and Frigates in Head Seas", International Symposium on the Dynamics of Marine Vehicles and Structures Waves, London, April 1974.
6. Van~ater, R.B.Speend andinWvs BasPn. er P.R., Zubal, o.,High,
"Hydrodynamics Ships", Davidson Laboratory Report No. 876, 1961.Waves",
22.
LodApl194 Chuang, S.L.,
"Slamming Tests of
Three-Dimensional Models in Calm Water and
NSRDC Report 4095, September 1973.
1961. -472-
_
I4 APPENDIX A IMPACT PRESSURE CALCULATION
r
Thy calculation of pressures is based on the statistical impact formulation of Ochi and Hotter". The probable maximum Slamming pressure at a given hull section in h hours of ship operation time is given by"
U
8o I
rrC
Go.-
'h koRtn3Annkn RV 1 - Rexp(-.52/c ' 2) where cRX and o are rms relative motion
and velocity, respectively, i is draft and
k is the form factor. k Is evaluated using the method proposed by Conolly 2 : the hull section is regarded as a truncated wedge with deadrise angle 8, as defined in Fig. 25. and k is calculated on-ina Z on slamming of wedge-shaped 22 bodies . Specifically, the following empirical expression is used in this paper: kIc
+
3Ii
O.15
.2
Z
X/L
1l - exp(-5B)l(.57cot8),
Equivalent deadrise angles for a typical V-bow frigate are shown in Fig. 25. Note that for the results presented in this paper, h 1.
.43
Fig. 25 Equivalent Deadrise Angles F
N.K.8
where R i an estimtor of seakeeping rank which varies from the order of 1.0 for hulls with very voor sea-keeping qualities 0 or hulls with exof to the order seakeeping quaalitiesCeIlent
OT
rL-.urdev describe a frigate and Mr.Schmitke hull design process w.hich efg ctively utiof ~ -~ies lizes state-of-the-art techroargies in thear areas of seakeeping and calm-water resistance. !v own work atet o relate hull geometry to seakeeping and thence to synthesize seakeeping optimum hull geometries " 2, though exclusively analytical, is predicated upon broadly similar reasoning; and leads to similar results. There are, however, significant differences. I con-18 sider a larger set of potentially limiting responses, and take them to be equally important. Mv rationale for restricting the trend identification process to long-crested, head seas is very different; and requires subsequent validation in shortcrested seas and at oblique relative headI take the viewpoint that good seaings. keeping should accrue as a result of qualitatively similar hull configurations for all classes of displacement monohulls, and base much of my work for destroyer-type hulls on earlier results for dry cargo shin hulls. Finally, I relate seakeeping to a different set of hull parameters. The last difference cited appears mst important for a formal discussion. In concluding their description of the NRC Series, Mr.Schmitke and r.Murdey observe that, in the course of standard series hull form changes, parameters other than those explicitly varied are also modified; and that these incidental modifications may inThis is, in fluence the resilts obtain.ed fact, probably the greatest danger associIf ated with standard series techniques. the incidentally-modified parameters happen to be those which govern the output of the series, direct interpolations can lead to erroneous design decisions. For a hull of given size, analytical procedures indicate that vertical-plane seakeeping responses are essentially governed by the distributions of beam, draft, and sectional fullness over the length of the hull. The conventional, naval architectural parameters which I think provide a minimal description of these distributions are the waterplane coefficients forward and aft of amidships. Cwp and CwA; the draft-to-length ratio, T/L; the ratio of the distance between the forward perpendicular and the cut-up point to lenuth, c/L; and the vertical prismatic coefficients forward and aft of amidships, Cv7pp and CVpA. In my work on destroyrer-type hulls, construed to include frigates, these paraeters are related to seaeeping as
An indication of the validity of t-he usig the results tor c -be otain est ;.i -cSre.F=Fiur h o that the ao-..licable form coefficients t as o are appro-xima as foi S Desin CT , CF Cnn . 0.7-
mnears
=
R =8.442 + 45.104 Ct
-
10.078 CWA
378.465 (T;L) + 1.273 (cAL) - 23.501 CVpF - 15-875 CVpA
-
-474-
12 !8 24
0.68
C.97
0.62 0.68
0.66 0.7
0.54 0-6I
0.92
0.72
0.58
As I understand the design of the series, these coefficients should be fixed for the associated columns in Table I- Frrther, c/L appears, again from Figure 1 and the series description, to be fixed at about 0.6 for all of the designs. Finally, a value of TIL for each design can be oemouted from LV1/T and B/T as used to define the ser-es. With these definitions, seakeeping rank can be estimated for any design in the series. For instance, it equals 6.1 for the 'parent" Design 6In Reference 16, Mr-H.rdev gives percentage changes with respect to the parent in pitch ar heave both as measured and as cwmputed for a critical combination of operatina conditions. The following i exhibits the correlation between the differ ences reported in Reference 16 and those computed from the estimator- given above. It is as ou ld be expected for a va - d but primitive and rather generalized es.t There is applied to particular conditions. considerable scatter at midrange. but the response extrtes are indicated with reasonable accuracy. The NRC Series has overall waterplane coefficient as a defining parameter. Overall block coefficient is als, used, and the discussion implies that this parameter is considered an adequate measure of sectional fullness. Draft and length are used- in defining the series but are always explacitly related to bean. The authors coment on the relevance of forehvdy versus after body parameters, but never make them explicit and do not make a distinction between them in defining the series. I would suggest that they make the forebody/afterbody distribution of form coefficients an integral element of the series deiinition, and that they consider the possibility of using the parameters in the estimator cited above. The following table recasts Table I using the parameters 'f the estimator to order the series desia.s. TI-he nubers given parenthetically after each design number are the average of the percentage changes in pitch and-heave from Reference 16 and minus the estimated rank difference
S S
for wth hatdesin espet t thc paentDesigns In this form, the averages of the
design. R
hull-to-hull differences in pitch and heave show moderate coherency. The estimator results indicate that the poorest seakeeping hulls in the series (lower right in the table) have been tested, but that those with the best seakeeping (upper left in the table) have not. With respect to the designs estimated to have extreme seakeeping qualities, it should, however, be pointed out that both T/L = 0.028 and T/45 = 0.045 are outside the strict range of the rank estimation equation. It will obviously be of considerable relevance to determine whether or not Designs 12 and 7, 24 and 19, et al., have similar seakeepting characteristics. Quite aside from the foregoing differences of opinion as to the hydrodynamic basis of good seakeeping, one must inquire as to the success of the final product. In the case of Design FDES, it appears that Mr.Schmitke and Mr.Murdey have negotiated the treacherous waters of design tradeoffs with minimal compromise of their hydrodynamic principles. The level of detail provided regarding the design is inadequate for a rigorous assessment, but from the description provided, my perception is that
inorder of Average Change from Reference 16
6
'-
-
$-,1 !
-
--
0
4
O
15 1LUI
*a
&
"8..,
2
1
____1______
0
29
o
' D A 0
V
-2
-0
VM
E3eitch Theory Heave Experiment
-
of e
__61
-20
Theory T Heavo
AG C
t____
-10
Design FDES probably has an estimated seakeeping rank on the order of ten. Such being the case, it will be, ton-for-ton, among the best seakeeping hulls afloat.
0
Pn
I wish that I could say that I had done
24
__
20o
10
30
Response froReference 16
Ccrretion between NRC series
as well with my own efforts to support the hull design process.
results and rank estimator
The NRC series ordered by estimator parameters
,2
B
TC,
BT
T
L
0.80
CWF/CWA -PP/CVPA
238
150
0.80
0.74
3.74
0.68/0.92
0.68/0.92
0.62/0.86
C.62/0.86
0.66/0.54
0.72/0.58
0.72/0.58
0.78/0.62
5.20
0.028
10
-8.4)
4.20
0.032
11 (0
-'.9)
3.28
0.036
5.20
22
-6.3)
-3.0)
4 ,
23 (-8.1. -4.8)
5 (*1 -1.5)
12 (-11.9, -5.4)
24 (-4.3. -3.3)
6 (0, o)
0.036
7 (A, -5.4)
19 (A, -3.3)
4.20
0.040
8 (,
-3.8)
20 (+5.4. -1.8)
3.28
0.045
9 (*, -2.0)
21 (+12.4, -0.1)
1 (, 2 3 (,
,
16
-1.0)
17 (+0.8, +0.5)1 18 (+7.6, +2.0)
0.0)
13 (*,+2.0)
+1.5)
14 (+15.2. +3.6)
+3.4)
15 (+22.7
+5.4)
_
* Value Not Available
___-475.
T __-
_
_
__
C.M. Lee(DTNSRDQ The subject treated in this paper is one of the eventual goals of naval ship designers and opens a new challenging avenue for ship hydrodynamicists. With a short duration of my experience in the parametric studies for the seakeeping hull designs for conventional catamarans, I can fully appreciate the extent nf endeavors involved in studies such as thob- iresented here. Contrary to ones investment of a great deal of time and effort, a subject like this would always induce diffe.ing opinions from the people who are also engaged in similar work. One of the major arguing points would be how one should define the seakeeping qualities and predict these qualities. For example, should the severity of slamming be represented by a point pressure or the integrated pressures over a certain unit area of the hull ? Or, even a question such as "Is th peak intensity of the slamming pressure during unit time of operation more important than the frequenc, of occurrence of the slamming with the lesser intensity in the peak pressure ?" could be agrued on. Again, even for the slamming alone, the criterion can depend upon whether the local impact or the broad impact causing the hull girder vibration should be our major concern, As described briefly above, unless we narrow down the criteria involved in seakeeping to a manageable level (may be an impossible task), the t-'ade-offs between the seakeeping and resistance could remain a formidable task to accomplish. However, a paper such as this seems to me very encouraging since it provides not only valuable guidelines for the future investigations in similar nature of works but also amplifies the necessity for changing the traditional design concept of "the resistance first and the seakeeping later." I would like to congratulate the authors for the pioneering and inspiring work they have accomplished.
This paper is significant, not only for providing useful guidance for the design of frigates with good seakeeping performance, but for developing a systematic way to show graphically the effect of important hull parameters on performance. pppreduction The paper stresses the importance of length as a seakeeping parameter and gives par-
Fig.9, which F4iports to show the affect of CB alone must involve a change in length, along with other dimensions , as well, to (Fig.14 makes use of® hold A constant. instead of L 2/BT.) This writer long ago recommended the use of L/V /3 as a parameter*, since in design one is coneerned mainly with ships of different lengths to meet the same requirements, and hence to have the same (approx.) displacement. Another consideration in that earlier work, not considered in this paper, was the influence of natural periods of pitch and heave on performance. It was shown* that the generally favorable effect of length, as ieported here, is the result of the influence of length on pitching period - provided the ship is in the subcritical range of operation. -TW paper also stresses the favorable effect of increasing CW on motions. Again as pointed out in my earlier work*-this advantage (greater damping) can be related to its favorable effect on natural periods. Hence, it would be of interest to make a supplementary study of the natural periods of all models tested. If for any reason it is difficult to determine these values experimentally, then they can be established by plotting motion amplitudes vs. speed with constant wave length (and height) and noting the encounter frequency corresponding to the peak response. The theoretical parametric study, showing results for irregular seas, is interesting in that it leads to similar conclusions as the model tests. But again no consideration is given to the effect of natural frequencies, and it would be of interest to plot the sea spectrum and the RAO's on a frequency base for a number of cases as a means of clarifying and explaining the trends shown, for example, in Fig.14. The summary under "Over-All Assessment" (p.467) seems to be an excellent guide for frigate designers. Exception might be taken only to the statement under (5) that the choice of @"should ....be dictated primarily by powering and arrangement considerations", since it follows an admission that "increases in length will always improve seakeeping". The only from the seaobvious reason for caution keeping viewpoint in increasing length seems to be to avoid reducing draft to the point that bow emergence and slamming become serious. Fig.14 shows an increase in bow pressures with increase in B/T, and hence in draft, but not with reduction in *
ticular attention to the problem of isolat-
ing it from other parameters. The authors favor the slenderness parameter L 2 /BT as a basis for plotting, since it does not involve CB. This parameter has merits, and in Fig.7 trends of length are easily shown. But since both A and CB are held constant, the figure could just as well have been plotted against®. Furthermore,
* E.V.Lewis, "Ship Speeds in Irregular Seas"
Trans. SNAME,(1955) **Hamlin and Compton, "Evaluating the Seakeeping Performance of Destroyer-type Ships in the North Atlantic",Marine Technology,Jan. (1970) -476-
_
T
-
-
-
In conclusion, this is a valuable paper that deserves a great deal of careful study. It may well be that the results have greater generality than the authors suggest.
May I add my congratulations to the authors on their very excellent paper. Such efforts are very important to ship designers who are faced with maintaining mission effectiveness (good seakeeping qualities) while attempting to improve fuel efficiency, especially at cruising speeds. The section which reviews the sources of information is a good "state-of-the-art" summary. What the overall assessment section points out, however, is that some of the most important parameters such as Cwp and CWA are frequently not tabulated in comparative seakeeping studies, including this one. K.Bales of DTNSRDC has suggested that Cvp F and CVPA are also significant for good seakeeping performance along with the after cut up ratio C/L. I would like to request that the authors supply an additional table which lists these important parameters for each hull form.
R eply Authior'sRel
G. Beflone (Canreu Navali Mundt) As we had available Dr.Schmitke's seakeeping calculation program, particularly fit for slender, high speed vessels, and frigates are our company's main production, we have carried out a systematic analysis similar to the one which has been described by the author. Here I want to show some of the results we obtained, most of which are in good agreement with those obtained by Dr. Schmitke. We checked the parameters indicated in Table 1, together with the range of their variation. The study has been divided into 2 phases, of which only the first (regarding CWL , C B , L/A /3 ) has been so far completed. In Tables 2, 3 and 4 indications about the basic form hull, which has generated all the others, the sea state representation employed in the computation, and the criteria which have allowed the formulation of an "effectiveness index" for the different examined solutions, are reported. For the different sea states, the above said criteria, have led to the quantification of a "maximum sustainable speed" which has been considered as the index representative of the good seakeeping behaviour of the different hulls checked. In Table 5 these speeds are shown for sea state 6 (for sea state 5 similar trends have been observed, while for lower sea states no limitations to maximum speed were found). As regards the conclusions that can be drawn from this study, only two points
_
difft from the observations made by the author: (a) Vertical accelerations at bow seem to us strongly affect the seakeeping behaviour of a ship and create the necessity of a speed reduction before the probability of slamming gets its upper limit. (b) The variations of parameters CB and L/A' /3 , according to our computations, have nearly the same effect on seakeeping. (The author,on the contrary says that equivalent percentage change in C B has far greater effect). Therefore the questions are the following: (1) Is it possible that these differences are due to the fact that we have chosen the probability of slamming at bow as a restrictive criterion, instead of the intensity of the slamming impact pressure ? (2) We would extend our parametric study to the longitudinal distribution of weights on board (pitch gyration radius Kyy), but the author, in this and previous papers states that the influence of this parameter is negligible. Could he explain more in detail what led him to this conviction ? To my opinion, based only on an impression, it should be a rather important element.
We thank Mr.Bales for his discussion, which is based on his own extensive work on this subject. We agree with him that it would be desirable to include more parameters in the series. However, we have already included four parameters, plus bow and stern interchanges, and this makes a total of ninetysix models. To include more parameters, although desirable, is quite impractical. We would like to add that before embarking on the series, we performed a comprehensive analytical study of seakeeping to provide guidance for experiment design. In this study, we investigated approximately twelve hull form parameters, of which the most important turned out to be L 2 /BT, B/T and CW. CB (or Cp) was found to be of secondary seakeeping importance, but was included in the series from resistance considerations. Mr.Bales mentions the possibility that direct interpolation of the series data may be misleading. In this context, we are happy to report that for model 316, the series interpolation agrees very well indeed with the model test data, for both resistance and seakeeping. We had intended originally to include these comparisons in the paper, but space was insufficient. We agree that CWF and CvpF are impor-
-477.
o
-.-
_
__
I
The effects of varying the wave period in relation to the RAO's have been studied, and the trends of performance found to be similar over relevant ranges of ship length and wave period. For this reason simple averages over wave period were used in the analysis. The conclusion that length should be selected primarily in the basis of calm water performance objectives rather than seakeeping was made directly from the analyses described in the paper, as shown for example, in Figures 13 and 14. We agree with Prof.Lewis that major increases inawill, at constant disriacement, lead to shallow draft for which bow emergence may offset the benefit of smaller -otions, but this does not occur for the changes of length considered in this piper.
tant seakeeping parameters. In fact, we have specifically said so for CVF. As regards CVPF, our statement that "best seakeeping is obtained with a hull combining high CWF with low CB" amounts to much the same thing as "keep Cvpy low". This is roughly the same message as one obtains from Mr.Bales's work. With regard to Mr.Bales's table, we are happy to say that testing of the models with B/T=5.2 is in hand. However, we do not have the same expectations as Mr. Bales with regard to the seakeeping pertormance of these models. While the trend of increasing B/T is to reduce motions and accelerations, it is to increase slamming pressure. In our view, the latter will outweigh the former. However, time will tell. Finally, we reiterate our gratitude to Mr.Bales for his comments and note that although our path is somewhat different from his in arri, .ng at a final hull form, we achieve very similar results.
Prof.Johnson asks for more data defining the hull forms. This paper is intended to give an outline of a particular approach to providing design data on seakeeping, and the data relevant to this approach are given. Other approaches will require different data, CVpF and CVPA used by Mr.Bales being but two examples, and it was not possible in this paper to give all the possibly useful numerical data. However he body plans, Fig.l, are given, and thes may be used to derive other coefficients, as has been done by Mr.Bales in his contribution.
We thank Dr.Lee for his kind words, As he points out, the selection of seakeeping criteria is not a simple task and, for particular applications, may result in a rather more extensive ;et than we have used in the paper. Howeve!r, we ciphasize that in frigate hull form design, the designer should, at a minimum, address slamming, deck wetness and vertical accelerations. Slamming may be treated in several ways, but it is our firm opinion that slamming severity ust be included. In paper, we have used most probable maximum slamming pressure, but on other occasions we have used slamming force. The latter is probably more satisfying intuitively, However, it is our experience in frigate studies that either criterion will push in the same direction, the ship designer with ful waterplanes is to V-bows that forward.
We thank Mr.Bellone for presenting results of related studies which he and h s h of re e erformed reus colleagues have recently performed. With regard to his query regarding seakeeping criteria, it is our opinion that his slamming criterion is not sufficiently stringent whereas the opposite is true of his vertical acceleration criterion. For evample, in frigate studies we have found the following criteria to work reasc-nably well:
We agree with Prof. Lewis that in Fig. 7 L2/BT may be replaced by®. However, similar data will be available for a different value of C, and in analysing the series as a whole L2 /BT is preferred. As mentioned in the pal-er, the cha.ige of CE shown in Fig.9 does irvolve a chazge of length from 394 to 383 ft. This change was shown to have little effect on the motions nor on the derived trends with CB . The importance of natural periods of pitch and heave is recognised. However, in this paper the authors did not take them into account directly, since they are difficult to estimate at the early design isthat changes in the stages. It is believed parameters selected imply changes in the natural periods. Furthermore, natural periods are greatly affected by weiqht distribution, which is not, within practical limits, dependent on hull form.
significant vertical accelaration at 0.2L = C.4g most probable maximum slamming pressure 40psi mam most probable maximum slamming lorce 0.15t The latter two are evaluated for a period of one hour. As to his query retarding pitch ra&ius of gyration, we did not include it in our the aship andrst econdsecause satde form it is not hull dorbecause studies first pazamnter and second because the ship designer hau little control over it. We do not mean to imply that its influence on ship motions is negligible, and we would certainly favour further work on this topic.
-478-
&
Optmizng heSeakeeping Performance of Destroyer-Type Hulls Nathan K. Bales D~~BZ-W TaliOf N'VzaI SIN9 Rese-arcth ana, Develpmer Center 8eesa. Matrd~ U SA
The Seakeeping Optim~um Hull in Waves: A Perfect Pit
-479;-
R
A model which relates ship hull geometry to an index of seakeeping merit is developed. This model is quantified for destroyer-type hulls of specified displacemont in long-crested, head srCs. The quantified model is validated by using it to define a destroyer-type hull with exceptional seakeeping qualities. Necessary conditions are demonstrated for generalizing the model to other displacements and headings; and for applying 4t under circumstances involving specific, mlssion-related criteria. Hypothesizing the exiztence of sufficient conditions where necessary conditions were demonstrated, the model is rewritten in a manner suitable for u3e in optimizing seakeeping performance uncer rather arbitrary constraints on hull geometry. It is shown that the optimization can be accomplished using non-linear programming techniques, and that appreciable improvements in seakeeping performance can be realized even when highly restrictive constraints are imposed. The limitations associated with the results obtained are discussed. It is concluded that these results provide a viable basis 'or earlydesign synthesis of destroyer-type hulls with superior seakeeping performance.
A A
V
Seakeeping rank Estimated value of RI
R
ABSTRACT
Significant single amplitude of vertical-plane ship-to-wave relative motion at Station 0 Significant single amplitude 20vert-cal-a o r-/3 eat shi-towv Sti ow 2 mtiione elave r Second(s) single mplitude of absolute vertical motion at Station 20
(r11 3 )0 (r
s
(si3)15 1 1 ( 1/3)0 T
Local sectional area Waterplane area aft uf amid-
Significant single amplitude of absolute vertical velocity at Station 15 s f s amplitude Signifian singl apie of absolute vertical accelera tion at Station 0 Draft amidships Modal wave period
(Tw)0 t
Metric ton(s) or local draft
V x.
Ship speed Underwater hull form parameter
z1 / 3
Significant single amplitude of heave
NC 1N3 A
Significant single amplitude of heave acceleration Ship displacement Significant wave height
(yw1 3
single amplitude
ships waterplane area forward of amidships
0significant 1/3 pitch
AX
Maximum of A Constant
P a
Linear ratio Density of salt water Standard deviation
B
Beam amidships
VA
Displaced volume aft of amidships
b
Local beam Vertical prismatic coefficient aft of amidships Vertical prismatic coefficient forward of amidships raterplane coefficient aft of amidships
VF
Displaced volume forward of amidships
C
CWA
1
INTRODUCTION
Seakeeping came of age as a discipline of applied hydrodynamics in the mid-1950's with the emergence of strip theory and
amidships
(C c
3
linear superposition. Subsequent advances and refinements led to the development of Slaming constant at Station 3 technologies which are ustf'J in the later stages of the ship design These from Station 0 totechnologies are now beingprocess. introduced into point the earlier stages of ship design and into
Fn
Froude number
L
Ship length between perpendiculars Meter(s) umber of slams per hour at Station 3
ship operations. There is, however, one critical need in the early stages of ship design which cannot be directly addressed by our mature technologies. That need is for a means of synthesizing hull geometries leading to superior seakeeping qualities. The need for a synthesis technology is widely recognized. Papers attempting
Probability of bottom slamming at Station 3
to develop relationships between hull geometry and seakeeping. the fundamental prob-
m (Ns) (P )
3
-480.
WS
-
a
lam associated with development of such a technology, have been appearing in the literature for two decades. These, however, have had little or no impact on the design process. Within the past few years, there has been a flurry of activity attempting to develop surface combatant hull forms with superior seakeeping qualities. For these ships, increasingly sophisticated weapons systems are demanding more stable platforms at the same time that economic pressures are demanding smaller ships. Thus, the payoff associated with good seakeeping is very high for surface combatants. The author has been sporadically involved with work in the area under discussion since the mid-1960's. His early
2
efforts failed in the sense of having little
or no impact on the ship design process. In the 1977-78 time frame, he enjoyed partial success with two efforts involving particular surface combatant designs. During this same time frame, he undertook a more generalized effort to relate seakeeping
to the hull characteristics of destroyertype ships. The latter effort was, in view of the historical difficulty of the problem, remarkably successful. Accordingly, the results thereof were used to develop a synthesisother t o whand, technology which is now being applied to the design of United States Navy ships. The development of the model relating hull form to seakeeping and of the associated synthe.sis technology are described hereinafter.
-
2. DEVELOPMENT OF A MODEL RELATING S.KEEPING TO HULL FORM
of a general model. The next two steps were to quantify the seakeeping index used for the model and its general-form coefficients. The quantified model then had to be validated. Finally, the model had to be generalized, i.e., it had to be shown that some of the specific assumptions made to develop and quantify the model were not necessarily required. Each step is described in the sections which follow. 2.1 The General Model To begin, let us postulate the existence of a comparative measure of seakeeping performance in head waves: say R for seakeeping "rank". Further postulate the existence of a small set of underwater hull form parameters, say x., i = 1, 2, ... , n, which effectively govern R for a homogeneous class of hull geometries. Then our immediate objective is to define R, the x. 's, and the functional relationship b tween them, i.e., to write R = f(x , x2 ... , xn) (1) in explicit Iterms. For R we need a robust, criteria-free index which is not dependent on specific details such as ship subsystems or the the selected index must ultimately be valid for a wide range of specific criteria and dependencies. This dilemma drives us to adopt an equal distribution mustgnopute astoa set ofetical We must compute vertical plane ship responses which is comprehensive in the sense that it includes all such responses upon which criteria are likely to be based. These computations must be promdfrcmrhnierne performed for comprehensive ranges of fsi ship speed and of sea condition. The response statistics resulting from the computations
The fundamental premise of the modeling effort was that a meaningful, comparative index of overall seakeeping performance could be defined and thence quantitatively For of conventionalaveraged. a small number related hull foto arae It wa furter the averaging process, all responses, ship aul m tha hte In an ther speeds, and sea conditions are taken to be assumed that both the index and the relacf It is the absence important. equally quantified adequately be could tionship ~any weighting or prioritizing in the final headsea. Th imlicaion using analytically-based ofaverage any results for longwhich gives rise "equalfnal wihi rit to thethe crested, headdistribtion of ignorance" descriptor. bethisadequately iont are that rolling motion can The basically philosophical definition ~of seakeeping rank just given is considered be aequtel cotroledby subsequent appendage design, and that coupling effects ofeaepnrnkjsginiscsdrd from the lateral modes at oblique relative adequate for development of the general short-crested seas sa will and/or in shrtcese l model. Specific definitions are dependent upon the ship class considered. Such a not significantly alter trends identified under the relatively simple conditions definition will be supplied for destroyerevaluated. Finally, it was assumed that type hulls when quantification of the model is discussed. only the characteristics of the underwater I Itisssed. is now in order to focus on selechull had to be considered. The implication of the underwater hull form parameters, tions of this assumption are that deck x. in the notation of (1), to be used for wetness can he limited to an acceptable tAe model. To make these selections, the level by subsequent design of the aboveauthor relied heavily on his earlier work water b w, and that abov'-water hull effects with analytically-defined seakeeping stan
.
Sheadings
=
will not distort trends
.entified on the
basis of underwater hull geometry. These hypotheses were formulated on the basis of
dard caries.
This work is exemplified by
his 1970 paper with Cummins*.
past experience, relevant literature, and
I
unabashed intuition. The approach which evolved included five basic steps. First was development
*
Z-481-
A complete listing of references is given on page 503
[
The a and Cumznins series is based on the fact that a viable approximation to the vertical plane responses of a ship antong waves can be obtained using a Lewis section representation of the hull. This concept is equally applicable to the problem at hand. Though a detailed mathematical representation of the variation of the Lewis section parameters over ship length, such as used in Reference 1, seemed - o complex for the present application, i. appeared possible that a carefully selected setdefine of hulltheform parameters might uner waterii.,s, effectively effetvelyefies te wseia funerwater profiles, and sectional fullness characteristics of a homogeneous class of ships, ships. Another insight gleaned from the standard series work was that it is very desirable to treat forebody and afterbody characteristics separately as the former have a greater impact on vertical plane responses than do the latter. The waterplane coefficient forward of amidships was found to be especially prominent. Increases in this parameter invariably improved seakeeping, and the magnitudes of the response changes which it caused exceeded those associated with any of the other coefficients considered. Finally, the standard series provided some insight into the comparative seakeeping characteristics of hulls with and without transoms. It was found that hulls without transoms generally exhibited seakeeping characteristics superior to nominally equivalent hulls with transoms. These differences were found to be attributable to the typical differences between the afterbody, underwater profiles of the two types of ships rather than to the transom itself, Parameters for the model were selected on the basis of the foregoing comments and of such other knowledge of the influence of hull form on seakeeping as the author could bring to bear. Appendix A provides an abbreviated rationale for each selection. Here, the selected parameters will be identified and briefly commented upon. A total of six parameters were selected for the model.
incredsilq Cw, tion forei
CWA,
nT
c/L and wntVJF
A
and
rogcCWF was considered the strongest in the set. That for improvement with increasing CpA was weakest. With the foregoing,
e
(1) can be written
, T/L, c/L, C C (2 ) WA VPF,CvPA) and it remains only to define the nature of the functional relationship to be used. I eiigti eainhp h In again defining thisheavily relationship, te author relied on his earlier R
f(CW,
seakeeping standard series work. A ruled surface function was used for interpolations over the standard series data base. Such a function consists of the algebraic sum of a constant, a linear term in each independent variable, and all of the linear, interaction terms which can be formed from the set of independent variables under consideration. Thus, a ruled surface equation for the six-parameter model now under consideration would have 64 terms. This was considered to be too complex a function for the present effort. A feature of the ruled surface function is that it reduces to a simple, linear equation when only one of its parameters is varied. In the course of the series work, this led to an exploration of using only the constant and the single-variable terms from the ruled surface equation to approximate the results of the latter when more than one parameter was chanqed. The exploration indicated that the linear approximation was correct in a qualitative sense whenever appreciable changes in the dependent variable were involved. The qualitative results, though erroneous, appeared to provide an adequate bases for tradeoff decisions. In view of this, it was decided to employ a simple, linear model, i.e., to write (2) as R=ao+aI(CwF)+a
2
_
(CWA)+a 3 (T/L)+a 4 (c/L)
+a 5 (CvPF)+a 6(C VPA)
(3)
They are: where a.,
1. Waterplane coefficient forward of amidships, CWF; 2. Waterplane coefficient aft of amidships, CWA: 3. Draft-to-length ratio, T/L, where T is draft and L is ihip length; 4. C,,t-up ratio, c/L, where c is the uistance from the forward perpendicular to the cut-up point;
i = 0,1,2, ...
,
6, are constants
to be d~termined. 2.2 Quantification Equation (3) was quantified using a data base ccnsisting of the geometric characteristics and seakeeping responses of an ad hoc selection of destroyer-type hull designs. A "destroyer-type hull" is taken to be a rather fine, naval hull intended for high-speed operation and a warfare mission such as antisubmarine or antiaircraft warfare. Classes of ships which satisfy this definition are frigates, destroyers, and light cruisers. In the preceding material, the idea of a "homogeneous" class of hulls was
5. Vertical prismatic coefficient forward of amidships, CVPF; and 6- Vertical prismatic coefficient aft * of amiuships, C VPA . Seakeeping was projected to improve with -482.
J
AW
invoked on several occasions.
responses in Bretschneider wave spectra with
The func-
F
i
base needed to quantify the general model relating hull geometry to se4keeping was developed, It was decided to normalize on the basis of displacement. There were three reasons for selecting displacement in preference to length as the normalizing factor, First, the author's past results indicate that displacement is, for realistic ship forms, better correlated with seakeeping performance than is ship length. Second, displacement is a more fundamental metric cf ship cost than is ship length. Third, and strongly tied to the second reason, is decisions as to ship in terms of displacement. size are madethatmosteary-deignof Twenty destroyer-type hulls were chosen for the data base. Care was taken to include hulls representative of a wide scope Care was also taken to of design practice. avoid inclusion of pathological hulls judged to be unbuildable or unlikely to be built. Each hull selected was normalized to a displacement, 4300t, considered to be typical of the class under consideration, The normalization was performed by multiplying the linear dimensions of each hull by the cube root of the ratio of 4300t to its as-designed displacement. Table 1 characterizes the 20 displacement-normalized hulls. Both the parameters included in the model relating hull form to seakeeping and the overall dimensions of the hulls are included in this tabulation. The symbol B represents beam amidships. All other notation in Table 1 is as previously introduced. Based on the general definition of seakeeping rank (R) given previously, it
numbers selected yield speeds covering a 5 to 30 knot range for each hull in the data base. Appendix B provides additional comments on the selection of operating conditions. The computations were performed using an upgraded version of the linear strip theory computer program developed by Frank and Salvesen 2 . A 20-station, close-fit representation was ased for each hull. The radius of gyration in pitch was taken equal to 0.25 L for all of the hulls. These computations produced 200 response statistics (5 modal wave periods x 5 Froude numbers x 8 responses) for each
the 20, data-base hulls.
Unweighted
averaging over modal period and Froude number, as advocated in the general definition of R, reduced this data set to the 20 x 8 matrix exhibited in Table 2. In represents significant this table, ' wave heiaht, and is used to normalize all of the simple, linear responses. These responses are in terms of significant single amplitudes (subscript 1/3), and are further subscripted, when required, by the applicable Station number. General notation used is 0 for pitch, z for heave, r for relative motion, and s for absolute motion. The response statistic designated (C ) in Table 2 is related to the piobabils 3 ity of occurrence of bottom slamming at Station 3, say (Ps)3 , by (P ) s 3
exp'-2(C ) /[(I )i] w'1/3 3
(4)
It may then be described as a slar-'ng in-
was decided (see Appendix B for rationale)
is cidence parameter. Appendix B de (Cs) using threshold relative veloi±ty as s defined by Ochi3. Let it be notei that (Ps) 3 is inversely proportional to (Cs)3 -
to compute eight seakeeping responses for each of the data-base hulls. The responses selected were pitch, heave, ship-to-wave relative motion at Stations 0 and 20*, bottom slamming at Station 3, absolute
Hence, in contrast to the other resp"'nses
vertical acceleration at Station 0, heave acceloration, and absolute vertical motion at Station 20. In two particulars, this set of responses fails to support the general definition of R. Heave was included, not because it is a criterion response, but because it seemed inadvisable to omit half of the modal motions relevant to the problem. Absolute vertical velocity, a criterion response for helicopter landings, was omitted. The omitted response will be utilized subsequently in the context of gener-
alizing the model. of R, it o i
I
modal periods, (Tw) , frcm 6 to 14s in twosecond incrementswaRd for Froude number, F , from 0.05 in increments of 0.10 to 0.45. CRaracterizing the wave spectra on the basis because alone is wave period modal of are Froude Theviable linearity. assulming we
tional similarity of frigates, destroyers, and light cruisers has led to their constituting a rather homogeneous set in terms of hull form. However, their sizes vary be size data ship the that required widely. This manner before in some onormalized
definition was decided to compute these cp tThese a i
considered, a large value of (C) 3 s a "good" while a small value is "bad"A few interesting facts can be gleaned from inspection of Table 2. Hull 06 is evidently a good one: it minimizes pitch, heave, both accelerations, and relative motion at Station 0. Hull 05 exhibits the best slamming characteristics. As to the Station 20 responses, Hull 10 is best in absolute motion whi le Hull 04 is best in
relative motion.
It is also wortnwhile to
observe that the total variability in the Station 20 responses is relatively small. vary by only 5 percent over the 20ship data base while the other responses considered vary by 8 to 28 percent. Prior to response averaging, each Table 2 datum had to be nondimensionalized.
* Stations are numbered from 0 at the forward perpendicular to 20 at the after perpendicular.
-483.
v
-
iniiulrospon~sos.
values
To accompliqh this, all of the (C~ -in
--
avrgdhvrthiihtrsoneporec
index.
characteristics of the DisplaceCW
01
0.0 .1
130
.
*"
14
4.3
634
128.2
14.04
4.20
0,501
04
17. 5401034 1.214.3
0434
@00
U21.0
U.0
4."1
C.410
1.11
0.0"M0
0.05
0.770
0.611
04
134.4
1310
4.26
0.041
0.721
0.0311
0.050
0.55'
6.036
07
118.7
1153
4.6
0."
0.2
C."
OO
On
0.6"
Of
114.8
24.46
4.05
0.67-
0.238
9.0120
0.0
0.7
0.9
00
11P..
14."8
4.10
0.104
0.02
0C01M
0.1150
0.732
0.414
10
1310.0
15.80
4.42
0.640
0.02
0.0250
0.000
9.124
11
110.5
14.20
4.14
0.011
0.012
0.435
0.90
12
124.0
14.50
4.42
0.414
0.146
MON0188
11
12.8.
13.123
3.14
0.374
0.9=1
0.450
14
111.0
U4.32
4.00
0.413
0.04A
0.0363
0.810
is
134.0
10.64
4.24
0.625
0.84
0.015
0.00
16
115.4
14.55
4.50 17
0.435
0.148
~
0.0"0
0.700 .0
17
1317.0
14.51
4.40
0.451
0.001
0.03150
0.00
0.07430
is
11.4
13.03
A.88
0.425
0.003
MOM35
0.4
0.70
0.3On
It
7IIA
14.O
A.44
0.062
0.01 1
0.0510
0.4W
0.555
0.571
Table 2
~~
37
0.5
0.3OIM
"4 0.309
0.0128
0.00
MICA0
~
~~~~~the Data
0.6
0.4
.1
C.001 .01
'0
~
WX
a
SI
IL
AIL
0.518
06
10.80
04007
0.07)
0.017
0.854
0.517
0.04"
0.10.10
:.M0
n5
t.15
0.487
0.054
0.010
*."a0
0.152
0.436
0.00"
0.300
15
0.0
0.40
0.441
0.0150
CAW0
0.477
0.50-
0.720
0.355!
10
0."0
6.641
0.922
M.0SS
CW0.08
454
0.00
a.50.
0.636
01
I.4.4
0.64
0.017
0.6367
0.MG
0.540
0.644
0.740 .0
0.584 .1 15.
.1 .91
004 0.654
004 0.934
001 0.011
.1 0.840
.4 0.740
.1 0.400
05
4.70
0.400
0.111
0.0M0
0.00
0.750
0.40
14
#.70
0.694
6.111
0.0106
0.010
4.720
0.6II
1
430
0.411
0.412
0.0132
0.00
0.13M,
.2.03
.91
0.015
0.6"
0.010
0.700
0.540
0.54
.54
0.104
*.all
0.9344
0.00
C.732
0.424
.2
.5
(1j5 IJ
~
(V ,1
~3 341.30 12
0.3
0.14
0.925
0.011
124
0.231
0.5-i
0.441
9.440
0.53L
0.1
0MW0
31n
0.520
0.414
0.4CA"ss
03
#-.35
0.14
0.013
0.80"
0
0.174
0.310
0.451
0.
01.540
0.004
0.01
:12
0.325
0,515
0.430
as
0464
0.144
0.351
6.1146
1TO
0.2
0.342
0.A$7
to0
0,450
0.an
CAR
0.745
M1
0.216
013
050.440.01 .00
0.404
6.438
31
.33
03)
0.51Table
115
6.314
6.01
0.4)
.550.44 0.535
0.31 0.405
0452ing 6.473
0.2141
0.900
MI$0 05)40.51 00801.1 0.M4
141
10
0.30
0.219
00 0.931
11
630
0.1-43
0.054
0.05
110
.245
O.JOS
0.42S
U,
0.54
0.141
0.004
0.010
1)
9.239
0.114
0.430
11 0.50 4 0.1
0.300 0..0
1.031 CV04
0.055 .010
127 14o
0.514 0.04
#. 0.511
0.454 41.44s
O.001
0.024
1I4
CA-#
0 0.4
.45
11
0.04
4.21:
0.0
0.054
1W
Car0
C.AP
0.0
00.01I
t94
0.0n1
D1O
0.300
0.58
e.410
.M294
1.00'
C.t.5
115
0.211
0.311
0.452
It.340M-1 It10 X
0
Seakeeping Ranks of thie Data Base Hulls
0.540
00
0.514
Table 3
08.3
~
-
involved were thought to be of
os~differences
0I
00
seakeeping qualities of a given hull Second, the resultant increases in the numerical magnitudes of the hull-to-hull
0.64
0 457.0081001002 0
~
10
at
45
~
715,
MIA1
admits quick judgments as to the relative
Base Hulls
'o0~L!
A1I
=
0.4
Average Response Statistics for
=
There were two motivations for this final step. First, use of the normalized scale
mw Cyr.
c1
1334 03
--
T~o IL
12.
--
the worst average being assigned the 1.0.
(4300t) Hulls
%A
This was normalization of the over-
all averages to a scale from 1.0 to 10.0 with the hull having the best ,highest) average assigned the 10.0 and that having
attainable range of variability of the
ment-Normalized
was felt that
it was decided to perform one additional operation to formally define the
ship.However, it can be noted that these average responses are implicitly weighted by the Table 1
it
this implicit weighting was compatible with general definition of R. An averaging procedure assigning strictly equal weights to all of the responses considered might have allowed those with little potential for improvement to drive us in the direction of little ovzerall improvement. The average responses derived by the n icse ecie poeuejs satisfy the general definition of R.
by the largest value Table 2 were divided esposesthe and he oheraverge terrseciemn tereof;ddb wer bdiide thir esectve inma, Thus, the "best" hull in terms of each response considered was assigned a 1.0 for that response while the other hulls were assigned proportionately lower values. These nondimensionalized values were then
1~ 2
:_
.4
4.41
.1
.8
CO
0.403
0.064
00ON4
2.700
W.51
8.611
0.00
0.050
0.080
0.743
COO)
0.81_
0.5251
0.5
".14
is)
.'
"
20
3.00
0.307
0.1181
0.0531
03
3.44
0.515
0.400
0.0m2
0.140
i.50
0.436
001
044
.74
040
.0
.5
10
5.45
0.543
0.15
0.010
0.6-M
W.50
15
1.40
0.554
0.9122
0.04%0
0.450
6.714
4
2.551 0.*
Coefficients and Characteristics
of the Parameters in
the Seakeep-
Rank Equation
(1a884W~x .842
.o
"s. =5 4
C40
04
*
Eft
ft..10
.
t40-08I
-
0.145
0.40%
0.
5.140
*.02
6.2"5
0.
031
0.50
0.000
0.W
Z1410.
155 .17
C111105
~4844
--
potential value for that portion of the quantification effort yet to be undertaken. Application of the procedure just defined to the data base hulls produced the R-value data presented in Table 3. This table shows the data base hulls in order of rank from best (R = 10.00) to worst (R = 1.00). The table includes the hull form parameters in terms of which R has been modeled. Gross tendencies toward increasing rank with increasing CWF and
-
with decreasing CVPF are discernible, but it is evident that no one of the parameters considered provides a reliable indication of the behavior of the index, To complete the quantification effort, a multivariate, linear regression analysis was performed using the Table 3 data with R as the dependent variable and the six hull form parameters as the independent variables. The regression analysis provided estimates of the constants a., i = 0, 1, Table 4 prelents the esti6, in (3). mated values. It also defines (from Table 3) the parameter ranges to which the estimates are strictly applicable. Finally, Table 4 gives the potential change in seakeeping rank associated with each parameter. These data are obtained by multiplying the estimated constant for a parameter by its range. 2.3 Validation Initially, it should be noted that the constants estimated by the quantification process are in broad agreement with the projections made in the course of developing the general model. The exceptions are
,
'I
--
-
,
/
/-I
Iof
-l
the effects of C and c/L. -le formrar VPA parameter has an effect opposite to that anticipated while the latter has a smaller relative effect than anticipated. Both reversals are probably attributable to the difficulties associated with defining afterbody geometry which are discussed In Appendix A. A simple validity check can be made by comparing predicted and direztly computed R-values for the data base hulls. Figure 1 presents the results of this exercise. The symbol R is used (-and will be used hereinafter) for the predicted values to distinguish them from those directly computed. Figure 1 includes lines defining a two standard deviation (o) bound about The magnitude of a the predicted value. is 0.508, and the largest single deviation is 1.073. It was felt that the most relevant validation effort which could be undertaken was to use the R predictor to difine a hull which had parameters withii the data base ranges thereof but better seakeeping characteristics than any hull in the data base. The form coefficients for such a hull can be obtained by inspection of Table 4. One simply assigns each palameter the data base extreme whicb totomaximize The transition fromwill this tend point a strin hull R. sufficiently well-defined to admit theory computations is rather tedious. Dimensional quantities must be introduced; and these, too, must be kept within their data base ranges. Finally, a "ship-like" configuration satisfying all of the imposed restraints must be developed. Appendix C describes the process in detail. Here the characteristics of the resultant hull will be presented and briefly discussed. The hypothetically optimum hull defined by the work described in Appendix C will be identified as Hull 21- It is 135.8 m in length, has a beam of 14.72 m and a draft 4.28 m. Its dzsplacement is 4302t. Its coefficients, as associated with the model developed here, are as follows:
0.690, C c/L =0.850,
SA
The , Zo '
/base /
0.915. T/L = 0.0315, 0 =0.689 and
0.552.
Sealkeeping computations werewere perfor-d for Hull 21. Thieso computations ideatical to those Derforme%,d for the data base-: hulls except that a Lewis section representation was used for Hull 21. Such a representation was the best which could be derived at the level of hull definition available, an- was thought to be adeau~te for the purpose at hland. The results of the Hull 21 seakeeping computations were used t recopute the
/ o1 0I ________________,_____________
Fig. I
F
volumes and areas are slightly different fromHul their ideal values, e.g., C_ is 0.008 Te 21characteristics involving less than the data base maximum. Al of these differences are such that the coefficients rat.ges.of Hull 21 fall inside the data
-4
f; ,/
ull 2
Estimated versus Direct-y-Comiuted Ranks for the Data Base Hulls -4-5-
seakeeping rank index, R, under the assumption that this hull was a member of the data base set. It was found that , had Hull 21 been a member of the data base, it wculd have attained the maximum rank of 10.0. Hull 06, which received the 10.0 rank in the actual data base, would have been second-ranked at R = 7.2. Hence, the R predictor has been used to define a hull that is within the geometric scope of the data base hulls but has seakeeping qualities superior to all of these hulls. The foregoing conclusion equates "seakeeping qualities" with the rank index, R. The equal distribution of ignorance approach taken to defining R is, accordingly, implicit to it. We must, for credibility if not for validity, inquire as to the degree to which the local seakeeping qualities of Hull 21 are inferior as a developed result of this. Figures 2 through 9 were for the inquiry. Each the eightresponses cited figures treats one of the of seakeeping considered
and lowest local responses of any hull in the data base. When no envelope is explicitly shown, the implication is that it is defined by either Hull 06 or Hull 13. Insights which can be gained from inspection of Figures 2 through 9 are discussed in the following paragraphs. Minor trend reversals in waves of six- _ second modal period are neglected in the discussion. These reversals involve relatively small response magnitudes, and are at least as much attributable to numerical procedures, e.g., spectral closure, as to realistic differences between the hulls involved. Excepting the slamming incidence parameter (Figure 6) and relative motion at the stern (Figure 9), the results shown by the fugures under discussion are remarkably consistent. 06 typically defines the best envelopeHull or is very close to it. Hull 13 is only slightly less consistent in approximating the worst envelope is Hull 06 in approximating the bestthan envelope. Hull 21 is generally equal or superior to Hull 06. In some instances when Hull 06 does not define the data base best envelope, Hull 21 is superior to the hull which
here. The response being evaluated is plotted as a function of modal wave period for each of the five Froude numbers evaluated, Five curves are shown on each plot. These illustrate the local seakeeping qualities of Hull 21 and of the data base hulls which are best (Hull 06) and worst (Hull 13) in the sense defined by R. They also Jefine the data base envelope, i.e., the highest
-A 0 "LIL L -DATA
=
}.
z
a
qpi
._"_______-__
SASE
I
I 0" '
!1Lat
a)-.-
WtnVE
.
=
0w",
-
Vl !
RZ
!I
- 01
r V.AG-
,,,as
S
E .I
.
+ __ .I
I
i
1
_L I
~Fig.
----
2
Pitch Comurisons between Hull 06, Hull 13, Hull 21. and the Data Base -eoEnl
IS
as
Fig. 3
Heave Comparisons between Hull 06. Hull 13, Hull 21, anid the Data Ease
o
HULLf9
0! 0U
0 4ULL* J A I.UL tZ &AM -OAT^
U'llSl-
*I 1
I -0
-
-
-
A
4. L2IO & SAM £fU 14
1
t 1.
14
04
I-'-
s
FcsOIs
140h
-0,5it 12020----
0
0
pais4
1A
ul0-
Hul 3 ne5p ong
Hl
'
-04
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13, ase nvelope Hull 21, and the H tion to the effect that a decrease in draftto-length ratio should improve sea' iepinQ common sense argument.- regard deinhedt. ocontradicted ing theeffect of draft on bottom slamming. It was noted earlier that the Station bsltAceeation iZa 14io 5nd12 Fig.enube e 5at atio of 0e variti10 the supportedniec the model e Quantification had0 relatively 20 responses considered However, the Figure 6 author's contention. and that the response averagA smallnges, the slamming iniecaaaee plots of Chsmyarcpomarionsebtwen Hull 06,ft -te against was biased gosdrd employed ing procedure ene
~Ig.
=the
responses with relatively little potential aReltie Corn4Satio exceptions One of theMionae for improvement. cited in the foregoing paragraph, relative motion at the stern, appears to reflect these circumstances. Such being the case, favorable results found for absolute motion at the same location must be considered fortuitous. However, the behavior of relative motion at Station 20 with variations in modal period and Froude nanber can be seen to be different from that of the other responses considered. This may account for the anomaly. In any case, Figure 9 indicates t.hat a reversal of the overall trends found occurs in the case of relative motion at Station 20. The roles of the best and worst data base hulls in an overall sense are reversed. Hull 21 is inter-mediate in most cases, but exhibits superiority in logwaves at the hig-hest Froude number considered. It is pointed out in the appendix
(which was the second response accepted fromigi Enveloey ootoa response of local o(s discussion the initial comparisons) indicate that the cormmon sense argument has merit in the limit as encounter frequency increases. When this frequency is high, most notably at combinations of low (Twin and high Fn' absolute ship motions are very small; and relative motion (in the kinematic sense, at least) is approximatel-y icdne eqatowvelain.Tnth of bottom slamming comes close to exhibiti:nc eirclcrelto ihdat l n~neof sla.Recalling that the Iiec ming is inversely proportional to (C5)3 Figure 6 is seen to reflect the situation in 6s waves, Hull 13 with. j15t described. T = 5.14 m is usually better than either 4.28 m). Hull 06 (T = 4.26 m) or Hull 21 (7 A 2s increase in modal period reverses the situation. As modal period continues to increase, Hull 13 converges to the data
on parameter selection for the general model
base worst envelope.
Then Hull 06 is
intermediate, and Hull 21 converges to the
(Appendix A) that the author's preconcep-
-487-
=
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haiti,,
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NB VO
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,-tS
Heave Acceleration Comparisons between Hull 06, Hull 13, Hull 21.
7
and the Data Ease Envelope
21, and the 13. Hull Hull 06, Hull ISO ,x Data Base Envelope
___as *
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52
Station 3 Slamming Incidence Parameter Comparisons between
)_
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aft
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data base best envelope, Since the magnitudes of (Cs)3 at the s m 6s modal period are relatively high (implying a lower incidence of the phenomenon), the reversal which occurs in that condition is not considered too deleterious. The low-draft ships which are good in an overall sense have good slamming incidence characteristics under those conditions in which slamming is most likely to limit operability. One other matter seems worth pointing out in the present context. Slamming is treated here purely from the viewpoint of its incidence, but the fine forebody sections implied by the low value of CVpF applicable to Hull 21 can be expected to minimize the deleterious effects of the phenomenon when it does occur. Overall, it is thought that the local comparisons presented in Figures 2 through 9 provide strong support for both the equal distribution of ignorance approach to defining R and the excellence of Hull 21. The only definitive reversal found is that of relative motion at Station 20, and it is minor in the sense that the attainable variation in this response is small. Perhaps the most exciting feature of Hull 21 is that its superiority is greatest when conditions are worst, i.e., at the higher Froude numbers in seas which maximize the responses of hulls in its size range.
assessment of the capabilities of the hulls considered to support antisubmarine warfare by conducting helicopter operations. The performance assessment used specific-criteria magnitudes, and required-consideration of a number of responses not considered in the data base computations. Hence,, this effort encompasses the initial generalizations. In addition, it tests the ability of the results which have been derived to yield a hull which exhibits superior seakeeping in a specialized situation. It will be recalled that it was the need for a seakeeping index that was very generalized yet applicable to specialized circumstances which motivated the adoption of the equal distribution of ignorance approach to defining the seakeeping rank, R, used here. The results of the initial generalization effort continue to be of academic interest because they were obtained without any complicating changes in the computational frame of reference used for the earlier elements of the investigation. Hence, these results will be briefly reviewed. The subsequent generalization to 'round-the-clock relative headings will then be described in some detail. The initial generalization effort indicated that the displacement-scaled versions of Hull 21 were superior to their existing counterparts except in relative motion at the stern (for which response the differences involved were small) and in slamming incidence at combinations of high F and low (T ) for which the incidence Of the phenomeRon was relatively small. Further, the magnitude of the Oifferences involved appeared to reflect R differences between the hulls which were computed assuming that the predictor could be applied independent of displacement. Absolute vertical velocity was found to exhibit typical trends with (T )0 and
2.4 Generalization The assumptions made in the course of deriving the general model relating hull form to seakeeping and of quantifying it for destroyer-type hulls are so restrictive as to make the results obtained useless in any practical sense. These results apply only to selected seakeeping responses of 4300t, destroyer-type hulls operating in
long-crested head seas.
and in hull-to-hull omparisonW.
Lifting the hull
type restriction was beyound the scope of the investigation being described. The possibility of relaxing the restrictions as to response, displacement, and/or relative heading was explored at the level of necessary conditions. Initially, Hull 21 was scaled (by the cube root of the displacement ratio as described in context of developing the displacement-normalized data base) to the size of an existing small frigate and to the size of an existing light cruiser. The seakeeping responses of the existing hulls and their Hull 21 equivalents were then computed using the same procedures as for the-data base. Absolute vertical velocity at the after quarter point, noted previously to be a criterion response omitted from the data base computations, was determined, The responses of the two pairs of hulls were then compared. Subsequently, the general approach of scaling Hull 21 to the sizes of two existent hulls was applied for 'round-the-clock relative headings in both long and short-
Prior to attempting the generalization to 'round-the-clock relative headings, a detailed design of Hull 21 was developed. The procedure followed is described in Appendix D. Here it should be noted that attention was given to minimizing resistance, that the author's procedure for computing minimum freeboard requirements' was applied, and that bilge sels were sized in accord with the principles set forth by Cox and Lloyd s . Further, the responses of the detailed design were computed by the same procedures employed previously to verify that its seakeeping characteristics had not been seriously altered by the detailed design effort. Figure 10 presents the body plan which resulted from the detailed design. In this figure and hereafter, this design is referred to as Hull 21D to distinguish it from the Lewis section version considered up to now. Hull 21D has the same overall dimensions as Hull 21, but it differs slightly from the latter in displacement and some coefficients. Specifically, it
crested seas.
displaces 4343t, and has CWF
This effort was extended to 489.
0.698,
0.560. 0.678 and CvpA A 0.922, CVpF =P W The coefficients remain within their respective, data-base ranges. Two existing hulls, both designed to support helicopter operations, were selected for comparison with Hull 21D as scaled to their respective displacements. The smaller of the two hulls selected is 124.4 x 13.8 x 4.52 m, and displaces 3595t. It will be designated Hull A. The larger, designated Hull B, is 161.2 x 16.8 x 5.94 m, and displaces 7948t. Hull 21D as scaled to the displacement of Hull A will be identified as Hull 21D/A. Its dimensions are 127.5 x 13.8 x 4.02 m. By corresponding indentification, Hull 21D/B is 166.1 x 18.0 x 5.25 m. The displacement-scaled versions of Hull 21D are, then, generally longer and shallower than the existing hulls. Breadth comparisons vary. This parameter is the same for Hull 21D/A as for Hull A, but significantly greater for Hull 21D/B than for Hull B. Six-degree-of-freedom motion computations for the two pairs of hulls just described were performed using the United States Navy's Standard Ship Motion Prediction Computer Program, SMP-79. This program is based on the linear strip 6theory of Salvesen, Tuck and Faltinsen with improve-7 ments in roll prediction based on Schmitke and on the Cox and Lloyd paper (Ref. 5) previously cited in the context of bilge keel sizing. Appendix E describes the computations performed in detail. Here it should be noted that speed polar diagrams, as introduced by Covich and Comstock , were used to assess performance. As mentioned previously, the performance of the hulls under consideration was assessed in terms of their ability to carry out helicopter operations. These operations were divided into three categories: launch, recovery, and support. The "support" opera-
I
tion consists of on-deck handling of the helicopter and its equipment. The categorization is required because each of the three operations involves different criteria variables and/or iragnitudes. Further, in order to obtain an operationally relevant assessment, criteria for avoiding hull damage and for personnel limitations were superimposed upon the criteria for each category of helicopter operation. Complete results of the performance assessment are presented in Figures 11 through 16. Each of these figures presents an Operability Index (01) as a function of modal wave period for a specified helicopter operation (and, implicitly, for the hull and personnel) and a specific significant wave height. The OI's are defined such that they equal 1.0 in a seaway so benign that the operation under consideration can be carried out at any relative heading and at any speed up to design speed (taken to be 30 knots for all of the hulls evaluated). For more severe seaways, the 01 is fractional : and when the seaway becomes sufficiently severe to make th operation untenable, the 01 is zero. For instance, an 01 of 0.40 indicates that, in the seaway to which it applies, the ship under consideration can perform the operation being evaluated in 40 percent of the possible combinations of 0 to 360 degrees in relative heading and 0 to 30 knots in ship speed. Larger Oi's are considere .4 to be indicative of superior seakeeping because they imply that the "window" in which the operation considered can be performed will be easier to locate initially and thence offer more flexibility in the speeds and headings at which the operation can be conducted. In this context, Figures 13 through 16 indicate that the displacement-scaled versions of Hull 21D are invariably superior to their existing counterparts. It can also be observed that, in the cases for which fourway comparisons are possible, the performance of the 3595t Hull 21D/A typically equals or exceeds that of the 7948t Hull R. This finding raises serious questions as to the validity of the prevalent conception to the effect that the gross size of a hull is the primary determinant of its seakeeping characteristics. Quantitative comparisons of competing hulls can be made either in terms of 01 ratios or 01 differences. Ratio comparisons are logical from the hydrodynamics perspective, but difference comparisons seem more meaningful from the viewpoint of ship operations as they are at, )lute measures of gain Ratio comparisons or loss in window s for the A-hulls in 4.0 m waves indicate that the superiority of Hull 21D/A over Hull A ranges from 50 percent to a factor of eight. Factors of two to three are common. Differ ence comparisons for the same data show Hull 21D/A to be superior to Hull A by 12
51 percent.
,to
Differences between 20 and
30 percent are prevalent.
Fig. 10
The remainder of the comparative data (for the B-hulls and for the A-hulls in
Hull 21D Body Plan .490.
6.0 m waves) cannot be rationally quantified because it contains OI's of zero. In retrospect, a 5.0 m wave height would have been a better choice to evaluate the B-hulls than the 6.0 m actually chosen. In any case, the superiority of Hull 21D/B over Hull B is evident in a qualitative sense from Fig. 14, 15 and 16. The foregoing results are held to demonstrate necessary conditions for several generalizations of the results which have been obtained here. The direct generalizations are with respect to hull displacement, seakeeping response and relative heading. In addition, the results show that a hull design based on the very generalized seakeeping rank estimator developed here can exhibit superior seakeeping performance even in a highly specialized operational scenario involving particular criteria. Finally, it is worthy of note that the superiority of the Hull 21D variants over their existing counterparts is well in excess of the incremental improvements so frequently associated with ship hydrodynamics. 3. OPTIMIZATION The definition of Hull 21 (Appendix C) was a very simle exercise in optimization. All hull parameters in and associated with the seeeping rank estimator R were -0-HULL
A
H HULLS
IILIDA3HULL 2101
o~1'
assigned the data base extreme values that would tend to maximize it. Simple algebra led to a unique solution. In a realistic ship design environment, it must be expected that some parameters will be fixed and others tightly constrained. Then the R equation will be overspecified. There will not necessarily be a unique solution, and obtaining any viable solution will require more elaborate mathematics. Though the hull form constraints explicitly imposed to define Hull 21 were lax, there were highly restrictive conditions implicit to the prccess. These were the limitations associated with the definition of seakeeping rank, i.e., selected seakeeping responses of 4300t, destroyer-type hulls in long-crested, head seas. The immdiately preceding section demonstrated necessary conditions for relaxation of all but the destroyer iyPe hull limitation. It also showed necessary conditions for the practical value of seakeeping rank in the context of a highly-specialized operational scenario. To develop a utilitarian optiomization technique, we must go beyond the mathematics of overspecified systems of equations. We must make a bold assumption to the effect that all of the generalizations for which necessary conditions have been shown are, in fact, universally applicable. In other words, we must assume sufficient conditions where only necessary con ditions have been demonstrated. 0
HULL21A
*
-0-HUJLL2101A
~~LONG-CRESTED SEAS
02-
HULLS
*HULL21OJ8
OGESDM
0.0 0.4
0
.
-
1.00lI -
4
X4 -
OA __
50.6
0*HRTCETES
.
N j_
7
Fig. 11 F
SEAS
EL L20.2
____Helicopter
SHORT=CRE
1
13
-0 -/.
is
0
Operability Index Comparisons for Launch in 4.0m Waves
Fig. 12
-491.
7
9
11
T3
is
Operability index Comparisons for Helicopter Recovery in 4.0m Waves
-0--
HULLA HULL21DA
OHULLS
-0-
HULLS
U
-"
HULL2101B
HULL 21D)
HULLA
3
HULL21D/A
1010 LONG-CRESTED SEAS LONG-CRESTED SEAS
i
0.4
.4A 0.2 0.0
-j
1.0
1.0
.
I
I
IS
17
SHORTCRESTEO SEAS os
SHORT-CRESTED SEAS
0.8
006
5
00
0.2A 0.2
0. 0
L-4
A
________________________
7
9
11
0
13
9
M.1see
Operability Index Comparisons for %clicopter Recovery 4n 6.Om Waves
Fig. 15 Operability Index Ccmparisons for Helicopter }A Support in 4.0m Waves
Fig. 13
-0--HULLS
*
HULLA
-- Q-HULLIDIB
-' LONG-CRtSTE t0.
0
"--
HULLB HULL210D M---
I
I
IHULL20IA
- / III,,o
,
13
11
Is
SEAS
HULLA HULL210IA
I
I
LOWI-CRESTED SEAS
0.2
A.606
-
O-
o
I
I
o-
1t
t 0
I
I
SE S SHOAT-CRESTAD
oA
SHORT.CRESTED SEAS
0.5
-[
0.6
0.0
a4
-
04
.
02 -
01
0.0
0
Fig. 14
9
11
6
o.===2t
12
15
0.00
17
Fig. 16
Operability Index Comparisons for Helicopter Launch in 6.Om Waves -492-
9
11
13
i5
17
Operability Index comparisons for Helicopter Support in 64Om Waves
-•07
Using a as specified in (7) and the
Even given this assumption, a philoso-essary
-=
phical dilemma remains. Having shown necand assumed sufficient conditions for the "practical value" of seakeeping rank in a specialized situation does not imply that we can obtain a true optimum for any particular ship mission. The formal optimization is strictly in terms of the rank estimator. The 5issumption only implies that optimization of R will yield a hull which is "good" in a particular sc enario. Hull 21D is optimized in terms of R. In terms of ability to support helicopter operations, we only know that it is good. With the above commentary in mind, it is in order to proceed with the details of optimization. The immediately following section outlines an appropriate methodology. Then this methodology is exercised to demonstrate its capabilities in a variety of problems.
117.8 < L< 135.8 m 12.68
0 200
0 -Fg.
400
60
Wae Length (m) Tansfer Function for SWATH Ne zero Spd us BOM Slower son1
2
:3
-2
6C at
SWATH 6C
~Mode!
Theory
- r
j ti-A
44
o
a,mornlde 0
force. Consequently, Viscous damping effects which may be iavored in modelling the vcriic! plane motions of conventional dsplocement ships become relatively invortiat for SWATH ships. The effect of speed on the wcve train is not modelled for either or conventional ships. For the 5#iLH ship the woke created by a strut will affect the flow at sections that this effect will be oft of it. It cam be expected greater for ships with two struts per hull than for ships with ane long strut and that slender struts wil! generate less wake than thicker ones. Corelation of theory with model exeriments has shown generally good agreenmt for a range of speeds and headings. A series of experiments for the SWATH 6 series was carried out by Kollio (Ret. 13). These were models for a 2900 tonme, 73 meter design with various GML's. The confi rations, denoted A, B, and C, hod the sane hills but different strut configurations. The 6A aid the 6B ore sngle strut configurotions wtreos the 6C is G twin strut configuration. Correlation between theory and rntxlel experimients is generally good; however, certain problems emerge. Zero speed prediict is are somewhat due to probletms inconsistent in quality. This is prtbtly in the general modelling of the nonirear viscous dcmnping. At higher speed, behavior is well descrted but for 3ne conditions some discreporcies in magniude exist. Comparisons of lfidolo's model expirnents and predfictions for the 6C configurations ore presented in Figures 20 to 23 for zero and 20 knot in bean seas. Heave and rel. ive bow motion resne per unit wave are given ma unction of wove lngth. At zero
ofheave is well predicted though i
40D
WaVe Length (m)
Fig- 21 .eao Tanufer Functionf lrSWATH Knots in Seem Smn
6C at 20
sedtep valtie is preicted to occu at a slightly greater wae length fourd in the measxemeats The discrepancy hi ti heave also seems to be reflected in the relative bow motion predictions. The =orrelation for 20 kcots is very good as shon i Figures 2D aid 27-
6.2 Excitation ForceCnAoisans for SSP KAIMAUNO
The g correlation with model sca transfer fixictionz for smAlh fiC hrqiies, that pmreictions for SWP KAIMALDID. also a twin strut design, should be no noxde! Lkorturitelyi there c~e M reomnxtle. with AIS4O scalaI- sor
*
SC
XoSWATH -F I-X/Th--
Y
S2Moe
J~~t~~ +
_
0
20D
40D iWa Length (M)
.by ig- 22 RAfive
Bow Motion Transf
Fnct ier
for
the head sea motion in Figures 4 and ,S,even though ata ,nsare the same, exciting forces ore not necessmily
4
C SATH SWATH6C e-Mode "
-- Tation
2
0
20o
40w
em
wave Length IN
+
However, model which to mim~ theoreticam orvios scale res.dts do exist for the wove exciting heave force Since they amre si hwportont tetlert a0d p4ipJi nimt in motiort prediction, they give one indicatiOnl of the relny of the motion preTiction for SP KA A .iO. The forces aid motnos on the model in head sea were obtained as part of an experimental pjr m documented Fein and Stall (Ref. 14). The model did not hove b bllnk the inluder j the effect of blistfr.- The blisters were shown to have small effect On
23 PRelw Bow Motion Trarfer Funcon for SWATH SC at 0 Knots inBoun Smft
the some. force WiT3eD) aid O~az The rnidimnionol heavr given in terms of ( with respect to the we eno eucounter frequency. The noncimentiofor encouiter frec
ieney zs
where %
L g
is the encomnier frequency in radlsec,
isrnidlength in fiiettanid
is gavitational constant in
lsec 2 .
°2
Nondimensional heave force is given by
300
F3oe) = F' LL =
0
(
-where
F30 is the measured force wmplitude in newtons,
"
n is mass of the ship in kilogramns cnd b is wave height in mters.
0
0
100 =
3
SSP Model
0
2
O
2
0e
Ook
1
0
j-100 -
o
Model meory
0
I 0
0
2
4
6
8
Fig. 26 Heave Force Phase Angle in Head Seas at 7
Knots
0
2
2
8
6
4
Fr24 theav Excitn Force i Heed Sas at 7 Knots -100 __
_
_
__
_
_
0
4
2
0
_
Fig. 27 Hem
O0
Force Phese Angle in Head See at IS
The results for wove excitation force and phase at 7 And IS.S knots are gen in Figures 24 to 27- These results ore typical of the correkf-licn at other siceds. Agree-ment between theory and model results for heave exciting force were quite good at all speeds. Magnitude and position of the peaks are well predicted in Figure 25 for the high speed. For the 7 knot speed, Figure 24 shows good agreement up toct = 2.6, which is equvalent to at encounter frequency of 1.7 ro_4sec for the ship. This covers the rnnge of frequencies found in most sea spectra. Phases agree closely for both speeds over the entire frequency range.
0
0 3 0
2
6.3 Transfer Function CawnTxisn
0
~~~~ht-A SFig.
;-e
2
8
Knots
4
0
6
for SSP KAIMALWJO
oavoilable
Although model sccle transfer functions re not for SSP KAIMALIN0, transfer functions car., in principi, be extracted from the full scale trial data, alti ,4t this is not an ideil aooch because of problems
1
l
ams:oted With wzve nasurements and detwenrn drectlionality
4
6
aspects of the seaway-
6
Prediction of the motions for SSP KAIMALO is
maore difficult than for ships in the SNATH -0series. The
rsults indcte that the prsene of I"o struts per
S6C
6~ He2 H , E xcm ig Force in Heed Sen at I.5 KnoKAIMALO
oF
does not in itself creall
SATH respms fo
)-"oblems inieFin
,lowever,a dominant feature c.
is s large aft m-urited hydrofoil The ser
an the3 W"1s~--dont=
A
M-_
K
separation between hulls and are much smaller in area relative to the ship size. Observations during full scale SSP KAIMALINO trials indicate that ventilation can occur as the height of the free surface over the foil varies at some speeds, resulting in reduction of the effectiveness of the foil. This is not modelled currently in the SWATH prediction programs. With these reservations, predictions for heave and pitch for SSP KAIMALINO travelling in beam seas are presented in Figures 28 to 34. Full scale results were available for this heading at 7, 10 and 15.5 knots for uncontrolled heave (vertical motion corrected to the CG) and pitch motion. Heave is presented in nondimensional form and pitch in degrees per meter. Both motions are presented as a function of the wave frequency. Predictions are shown with solid lines, full scale results with dashed lines. The degree of correlation for heave is not as good as for the SWATH 6C. The heave predictions fur SSP KAIMALINO are heavily damped whereas the full scale results arethenot.modelling This predicted characteristic probably due to of the effect of the aftis
Beam Seas 10 Knots -. Trials Th Theory
1.5 . E >
/
1 1.0
E >
O 0.5
foil. The lift curve slope values were estimated by theory and confirmed by model results. Degradation in lift due to in theFor free 10surface included in changes the theory. knots conditions two sets are of not results for 0 I nominally the same trial conditions are given in Figure 0 0.5 1.0 1.5 29. The trials were conducted on the same day but cue separated by a number of hours. The results differ Fig. 29 Heave Transfer Function forSSPKAIMAUNO somewhat in magnitude, although they exhibit the some trends. The major differences occur over the frequency at 10 Knots range where there was little wave energy, illustrating the difficulties in obtaining transfer functions from full scale trial data. The correlation for pitch motion of SSP KAIMALINO Beam Seas is better than for heave motion. The correlation for zero and 7 knots is quite good, though again the agreement 15.5 Knots between theory and trials at 10 knots is not. For this -- o-- Trials speed a large free surface deformation was observed d Theory during the trials. The 10 knot results in Figure 33 show1.5 a. E Beam Seas 7 Knots
1.0
0
1.5
-
=Trials Theory
-o C. E
a.
E
f
0
0.5 *31.0
E
o.
0.
"X
1.0
0.'
0
0.5
1.0
1.5
Fig. 30 Heave Transfer Function for SSP KAIMALINO at 15.5 Knots
1.5
0the
We Fig. 28 Heave Transfer Function for GSP KAIMALINO at 7 Knots
differences between the two experimental conditions as large as the differences between the theory and one of trial results. The 15.5 knot corielation is better than at 10 knots. The overall accuracy of the SWATH ship motions program has been established using model experiments. Its applicability to SSP KAIMALINO motion prediction can be inferred from the reasonable correlation in the cases given. .544-.
4
Beam Seas Zero Speed
--- Theoy Trials>
(---Til
T
- Trials Irials Theory
\-.
3
y
4
rI/
Beam Seas Knots
015.5
E
0
A0+z-Y
1
05
E
period, Te =
Z-*pitch
E
0a-______
2
n[NKh
0
9V I
(4)
and roll period, we
T+=2
Pitch Transfer Function for SSP KAIMALINO at 7 Knots
Fig. 32
19
" 2 E E E 1
0
Fig. 33
0
0.6
1.0
(5)
I
is the waterplane area in m2 , is the ship displacement weight in kilo.grarns, is the gravitational constant in m/sec is the density of water in kg/m 3 , isthe added mass coefficient, CO is the added pitch moment of inertia coefficient, C¢ is the added roll moment of inertia coefficient, is the pitch radius of gyration in meters. K o is the roll radius of gyration in meters and d is half the hull centerline separation in meters. As these equations indicate, several factors affect the natural periods. The added mass and moment coefficients are a function of lower hull geometry. The waterplane area and metacentric heights are a function of the geometry near the waterline. The radii of gyration are dependent on the ship's mass distribution. Equations 3, 4 and 5 ignore coupling between modes and damping but do provide ar. estimate of natural periods. Assuming simple geometrical Ldded mass and moment values for SSP KAIMALINO of Cz = C0 -)Co 0.9 as an approximation following Numata (Ref. 16) and
Trials
0.
GMT
1'I
Aw A g p
Beam Seas 10 Knots
3
2
where
4 0
UK2+
1.5
Pitch Transfer Function for SSP KAIMALINO at 10 Knots -545-
__~~
-= --
-w-
-2
30
using the data in Table I for the as-tested ship gives: Tz = 8.6 seconds, T= 9.7 seconds, and T = 15.8 seconds. The periods also were calculated usingi=quations 3. 4 and 5 in which Cz, C0, and C0, the added mass and added inertia coefficients, were obtained from strip are a4 theory. C 2 increases with speed while C, and independenI of speed. These values are plotted in Figures 35, 36 and 37 along with the natural periods obtained from the 1976 trials and the 1979 control evlluation trials. The ship had buoyancy blisters for these trials. The trial natural periods are the periods associated with the peaks of the transfer functions. Analysis of data for some conditions resulted in different natural periods. These have been included in the figures to demonstrate the accuracy with which natural periods may be obtained by this technique. Natural periods obtained in 1973 by applying impulses to a scale model of the original design at zero speed also are included. Heave natural periods as determined from the trial data (Figure 35) are foir!y constant with speed as theory predicts. The slight variations between theory and trials may be due to changes in the waterline at various speedS which are not accounted for in the theory. Pitch natural period (Figure 36) tends to increase with speed. The period based on added inertia from strip theory shows this speed effect reasonably well. The roll period in Figure 37 is predicted by theory to be 18.5 seconds, irrespective of speed. The trial results for roll are sparse since at these long periods there was very little wave energy and generally unreliable data. The trial data that are available agree fairly well with the theory. The model test result is lower than the trial value. The model represented the original design configuration, and does not fully represent the ship as tested. The estimated roll period ,slow. In this case the value of Co is based on the assumption that rolling is equivalent to the heaving of one hull up while the other moves down. There are certain static errors inherent in comparing natural periods from trials with predicted values. One is uncertainty about the exact fuel and ballast load during the trial. Though the ship was ballasted to the same condition every day, fuel use and weight shifting can occur. This effect on metacentric height and moments of inertia cannot be estimated. Also viscous damping has a strong influence on pitch natural period underway. A better method for obtaining natural periodt is to apply a force impulse to the ship while underway, which unfortunately has not been done to SSP KAIMALINO since the addition of the blisters. 30 O Estimated o Model Experiment (1973) Full Scale Trials (1976 and 1979) V Theory 20 eto
(1973)
Theory 20 0 )
10
0
10 Speed (Knots)
15
10 Speed (Knots) (
15
36 Pitch Natural Period for SSP KAIMALINO
Fig.
O -
30 30
Q V
Estimated Model Experiment (1973) ) ModeE rimen (1973 Full ScaleTrials(1976and 1979)
-,Theory
V
20V 'A
VV o
10 10
0 0
5
10
15
Speed (Knots) Fig. 37 Roll Natural Period for SSP KAIMAUNO 7.2 Natural Periods and Design The advantage of the long natural periods attainable by SWATH ship designs is that resonant conditions leading large motions and degraded operability will not occur for most wave conditions. The designer desires platforming conditions to minimize pitch and heave motion in head seas. If the natural periods in pitch and heave are about 20 percent greater than the encountered modal period of the secway this supercritical behavior can be accomplished. ifnatural periods are sufficiently long the extreme motions associated with resonance will not lead to severe velocities or accelerations since encounter periods would be low. This was found to be true in the case of SSP KAIMALINO when operating in following seas at 15.5 knots without control. Pitch motions of 20 degrees crest-to-trough were recorded but operability was not adversely affected as long as the propellei remained submerged and the bow was not impacting. Simple
V
0 5
5
0
V
7
Estimated
" Model Experiment V Full Scale Trials (1976 and 1979)
0 1-
o
1
atomatic or even manual control could easily prevent
Fig. 35 Heave Natural Period for SSP KAIMAMNO -546-
such extremes of motion. Similar thinking applies to beam sea rolling. A very long roll period assures that resonant conditions will not occur in beam seas under almost all conditions. Resonant conditions that might occur in following seas at moderate speeds can be controlled by automatic motion reduction. Thus the trend in SWATH ship design is to take advantage of long natural periods for good inherent seakeeping in most sea conditions while relying on automatic control and other means to decrease motions when resonant conditions are reached in followino seas at moderate speeds. The SWATH !i.ip designer must also be aware of the interrelationships among the v'irious natural periods, so that be separated should periods hat rol epartedso Pitch and roll Pitc priodoa shuldbe in occur not will motions uncomfortable "corkscrewing" quarteringsituations. knots when To se(166 seconds) and To ('41 ds) of (SP seconds) seIonds) were . knorthen n ocand
both excited at the same time. The change in periods with
its capability for naval operations such as aircruft landing and takeoff, weapons firing, and towing sonar. 7.3 PREDICTION OF EXTREME MOTIONS Inprevious sections ,re motions of SSP KAIMALINO have been presented as obtained in seas o a specified severity characterized by the specific wave spectra given in Figures 17, 18, and 19. The shape of weve spectra, however, can vary considerably (even though the significant heights are the same) depending on duration and fetch of wind, stage of growth and decay of a storm, her lifetime, duringvariety swell. Thus, or the existence a vast of wavea ship mayof encounter SWATH e threfor enconserat For SiATi For design consideration, therefore, it is wave severity important to examine the effect ofonhe aswvspcrlfmutin agtde as wellf as wave
spectral
formulations on the
magnitude
of
balance between the above requirements. Other issues that will come into the decision ore structural and arrangements considerations. Longitudinal metacentric height is a function of the distribution of the waterploane area. Increasing GM decreases the pitch natural period but provides a "stiffer ship in terms of pitch restoring force for surviving in extreme seas. Increased clearance height tends to decrea structural weight penalty. Platforming (not
responses of SWATH ships. For this, computations of the probable extreme values of heave and pitch of SSP KAIMALINO in a seaway have been carried out in three different mathematical spectra following the method for short term response prediction presented in Reference 18. The probable extreme value is the largest value likely to occur in a specified ship operation time, here taken as the duration of the sea condition. The Bretschneider, Pierson-Moskowitz and the Ochi six-parameter spectral formulations were used in the computations, and these were applied to the transfer functions obtained from an analysis of SSP KAIMALINO data obtained in beam seas at ospeedof 10knots. Computations using the Bretschneider and the Ochi were made for aa family six-parameter s Co p representations tations erehader the oof wave spectra from which the upper and lower bounds of responses with confidence coefficient of 0.95 were determined as well as those responses most likely to occur; that is, the responses in the most probable wave spectrum for a given sea severity. These responses were then compared with those computed using measured spectra at Stction India in the North Atlantic (Weather bounds cover howmeasured well the sectra. to determine I) in order Station The in the of responses the variation sp results ore shown in Figures 38 and 39 for pitch and in Figures 40 and 41 for heave for wave heights up to 4.9 meters. Included also in the figures are the responses in the Pierson-Moskowitz spectrum. Wave heights greater than 4.9 meters (Sea State 6)were not investigated since craft linearity has not been verified in seas of severity greater than this. The scatter of the responses computed in the measured Station India spectra shown by the crosses
responding in) all waves would require a high clearance, This is the approach taken by designers of column stabilized platforms. Changing from platforming to
indicates that the probable extreme values of both pitch and heave amplitude vary considerably for a given significant wave height. The Pierson-Moskowitz spectrum
contouring as seaways get higher with longer associated periods is the desirable approach for SWATH ships. Th*s puts an unper limit on the desired natural periods control and bow up trim are further Automatic considerations that may influence the final clearance
while wave inheights underpredicts the motions in the lower high the higher it tends to predict values somewhat the from apparent also is It heights. significant wave figures that, in general, for both pitch and heave motions, the upper and lower bounds of the values computed using the six-parameter spectral formulation better encompass the data from the measured spectra in the North Atlantic than do those obtained using the Bretschneider formulation. While the former appears to cover the range of magnitudes computed using the measured spectra reasonably well, the latter tends to overpredict the lower bounds and underpredict the upper bounds of both modes of motion. The six-parameter spectral formulation can better describe the shape of spectra (e.g., double peaks)
speed underlines the importance of having a good motion prediction technique that can 'Le used in the early stages of design. Roll and heave periods and pitch and heave periods should also be separated if possible. The periods is interrelationships among the natural particularly important at low speeds where control fins are not effective in modifyir.g motions. The choice of natural periods may be dictated by low speed behavior, particularly if the ship's mission requires a long duration at low speeds. The natural periods are closely related to three other design issues: the amount of waterplone area, the bridging structure clearance height and the metacen-ac heights. The heave natural be increased also will Such acandecrease waterplae area. period decreasing by decrease the heave exciting force in waves, and probably decrease resistance as well. It must be remembered however that as the waterplone area is decreased, the restoring force also is decreased. This can result in the situation that, should resonance occur at these longer periods, the motions can be rather large even though the exciting force is small as shown in Pien and Lee (Ref. 17). The final selection of waterplane area must reflect a
design of a SWATH ship. Overall sekeeping is both a constraint and on opportunity to the designer as hydrostatics, structures, arrangements, powering and other factors are balanced to obtain a good design. The dependence of seakeeping on metacentric heights, moments of inertia and clearance points out that the usual weight growth of ships con be a serious problem for a SWATH hull. Thus, until experience
than the Bretschneider formulation.
is gained in design and construction, the SWATH ship will
This allows for a
better description of some seaways, and probably accounts for the good correlation between the responses
lorger margins than conv.ntional hulls, require Nevertheless the SWATH concept offers the opportunity to design a hull with low motions and therefore enhances -.547.-
-
r 28
SSP KAIMALINO Beam Seas 10Knots Station India
8 SSP KAIMALINO Beam Seas 10 Knots7
7
Station India
16
6
Emaximum
+
+
MaiumL Maximm
+
+
E 4+
Probable
Minimum ++ ,+4
CO
2+3
+
8
- Most Probable
0
Minimum
44
* +
+
bMost
2a
U 12
+
5-
20+
4
4* +
*1
4+
0 ++ 0
6
4
2
28
1
, 1
4
7
+ Maximum
0
16
+54 ++
E+
Most Probable
Maximum 2
0~1
-< +
+
imum
MosM39 Probable
8
+
C-
+
12
4
E
Minimum
44*+
4
*
4
ia
1
SSP KAIMALINO Beam Seas 10 Knots Sj-. India Station
20 -6
CL
1 6
Significant Wave Height (m) Fig. 40 Probable Extreme Heave Amplitudes for Measured Spectra at Station India, PiersonMoskowitz and Bounds of Bretschnelder Spectra
SSP KAIMALINO Beam Seas 10 Knots Station India
24
1 2
01.0
Significant Wave Height (m) Fig. 38 Probable Extreme Pitch Amplitudes for Measured Spectra at Station India, PiersonMoskowitz and Bounds of Bretschneider Spectra
I
i
1
Pierson-Moskowitz
64
3N
Pierson-Moskowitz 0' 0
6 4 2 Significant Wave Height (m) Fig. 39 Probable Extreme Pitch Amplitudes for
Measured Spectra at Station India and Bounds of Ochi 8-Parameter Spectra
- 48-
:
*2Piersonl-Moskowitz 0 + 0
6 4 2 Significant Wave Height (m) Fig. 41 Probable Extreme Heave Amplitude for
Measured Spectra at Station India and Bounds of Ochi 6-Parameter Spectra
based on the six-parameter formulation and the Station India data.E
3.0
a
2.5
Iddt2Design
While the most probable extreme value of a motion in a specified seaway is that which is most likely to occur, the probability that the extreme value will exceed the most probable value is quite large (theoretically 63 percent). Hence, it is highly desirable to predict the extreme value for which the probability of being exceeded is a preassigned small value fl. Ochi (Reference 18) developed a formula for predicting this extreme value which F-e calls the "design extreme value," and it is expressed as follows: Yn(P) =V2
_
In' n(fl/k)
Me
Extreme Value (p--.01) -
-
Height of Bridging Structure Clearance (1.83m)
E 2.0 .2 Z 1.5 0 a *)
m2"e' (6)
1.0
Q:S0.5E
where
x
is the observa tion tim e in hours, m O is the area under the response spectrum, m2 is the second moment of the response spectrum, 3 is the risk parameter and k is the number of encounters with a specified sea in a sh;p's lifetime. The risk factor, 3, can be assigned at the designer's discretion but is given a value of 0.01 if it is desired to design for 99 percent assurance that this (single) The number of amplitude will not be exceeded. encounters with a specified sea, k, involves the ship operation time at the specified speed in the sea considered and the maximum duration of the sea. As noted previously, an important consideration in the design of SWATH ships is the sizing of the bridging structure clearance height, since it must reflect a balance between requirements to limit weight and to assure adequate transverse metacentric height, and the requirement for sufficient clearance to minimize water contact and slamming on the bridging structure. Therefore, it may be of interest to examine how the extreme relative bow motion that may be experienced by SSP KAIMALINO without any form of control relates to its clearance height of 1.83 meters. Computations usirg Equation 6 were therefore carried out for a head Sea State 5 (significant wave height of 3.05 meters) at a speed of 7 knots using the B.etschneider formulation for a range of modal periods appropriate for the specified wave heiaht. The risk factor 3 was taken as 0.01 and the lifetime exposure, k, or the number of encounters with the specified sea, heading, and speed was assumed to be approximately ten. The calculations we:e made for the centerline of the cross structure at the forwardmost location of the flat bottom just before the start of the curved bow. The results are presented in Figure 42 where they ore shown as a function of operation time. It is seen that the extreme value increases significant;y during the first several hours and thereafter increases very slowly throughout the 44 hour period which is the estimated duraition of a sea of the specified severity. The figure a~so shows that the extreme relative bow motion of the ship will exceed the cross structure clearance height within four hours operation time if no motion control is used. However, its performance during the remainder of the storm (about 40 hours in this case) should not degrade much after the initial time period, Although water contact can be expected it does not necessarily imply a slam. Indeed, experience aboard SSP KAIMALINO during the seakeeping trials did show that whi!e there was a fair amount of water contact with the bridging structure in head Sea State 5, these contacts tended at the lower speeds to be gentle wave slops that
"
0
_ _
0
_ _
10
_
_....._
30
20
40
_ _
50
_ _
60
Time in Hours Fig.
42 Extreme Values of Relative Bow Motion of SSP KAIMAUNO in Head Sea State 5 at 7 Knots
did not impart an arresting motion to the ship. Though some variability may exist between the assumptions made here regarding SSP KAIMALINO operations and the actual operational experience over her lifetime, this analysis does demonstrate the importance of the relationship between clearance height and relative bow motion. 8.. CONCLUDING REMARKS The results of the full scale trials of SSP KAIMALINO demonstrate the good motions characteristics of SWATH ships and the benefits of automatic motion control. The agreement between the significant values of motions from mode; and full scale results is reassuring for the application of future model scale results. The theoretical predictions agree with the twin-strut SWATH 6C model data. For SSP KAIMALINO the agreement between theory and model exciting forces is reasonable, while the transfer functions derived from the trials data have an inherent uncertainty that makes firm conclusions difficult. The motion results for SSP KAIMALINO illustrate the importance of separation of natural periods and the influence of these periods on the motion response. The extreme value predictions offer a means of applying experiments and theory to design and operational problems. The SWATH ship offers great potential for achieving good seakeeping. To the designer this means increased operability and mission effectiveness. The small waterplane area alters the seakeeping characteristics and allows for a relatively small control force to make large changes in the motion responses. The SWATH concept lacks the long design history of monohulls; however, the tools required for predicting motions, natural periods, and extreme values are well developed. Through these prediction techniques, the effects on seakeeping of natural periods, metacentric heights, waterplane area, and other parameters are becoming understood.
-549-
1v7 =-=
--
_
REFERENCES I. Lamb, G. R. and Fein, J. A.. "The Developing Technology for SWATH Ships", presented at AIAA/SNAME Advanced Marine V+.hicles Conference, Paltimore, Md, Oct. 1979.
14. Fein, J.A. and Stahl, R., "Head and Following Wave Exciting Farce Experiments an Two SWATH Configurations", SPD 0928-01, June 1980, David W. Taylor Naval Ship R&D Center, Bethesda, Md.
Lee, C. M. amd R. M. Curphey, "Prediction of 2. Stability and Wave Loads of Smol!Motion, Waterpl-ie-Area, Twin-Hull Ships", Trans. Soc. Naval Architects and Marine Engineers, Vol. 85, 1977, pp. 94m_130.
IS. Comstock, J.P. (ed.), Princiles of Naval Architecture, Society of Naval Architects and Marine Engineers, 1 77, pp. 659-670.
-
Hydrodynamic E., "Predicting 16. Nurnata, Behavior of Small Waterplane Area, Twin Hull Ships", Presented to the New York Section of Society of Naval Architects and Marine Engineers, May 1980.
3. Dolzell, J. F., "A Simplified Evaluation Method Speed, Seakeeping Plane, Zero for Vertical Characteristics of SWATH Vessels", DL-78-1970, July 1978, Stevens Institute of Technology, Hoboken, N.J.
17. Pien, P.C and Lee, C.M., "Motion and Resistance of a Low Waterplane Area Catamaran", on Naval at the 9th Symposium Presented Hydromechnics, Paris, France, 1972.
4. Lee, C. M., "Approximate Evaluution of Added Mass and Damping Coefficients of Two Dimensional SWATH Sections", 78/G84, Oct. 1978, David W. Taylor Naval Ship R&D Center, Bethesda, Md.
18. Ochi, M.K., 'Wave Statistics for the Design of Ships and Ocean Structures", Trans. Soc. Naval Architects and Marine Engineers: Vol. 86, 1978, pp. 47-76.
5. Oshima, M., H. Narita and Y. Kunitake, "Experiences With 12 Meter Long Semi-Submerged Catamaran (SSC) "Marine Ace" and Builaing of SSC Ferry far 446 Passengers", AIAA/SNAME Advanced Marine Vehicles Conference Paper 79-2019, Baltimore, Md, Oct. 1979. 6. Long, T. G., Hightower, J. D. and Strickland, A. T., "Desi.n and Development of the 190-Ton Stable Semi-Submerged Platform (SSP)", TP 397, July 1974, Naval Ocean Systems Center, San Diego, Ca. 7. Higdon, D. T., "Active Motion Reduction in SSP KAIMALINO in a Seaway", AIAA/SNAME Advanced Marine Vehicles Conference Paper 78-740, Son Diego, Co., April 1978. 8. Higdon, D. T., "Active Motion Reduction for a 3000 Ton SWATH Ship Underway in Regular Head and Following Waves!', 80-05-01, May 1980, Seaco Inc., Kailua, Hawaii. 9. Stenson, R. J., "Full-Scale Powering Trials of the Stable Semisubmerged Platform, SSP KAIMALINO", SPD 650-01, April 1976, David W. Taylor Naval Ship R&D Center, Bethesda, Md.
F
10. Woo, E.L. and Mouck, J.L., "Standardization Trials of the Stable Semi-Submerged Platform, ESP KAIMALINO, with a Modified Buoyancy Configuration", 801049, April 1980, David W. Taylor Naval Ship R&D Center, Bethesda, Md. 1. Kallio, J.A., "Seakeeping Trials of the Stable KAIMALINO)", Platform (SSP Semi-Submerged SPD-650-03, April 1976, David W. Taylor Naval Ship R&D Center, Bethesda, Md.
4
12. Woolaver, D. and Peters, J.B., "Comparative Ship Performance Sea Trials for the U.S. Coast Guard Cutters Mellon and Cape Corwin and the U.S. Navy SWATH Ship KAIMAL INO", 801037, March 1980, David W. Taylor Naval Ship R&D Center, Bethesda, Md. 13. Kallio, J., "Seaworthiness Characteristics of a 2900-Ton SWATH", SPD-620-01 Sept. 1976, David W. Taylor Naval Ship R&D Center, Bethesda, Md.
-550-
Disc
sio
producing funcresidual risks inherent in tional SWATH ships such as the MESA 80 built by Mitsui Engineering & Shipbuilding and the Dutch DUPLUS. I would like to thank the authors for taking this important step in that direction.
C. KeneU (Nava Sea System ommand) The introduction of a new ship concept is a high risk venture due to the high cost of today's ocean vessels and the vulnerability of man and his machines to nature's forces. These anxieties can be reduced significantly by demonstrating that the behavior of a new vehicle can be accurately predicted prior to committing large sums of money to construction. This paper demonstrates such a capability for some of the most important aspects of SWATH ship seakeeping. The correlation of the authors' analytical predictions with test results from the three meter model and the 25 meter model is most reassuring. Documenting the wave spectra "used" for the large model tests lends a sense of completeness to the paper. A curious feature of the motions data in figures 4, 5, 8, and 9 is the noticeable scarcity of data for speeds from 10 to 16 knots. This part of the curve is particularly interesting since both pitch and roll responses show peaks for these speeds. The authors discuss the role of the ship's bow wave in dampening pitch below 10 knots. Perhaps at higher speeds the ship is balanced on its bow wave and is not sufficiently stable to allow data collection. Elaboration on this point would be appreciated. A related aspect that is puzzling is the apparent lack of influence of the buoyancy blisters on motions. The location of these hull appendages representing about 15 percent of the displacement sho.ld lead to cancellation of hull and strut generated waves, including the bow wave. As a result, the vessel with blisters should be less sensitive to sinkage and trim effects. However, motions at 10 knots are the same with and without the blisters indicating the same heavy pitch damping due to the bow wave. The scarcity of data above 10 knots limits further speculation. Perhaps the authors can provide some edditional insight into this matter. Heave reponses for the SSP Kaimalino are conspicuous by their absence. The reasons for not collecting such data are of interest. The reasons for selecting beam seas for extreme motions predictions rather than the more conventional head seas would be appreciated. This fine paper is clearly a product of the R&D community. However, by assembling such a comprehensive tre -ment of SWATH motions predictions, the authors will find that the importance of thair work extends well beyond the technical community. Demonstration of technical capabilities in this discipline as well as many others may well convince decision makers t- accept the i
N.TOM(MHO In DTNSRDC, very sophisticated researches have been performed on SWATH ships, and today, the authors added another valuable report. At first, I would like to express my highest regard to your vast studies on the seakeeping characteristics of SWATH ships. In Nagasaki Experimental Tank, we have studied this type of ships referring to the papers of DTNSRDC, and performed several tank tests both in calm water and in waves. On the basis of our experience, I would like to ask the authors' opinion on the following points. 1. In the present paper, the authors mentioned "Largest motions were found in following seas when ship speed was close to the wave speed." Reading the same comment in the earlier paper of DTNSRDC, we have been thinking over the cause of this phenomenon. The authors considered that large motions were activated mostly by a full span aft foil of SSP KAIMALINO. From our study, however, contribution to these large motions should be shared by several factors: such as, fore-aft asymmetry of the hull represented in the value of LCB-LCF, Munk's unstable moment which reduces the stability of pitching, and of course, the arrangement of foils. Let me explain more in detail. We have been using the following formula to estimate the motion amplitudes of SWATH in the specified condition, i.e. we = 0 in fol -owing waves, neglecting all of the terms expressing the dynamic effect. R
F
z
, R 3 5
3 3
= ,
R
5 where elements of R3,3, R3 ,5 , R 5 ,3 , and R5,5 restoring coefficient matrix z, e complex amplitudes of heave and pitch F 3 , F 5 : complex amplitudes of wave exciting heave force and pitch monent
-551-
R3 ,3 = R
=
oU2
gAw(LCB-LCF) +
3,5 R ?gAw(LCB-LU) R5 ,3 .5 gVGML - Y2 Cz 5,5i=li( F= 3
alfgAwA
-
jBfwOUA
p
i
N E A.)
N N iA
1 (f)CLOLi
N E 1u2
(l
a
L Lc i
E
F5
ja 2 ?gVGMLkA + 82y
0
UA
N Z liAi(f)CLai i=l
Ai(f) : projected area of i-th foil to horizontal plane ii : x-coordinate of quarter-chord point of i-th foil CLai : lift curve slope per radian of i-th foil al,a2,81,82 : correction factors The other nomenclature is same as the authors' one. Making use of published data of SWATH 6-A given in Ref.(2), I would like to explain the causes of large motions in following waves. The value of (LCB-LCF) is assumed to be 3 m. The amplitudes of heave and pitch -ere calct'lated in the case of we = 0 where X/L = 1.0, 8 = 00, and U = 20 knots. In this condition, CLl,(12,81, and B2 are assumed to be 0.1, 0.3, 0.1, and 0 6 res-aectively, based on our experience. Calculations were made for the various hull configurations as follows. The results of calculation are shown in the table. 1) Original SWATH 6-A 2) SWATH 6-A with foils, each projected area of which is twice as large as the original one. 3) SWATH 6-A with foils, each projected area of which is three times as large as the original one. 4) Same SWATH 6-A as 3), but assumed LCB-LCF = 0. Same SWATH 6-A as 3), but the effect of Mul.K's unstable moment is not considered,
I)
ler. We were afraid that in waves severe slamming might occur on the bottom of bridging structure, and performed a series of tank experiments. As the result we found there is rather high possibility of slamming, if the SWATH keeps its advance speed at the specified value for a long time. On the other hand, we hope there might be no fear of slamming in the practical operation, because we know that keeping its advance speed at that particular value is very difficult and advance speed deviates easily. Hentce, we have not arrived at a conclusion. For SSP KAIMALINO, Froude number nearly equals 0.35 when its advance speed is 10 knots. If I could ask the authors for any further explanation to the point in relation with the sea trial of SSP KAIMALINO at 10 knots, it would be appreciated very much. 3. During the tank test of our SWATH model in regular waves, I have noticed sevral times the phenomenon which seems to be the so-called "unstable rolling".1) The phenomenon occured in the cases where encounter period nearly equals the natural period of heave-pitch coupled motion, and the natural period of rolling is twice as long as the period. In this case, incident wave activates rolling motion of its natural period together with resonant heave-pitch coupled motion. This phenomenon is resonance of rolling motion with 2-nd order hydrodynamic force generated by large amplitude vertical motion. In the authors' paper, there are some expressions suggesting the existence of "unstable rolling". Therfore, I presume that SSP KAIMALINO also have had the ex-
ZA/A
LeA/2A
perience similar to the one appeared in our model test.
Original SWATH 6-A meaurenat DWTSC 6A measured at DTNSRDC
0.6
2.0 I
ip
iR!1) Original SWATH 6-A
I2 Foil area IE3) Foil area iT' times 4
F
twice
0.89
3.47
1.46
3.52
1
three 2.09
3.67
a
times LCB-LCF = 0 R li 5) Foil area : three .Utimes without L Munk's moment u s e
__0.9
1
1) Paulling, J.R. and Rosenberg, R.M.: On Unstable Ship Motions Resulting from Nonlinear Coupling, Journal of Ship Research, Vol.3, No.1 June 1959
. H. fauo(Yoma N.Univ.)
0.77
1.25
From this calculation, it is understood that following three factors, too large foils, fore-aft asymmetry of the hull, and Munk's unstable moment, equal"' contributed to invite large motions in the case of w e = 0. 2. According to our experience, in the region where Froude number nearly equals 0.35, rather high transverse wave is generated by bow in the part from midship to square station 3 inside of two hulls, and make the bridging structure clearance smal-
~-552-
Any comments by the authors on
this phenomenon would be most instructive for us in both theoretical and practical viewpoints.
I appreciate the authors' presentation
of full scale performances because existing data for this kind of hull configuration are rather seldom. I would like to make a brief comment about the Hull shape of Kaimalino shown in Fig.!. I don't think the hull configuration is not oltimized from the point of view of wave resistance theory. According to my earlier study, effective cancellation of waves generated by the lower hulls and those of surface piercing structure cannot be realized by this configuration.
R.T.Schm~he (OtR)A We are thinking of initiating a methodical model series for SWATH ships, aimed
primarily at the 2000 - 10000 ton size range. We are thinking in terms of roughly 10 models. The series would address both resistance and seakeeping. Would the authors offer some guidance regarding experiment design ?
nil center of gravity to located In the open ship well area. The vertical motion data and vertical acceleration data from transducers located close to the LCG presented in Figures 6, 15 and 16 provide an indication ot heave behavior. We agree with Mr.Kennel that heave motion is an important parameter in the correlation with SWATH predictions. The use of beam sea conditions for extreme motion predictions reflects our belief, based on other SWATH model data, that unlike conventional monohull ships, beam sea heading appears to be better for
survivability for SWATH than head-into-thewave conditions. We agree with Mr.Toki that the fore-
saft
hull asymmetry and Munk's unstable moment can lead to large motions along with the large foil near we - 0. However, model tests and analytical results addressing other similar hull shapes suggest smaller motions in following seas than
J.A. Fein, M.D. Ochl and K.K. McCrelght (DTNSRDC)
experienced by SSP KAIMALINO. Further, a comparison of motions computed by our colleague, Dr.Lee, at the Center at we 10, assuming only the presence of static terms, with model results shows that the static assumption may not be completely adequate for describing the motions in this region. Therefore we are not sure that an evaluation of the contribution of the various factors to the motions in this condition can be adequately made from such an approach. Regarding the cause of the unstable rolling motion we agree that the Mathieu type of coupled instability could exist under certain conditions. We have not observed this behavior in our model tests so far. However, we have observed some roll motion instability in waves whose period was one-half the roll period. Roll motion would become large with rolling occuring at its natural frequency. Thus was observed on our SWATH 6B nodel. Pitch and heave, however, were not large and the encounter frequency was not synchronous with the pitchheave coupled motions. Similar results were observed during model tests at the Davidson Laboratory of SIT. Dr.Lee of DTNSRDC has carried out an analytical study which suggests that waves having a natural frequency twice the natural frequency in roll can cause a roll instability due to the presence of nonlinear behavior occuring in the restoring force. However, we are very interested to learn that Mr.Toki has also observed unstable behavior of a dynamic origin and feel that future investigations are warranted in this area. Finally, on response to Mr.Schmitke we wish to encourage his country's efforts. Areas where series experiments are needed include fin size and location, rudder configuration, and upper hull bow design. In addition, the distribution and amount of waterplane area, which influences natural periods, metacentric heights, and LCB-LCF differences, could serve as a framework for the parametric variations.
The authors wish to thank Mr.Kennel, Mr.Toki, Mr.Schmitke and Professor Maruo for their very valuable discussions. It was of special interest to learn about the Mitsubishi SWATH research from Mr.1'oki. Both Mr.Kennel and Mr.Toki have inquired about the motions data above 10 knots. From just over 10KTS to 13KTS the ship is overtaking its bow wave and thus constant speeds in that range are difficult to maintain. The ahip generated bow wave does reduce the clearance height of the SSP in the aft portion of the ship in this speed range as Mr.Toki has pointed out. This is a phenomenon that must be considered in the operation of the ship, however, since sustained operation in this transition speed range is difficult to maintain increased slamming frequency in this condition is not expected to pose a problem in practice. The scarcity of data above 13KTS is unfortunately due to a lack of sufficient operating time with the instrumented ship as well as speed limitation of the ship. The lack of influence of the blisters on the pitch motions we attribute to their location near midships on the inboard side of the lower hulls. The blisters attached to the ship during the trials were smaller than those now installed and were nct designed to cancel ship generated waves, Thus we agree with Professor Maruo's comment regarding hull optimization. The SSP does not reflect current design philosophy in the United States but it does provide a vehicle for assessing some of the merits of the SWATH concept. We are currently involved in an effort to increase SSP displacement in which we are giving consideration to certain parametric changes for improved performance from both the seakeeping and wave making resistance point of view. The pure heave motion of the SSP was not measured directly since the longitudi-
-553-
Hull Form Design of the Semi-Submerged Catamaran Vessel Yuzo Kusaka, Hiroshi Nakamura and Yoshikuni Kunitake Mitsui Engineenng & Shipouilding Co. Ltd Japan
ABSTRACT
Rr
Residual resistance
The hull form design procedure of the SSC (Semi-Submerged Catamaran) is discussed from the viewpoint of minimizing wave re-
S( C(8)
Amplitude function of sine wave AmJlitude function of cosine wave
p
Fluid density
sistance.
I
The effects of hull configuration
of the SSC on wave resistance are briefly m Density of source distribution presented using the results of series model Density of line doublet distribution tests of hull elements. A mathematical model for wave resistance based on the thin M Density of normal doublet ship theory has been developed, in which distribution s S Michell's source distribution is used to Ship speed V represent the strut's geometry and the . INTRODUCTION equivalent line doublet distribution to the lowerhull is introduced. The correction factors included the fom mathematic-, a attempts to develop a new type of are inompaiso eried of : h model et-Many are romcomarion eried f t! tL 'etmarine vehicle have been made to break the ical results with the measured data. The various performance mlimitations imposed design procedure to obtain an SSC hull form the conventional on conen-by conventhe and monohull tionanvcatamr described, are resistance wave tional catamaran. The Semi-Submerged with minimum and examples of the optimized hull form are Catamaran (SSC), which the U.S. Navy calls presented. Design andSSC sea trial80'results of the first commercial commrcil SC 'MESA 'ESA 0' are re presres SWATH, is one of such high performance etedfiradvanced ntdwhich NOMENCLATURE
L
Length of lowerhull
D
Submergence of lowerhull Diameter of lowerhull W&
U Fn g
Displacement volume of ship Velocity of fluid Froude number Gravitational acceleration constant
K.
Wave number
Rw
Wave resistance
vehicles, and has two lowerhulls; account for a major part of the displacement below the water level, and this
submerged part and the deck structure are connected with streamlined struts. As well as the extensive researches on
this type of vessel carried out by the U.S. Navy [l],[2],[3],14],[5], Mitsui Engineering Shipbuilding Co., Ltd. to which the authors belong, has done a lot of deveiopmental work on the SSC since 1970 [6]. Encouraged by the satisfactory performance achieved by the experimental SSC 'MARINE ACE' (Phote 1) built in 1977 which is 12 meterz long, the first commercial SSC prototype 'MESA 80' (Photo 2) was built in 1979 as a passenger ship carrying 446 persons. The extensive sea trials of the 'MESA 80' in 1980 proved the high performance of the SSC and the great potential for
.555-
.
i
quired so as to attain a well-balanced SSC design. As part of such useful design technology, the authors have developed a mathematical model for wave resistance which shows a good agreement with the model test results. This mathematical model is presented in chapter 3. Also in chapter 4, a hull form design procedure for jinimizing wave resistance is presented. Design and sea trial test results of the 'MESA 80' are briefly presented in chapter 5. 2.
Photo. 1
'MARINE ACE'
--
___ _____
____
-of
Photo. 2
'MESA 80'
its application in the field of marine transportation and ocean development. In order to expand the SSC's application and to enhance the operational economy of the SSC much more, possible reduction of required power in calm water will be required in addition to the high performance in waves on motion and speed which has been already proved by many full scale test results. As the wetted surface area of the SSC is almost twice as much as the muohull, which results in some penalty from frictional resistance, great attention. in the hull form design of the SSC must be paid to reducing wave resistance as well as to attaining high propulsive efficiency, Due to the complicated combinations of hull elements of the SSC, there is more degree of freedom in hull form design than in that of a monohull. Therefore it is of prime importance to understand the fundamental wave resistance characteristics of the hull elements. Typical experimental results, varying the lowerhull proportion and it's submergence etc., are presented in chapter 2 of this paper. In designing the hull form of the SSC taking into account the various design limitations as well as the basic design requirements, design synthesis technology is re.556-
FUNDAMENTAL WAVE RESISTAjCE CHARACTERISTICS OF HULL ELEMTS
In the initial design stage of the SSC, prior to deciding hull dimensions, the preliminary design work for selecting basic hull configurations, hull material, propulsion system and machinery arrangement, etc. shall be made from the viewpoints of hull weight and cost so as to keep the overall erformance of the design within the speci fied requirements of the performance suc h as capacity or payload, speed and motion. However, due to not only lack of accumulated design data for the SSC, but also the sensitivity of the hull elements such as lowerhull, strut and fins on the SSC's performance, the hull form design of the SSC seems to become more difficult thi~n that a conventional ship. Under these circumstances, understanding of the hydrodynamic characteristics of hull elements is necessary to obtain a well-balanced SSC design. As is well known, the SSC has about twice the wetted surface area of a conventional monohull of equal displacement, which results in some penalty in the total resistance. Accordingly, design considerations must be rainly taken care of realizing the SSC hull form. with minimum wave resistance. (1) Lowerhull Series model tests of the lowerhull to investigate the characteristics of wave resistance were carried out at Mitsui Akishima Laboratory, changing the submergence to diameter ratio f/D and the length to diameter ratio L/D of the lowerhull. Models were manufactured so as to have the same sectional area curve and displacement, and each model had a syu-metrical shape lengthwise and the parallel part was 60 percent of the total length. The wave resistance of the lowerhull was obtained by subtracting the total resistance measured in the fully submerged condition where the wave resistance is very close to zero from the total resistance measured at a given submergence. Figure 1 shows typical results of the wave resistance of the lowerhulls with dIfferent f/D. As seen from this figure, the wave resistance coefficients of the lowerhull remarkably change by Froude number. As the f/D of the lowerhull decreases, the coefficient increases over the whole speed range, specially at hump speed. In Figure 2, the wave resistance coefficient denoted by displacement are presented for different
length to diameter ratios of the lowe-hull. Tne effect of L/D of the lowerhull on wave resistance is quite different between higher speeds over Froude Number 0.4 and the speed
Cr
range below that Froude number. In the case of a high speed SSC, therefore, it will be desirable to have a larger L/D of the lowerhull than about 12.
Rr I!VL
x 10
0.072 /L-.0 5-
103
Cv~ ~" 1
0.2
f /D ",0.75
Fig. 3
0.2
2
0.3
0.4
0.5
0.6 Fn
- "/2 _r 5
0.7
0.3
0.4
0.5
Fn -Vlr%'T-
0.6
0.7
Residual Resistance of Struts with Different t/L
SX 10 IL
VAilI
0.15
Fig. I Wave Resistance of Lovrhull (Effect of Lowerhull Submergence)
0.2
LID
2-
?i'-. 4
7
0.3
0.5 0.4 F2 V14
0.6
0.7
Residual Resistance of Struts with Different Depth (t/L=0.036)
(3)
00.5
Twin Hull As well as the wave resistance characteristics of both lowerhull and strut, it is necessary to investigate the interaction
0.6
effect of a twin hulled SSC for the hull form design. Figure 5 shows the estimated breakdown of the wave resistance of the *MESA 80'. of which the design is presented in chapter S. The Proude number at the design speed is about 0.7.
Fig. 2 Wave Resistance of Lowerhull (Effect of Length to Diameter Ratio)
th e the wave resistance of the iowerhulis largest among other components, and that of the struts is the sallest at the design Froude number. While, the magnitude of the wave resistance caused by the strut-lowerhull interaction is almost the sare as that of the lowerhulls, the resistance caused by the port-stbd. interference is small at the design speed. Since there is hardly any possibility of utilizing the interaction effect well for the SSC designed at a high Froude number, it becomes noze essential for such an SSC to make both lowerhull and strut as slender as possible to reduce wave resistance.
In order to investigate the fundamental characteristics of the strut on wave resistance, some strut models were tested in the towing tank of Akishima Laboratory. Figure 3 shows the residual resistance of struts with different thickness to length ratios t/L. Also in Figure 4, the effects of strut immersion are shown. As seen in these figures, it can be found that the strut with smaller t/L and smaller immersion will be more favourable in the speed range over Froude number 0.4. However, the magnitude of the wave resistance caused by the strut is much smaller than that of the lowerhull in the case of 'MESA 80'. 5S57.
m(x) --
3 [CV
-
0
I1 w
XV', 3
m____
TTAL
.. SnrS uuu R MILL
---
ix)
= -2
(2)
-
Then the wave resistance coefficient Rw-/l/2oU2 L2 is given by Cws =, --
s](Cevjte) Jlcos-eae
(3)
,
where rf--j(x)dxdzsec3 E exp(Kozsec
SS(6)
sin (Kxsec),
......------------------ '----------------0. 0.=
r
0-5
0.-
0.3
0.2
0.6
0.7
)
Cs()
""(x)dxldzsec6exp(Kzsec~1
2 cos(K~xsec)
2[VSTR-?!
-
.3 .3
02
.2
LOMER HULL INTERACTION
.. C.4
0-5
PORT. - ST75.
I CW
0-3
0.4
0.6
3
0-7
iNTERFERM"CE
--
0.,
Components of the Calculated Wave Resistance of "*)-SA eO'
Lowerhull
Assuming that the lowertal! is a fully submerged revolutional body, the equivalent line doublet distribution representing the lowerhull is described as folIows by usino the sectional area cur-e of lowerhull ACx)-
These
0
U-t
Fig. S-
3.1.2
ine doublets are placed on t e
centerline of lowerhul. And the wave resistance coefficient is =
C 2D where So1
2-
s;(a)+C!eziCos
e e- ,
=
-i.x)dxsecexo(Kzsec_) cost. =xsec)Z,
= -
2Ixxsec
-
MATHEMATICAL MODEL FOR WAVE RESISTA CE
(7) 3.1
C(-
Theory
A Ict of sophisticated theories of wave resistance have recently been developed for conventional ships. However, considering that the SSC consists of a oair of demihulls with much smaller bea--length ratio than that of a conventional ship, the classical thin ship theory is still effective for esti-ating the wave resistance of a shin with a slender hull forn like the SSC 171, [8), (91. The mathematical model for wave resistance described in thin paper is based On the thin ship theory applying aichell's source distribution for struts and tre line doublet distribution for loverhulls. A coordinate system We. x, y. z) mving with the ship is set up on the demihull, in whicn t ne x axis is positive to the direction of motion. Both struts and lowerhulls are assumed to be sy- -mtric about their centerplane v 0.
sin(K-xsec-)
-
The wave resistance of a t n can be obtained by Eqs. (4) and (7). Taking the distance of hull separation as 2bt the total wave resistance co-effici-ent is described by,
r..
s~p~dcsbe
'_
T
where SyjE)
2Ss(2)cos =2S
Cr(1-) =2Cs(e)cosrz
() |(}cos, t cosu
_ can be written in another form ,
Str-t_
Considering that the strut has a wallsided hull surface, the hull geo=etry of the strut is described by Y
)
Win Hull
3.1.3
Eq. (8) 3.1.1
exp{.:scc
+f W(x)
(1)
The source distribution rn(x) equivalent to this hull form is obtained by Eq. (2) from thin ship approxi tion,
C2a =
2-S2Vos'=l{[s(e)4Cse)1 +
t(Z)4C(-)i
+2t-S()S(e
)-CSci)C.AeKO1
= Cwstrutsitw(l
(eintera-tion)
.353.
trhuil5) .
(1 1
h., the factor 2(1'cos2_4 indicetes thle interference effect of tne twin hull.
HUwCvL, it is truublesume work to evaluate the density of doublet distribution equivalent to the effective camber. Also,
3.1.4
the camber effect appears more influentially
Camber Effect
The theory mentioned above is oased on the assumption that the flow around the demihull is symmetric. However, for the twin hulled SSC, it may be easily found that the demihull will actually be placed in a curveO flow due to the deformation of the flow by the presence of another demihull. In order to reflect this phenomena in the theory, it must be considered tnat the symmetric demihull has an apparent camber corresponding to the effect or the curved flow. This effective camber can be expressed theoretically by the normal dcublet distribution on the centerplane of each demihull. A strut with geometric camber is introduced to investigate the camber effect on wave resistance. Amplitude functions of the wave generated by the thickness of the strut are given by Eq. (4). And, denoting the equivalent doublet distribution to the camber as M(x), the amplitude functions by camber are Sc(O) = K-JM(x)dxfdzsecsesine exp(Kzsec) 's(Ke, cficients c.fM(x)dxfdzsec'OsinO -least 21 x exp(Kozsec2 0) sin(KoxsecB)
Cc(S)
= Ir
{[Ss(8)+Sc(6)] 2 +[Cs(e)+Cce)] 2 } ,results, coslede. t2)
22 o
X
J -,
Where, Ss(S) and Cs(S) are the even function of 5, on the other hand Sc(S) and Cc(S) become the odd function due to the term sinS in the Eq. (11). T'erefore, the wave resistance can be described by
xXcos'ede os dRw =
3
on the whole, seem to show the possible use of the thin ship theory to estimate the wave resistance of anSSC. However, as seen in the figure, the difference in phase and magnitude will not be desirable for ship designer. Tha may cause a serics problem in required poe estimation.
]+[Sg(S)+c"()]
Cws =27f2{ [sA(e)+CS(e) i
validationStudy Theoretical calculations were made for a typical SSC in order to confirm the validity of the mathematical model. In this calculation, singularity distributions as shown in Eq. (2) and (5) are expressed in a form of the polynominal function. The coefof the function are decided by the 3.2
square method from input data of the strut offset and the sectional area curve nf lowerhull. Amplitude functions of Eq.(4) and (7) are analytically calculated for each elementary wave angle e, then the wave resistance is obtained by Eq.(8). Figure 6 shows the comparison between the the SSC. measured resistance calculated coefficientand ofan The wave calculated
= -
Then the wave resistance coefficient of the strut is given by Cws
in the magnitude of diverging waves than in transverse waves due to the term sin 6 in Eq. (11). Therefore, as regards the wave resistance, this camber effect is not very significant as compared with the effect of thickness because the diverging waves contribute much less effectively to the wave resistance than the transverse waves. For these reasons there is no consideration of the camber effect in the mathematical model. Nevertheless, it can be easily expected that the properly selected geometric camb of the strut may cancel the camber effec\, which diminishes the last two terms in Eq. (14). As one approach to obtain the optimum geometric camber, Pien [10] proposes the rational scheme to deform each demihull along tne streamline generated by the opposite demihull.
i
x0
-
X
3
Cw(tnickness)+Cw(camber)
(13)
F=
2
10
Eq. (13) indicates that the thickness and the camber of wave the strut contribute independently to the resistance. For the struts placed in parallel ,the wave resistance coefficient is given by --
Cw = 2nfy{2(1+cos2w) [sA(S)+cA(S)] +2(1-cos2w)
(Sc (S)+Cc
(e)
MEASURED CALCULATED
]]1
i
-4sin 2 wEss(S)Cs(S)+cc(S)Sc(e) ] 1
1,,
0.2 =
cos 3 8de Cw(thickness) +Cw (camber) +cw(interaction)
.
0.3
Fig. 6 (14)
0.4
0.5
0.6
0.7
=V.g
Comparison of Wave Resistant.e of an
SSC between the Measured and the Calculated -559-
NOI -
~
-~;-u
-
As described in chapter 2, the wave resistance by the lowerhull has a fairly large portion of the total wave resistance. Therefore it becomes of prime importance to carefully compare the calculated wave resistance of the lowerhull with the measured one, in order to obtain good agreement between them. Figure 7 shows example results of lowerhull series tests presented in chapter 2, associated with its theoretical results, and there is still a little disagreement both in phase and magnitude Letween the measured and the calculated values.
where geometric submergence
f
Results calculated by using these correction factors are shown in Figure 8. The good agreement between the calculated and the measured wave resistance suggests that this method brings a better prediction of the wave resistance of the SSC and by which an optimized hull form will be developed. 0 RW 172p
5C1
1.0 MEASURED) CALCULATED-WITH
f/D
Do.
l2----
LD
CORRECTION
CALCULATED
0.2 0.3
0.4
CALCULATED CORRECTION R----- f/D - 1.5 MEASURED CALCULATED - - Do.
- 1.5 MEASURED
Do.
0.2
1.0MEASURED
Do.
3'WITH
£/D
0.7
0.5 0.6 Fn - V//r
Fig.
0.3
0.4
0.5
0.6
0.7
8 Comparison of Wave Resistance of Lowerhull between the Measured and the Calculated with Correction
Fig. 7 Comparison of Wave Resistance of Lowerhull between the Measured and the Calculated without Correction
4.
As regards these problems on wave resistance, Inui [il has found several correction factors for the displacement ship due to precise observation of the wave pattern and analytical investigations. These factors are corresponding to the so-called sheltering effect and the viscous effect. From the data obtained, the authors have found that the phase difference could be expressed in the following form in case of the SSC.
The theoretical design procedure to obtain the hull form with minimum wave resistance has been achieved by many hydrodynamicists for the conventional ships, but the effort is very limited for the SSC hull form [12], [13]. This paper presents the design procedure to obtain the optimum hull form of the SSC generated from combinations of a source distribution on the centerplace of strut and a line doublet distribution of the centerline of lowerhull, using the thin ship
~theory.
KoL(measured) = KoL(calculated) KL
F (KL, f/D)
-
AKoL 4.1 Singularity System and Restrain (15)Condition Singularities composed of the sources m(x) and the line doublets )i(x) are expressed by the polynomial functicn as follows,
where AKOL
phase difference
KL
l/Fn 2
F
linear combination
N r,(x)
a ax i M P(x) = b x ±=l
Also, the authors found that, instead of the geometric centroid of the lowerhull, the equivalent centroid for wave making effect should be adopted into calculation and the submergence of the lowerhull had to be reduced by the factor a which is the function of f/D to get more precise value in magnitude. The equivalent submergence is expressed by f=
f,
HULL FORM OPTIMIZATION STUDY
i-( ,(17) (18 .(18)
As the singularity distributions described in Eq. (17) and tl8) represent the 'single strut per hull' configuration without having any parallel part of the hull, other singularity distributions are also introduced for the different hull configurations such as the 'tandem struts per hull' configuration.
(16)
-560.
From the following variational principle
Restrain conditions for the principal particulars and hull geometries can be expressed in the following forms. "a (1) Displacement: N aiffx dxdz Vs = i-l VD =
i=l
= 0,
bj = 0
(25)
the Lagrange optimization equation can be expressed in the form, A X= B ,
for strut, (19)
bIx idx
for lowerhull.
(26)
where A is the symmetric matrix, 211
P NK
0
p
0
0
0
PKN 0
QLM
0
0
21
A
(2) Midshipsection area: N
As =
I aiffxi-ldxdz
for strut,
i=l
M b X(idship)
AD
,AW
211QML (20)
for lowerhull.
(3) Waterplane area: = N [ aijxidx 1= l
for strut.
and B is the column matrix defining the restrain conditions,
(21)
B = (0, ...
Other constraints can be written in the same way. 4.2
4.3
Then the wave resistance can be written as follows from the equations (4), (7), (17) and (18). N M N N a b II aa I +2 I CW = Si-lj j=li j ij
...)(28)
Numerical Examples
The computer program for the hull form optimization of the SSC was developed by the authors based on the abovementioned theory. This optimization program is available for the various hull combinations as well as for the hull element. Numerical examples are illustrated graphically. Figure 9 shows the optimized sectiona. area curves of the lowerhull with f/L = 0.08 and the equal displacement for different midshipsection area As at Fn = 0.32.
M 111b
0,?s,As,Aw,..,VD,AD,
The solution of the simultaneous equation (26) gives the optimum coefficients of singularity distributions.
Optimization Scheme
ilj1 j
(27)
(22) (2
where I Iij= 21TfT{Si(O)Sj(e)+Ci(6)Cj(e))cos3e x (l+cos2w)d, Si (e) = 2fxildxdzsec'oexp(Kozsec2e) C ci (e) =
x sin (KoxsecS), 1 if (23) 2 fxi oexp(Kozsec 8x e) SECTIONAL AREA CURVES x cos(Koxsec8).
and II., III.- are given by the same way. Tkan the 6ptimization functional p is written by, N
N
Rw T/-2p-VL
Cw
2
× i01
1"". /
M
N
i=lj-l
/~jl'
M M ±=lj-l -
'
1
M
L
-~
+
b.b III
+
+
........
X +K
K
N .
k=l
1
Pkai-rk)
-\
Af 2 = 0.006
I
-ik
+/ /
Q lb.-rI+K),
J=l
, "-
(24)
.
DEsIGN 0 SPEED
2 tAq(L - 0.007
--.
A/L
Fn
V/,/W
2
6
where X : Lagrange multipliers,
r : restrain conditions such as 7s, VD and Pik' Qil are given by Eq. (19), (20), (21).
Fig. 9
-561-
_0.008
Optimized Lowerhull for Different Aa/L2
It can be seen from the calculated wave resistance shown in this figure that the case of A,/L 2= 0.008 gives the minimum wa.'e resistance and the optimum prismatic coefficient. Figure 10 shows the optimized hull form of the lowerhull with f/L = 0.1 for different Froude number under the combination of the given strut as shown in the figure. With the increase of design speed, the end shape of the lowerhull tends to have larger bluntness and the midshipsection area of the lowerhull becomes smaller. The wave resistance coefficients for these optimized configurations are shown in Figure
11.
5.
DESIGN AND SEA TRIAL RESULTS OF THE 'ESA 80'
_1 Desia n
'MESAThe 80' first being commercial capable ofprototype carrying SSC 446 passengers was designed at her service speed of 23 knots and required to be stably serviceable even in rough seas with the significant wave height up to 3.5 meters. Under the basic design requirement briefly mentioned above, a lot of preliminary designs were compared in the early design stage, changing hull configuration, hull material and propulsion system, etc. to attain the reasonable SSC design. Then, the anti-corrosive aluminum alloy was chosen as hull material so as to reduce the structural weight. Although a gas turbine was considered as the prime mover, the diesel engine with light weight was selected because of its cheaper operating and maintenance cost in spite of its heavier weight and larger size than that of a gas turbine. Since the main engines were located on the second deck, the newly developed power transmission system of so-called Z-drive type was adopted, which has higher transmission efficiency and reliability than the other transmission systems considered. As regards the hull configuration, the 'single strut per hull' was considered to be more favourable than the 'tandem struts per hull' from the viewpoints of space availability, stability and speed.
GIVEN STRUT
OPTIMIZED AT En
=
0.36
OPTIMIZED AT F
=
0.32
--.
-----OPTIMIZED AT En
0.27
Optimized Lowerhull with Given Strut for Different Froude Number
Fig. 10
Rw / 1.
Sometimes, an optimized but unpractical hull form of the lowerhull is obtained at the higher Froude number than 0.4. In such a case, necessary restrain conditions for obtaining the practical hull form must be introduced into the calculation. Also, in designing the practical hull form, the aftbody shape of the lowerhull shall be modified from a viewpoint of the propulsive performance.
L2
Table 1
iO'
OPTIMIZED AT En 0.36 --- OPTIMIZEDLength 0P lzAT o.32•
-
--- OPTIMIZED AT Fe
Length overall b.p. Breadth (mid.), Max. Depth (mld.)
/
0.27
/
/
m m m m
Maximum speed at design draft 27.1 knots Main engine: Fuji-S.E.M.T. marine-diesel Max.continuous output: 2 sets 4,050 PS/1,475 rpm
0.4 Fn
Fig. 11
about 35.9 31.50 17.10 5.845
Design draft (mld.) 3.15 m Maximum design draft (mld.) 3.65 m Passenger capacity (all seated) 446 persons Crew complement 7 persons
/
0.3
Principal Particulars of 'MESA 80'
V/=vr
Wave Resistance of Optimized Lowerhull (Shown in Fig. 10) -562-
-
-
-
-i -
=-
--
-,
WTTFF
u
L Fig. 12
General Arrangement Plan of 'MESA 80'
W_
After the candidate design with its displacement of about 370 tons was selected, detailed parametric studies on hull dimensions were carried out, and the final design was decided. Principal particulars and the general arrangement plan of-%,the 'MESA 80' are shown in Table 1 and Figure 12 respectively. In deciding the hull dimensions of the 'MESA 80', a lot of design trade-off studies were made. As a result, the proportion of the lowerhull and its submergence were finally set L/De of 12.1 and f/De of 0.99 for the maximum design draft, where De means the equivalent diameter of the lowerhull. _ _"__ " with _ _ _ large L/D ratio was not The lowerhull adopted because of its weight increase, although only a little gain in resistance was obtained as compared with the lowerhull adopted in the 'MESA 80'. Selection of the submergence of the lowerhull was restricted by the maximum draft of 3.8 meters, desirably 3.65 meters. Also the elliptical cross section of the lowerhull was considered to be more favourable than the circular cross section because of the greater f/D, easiness for the reduction gear arrangement and the larger damping force of motion. The amount of the waterplane area and the distance between demihulls were decided mainly from the motion performance and the requirements on stability. These dimensions were selected so as to have enough large natural period of motion except heaving to avoid notion resonance for the design wave with its period of about 6 seconds.
The designed natural periods of motion of the 'MESA 80' are shown in Table 2. Despite of a short natural period of the heaving motion, the acceptably small amplitude even in the resonant waves was confirmed in the model tests because of increasing the motion damping forces by fins and the lowerhull with elliptical cross sections. Tlable 2
Natural Periods of Motion of 'MESA 80' at Maximum Dsg rf
I
5.2
Heave. 6.2
-.
T(sec)
Pitch " ' 9.5
Roll "
10.9
Typical Results of Sea Trial Tests
From September 1979, the extensive sea trials of the 'MESA 80' were conducted to confirm the full scale performance and to evaluate the theoretical and/or model predictions. Especially, the propulsion tests and the seakeeping tests which evidently reflect the effort of the hull form design, were extensively carried out several times at the different sea conditions. During these test period, the highest significant wave measured by the wave-rider buoy was 2.6 meters
-563-
~J
and also the wave height of about 3 meters in significant value was visually observed in another test. The speed tests in the calm sea condition were conducted by use of the mile-post in Tokyo Bay, changing ship's draft, trim and engine load. The maximum speed attained at the design draft was 27.1 knots. This result showed the good agreement with the predicted speed from the model propulsion tests. While, the speed in waves was measured on the maximum design draft by using the electro-magnetic type log. Figure 13 shows the measured speeds at service power for different sea conditions. Also in the figure, the measured speed are compared with the speed in the calm sea condition in a form of the ratio. It can be founO from this figure that the speed loss of the 'MESA 80' even in high Sea State 4 is only less than 2 percents, which is much smaller than predicted. 215 1.0
-- 24
r_
It was predicted in advance frcm the model test results that the large secondary humD of the wave resistance coefficient of the 'MESA 80' as seen in Figure 5, would cause the unstableness in the speed-power relation. Therefore, the speed measurements in the low speed region from 10 knots to 13 knots were carefully carried out. The results of these tests are shown in Figure 14 associated with the predicted curve from the model tests. Althouqh it is a minor problem in the case of the 'MESA 80' because of her sufficient engine power to get over such unstable speed region, careful attention must be paid for the hull form design of the SSC with design Froude number up to 0.4. The seakeeping tests were also carried out extensively, changing the ship's speed and the ship's heading to the wave. Typical results of the measurements in the sea of 2.4 meters' significant wave height are shown in Figure 15. The automatically controlled fins were used in the tests. Figure shows the pitching response (single significant amplitude) for different heading angles. As can be seen in the figure, pitching angle is less than 1 degree for
all headings.
1 23
0.95 o0.90
2
=
2
4
3
--
50
SEA STATE
4 -5
4
WAVE HEIGHT
2.4 m
SIGNIFICANT
z3
5
SEA STATE2 13 Speed in Waves of 'MESA 80'
Fig.
z
FULL SCALE TRIAL(MILE-POST) (Ei.ECTRO-MAGNETIC LOG) PREDICTED FROM MODEL TESTS
O * -
HEAD
Fig. A/
i
"
NF ,a4-
/
FOLLOWING BEAM WAVE DIRECTION
15 Pitching Amplitude of 'MESA 80'
The sea trial results of the 'MESA 80' briefly presented above have shown fairly agreement with the predicted results from the theoretical calculation and the modal tests. Also, the data obtained by these full scale tests gave a lot of useful information towards establishing the technology for future SSCs.
4good
1,500 :E -
500[
6.
.[004
CONCLUDING REMARKS
B.H.P. Owing to the efforts on SWATH by the U.S. Navy as well as our developmental work on the SSC, recognition of the SSC is now gradually spreading in the field of marine and ocean development. Although it was the most essential outcome to confirm her various advantageous performance in actual seas at the developmental stage of the SSC, the design te.hnologies for the hull form optimization is nowadays being required.
[ 500 -transportation 10
15 SHIP SPEED(KNOTS)
Fig.
14 Speed-Power Relation of 'MESA 80' in Low Speed Region -564-
In the paper presented here, a theoretical approach for minimizing the wave resistance of the SSC has been made based on the thin ship theory with some correction factors, and it has been found that the mathematical model newly developed was practically quite useful in designing the optimum hull form of the SSC. Finally in order to expand SSC applications, it will be necessary to establish the design synthesis technology to obtain a highly economical SSC.
Seminar on Wave Resistan.;,-t. Design", 1976. 1i. Inui, T., Iwata, T. and Sen, P., "On the Humps and Hollows of the Wavemaking Resistance of Ships", JSNA jAPAN, Vol. 97, 1955. 12. Lee, A.Y.C., "Source Generated Ships of Minimum theoretical Wave Resistance", RINA Vol. 111, 1969. 13. Maruo, H., "Theory and Application of Semi-Submerged Ships of Minimum Wave Resistance", Japan Shipbuilding & Marine Engneering, Vol. 4, 1969.
ACKNOWLEDGMENT The authors wizh to note that this paper is based on the efforts of many persons concerned with the SSC projeet at Mitsui Engineering and Shipbuildii.g Company. The authors are deeply grateful to DrT. Tomita, Mr. M. Oshima and Dr. H. Narita who are promoting the SSC project at Mitsui for their encouragement in carrying out the R & D work for the SSC. REFERENCES 1. Lang, T.G., Hightower, J.D., Strickland, A.T., "Design and Development of the 190-Ton Stable Semisubmerged Platform (SSP)" Transactions of the ASME, ASME Winter Annual Meeting, Paper No. 73-WA/Oct. 2, Detroit, Mich., November 1973. 2. Hightower, J.D., Seiple, R.L., "Operational Experiences with the SWATH Ship SSP KAIMALINO", AIAA/SNAME Advanced Marine Vehicles Conference, San Diego, CA, Paper No. 78-741, April 1978. 3. Hawkins, S., Sarchin, T., "The Small Waterplane-Area Twin Hull (SWATH) Program --- A Status Report", AIAA/SNAME
Advanced Marine Vehicles Conference, San Diego, CA, Paper No. 74-324, February 1974. 4. Lamb, G.R., Fein, J.A., "The Developing Technology for SWATH Ships" AIAA/SNAME Advanced Marine Vehicles Conference, Baltimore, Maryland, Paper No. 79-2003, October 1979. 5. Lamb, G.R., "The SWATH Concept: Designing Superior Operability into a Surface Displacement Ship", DTNSRDE Report No. 4570, December 1975. 6. Oshima, M., Narita, H., Kunitake, Y,, "Experiences with 12 Meter Long SemiSubmerged Catamaran (SSC) 'MARINE ACE' and Building of SSC Perry for 446 Passengers", AIAA/SNAME Advanced Marine Vehicles Conference, Baltimore, Maryland, Paper No. 792019, October 1979. 7. Chapman, R.B., "Hydrodynamic Drag of semisubmerged Ships", Trans. ASME, J. Basic Eng. Vol. 94, 1972. 8. Lin, W.C., "The Force and Moment on a Twin-Hull Ship in Steady Potential Flow", Tenth Symposium onNaval Hydrodynamics 1974. 9. Beasho, M., "On the Wave Resistance Theory of a Submerged Body", JSNA JAPAN Vol. 99, 1956. 10. Pien, P.C., "Catamaran Hull-Form
*565.
Discussion
=The
with those of the authors' as shown in Figures 9 & 10. For Froude Number greater than 0.4, the most important consideration in wave resistance is "thinness" (ie low hull diameter and low strut thickness). C.Kennel(NavalSea System Commaj C KeneIUav~!ea~s~emomrnnd)At lower speeds, "slenderness" is most important (ie low hull prismatic coefficient Semi-Submerged Catamaran is a promising displacement ship concept that 2 allows the naval architect new freedom to C 'VLH /= design more efficient ships to challenge the worlds oceans. Mztsui Engiroering and low waterplane area coefficient Shipbuilding has been advancing Jur understanding of this new tectnology in a most = w.P. Area/t,.L Cwp impressive manner. This paper marks yet another significant achievement in this difficult effort. I would like to congratThis later conclusion should not surprise ulate the authors on their interesting and devoted readers of old naval architecture thought provoking paper. There are a few books which refer to the wave drag hump at questions 1 would like to ask the authors Fn.= 0.3 as the "prismatic hump". I would to better appreciate their fine work. like to ask the authors if they have done The curves for lower hull wave drag similar optimization studies for tandem in Figure 1 were derived by subtracting strut forms ? the total resistance at deep submergence While the quality of SSC technology from the resistance at shallow submergence. being developed by Mitsui is evident from I would like to know how deep the deep this paper, the true value of Mitsui's submergence was and how the effects )f effort can best be appreciated by observing model supports were treated. their ship MESA 80. I am extremely grateful Figure 4 presents strut residual reto Mitsui for allowing me to inspect MESA sistance curves for two submergence depths 80 while underway last week. This high (10% & 15% of strut length). My design speed ship is indeed impressive. Although experience has convinced me that submerthe waves on Tokyo Bay were small during my gence depths of about 5% of strut length voyage, data I have seen and movies showing are best. In addition, Mitsui's MESA 80 MESA 80 traveling at high speeds in a statS has a submergence depth under 5%. I would 5 seaway show that the Semi-Submerged Catlike to hear the authors' opinions on the amaran concept works in a most impressive effect of these much smaller submergence manner. Mitsui Engineering and Shipbuilddepths. Also, I would like to know how ing and the authors are to be congratulated strut end effects were treated. Figure 6 fcr their significant achievements. shows excellent correlation between measured residual resistance and calculated wavemaking resistance for a SSC. CompariC.M. Lee(DTNSRDOC) sons I made in 1973 with model test data for several hull forms showed similar The authors presented a procedure how agreement in curve shape, but, the difto optimize the hull form for semi-submerged ference between theory and experiment varied foi the different cases. I finally catamarans based on wave resistance reduction. As is well known, the semi-submerged concluded that a correlation factor twin-hull concept has evolved mainly die to its advantages on seakeeping qualities. LRR 1 0.0005 Thus, any optimum-hull design for SSC R should pay an equal attention, at least, to 2 the seakeeping performance. It appears that MESA 80 is a fairly was a good value to add to wavemaking resuccessful design in the viewpoint of both sistance for design estimates. This seems resistance and seakeeping performance. I to be roughly in agreement with the auwould like to know how the seakeeping asthors' data. I would like to know if the pects of that ship was reflected in the authors have made such comparisons for hull design procedure. other configurations (particularly tandem At DTNSRDC, an optimum hull design strut formz). If so, is the correlation methodology based on both resistance and similar ? seakeeping has been sought, and I would My approach to hull form optimization say, a certain degree of success has been is to treat the problem as a total ship achieved. As P part of this effort, a design rather than a resistance optimizasimplified prediction of motion in waves tion problem. As a result, realistic has been developed by using approximate limitations imposed by machinery arrangemeans of obtaining the added mass, damping, ment, intact and damaged stability, strucand wave excited coefficients. The comtural design, and access requirements must putation time for motion has been signifibe observed. Nevertheless, my conclusions cantly reduced, which allows a more effeeappear to be in completement agreement tive iterations necessary in the hull
-566-
design.
is fully examined.
I would like to end my discussion with a sincere congratulation to the authors for their valuable and successful worK.
On comparing with the above mentioned investigations, the result described in the present paper seems to be far less extensive and less complete. Nevertheless the present authors have not given any referto the above achievements. I wish to ask if the authors do not have any knowledge about the above investigations at all, have ignored them intentionally by some oter reasons.
Hence H.Tanaka (SRI
I-
cor the authors on their I congratulate fine works and have great respect for many efforts in the long time. My simple asking is concerned with the principle of linearity
in a resultant wave. The authors prediction method of the wave resistance, essentially depends on the linear combination of waves due to hull elements. Unlike conventional ship hull forms, complicated hull forms in a semi-submerged catamaran (ssc) seems to be difficult to realize the linearity in the synthesis of the ship waves. I can imagine that the ]inearity is spoiled by characteristics of a submerged body with shallow submergence which generates a couple of vortices as a trailing vortex of a wing. Furthermore, concerning high speed catamarans, the interaction of ship waves between both demi-hulls are left unsolved. Studying their paper, I hardly confirm about realization of the linearity because none of wave patterns or wave spectra were shown in this paper.
For the optimization
of special hull forms such as ssc's, checking on realization of the linearity plays important role at the beginning cf studies, I would like to have their experiences or opinions on this problem,
C.C.Hsueh ouazfogInstof);
I'm glad to hear the theoretical approach for minimizing the wave resistance of the SSC, because I am investigating a method of minimizing the wave resistance with the shiDform as follows (Ref.l)). A new shipform of flat bow and cochleachanneld stern is described whose excellent performance is compared with those of conventional shipforms and the fundamental principles are tentatively explained. The effect of the bow and stern on ship performance and some abnormal phenomena observed during the investigation are discussed. Comparative tests on the ship models show that when(®= 2.8 v 1.7: thp rpqi-
tance of the new ship is decreased by 25 40%, the wave height from the bow is decreased by 60%, propulsive efficiency is increased by 35 . 45% and horsepower of the engine can be saved in the amount of 35 n 45%.
The first full-scale ship's trial trip has also been discussed. The facts that the design of the con-
H.MarO (Yhamafl oiv)
ventional ship is finalized and that thi.
NJ
Extensive investigations into semisubmerged hulls have been carried out by Ishii and a synthetical report of the investigation was submitted to Osaka University as a thesis in 1967. Several results of this jtudy have been published partly in other publications from time to time. The research involves a thorough examination of the condition of optimal hull configuration, and feasibility study of the semi-submerged hulls including catamarans in the practical operation from the point of view of design practice. The principle of the interference between waves generated by the underwater main hall and those by the surface piercing strut is fully exploited to minimize the wave resiscance, and model tests have proved successfully the validity of the theoretical assessment. From the theoretical point of view, the optimization of the semi-submerged hull form can be attained only by taking advantage of the above principle, and the method described in the present paper can never iealize the true optimization. The investigation includes also a study of the effect of mutual interference between each demihull of a catamaran and the optimal cond'.tion on the spacing between demihulls
type of ship is still being built make it appropriate to be chosen as a typical shipform for comparison. Let me summarize as follows: (1) In spite of the fact that the cochleachannels have given the new ship added resistance and the wetted surface is also rather large, this shipform still has fairly low total resistance for medium and hich speed ships because it can considerably reduce residual resistance. (2) The new ship has relatively large wake fraction, small thrust deduction fraction, and reduce energy loss of the rotating race, thus possessing better than average propulsive coefficient. (3) The new ship is a promising inland river new shipform because it has good wavedissipating capacity, appreciable increase in speed, evident improvement on stability, manoeuvrability and anti-flooding. (4) Such questions as the resistance, propulsion factors and slimming of new ship are to be studied turther. REFERENCES 1, Hsueh, Chung-Chuan et al., "An In.567-
-~~~
T
---
~ ~
-
-
vestigation of a New Shipform", Huazhong Institute of Technology, Wuhan, China
tance of each model support. In the resistance tests of struts, no special consideration was paid for strut end effects. Regarding Hr. Kennell's contribution on the optimal depth of strut, the authors should say that the submergence depth of strut at design draft condition has to be decided so as to minimize the total resistance, and the strut depth should be minimized in the high Froude number region where the wave resistance of lowerhull may be fairly small as compared with the total
o Reply Authors Re y Y. K(take (MtsuiE&S)
resistance. Since researches on the SSC's hull form by the authors have been carried out
The authors are extremely grateful for valuable discussions on the paper which describes a hull form design method of the SSC and the design procedure of the MESA 80 as well as her trial results. Unlike a conventional moaohull ship, the SSC has so complicated hull form that the change of a design parameter of the SSC influences on various ship performance much more sensitively than the conventional ships. Therefore, careful design synthesis has to be conducted for hul] form opt4 .mization from the view poiric of the total ship design on which Mr. Kennell mentioned in his discussion. Also, the remarks of Dr. Lee on seakeeping performance in the SSC design work are quite agreeable with the authors. The authors should say that great attention for seakeeping of the MESA 80 as well as for wave resistance was paid in selecting the sectional sbape of the lowerhull, amount of the waterplane area and distance between demihulls, etc. The authors are very pleased to answer Mr. Kennell's questions which make the author's paper more precise and understandable. AS to his question on Figures 1 & 4 of the paper, the authors should say that the submergence depth of lowerhull in fully submerged condition was four times of diameter of the luwerhull, and the difference of resistance between the model supports with different depth was eliminated by using the calculated frictional resis-
-568
mostly for the 'single strut per hull' cc-figuration, the authors have not yet enough informations for hull form optimization of the 'tande strut per hull' 7onfiguration. However, the authors would like to agree with Mr. Kennell's suggestion on the hull form parameters which strongly influence to wave resistance reduction. As Dr. Tanaka points out in his discussion, it seems to be very important to consider precisely the actual hydrodynamic phenomena of waves generated by hull elements of SSC into the mathemtical model for better prediction of wave resistance. It is observed in the model tank tests of the SSC that the waves inside the both demihulls show complicated behavior which suggests the presence of the nonlinearity. Although the measured wave resistance shows a fairly good agreement with the calculated value by the mathematical model presented here, which is based on the assumption of the linear combination of the waves, the authors will intend to refine further our mathematical model so that the actual phenomena can be more exactly described. In reply to Prof. Maruo, the authors should say that the interference effects of the hull element on the wave resistance are fully included in the optimization described in the paper, and Dr. Ishii's work will be kept in mind for future work on this problem. Prof. C.C. Hsueh's contribution is also welcomed.
I Theoretical and Experimental Investigations of Non-Equilibrium Jet of Air Cushion Vehicles Tao Ma, WeI-Un Zhou and Xiong Gu ShanghaiShip Design and Research Insttute Shanghai. Chnma
ABSTRACT
t -jet thickness. 1 -finger length. h -hover height e -finger inclination. ? ---Mtion viscous coefficient. -air density. T -wall tangential stress. P -pressure. Q -flow rate. M -jet momentum. N -hovering power. T -non-equilibrium flow coefficient wh1pm-derivatives of non-equilibrium cushion flow versus hover height, cushion pressure and finger inclination respectively Ce -cushion flow escape coefficient.
Mathematical models of two types of cushion configration, i.e. wall jet and olenum charber, are established fo bag-finger skirt system under equilibrium, overfed
and underfed operating conditions. The wall jet theory is based on turbulent jet theory and boundary layer theory for the part of wall jet, and on momentum equation for the part of jet curtain, similar relationships are set up for plenum chamber theory according to Bernoulli's equations, thus yielding general theoretical expressions for bag-finger skirt systems. Agreement between theory and test results is verified through nonequilibrium jet test performed in a skirt test rig. The present theoretical formulae are applicable to the determination of general hovering parameters and the prediction of such motion characteristics as stability and seakeeping quality. NOMENCLATURE
Foot Notations m- maximum value j- nozzle value t.c-under pressure, respectively representing bag and cushion pressure e-escape value ea-escape into the atmosphere c- under Q, representing underfed or overfed flow out of or into cushion 1. INTRODUCTION
X.Y.-wall jet coordinates. X is alongside the wall and Y is perpendicular to the wall. X0 distance from jet core to jet exit. -boundary layer thickness, -5" momentum loss thickness. b jet mixing layer thickness. bt -coordinate of escape flow boundary streamline. , - non-dimensional boundary streamline coordinate in boundary layer and jet mixing layer respectively, ,' -integral functions defined by equations (15) (17).
In the field of basic theoretical research of air cushion principle, there have appeared various mathematical models since 1960's on the basis of large amount of experimental and theoretical investigations, of which thin jet theory 1I)and exponential theory [2) for peripheral skirt, wall jet
-569-
_-
_-__
_
-
-_-__
-
'
-
1_-
-
-
Pt
PC
Qe3e ____
____
Qe
___Qj
Equilibrium
Peripheatl Jet
h>0
Wall Jet
hQe
Qjy
A =0pressure O
-606
-
-r
vide K- E distributions appropriate to next downstream plane. (e) Then, the same manner from step (b) to (d) are repeated at all the downstream plane in the flow field. Reaching the exit plane, one marching is completed. (f) The marchings from inlet plane to exit plane are repeated several times, and the pressure in three dimensional array is improved at every marching. (g) Finally, the calculation is terminated when the pressure of the flow field in current marching do not alter comparing the pressure in previous marching.
M where ',.Ax is the maximum radius of the body. The inlet plane is located at 0.5L from the leading edge and the exit plane is at 0.18L behind the body. The grid nodes at r-6 plane are l2xl2 whose configuration is shown in Fig. 3(a). The grid lines along x-direction are 35 whose interval is every 0.02L. The grid configuration of x-r plane in stern region is shown in Fig. 3(b). 0=0
3. NUMERICAL RESULTS Calculations are performed for two models by using the method in the previous section. The first one is an axisymmetric body which is named Afterbody 1 by Huang et al. (1978). The second one is a liner ship model which is named Model B (Muraoka 1979). Comparing the flow properties between calculation and experiment for the axisymmetric body, the basic possibility of the present calculation method is examined. The validity of the method to the complicated flow around actual ship is tested by calculating the flow around the liner ship model.
CY -
1
Fig. 3(a)
o3.1 Calculation for Axisy'mmetric Body The principle particulars of the axisymmetric body are shown in Table 2 and the afterbody configuration is shown in
Grid configuration of r-e plane for axisymmetric body fraiyxercbd
x/L
Fig. 2. Principle Particulars of axisymmetric body
Table 2.
L(m)
3.066
rrmx (M)
Re
0.1397 6.60x106 ..
Fig.
3(b)
Grid configuration sf x-r plane for axisymmetric body
V 0O.5 01 0.5
0.6
0.7
0.8 Iexperiment.
Fig. 2
0.9
1.0
Aftbody configuration of axisymmetric body
The numerical calculation of the flow around the body is carried out in the following way. The location of outer boundary in the present calculation is assumed to place at the constant distance from the hull,
-607-
The inlet conditions are shown in Fig. 4 where the circles indicate the values at grid nodes. The comparison of calculated and measured velocity components, u/V and -v/V are shown in Fig. 5 and Fig. 6, respectively. in Fig. 6, the calculated result of -v/V at x/L=l.057 is nearly equal to zero and is slightly different from the The calculated velocity component by the present method have the save high accuracy as the numerical results of boundary layer theories of Huang et al. (1978) and Soejima (1980). In Fig. 7, the pressure variations in the flow field in interest are shown comparing with the measured ones and the numerical results of potential flow. The viscous effect on pressure is appaared only near the vicinity of hull surface around the. sterri region and the calculated results agree with the ex-
periment except at x/L=l.057.
Fig. 8 shows the comparison of turbu-
Calculation
The measured K is deter-
lent properties.
mined by the fininition I (u, 2 -t v, . w,) where u', v' and w' means the velocity fluctuation in x-, r- and (-directions.
0
o
.
9;
ExerimentbyHuanetat.
0 ; Grid node
1.0"
0.1 3
4-
0
00
x/L - 0.846
0
20
0
0
x10"
0.5-
x/L = 0.755
0.5
(r - r)/rmax
0
0 x/L =0.934
X1
0
S05
0x x/L-. /L).O Mx
T
0 (r-rs)irmax
0
4
(r-rs)/rmax
0
0-5
0.5
X
Boundary conditionsI0Fig. at inlet plane
for axisymmetric body
ex/L=0.5 x/L =1.057
0
xlL =0755
6
IL0,6Fig.
0.75
Comparison of velocity component
The measured t is determined by Vt/c'1 while the calculated £ is determined by >C k/. The coincidences of turbulent properties between calculation and experiment are fairly good, although the accuracy of calculation becomes worse as x/L increase. The comparison of turbulent kinematic viscosity 14 are shown in Fig. 9. this figure, the values of Ur, Sp and 0 are all derived from the experiment of Huang et al. The numerical results of Hluang et al's method are coincident with one line by nondimensionalizing because of 0-equation model of turbulence, but the experimental values are not coincident with one line and diminish towards the stern. On the other hand, the numerical results of the present method present fair agreement with the experiment at each section- From the above examination, it can be said that in evaluating the turbulent properties like K, t and W around the axisymmetric body, the K- E model of turbulence is effective.
x/L=0.934
xJ'L1.057In
0 Calculation S
0.5
(r- rs)/rmax"
1.0
0
0.25
Exerimen byHuange.
3.2 Calculation for Liner Ship Model
In the previous section, the validity of the present method for axisymmetric body 0_0
was confirmed by comparing the calculation
0.25
0.5
of velocity profile, pressure distribution and turbulent properties with experiment.
0
(r-rs)/rmax
Here, the present method is tested to the
liner ship model B (Muraoka 1979).
Fig.S
The principle particulars of the model is shown in Table 3 and the body plan,
Comparison of velocity component u/V for axisymmetric body -608.
~-
-_
k
Calculation of Viscous Plow
Calculation Experiment :)y Huang et al.
ou*&O -
Calculation of Potential Flow
-
o0 t0
3
1ix0
Experiment by Huangestal.
A9
xIL
=0.755
1xIL 0.846
-01-x/L
.755
2-
0
..0 . ..... xIL .846
4 0-
-0 .05
0.1
xxL.934
'0o,
0.
0 93
0
J
0
~
XIL-.964
0-0
0.1/
0 0
Fig. 8 0.05105
0
Comparison of t-arbulent properties for axisynuetrii. body
Principle particulars
HuagIal
Caiculaoion
-
B(m)
CB
1i~o~r~
x10~~
0.9231 0.3231 0.5720
10. The outer boundary in the present calculation is placed at the cylin~drical shape of constant radius lr*/ 2 1.'7 . The inlet plane is located at St. 5 and the ship. exit plane is located at 0.lL behind The grid nodes at r-a plane are l2 xl2 whose configuration is shown in Fig. 11(a). The grid lines along x-direction are 31 whose interval is every 0.02L. The grid configurations of center vertical plane and water plane in stern region are shown in Fig. 11(b) and Fig. 11(c), respectively. The stern profile used in claculation is different fromt the actual ship model as shown in Fig. 11(b). The calculated wake distributions around stern region from St. 1 3/5 to St.
15]
6 10 50
~
J
A A
0.5
(r -rs)[6r
1.03 x10 7
stern and stern profiles are shown in Fig.
15-
Fig. 9
I
.28 ;
__p
Re
0
6.00
Lrn W
d (inm
4'10-
0.5
rs)/rmax
-
Experiffmni _____
(r - rsJ/rmax
liner ship model
Comparison of pressure variations in the flow around axisymmetric body
.75S
0
0.5
0.5of (r
Fig. 7
(r - rs)IrM#X
3.
_______________________________Table
0
XI1.057
-
1.0
Comparison of turbulent viscosity for axisynmetric body -609.
A.P.
C.L.
F.P.
.40
~~St. 1
0-0
ru o
St
..
S.
~
0S.-
fi.1 oypase liner n trship rfls of model 11() planefor
Grid tFig.configuration of liner ship model
waerpln
.3
at St. 3/5 0.2
I-UN .1
0.5
Ship
f Fig. 11(b)
0 /
Grid configuration of center vertical plane in stern region for liner ship model
1.0
vlv
Fig. 12(a)
-610-
B.L. 1.0
Calculated wake distribution for liner ship model7 x 107V at St. 1 3/5 and Re =1.03
M
-2/5 are shown in Fig. 12(a) through Fig. 12(e). The recirculating flow is seemed to occur around the bilge at St. 3/5, and to develop towards ship stern. In Fig. 12(e) at St. -2/5, the ring shape of equivelocity co~itour is observed near the cross
Fig. 13, the measured wake distributions at St. 1/5 is shown. Although the calculated radial velocity near center line is large than the measured one, the calculated wake L C.L.L 03
L.W.L. 0
St 1
0.2
0.3
0.5
.
0.2 1.0
0
1u0.-
j-.5
-!V
j
L
/~
0
1.8.1 /
Fig. 12(b)
v~w'/vfor
1.0
Fig. 12(d)
Calculated wake distr.ibution liner ship model atl St. 1/5 and Re= 1.03 x 1-0
Calculated wake distribution for liner ship model 1.03 x 10 7C.L at St. 1 and Re
St. -2/5
i0
S1.4
0.3.
0.2
C.L. -
St. 3/50 0.4
0.5 0
02 I -uN
L
0.5
0.1
1.0 -
0.2
B.L. 0
i1.0
1.0
-
Fig. 12(c)
B-L 1.0
1
Calculated wake distribution for liner ship model 1.03 x 10~ at St. 3/5 and Re Fig. 12(e)
Calculated wake distribution for liner ship model x 10~ at St. -2/5 and Ite =1.03
distr ibutions are seemed to show qualitatively the tendency of the flow behind ship stern. Fia. 14 cl-ws the calculated viscous pressure distribution on the hull surface from St. 5 to St. 1/2 along five diagonal lines con paring with the pressure d the experiment0 of the potential 1.z(Wamimatsu 1976: this ship is named M-7). The calculated viscous pressure from water plane to 0=45"1s diagonal line are closer to the measured one than the Pressure of the potential flow. on 0=673"0 s diagonal line, the calculated viscous pressure is different from the measured one.
Calculation of Viscous Flow fPtnilFo
aclto
-
frmEpret
siaefnxprwt .2-
01 0-
L.L.
~ 4c
.0 5
0.52. 0.3 -j
OA
(I00
0,
.-
0
/
aI
00
I -=I
*
Fig. 13 rrss
0 -/
14.D
01
.L.
Measured wake distribution
r
Fig. 14
liner ship model
at St. 1/5 and Re
Table 4
Compariso~n of pressure distribution on hull
at Re
1.03 x 10'
=1.03
sracof linberrhpoe x 107
Resistance Components of Liner Ship Model at Re
Resistance
Calculation
Conponent
(kg)
1-03 x 107
Eprmn ikg)Reak
Viscous (Rssace
0
j
146 (Rvp)pattern
Frictinal Resistance 4.17 (Rf)Exp.;
tICal.; J
From surface integral of P-P0 on the surface of afterbody
Exp.; Pressure resistance minus wave
08
resistaan
Cal.; From Cumpusty UIead method for Jforebody and from present method for *
4.29afebd Total resistance minus pressure
a
___________________________resistance
ViscousI Resistance
5.63
(Rvp+Rf) Do.~~~~~ () 65
I_____ 1
j
5.16 50
____________________
Frm the law of momentum by using wake distribution at 0.3L behind ship. -612-
V C.L. IL.W.L.
CI LW.L.
S t1
St.
mdlSt.
(o-e)
o
CO =.
B.LB.L.
Fig.
15.a)
Eih. 15(c)
LWIL'
-
,=0
Fig. 15(b)
%,.,.
.
respect of economy.-
(3) To make reflinement of arid configuraton' in order to obtain stable convergence. (4) To test other coordinate system in order perform item ()-
\to I
-1-03 x10'
thin boundary layer theory in order to improve the inlet conditions. refine the calculation technicues in To 2)
St 5-
t
Calculated distributions of 2 and *4'c for liner shin model at St. -2/5 and Re
LW.L.
1.700 "i
et1
x
C.L(
CL S/
:
L.B
KIV1O
Calculated distributions of K/2nd4for liner ship model at St. 1 3/5 and x lJ Re =1.03
K.1V
- 1100 1
-.
10
104r.P
(St. -2/5) an
10
(S.x2/)10'"
I KN
C.L.
C.L.
,bolic
4o CONCLUDING REMAPRRS b cOn the assumption of odrtialv mara-. _del of turbulence, the flow and
calculations a-e performed for an axisvmtric body and a are liner ship model. ThW derived: results )following ! The turbulent properties around the axi() sy-ntric body can be predicted well. Calculated distributions of 2 2)The calculated wake distributions fr for liner ship and K~/V
:=
liner ship model are qualitatively coEspecia- t wL Lthe isphenomena. se-d to occur. vortex the bilgew
model at St. 3/5 and Re = 1.03 x 107
(3) The evaluations of the pressure, turbulent properties and resistance com-pnents for ship are possible. Thus, the present method is made clear to be appropriate for the calculation of thick y layer and wake of ships thruch bn numerical examples. Further refinement of the present method is necessitated in order to improve the accuracy of calculation for practical use.
Table 4 shows the viscous resistance and viscous pressure resistance derived from the calculation of the same ship under the similar condition to the present calculation (Muraoka 1979). The calculated viscous resistance Rv is reasonably coincident with the sum of calculated viscous and calculated pressure resistance R , and the numerical frictional resistancei results show roughly coincident with the experiment. In Fig. 15(a) through Fig. 15(c), the calculated distributions of K/V2 and */v are shown where Vc denotes the laminar viscosity. The turbulent kinetic energy K at St. 3/5 in Fig. 15(b) is very intensive at the bilge part. The turbulent properties are high only within the boundary layer and the wake. From the above calculation, it is confirmed that the wake distribution around and behind ship can be predicted reasonably with the viscous pressure and turbulent properties. However, in order to make the present method more practical, the following items are required: (1) To utilize the differential method of
ACKNOWI- DGMENT Before closing this paper, the author expresses his ureatful thank to Prof. R. Yamazaki of Kyushu University for his advice and encouragement. The author would like to express his gratitude to Dr. - . Tasaki and Mr. M. Namimatsu of IshikawajimaHarima Heavy Ind. Co. Ltd. (1H) for their encouragement. The author also expresses his heartful thank to Dr. T- Tsuts.um of IHI for his stimulating discussion and suggestion. Thanks are extended to Mr. Y. Shirose of IHI for his cooperative wiorks.
-613
-
REFERENCE (1) Abdelmequid, A. M., Markatos, N. C., Spalding, D. B. and Muraoka, K., "A Method of Predicting Three-Dimensional Turbulent Flows around Ships' Hulls", International Symposium on Ship Viscous Resistance, SSPA, (1978) (2) Bradshow, P., Ferris, D. H. and Atwell, N. P., "Calculation of Boundary Layer Development Using the Turbulent Energy Equation", Journal of Fluid Mechanics, No. 28, (1967) (3) Cebeci, T., Kaups, K. and Moser, A., "A General Method for Calculating Three-Dimensional Incompressible and Turbulent Boundary Layers. ThreeDimensional.Flows in Curvilinear Orthogonal Coordinates", Douglas Aircraft Co-, Report MDC J6867, (1975) (4) Hatano, S., Mori, K. and Suzuki, T., "Calculation of Velocity Distributions in Ship Wake", Journal of the Society of Naval Architects of Japan, Vol. 141, (1977) (5) Himeno, Y. and Tanaka, I., "An Exact Tntegral Method for Solving ThreeDimensional Turbulent Boundary Layer Equation around Ship Hulls", Journal of Kansai Society of Naval Architects, Vol. 159, (1976) (6) Hoekstra, M., "Boundary Layer Flow vast a Boulbous Ship-Bow at Vanishing Froude Number", International Symposhum on Ship Viscous Resistance, SSPA, (1978) (7) Huang, T. T., Santelli, N. and Belt, G-, "Stern Boundary Layer Flow on Axisyrmetric Bodies", 12th Symposium on Navel Hydroaynamics, (1978) (8) Larsson, L., "Boundary Layer on Ships, Part 4", SSPA Report No- 47, (1974) (9) Launder, B. E. and Spalding, D. B-, "Mathematical Models of Turbulence', Academic Press, (1972) (10) Mori, K. and Doi, Y., "Approximate Prediction of Flow Field around Ship Stern by Asymptotic Expansion Method", Journal of the Society of Naval Architects, Vol. 144, (1978) (11) Muraoka, K., 'Calculation of Viscous Flow around Ship Stern", Transactions of the West-Japan Scciety of Navil Architects, No. 58, (1979) (12) Muraoka, K., "Examination of a 2Equation Model of Turbulence for Calculating the Viscous Flow around Ships", Journal of the society of Naval Architects of Japan, Vol. 147, U!980) (13) Nagazatsu, T., "Calculation of Viscous Pressure Resistance of Ships Based on a Higher Order Boundary Layer Theory", Journal of the Society of Naval Architects of Japan, Vnl. 147, (1980) (141 Nu=im.tsu, M., "A Measuring Method of Hull Pressure and Its Applicatioa'. Journal of the Society of Naval Architects of Japan, Vol. 139, (1976) (15) Okuno, T., 'Distribution of Wall Shear Stress and Cross Flow i= Three-
(16)
(17) (18)
(19) (20
(21)
Dimensional Boundary Layer on Ship Hull", Journal of the Society of Naval Architects of Japan, Vol. 1391 (1976) Patankar, S. V. and Spalding, D. B., "A Calculation Procedure for Heat, Mass and Momentum Transfer in ThreeDimensional Parabolic Flows", Journal of Heat and Mass Transfer, 15, (1972) Pratap, V. S., "Flow and Heat Transfer in Curved Ducts", Imperial College Report HTS/75/25, (1975) Raven, H. C., "Calculation of the Boundary Layer Flow around Three Ship Afterbodies", international Shipbuilding Progress, Vol. 27 (No. 305), (1980) Rotta, von J. C., -Turbulence Stromungen", B. G. Teuber, (1972) Soejima, S., "Calculation of Thick Boundary Layers around Ship Stern", Trans. of the West-Japan Society of Naval Architects, No. 60, (1980) Spalding, D. B., "Calculation Procedure of 3-D Parabolic and PartiallyParabolic Flows" Imperial College Report HTS/75/5, (1975)
-614.
U
DiscussionKMoi TT Huang wDmSoo Could the author give the specific iterative pocedure used in statement Cc) of 2.4 Calculation Procedure ? Why the computed mixing length paramemter t(m) shown in Figure 8 does not approach zero when the value of r becomes Qreater than the va-ue of (+ + rs* 7
=
G.E. Gaddt'M0e Althouch there tative .imilarities predictions and the clear as the author
are encouraging qualibetween the theoretical measurements, ii admits that further
refinement of the calculation method is
needed. When fig.12d is compared with fTi. 13 the most obvious deficiencies of the theory are seen to be: (i) insufficient orediction of the vortex effect (2) failure to predict the -iah shear region, with closely spaced veoutside a reion locity, contours, ousd ein of stagnant flow. Obviously the grid configuraticn shown in fio.lla is far from ideal for the flow near the stern. Does the author think that this may be a contributory factor to the defects of the prediction ? Also, is it possible that the turbulent shear stress model may become inadequate in such a situation 7
M.Hweksaf(v Just two orestions and one cocent.
(I) How did you harle the kink in the coordinate lines at the tail of the bod of revolution 7 (2) Did you experience in this wrk or in arn other study with the K oe any trouble with the length-scale equation
(w-equation) at the outer boundar. of the
computation d-- in 4r Wthin-k that the author's conclusion about the adequacy of thea-c-e1 for the flow past axismmetric bodies is premature. The present annlication is on a bodv with high lennth-dia=ezer ratio so that longitudinal curvature effects are weak. I suspect that the nredietinns fur fuller bodies would have been worse becanase the K - - model does not include prcperly the eflects of lo.n itud inal strea line curvature on turbulence.
the (1) Through the calculation axisymmetric body, the author concluded that the validity of the present -eho- is confirmed. ut, in case of 2-D f!Owb, the partially parabolic epproCimation reduces to the boundary laver approximation wi' higher orders. So the discusser dcoes not think that the partially parabolic anroximation is checked by the present calculation for the axisvmmetric body. (21 The partially parabolic anroximation neglects the diffusion in tne main flow direction and keeps that in the gi. hdirection. The discusser still can convince neiLher that the Iatter is more imortant than the fcrmer in cases of ship-like bodies nor that such a complicated approximation is necessary even for upstream region like S-S1 3 /5 or I. The results of the ax's'metric body which show goo agreements witn experiments for x
different from the pressure in the two other miom. ntum equations. This is a deli------berate inconsistency introduced into the
028
treatment of pressure and its purpose is to decouple the pressure gradients. By this means, marching integration can be employed and two-dimensional computer storage used all.'ariables {3). Often the pressure
in the z-momentum equation, (T), is taken as the space averaged pressure over a cross is assumed section, and the gradient ahe to be known (or calculated) before calculation of the lateral gradients tp/an,
and apaP
F -
ME MY
SOU
This practice is implicit inSOT
two-dimensional
V
F
boundary layer
theories
aloin the case considered here, the presence of the hydrofoil in the flow field will cause non-parabolic effects. These
Fig 3
Three-Dimensional Flow Domain
are taken Into account by employing the twodimensional pressure field calculated by the partially-parabolic procedure of Section 2.1 in calculating the axial pressure gradient (and only the axial pressure gradient) for each marching station. This decouples the pressure in the momentum equations and renders the latter parabolic, The boundary conditions for this stage of the investigations were selected to repthe hydrofoil mounted normal to the resent floor of the Circulating Water Channel as
The boundaries of this domain are the free boundaries (North and East), the solid boundaries (West, along the aerofoil and South). and the axis of symmetry (West, before the leading edge and after the trailing edge of the foil). The conditions given below: at these boundaries are -
described in Section 3. Inlet conditions: At the inlet to the solution domain (a distance of 0.8 chord from the airfoil leading edge) plug profiles were assigned to all variables, except the longitudinal velocity w, for which experimental profiles in the absence of the foil were used. The u and v components were set to zero. Two values for the mean level of inlet turbulence were used, namely, 5 and 10, of the mean inlet velocity. The dissipation rate of turbulence at inleL was specified according to a prescribed length scale distribution, and wa- given by:
West boundary: For the foil solid boundary, the no-slip conditions and "wall fu:.ctions" were employed. For the symmetry planz zero-gradient conditions were assigned. North boundary: Far from the floor, the flow is 2-D, therefore the results from Section 2.1 were used as boundary conditions, e.g. w = w 2 D, u=u 2 D, P=2D ak and
32-Because t=f
k
/
(34)
where f = 1/(0.0711)
South boundary: u=v=w=O. The values of k, c were obtained from the "wall functions".
(35)
and H was the total width of the domain. The nature of the boundary layer formed around the foil and along the floor at the flow channel, implied that the flow was expected to be 3-D in the vicinity of the corner formed between the foil and the floor. Far from the fin or the floor, the flow was expected to become two-dimensional.
-637-
a
0. 0.
(36)
(7
of entrainment at this boundary, the v-velocity was calculated by satisfying the local continuity at each calculation station. This velocity was used only in the continuity equation, whilst the momentum equations assumed zero North boundary v-velocity. East boundary: Far from the aerofoil the flow becomes 2-D, so that normal gradients of the variables were taken as zero across the boundary except for pressure which was assigned its known value at the North East corner of the domain. It should be noted that both the North and East boundaries are physically the
-
condition. Close to the surface a modified wall function accounting for axial pressur4 gradient was employed. This function is given by:
same, i.e. free, yet they are treated differently. At the North boundary inflow and outflow is allowed, whereas the East boundary is considered as a symmetry plane. Thus, the latter is consistent with the boundary conditions used in the two-dimen-|E sional calculations. As already discussed the pressure
p(T0 =
q
n
F L
(38)
U
gradient calculated for the two-dimensional where qp and yp are the total velocity at
case was used as the pressure gradient in the downstream (z) direction. In the direction normal to the floor (y) the pressure was assumed to be uniform and equal to the pressure at the North boundary-
the near-wall point P and its distance from the wall, and F is the modification factor given by:
2.2.2 Solution procedure. The grid layout used was a "staggered" grid system as before. Integration of the partial differential equation governing the transport of each variable yields an algehraic equation for each grid location, representing the discretized form of the balance of the varnable, over the control volume corresponding A special treatment is to that location. applied to the momentum equations, deve-
loped in (3. equation which is presre used and
yp a F = (+1
z9)
The point P must be located in the fully turbulent region, i.e.
+ y
P 11.5
(40)
The turbulence kinetic energy at the
This yields a Poisson
at a value prowas fixed point near wall to the dissipation rate /,and rt t that issipato o-tirnal turbulence at that point calculated fro a linear length scale distribution. Svzmetrv-plane conditions Downstream
th fo orrctin"po-tional correction-pressure the "pessre for equtio used to update the velocity and fields. Upwind differencing isof the difference equations are
solved in turn for each variable by the application of the tri-diagonal matrix algorithm. More details on the solution method may be found in the above reference. The ethd tothe inrodced here ereaxial introduced the method innovation toinnoatio is that iteration is provided at each axial station of the marching integration, in --order to improve the convergence and the Saccuracy of the results.
c as eros the r the body viously the Patal body becomes a symetryplane. Free-boundary conditions: At the freeboundary of the calculation domain, the ouavftecluaindmith velocity, w, and the static pressure a e p ere given the values obtained from a potential flow solution along that partiThe mass flow rate across cular surface. the boundary was computed from local con-
tinuity during the calculations. conditions: Calculations were Inlet starting either at half the body performed ~length from the nose or at 15S of the the nose.tthe hflth length frm 2.3.1 Theoretical formulation. The oi teno.t the alflegt pste basic geometrical model for which twot b pa dimensional calculations were obtained to be zero and the boundary layer paraduring this stage of the work was the body taken by the method e by by Patel Pmeterset al {S:. was considered The flow of Myring {11.from Thepredictions velocity profile the of edge th fro ine sl the using computed over and behind this was the edge of the scaled linearly tofrom in procedure described partially-parabolic the free boundary. layer mde.boundary to-dmenionl mode,.ublnekntceeg n is Secton ob a ae The its two-dimensional Section 2.2 in k was taken to be energy kinetic turbulence gcverthe and system The coordinate uniform and equal to a percentage of the ats s ad the gcve Thecrdin For the ning differential equations are the sesquared. rFor ula frsstraeocth s ue described in Section 2.2 but omitting the d9, was used to calculate the distribution third dimension, -. The dependent varnof length scale tt ables considered are also the same except When the calculaions started ahead of for the disappearance of the u-velocity component. It should be noted that for the the body, uniform values of v=0 and w=12.192m1s were used at inlet. Other conpredictions obtained during this stage, no ditions remained Abe same as given above. parabolic assumption is made concerning Because of Exit boundary conditions: The two-dimensional pressure prcssure. te so boundarabonintuof Ex t distribution is the outcome of the compunature of the soluthe partially-parabol-c tion procedure an additional boundary tational procedure which now operates in condition was needed for the pressure, at its partially-parabolic mode, the exit boundary of thez calculation domain The boundary conditions for these 2.3
S
Prediction of the Turbulent Flow Field Around An Axisymmetric Body
(i.e. 1/3 of body length downstream of the stern). It was assumed that the exit
calculations were as given below:
Wall conditions:zerofro At thethe bodyossurface,, the aelocitinwere the velocities were zero fronni the no-slip
sen. I a sue at the htteei value given by uniform, pressure was the potential flow solution. -638-
-
i
The height of the NACA 0017 seclion foil. foil was such that the outer sections were well clear of the surface boundary layer and the flow could be regarded as twodimensional. Measurements were taken at 6 6 and 0.5x10 , Reynolds numbers of 2.0x10 based on the foil chord of 450mm and in the !atter case an adverse pressure gradient was generated by inclining the plane surface 2.5 degrees to the flow direction. The non-dimensional boundary layer profiles, on the plane surface at the position of the fin leading edge but in the absence of the foil, are given in Figure 4. The longitudinal variation of static pressure coefficient with the inclined surface, again in the absence of the foil is shown in Figuie 5. This pressure variation has been non-dimensionalised using the velocity outside the boundary layer at the position of the foil leading edge. The procedure described in Section 2.1 has been used to predict the two-dimensional wake behind a NACA 0017 section foil and a flat plate in uniform flow. The variation of the wake centre-line velocity with downstream distance is given in Figure 6, for the NACA 0017 foil and in Figure 7 for the Also given for the foil are flat plate. values from the CWIC experiment, and from an empirical curve based on the correlations of Spence 10) and Lieblein and For the flat plate the Roudebush {11}. experimental results of Chevray and Kovasznay {12) have been included an Figure 7.
Initial guess for pressure: The pressures prevailing at the free-boundary were taken as tne initial guess to the pressure field. 2.3.2 Solution Procedure. There are two basic differences between the sourion procedure used and that described in Section 2.2.3. Firstly, it requires several sweeps of marching integration, from the upstream to the downstream end of the calculation domain, thus, it is an iterative procedure. Secondly, full account is taken of the effects of pressure, on axial velocity, This allows pressure effects to be transmitted in both upstream and downstream directions. More details on the sequence of calculation steps may be found in :6}.
-
. L
3.
COMPARISON BETWEEN THEORETICAL PREDICTIONS AND EXPERIMENTAL RESULTS
In the computations which are discussed below finite-difference grids were used which ensured that the results were For the tuo-dimensional grid independent. flow over zhe hydrofoil, the grid possessed 36 points in the longitudinal direction and For the 16 in the lateral direction. three-dimensional flow a grid of 12 nodes in each of the lateral directions and 40 in the longitudinal direction was adequate. For the half Patel body, a grid of 20 nod.!s radially and 30 n3des axially was used. III the case of the full Pate] body the number of nodes were 17 and 37 respectively. The grid spacing for all cases
EvEL FLOOR, ,
was non-uniform, the grid lines being more closely spaced near the wall and in the nose and tail region of the body. These grids were determined after calculations had been carried out with different grid Fro-t these studies, it was condensities. cluded that grids finer than those above led only to marginal changes in the results but to considerably increased cost. Convergence was monitored by observing, changes from aweep to sweep for flow variables at particular locations, at.d by computing -ass cont:nuity errors. in general. convergence was fast and monotonic. After about 50 sweeps. variabies were changing by less than 1 and the maximum integral mass imbalance at any plane was less than 0.01% of the inlet mass flow rate. 3.1
A
-----
C-IE
Z
0-
V.I o
FSooRl=o-S
10 G
IGO
E
I ,O00
.
In
I 40
Two and Three Dimensional Wake Behind a Hydrofoil Prior to cmnmencing the theeretical
studies described in this paper, an experimental investigation had been carried out at AMTE(H) into the wake generated behind a hydrofoil protruding through the boundary layer developed on a plane surface. These experiments were carried out in the Circulating Water Channel (CWC) at Haslar and consisted of velocity suirveys at a position 2/3 chord aft of the zIailing edge of a
-639-
-
A
0 0-4
0-
o.
o L V LOC eEC STREAM VELOCtTY Fig 4
Brundary Layer Profile at Foil Leading Edge
L -
--
PEIC-rI0N
IL
00
--
U,0-1
,W
(
ifw~
30 zul1
0 U-
III
~-
0
DISTANC e
Fig 5
I
600
400
P-00
O)STANCE
AFT OP FOIL- LEADING EDGE
Fig 7
Variation of Static Pressure in the Circulating Water Channel
a
Z xtO~
R1 .=
-_reduced.
1-2
RN=SXO
I
.
-
Wi > >I
..-
[
Variation of Wake Centreline Velocity Behind a Flat Plate
It can be seen from this Figure that the predictions for the NACA 0017 foil at high Reynolds number are in
_incidence.
>
GE
The empirical curve for aerofoil sections given in Figure 6 was based on from tests run at Reynolds numbers 6 6 and conducted on foils from 0.5x10 to 5x10 of varying section shape, thickness and
!data
U
-O
T*AILIN
lower Reynolds numbers and hence less rapid mixing of the flow in the near wake region. At a position about one chord aft of the foil trailing edge however, the effect of Reynolds number is very much
RESULTS
C WC.
AFT o L P EAT
0-8
0.G
It can be seen from Figure 6 that the effect of varying Reynolds number is most close to the trailing edge of the foil and that reducing the Reynolds number produces a slower recovery of the wake centreline velocity. This is the at result of less turbulence in the wake
_significant lCTlONS ----p -EKPEiIMENT ( i i 10
-
0-4.
c
B0OO
|
/good
(
X,0
f!t4=
agreement with the empirical curve; however, the predicted variation with Reynolds number is considerably higher that. that of the data used for the two corre-
O-
_0_____ 0-2
0
F DISTNCEAFT
lations. 0-G
0.4
0-
ship appendages.
FIL TAILNG EI~E
CNORD LE FOL.
GT---predicted
Comparison between the
centre-line velocity at a position 2/3 chord aft of the trailing edge and the CWC results show the predictions to be 9% low. Generalisation from these results is difficult; however, it appears that the predictions give a somewhat slower wake recovery than shown in the experimental data.
Variation of Wake Centreline Velocity Behind a Hydrofoil
Fig 6
In particular, the lowest
Reynolds number prediction shows a much slower wake recovery than the experimental data; however, this Reynolds number is well below the full-scale value for major
_
_
daaThe prediction of the wake behind the
IA_
thin flat plate is a less severe test of the finite-difference grid employed in the theoretical model than the finite thickness _640-
_
_
_
_
_
_
_
_
foil case. This is reflected in the very good agreement between the experimental results and The predictions for the flat plate given in Figure 7. Comparison between the results
PREDICT mN ,r the
hydr-foil and the flat plate suggest tnat the ess satisfactory prediztions in the former case may be associated with the specification of the finite-difference
,,
i
grid. The above results were found to be
i.,
insensitive to the variation of inlet turbulence produced by a 70% increase in the inlet mixing length and to the change in the wake mixing length law resulting from a doubling of a. (Section 2.2.1).0The three-dimensional solution described in Section 2.2 was used to predict the axial velocity profiles behind the NACA 0017 foil under the conditions of the tests in the Circulating Water Channel. The
HEIGIITAOYE SUR ACE(MM)
i
i
.
,
-
I
20
10/
I,0
results of these predictions are given in
_
_
_
Fig res 8 to 11 together with the exnerimen al data.
0-9
The influence of Reynolds number on the predictions is shown in Figure 8 from which it can be seen that near the channel floor the highest Reynolds number leads to the highest velocities. This is consistent with the thinning of the boundary layer on the channel floor as the Reynolds number increases. Clearly as the distance from the floor increases taiis effect
-
Ri =
-.o o-9 J I J U >-0 0
S %IO
SI 1
8
2a-
foil (Re~lx1O
>I, 08
beoe i
:
' i'
Oj l __,____
_____
0-7" !
,:' ":
160
Bc
0
80
l
:40nxn itually i ZS
ls
6
, Level Floor)"
iniiat
ota
b
bu
from the floor the results are virindependent of Reynolds number. .0 4( ThiS distance compares with a channel wall boundary layer thickness of the order
I
of
t-Direct i '
dicted and measured velocity profiles
100ra
(Figure 4). comparison between the pre-
(Figures 9. 10 and 11) is hindered by the-
non-sllrnetry of the latter; however, the agreement between the two is generally
160l
L-TCRAL,,_ t-os,-TIoNl(a
Fig 8
""
0_
**Y"'bU.
a Oa
-
J
II
.-
-
good. For all three cases, two Reynolds numbers with zero pressure gradient, and one adverse pressure gradient condition, the predictions appear to be most accurate near the channel floor. This is
Effect of Reynolds Number on Predicted Velocity Profiles (Level Floor)'
-641-
-
par icularly encouraging because this is the region where there are significant three-dimensional effects. As the distance
PREDICTION EXPERIMENT
-
from the floor increases and the flnw becomes more two-dimensional the wake defect
H-vEIGHT ASO
suFAeL=-nw)
is generally over predicted, as was the case
_
1-
for the purely two-dimensional predictions. This is especiallv true for the averse pressure gradient case. however. the test
o --
-
flow conditions were rather un.:table in
.
'
-
I
this situation and the measured data is thus not as reliable as that for the zero
_
pressure gradient condttion. The predicted velocity profiles near the channel floor clearly demonstrate the change in
_.O
tiow conditions which occur in the presence of the adverse pressure gradient caused b the inclination of the channel
floor.
-
is considerably reduced.
-
_
1'--0Sv
As in the
:
41
o
L It
technique enployed represented a sound framework for the t-rediction of the threedimensional flow behind a hydrofoil.
0-S
o,
.
0-7 PREDICTiON
i . . . ..
IG0
1-
Fig
. 0n -
-3.1 1fl-S _axisrunetric
0_j
-of
0UW 009
I
Ji
I1z-S
_____
__=
ia I
IGO
80
0
s0
AEAL
OE
:42o.___
0J
80
A IGO
LATRItAL. POSi'TION (rmm)
0
so
160
OI~Q
I1
3.2
Velociv Profiles foil(Re=~~x10 O.0
nid a Hvr Inclined Fior)
Flow Over the Tail Region of
Axisv=-ettic Body correlation between the nredictions and measurements described in Section were regarded as sufficient to warrant extension of the prediction method to an body. The most extensive set exoerimental data available for this was considered to be that of Patel et case al (8S1 and hence predictions have been made for comparison with the data given therein. The method of solution described in Section 2.3 was used to predict the flow varameters over the tail region of this body. The free boundary conditions wre determined from potential flow siution for the co=olete body. The inlet boundary conditions were determined either at the bodv mid-length from predictions of the bound-ry layer parameters given by the method of Myring 1l} or at a position 15% of the body length ahead of the body from the tondi-
Fig 10
Velocity Profiles Behind a HydroG , foil(fl=0.nl0~Leve Flor)tions foil (Re=O.5xlO Level Floor)
lengt
ae f thelodi of uniform inflow. -64'---
Nai
i-
SuRPAcCinift)
I
*.
-
-
-E-xPRMENT
0
;
>1 K . 0
w-. two dimensional case this variation was found to have little significant effec! upon the predicted velocities. The result, discussed above were considered to have demonstrated that the
T
_
0-' 08
The influence of inlet turbulence on
'!
_
1-0
the above solutions¢__ was investigated using
of 0.05 and 0.10
_
.--
In
the latter case the variation in velocity
values of
__
0-8
Initially studies were carried out using the half body inlet conditions to determine the sensitivity of the solution
BZL!Sh PR&DICTIO i#LET TJRSULEN'CE
xEPERIMENT
to the position of the free boundary, to the foim of the finite-difference grid and to the inlet turbulence level, and the results used to select these paramters. Predictions were then ^arried out for the .stern half of the Patel body and results are cntpared with the experimental data in Figures 12 to 17
I By,
o
do o
0
==
O-z
004
0-004
.=o;ttTboundarye
00t
ALX&AL
Fig 12
07
This is
71
that quence of the inlet plane assu~twOn the flow 's ful1y turbulent all thee way nis tne outer houndar , from -he odv not a reasonable assutico evidnly ..- undarv is located far since the uter zror the boundar" lave-. edte, and the experi=ents indicate that -he turbulent
z
o L14
r
Variation of Turbulence Kinetic Energy at 965 of Body Length Fr-- Nose but indicate a resioual of 2% turbulence U Fig 13
•
I 08
.c oE DIST^MCF- I=W0%.% r=" A 'r 600V
region is .Confined writhin the bound. ry etr_ There-fore. in future ok -ay:_r.-trc-! lenc- quantities s-Ivr L ently inside and u; -ie at the inlet piane. r bnd . Ti- axial variatio- of the ayer thickness is given in rigtre 14 fr.c which it is clear that there is very close etween the predicted and eagree-rment mental values. ne axial variatinof the static pressure coefficients at the wal and at are give. in layer cofficients of the the edgeISFigure Tneboundary ¢=...rimentai usedre tunnel wall pres re as reference free trem static udhich was oer than r wee r pressNre. hs the total head coefficient
Axial Variation of Skin Frction
Coefficient
--
'
:3
0
-
*-
4)-
-LET NCA TU_ tUL.E
IL
=
-
I
I'h
U.
Z
o8
The prdicted axial variation of the skin friction coefficient is given in Figuze 12 for two inlet turbuicnce levels, since this was the only nara-eter found to e significantly dependent upon inlet 05 turbulence level within -he range cf £ tht ...... it15 to w -04 the lo-er level of turbulence leads to predicted values of skin friction coefiicient which are very close to the m-easured values.
approxi--tely outside the boundary layer w 1.06 instead of the ideal value Of 1.0. T-ne predicted pressure coefficients have therefore been increased by 0.06 so that they are cctsistent with the experimetal correction was --de by A siilar data. it is Dyne -13' azd was approved by Patel. evident frm Figure 15 that the static pressure gradient is under-predicted betof the body we- approxi=ately 80% and 91% length but considerably over predicted for the last 10n of the body length.
The variation of turbulence kinetic velocity at t where w w, eey( energythe edge of the boundary layer) in the direct ion perpe-ndicular to the body at 98% of body length free the nose 1s given in Figure 13 for 8% and 15% inlet turbulence. it is apparent that the predicted turbulence level at this axial location is independent of inlet turbulence specification. Furthermore. the predictions are agreement with Patel*s measurein fair ments (81 over the boundary layer thickness
.643
1
X
tL
NU
r
0
01_
-EDICTIOo
E'(PERIMETNTC
Laof
Thdns
.Z
E~W%4
-
4----OS.
MY
fBd
Lnt
N --
P
EOIC-ION
rmNs
000 06S
FR0%A
AXIAL- flSTAM4CG
SOONTOTAL-
a as Va-OCTY
FREE STREAW
Fig 14
Fig 16
Axial Varia' ion of Boundary Laer Thicknes .. -
tEOICTiON
eKPEtIENT
O-S -
o *
-(1
,
.
_p
we.~ -
-becoming
i
a.
_
ui 0
..
-
-
3
The total velocity profiles 'or axial locations 66.2 and 991 of ihe bCy lenh from the fore-end are ivefl Figures ~and 17 Also shown for fte mostf0; location is the velociv -red;Acted v th yrxn| can hich was used to detmine the inlet b-.ound-r A . Oebz~j iengtn oosii. Fro-. *hcs 1r~c.1:-----------t the -ec-& profiles a-e over-predicted, tnL cr-e r larger with distancru alo'ng the tt. so that at 99 of body length errors up At the iornrd approximately 55' occur. position the maximum error can ' seen to be enlv about 6 . From- comparison between this piz dicticn and the Myring estimate it to a similar order of over-prediction present in the data -sed to svecify tne conditions. Thus. to some degree the over-prediction of the total.'iocitv is the result of incorrect inlet condiz.ns:
,*t o oa
_____inlet
I f3 -
6
however, this clearly would not account
01
for the large errors near the tail of the
0
body.
-Ckt -07
O-
AXIAL bISTANCE SonvY
Fig 15
Total Velocity Profi le at 66. 2' of Body Length Fromz Nose
is consideaed that this error is probably
*due *
VELOCITY
0-S
IROM NOSE
LSNGTH
Axial Variation of Static Presure Coefficients
LEE4
io
The results of predictions of the total velocity profiles at 50c and 96- of the body length and the axial variation of skin friction coefficient and wall pressure coefficient for the full Patel body are given in Figures 18 to 21. It is apparent from Figures 18 and I that the velocity profiles are further over-predicted cmpared with the half body calculation. This is possibly the result of the mechanism which leads to the ov.-r-prediction operating over a larger range in the full-body case. The skin
friction coefficient (Figure 20) can be seen
to be higher than for the half body calculations at all stations and hence further
---
from the experimental data. The wall stacic pressure (Figure 21) shows a steeper pressure gradient at the stern for the full-body case although this difference is not large.
HALF
-ULL
>
MODY 92REOICTION BODY FEDICTON
0 to 00
77L
_
uW
ION
- EICT
00G{
01 008 31iJ004-00
0
OGa~l
0
r-~
Zr In Z
al
, J
VELOCITY FeEIS STREAM VELOCITY
19
1Fig
Total Velocity Profiles at 96% of Body Length From Nose
S
0.4
aA
TOTAL 'VELOCITY
FREE
Fig 17
HAIF
---
P ULL eBODY F-REDICTION
,ODY PREDICTION
STFEAM VELOCITY
B.ZiNLaT
JTUveULENCE
Total Velocity Profiles at 99% of Body Length From Nose --
(MYRING PREDICTION BODY PREDrICTION
+
lPFULL. 0
---
0
.t
z
IL
I
U
"
z
oo s
-
.--
-
I-I (w
-
--- -
oOKO of
D-e000
L-
.01 07
Il
0/
0.00
W________ 0
I °.
0
- 0
-----
AXIALDISTANCE
"--BODY
o.G
o.8
Fig 20
-645-
___-
10
NOSE
TH
Ila
Total Velocity Profiles at 50% of Body Length From Nose
________
FROM LENG
1-0
TOTAL_VELCCTY FREE STIeAM VELocrITy
Fig JG
0-9
0a
-
-
-
Axial Variation of Skin Friction Coefficient
the turning of the flow; it was precisely at these planes where the maximum inaccuracies occurred. A possible way to eliminate this source of inaccuracy is to solve for a velocity component which follows the grid-lines, instead of the axial velocity, wnich was
---- HALF SOMY PREDICTION ---UL BOb' PREmCT ION EXPERIMENT ---ZI 0 5,.
I
o04
I
used in the prev.ious analysis. - /
I
W 0 a . u0
I -need
3by
00 I
W 0
-
U -01Removal
F
of current work, together with the use of a velocity component which follows the grid lines. The basic concept is that the solution domain is extended to very far from the body in all directions, and an additional equation is introduced, namely, that for the velocity potential. The latter will be solved over the whole calculation domain, the rest of the equations being solved only over the narrow viscous region around the body. The boundary between the viscous and inviscid regions
lject 0.7
0 U
O-.
to
B00Y LE-NGTH Fig 21
--
=
The advantage
is that this velocity is simply oriven by ti, pressure gradient term between its upstream and downstream grid points. The most serious limitation to the potential of the described method, is the to prescribe boundary co.ditions at the free-boundary. This is done at present utilising the potential flow solution. This itself is a source of further uncertainty, and leads to the need for a different potential flow solution for each new location of the outer boundary and enach new body. Of Ellis limitation is the sub-
Axial Variation of Wall Static Pressure Coefficient
It was ccncluded by Patel et al (8) that in the tail region of a body of revolution, there is interaction between the turbulent rotational flow within ti boundary layer and the potential flow outside, so that neither can be calculated indenendently of the other. Since the potential so-ution for the body alone was used to specify the pressure at the frLe boundary in the above calculations this may be a significant factor in the poor prediction city and pressure. of total v The a. .1e results hate demonstrated the capability and limitations of the k-c prediction technique when initially applied to an axisymmetric body. It is encouraging to see that the computed results irlicate the correct trends, and that only a few minutes are required to produce solutions for different input parameters. It is disappointing, however, to see that better agreement could not be secured between predicted and experimental data. The space here is too limited, and the information presented too little, for a rigorous analysis of the causes of disagreement. The study has highlighted, however, the following Problem areas, Firstly, there is the difficulty of formu-
will be determined by successive applications of the potential and viscous solution procedures. Current work also includes extension of the above methods to consideration of the flow around appended bodies.
lating the finite-difference equations, with due allowance for non-uniformities of
found to be 9% below the experimental value.
cell shapes, so that the proper relations between pressures and velocities are orecisely preserved. This was apparent in the calculations performed which sometimes exhibited stagnation-pressure distributions which were inaccurate. This effect is particularly significant at the planes of the nose and tail of the body, because of
Comparison between predicted values of the wake centre-line velocity behind an isolated flat plate and experimental data shows that in this case the method gives a very good prediction of the wake recovery. Good agreement has been obtained between predicted and measured velocity profiles at a position 2/3 chord aft of the
4.
CONCLUSIONS
Methods have been derived using a twoequation turbulence model (k-E), to predict the two- and three-dimensional wake behind a hydrofoil and demonstrated to have the potential to warrant extension to consideration of flow prediction around an appended body. Comparison between predi ted values of the wake centre-line velocity behind an isolated hydrofoil section and 2xperimental data shows that the prediction method gives a somewhat slower wake recovery than occurs in practice. This discrepancy is largest at low Reynolds numbers and close to the foil trailing edge. For a NACA 0017 foil. at a Reynolds number of 2x10 6 , the predicted wake centre-line velocity at a position 2/3 chord aft of the trailing edge was
-646-
trailing edge of a hiydrofoil protruding through a plane boundary layer. The predictions are found to be most accurate close to the plane boundary where tl'e thre*-dimensional effects are most significant. The effect of an adverse pressure gradient on the flow in this region is also well predicted. Initial application of the k-c prediction technique to an unappended axisymmetric body has shown promising agreement with experimental data. The predicted axial variation of skin friction and boundary layer are close to measured values. The static pressure gradient and total velocity are, however, both considerably overpredicted in the tail region. The use of a potential flow solution to prescribe the free boundary conditions and the definition of the finite-difference equations at the body nose and tail are major factors influencing the accuracy of flow predictions around submerged bodies. Further work is in progress in these two areas in order to improve the accuracy of the prediction technique.
10. Spence D A: "Growth of the Turbulent Wake Close Behind an Aerofoil at Incidence". Aero. Res. Council, CP No.125, 1953. 11. Lieblein S and Roudebush W H: "Low Speed Wake Characteristics of Two-Dimensional Cascade and Isolated Airfoil Sections". NACA TN 3771, 1956. 12. Chevray R and Kovasznay S G: "Turbulent Measurements in the Wake of a Thin Flat Plate. AIAA Journal, Vol 7, No.8, 1969. 13. Dyne G: "A Streamline Curvature Method for Calculating the Viscous Flow InternaAround Bodies of Revolution". tional Symposium on Viscous Resistance, SSPA, Goteborg, Sweden, 1978.
REFERENCES 1. Myring D F: "The Profile Drag of Bodies of Revolution in Subsonic Axisymmetric Flow",. RAE TR 72234 (unpublished). 2. Moore A W and Wills C B: "Theoretical Computation and Model and Full-Scale Correlation of F~owv at the Stern of a Submerged Body". Proc. cf 12th Synposium of Naval Hydrodynamics, Washington, USA, 1978. 3. Patai-kar S V and Spalding D Brian: "A Calculation Procedure for Heat. Mass and Momentum Transfer in Three-Dimensional Parabolic Flows". Int. Journal Heat and 4. Launder B E and Spalding D Brian: "The Numerical Computation of Turbulent Flows". Computer Methods in Applied Mechanics and Engineering, Vol 3, 1975, p 269. 5. Pratap V S and Spalding D Brian: "Numerical Computations of the Flow in The Aeronautical Quarterly, Curved Ducts". Vol XXVI, 1975. 6. Abdelmeguid A M, Markatos N C G and Spalding D Brian: "A Method for Predieting Three-Dimensional, Turbulent Flows international Around Ships' Hulls". Symposium on Viscous Resistance SSPA, Goteborg, Sweden, 1978.
7. Schlichting H: "Boundary Layer Theory". McGraw-Hill Book Co, Inc, 1960. 8. Patel V C, Nakayama A and Damian R: "An Experimental Study of the Thick Turbulent Boundary Layer Near the Tail of a Body of Revolution". No. Institute 142, 1973. of Hydraulic Report Iowa
~Research,
LSection,
9. Escudier M P: "The Distribution of Mixing Length in Turbulent Flows Near Imperial College, Heat Transfer Walls". London, Report TWF/TN/l, 1966.
*
-647-
Discusiontion. Discussion
The 3-D flow configuration consisting
G.E. Gadd (NMI) Fig. XII.34 of Thwaites' book "Incompressible Aerodynamics" shows that a circular cylinder protruding through a boundary layer on a plate produces a horseshoe vortex trailing downstream with counter rotating vortices outboard of it. This vortex system causes spanwise variations of the boundary layer thickness on the plate rather like those shown in figs. 9 and 10, so presumably the hydrofoil must give rise to a similar pattern of vortices. Are these evident in the numerical solution ?
M. Hoekstra (NSMB) The present users of the partiallyparabolic method are on the right track now, I think, in studying carefully the performance of the method in comparatively sj,,.ple external flows. I like to encourage the authors to continue this work. What astonished me is that the authors apply their for method, which is essentially inadequate separating flows, to the case of a hydrofoil on a flat plate which is known to give separation at the root of the hydrofoil. What kind of results were obtained in this root region ? Can the authors comment ? According to eq. 40 the authors match their solution to a wall law at v+ ! 11.5. How is the actual boundary value of y+ chosen and is there any influence of the choice on the results ?
LLarson (SSPA) Spalding The methods developed by Prof. and his group have been very successful for internal flows. However in external applications several new problems appear. It therefore seems like a good idea to apply the methods to fairly simple cases before turning to the complicated flow around a ship's stern. My first question concerns the 2-D case: Rather than matching the viscous calculation to an external potential flow at the edge of the BL an artificial boundary condition has been introduced 2.5 chords from the foil. Judging from Fig.9 this disfance corresponds approximately to 100 times the BL thickness at the trailing edge. I doubt that an efficient use can be made of a viscous flow program in this way. How many grid points were there in the BL ? You use 16 altogether in the lateral direc
-648-
--
-------r= --------
of an obstacle protruding from a flat surface has been used frequently as a test case for 3-D BL methods. The interest has however mostly betn directed towards the region in front of, or beside the obstacle. In this paper the flow behind the obstacle is compared. As is well known this region is highly vortical, containing strong longitudinal vortices. Accurate predictions could therefore hardly be expected if the pressure is not allowed to vary in both lateral directions. W.-D.XuHatbinShipbuildngEngnst)
For calculation of turbulent boundary layer, the empirical formulas used may be examined first. For example, in order to solve the energy-integral equation, we may first find out the velocity distribution and thickness of different regions as shown in my paper titled "Velocity Distribution and Energy Equation of Flat Plate Turbulen' Boundary Layer". After analysing twentyone u+ , y+ curves converted from experimental data of Klebanoff & Diehl, I obtained the following results by Method of Least Squares. The velocity distribution of inner region well mic law:(I.R.) U+ = is 5.75 log known y+ + as B. the But logarithI found that B is no longer a constant, it is a function of Re : B 2.086 log Re - 3.225, i.e. when R 8 = 104, u+
5.75 log y+ + 5.12 (for I.R.).
(1)
For more easily integrating the energy equation, I found that velocity distribution of outer region (O.R.) can also be expressed by the above logarithmic law but with a different slope : u + = 10.4 log y+ -D. The term D is also a function of Re D = 1.973 log Re + 0.879, i.e. when Re = 104t u= 10.4 log y+ - 8.78 (for O.R.) (2) Of course, for laminar sublayer (L.S.): u+ = y+. From the intersecting points of these three expressions, the thickness of different regions can be determined Ye =(e y
+ 341)/40 (for L.S.)
(3)
= 0.837 log Re - 0.372 (for 1.R.)(4)
when Re = 104, y+ = 11.03, yi/ 6 = 0.257. Then the friction -term of the energy equation can be calculated for each region, finally I obtained the following Rx Re relation 63 R9 11/8 R
0 .9 4
It agrees better with the experiment of Klebanoff & Diehl than other formulas often used as shown in the Fig.*(one of them is deduced from 1/7 law).
that the current predictions in the wake are in fair agreement with the experiments, indicating that the effects of vortices may be localised and not very significant in the wake of the hydrofoil considered.
Eq.(2-25) is dedugqd 4 from the 1/7 law Rx = 63.771 R5a Eq.(2-27) is as follows 6 R=
Reply to Mr. Hoekstra
143.11 R7/
I thank Mr. Hoekstra for his encouragement. I would like to point out that our method is adequate for separating flows, and the code includes all three options, namely parabolic, "partially-parabolic" and elliptic. Indeed, it was the last version that was reported in the slide panels of
X0 2
63R_
I
R 3x
R a
A / /
i)Y8_
my presentation.
* (togRe*0.1 ,wake.
AA Klebanoft & Diehl a
0 U U= 35 ft/s BE 55 108
_
10
_
_
Simply, we did not at-
tempt to predict the separation adequately, and attention was not focused at the root region of the hydrofoil, but only in the The apparent fair agreement of the predictions with experiments in the wake tends to indicate that effects of vortices may not be very significant in the wake region of the hydrofoil considered. Concerning the second question, the first grid line next to a solid surface is set at such a distance from the surface, y, so that the calculated y+ at the first grid line is in the range 30 < y+ < 120, over which it is reasonable to assi.me that the log law prevails. An initial guess and subsequent adjustment may be necessary.
_t
12 -6 14 RxX 10
Within the above limits the results were virtually uninfluenced by the precise value of y+. A few runs with varying distances of the first grid line from the wall are therefore in general sufficient to ensure results independent of y+.
Author's Reply
Reply to Dr. Larsson Concerning the first question, the figure 9 mentioned refers to the threedimensional case, for which the North Boundary was located at only 0.228x chord from the floor. For the 2-D case the outer boundary set at 2.5x chords from the foil, was but indeed this is only 15-20 times the
N.C. MarkatoS (CHAM Lid) Reply to Dr. Gadd Vortices are not evident in the sclution, because, in the reported calculations, the pressure field was derived by considering an isolated hydrofoil (see text) ; and that two-dimensional field was applied uniformly all the way down to the plate. The procedure, used is not, therefore, a full "partially parabolic" one. The latter would have failed near the plate, in the presence of vortices, We did not use such a procedure because .o fuil analysis was intended at this stage. If we were to carry out a complete analysis we would have used the elliptic version of the code (reported in the slide panels of my presentarion). Therefore, the absence of vortices in the numerical solution is a consequence of the way we chose to treat the problem. It is interesting, however, -649-
B.L thickness, if I recall correctly. The general policy is to locate the outer boundary far enough from the viscous region, so that known boundary conditions can be applied, and then to investigate the sensitivity of the results to the grid both inside and outside the viscous region. It is because we performed the second stage and demonstrated grid independency that we ased such a grid distribution. The grid points in the B.L were 10, at the trailing edge. Despite the above considerations, I do realise that this is not an efficient use of a viscous flow program. This realisation is among several reasons that led to the development of our new approach which was explained in my presentation. This approach consists of using two calculation schemes for the irrotational external
Iin
flow and the rotational flow close to and the wake of the body, full account being taken of interactions. The answer to the second statement is the not same intend, as for Dr. question. did and Gadd's have not predicted We the presence of vortices,
Reply to Prof. Wei-de Xu I thank Professor Wei-de Xu for supplying me with his interesting modifications to the for log future law. Weinclusion will study thoroughly in them our wall functions.
-650-
-_
_ _
- --
-=
- - -- - - _
Effective Wake : Theory and Experiment Thomas T- Huang and Nancy C Groves B
=- M
" USA
ABSTRACT
dynamic theory. Until recently, detailed velocity surveys in the presence of an operating propulsor were not available, and the actual distribution of effective velocity into a propulsor had not been examined fully. Providing the correct distribution of effective inflow for w-ake-adapted propulsor
An improved theoretical methd is presented for computing the effective wake of propulsors operating in thick stern boundary layers on axLy mmetric bodies. The hydrodynamic interaction between the nominal velocities upstream of the propulsor and at the propulsor location is assumed to be in iscid in nature and the total energy is assumed to be conserved along a given streamline with and without the propulsor in operation. Theoretizal predictions using the method are compared with experimental data obtained in the United States and Japan for five different propulsorlaxisymnmetric body configurations. For all five cases examined, the computed total velocity profiles immediately upstream of the propuisor (with the propulsor in operation) are in good agreement with the mea.ured values. In addition, the volume-mean values of effective velocity profiles computed from the measured nominal velocity profiles are in good agreement with the measured values of the Taylor wake fraction (l-WT) for all five nominal wake distributions over a wide range of propulsor thrust loading coefficients.
design is emsential to meeting the ever increasing demand for improving propulsion performance and energy conservation. In order for L propulsor to produce a required thrust to power a ship %ith minimum power and minimum cavitation at a prescribd propulsor rotational speed, the effective velocity d-tributi-m used in the propulsor design must be very accurate. In 197C, a L-a-ser Doppler Velocimeter (LDV) was successfully used by h,: "_nget alA to measure velocity profiles very close to the propulsor. The measured velocity profiles, stern pressure distributions, and stern shear stress distributicns with and without an operating propulsor provided the necessary clues to the proper understanding of the interaction between a propulsor and stern boundary layer on axisymmetric bodies. The influence of propulsors on axisymmetric stern boundary layers was found to be contained within a limited region extending two propulsor diameters upstream of the propulsor. The propulsoristern boundary layer interaction was found to be inviscid in nature. The inviscid approximation computer program developed by Huang et al.1 predicted very well the measured total velocities in front of the operating propulsor. Subsequent detailed measurements of the velocity profiles with and without a propulsor operating in two axisymmetric wakes were made by Nagamatsu and Tokunaga. 2 An invscid approximation again predicted well the measured total velocities in front of the operating propulsor. &chetz and Favinl-; have formuuated a numerical procedure based on the full Naier-Stokes equations to compute the flow near body/propulsor configurations. The computed axial velocities at two propu!sor diameters downstream of the propulsor compared satisfactorily with the measured results. This numerical procedure has not been applied to. the cornputation of effective wake in design of wake-adapted propulsors. The influence of a stern-mounted propulsor on the flow field past bodies of revolution and a flat plate was measured by Hucho 5&- although no attempt was made to -
1. INTRODUCTION The velocity profile at the location of the propulsion device in the absence of a propulsor is called the nominal velocity profle. The effective inflow velocity distrbtion experienced by the propulsor depends on the mutual interaction of the propulsor Coading distribtion and geometric characteristics) and the stern boundary layer. It can. be significantly different from the nominal velocity distribution. In the design of a wake-adapted propulsor, the radial distribution of effective inflow velocity is often estimated by ratioing the measured nominal circumferential-mean axial profile up or down by a constant factor. The factor is sometimes taken to be (l-wTYl-w.), where wT is the Taylor wake fraction and w\ is the measuied volume-mean nominal wake. Naval arch'tects derive the Taylor wake traction from propeller open water curves on the basis of thrust identity between propeller powering experiments in open water ant: bwhind the ship model. The constant-factor empirical approach for obtaining effective inflow distribution is n-t based on a rational hydro- 651 -
tually calculate the effective w-ake distribution. Other inves-
where p is the stream functiorn for an incompressible --.risym-
tigations related to this subject were made by Wertbrecht,F1
rietric flow defined by
Hickling.Y Tsakornas and Jacobs,"' and Wald. 1 Methods to estimate effective wake were proposed by RacstadY Nagaxnatsu and Sasajixna.13 and Titoff and Otlesnov. 14 The only known previous effort to theoretically address thi problem is due to D.31. Nelson* who developed a computer pro-1 gramn for calculating the effective wake from the measured nominal wake and static pressure distribution acrorss the boundary layer. in the present paper, an improved method foe- computing effective wake distribution from the measured nominal wake distribution is derived. Serious effort has been made in this work to compare the thcretical velocity predictions with velocity distributions meas-ured by an LDV oyr z.five-hole pitot probe immediately upstream of an operating propul.-e-. For all five cases examined, the computed total velocities up-------s-eamn of the operating propulsors are in good agreement
, =
--
-
r 3
Sin-e the flow viocizies are irncrewsed due to the ation of the propelsor. strea'n surfaces are snifted c' oser to the body surfaceo. AR, show-. in Figure !. a typical stream surface moves inward from r to r , while the resultant velocitv is higher than the nominal velocity. The resultant velocity u. as wot.ld Le mieasured in front of the propulsor. will be calle the total velocity. -- -- -- ----0o Eze~ PtUn
with the measured values.
The computed volume-mean effective wakhe distributions are Shown to compare favorably with the measured Taylor wake fractions derived from sielf-propulsiun exp !riments of five different nominal wakes. A cojnsiderable amount of experimental data and relevant computational results are tabulated to permit the independlent awsessment of the present method by other investigators.
Ia r or
U
PutIni.L oAit 2 ,~j
I
__-----_-----
-
V2z -
~
-
I
SraewtotK
Sufet Withou
Srm
stream Surfac with Ptpiso in Oprailmn
2.THEORY 2.1 PropulsoriStern Boundary Laver interaction
-
The experimental data given in References I and 2 allow one to conclude that the influence of propulsors on upstream stern boundary layers is detectable only within two propulsor diameters upstream of the propuLsor. Upstream of the propulsor. the mean -ircumferential velocity, vs, is ideatically equal to zeto on an axisymnzzet-ic body both with and without the propulsor in operation. The followuing assumptions are made to derive a theoretical approximation of the hivii-dynamic interaction between a propulsor and a thick stern bounmdary layer upstream of the propulsor (a) the flow is axisyninetric and the fluid is incompressible: (b) the interaction of propulsor and nominal velocity profile is considered to be invscd p opuso-:n i ntur; ucd hus vsc us oses ndIOH turbulent Rey-nolds stresses are neglected: (c) the conventional boundary-laer assumption, Ov.Jax '*
"_ .-
ofteFig. 9 is one of the examples which shows the components ofteC-region flow. The interaction term. u,. v, is not so
________2
stnall _ while the Dregion contribution is s all.,
..
s
.
total velocity
Z~'u/U
~Assumed MSL
PxX/1,=1. 3
SD.Region
E-2
- -- EM*-120
M. .,easured "
d
""
F
0..1
1
2tU.ois
0 .L-e on coutritutl
-. 5 -5-200
K/
X/tL=.
Fig. 9 Decomposition of Velocity at xi= et
1.0 (EM-200)
Fig. 10 is the additive source distributions which induce the interaction componerts. They do not inerese monotonously like displacement thickness but they get maximum around the separation positions and once decrease, then increase again. The present results seem reasonable because boundary lyer developments get significantly less by separations.
ig. 11 Vo
tiityDistributions
W1i = 1.1. 1.2. 1.3)
Comparisons between models show just dislocation of the peaks but not significantly in magnitudes. The dislocation means the wake of EM-300 is wider than others and eventuallv the head loss too. The measured distributions arz more mild than .alcu. lated; the peak is less and the wake w.dth is wider. This means the diffusion is more intensive in real flows. This discrepancy seems to be removed by choosing a larger diffusion constant than 50P. but the value of the diffusion constant does not affect so much. For the improvements some of neglected terms should be revived, which is left for future works. Any way we can safely conclude that the present results arc applicable to the resistance estimations with required accuracy. 4.3 Viscous Resistance Before the calculation of viscous resistance, pres.ure distribution!, which are nees for it. are determined by Eq.
_738-
(28) for the C.region and by Eq. (8) for the D-region. In Fig. 12 calculated results are compared with measured :n case of EM200. The pressure is measured by a static pressure tube which is of 1.2 mm d.ameter with ,wo 0.4 mm 0 hole1 on diametricaliv opposite sides. (The accuracy of the mcasule
PPO)
C
L
I
j
mients is checked by measuring the pressure field by towing a model astern: they show good agreements with potential calculations as shown in Fig. 12.) Though discrepancies still exist in the outes parts of wake, they are rather well ?cpioduccd. In thcalculation, P. is assumed 104 times of the molcular viscosity which differs ziuch from that used in the diffusion calculation. This may be open to discussions, but the present author is of opinion that v, can not be always related to a single constant but it may depend o-i the related physical variables, e.g.. pressure. vorticity and so on. The viscous resistance calculation can be carried out now by invoking the previous results. In the present calcuLtion, the integrations in Eq. (19) are carried out on more than a sing!e
Calculated by Present
d by Potential Theory Moo easured 41Do, Towed Astern
~
chosen around Si= 1.2 x/E in calculations. Experiments control surface are carried and averaed; twofor or EM-125 three control surfaces are at f2o far downstream: = 1.9 and EM-200, xle=
0out 0L .
2.8 fG- EM-300 respzctively. measured total head losses are shown in Fig. 13
. .The
02 -I
--
_eg ( - -H)
J
.,easured
x/L o .
0-
,
|
1
"61o
2_2 (x-I lxIR=.. 1.15, 1.2. 1.25)-
{
N~2 oO .
110
"l5Cous Resist.
~ ~ ~
Fig.
EM-300 {'
-,"/
~
~
f
~M20 ~
Ek-20o
Et4-15
0
.
1.9
1.15
~0103
20
.2
EM125 02._.
9 o
Fig. 13 Total Head Losses
0
~ 12Peesisstiuiosit.k rctoo
0
EM-12
A
/0
,
-200 -
20
Rnxle
°" 00
OA
j
'o
EM-
00
2-0-
t _
__
__
I !
-739 _
_
_
_
u
_
_
_
_
_Ic
_eit
toge-ther with the calculated result of EM-200. Their integratiors
than those of inviscid but no significant shifts in the hump-hmllow
the calcu.ation is carried out just behind the stern, the head loss as keenly concentrated (the coordinate should be chang-d from the right to the left when y approaches to zcro. Measurements are carried out by a pitot tube with enA-piaesFt. 14 shows comparisons of the viscous resisance. C-.kulated results agree well with measured ext-en EM.300which
cerned (we should remind here the wave resistance theory on which we have entrained -sbased on the low speed assumn 'o'. the present results zre well reproducing relatively to the luvncid cases. So it may be safely concluded that the neglecton of vi cosity is partially responsible for the discrepancies b-w th inviscidly calculated results and experiments.
same reason as mentoned in 4.2-. In Ft. 14 the frictional resistance curves are also shown which are obtained by Eq. .23) where skin frictiom of the boundary layvr calculation are used. Subtractions give the viscous pressure resistance indirectly. As far as the present results are appreciated. it is safely concluded that the viscous resistance can be proportional to
point of views. The total resi tac curves sh.w us that the humphollow phases are shifting to lower speed side in the or er o_EM-300. -M-?00 and BM-125. This means the cffective length of ships. the kngthl between the bow and -he ster wave orgins. gets shorter in this order, in other wods, the stern wa-e orins move forward irn this otde:. And if expermcntal results ae plotted
the frictional resistance. Of cob;se, this may be limited to the case where significant free surtacc effect, are not existing,
In the present case. because awiaz at z 0 va iishes for all the used models, the second term of Eq. (31) disappears.
against Frou-dc numbes based on the effectivc lengh. the phase shift will disapper:. It is early gruese-. as shown in Fig. 2. the stern waves are generated around the separation positons. So phse shifts, which are giving fatal discrepancies between trnv= si cakula:sons and exprimentL. are related to -i.he sip"--an. in other words. related to the siscosity. Aparing from the present discussions. i is ineresic tihe
For the velocity distributions far behind in wakei. namely, x 1.20. 1.25 and 1-35 for EM-125. .M-200 and al-300 respectiely. similar profiles ae assumed; the maximum velocity defect at y = 0. U-u jx. 0-. is assumed to be proportional to x-*h while the breadth of wake spreads to proportionally to xu-S. In Fig. 15. tl:e calculated wave rcsistance is compared with experiments in the total resistance form after adding it to the viscous rcssancr curves. This is because some difficul;ic are nd in determination of the form factor K 'experiments are carried out 3-dirnenaiorolt and the viscous resistance is differmg
the wave resistance does not inmaa so much against the seed (we havc found there difficulties to dete-inr the form factor In case of EM-300 more intensive interactions between Wave and wake mat be existing than those taken k-:o account us the -r-sent paper -f.cr surface effects on wake flow or separation are not taken into account heret More prcise experimental st-dies l be necessary. In F_. 16. typicai exampic -if nro t-u"de fuwtion is snow comparing with inviscid and visc-s . ts Fs = Wi" F-S-200. There sknificant redutions in a3r.1rtdh ii trazverse wa-C
ftra.the results of the previous section). The e-us of viscid calculations are less in magnitude
components are obscrad which nlv contrwiutc to the r uc tion of wae -eisance. But -o sinit ,.- pha nit_ a ' obse.v'd.
phases are still obsered. But, as far as low sped ranges are con-
provide the viscous resistance, the first term of Eq. (19). Because
is unnaturally smaller than measured (this -may be due to the
T1he above concluion can be supported from another
4.4 Wave Resistance
rcsitan~ce cur%- of FM-300 is s*- 'kantly diff-rent fromn ot+hcts-
Calculated -!ccty disributions arc substituted: into E4. t31..
ia---
4-iT--, jTowi1nkg
£4 @
+
Fg. 13 Comparisons of Wave Resistance
:'.:
P."0
sy
I
:x
I
ai
"°
U TOM st
0-
.
-
/ *.
0/r
'/
-a
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ir
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020
025
020
I
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o. Hiosna Unirersit toar :ntoul % the prescnt work.
X1 2-Dr
L
prera Md.
-by
zscusstons an! cntraSeflftl
All tHim cncntr'osaeac Hmnua Lwrs.i Crnput
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nif
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c
wFfA
-by Potenia Dhory -
REFERENCES
-
Fn=-0.17
1. Hatano. S.. Momr K.. Fwus:imna_. m.. andA Ya-ank It.. 'talculion of Veloit Dnn- muiow.s in Ship Wake-- j=-_!a of The So-iw Of N1arn. Arciet of -: apan. Vol. 139, Dec. 197F5 . 55-63. Do. '_ -Apprxintac Prediction- 0! 2.Dci. K. Fk-w FCIA round Sim- Strn syprtic nn ExpnsionMthd Iirral Of rhe Socier.. of Naval Arcnnec-a~ Oi Japn V01 2!44 0cc WC=
7 IS . m ItIl-20 3.M-uk.K
O~q)
Sht
Sem
CA
no
Awms.L
How atd Stmn* r-v Coro-r Hr-:doavnaina ONT. 196-3 r 1.MA~ H. andl
5.CNCLUDING RLMARKS Forcrmw.aks can be -itbne s t-UMHUzni' 1.The necsen nrhod -mdia-was of=S near j nieiw is
Re=titanc
in
Arcsit.
t.
~S
:csiianee. As to he zf
--
r~-vs-
Cin-cSC
!a ti
pi-trional to- she fmz-i---' resisanc. 3's The- were. -cS.=-e :s -sra c inchnin vic c ec. a doeCs ,r- memat. tn tunt~ A -mm-s Thouh-.-ot znt -ums enCKeS -ne-vt al :hccrrv. ntcus rteets
io the mntess:ncrt utwhm can reate the
=4 _
tinr ptr.
t6..
mm'r =k
-cr' tl= :Z.
~zzr
fkvw.~-
rlhr expi:;:-
-U
-b
'LHects
ledcn
*mt Wav
Framework of Loww SpeedWare Renae h-r.con -he acu-= t E nn Un sir. VolI. 7. N =--,.197.9. 9-1.9 La. ad Takes l a .K_ -A'mie- on Free '-'ace h F nInc ..aj Vk~ AND--4.-iBo aA-=# Suctrv ot Naval Ar-cs-ects w apan. Vol 137 lbn Is97 5
atcr
Ohmunaan
1c-r
t!
ORTI
=jc n a NbAw nhiuh
c-ircd.: cspcily rWDc.
-
t
CoeCdhi IsprerAw~n sh4U te remnded
tso
Peat. Pana(d
cwrr
pt
+m
(-
1
-4q:c(afrCnfq
Iq
Wb
are apjw-ecmnd tfoe thei coc-pen
W
..
flow or sepaaion whn n mar he
7
he t~p:r 6 ret m Cato
to the vimen
E4UATKY\' -AM,
NQ
&H yodw-wneiiosciaosc ra micl are aw~.
wk;ax7'2 ii
b C-ort rrssure
nennymuatrcsn of -=trzs --cated := to yr strs. F-A.. -awwpoint 46lou e1 nment6mneatithtzte works not take-- in the wmn: stodwa Z: ml twe fceTa_ r4 zbw
cm-
REYW'M
&. v-zi
abkl
face
APP t N JI
.44
om
he pointed- out: s-tnc
r:.mor rewd" 5law uc--s
al
K
=
Eeidat Shi
n-c
onrt 2-11 amd xt 3D flowsa and the an-w±n te-rcn ronrhfed wo'he~ Si_ wae rrcuni's' aw-"sttrr
:C-
oc m O fla
tu orh 14=2
7pneiak
are
rowl
F---
Pap-as o* 5- T. 4ar-_ct 0on 39844107 1Shp I-Wat.±S and Ware-
Ja.-a...
V
-h.hetmnai-z
a- Visou
'
ai
i. .j-. Dec. 1- . 9 t .4=...t. Y__l-imAiement Lffer al Thre4)imc siu-n= Tus-' .. - B-auMr Las and. Wake -- Ship-. Poeed in n~r of nn Sen inar- on Wart ResiSL2=nce. Fc a19T
but the precrem dinxrilcn ane well rewdcuL Thecneocs rei-----aanee is ennnat-d-ed acctL
Viicoos Flaw ar- nd
Trsacttrs o. Thc Wea lapan So..rtv o 58 *u S17) p .52-
U
-2Kwjjy+2Knj
arada4M.2 mSa, +~t
qII r2
t
).+ K
K,3) 1, K3, K,j-i
~( +
-- -3 ,: .-I-(K&- -q %2 (Knq,)q,
-
hS
8K32
'
V
2 ,x(W
)q,eSx -~ 3M
n
' Kn(K+
8,
XIS)
+K
=
an= ;-r-hOx
2Km: -h'q +2K.,84
Is
&--q'F" (T)e
opcrata- v
8
-z
-K,
a:
a:
n
a
0:TX
sX, +Kig,±Kn-K 2 ;. ,'x
3-Kz.ttr7r+nKit
a-, 5q_-
, -- KKq'+
?
K
-
.,
kgxV2ff Jq
+2K
_.2Kx,,.G'q- 9.I~tqT
hCan-n
I=
(Knar
EJ3' 8K, 1
8q-
rw ___
__
____'A-3
if 742.n
dd furo ara.wr th
nmc
iosyap~xiai.,i vw
iit
dfflsoz
c=-5
3**
=
(A-10)
W3W
(hIqa-!(~3
I2 ________________q
-
-
--
2
--
(h
-
-q-l)--
Discussion
It seems to be quite valuable to de-
V.C. Patel (unv of Iowa) Professor Mori's model of the flow over the stern, as depicted in Fig.l, appears to be valid for a two-dimensional bluff body where a region of reverse flow can usually be identified. Figure 2 for EM-200 and similar pictures for MS-02 (whose aft section also had elliptic framelines, see Mori and indicate Hotta : '-th thatSym. slender, NavalHatano, Hydrodynamics)
velop the calculation technique of the vorticity diffusion region further by using experimental data of simple cases, without D region before the theory is extended to more complicate or three dimensional cases for practical purposes. Vorticity diffusion method might be promising to handle the thick boundary layer and near-wake in such cases. Velocity fields around the have been measured in the windthin body tunnel by Rang 2. When we estimate the wave-making resistance, both viscous effects and wave are important of 3the same potential range eveneffects in the low-speed order
nearly two-dimensional formns also lead to pockets of reversed flow. However, such flow reversals are usually not observed on the highly three-dimensional sterns of real ships except perhaps at the free surface. Instead, one observes reversal of the crossflow within a thickening boundary layer and the formation of a longitudinal vortex, Would the author comment on the possibilities of extension of the proposed model and solution procedures to such cases ?
i
i
Therefore corrections for the body boundary condition violated by the wave-potential near the bow will be good for the estimation of wave resistances. The author decomposes the total velocity into three components ; pure potential, viscous, and interaction term. The last one is obtained by distributing additional sources on the body-surface to satisfy the zero-normal velocity. It is still not clear whether we should keep the non-slip condition/ or zero-normal velocity on the surface. If we consider non-slip condition to KtoaUvbe more reasonable, I'd like to remind the N Kato (Toka,Univ) author a vorticity theorem suggested by I p a h t ' p hLandweber 4 as follows. I appreciate the author's paper whichConsider a stead mean flow of an inin atfres teady a s Coyside of analysis theoretical the to contributes in at rest a body about fluid cor,?ressible vortices. separated wake eincluding near 1nhear iludinse e v . an unbounded fluid, with mean vorticity I have three questions ; present in the boundary layer and wake, (1) Although the autnor derived the BLW, bounded internally by the body surface three governing equations from the NavierS and externally by the surface T ; see Fig. Stokes equations by the use ,f small paraThe boundary conditions to be satisfied meter c defined in the equation (1), the are the non-slip condition on the external author made an additional assumption of the side of S and the irrotationality of the location c the dividing streamline which flow exterior to T. Then the disturbance is ought j be solved essentially in a flow exterior to S can be generated by the closed procedure. I would like to have the vorticity in BLW only. author's comment on this problem. Therefore viscous components by Eq. (2) The author expressed that the dis(13) in the paper already include the others agreement of the calculated results about implicitly except the incident uniform flow, the velocity distributions of the near wake if correct vorticities are -btained. We flcw and the viscous resistance of EM-300 can find the summation of vi3cous component model with measured is due to an unsteady and D-region contribution make very amount wake flow of the model which is not taken of the velocity defect of the total velocity into account in the proposed method by the profile in Fig.9. Vorticity distributions author. But I think the disagreement is have already taken account of displacement ~~also due to the assumed location of theefctofhebudrlaranwk. dividing streamline which is influential on the solution. May I have a comment of the author ? 13) I would like to ask if the author has any idea to develop the proposed method to analyse three dimensional separated flows T including longitudinal bilge vortices as UBL observed in the wakes of full ships. S.H. Kang (KRIS)
I deeply appreciate the research of near-wake and resistance calculations by Prof. Mori. Congratulations ! Here I would like to make two suggestions and one question.! -744-
, Figure
(1) Your suggestion to carry out for simple cases without P-region flow is quite important to check the validity of the present method. It has been already carried out in case of a flat plate in our paper of 1978. See Fig.Al. The second iteration is also carried out there. (2) Unfortunately I have not yet read Dr. Kang's paper. But it may be true that the introduction of the exact hull surface condition including the free-surface effects may bring forth improvements to some extent. The author, however, is convinced that neglect of the free surface effects may be within the order of our approximations (see, ref.? in my text). Apartina from the viscous flow problem, neglect of the free surface effects in the hull surface condition does not matter so significantly (see the results of the Washington workshop results). (3) It may be true that the vorticity in BWL is enough to produce the distributed velocity deffect, if and only if the vor-
REFERENCES 1) Weinblum,G.et al, "Investigation of Wave Effects Produced by a Thin-Body-TMB Modal 4215," DTMB Report 840, (1952) 2) Kang,S.H. "viscous Effects on the Wave Resistance of a Thin Ship" Ph.D.Thesis, The Univ. of Iowa, July, (1978) 3) Kang,S.H. "Second-Order Wave Resistance Calculation of Thin-Ship" Vol.16, No.3,JSNA Korea, Sept. (1979) 4) Landweber,L. "On Irrotational Flows Equivalent to the Boundary Layer and Wake" DTNSRDC-78/1, Nov. (1978)
Author's Reply
[
K.Marl (Hiroshima Umv)
t -ity distribution has been obtained to
Thank you Prof. Patel for your discussions. Prof. Patel's understanding is trve for the cases of real ships. In my scheme, the longitudinal vorticity component wi is assumed to be of higher order by e than that of the girth direction component W2. So eventually the diffusion of w, is neglected. But if we make another assumption that the order of w, is the same as other components, we may be possible to predict the ship stern flow. In that case near the separation position, the boundary layer equation should be solved appreciating even reverse cross-flows; unless, the revival of w, is meaningless.
satisfy the hull surface condition. In my present scheme, the vort~city distribution is solved without hull surface condition at the first stage. Unless it may get very difficult to solve the diffusion equation; one of the possible way is to solve by the time-step method. The mirror image of vorticity is determined, at the second s=age, in order to satisfy the hull surface condition. So, hopefully, the final results may be the same.
y 0.03
Of course, significant interactions between the longitudinal vortex and the outer flow can not be presented by the pre-
......
sent scheme generally.
CALCULATED BYFRST IIERATIO4 00. BY SECOND ITEATION B CHMVY * BASUYED AN KOVASZA (1969).
c
*
Thank you Dr. Kato. (1) It is true that the dividing streamline is ought to be solved in a closed procedure. The author supposes that the by using the assumed velocity field obtained corresponds to that of dividing streamline tr the first iteration. Frtescn ation it can be determined from the velocity field of the previous iteration. Such a procedure to assume the direction of the
10
A i .5
0.0
X-0.083
-0.0.3
X-0.021
-,
vortcity shedding is very common in the air-
0
foil theory.
,i"
.
(2) The author has attempted velocity
predition for another dividing streamline. But, unfortunately, no significant improvements can be obtained. (3) The answer to Professor Patel may be referred.
-.
*."
P
ig. Al
.f' 0.7
0.9
o.7
0.9
0.8
Comparison of Velocity Profiles in Hear Wake Flow of a Flat Plate. (x is nondimensional length divided by the plate
length, which is 240 cm. at the trailing edge.)
Thank you Dr. Kang for your valuable Ssuggestions.
-745-
Its origin is
IA.
GEMAK-A Method for Calculating the
Flow around Aft-End of Ships A. Yocel Odabai The Buish Ship Research AssociatiOn Wallsend. United Kingdom Oner aylan The Technical Univesity of Istanul. urkey
=aft-end i'
ABSTRACT
propeller and hull designers need to know
This paper summarises a method at developed for the calculation of thi development around the aft-end of ships at the hipReserchAssciaionimprove ritsh Association Research the British Ship (BSRA) Hul within the context of the PropellerntrctveVbrtinExcitation Hull Interactive Vibration T Project. mPHIVf) Tfi e method consists of four parts. The first part is the computation of the potential flow with surface singularity distribution. The second part calculates the development of the boundary layer by means of a relatively simple integral method
the velocity distribution in the propeller plane to distinguish between acceptable and unacceptable wake flows and where ncsayt le h ulgoer oi Studies BSRA during to hullatgeometry alter the necessarytheto flow. the flow. Stu it bVibration) rin the recent PEV (Propeller Excited Project indicated the importance of the knowledge on the flow around afterbodies of a ship, as far as the cavitation induced vibratory excitation is concerned, cf. Odabasi and Fitzsimons (Ref. 1). Additionally, it has been known for some time that the existence of an early separation and heavy vortnx sheddiag
which provides approximate information on
increases the total resistance of ships
the outer edge of the boundary layer as well as the numerical values for integral flow parameters, such as displacement thicknesses, momentum thicknesses, etc. The third part modifies the potential flow calculations due to the displacement effect of the boundary layer and wake and provides velocity and pressure values at the edge of the boundary layer. The fourth part calcu-
whereas light vortex shedding can contribute towards the improvement of the wake. Knowledge on the flow around the aft-end of ships, therefore, can also provide useful information for reducing the total
ates the boundary layer development at the by using a modified version of
prepare guidelines for aft-end design of ships it is considered necessary to be able t rdc h lwaon h f-n n
~Bradshaw's The
To reduce the vibration excitation due to propeller-hull interaction and to t
turbulent energy method. turbuleenegy
opeittefo rudteatedadi work towards this end has been an important part of the PHIVE (Propeller-Hull Inter-
th
The program developed for the imple-
I
_
calcuateono th boundary layr d et calculate the boundary layer development of~~compared a ship with modelexperimental and the results data.are SomedelodinBA conclusions are drawn and areas for further progress are proffered. 1 1.
IRUIproduce INTRODUCTION
ation: Vibration Excitation) project. The work reported summarises the method oalutea -nd= developed in BSRA to calculate aft-end flow and near wake of ships, including the effect of propeller-hull interaction. Since the aim of the PHIVE Project was to methods and procedures to be used by the industry rather than a purely theoretical research, development of the
The need to predict the flow around
BSRA method has been based on three
ship afterbodies, in particular ship wakes arises from the fact that in order to avoid unsteady cavitation and consequential
prcpe
s
high levels of vibration and noise, -747-
II
-il2
-
II
-~
_M
I
(1) making best the existing capaand computer suchuseas ofmethods bilities
layer, requires integral experience,compuin our sequential Since, tation. which
programs,
methods produce sufficiently accurate
(2) making the method, and hence the program suite GEMAK, an economically viable proposition by adopting programs which require less computational cost, and (3) making sure that the final output correlates well with che experimental
Larsson (Ref. 5), Himeno and Tanaka
data and that the predictions made by the method can provide reasonable estimates for ship wakes, To fulfil the first requirement an extensive review study has been conducted to assess the feasibility o: alternative
(Ref. 6), it was understood that, at present, integral methods could not produce the desired result and hence a differential method had to be used. It was also clear that almost all of the available differential methods were suitable mainly for
methods calculating the flow fields around
simple shear laycrs and hence some
ship hulls and the general outline of the BSRA method has emerged. The second requirement n-essitated some numerical experimentation in order to optimise the
additional modifications would be necessary independent of the choice being made on the method. As will be discussed later, analysis led to the belief that the
numerical computation scheme3, and the last
Bradshaw's turbulent energy method (Ref. 7)
requirement gave us the opportunity to re-examine the various schemes employed in the method (in the light of the detailed boundary layer and near wake measurements conducted by NMI) and reformulate some of them by making a term-by-tcrm order of magnitude analysis rather than a hypothetical order of magnitude analysis. The final outccme became a hybrid method as illustrated in Fig. 1 which will be discussed in more detail later, The method has been tested against a number of model test results for which the boundary layei -esults are available, and to illustrate its application a comparison of computed results and experimental data for the Hoffmann's tanker form (Ref. 2)
was the best choice for this particular flow. The next consideration was the determination of initial and boundary conditions for the boundary layer calculation. The initial conditions were obtained from Gadd's integral method (Ref. 8) by employing it for the forward part of the hull. To obtain the boundary conditions at the edge of the shear layer a reasonable estimate (not necessarily very accurate) of the boundary layer development at the aftend was required in order to account for the boundary layer displacement effect. Gadd's method (Ref. 8) was again used to determine both the displacement and the boundary layer thicknesses from which the velocities on the matching surface (which
and for NMI tes forms are presented together with some discussion.
encloses the boundary layer) were obtained. There were still further problems
The final part
of the paper discusses the merits and the shortcomings of the
regarding the choice of the potential flow
method developed with some comments on the scope for possible further improvements,
strain rates and normal pressure gradient, the choice of the coordinate system and the numerical scheme for the shear layer calculations which is discussed below. However, the scheme of calculation, as shown in Fig. 1 emerged at this state of
2.
I
calculation method, inclusion of the extra
PROBLEM FORMULATION AND METHOD OF SOLUTION Fcllowing the initial revi..
Slayer.
result for the initial 75 per cent of the ship where the flow is a simple shear layer with the exception of the flow around the bow, we concentrated only on the aft-end flow. In the light of the experience of other researchists, cf. Landweber (Ref. 4),
study it
the investigation.
became evident that a number of choices had to be made for the solution of a number of sub-problems. The first question was on the type of flow around aft-end of ships. Since a large amount of extra strain rates are present in the flow it was necessary to decide whether the flow should be considered as a thickening shear layer or a distortion
This choice is crucial because to
3.
CALCULAIION OF POTENTIAL FLOW
As is well known flow around a body, moving with a constant velocity on the otherwise undisturbed free surface of a liquid, can only be computed by adopting certain assumptions. Although the basic
assumptions allow us to formulate the
use an ideal vortical flow approximation
problem within the framework of the classi-
would be more appropriate if the flow was considered as a distortion layer. From a study of a limited amount of lata we concluded that the role of the Reynolds stresses in the aft-end flow was still important, and hence it should be treated as a complex turbulent shear layer.
cal potential theory, the existence of a free surface and the representation of the body surface create additional problems, which necessitates some further simplifications. For the solution of the irrotational flow about a ship form, a number of
The next task was to devise a methodology and a computational scheme suitable
potential-flow methods have been developed. In reality however all of them are
for the calculation of a complex shear
restricted due to the complexity of the two
- 748 -
E LL__-
_
---
--
.
major boundary conditions, namely the body surface and the free surface conditions, which can be treated in their linearised form. As is well known, from theoretical wave resistance studies even for the time consuming second order theory, the agreement with experimental results is not
becomes particularly significant at highly curved regions such as the aft-ends of ships. To avoid such errors it is possible (1) to increase the number of elements and hence to reduce the element sizes or (2) to employ curved surface elements with variable source density as is investigated
particularly good. Therefore in recent years methods satisfying the exact body surface condition have received much more attention. Following the first success of Hess and Smith (Refs. 9, 10), the methods of Surface Source Distribution, (SSD) have been applied to various problems. The original approach by Hess and Smith does not include the free surface effect and hence gives the solution of Neumann problem for a given form and its image, (i.e. Double model in an infinite fluid), In order to improve the accuracy of the results obtained from the Neumann problem, Brard (Ref. 11), and many others studied the Neumann-Kelvin problem which again takes the exact body surface condition into acc3unt and includes the effect of free surface in its linearised form. In an attempt to include the non-linear free surface effects, Gadd (Ref. 12) introduced the idea of additional source distribution in the region surrounding the double model in the undisturbed free surface plane; recently he has improved his method by extending the free surface panels to cover a larger portion of the disturbed free surface region and by adding a new distribution of Kelvin sources along the axis of the hull at one third of the draft, (Ref. 13). Critical evaluation of existing methods can be found in the review studies of Wehausen (Ref. 14), Tinmnan (Ref. 15), Chapman (Ref. 16) and many others. A number of sophisticated SSD methods are available, but even with current computers, the computation of the flow around a threedimensional body requires a large amount of computer time and also careful thought for the rational discretisation of the body surface; that is, to avoid unrealistic results and numerical instabilities, the size, shape and the number of surface source facets have to be carefully selected beforehand. In the early stages of the PHIVE project extensive work was carried out at BSRA to find the optimum solution of various discretisation problems and to reduce the computer costs for the potential flow calculations of a given three-dimensional body. In this work classical Neumann and Neumann-Kelvin problems have bten solved. In most of the SSD methods, the body surface is replaced by quadrilateral elements or facets. One of the major drawbacks of this approximation is that the planes formed by all four corners of each element do not necessarily match the real body surface hence, either a discontinuity will occur on the source surface or the centroids of each element (so called control points) will form a different body shape than the original one. This statement
by Hess (Ref. 17, 18 and 19), or (3) to use only triangular surface elements, cf. Webster (Ref. 20), $aylan (Ref. 21). As is expected any increase in the number of surface elements will increase the computer costs dramatically. The second alternative. the use of higher-order surface elements, has also its own drawbacks. Hess was the first to show that the effect of surface curvature and the effect of the variation of singularity distribution are of the same order of magnitude. Therefore inclusion of only one of them is inconsistent and also does not improve the accuracy of the results sufficiently. Use of the consistent higherorder formulation, such as paraboloidalpanel linear-source or cubic-panel quadratic-source, is generally not feasible because of the complication of problem formulation and because of the much increased computer time requirements. The last alternative, to increase the accuracy, that is to represent the real body surface more correctly by the use of the triangular surface elements, has been investigated by BSRA. Although the control points of triangular elements are closer to the real body surface and all three corners of an element will lie on the same plane, it is found that if the characteristic length of an element is larger than 2.5 times the distance between adjacent control points numerical computations become unstable. This limit is generally exceeded when only triangular elements are rP-sidered and some undesirable waviness appears in the final results. Figure 2 illustrates this type of waviness for a simple axisymetric Huang form, cf Huang, et al. (Ref. 22). Another drawback of the triangular discretisation is that the control points generally do not lie in the same vertical plane which makes additional interpolation routines necessary to calculate the desired properties at given vertical positions. Having considered these arguments it has been decided that body surface should be discretised by using quadrilateral flat elements (each having approximately the same size and area) and triangular elements should be used only where necessary. As an obvious result of this choice higher computational costs become almost unavoidable. In other words, for better representation of body surfaces and for better results, the total number of elements should be kept as high as possible, or else, to fulfil the requirements explained in the first section, a different approach nas to be found. During the PHIVE project, following the choice of element shapes, a new discretisation technique has been developed and a pronounced saving on computer costs was obtained with no appreciable decrease in
-749-
accuracy. For ships or ship-like bodies discretisation is made by using an expanded
surface and in the data generation stage of the potential flow programs (Double model aitd N-K programs) special emphasis 'as been attached to the fore- and aft-ends and to the regions where the curvatures change suddenly. It is considered that the user should have the opportunity of using his engineering judgement in the preparation of the surface elements. Although the data generation takes longer compared to the standard process, the flexibility introduced in this way is worth the extra effort. In addition to the shell expansion type discretisation, separated modules are formed for the bottom part and the side walls and triangular elements are also used to provide a better approximation to the hull geometry. Figures 3 and 4 give an idea about the accuracy of the results obtained
from double-model program.
following findings can be expressed: (1) Following tae procedure explained in
Ref.(21), the results obtained from
(2)
(3)
As can be seen
the (N-K) program agree well with the results of the double model program for very low Froude number (Fr 1 0.01). This is illustrated in Figure 8 and gives the confidence for the use of the (N-K) program. If the free surface effect is considered the non-dimensional pressure distribution changes from that obtained for the double body, cf. Figure 9. In this Figure the experimental results of Huang and von Kerczek (Ref. 23) and the theoretical results reported by Adee, et al (Ref. 24, 25) are also included for comparison. In the propeller plane the effect of a linearised free surface appears to be important. For the Huang body No.1,
the differences in axial velocities
from these figures, for the Hoffmann tanker form, it is possible to achieve a reasonable pressure distribution with fewer number of elements, As the main purpose of the study is to provide information on the flow around the aft-ends of ships, possibilities of using a simplified representation of the forebody have also been considered to reduce the number of surface elementz. This idea becomes more meaningful if the variation of pressure and streamlines along the parallel middle body is considered. Having investigated the effect of forebody it has been found that, to compute the aft end flow, the front 25-40% of the ship can be neglected or at least fewer elements of larger size can be used to represent the forebody. Figures 5 and 6 illustrate this feature for a Huang body and for Hoffman's tanker form. As an overall comparison of BSRA's discretisation method, the streamlines around the same tanker form have been calculated for a total 74 surface elements and are plotted in Figure 7 against Hoffmann's calculated streamlines
are found to be very small, however in tangential and radial velocity components there is a significant effect, cf. Figures 10 and 11. These results confirm the belief that all three components of the velocities in the propeller plane should be measured in order to make meaningful comparisons. As is shown in Figure 1, within the PHIVE project, potential flow calculations have been used only to provide the necessary information for the three-dimensional boundary layer calculations and, as is explained in Section 2, for the same reason a multi-stepped procedure has been designed including Gadd's integral method and the program unit for the boundary layer displacement effect, (blow-out program). The linearised free surface effect has been excluded from the fina: program package in order to kcep computer rzquirements within reasonable limits, but the experiences gained from the (N-K) program and surface discretisation technique have been widely used through the whole hull flow suite.
for 414 elements, (Ref. 2).
4.
In the literature there is a large for the number of published information double body problem but, in the case of the (N-K) problem, both theoretical and experimental results are rather scarce. In employing the (N-K) approach the biggest difficulty is the relatively large computer time required. The reason leading to the development and implementation of the (N-K) program is that the results of the double body approach are only valid for very low Froude numbers and inclusion of the linearised free surface condition is expected to provide a larger range of applicability. On the other hand the computer time required for the numerical integration of the free surface related terms increases considerably, i.e. for a ICI. 1904-S computer the elatsed time is found to be nearly 21 hours for a ship form presented by 60 surface elements. From the limited studies conducted at BSRA the
--
~~M
-
=-
CALCULATION OF BOUNDARY CONDITIONS ON MATCHING SURFACE
The potential flow solution gives the velocity and pressure distribution around a three-dimensional body, together with the characteristics related to body geometry, i.e. coordinates of the control points, areas, components of the unit normal vectors etc. In the BSRA program suite GEMAK, this information was used both for computing the boundary layer development by Gadd's program and for determining the displacement effect by the blow-out program. Gadd's Integral Method: To compute the boundary layer developmert by an integral method, Gadd has solved the streamwise and transverse momentum integral equations together with an entrainment condition at the edge of the boundary layer, cf. (Ref. 8). The formulation of
dicular axis system, for this transformation the angle between the transverse and the perpendicular (normal to the potential flow direction at each control point' directions is assumed to be small. It has been shown that, for ship hulls, the results obtained from this method agree well with the experimental results specially for the regions where the boundary layer is thin and the body curvatures are not high. Near the ship sterns however the method generally overestimates the boundary layer development. To avoid such unrealistic results Gadd assumed that the velocity potential for the flow outside the boundary layer can be approximated as:
these eqaations has been wade by considering a locally rectilinear but non-orthogonal axis system that is, at each control point, x1 is taken in the direction of the potential flow, y' normal to the hull surface and z' parallel to the transverse line of the control points. Since the integral methods predict only the integrated properties at points on the body surface, the boundary layer integral quantities are defined as: 6
[0
u, - u' dy'
1 =
,
(streamwise displacement thickness)
o wj -
(6
w'
= U.x - cos kzeky
U
02 =
dy' and the displacement effect of the boundary layer can be modelled roughly by altering the w components of velccity by U.G M Z(a
o (streamwise dy' momentum U2thickness)"
u'(ui 2 - u')
I
"
where G is the length of the surface from waterline to keel. However, during the PHIVE project a different approach has been adopted for the calculation of
o
022
=
I
w'(w
-
')
dy'
--
I
z(--)
U
displacement effect.H
0
12
=
J
A w'(u
J
-
u')
2
Blow-out Method: It is known that a boundary layer displaces inviscid flow outward from the
dv' -
body creating effectively a semi-infinite pseudo-body (Fig. 12). When the boundary layer is thin this displacement effect is small and its influence on the boundary layer calculations is negligible. For thicker boundary layers, however, this statement is not true and the displacement effect manifests its influence en the boundary layer calculations in two ways. Firstly the streamwise znd the spanwise pressure gradients differs from those calculated from the potential flow around the naked hul] due to the differences between the geometries of the hull and the pseudobody. Secondly, the assumption tnat pressure is constant along the normal within the boundary layer (i.e. the normal pressure gradient is equal to zero) is no longer valid. As is well kLown, one of the assumptions employed within all the boundary layer theories is that the flow outside the boundary layer is equivalent to the irrotational flow of an ideal fluid. This assumption immediately suggests that if the flow around the pseuao-body is solved as an external Neumann problem then determination of the improved external boundary conditions for the boundary layer calculations becomes a routine task. The first person known to apply this
o
u'(wj 2
1U2
-
w') dy'
o where u' and w' are the velocities in the x, and z' directions, U is the external velocity and u', w' are the velocities at the edge of th bondary layer (y' = 6). The number of these unknowns are reduced by the choice of velocity profiles. In the streamwise direction velocity profiles are assumed to belong to the Coles profiles cf. (Refs. 26, 27), whereas in the transverse direction Mager's form has been chosen (Ref. 28). The local skin friction coefficient in the streamwise direction (C ) is determined as a function of the shipe parameter H and the momentum thickness I U.V 1 1 /), Reynolds number R. , (R 11 ll and the Ludwieg-Tillmann relation (Ref. 29) has been used for the initial estimate of C . Finally, Gadd has proposed a new formuUtion for the entrainment coefficient Cv: CE = 0.Ol.H (1 + tanh (5H-7))
idea is Stuper (Ref. 30) who made an
H-11"
attempt to calculate the lift of a Joukowski aerofoil. He was able to estimate the di.~ribution of the displacement thickness
compare the results with experiIn ordertheto integral ments, quantities have been transformed into the streamwise and perpen-751 -
--
over the wing. He then replaced the wing by a new boundary displaced out from the wing and calculated the potential flow about the new shape. Preston (Refs. 31,32) provided a more rigorous derivation of the equivalent flow for two-dimensional flows, Moore (Refs. 33, 34) generalized the idea to three-dimensional flows in general orthogonal coordinates. Lighthill (Ref. 35) proposed four alternative methods for the boundary layer displacement effect for both two and three-dimensional flow problem. The blow-out method (or equivalent source method in Lighthill's terminology) received a wide acceptance in the field of aeronautics and has been used extensively, cf. Giesing and Smith (Ref. 36), Jacob (Ref. 37), Rubbert and Saaris (Ref. 38). Recently Landweber (Ref. 39) critically reviewed the methods proposed by Preston and Lighthill and proposed a centreline singularity distzibution. The method adopted by BSRA follows mainly the approach proposed by Lighthill (Ref. 35) and, due to the coordinate transformation, have been made. only minor modifications
ment thickness to the body and the ideal flow is computed around this new geometry. To implement the second approach, let us consider the mass transfer at two sections in the boundary layer (see Fig. 12). At the first section the potential flow outside the boundary layer will be moving with a velocity U1 and the mass flux per unit span will be U 6 At the secona station the flux caA he I-itten as:
In order to improve ones physical appreciation of the blow-out method, one can consider two-ditransional flow problem for a (s-n) coordinate system, where s is measured along the body contour from the front stagnation point and n is measured along the nor'il; for this the difference between the external stream velocity and the velocity component within the boundary layer (U-u), represents the reduction in flow velocity due to the presence of the boundary layer. From the continuity equation there must be a constant volume flow per unit span between the surface and any streamline just outside the boundary layer. There is however a flow reduction due to the above-mentioned velocity defect which is expressible as:
surface, (i.e. -
(U
- u)
U1 61 (U1 61) The difference between the two stations will be (U 6 s 1 This difference in flux can be supplied by blowing from the surface with a velocity vw w , such that: v
w
=
U. .61)
a
This blow-out velocity is then used to modify the boundary condition on the body v + vw). As the an the potenfor matrix influence coefficient tial flow was already calculated to determine the initial flow, determination of the total source strengths amounts to the solution of a system of linear equations with a new right hand side. In three-dimensional flows this procedure requires some modifications due to the existence of cross flow within the boundary layer. Following the notation used by Gadd (Ref. 8) one can consider a locally orthogonal rectilinear axis system, centred at the control point of a surface element, with axes x in the stream direction and y normal to the element plane and z in the crossflow direction, then the streamwise and perpendicular displacement thicknesses can be defined as:
dn. 6
~given
The constant volume flow is obtained if the flow reduction inside the layer is compensated by an outward displacement of the body coutour by an amount 61 which is as:
( uU)
dY
u
6 62
=
w o0
6
U
(U
- u)
dn.
where 6 is the displacement thickness of the bouAdary layer. Detelmination of the strengths of the equivalent sources on the body surface is made either by using an equivalent body and solving a new Neumann problem or by using Gauss' flux theorem. In the first approach the physical body is replaced with a pseudo-body by adding the displace-
where u and w are velocity components at the edl of thU boundary layer and U is the total The external velocity (w = 0 and u = U). equation of contiluity takes the form: av Ty-
au IW-
:ntegrating this boundary layer: h 1 V |-
aw
+
equation across the
au (-
aw -)dy
h
boundary values for the final stage of the boundary layer calculations. For Hoffmann's tanker form, the effect of blow-out on the source strength distribution can be seen in Fig. 13.
)] dy 0fh
au
aw
J a'x
- ( --
aw-l) dy
5.
3z-
where, for h > 6, the upper limit of the integrals can be replaced by -, cf. Lighthill (Ref. 35), and hence: " v
TURBULENCE MODEL As is well-known, due to the present
o
(U.
)
(U.61 ) dy +
[ i
(U.62 ) dy
0
3U Y 3-x In this equation the last term alone comes from the normal potential flow calculations. Therefore the first two terms will give the additional blow-out flux due to the boundary layer, as if there were an additional source distribution on the ship surface. The strength of this imaginary distribution can be expressed as: a S (U.6 (U.62 )In the blow-out program, streamwise and perpendicular displacement thicknesses have been taken from Gadd's calculations and for the numerical solution of the last equation, use has been made of an adopted numeric derivation routine based on a least squL. scheme. As is already mentioned for the twodimensional case, solution of the problem is trivial since the influence coefficient matrix is available from the initial potential flow calculation. As the boundary layer extends beyond the body to form the wake the blow-out scheme is not confined to the body surface, it has to b3 continued sufficiently lAr downstream. Experience gained in potcntial flow computations shows that this distance should be of the order of five times the boundary layer thickness at the aft-end of the ship. However, in practice the displacement thickness of wake is hardly ever computed. This poses a serious problem in the inclusion of wake effect on the boundary values at the edge of the boundary layer. Normally the displacement thicknesses are expected to decrease in the near wake as the momentum thickness remains approximately constant. When there is no information on the wake flow, it may be a reasonable approximation to use the displacement thicknesses at the aftmost position of the hull as the displacement thicknesses of the computation region in
the wake.
speed and storage capacity of computers one is not able to solve the N-S (Navier-Stokes) equations and some form of approximation has to be made. A convenient method of approximation is the "turbulent shear flow" approximation where the principal quantities of the N-S equations are assumed to consist of a mean and a fluctuating part and the governing equations are obtained by applying time averaging, cf. Townsend (Ref. 41). Further simplification results in what is known to be "the boundary layer" or more correctly "the thin shear layer" (TSL) approximation. As the name implies TSL approximation assume some form of thinness for the shear layer thickness (6) compared to the principal radii of curvature (R) of the streamlines. However the restriction can be relaxed in the "fairly thin shear layer" (TSL) approximation where a6/3x1 is assumed to be small, x being the main stream direction, cf. Biadshaw (Ref. 42). The equations obtained with these approximations are incomplete and the difference between various differential methods appears in the choice of the closure equations which are introduced to model the transport of Reynolds stresses. In our assessment we considered three differential methods; the zero equation model, cf. Cebeci et.al. (Refs. 43, 44), the one equation model, cf. Bradshaw (Refs. 7 & 45), and the two equation model, cf. Rastogi and Rodi (Ref. 46) and Abdelmeguid et.al. (Ref. 47). The merits and shortcomings of these methods were compared on their physical and mathematical basis, and on their evolutions, i.e. the type of data used in the determination of empirical constants or function. The conclusion was that, accepting the fact that none of the methods was universal, the Cebeci's method was easiest to modify but such a modification would require a large amount of ship boundary layer data which was not available, and Bradshaw's method was both physically and mathematically more correct than the others for the aft-end flow. Within the TSL approximation the governing equation and the Reynolds stress transport equations for the selected turbulence model are: Momentum transfer equations for i=l and 3
U
The resulting source strengths have been used to evaluate the velocity distribution at the edge of the boundary layer (matching surface) to provide the improved
av.i
Uj ax.
-753-
x.
ij
o
i
(1)
A
Bi Momentum transfer eqaation for i=2 p 0 (2)
wall, and furthermore the contribution f2 rom the outer edge of the boundary layer to the generation of shear stress is quite small. The law of the wall used in BSRA's method is due to van den Berg (Ref. 49) and his results are modified for the extra strain rates as proposed by Bradshaw and
Maroney (Ref. 50).
I 0(3)
ax.
Reynolds stresses transfer equations for i=l, 3 U. 2a
a-,
a i =
ITI
-
gx '21
(x)
LAt
[G( I-tm 1/) .] (4) 2x where Xl, x2 , x. are, respectively, the coordinates in treamwise, normal, and girthwise directions*, U. and u. (j=1,2,3) 3 are, respectively, the mdan and the fluctup are, P and components, velocity ating respectively, the mean and the fluctuating v tpation dynamic pressures, T. (i=1,3) are the Reynolds stress, p is the density of fluid, ITI is the magnitude of the total shear vector is the maximum value of oetside and the z icous sublayer. This formulation contains three empirical quantities; a1 ( Tjr/q ) the stressenergy ratio whea q is the turbulent kinetic energy, L the dissipation length scale, and G the entrainment function. As a comparative study of aprelta ofthe -
expeimetaldat, Brdshw (ef.48)has
showed that the motion within the boundary layers consist of an "activei universal component scaling with (T/-)- and x which produces the sh..i -tress, ang an "inactive" compone- imposO. by the eddies and pressure fluctuations in the outer part nf the boundary layer, which does not procuce shear stresses and can be regarded as a quasi-steady oscillation of the inner layer ilow. Measurements indicated at that time that a suitable value of a was 0.15. Later experiments, however, indicate that a takes smaller values near the wall and n~ar the outer edge of the boundary layer and remains almost constant between 0.2 x2 /6 and 0.8 x 2 /61,the value of the constant being between 0.135 and 0.18 depending on the flow configuration. Computations at BSRA, however, indicated that the effect of change of a1 on the overall turbulence structure is only marginal. This is due to the fact that the flow in the viscous sub-layer is governed by the law of the Equations are written in Cartesian tensor notation and conversion to any system or coordinates can be done by the *
relationship x
=gi.C
the metric tensor and t the new coordinates.
where g..
is
(j=l,2,3
are
The dissipation length scale parameter L is defined by Che relationship L = (T/0) 32/ where c denotes the viscous dissipation. This parameter is a measure of the scale of eddies which contains most of the kinetic energy and which are resp'onsible for the trauspurL of momentum. high Reynolds numbers the rate of energy dissipation is equal to the rate at which energy is transferred to progressively smaller eddies, and this rate is determined by the structure of energy containing eddies. Therefore, the dissipation rate is not a function of viscosity. In the Bradshaw proposed (Ref. 49) function method original a universal distribution tion Lnivrsaldiorit var L L = L (x2/6) for the variation of dissilength scale across the boundary layer, and in the later version of the method (Ref. 45) a transport equation is used to determine L. Calculations carried out at BSRA indicated that this is the most important quantity, as far as the velocity and the shear profiles are concerned, and in agreement with Bradshaw (Ref. 42), the existence of extra strain rate, i.e. streamline curvature, divergence, swirl, etc., can produce significant changes in dissipation length scale. Although it been argued that the change in L due t~ er strain rat is caused case by by a a tu extra strain rates is reduction in the production of turbulent kinetic energy rather than with an increase in dissipation, cf. Ref. (41) p.74, since as /2/E a ratio L expresses 3/ 3 (T032/ (a/o) 2 /= af/ 2 , this argument does not alte# the final outcome. Finally, Bradshaw's empirical function G which is also known as the entrainment function represents the diffusion of turbulence and again is related to energy spectra of turbulence. This function is used to model the effects of pressure redistribution and transport of turbulent kinetic energy in the normal direction and is defined by:
max G T P This definition is consistent with the structural similarity assumption, cf. Townsend (Ref. 41) p.121-129, and it also corresponds to the observations that the energy transfer depends mostly on flow in eddies large enough to span most of the
boundary layer, taking the form either of
convection of smaller eddies or transfer of energy from one part of the large eddy to another. As a result of the observed effects of extra strain rates one would
_754.
_______-
expect that the entrainment function too should be modified since it is known that negative extra strain rates increase entrainment whereas positive extra strain rates reduce the same. During the PHIVE project such modifications have been implemented in BSRA. The changes due to
The second alternative is due to Bradshaw (Ref. 52). He proposes that the guiding principle in a pragmatic approach is to inspect the terms which are present in the N-S equations but are neglected in the boundary layer equations and devise approximaticns for these that are incon-
these modifications, however, were less
venient to retain in exact form.
spectacular than the effect of the changes
improvements in the boundary layer equations
in the dissipation length parameter,
can be vade if a degenerate form of the normal component momentum equation is included to predict P(x2 ) reflecting the
Improved Turbulence Model
missing upstream influence. The description of and the improvements on the method given so far is valid for TSL approximation and to make it suitable for the aft-end flow calculation further changes were necessary. These were the incorporation of extra terms from the FSTL approximation, normal pressure gradient, effects of extra strain rates, and the choice of a convenient coordinate system. The first modification is fairly straightforward involves some curvature terms and and mainly turbulence production terms. In their inclusion into the governing equations the order of magnitudes of each term were also checked against the boundary layer measurements made by NMI. Incorporation of normal pressure gradient was more troublesome because f the numerical problems. Within the TSI. approximation pressure is rendered constant across the boundary layer. Although this assumption holds approximately for a large majority of the boundary layers of practical interest it becomes unsatisfactory when the boundary layer becomes thicker especially in regions of high curvature. The use of the blow-out scheme and the pseudo-body concept certainly help towards the improvement of the solution but only by improving the boundary conditions at the outer edge of the boundary layerThe assumption that, on the normal, pressure is constant still remains. Accepting the fact that the full N-S equations cannot be solved, there were three alternatives: (1) Use of higher order boundary layer approximation. (2) Refinement of TSL approximation based on experimental evidence. (3) Empirical modelling of the normal pressure gradient. The first alternative is certainly the scientifically consistent approach. However, from an examination of the second order governing equation it is seen that due to both the additional terms and the elliptic nature of the equations computations become prohibitively expensive. Furthermore, the only additional terms in the normal momentum equation are curvature related terms and the terms related to the normal velocity and the Reynolds stress gradients do not appear, which are known to be important from the analyses of experimental data. To pursue this alternative any further was therefore not considered worthwhile,
.755 -
Worthwhile
Inclusion of
this equation into the governing equations still makes tne system elliptic, necessitating an iterative calculation scheme. However, the computation load may be reduced by solving the normal component equation separately, as done by Pratap and Spalding (Ref. 53) and Mahgoub and Bradshaw (Ref. 54). Although this alternative looked very promising considerations of the computation cost has meant that it has had to be kept for possible further development. The last alternative is the use of empirical normal pressure profiles which, considering the empirical information used in both the integral and differential methods, is justifiable provided they are based on experimental evidence. Inspection of experimental data indicates that pressure variation in the normal direction is mostly regular, cf. Patel and Lee (Ref. 55). and a quadratic expression may be sufficient to approximate this variation. Given both the physical appeal and the relative success in its use, the centrifugal approximation was chosen for the BSRA approach and correlation with experimental data was achieved by introducing a new velocity scale. Although the early theoretical and experimental work on the effect of extra strain rates go as far back as 1929, cf. Prantl (Ref. 56), Wattendorf (Ref. 57), concerted efforts in this direction appeared only after 1960. It is common practice that when the flow around a curved object is studied analytically a body oriented curvilinear coordinate system is employed. Adoption of such a coordinate system produces some extra terms in the boundary layer equations which are related to the curvature of the coordinate lines. If one carries out an order of magnitude analysis in laminar and turbulent boundary layers by using 6/R, R being the radius of streamline curvature, the extra terms that appear are of order 6/R times the largest existing terms, cf. Nash and Patel (Ref. 58). Although this result seems to be confirmed by experiment for laminar boundary layers, experiments on turbulent boundary layers indicate that the change in the Reynolds stresses due to the extra strain rates is roughly ten times the prediction obtained frcm a straightforward order of magnitude analysis. In the light of this experimental evidence it is imperative that special emphasis should be placed on the inclusion of the effects of the extra
strain rates in the calculation of the boundary layer development around the aftend of ships. In 1969 Bradshaw (Ref. 59) proposed an engineering calc'alation method by using an analogy between ruoyancy and curvature effects. Although the present state of knowledge on the subject is far from complete, experiments carried out by Meroney and Bradshaw (Ref. 51), Patel and Lee (Ref. 56), So and Mellor (Ref. 60). Castro and Bradshaw (Ref. 61), Cebeci (Ref. 62) and Hunt and Joubert (Ref. 63) indicate that Bradshaw's method produces reasonably accurate results. In fact recent experiments by Snits, Young and Brndsha (R.. 64), and Smits, Eaton and Bradshaw (Ref. 65) demonstrated that the method also works for curvature impulses and for three
restrictions on the coordinate system could further be relaxed. This last point wa-s also important in relation to the numerical computation. In the numerical computation side, the problem was not only the choice of an integration m'_thod, but also the choice of the integration region. After a review study the conclusion wds reached tat implicit methods, in spite of their unconditional stability, were not suitable for BSRA's purpose due to the demand for inversion of fairly large matrices to determine the flow quantities which was too costly in BSP's ICL 1904S computer. Amongst the available explicit methods we tested an extended version of Lister's method of characteristicc- 'Ref. 69) and Lindhout's version of Cc. t-Isaacson-Rees
dimensional curvatures. An excellent review on the subject is available by Bradshaw (Ref. 52). The essence of the method is to modify the dissipation length scale by
scheme (Ref. 70). After some testing we decided on the former method due to its flexibility. Stability of numerical integration was ensured by using a modified Courant-Friedrick's-Lewy stability criteria, cf. Raetz (Ref. 71). The complete program suite GEM.AK has been designed to consist of component programs which are interlinked as shown in Fig. 1. Discretised hull geometry and ship forward speed are used to run the potential flow program which also drives Gadd's integral boundary layer method program. As the computation of the free surface contribution (Neumann-Kelvin problem) is quite time consuming and costly, the double-body (Neumann problem) computer program is being used at present. The output of this stage contains integral boundary layer parameters as well as velocity profiles. rPplacement thicknesses obtained in this way are used to re-evaluate velocity and pressure distribution at the edge of the matching surface is determined by a special smoothing of the edge surface of calculated boundary layer. This task is achieved by the explained blow-out scheme. The output of this stage together with the discretisedhull geometry provides the necessary information for the running of Bradshawts turbulent energy method program. The output of this program produces the velocity and shear profiles, and the internal parameters along the boundary layer and near wake for a given ship form and speed. Since the computational domain for the turbulent flow calculations is now decided beforehand by the choice of the matching surface numerical integration the gover-ning equations has to be of made in a
L Leff
1 + (a e) eff
where L~
is the effective dissipation length Ale, e is the extra strain rates obtained by using stress Richardson number, a is the empirical magnification constant, and (ae)eff is given by d X ! 0 eff oe - 0 eff where X z 106 and this relationship allows ?or the effect of memory to be taken into account. The effect of extra strain rates for moderate curvatures on streamwise velocity component and Reynolds stress is illustrated in Fig. 14. D Nwhich 6. COORDINATE SYSTEM AND N MRICAL SOLbTION
"
At the beginning of this research we were made aware of the difficulties on the choice of the appropriate coordinate system by the conclusions of the previous investigators, cf. Mangler and Murray (Ref. 66). Therefore, a considerable amount of effort was spent for the design of a method for the automatic generation of a suitable coordinate system and a program suite, called "A?-ACAW1', was developed to achieve this end where the mapping method of Tuck and Von Kerczek (Ref. 67) was used to determine approximate streamlines. As one gained more experience in turbulent flow computations it was realised that provided; (I) one of the coordinates is nearly aligned with the normal (see Wesseling (Ref. 68) for the need to have this condition), and (3) the marching direction, i.e. x -axis, does not make an angle more than thirty degrees with the flow-direction, any coordinate system is equally acceptable. Furthermore, if the concept of matching surface is adopted
M
I
somewhat different fashion especially since it was desirable to avoid a complete iterative scheme as used by Abdelneguid et.al. (Ref. 72) and Mahgoub and Bradshaw (Ref. 54). The use of smoothed boundary layer edge surface in place of a semi-arbitrary matching surface he-ps tow2rds the achievement of this purpose but in itself it is not sufficient. Therefore, a procedure called "'stitching" has been developed by the authors. In this procedure calculation starts from the matching
U
J
surface towards the wall, after wall boundary condition is satisfied the computation is carried out from wall to the matching surface. if the velocities agree then the computation goes to the next step otherwise the velocities and Reynolds stress are updated by using an underrelaxation scheme and the boundary conditions at the matching surface is updated and the computation restarts from the matching surface as before and the procedure is repeated until convergence is achieved. 7.
NOMINAL AND EFFECTIVE WAKE CALCULATION
7.1
Nominal Wake Most of the present
boundary layer
calculation meliods attempts to solve for the near wake either by using a fictitious flat plate at the centre plane or terminating the boundary lzyer calculatJo- from this poin .wards. In the first type of calculat., velocities near the centre plane are underestimated as a result of the wall boundary conditions and esclusion of mixing. In the latter case. the assuption of the mixing layer does not strictly hold because in the near wake some of the boundary layer flow behaviour is still observed as a result of the memory effect of the flow. To find a satisfactory solution to the problem two alternatives were investigated: (1) the use of Eradshaw's intcraction hypothesis (Ref. 73). and (2) derivation of new boundary conditions for the wake centre plane. Efforts were spent in both directions, but because of its success in airfoil and axisysetric flow applications cf. Morel and Torda (Ref. 74). Htuffcn and Ngv (Ref. 75). Andreapoulos (Ref. 76). the former approach has been adopted. According to the interaction hypothesis. the near-wake consists of three regions (see Fig. 15); the inner wake region is the place were the mixing takes place. The second region is the extension of the wall layer of the boundary layer which now matches to the edge conditions of the inner wake layer, and the last region is the continuation of the wake region of the boundary and it is the least influenced part by moving from the bonay layer to wake. Since most of the structure of the near wake is similar to the boundary layer with the exception of the inner wake layer, interaction hypothesis suggest that it can be treated like an overlapping boundary layer, and overlapping takes place in the inner wake region. So far as numerical compuation is concerned the change to be made to the boundary layer calculation scheme is "inimal and the change from boundary layer to near wake is identified by using a logical control statement.
Propller-Hull Interaction and Effetive Wake Calculation From the practical point of view the determination of effective wake distribution was of a paramount importance both for the propeller design calculations and for providing the necessary input in the cmoration of the cavit'ition induced hull surface forces. Our preliminary studies revealed that most of the relevant anaiytical work was based on the studies of Dickmann (Refs. 77-78) an4. the recent research efforts were mainly aimed at deriving an empirical or a semi-empirical correction method for t _ de-ivation of effective wage from the easured nminm-adel wake by using available d-ata. cf_ Sasaitna and Tanaka 'Re'. 70%. _ (Ref. SW). The work carried out at O-TIhSRIC by Hung and his co-worters (Ref. 81) on axisvmoetric bodies -s not in line with the writers' method -f approach. because he used . inviscid flow model (vorticity include-!), hence assumed that the flow at the aft-end of ships is a distortion laver rather thann a ccmplex turbulen shear layer. Since -ther "-T clear indicain on the meth .od b adopted. the witers afer cre4u1 examinatzion of availble data, decioc to employ an improved version of DC h's t conception wi hot rrortir tt separation of the flow itno _orcnni-i1 visrous. and wa- making comz.nn-s. Since the -,A approach c--siss o Tsteps the tffect of the prcnvzier i-neration on the .- ow dve R h taken into account in each st-. In -he first step the effect of pr. ller is assumed to be potential in a- -i'n ar D c thz source sre.h:__ in t oot i l flow programs ar- modifiedee to ace--' f'_ t ieoc'ies propeller induced norm modified potential flw is t-hen use--d . drive the integral calculzion metod fthe estimat ion of f- ,tdarv layer a aisplac ent thicknesses. oe to the Change in the external otential field the new l calculated boundary layer differs "rem nominal flow boundary layer. ard ccnse quently the region oi integration or th detailed boundary layer caki-ulatio aischanges. The next step is t -c "xi cation of the boundary conditims the matching surface. This modificatio-;a s place because (I) the new matching surface location and the new source strengt are different from the necmil 'lo cse, and (2) on the =atching surface pr-pller induced velocities are linearly added to the velocities induced by the pseuc-bAy (hull plus displacement thiknes). Following the re-determination of the camputational region and the boundary conditionE, the turbulent flow and n-ar wake calculat-ons reains -he sc as in the nminal flow caiculation sin" the form o? the governing equations for the field does not change with the presence of the
-
propeller-hull interaction. A flow chart illustrating the sequence of operations in the computation of effective wake is illustrated in F~g. 16. 8.
the sensitivity of the method to the variation of initial conditions and of the built-in empirical correction factors. These tests indicated that the sensitivity to the variation of the initial conditions was very high, indicating that the integral parameters employed to start the computations must be chosen very carefully. Variation of the pressure rise reduction factor (an empirical factor to avoid the formation of a stagnation point at the aftend) and the blow-out scheme showed that this parameter had a controlling influence on the development of V'w around the stern and in order to obtain . reasonable agreement with the experimental data it was necessary to change this parameter for differing hull forms. The method, generally, gave good agreement with experimental data for the forward 80 to 85 percent of ships and for extreme V-shape hull forms this distance reached up to 90 percent of the ship length. For the last 10 to 20 percent of the ship, the method generally overestimated the boundary layer development. This, however, was a useful feature so far as the BSRA computation scheme is concerned since it was assumed that the matching surface obtained from the application of Gadd's method (Ref. 8) would enclose the real boundary layer. As explained within the text the blow-out scheme was achieved by a modified use of the boundary layer program and the only tests to be carried out on this part was the comparison of panel source strengths .;ith and without the blow-out scheme to make sure that when the boundary layer is thin the values obtained are close. Verification of the program BRADSHAW was the most time consuming and important part of the total program verification process since the outcome of this stage would decide the success or otherwise of the program su-te GEMAK. During the program development stage the initial version of the program was tested against the published experimental data on wings giving quite satisfactory results. The next verification stage was to compare the fully developed program results with the wind tunnel data for scaled ship models.
RESULTS AND DISCUSSION
During the development of the hull flow studies three different types of application have been considered. They are: (1) application for program control purposes. (2) application for validation 9gainst the experimental data, and (3) applicatioi with the systematic changes in the design variables to provide guidance for the design and modification of aft-end forms of ships. In the following part of this section results of these applications are given. Application for Program Control
I-
Since the computers obey only the commands issued in the program as specified by the programmer various routines employed in the program suite GEMAK for the purposes of numerical ingration, differentiation, smoothing, interpolation etc., have to be checked against numerical instability, artificial viscosity arising from numerical differentiation, and the usual execution errors such as overflow, negative argument in logarithm etc. Grave consequences of neglecting this vital check in numerical fluid dynamics is best illustrated by Roache (Ref. 82) and Holt (Ref. 83). Application for Program Validation Tfgequite The aim of thi. stage was to establish the credibility of c. h individual program separately and the program suite GEMAK as a whole. Apart from the intermediate data handling program there were three main programs involved in the suite and at the irst instance each program has been tested separately. Since the comparison of the potential flow calculation program with experimental data and with the published results of other investigators have already been discussed before they will not be repeated here. These tests, however,
indicated that the potential flow compu"
The published data on a full form (C
tion schemes were quite satisfactory at
tanker model has been chosen for thig
the hull discretisation scheme devised
purpose.
within the project was efficient, Another important result of these numerical tests was that so far as the aft-end flow calculations are concerned, depending on the hull shape, between 25 and 40 percent of the forward part of the ship can be excluded from the potential flow computation without incurring any significant error in the calculated velocity field, The .. integral boundary layer calculation method program was already tested for a number of hull forms by Gadd (Ref. 8) and his results indicated that it was at least as good as any other available integral method. The BSRA testing on this program was essentially confined to assess
form (Ref. 2) was limtied to mean velocities and static pressures on a number of points on five stations along the ship's length and only the last station (x/Lw = 0.916, x/ = 0.942) was of interest ior the BSRF computation scheme. Hull geometry around this station and the measurement points used for comparison are presented in Fig. 17 and a comparison of measured and calculated streamwise and crosswise velocity profiles for the measurement points 184 and 185 are presented in Fig. 18. As can b3 seen from these figures the agreement for the streamwise velocity component is very good and for the crossflow components is fair.
-758-
-
~~~ -
-
__
_--
=
.85)
The data on the Hoffmann tanker
=
It should be mentioned that for the point 185 an integral method would provide the same level of agreement since it is a smooth flow location. As the flow becomes more complicated with severe velocity graden om near cate gradient the hultheve hull the advantgelthat advantages of using a differential method such as the one used by BSRA becomes apparent. Since the most severe measured flow condition for the Hoffmann tanker form occurs at the measurement point 183 this point was chosen to test the BSRA method against both the experimental lata and Gadd's integral method (Ref. 8) and the result is shown in Fig. 19. Here the strong flow retardation near the hull surface is quite well predicted by the BSRA method whereas the integral method, due to the limitation in its assumption, predicts still a fairly smooth flow condition. Although the amount of flow retardation around y/6 = 0.25 seems to be over-estimated by the BSRA method there is also a considerable amount of scatter in the data in the same location when different measuring arrangements were employed. The intention for the comparison of measured and calculated wake was not realised because of a flow peculiarity
first indicated by the program BRADSHAW. During the test runs the program indicated negative wall shear stresses in some of the calculation lines which in the usual turbulence understanding indicates separation and consequent flow reversal. However, shear stresses away from the wall were all positive. To ensure the continuation of the flow computation as far downstream as possible an artificial scheme called "bubble capturing" has been developed with which hull boundary is artificially displaced to the first point on the normal with posititve shear layer stress, resulting in some a form of boundary shrinking. With the application of this technique the following computational steps indicated that the wall shear stress became positive again further downstream but within the boundary layer there was a rrcion starting on thaeshull surac awayssary on the hull surface and mov.,. away from the hull in effetivly further ormng downstream pocet.Theunuual direction, effectively forming a pocket. The unusual side of this phenomenon was that the mean flow did not change its direction but attained values as small as 9 percent of the forward speed within a distance of 10 to 20 percent of the boundary layer thickness. Eventually, when the negative shear awayany ivaidate hypeolic loe e calculation ulton invalidate the the hyperbolic flow assumption, computations on that calculation line had to be stopped, A check on the experimental data from NMS indicated that our results were quite in agreement with the data and the start of negative shear predictions was accurate within 2 percent of the ship length. Evolution of this phenomenon (which we called "shear separation") is schematically illustrated in Fig. 20.
-
_
~the
This result is particularly important for it is probably one of the crucial
reasons in the formation of severe wake gradients. Obviously it is quite possible the full may scale Reynolds number such afor phenomenon disappear. If, however, computations indicate the presence of shear separation it is advisable to alter the hull form. Application for Design Guidance At the beginning of the project it was believed that such guidance would be obtained by a systematic calculation scheme with varying hull forms. However, as the understanding in BSRA on the turbulent flow development increased a great deal of useful information relevant to the aft-end design have been gathered either by the study of the existing data and the critical examination of the governing equations employed in the calculation method, or by making small changes in the flow parameters for the same hull form.
The main contribution of the developed analytical aft end flow calculation scheme is to identify trouble region andof to provide insight the on the consequences possible hull form alterations. Ia this respect, it may be useful to re-identify the differences between the differential method used by BSRA and the integral methods. By the use of the differential methods detailed information on almost all the important flow quantities are calculated by taking into account local variation of the hull form with its consequences on the flow including those arising due to the extra strain rates as well as the continuity of the fluid flow. Therefore, if a hull form appears to be likely to have severe velocity gradients with low eoiygainswt at o velocities eoiisa the top of the propeller disc either from the result of the model tests or from the hull flow and wake calculations the method explained here can be used both to trace the origin of the undersirable flow, i.e. where it starts and to indcat indicate the the-necessary changes in the aft-end geometry. To have a better appreciation on the effect of form
changes on the flow development around the aft-end of ships it will be useful to indicate the effects of external boundary conditions and extra strain rates, i.e conditions an e train ratesre. curvatures, on the turbulent flow structure. (1) Change in the velocity without change in the pressure gradient. If the the boundary l y velocities r are a e uniformly layer u i oatm the y increased i edge c e sofd or o decreased e r a e the change in the total flow structure is only marginal, since this will mainly change the velocities at the outer edge of
the boundary layer. As a rough indication the change in the streamwise velocity components due to a small change in the external velocity changesknloclM in the normal direction as e where k = -11.5.
ormaldiretion7s59.
r+ (2)
Change in the pressure gradient.
The method reported here can be used
A change in the streamwise pressure
both to identify the trouble area and the
gradient produces two main effects on the turbulent flow. They are: (1) a change on the wall shear stress and the velocity profiles in the wall region (y/S = 0.0-0.20) and (2) a change in the maximum shear stress, The effect of the first zhange is quite apparent since the all important very low velocity values and steep velocity gradients are formed in this region. The second
amount of pressure gradient and extra strain rates to be compensated. Figure 21 schematically illustrates the determination of the region of flow improvement which can be determined by the program suite GEMAK. The information on the incoming and outgoing characteristics and on the local pressure gradients and extra strain rates are contained in the output of the
change on the other hand determines the
BSRA method.
amount of energy loss and hence the change
L
in ship resistance, because a change in the streamwise pressure gradient determines
9.
the magnitude of the maximum shear stress value in the outer layer of the boundary layer (y/6 = 0.25-1.00) which in turn determines the diffusion of turbulence and the length scale of turbulent eddies, and
This paper summarJses the work conducted at BSRA on the development of an analytical calculation method for the determination of flow around the aft-ends of ships within the context of the PHIVE
hence the amount of energy lost in this
Project.
region.
method consists of a number of steps as
It is for this reason that some-
CONCLUDING REMARKS
As explained within text the
times optimal hull forms from the naked
shown in Fig. 1, and in every step the best
hull resistance point of view (such as V-forms) are not so desirable when a more uniform wake distribution is required. This effect however is generally coupled with the curvature effects, (3) Change in streamline curvatures. As mentioned in section 5 and Appendix 5, extra strain rates resulting mainly from the streamline curvatures play a significant role on the structure of the turbulent flow. Since as a rough approximation flow streamlines on the hull can be represented by equal curvature lines, the amount of extra strain rate can be directly related to the variation of the hull form geometry.
available techniques are used to produce a reliable and cost-effective computer program suite. Based on the experience gained so far the following tentative conclusion may be drawn: (1) To the knowledge of the authors this is the first hyorid method developed for the hull flow calculation; in this method the flow up to 80-85 percent of the hull is calculated with the Gadd's integral method (Ref. 8) and a version of Bradshaws turbulent energy method (Refs. 7 and 25) is employed to determine the flow around last 15 to 20 percent of the ship.
It is known that positive curvature (i.e.
The method takes the displacement
convex) produces positive extra strain rate and negative curvature (i.e. concave) produces negative extra strain rate and as a rough guide the extra strain rates
effect of boundary layer and the effects of normal pressure gradient and of extra strain rates arising from the streamline curvature into account. The "stitching"
changes the velocities near the hull
process to satisfy both the hull
surface as
where U and U* are, respectively, the
surface boundary condition and the matching surface (obtained from a preliminary iteration with Gadd's method (Ref. 8)) boundary condition and the "bubble capturing" technique to
streamwise velocity components without and with the streamline curvature effects, e is the effective extra strain rate and n is the power law exponent for the velocity profile near the hull surface, and it takes values between 5 and 7 for concave surfaces and between 2 and 5 for convex surfaces. In the outer part of the boundary
calculate the flow after the shear separation are some of the procedures developed by BSRA which has extended the current capability and understanding on the turbulent flow computation. Comparison of results with the experimental data for the Hoffmann tanker form (Ref. 2) showed very encouraging agreement even for the severest flow
U*
U/[
+
{n/(n+l)}e]
(2)
layer their main effect is seen on the eddy
conditions as demonstrated in Fig. 19.
length scale and positive curvatures tends to reduce the boundary layer thickness and hence reduce the loss of energy giving rise to steeper velocity gradients near the hull whereas negative curvatures increase the loss of energy and hence the resistance. Figure 34 is given as a rough guide to compare the velocity defect profiles for convex and concave hull surfaces.
(3)
A mechanism giving rise to severe flow retardation and high velocity gradiert has been identified as shear separation in which the shear stress reverses without a mean flow reversal. Although this was firsz discovered during the development of the program suite GEMAK, the wind tunnel data of NMI has confirmed this finding.
-760-
I
(4) As a result of the studies conducted
6. Himeno, Y., and Tanaka, I.
to provide guidelines for the aft-end design, information on the role of various parameters in the development of aft-end flow has been gathered and presented, together with the role the BSRA method can play in the identification of the trouble area. (5) From the results obtained it appears that a fully computerised scheme to modify the aft-end of a parent form to produce a more desirable wake may be designed by an extended use of the program suite GEMAK. (6) The methology employed in the development of the program BRADSHAW can further be utilised to devise a more rational scheme to account for the scale effects during the scaling of the model wake for cavitation testing. ACKNOWLEDGEMENT The authors express their thanks to the Council of the British Ship Research Association for permission to publish this paper, to The Ship and Marine Technoligy Requirements Board as the major financial sponsor of the work, and to British Shipbuilders, The General Council of British Shipping, Lloyd's Register of Shipping and Stone Manganese Marine who provided financial support as the Industry sponsors. Thanks are also due to Professor P. Bradshaw of Imperial College for his help and valuable suggestions, to Dr. G.E. Gadd and Dr. M.E. Davies of NMI for their stimulating discussions, and to Professor T. Cebeci of California State University, Professor V.C. Patel of Iowa University, Dr. B. van den Berg of the National Aerospace Laboratory NLR, Professor W.C. Reynolds of Stanford University and Dr. F.G. Blottner of Sandia Laboratories for supplying invaluable information and advice, REFERENCES 1. Odaba:i, A.Y. and Fz..simmons, P.A. Alternative Methods for Wake Quality Assessment. International Shipbuilding Progress, Vol. 25, no. 282, (1978), p.34. 2. Untersuchung der 3-dimensionalen turbulenten Grenzschicht an einem Schiffsdoppelmodell im Windkanal. institut fur Schiffbau der Universitat Hamburg, Bericht Nr. 343, (1976). 3. Bradshaw, P. (with an Appendix by V.C. Patel). The Strategy of Calculation Methods for Complex Turbulent Flows. Imperial College of Science and Technology, I.C. Aero Report 73-05, (1973). 4. Landweber, L. Characteristics of Ship Boundary Layers. 8th Symp. on Naval Hydrodynamics,Passadena, California, (1968). 5. Larsson, L. Boundary Layers of Ships, Parts I, II, III and IV. SSPA General Reports, Nos. 44, 45, 46 a-- 4 7, (1974).
-76i.
An
Integral Method for Predicting Behaviour of Three-Dimensional Turbulent Boundary Layers on Ships Hull Surfaces. Technology Reports of the Osaka University, Vol. 23, No. 1146, (1973). 7. Bradshaw, P. Calculation of ThreeDimensional Turbulent Boundary Layers. J.Fluid Mech., Vol. 46, p.417, (1971). 8. Gadd, G.E. A Simple Calculation Method for Assessing the Quality of the Viscous Flow near a Ship's Stern. Int. Symp. on Ship Viscous Resistance, SSPA, Goteborg, (1978). 9. Hess, J.L. and Smith, A.M.O. Calculation of Non-Lifting Potential Flow about Arbitrary Bodies. Douglas Aircraft Co., Report No: E.S. 40622, (1962). 10. Hess, J.L. and Smith, A.M.O. Calculation of Potential Flow About Arbitrary Bodies. Progress in Aeronautical Sciences, Vol,8,(1966). 11. Prard, R. The Representation of a Given Ship Form by Singularity Distributions when the Boundary Condition on Journal the Free Surface is Linearised. of Ship Research, Vol.16, March, (1972). 12. Gadd, G.E. A Method of Computing the Flow and Surface Wave Pattern Around Full Forms. Trans. RINA, Vol.11, p.207, (1976). 13. Gadd, G.E. Contribution to Workshop on Ship Wave Resistance Computations. Workshop on Ship Wave Resistance Computations, DTNSRDC, Bethesda, Maryland, (1979). 14. Wehausen, J.V. The Wave Resistance Advances in Applied Mechanics, for Ships. Vol.13, p.93, (1973). 15. Timman, R. Numerical Methods in Ship Hydrodynamics. First Int. Conf. on Nm. Ship Hydrodynamics, p.1 (1975). 16. Chapman, R.B. Survey of Numerical Solutions for Ship Free-Surface Problems by Surface-Singularity Techniques. Second Int. Conf. on Num. Ship Hydrodynamics. p.5 (1977). 17. Hess, J.L. Higher Order Numerical Solution of the Integral Equation for the Two-Dimensional Neumann Problem. Computer Methods in Applied Mechanics and Engineering, Vol.2, No.1, p.1, (1972). 18. Hess, J.L. The Use of HigherOrder Surface Singularity Distributions to Obtain Improved Potential Flow Solutions for Two-Dimensional Lifting Airfoils. Computer Methods in Applied Mechanics and Engineering, Vol.5, No.1, (1975). 19. Hess, J.L. Status of a iligher-Order Panel Method for Non-Lifting Three-Dimensional Potential Flow. Douglas Aircraft Co., Report No.: NACS-76118-30, August, T977). 20. Webster, W.C. The Flow About Arbitrary, Three-Dimensional Smooth Bodies. journal of Ship Research, Vol.19, No.4, p.205, (197b).
L
|
i
1 A 1
'I
21. $aylan,O. Theoretical WaveResistance of a Ship with Bulbous Bow. Istanbul Technical Uni., Faculty of Shipbuilding and Nav. Arch., (1977). (In Turkish). 22. Huang, T.T. and Santelli, N. and
Inc
Belt, G. Stern Boundary Layer Flow on Axisyetric Bodies. Washington 12th Symposium on Naval Hdonaics, D.C.,Mehd Naval. HrAIAA 239). ae39. 23. Huang, T.T. and von Kerczek, C.H. Shear Stress and Pressure Distribution on a Surface Ship Model: Theory and Experiment. 9th Symposium on Naval Hydrodynamics, Paris, (1972). 24. Adee, B.H. and Harvey, P.J. An Analysis of Ship Resistance. Dept. Mech. Uni. of Washington, Report UWE-BHArch, (1975). 25. Adee, B.H. and Fung, K.L. An Investigation of the Fluid Flow About Ships. U.S. Dept. Comm., Nat. Techn. Inf. Service, PB-261084, August, (1976). . D.E. - s The Law of the Wake in 26. Coles, turbulent Boundary Layer. Journal of Fluid Vol. 1, Part 2, p.191, (1956). Mechanics. 2--7.Coles, D.E., and Hirst, E.A. ~oe s Comuan E Hrt Turuent Proceedings: Computation of Turbulent Boundary Layers - 1968. (Editors), AFOSR-IFP. Stanford Conference, Vol. II, (1968). 28. Mager, A. Generalization of Boundary-Layer Momentum-Integral Equations to Three-Dimensional Flows Including K those of Rotating System.
37. Jacob, K. Computation of Separated Jrssb Fo Aou n oi ard
Determination of Maximum Lift. AVA Report, 67A62, (1967). 38. Rubbert, P.E., and Saaris, G.R. Review and Evaluation of a Three-Dimensional Lifting Potential Flow Analysis Methods for Arbitrary Configurations. oAriayCnfgatns Paper, 72-188 (1972). Landweber, L. On Irrotational Flows Equivalent to the Boundary Layer and Wake - The Fifth David W. Taylor Lectures. David W. Taylor NSRDC, Report DTNSRDC-78/IIl, (1978) 40. Myring, D.F. An Integral Prediction Method for Three-Dimensional Turbulent Boundary Layers in Imcompressible Flow. Royal Aircraft Establishment, Technical Report 70147, (1970). A T 41. Townsend A.A. The Structure of Press, Cambridge, 2n Ed., (1976). 42. Bradshaw, The Understanding and Prediction of P. Turbulent Flow. Aero739, p.40 ,-972). No. , J., Vol, nautical 4-3. Cebeci, T., Kaups, K., and Ramsey, J. Calculation of Three-Dimensional Boundary Layers on Ship Hulls. Proc. 1st Int. Conf. on Numerical Ship Hydrodynamics, Bathesda, Maryland, p.409, (1975). 44 . Ceb la, . 409, (1975). 44. Cebeci T., Chang, K.C., and Kaups, K. A General Method for Calculating Three-Dimensional Laminar Turbulent on Ship and Hulls. Proc. 12th Boundary Symp. on Layers Naval Hydrodynamics,
T A
NACA Report 1067, (1952). 29. Ludwieg, H., and Tillmann, W. Untersuchungen45. nespannung in turbulenten eibungsschichten. spIngenieur Archiv, Vo. 17, p.288 (i949), as NACA TM 1285, (1950). English 30. Translation Stper, G. Auftriebsverminderung
ryland, p.188, (1978). Bradshaw, P., and Unsworth, K. Computation of Complex Turbulent Flows, in Review-Viscous Flow. Lockheed Report L77EROO44, (1976). 46. Rastogi, A.K., and Rodi, W. Calculation of General Three-Dimensional eines Fl geis durch seinem Widerstand. eieshr FlugesdurhseinmiudrstanTurbulent Boundary Layers. AIAA Journal, Zeitschrift fUr Flugtechnik undVo.1,p5,(97) Motorluftschiffahrt, Vol. 33, 439, (1933). Vol. 16, p.5l, (1978). 47. Abdelmeguid, A.M., Markatos, N.C.G., The Approximate 3. rH. and Spalding, D.B. A Method of Predicting Calculation of the Lift of Symmetrical Three-Dimensional Turbulent Flows Around Aerofoils taking Account of the Boundary Ships' Hulls. Int. Symp. on Ship Viscous Problems. Control to Application with Layer, A.R.C. Resistance, SSPA, Goteborg, (1978). ARC. Reports Rep48. and Memoranda No. 1996, Bradshaw, P. The Turbulence 32. Preston, J.H. The Effect of the Structure of Equilibrium Boundary Layers. 32..H.The reson, ffet oftheJ.Fluid
Mech., Vol. 29, p.625,
(1967).
,B. Ivsgtno Van D B . Boundary Layer and Wake on the Flow Past B. Investigations of 49. Van Den Berg, Incidence. Aerofoil at Zero a Symmetrical Three-Dimensional Incompressible Turbulent A.R.C. Reports and Memoranda No. 2107, A- C. RBoundary Layers. National Aerospace (1943). MLaboratory NLR, The Netherlands, Report 33. Moore, F.K. NLR TR 76001 U, (1976). Journal of the Flow. Three-DimensionalNLTR701,(96) Boundary-Layer Beonary-Laylowee . urnal o te 50. Maroney, R.N., and Bradshaw, P. Aeronautical Sciences. Vol. 20, No. 8, Turbulent Boundary Layer Growth over a p.525; (1953). Longitudinally Curved Surface. AIAA 34. Moore, F.K. Displacement Effect Journal, Vol. 13, p.1448, (1977). of a Three-Dimensional Boundary-Layer. 51. Bradshaw, P. Effects of Streamline NACA Report 1124, (1953). 35. Report 4, (M.J. . OnDisplacemCurvature on Turbulent Flow. AGARDagraph 35. Lighthill, On Displacement(1973). Thickness. Journal of Fluid Me'snics, 19 17) T .i 52. Bradshaw, P. Higher-Order ViscousVol. 4, Part 4, p.383, (1958). Inviscid Matching. Imperial College of 36. Giesing, J.P., and Smith A.M.O. Science and Technology, I.C. Aero. Tech. Potential Flow about Two-Dimensional e 78-103, (1978). Hydrofils. Journal of Fluid Mechanics, Vol. 28, p.113, (1967).
-762-
___
1
53. Pratap, V.S., and Spalding, D.B. Numerical Computation of the Flow in Curved Ducts. Aero-Quart., Vol. 26, p.219, (1975). 54. Mahgoub, H.E.H., and Bradshaw, P. Calculation of Turbulent-Inviscid Flow Interactions with Large Normal Pressure Gradients. AIAA Journal, Vol. 17, p.1025, (1979). 55. Patel, V.C., and Lee, Y.T. Thick Axisymmetric Boundary Layers and Wakes: Experiment and Theory. Int. Symp. on Ship Viscous Resistance, SSPA Gote'org, (1978). 56. Prandtl, L. Effect of Stabilising Forces oii Turbulence. NACA Tech. Mem. No. 625, (1931), (German original appeared in 1929). 57. Wattendorf, F.L. A Study of the Effect of Curvature on Fully Developed Turbulent Flow. Proc. of Royal Soc., Series A, Vol. 148, p.565, (1935). 58. Nash, J.F., and Patel, V.C. Three Dimensional Turbulent Boundary Layers. SBC Technical Books, Atlanta, (1972). 59. Bradshaw, P. The Analogy Between Streamline Curvature and Buoyancy in Turbulent Shear Flow. J.Fluid Mech., Vol. 36, p.177 (1969). 60. So, R.M.C., and Mellor, G.L. An Experimental Investigation of Turbulent Boundary Layers Along Curved Surfaces. NASA-CR-1940, (1972). 61. Castro, I.P., and Bradshaw, P. The Turbulence Structure of a Highly Curved Mixing Layer. J. Fluid Mech., Vol. 73, p.265, (1976). 62. Cebeci, T. Wall Curvature and Transition Effects in Turbulent Boundary Layers. AIAA Journal, Vol. 9, p.1868, (1971). 63. Hunt, I.A., and Joubert, P.N. Effects of Small Streamline Curvature on Turbulent Duct Flow. J.Fluid Mech., Vol. 91, p.633, (1979). 64. Smits, A.J., Young, S.T.B., and Bradshaw, P. The Effect of Short Regions of High Surface Curvature on Turbulent Layers. J.Fluid Mech., Vol. 94, Boundary p.20 9 , (1979). 65. Smits, A.J., Eaton, J.A., and Bradshaw, P. The Response of a Turbulent Boundary Layer to Lateral Divergence. J.Fluid Mech., Vol. 94, p.243, (1979). 66. Mangler, K.W., and Murray, J.C. Systems of Coordinates Suitable for the Numerical Calculation of Three-Dimensional Flow Fields. Royal Aircraft Establishnent, RAE TR 73034 (1973). 67. Tuck, E.O., and Von Kerczek, C. Streamline and Pressure Distribution on Arbitrary Ship Hulls at Zero Froude Number. J. Ship Research, Vol. 12, p.231, (1968). 68. Wesseling, P. The Calculation of Incompressible Three-Dimensional Turbulent Boundary Layers. Part 1. Formulation of a System of E uations. National Aerospace Laboratory NLJ, The Netherlands, Report AT-69-01, (1919).
-763-
69. Lister, M. The Numerical Solut n of Hyperbolic Partial Differential Equation by the Method of Characteristics, in Mathematical Methods for Digital Computers. ed. by A Ralston and H.S. Wilf, Wiley, p.165, (1960). 70. Lindhout, J.P.F. An Algol Program for the Calculation of Incompressible Three-Dimensional Turbulent Boundary Layers. National Aerospace Laboratory NLR, The Netherlands, NLR TR 74159U, (1975). 71. Raetz, G.S. A Method of Calculating Three-Dimensional Laminar Boundary Layers of Steady Compressible Flow. Aeronautical Research Council, A.R.C. 23634, (1962). 72. Abdeleguid, A.M., Markatos, N.C., Muraoka, K., and Spalding, D.B. A Comparison between the Parabolic and Partially Parabolic Solution Procedures for ThreeDimensional Turbulent Flows Around Ships' Hulls. Appl. Math. Modelling, Vol. 3, p.249, (1979). 73. Bradshaw, P. Prediction of the Turbulent Near-Wake of a Symmetrical Aerofoil. AIAA Journal, Vol. 8, p.1507, (1970). 74. Morel, T., and Torda, T.P. Calculation of Free Turbulent Mixing by the Interaction Approach. AIAA Journal, Vol. 12, p.533, (1974). 75. Huffman, G.D., and Ng, B.S.H. Modelling of an Axisymmetric Turbulent Near Wake Using the Interaction Hypothesis. AIAA Journal, Vol. 16, p.193, (1978). 76. Andreapoulos, J. Syzmmetrical and Asymmetrical Turbulent Near Wakes. Imperial College of Science and Technology London, Ph.D.Thesis, (1978). 77. Dickmann, J. Schiffskorpersog, Wellenwiderstand eines Propellers and Wechselwirkung mit Schiffswellen. Ing. Arch., Vol. 9, p.452, (1938). 78. Dickmann, J. Wechselwirkung zwischen Propeller und Schiff unter besonderer Berncksichtung des Welleneinflusses. Jahrbuch der STG, Vol. 40, p.234, (1939). 79. Sasajima, H., and Tanaka, i. On the Estimation of Wake of Ships. Proc. l1th ITTC, Tokyo, (1966). 80. Hoekstra, M. Prediction of Full Scale Wake Characteristics based on Model Wake Survey. Symp. on High Powered Propulsion of Ships, Wageningen, (1974). 81. Huang, I.T., and Cox, B.D. Interaction of Afterbody Boundary Layer and Propeller. Symp. on Hydrodynamics of Ship and Offshore Propulsion Systems, Paper 4/2, Oslo, (1977). 82. Rozche, P.J. Computational Fluid Dynamics. Hermosa Publishers, Albuquerque, Revised Printing (1976). 83. Holt, M. Numerical Methods in Fluid Dynamics. Springer Verlag, New York (1970).
7
2
a
I
(
Di~reoty andDShipofSpeed
Potential Flow Calculation: Neana-Kelvin problem or Neumann problem
I
TN Integral Method for the Initial Boundary Layer Development Estimation
I =-:
]
Blowa-out Method toVeoiyadPsur
Data at the Edge of of dge the the Boundary Layer
Produce the Boundary Conditions at the Outer Edge of the Boundary Layer
Produce Metric Properties
of the Surface
Bradshum's Turbulent Energy method for Boundary Layer
I
and Wake Prediction
Fig. 1 Flow Chart of the BSRA Computer Program Suite for the Calculation of Boundary Layer Development on Ships
I~ 1 7,i -I J-i i
7.-._-
I
4d -
"'
:
Fig. 2 Comparison between Triangular and
Quadrilateral Elements
-
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is*&iCS
Om-~mXL
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-
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.1
.f.
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. .
.-
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n~
O
6
-"I.j .1NA. C, IS0
30,
1
0
0.1
-
0'~r.
*tl
.
~
IDa
.9
Y
0 0
.. I
I
?
~I~
-10O
I
S e
0.2~~~0.
*0
0.
,
0.3
..........
44
-
9
T
(2--i .16&
.1
0.
-
~j0!L.
!31w (L4wrf
01
04
1
0
0
0
0
3
0
Fig. 6 Effect of Forebody
Fig. 4 Comparison of Different Methods
-765 -
_
_
04
-
AL
0.3
a-
0.1
-----
fr7
Reut ~--.~ff
Eeet BSRa
LL
t
---
f4
Fi.8F
10 Tang.rsn
ofoite Streamlineerforane
Fig.
Rfesuts for
eSufae
414 Eluemet
Number___
a
-~-,.-----
I
I
--
.4. j
(ihFree Surface f)
-- -- - -
II "-1.7
1.-1
Iyzjt
if/i ,
,
;'//
i i
>11
M, the optimal controller given in Section 2 will ensure 100%
radiation potential amplitudes, P+ and P-, derived by considering unit forced oscillations of the device in otherwise calm Newman has shown that fluid.
efficiency provided M=2. This result has 3 been confirmed experimentally by Evans( ) who tested an immersed cylinder controlled in both translational modes of motion (heave and sway) to ensure perfect wave
Ip
N(j)
2 +
tp-I21
3.2
and
Non-Optimum Operation The optimal controller derived in
Fo()= pgA P+ (4) where A is the amplitude of the incident wave. Therefore normalising the absorbed
Section 2 necessarily incorporates a substantial reactive component whereas most experimental work has concentrated
power per unit width to the corresponding
on simple resistive loads.
incident wave power, Pw' where
equation (1),
Pw
=4"
2
ext fe(t)
pgA
+
Referring to
the case of = -
gives a harmonic solution whose amplitude can be determined from equation (1). The absorbed as power in this case can be calcul-
gives a maximum efficiency of +2 , ,2 2ated
ip+1j
Ip-2
1 Re
and to evaluate this it is necessary to determine the hydrodynamic added mass and damping matrices and the exciting forces by solving the fluid dynamic equations.
for high efficiency.
This is achieved in two dimensions by
Moreover, if N is greater than unity
approximating the device cross section as
equations (3) and (4) can be generalised
a polygon and placing a point source of
as
waves at the centre of each line element.
F
1
2
This has been derived by a number of authors (e.g. Evans( 5 )) and demonstrates that the ability to generate waves in a single preferred direction is advantageous
T 1 w2 I P (p) + [N(s) mn L mn L n for mn =1,2...N, and frmm +
--
absorption.
(3)
+
-and
+
If this analysis is extended any further to cases involving more than two modes of motion which are all controlled,
0
and the absorbed power is
--
-
l p2
This latter condition ensures that both modes are not entirely symmetric (P+ = P) ) m or asymmetric (Pm = - Pm
to give
:
in each separate mode of motion. Using expression (2) for the maximum absorption, gives an efficiency of 100% for N4M2, provided that
These are given strengths which are chosen to satisfy the prescribed boundary condio. the body surface. Calculations for the Salter Duck (6 ) operating in a single degree of freedom
T (p m(n]tions
pgA P
m 0 where now the added damping is a matrix the potential amplitudes constitute a vector corresponding to forced oscillations
-819-
(roll motions only) agreed very well with experiment and in Figures l(a) and (b) results are shown for the performance and translational forces that were monitored during these tests.
-
1.0
1.0
--
0.8Theory
0.8 L
S00..6 0.6
Experim0.6
---
0.4
wU
L 0.4 .0.2 __0.2
%5
0.5
Figure l(a).
1.0 1.5 Normalised frequency
The Performance of a Salter Duck - Theory and Experiment. I1: Sway force
0.6 .
.0.5 0
1.0
1.5
2.0
Normolised frequency
2.0
Figure 2.
The Performance of a Raft Pair Theory and Experiment.
-
The second device has been conceived through the desire to utilise air as a transfer medium between the wave and the mechanical
*
2:' Heave force +31 •simply Experiment
power extraction system. A device that is a shallow draught section with an enclosed water column in its front section has been shown to work efficiently and resulLs are given in Figure 3.
0
04
1.0
ID0.3
0o + 0 0.3 a
0.8 +
Z 0.1
0.6 -Theory
0
1.0
1.5 2.0 2.5 Angular frequency
3.0
=
E
e
0.4
Figure l(b).
Vertical and Horizontal forces measured on a Salter Duck. Further results in two dimensions were more
0.2
satisfying since the analytical work pre-
00.5
ceeded the experimental results. The two devices considered are very similar, the first being the Cockerell Raft which was originally conceived as a large number of hinged floats that articulate in response to the waves. The engineering problems associated with each hinge created difficulties and theory demonstrated that a single hinged system with a fixed rear float would be as effective as, say, a duck and this confirmed experimentalFigure 2). ly. was (see subsequently ysingle
_
_
_
_
1.5
2.0
Normalised frequency
Figure 3.
The Performance of an Oscillating Water Column - Theory and Experiment. It is important to note that all of te is are deine to t operate at in aa in are designed these devices mode of motion and therefore are effective unidirectional wave generators
-820-
_
1.0
-T
when driven in reverse. They have also been designed to ensure tnat the frequency of peak efficiency coincides with the prevailing wave frequency. Practical Problems
3.3
A major problem with these devices has been to restrict the translational (heave and surge) motions in a floating system since calculations demonstrate that if no restraint is applied to these modes then in plane waves the performance is significantly reduced. Salter, however, has suggested that such devices should be mounted on a long common element, called a spine, which will be stabilised by virtue of the phase cancellation of the wave forces along its length due to the directional spread of ocean waves. Moreover, it is suggested that if by either a wise choice of material, or controlled joints along this spine and the appropriate tranzlational spring stiffnesses achieved it is possible to improve upon the performance of the device operating in a single degree of freedom. It would appear that all floating terminators operating in a single degree of freedom therefore require a spine of some form in order that a compact structure can be efficient. Evans, on the other hand, whilst still wave genCombining generathat is a e good using has a device usig pdevice ta si a sgoodwave tor, has produced a simpler structure which needs control in two modes of motion. Therefore, there appears to be an interchange between civil and mechanical engineering complexity and it is not clear which route is preferable. In both cases the ability to simulate reactive power take-off has the effect of reducing the size of the structure although the mechanical achievement of such control may well be extremely
difficult. 4.
r 1
2
a 0 0 ' 0 and the maximum capture width is
XP
( 2
cos2o) 1 the orthogonal horizontal mode to
the sway would make this expression independent of angle and give a value 3X/2v. It is interesting to note that in three dimensions the added damping matrix is not singular for N>2 as could have been deduced from the representation as an integral of rank one dyadic forms which can have arbitrarily high rank. 4.1.2 Arrays of.Devices Perhaps the most interesting use of the three dimensional analysis is to
THREE DIMENSIONAL THEORY
4.1.1
Now the exciting force, F (0) is directly proportional to the radiated wave in a direction 0 created by a forced oscillation and this expression clearly demonstrates that high energy capture is obtained if the device radiates in a narrow band of angles but at the expense of being extremely directional. Normalising the above expression to Pw' gives a capture width for the device and in the case of a radially symmetric device vertically oscillating, F (0) is independent of angle this reduces t8 x/2, . If horizontal (sway) oscillations e ies apture width is 0vdoubled to c ct If, further, a combination of vertical and horizontal oscillations are considered (N'= 2) then the cross coupling terms vanish in the added damping matrix since the integral of cosO is zero. Letting Fol(o) = a and F 2 (o) = cose then N(w) is given by the matrix
Single Absorbers
examine arrays of devices which has(7 ) (1980). previously been considered by Evans He has analys .d configurations of axially symmetric devices that are each constrained to vertical (heave) oscillations and assumed that the devices were small in
In three dimensions there is again a simple relation between the added damping coefficients and the wave exciting forces such that 21
relation to both the incident wavelength ["'('Jmn
=w
1 8%Pw
m Fo ()
and their spacing so that the form of the exciting force, F (0), was not influenced by the presence o? the other structures. He therefore assumed that
o
(e) de _2 where the F m(6) is the force on a fixed body due to a wave incident from an angle of a and
1
o
L
F m(0) = Fe iklm cos(o- am )
22
where 1 is the distance of t he mth device from t waethe origin ad am is the angle the line 1 makes with the positive x axis. m The latter term merely represents a phase change due to the position of device m. In this case the added damping matrix reduces to a particularly simple form
w 4w which is the incident power per metre of wave frontage. In the particular case of a single body the maximum energy for operation in a single mode of motion can therefore be derived as 2itFo()iid2 .PwF°(0l{2 w 0 0
Jo(KRmn)FF/4XPw where R is the distance between the mth and nthmaevices and J is the zero order 0
-821-
-
Bessel function. Therefore width may be expressed as, _T
where J
Normalised d1aeter
0
the inverse of the matrix
) and L is eits Wpe j(KI
L
1.2
-1 is
mM
the capture
1.Nu1rcal
a vector whose compon-
iklm C0(O-
E
Now for N devices, their summed capture width working in isolation is N.1/2r and a q- factor can be defined as the change 7
0.95
.
-
< 0.9
in performance due to the presenc'e of other devices and is given by
q(
N)= L L
0.8 o
It may appear remarkable at first sight that any inceraction factor exists since it has been assumed that the exciting force, and hence the radiated waves are not changed by the presence of the other devices but apparently the energy capture is influenced: The explanation for the q- : factor lies in the fact that much of the structure of the hydrodynamics has been incorporated, particularly in formulating the added damping coefficients which estimate the forces induced by moving a device, both on itself and the other bodies. In this analysis it is assumed that the forces induced on body n, by motion of body mare due to the radiated wave of device m and the diffraction of this wave by all other devices is neglected. The q- factor therefore arises since there is an induced force coupling the devices together and depending on the magnitude and phase of these forces energy capture can be advantageously or adversely affected.
Figure 4(a).
spacing d. normalised by the wavenumber K = 2v/1. Evans assumes a constant value of unity whereas it is apparent that the wave diffracted from the other device causes this value to oscillate about unity. These calculations are carried out for device of normalised radius 0.47 and changes of up to l0% in the added damping coefficient can be observed for For spacings less than 5 diameters. larger spacings the wave scattering becomes less significant
4.0 6.0 8.0 Norodipdcing,
10.0
The Diagonal element of the A Added Damping Matrix for a
When considering the coupling coefficients where the force on body n is evaluated due to a forced movement of the mth device with all others held stationary then the diffracted wave field of the nth body will be of the same magnitude as the radiated field at this Therefore the simple form of position. Jo(Kd) will not represent the cross coupling terms and a comparison with the numerical calculations is shown in Figure 4(b) for a device pair.
4.2
Numerical Calculations Evan's assumptions can be tested using numerical calculations to determine the added damping coefficients. The technique used to solve the hydrodynamic equations is to partition the device into plane facets and position a point source of waves at each facet centroid. Results for the added damping coefficients are given for a pair of devices which are In cylinders with hemispherical bases. Figure 4(a), the ratio of the diagonal damping coefficient to that of a single body is plotted for various values of the
2 2.0
0
1.0_ X
__J_(Kd)
.(Kd) Norn.lised diameter
"
0.5 . 0
NumuAcl
%
0.95
sjlts
0
4.0
2.0
-0
%
8.0 /
r
-0
TjJ>0 for the former and 0 > P > TV for the latter, where U is the free stream velocity relative to
Basic fluid dynamic principle is applied to develop a mechanical device involving lifting surfaces in unsteady motion by which energy can -e extracted from a prevailin stream such as wind, current in the ocean, and tidal and river currents, The fundamental care considered here is based on pitch-and-heave motion of a single airfoil-shaped blade mounted on a vertical axis Ad moving n a uniform stream. for which motion the hydrodynamic force and moment acting on the blade and the energy relationship are evaluated on unsteady wing theory. The operation is optimized to give the highett possible efficiency under the isoperimetric condition of fixed specific loading (in proportion to the reactive drag the blade must withstand): this optimum motion corresponds to the least possible loss of flow energy while a required arnw-nt of stream power is being extracted. Ti e present simple case is explored with prospects for further generalization to systems of multiple-blade configurations.
...
.
-
-
_
[
the wing.
y
The optimum motion is found to depend
0, so that, alternatively,
on two dimensionless parameters: 'he reduce' frequency a( wi /U, where w is tiw angular frei quency of wing oscillation, I the sem -chord length) and the specific-loading factor, CT o (a coefficient form of T rated at unit amplitude of heaving). Detailed investigation of the characteristics of the optimum motion and pertinent physical quantities in terms of these two parameters are carried out, with the aim 1. to examine feasibility of the high fluid mechanical efficiency predicted by the optimum solution, and 2. to explore the range of a high rated power output (based on available incident flux of wind cnergy) to see if the limit imposed by Betz (1926 [1]) can be approached. These two factors are essential to a successful development of thi;, type of stream-power generator (especially for inlustrial applications) and require further study. The present rather simplified case is chosen to elucidate as much as possible the basic princirles involved and to provide useful results as a guide to more complicated situations, with prospects for further generalization to systems of multiple-blade configurations. Many important physical effects, such as those of flow shear, unsteadiness and turbulence that can arise in practice, are not considered here. However, from the previous studies of Wu (1972 [8]), Wu and Chwang (1975 [9]) on the effects of stream waviness for the self-propulsion mode, it may be expected that extra energy associated with a wavy stream can also be extracted (in Lddition to the mean-flow energy) with suitable modification of the present solution.
h)(x-b)]e
where ho o
[(i o' b4)
(.+la , h)
(Z)
+ (bg)
o lb/ b
tnh -=
_ t)I/Zan
2
+
(3)
goo/(t+ bg 1 )]
IP Clearly, the heaving phase angle cth in (2) is by itself irrelevant since it can be eliminated by a shift of the time origin. If b is further taken as a free parameter, then the three original parameters, ) I = 42, and the three new ones, (ge. (ho , u, ep -t h) m , are related by (3), or symwhich has a unique inverse, bolically r 1 (;b), = (i;b) say, provided that 4 0 ' 0, in which c'as tle two representations are equivalent to each other. As a point of further clarificaton, we remark that if the solution to the irrotational flow problem is obtained iii terms of C for the wing motion prescribed by (1), it can r7aadily be converted to an expression in 1 with a free choice of b, and vice versa. The ch^(ce of b is not the only free one that keeps (1) and (2) equivalent. In fact, p - h as arbitrary one may take (-i < o ICpj, implying that while energy *_sbeing extracted from the main stream, the wing - ast withstand a mean drag, D - T > 0, which is reactive instead of resistive in nature, and, by (8) and the property E 0, cannot be less than LPLIU for given power extraction, -P > 0. When P , 0, a fluid mechanical efficiency of extracting flow energy can be defined as
!3(()(4,O
C~
),
icide of available avaabe ergyt in a prevailing prevalin d incidentfflux of energy
stream, we will also introduce a 'rated powerto that first introduced by output-ratio', Betz (1926 [1 ]),similar as 1
_ dU 34o= 2
-r/
P/pay v=
IB
)C
,
(16)
(power
in which Pav is the total power available in the incoming stream, equal to the incident flux of kinetic energy across an area swept by the wing of unit segmental length in heaving. For the present pitchand-heave type of stream-power generator, this ratio is related to the mechanical efficiency
(12)
b
)C
11B= "*
is regarded as a vector,
T
(CT
T
CI, o> -1 say). Within this frequency range, the optimum nrotion is given by (20), (24) and (25), and it has two branches, In the first of the two, X = %It which is always negative and numerically smaller than , and corresponds to the highest efficiency attainable under condition (19), Cp
CE)i T
a(P
22
X
+ 2) 2
T
(0 < a < a,)
(27)
mode. First, the singular nature of the problem as
The other branch, with
noted earlier (ascribable to the property that the matrix Q in (12) ha, - zero eigenvalue, see Wu (1971 [7]) is removet .y the transformation 2 2 (4- a= ; ,1 2a 1 0 (20) which reduces the order of the quadratic form CE by one, giving oto by one, giving 2 2 CE ) (1 CE B(a) Q22(;, + C
2 a tP 2 2 (41 +
A1
PI
A2
P71 Q3 3 - Q13
2
02 2
2
+ 24o(A 1 4l+
Q 1 2 P2 2 33
A2 ; 2 1
,
2t +2+this
,
(23)
X2, gives the lowest
externally supplied power (Cp > 0) while still withstanding a reactive drag. It remains to be seen if particular mode of motion might be utilized in nature, such as by a bird flapping its wing forward to retard itself quickly for landing on a tree branch or by a fish rowing its pectoral fins broadwise to stream to stall. In the sequel only the maximum efficiency branch of the optimum motion will be discussed.
-832.
X
efficiency (with X1 in (27) replaced by %2 ). The numerical results of these two extreme il's are shown in Fig. 1 for several negative values In contrast to the propulsion mode of CT, o ( T 0> 0, in which case the optimum motion has a low-frequency cut-off), the optimum motion of the power-extraction mode (Cp < 0) exists for all frequencies up to a high-frequency cut-off (all too high appear in the figure), though the maximum efficiency decreases appreciably with increasing magnitude of CT, . The lowest efficiencies are seen . signifies . to be all negative, which the need of an
to
l
10
Fi.
axmm
I0
0
O
~x b
ndmniu
1*h
efiince
o
Fig.1 an minmumeffiienies Mximu f aficient flow-driven oscillating wing (strewr -power
C5
T,
OCT 1
to drag coefiicient T, the reactive 0 CT c ) in the former (CT < 0). The optimum feathering is nearly perfect (1-< 0)-I) I- 0), but for vanishingly light loading (IC, To
nificant because the b*-axis may be regarded as a good approximation of the effective center of pressure evaluated on the minimum root-mean- square basis. On this argument we may conclude that it
30
,
,'..
., ,"C4T,
Fig. 5
Variation of the feathering parameter 6 with thereduced frequency a and loading factor C 0ual T, o
006o h admvsollength) withlgtybcwrs( increasing loading - it is sig-
-834-
_
_
-
I=
_
-
F7
r 9
_
.-
__
original case of b
0
(see Fig. 4).
--
C 0.
"
a.4
Fig. 6
The optimum location of the axis of pitch, x = b, y 0, about which the moment of force of the optimum motion has its meansquare value minimum,
The phase lag of pitching (about the is) behind the phase of heavir.g.
Fig. 8
b 0 -a
With the axis of pitch changed from b -- 0 to b b* or b = -1/2, the accompanying changes of the moment are shown ir. Fig. 9 for the opti-
is advantageous in practice to choose the axis of istchvatntaeous inp chot oe the i owepitch at the 1/4-chord point for the optimum powerextraction motion (or at some point between b -1/2 and b for moderate tv high loadings, but the difference in nerformance due to such more refined drecin p is small), r d relocation of b is small). The kinematics of the optimum moticn expressed here in the new form with b = bv" is given by (2), which is now characterized by two new parameters, the pitch-to-heave amplitude ratio, Z ct[h* = Zn(go/2h°), and the phase difference becag , in which tween pitch and heave, at*= 1Yboth h o and CIA are given by (3), now ir.corporated with the optimum values of go, g 1, g and with b = b*. As can be seen from Figures 7 and 8. the
mum motion at three values of the loading facto:CT, 0 and at their corresponding frequencies cf am. The force and the moments are represented by their specific-coefficients defined as -
= .
UI
(31a) a
'
(
o 0
)
'
M
1/4
(M, M + b IL,
M-olL)/(-p 4
-jb
0
(31 b
-2.
These coefficients are referred to a unit amplitude of heaving of the mid-chord point as a common basis. Sev.rai interesting features are noted frow this quantitative comparison. First, for b = b", & attains the smallest possible ampitudc of all the moment coefficients (with respect to changes in b) and a nearly constant phase difference of 90 from that of ZT. whereas the phase
.Ci-2
difference between CL ank"
is appreciably
smaller (from 60 down to 45°). o Further, the difference between GM and (JM 1/4 is seen to be very small at light loadings, and they become
more differentiated only at higher valucs of
_. ..
Fig. 7
.
T.- o,
This small difference makes the 1/4-chord point a desirable choice as the design axis of pitch since it satisfies the dynamic stability criterion at al levels of drag loading. To exhibit the overall behavior of the lift and moment that result from the optimum motion at arbitraryvauso frequencies, i'nd G7*M the areroot-meanshown versus square ar in
j
The pitch-to-heave amplitude ratio, Z, b0 -axis. when pitching is about the b*-axis,
values of
ad
difference between Zi and Z is very small along the a =um line. At the same time, the phase difference between heave and pitch about the o 0 new b--a-.is is found to vary from 90 to 120 along the a = um line, as shown in Fig. 8, a variation which is somewhat more appreciable than in the
-835
.
i
reshow
verss a in decreases whereas *M increases as the reduced frequency a is ncreise'd across a , which infers that the component of power delivere by the lift decreases whereas that by the moment increases
Figures 10 and 11.
This result indicates that.
as a is increased in the neighborhood of am. This naturally leads into the second question concerning whether the moment iz passive or
-
;7
. C 10
"3,
.0.
~
0'
p
o~~
CFig. 10
--
TIR
motio
opiu
0
t0he0:5
To
oractiv.
0xis.
(aoth
M 1
and
i.T
whr
inb
2)
C of rime-mean-sqalue
he
1o
rtopti
M h eops actile Tofbyh
h
dfn
ofiin
of theenpwr
Fig. 9P Thea spcielmoet-oefieet L w nwr or~~~~~~~~~~~~~~~t
I.~
h
.
0:
(33
,
1-
edcops
fidteaswr
oin
ofiin
N4
YITC)
a-
at
of the root-mean- sq-.are of the edd the-frequeneycy b-s vesu th ce aito
'
LP
itheedb
ro
t'D
rbietdgt thetmea prodwer efiintd!dpro
pcfc
-836.
__
In Fig. 12, the numerical results of CPL L3PM and Vp are shown for b = be and o = a M corresponding to three values of C'T oL We note that the power components (L L an ePM are almost in phase, while the ampitude ratio of 1 ApT I /I p increases with increasin loading (eoual to 0.05, 0. 11, 0. 24 at CT o 10-, 10- 2 . I0-, to respectively). More significantly, with b 'pM is virtually all positive, implying that the
--
___
____
[-y ..._____
-
.
corresponding moment is never .assive (not flow driven) because external power is required to provide the given rate of working against the fluid dynamic moment of force. We can therefore conclude that the pitching of the optimum motion is stable -hen the axis of pitch is taken at b = b. This sitwation is scarcely changed when pitching is shifted to the 1/4-chord point but is somewhat weakened when pitching is kept at the mid-chord point (results not shown). In practical application, the power required for maintaining pitching. p., a sp plied by diverting a par! of the powerabsorbcd by oscillating lift, 0 PL. to offset LPM, and tis at no expense to the total power output as previ:.determined.
.the
%is
5.
/
/Our
. -
ultimate interest is of course to ascertain a theoretical upper limit to the amount of power that can be extracted from the available stream energy. To ir.vestigate this important limitation we first rewrite the relation (17) for the optimum motion with a = Cm(CT, o), and notin; C -r C , 50 so that
/
3/
t, .. ,i
/
'c' Oti.
!
-XiZ "
"
l"il
-rince
.
/
~.
B
•and 4.staoptimnumemotion
-
-
/.
4Under
A C,.
/
fl52c
I.
8) 5
0
McT
0
< 1
(36)
this condition, it appears possible for to approach the hydromechanical efficienc TI. However, the validity of the inequality (36) seems to question in view of its over-simplified assumptions used. an example to give a plausible estimnate of the upper limit to 'qB'we provide the iollowing
5 /open
P '\c~e.otao /As
as.l
__________________________________table IA
Fig- 12
T, o
t r similar to that used by Betz (1926 [ 1). otnsimplifying assumption that the surface, of aea Zx 1. swept by the heaving wing of unit m.ae readda nactuated disc across wihauniform pressure jump can be sustained, thnscan redy be shown, the mcar. reactive dra, - .cannot exceed 0lI2) pU 2 g . Putting this uprlimit in the coefficient form % hae
....... ......
.-
CT,o o
~~no: too large, and sinceC, i ydei independent of for the otmm motion, it thiat the rated-oower -output ratio, 113, is proportional to IR This seems to taking greater could be increased imply that greater values of g I . However, the present lnear theory is not suffocient to determine if there exists an upper limit to , beyond which the becomes infeasible.
plinearly
I
"
CT,... riaT"".o 'n
CT,0
I:tion
~
-
..-o
-
q(G1 .. CT, is seen (from Fig. 1) to be a slowly varying furcino .~~ true at least for
rfl..4...
,follows
/
=
(C " T~o
/~
-
£
.
THE RATED POWER OUTPUT
*.
~
er L:PL +' LpW contribution to the instantaneous specific-power-coefficientT frorn the lift and moment (about the b ns yce oeraf heopimm-0. motion with the scale referred to ICI
with entries derived from Fig.
tion (35) together with the values of in line with (36). C T, 0
05 -0. 1 -837-
I
o T' I
1. 2 and equa-
Iiassumed
n
0
1
'B
0.37
0. 53
0.84
7-.6
0.5
0. 44
0.68
0.-77
5
0.6.
-
In this connection we may remark that in the experimental study by Lang and Daybell (1963 [51) on the tail fluke movement of a cruising porpoise, values and they observed, werefor 8.3 [71) g /i of were usedasbyhigh Wu as (1971 evaluating Lsquarethe proPulsive efficiency. In the present discussion on power extraction. the effects of viscosity and nonlinearity, and those other factors that may play a significant role in causing leading-edge separation under high loadings, and possibly imposing an ultimate limitation to qB, have not been pursued. The importance of
L
I wish to acknowledge with deep appreciation the benefit I have derived from the sessions of valuable discussions with Dr. George T. Yates and his expert assistance in making available the numerical results for this study. This work was partially sponsored by the Office of Naval Research, under Co- tract N0001476-C-0157, NR062-230, and by the National Science Foundation, under Grant NS' CME-77-21236 A0l. It is gratifying that the basic studies carried out earlier under these supports have inspired potentialy useful applications. REFERENCES 1926 Windenergie und ihre Betz, A. 1. Ausnutzung durch Windmiihlen. (Vandenhoeck t~ld Ruprecht, Gttingen. ) 1979 Fluid Dynamic deVrie, 0. 2. Energy Conversion. AGARDograph Wind AGARD. Aspects 243, ofPubl. No. 1935
Airplane Propellers.
=
MM I/
M
-
PL- P-M
-
Pa
-
-
-
S T t U (x.y) 20/ Z eh
Propulsion of High Hydromechanical rfficiency. J. Fluid Mech. 44., 265-301. 7Wu. T.Y. 1971 Hydromechanis of Optimum Some 2. Part Propulsion. Swimming 46. 521-544. Mech. J. Fluid Shape Problems.
p
o
a h,
0.'
l
9 7g
Res. 14, No. 1, 66-78. 97 Wu, T. Y. & Chwang, A. T. 1975 Extraction of Flow Energy by Fish and Birds in a Wavy Stream. In *Swimming and Flying in Nature" Vol. IL 687-702, Eds. T. Y. Wu, C. 3. Brokaw & C. Brennen. Plenum Press, New York/London.
in (13) . leading-edge suction force thrust force, T its mean value the time variable free stream velocity a rectangular coordinate system pitch-to-heave magnitude ratio, ihb=b vleo about mid-chord point
fluid mechanical efficiency -
-
-
.-&As.
power L3. available in incident flux of flow energy 3x3 matrices with elements
par.ameters defined by (20)
=
P/TU _ 'B -P]Pav 4= UC-/ h o t(a) F+iG 'Al k A p a= WI/U am (CT,
least root-mean-square of M b Power required for maintaining wing motion; P, its mean required associated with power LM
p - angular amplitude of pitching -phase angle of heaving, pitching parameters of wing motion
. ' u, "T.;.w
J. Ship
F - (F 2+G 2 ) mean-value coefficients of energy loss, power, and thrust, respectinely
_
.two
1977 Optimum Wind4. Htftter, U. Energy Conversion Systems. Ann. Rev. Fluid Mech. 9. 399-4195.- Lang, T.G. & Daybell, D. A. 1963 Porpoise Performance Tests in a Sea-water Tank. NAY WEPS Rep. 8060, NOTS TP 3063. Naval Ciornia SstationChina Lake, 0pitching Aquatic Animal 1970 Lighthill, M. J. 6.
2
functions of a defined by (Z3)
specific lift and moment coefficients, defined in (31) instantaneous specific power L"pm eP' 4 -coefficients, defined in (33) C mean specific-power coefficient CS1 2 ), mean leading-edgeti o CT. o/2, specific-loading factor CT, o energy imparted to fluid; E, its E mean real, imaginary part of u(c( F(a), G(a) wing displacement, in y-directS on hix, t) h heaving amplitude of the b-axis of pitch when b =- b( value of h h length of wing (I = 1 ) -half-chord L -lift M moment of force about mid-chord point momert about x = b, y = 0 bL -M
In Aerodynamic Theory IV, ed. W.F. Durand, 170360. Berlin:Springer.
Energy by a Wing Oscillating in Waves-
-
M
o
ACKNOWLEDGMENT
Glauert, H.
1
A, A 2 B(a) CE.CP.CT
these effects cannot be assessed in full without further study.
-
NOMENCLATURE a o ,b n - complex coefficients b eb, yan - variable pitching axis, x b bq ae ofof theenomentg menmeau the moment tM b minilum A = A' + AA2 A --
ratcd power-output ratio proportional feathering parameter the Theodorsen function Lagrange multipliers real coefficient, defined in (26) fluid density reduced frequency value of a for minimum leadingedge suction at given CT, o circular frequency of wing oscillation
Discussion W. van Gent (NSMB
The author is studying the power extraction from flow by one lifting surface, This is meant to be a starter for multiple blade systems. Does not he think that the wellknown cycloidal propeller with many blades (vertical-axis propeller) is a much closer approach to this problem? I think in the present environment of ONR the knowledge of this type of propulsion is also readily available in the turbine-mode. is the author willing to comment on this ?
J.N. Newman (MIT) Professor Wu has treated us to a brilliant exposition, describing the absorption of energy from a steady moving fluid by an oscillatory lifting surface. The converse problem arises in the field of weve energy, where a Wells turbine is used to extract energy from an oscillatory air stream, Idealizing the latter problem, we may envisage the steady horizontal translation of an uncambered lifting surface, through a fluid with vertical oscillatory velocity. The resultant oscillatory inflow angle generates a mean positive thrust on the lifting surfacei with resulting power output. l'iofessor Wu's comments on this converse problem would be most enlightening,
Author's Reply T.Y.Wu( Ci I very much appreciate Dr. van Gent's comments regarding possible variations of the pitch-and-heave mode that I have just discussed with an intent to have it serve as a basic case study. As briefly mentioned in the paper, our research is continuing to explore other alternative modes of operation, including rcking blades, multiple oscillating blades and verticalaxis rotating-proveller-like stream power generators. The iast mode is being investigated by my student Jean-Luc Cornet with primary objectives to analyze the optimum motion for stream power extraction and to evaluate the stability and control of the
motion. With special reference to these aspects and the previous work by Edwin C. James (Ref.l) , new operational parameters arise in our investigation of the power extraction mode a:cd appear to require further study. In the present state of the art, I feel that our knowledge is stili insufficient for us to draw an overall judgement on the superiority of any one operational mode with respect to such major factors as the hydromechanical efficiency, cost effectiveness, and safeguard (which we wish to emphasize for operations in water streams) of both marine lives and generator blades. In regard to Professor Sparenberg's theory of actuator-disc optimum propulsion, I think that practical applications of his basic theory to power generating operations certainly seem to be feasible, but would require further effort in design and engineering development. Professor Newman's question brings up a valuable extension of the present scope to consider the performance of . Wells turbine operating in an oscillatory unidirectional stream.. My first reaction to this problem is that the hydromechanical efficiency of the "Wells-type turbine" can be cunsiderably improved by giving the turbine blades an additional degree of freedom in pitching (which it now lacks) and by having the pitch motion optimized with respect to the oscillatory stream. I am greatly indebted to Professor Neuman for bringing this interesting problem to our attention and wish to offer my optimism, to those interested, that one cnobtain the solution to the optimum motion for this case by following the approach of this preliminary study and an earlier paper by myself (Ref.[8], cited in text) which deals with the optimum extraction of flow energy from a wavy stream for the purpose of propulsion. Finally, I think it is important that our Session Chairman, Professor Ogilvie, raises the question oni the value of studying the optimum motion, for this immediately brings together the hydromechanical problem with the economic aspects of a new industrial development whose success must depend on its prospect in winning competition over existing sourzes of commercial power. Further, it is also essential that we should tap energy from natural resources with the least perturbation of our natural environment, lest we may face, in time, some undesirable, unforeseen, but inevitable consequences to inefficient handling of natnral resources to such an extent that results in unbeneficial changes in the climate, land conservation and the courses of ocean streams.
M
REFERENCE I) James, E.C. "1970 A Small Perturbation Theory for Cycloidal Propellers", Ph.D.Thesis, California Institute of Technology, Pasadena,California -839-
UE
FI Characteristics of New Wave-E nergy Conversion Devices Masatoshi Bessho
I
Natioral Defense Acaderrns. Yosuka
Osamu Yamamoto. Norihisa Kodan and Mikia Uemnatsu ReslearCfl Laboralorc-s of Nipoon Kokan K.K. Tsu TSu. japan
IiABSTRACT unit of imaginary
-.
wave number = 21C/N. z ass of the body norma vector ieto oc nit rato ~ * absorbed power in regular wave V** power of incident regular wave PZ absorbid power in irregular wave ?' o oer of incident irregular wave pressure on the surface of the body p incident wave energy spectrum S (%o) restoring coefficient ef energy absorbing system t ith entofhebd
A theory is given herein for thie absorption of wave energy by means of a sm metrical body with wave energy conversion systems oscillating in asinusoidal wave. conditions for achieving a complete wave energy absorption were derived from the e.:pressions for complete wave absorption. moreover, as a practical application of the theory, two new wave energy absorbing devices were proposed and theoretical calculations and model tests were carried out on the wave and wave energy absorption of regular and irregular waves. Results ofti~ the experiments agreed well with the theo~~retical calculations. Moreover the results of the model tests verified a comnplete wave energy absorption.
14
NOMECATURE
S
I
AR
A-r
B Ca
W . Pl
of incident wave di-lPlacement of i-th mode oscillaif iXCelvto
trnsmisio coeficint coefficient of Eq. (12) group velocity =
li C*= C2 , C3
=
Fq ,f i
=
g
= aravitaticonal acceleration = imtacentric height above C.G. = Kochin function of i-th modes
It~
restoring coeffi-
cient D, D~f 03 = determinant of Eq. (12) Ei=wave exciting force in i-th direction
hydrodynamic force in i-th direction caused by j-th mode' oscillation, Eq.
!(K)
distance between two wave height mneters (Fig. 4) Mass or mass moment of inertia of amltd
reflection coefficient
=ydrostatic
-electric-power absorbed by generator apideoi-hmeoslatn displaced volume
energy conversion system
= amplitude cfreflected wave = amplitude of traxnsmitted wave = breadth of floating h'xly =
K
fficie-ncy of wave power absorption -CV efficiency of wave power conversion z:damfping coefficient of energy conversion system radlius of gyration in rolling osciIlation wave leng th Mass density of fluid circular frequency
(10)
osilatoncor 1--ass moen ofierihf body
- XNFIUTOA Wently many studies have been made Cernizg the ut4 lization of wave eneray. wavC-ictivated Generator is one p-ractical application of wave energy. Masuda li]
he
-841-
invented this device in 1965 and Ryokusei-
results of theoretical and experimental
sha put it into practical use. McCormick [21, (3l developed theoretical analysis of the pneumatic wave-energy conversion bouy system and the results of his analysis agreed with the experiment tesults of Masuda. Contrary to these studies, Isaacs [4! invented a wave powered pump which converts wave energy into water pressure that accumulates in an accumulator tank and which activates a hydraulic turbogenerator when the water pressure is released from the tank. Independent of these practical applications of wave energy, Milgram 151 investigated the problem of absorbing twGdimensional water waves in a channel by means of a moving terminator at the end of the channel. Though the object of his study was to devise and develop a selfactuating wave-absorbing system, his idea, to generate a radiation wave that is exactly opposite to the reflected wave, could be easily extended to the problem of achieving wave energy absorption through the use of an oscillating body with wave energy conversion systems. As just such an extension of Milgram's idea, Bessho developed a theory for achieving wave and wave energy absorption by means of a symmetrical body with wave energy conversion systems oscillating on a free surface and his theoof second subcommittee theCor~mittee wasTowing read at ry in 1973. Tank Japan In 1976, Evans 161 carried out a theoretical study on predicting the absorption
studies conducted on the two new wave energy conversion devices which were developed as a part of results of Bessio's theory. On the contrary to other theories, his theory was developed by using the condition for corplete wave absorption. which is substantially similar to the condition for achieving 1001 wave energy absorption. Following to his theory, it is shown inj2 that the complete wave and wave energy absorption can be attained in the cas 3 when the floating body with energy oscillates in its natural convtersion syst period and the damping coefficients of the energy convezsion systems are the same as the one of the wave diffraction. One of the two new wave energy conversion devices is noored at dolphins in shallow waters (abbreviate as "Dolphin 'ype') and the other is oored by chain in the deep sea (abbreviate as "Chain Type). In the case of the "Dolphin Type", a halfinmer-_d syrzetrical body oscillates in only two modes, heaving and rolling, and the swaying notion is constrained so that 1001 wave energy absorption =ay be attained. In the latter case, since the floating sy=metrical body oscillates in three modes, ccilete wave energy absorption is not yet completed. The results of the theoretical two these studies and experimental Fv~e 53 andon14. in made are described devices absorption achieved by the "Dolphin Type" in the case of irregular waves is discussed
of wave energy by means of a damped, oscillating, and partly or completely submerged bdy. He derived expressions cn the efficiency of power absorption for the case where the body is a two-dimensional cylinder oscillating in either a single mode or in a ertain coiination of two modes and when the body is a heaving halfimmersed sphere. In 1978, Count 17) applied Evans' theory to a two-dimensional asymmetrical wave power device and made a comparison between the calculated and measured efficiencies of a Salter duck. Recently Srokosz and Evans [8) investicated the problem of two arbitrary cylindersoscillating independently and capable of absorbing energy in a single mode from a given incident wave. Contrary to these studies on the twodimensional problem of wave energy absorption, the recent theoretical studies were directed on the efficiency of three dizen-
in 55.
The experimental data is also given
2. THEOfIX OF WAE-EERGY ARSORPTION 2.1. Formulation Two-ditensional motions are considered and cartesian co-ordinates Cx, y) are chosen as shon in Fig. 1. -t is assured that the fluid motion is inviscid and incompressible, and that the resulting oscillatory notions are linear and harmonic.
a -
S
ig.
sional wave energy absorption. Budal and Falnes [91 began a study on a resonant
point absorber and defined power absorption length in 1975. Bdal [10] also theoretically analyzed a wve power absorning systcws that consists of a number of rigid, interspaced, oscillating bodies in 1977.
Using slender-body approximation,
N.e.eman [11] provided a ranch analysis of the maximum rate of wave energy absorption by a flexible or hinged raft. The present paper is intended to outline Bessho's theory and to show the
o _1-
-COrat
a X
n
syste
When a body oscillates in a incident wave of unit amplitude, the amplitudes of the reflected and transmitted wave far fro the body can be expressed as: K
Atw K
-
-cclo
IA}
ATr=Li4KtxiHliOQ I" -here K = wave numbe = 2u.
-842
(1)
-
where 4 is a displacernt of i-th mode's mtion. Pq is hydrodynamic force in i-th direction caused by j-th mode of motion,
3X= amplitude of 5-th mode oscillatio.
is restoring coefficient, El is wave
X4 =Ci
exciting forte in i-th direction, and Ri is reaction force in i-th direction acting ab::n the floating body from the energy sorbing system. 4., F# , and Ei can be written as follows: X i Ctm1
H: (K) = fochin function of j-tb rode oscillation Here j = 1, 2, 3 refer to sway, heave, roll, respective.y, and j = 4 refers to diftraction of incident wave by the fixed body. when the body is fixed in a incident wave, the expressions for the reflected and
SceX 14e
traiasnitted wave (1) and (2) become:
(j
S
=
(10)
jyait
It is assumed that the wave energy absorbing system operates independently with respect to the sy-neLrical and asynetrical mode of motion, the wection forces Rj can be expressed as follovirr.-:
These waves may be called as the scattered waves and can.be considered as the sun of the sywetric'al and asymmetrical waves progressing to far away from the body. On the othe. band, if the floating body has a s--ywetrical sectional form, the heaving motion generates a symetrical radiation wave and the swaying or rolling mo-ionS~.generates an asyietrical radiation wave.
Xt1?
(11)
Using the chbaracteristics of the Kochin functions for a synetrical boRK (See Appendix 1, (A.14) - (A.17)), the syxmtrical and asymetrical coponents of the scattered wave can be expressed as following.
IH
tX, 4~AajZ~
ZI21
(5 LA X;
A- -
0 =
8)
(3'
I l~' Khtew
,
..
r.AaictI
The solutions of equations (12) can be easily obtained and written as:
interference. In other words, the incident wave is comletely absorbed and consequently wave energy absorption can be cnmletely attained.
KX. "a/D
A n /C
Kx=H
i"otin
D
that the resUnder the assuptio are tIar and harmonic and that
(14) (14) (13)
wdere
Le sectinal form of the floating body is the equations of otion can be written as:
iXr
--- OIK
(s4 jnIi)I
j.$
system, the syrmmtrical and asynptrxcal radiation waves cancel out each coroneiits of the scattered wave respectively due to
of
=
V/W
If we can realize such mtions that
2.2. Equations
C 3K 3
=.,,v+L+ - cS./c-afri-fC4
(7)
satisfy the equations 17) and (a) by add--ing suitable van-energy conversion
e
*
lete wave absorption are:
Alt+ATr
(12)
ae*jIi(0K
CXY
(6
As above mentioned, the conditions
for the c
-tCmy€-)/K 3
Cf tZiK4HJ
Aatry
F
Substituting the expressions 1 3 and (11) into the equations (9), and abbreviating the term ezt, the equations of motion becove:
IC"
C" 4(g)4,R cst
F 443-
as to satisfy (17) and (21) simultaneously,
2.3. Wave Absorption At first, by substituting the solutions of the equations of motion into (7), the condition that should be satisfied for absorbing the syrmetrical component of the scattered wave is obtained as follows:
C=
-
Z
(16)
WooI1Q
0 3the
=I$:*(KIa n
out.
One of these is a syster using two
modes of motion (heaving and rolling) and swaying motion is restricted by dolphins. The other is a system that freed all 0odes of mot.on and moored b) chain. The former is desc:-ibed in the following section and
or -!t'7)
the transmitted wave and reflected are diminished to zero and the inciwave powcr is perfectly absorbed by wave-energy conversion systen. To verify these theoretical results, two kind of wave-energy absorbing systems were made, and the tank tc-st was carried both wave dent this
latter
It is known by (17) that if we have the system resonates with the incident wave in heaving motion and the magnitude of the damping coefficient of the energy conversion system is equalized with that of wave making heave damping of the body, (A R + AT) becomes to zero. On the other hand, the absorbed have
power in the aforementioned condition
in
34.
3. WAVE ENERGY CON'ERSIaN DEVICE MOORED AT A DCL'HIN 3.1. Model Experiments (I) Madel C'aracteristics The principal dimensions and character.istics of the model are shown in Table 1.
becomes:
3z -here Pw represents the incident wave power Expression (18) shows that half of wave power is
theincident system.
copofied for absorbing the yietrical be ntiis )L sould
tt
w2
tami
~
___________
k
roll
main
of (ath in
thiscase.
bod, AT) te
rol
_
0.15m
0.300i
(0)
0.451
n,
00
2.960 m'
amGU
Of
'
0 .3 3 2 1-35
m
sec
f(21
s
damping-
damping
ecoms tozero absored
_
'
resoaee thatriInbesthis case, the absorbed wave the inmtininecs part of u wa, t _u{!o known by (2)F.2 ten of eanergy conversion system solz with that of wav
n
444.I2Kg
t
coafc
I~aIL'1q2
as
oceflef
03
J J~a4in.-&)A,0fotiti ,
with pd that of
0.300
Dft •
heigt (G-)
(19) can be simply rewritten as: iHi0 0 en
___.9_0_M
of govity
(6) are not always necessarily employed. if the swaying notion is restricted: Eq.
ZtIe
Unit wo.
O____(B
both
be noted here that
rolling and swaying motions as known by
-=therieforwe
f model ______
nent of the scattered wave is obtained as:
or
I Choroctwistics
absorbed by te
Secondly, let us investigate the conthe equation that satisfies dition into (14) (8). the expression B substituting
it
T0e
wav
This 1:10 scalem sa
hasa
ewsor
8/2
O~m
A OIi L
Lewis HO
Fig
Form 8
=
.
/2T =1.667
_
_
2
Section of model
=
revolve in one direction despite the alternative motion of connecting rod. The gears multiplied the rotation 300 times for heaving motion and 200 times for rolling motion. g Since it had been estimated that, in this experiment, the maximum incident wave power would be about 160 watts (wave length 3.Om and wave height 0.2m), two 80 watt direct current generators were selected for heaving motion and rolling motion respectively. (3) Measurement The following items were measured: -Wave Incident height wave height and reflected wave height were measured by means of capatype wave height meters W1 and W2 which had been arranged as shown in Fig. 4. Transmitted wave height was measured by W3 (see Fig. 4). The method used to calculate the incident wave height and reflected wave
__citance
height is described inAppendix-II.
(2) Experimental Apparatus The experiments were conducted in 60m (length)x3m(width)xl. Sm(water depth) wave basin. The basin was equipped with a piston type wave maker which could generate regular and irregular waves. The maximum wave height is 0.5m and the range of wave period is from 0.5 to 4.0 sec. The model was set at the center of the basin and was connected to a beam, which had been built over the basin instead of the dolphin by three connecting rods with hinges to permit free oscillation. The arrangement is shown in Fig. 3.
-30m I
-3m
wave maker W
W
/ h0.1
Ooting wave
s:n
(WI ,W2 ,W3 ; wave height meter)
Fig. 4 Measuring of wave height electric generator
-Motions of the floating body The motions of the floating body were measured by means of a displacement meter which was composed of potensio meters, with 6-degrees of freedom. The measuring system is shown in Fig. 5. The measurinv arm was
electric generator _X7
\flocting body connecting rod /"YA
attached just at the center of gravity of
tthe
floating body.
displacement meter
^¥\Y/'Y/
PH
Fig. 3
Experiment apparatus floating body
In this arrangement, the relative motions of the floating body and connecting rods and of the connecting rods and beam (dolphin) drove the electric generators which were located at both ends of the center rod. In the external circuits of the generators, the variable electric resistors were connected so that the damping factors of the wave energy conversion devices may be adjusted to the wave generation damping factors And then one-way clutches and gears were set up at the each electric generators so that the electric generators could
P_
Potenti0 meter for heovng Potentio meter for rolling
G
center of gravity
PH
5 Measuring of motions
Fig -845-
-
-
-
_
- _
-Generated electric powcur The voltages which were generated by the heaving device and rolling device were measured by means of volt meters (see
2D 0
g
FiFi.
S
flooing body
the experimental data, and the full lines the theoretical values. These values are divided by the incident wave amplitude (3a) or wave slope (1Kla). The abscissa represents the ratio of the wave length (XQ to the width of the floating body (B). It is well recognized thattheoretical the experimental agree with the results.
Fig. 6 O)omping devices cdata The circuit current was calculated by dividing the voltage(Vi) by the resistance of the external circuit(Rz). Then the electric power was calculated as the product of the circuit current total resistance which and is the the square sum of of the the external and aninternal.resistance {t < .sec Fi. 6). oto ftefloating 1body. The measured values were then recorded on an analogue data-recorder and analyzed with the aid of a electric computer. (4) Wave Conditions r 2 incident waves The were regular ones.
(2) Characteristics of Wave Transmission and Wave Reflection
Voln
-
I_ Fig. ________________ sma Woveircle
F,
02c The
05
O
Results of the Experiment
0
(1) Motions Figs. 7 ofandthe8 rloating show the Body heaving motion
1.00_
Fig.
_.....
efficeireent
6
ge the and of wave to wave heights period was from from 0.9 0.1 to 2.82
0
_3.2.
in
C C
Ssec.
-
m o
The small circles in the figures represent _________represent
-
'0
0
0
7 Heoving motion
0 0
/ 3
4
/B
8
.
Fig.
I
10
0
I
_
8.9
00 0
/ .
7
Wave
9
..
trnsmitnion coefficient
- 846 -
A
amplitud
(
or wa
......... Th
1
cient (CR) and the transmission coefficient (CT). CR means the ratio of the reflected wave height to the incident wave height, and C1, the ratio of the transmitted wave height to the incident wave height. (The method used to calculation Cp and CT is described in Appendix-II.) In these figures, CR, together with CT, has the minimum value (nearly equal to zero), when;.,/B=2.5 • that is, when wave length reached 2.5 times as long as the width of the floating body, the incident wave energy is almost completely absorbed, Hence, a calm sea appears behind the floating body. When the incident wave length is less than 4.7m (X/Bi4.7), the transmitted wave height is reduced to below 50%.
Pw
0
=
2
'
JC
I
0
where, W means the electric power converted by the electric generator, Pw means the incident wave power and L means the length of the floating body. In regards to wave absorption efficiency J,, it is evident that the wave energy is completely absorbed by the device when the refltcted wave and the transmitted wave are diminished simultaneously (VB= 2.5, see Figs. 9 and 10). On the other hand, the experiment results of energy conversion efficiency ?Ewere short of the theoretical values by 20-30% (see Fig. 12). This loss may be composed of mechanical loss (mainly in the gears and clutches) and of scattering waves of higher order. To confirm this point, free oscillation tests of the wave energy absorption device were carried out, in which the electric resistors had been removed so that the generated power was not consumed, and about 20-30% absorption of wave energy was verified.
(3) Efficiency of Wave Power Absorption The efficiency of wave power absorption is shown in Fis. 11 and 12.
00
Sw
M-o(R ) (Vi(t))a/R2dt ,
(W =
Figs. 9 and 10 show the reflection coeffi-
4. WAVE ENERGY CONVERSION DEVICE MOORED BY 0
CHAINS
4.1. Model Experiments (1) Model Characteristics The principal dimensions and characteristics of this model are given in Table 2. This model has a Lewis-Form section, as shown in Fig. 13.
O O 0 0
_
_
_
_
1
2
3
4
5
6 A/B
7
8
10
9
Fig. I I Efficiency of wave power absorption IC
Table
-.
2
Characteristics of model Item
o1 0
o 0
Length
2.900 m
Width at water plane
1.000 M
Draft
0.500 m 8700 Kg
Displacement
0
70 8M9etacenter
Center of gravity (KG)
Fig. 12 Efficiency of wave energy conversion
Radius of gration Naturiperiod of heaving
In these figures, wave energy absorption efficienCv qw is defined by equation (28).
W
0=- - CR
Wherea., energy conversion efficiency expressed by equations (29) and (30). 1 E-W /Pw
height (GM)
motion
Natural period of rolling motion
(28) is
(29) -847-
0.200 rn -
0.418 m --
0.350 m 1.56 1.85
sec
se
pulleys and a damping device, and was attached to the counter weight (30 kg). The gears, clutches and electric generators were similar to those used in the "Dolphin Type" device tests. In order to control the damping factors 70Stelectric resistors were connected to the electric generators in series. The electric circuits were also similar to those used in the "Dolphin Type" device tests. The measured items were wave heights, motions of the floating body (heaving, rolling and swaying motion) and generated voltage, and these were measured by systems similar to those used in "Dolphin Type" device tests.
-i
05m
-B/2
--Sectional area
1A
E
Lewis Form Ho=B/2T--O a=..A/8r06
'_(3)
Wave Conditions The experiments were performed in regular waves. The wave height was constant, 0.1m, and the wave length varied from 1.3m to 9.3m. 4.2. Results of the Experiments
Fig. 13 Section of model.
(1) Motions cf the Floating Body Figs. 15, 16 and 17 show the heaving, rolling and swaying motions of the floating body.
(2) Experiment Apparatus and Measured Items Experiments concerning the "Chain Type" device were conducted in the same basin that had been used to test "Dolphin Type" device.10 The arrangement of experiment apparatus is shown in Fig. 14.
1.0 Za To I
4-
wovei\ generators
O
wave
O
pul ley
cOI 5P~ifl
mOorneg
I.I
counter
010
we ihty tension
j
wire
I
Fig. 14 Expenirnent
"
I
°
2
o°0
3
V
4
5
6
7
8
9
10
7
8
9
10
Fig. I5 Heaving motion sinker
K o}
apparatus
The floating body was moored by 4 sets of stainless steel wire and coil Spring, instead of by chains. The anale between the mooring wire and vertical line
o
!0 o 0o
was 550 . The spring constant of the coils was about 1.0 kg/cm. The damping devices were, as shown in rig. 14, composed of electric generators, pulleys, sinkers, tension wires and wire counter wei~hts. One end of the tension was connected to the sea bed (this is, to the sinker), and the other end was led into the hull of the floating body via
0.0 0
1
2
3
4
Fig. 16 RoIling
0
5
6 1/B
motion
-848-
NVL
7-
___
1.0
As for CR aind CT, the results of the experiment agreed well with the theoretical values, and it was thus confirmed that the theory was reasonable.
0
Xa 5a
(3) Efficiency of Wave Power Absorption Wave energy absorption efficiency~wand energy conversion efficiency IE (defined by equations (28) and (29), respectively) are shown in Figs. 20 and 21.
0O5
70 0
MB0
Fig.- 1
Swaying motion
0
In the figures, the small circles repre-
BF
V
sent full were tude
thp =jults of the experiment and the lines the theoretical values, which divided by the incident wave ampli(3a) or wave slope MKa).
o
o
Close agreemrent between the results___________________
of the experiment and the theoretical values was observed.
0
3
4
5
6
7
8
9
10
Fig. 20 Efficiency of wave power absorption
aesoe(~) 3.).o tude of Wave Transmission (2) characteristics and Wave Reflection Figs. 18 and 19 give the wae reflechersut age1.0btwe tion Cls coefficient C and the wave trans-
mui sion coefficientCaC
2
1
0/
0
-
is about 0.5
for those waves with 3:8m wave length
d
CR cefiin
Wave ref lection
Fign1
f
thasrthemaximm 2 3
4
5
6
71
8
9
10the
value for
/ 3
the theoretical value being 76% and experi
ment value 80%.
The theoretical results
Fig relecioncoeficentand lB Wav the experiment results agreed well. t1n
oreflected
Svalue
s
ofas
formaximum
is about 55%.
Iconsidered
as that the swaying motion of
0
0
I
25
It is believed that
the difference between and ew avwas due to the mechanical loss in the gears. thtCompared with that of the "Dolphin Type" device, 1 w of the "Chain Type" device is somewhat inferior. The reason may be
0
0
(Scattering of I?wwhen N/B=3 were caused by errors aade i9 the calculation of the wave heights.) shown in Fi. 21, also has a value forwi/B=3.t; h.wever, the
the floating body for the "Chain Type" device is not controlled to absorb wave energy, though the swaying motion of the Ty:pe" device is restrained. Therefore, the reflecting energy caused by the non-controlled swaying motion makes efficiency low.
10
6
YB"Dolphin g19 Wve transmiton coefficient Fig.dabsorption .8-9.
5.
WAVE POWER ABSORPTION OF IRREGULAR WAVES
It is common to adopt the linear superposition theorem for predicting some phenomena that occur in irregular waves; for example, ship motions in a random seaway. In this section we shall consider
2.5
S(u))x 16(m 2 -Sec)
1
2.14 (sec) =015 (m)
H
I
2.0
(W)
the applicability of this theorem for pre-------
dicting wave power absorption of irregular waves in the two-dimensional case. The power absorbed by a wave energy conversion device in irregular waves can be represented as the difference between incident wave power and the total power of reflected and transmitted waves. If we adopt the linear superposition theorem to predict reflected and transmitted waves, then we can denote incident wave power P reflected wave power P; and transmitted wave power P, as the following:
------
I .5cz
SR (W) SI Wx
1.0.
0.5-
~% "P
C6S(,)
z
,
(?)
0
P;=2'.": er( $ ) 4 where S(W) is an energy spectrum of an incident wave. Absorbed power 1; is described in the followir.g: 1Q''.
-1*41 -Gr,),() 0
2
4
6
8
I0 (rod
12 14 sec)
Fig. 22 Energy spectrums of incident wave and reflected wove
Pi IN au) (32) + T
(rn2-sec)
=.23)T=.14 (sec) TH ,3= 0. 1,5 (m)
Consequently, effiziency of wave power20" '
0
21
absorption is
1.511 5,]
Sf
:w
CMT, -
1S
(W) I
S T (IU)I 2
' CT
(J3,
1
If reflected coefficient Cp () and transmittion coefficient CT (uo)in regular waves are known, absorbed power P and efficiency of wave power absorption ? can be easily calculated by usina equation (32) and (33). In these equations energy spectrurs of reflected and transmitted waves are expressed by the product of the incident wave energy spectrum and the square of the reflection or transrission coefficient in regular waves respectively. In order to verify tnese relations, we conducted an experiment using a rodel of the "Dolphin
I I 0 0.5
w (
W
/ sec)
(rod!sec)
Fig. 23 Energy spectrums of incident wow and tmrnsntlied wave
Type" at the wave basin. Characteristics of the model, experiment apparatus and measurin, systems are described in §3. A typical example of an incident wave energy spectrum, reflected wave energy spectrum and transmitted wave energy spectrum measured during the model test is shown in Fig. 22 and Fig. 23.
The full lines in these figure represent the results of the model experittent and the broken lines represent the results of calculation in which theoretical values were used for the reflection and transmission coefficients and experimental values for the incident wave energy spectrum. These figures show that the results of calculation agree well with the experimental results on the reflection and transmission coefficients in irregular waves. From this result, we may conclude that equation (31) is useful for predicting the reflected and transmitted wave powers in -850.
-
~AM
irregular waves. Therefore we can reascnably predict absorbed power and efficiency of wdve power absorption by using equation (32) and (33).
AKNOWLEDGEMENT we wish to acknowledge kind advices that we received from Professor S. Motora and his constant encouragerent.
6. CONCLUSION
~REFERENCES
A theory for predicting wave energy absorption by using a two-dimensional, half-immersed, symmetrical body with wave energy conversion systeems is outlined in%2. The theory was developed by the use of the condition for achieving complete wave absorption and led to the conclusion, that the complete wave absorption can be attained if the natural frequencies of the symmetrical and asymmetrical oscillation modes arp tuned to the desired frequency and the
1.
2.
3.
damping coefficients of the wave energy conversion systers are the same as the wave generation damping coefficients for the symmetrical and asymmetrical oscillation modes. Furthermore it was showed that, if the above conditions are satisfied, a complete wave energy absorption can be also attained and each half of an incident wave power is absorbed by each energy conversion systems for the syr=etrical or asyr'etrical oscillation rodes. Applying the conclusions of the theory, we proposed two new wave eneray conversion devices and carried out theoretical calculations and model experiments. The theoretical and experimental studies concerning motion and wave absorption showed good agrecment with each other. The results of model experiments conducted on the "Dolphin Type" were especially good as they verified a complete wave energy absnrption which had been predicted by the theoretical calculations. On the other hand it was difficult to attain a complete wave energy absorption in the case of the "Chain Type". Because the floating body oscillates in three degrees of freedom, two asy-metrical radiation waves are generated by the s:ayina and rolling motions and, although the natural period of rolling can be easily attuned to a given period, it is very difficult to attune the natural period of swaying in the same manner because the restoring forces of the mooring lines are too srall. For these reason, in the present study we could not attain complete wave energy absorption. Thus, the maximum efficiency of wave energy absorption for this type device was only 80% both in the model exeriment and in theoretical calculations. in 5, the prediction of wave energy absorption in irregular waves was considered and it was confirmed that the linear superposition theorem is useful for this purpose. We also showed that the energy spectrums of reflected and transmitted waves can be predicted as the product of the incident wave energy spectrum and the square of the reflecti-,n or the transmission coefficient respectively.
4.
5. 6.
7.
8.
9. 10.
11.
12.
13.
14.
15.
Masuda, Y., "Wave Activated Generator", international Colloquim on the Exposition of the Oceans, Bordeau, France, March 1971. McCormick, M.E.,"Analysis of WaveEnergy Conversion Bouy", Journal of Hydronautics, Vol. 8, No-., July 1974, pp.77-82. !IcCorrick, M.E., "A Parametric Study of a Wave-Energy Conversion Bouy", Proceedings of Offshore Technology Conference, Paper No. OTC2125, May 1974, pp.959-966. Isaacs, J.D., Castel, D. and Wick, G. L., "Utilization of the Energy in Ocean Waves", Ocean Encineering, Vol. 3, 1976, pp. 175-187. Milgrar, J.1!., "Active Water-Wave .bsorbers", Journal of Fluid Mechanics, Vol. 43, Part 4, 1970, pp.845-859. Evans, D.V., "A Theory for Wave-Power Absorption by Oscillating Bodies", Journal of Fluid Mechanics, Vol. 77, Part 1, 1976, pp. 1-25. Count, B.M., "on the Dynarics of WavePower Devices", Proceedings of Royal of London, Vol. 363, Series A, 1978, pp.559-579 Srokosz, ?1.A. and Evans, D.V., "A Theory for Wave Power Absorption by Two Independently Oscillating Bodies", Journal of Fluid Mechanics, Vol. 90, Part 2, 1979, pp. 337-362. Budal, K. and Falnes, J., "A Resonant Point Absorber of Ocean Wave Power', Nature, Vol. 256, 1975, pp.478-479. Bual, K., "Theory for Absorption of Wave Power by a System of Interacting Bodies", Journal of Ship Research, Vol. 21, No.4, Dec. 1977, pp.248-253. Newman, J.N., "Absorption of Wave Energy by Etongated Bodies", Alied Ocean Research, Vol. 1, No.4, 1979, pp.189-196. Kotik, J., "Damping and inertia Coefficients for a Rolling or Swaying Vertical strip",Journal of Ship Research, Vol. 7, No.2, Oct. 1963, pp.19-23. Wu, T.Y., "Extraction of Flow Energy by a Wing Oscillating in Waves' Journal of Ship Research, Vol. 16, No.1, Mar. 1972, pp. 66-78. Bessho, N., "On the Theory of RollAn Motion of Ships among Waves", Scientific and Engineering Reports of-the Defense Academy, Vol. 3, No.1, May 1965. Bessho, M., "On the Theory of Rolling Motion of Ships among Waves", Scientific and Engineering Reports ofthe Fense Aca , Vol. 3, No.3, Jan.
-851 -
-
ja-
described as:
Y., 16. Goda, Y., Suzuki, Y., Xishinami, of Incident Kikuchi, 0., "Estimation and Reflected Waves in Random Wave Experiments", Technical Note of the Port and Harbour Research Institute, No.248, 1976. APPEINDIX
(A. 11)
Os
The corresponding Kochin functions are written as: (A.12)
". K),,%
.
I
H(cA
H1c3)
H
A.13
We denote the velocity potential l(x,
and
ta
If the floating body has a symmetrical sectional form, the following relations are obtained. aebid (A..14) s nHa 1 (K)
t
(
,I,Q.
=-R-
'o)eCj
(.
(A.1)
where (x, y) satistied the iollowing linearized free surface condition.
M
+)
.
-
o
o
vhere X = wave nurber = 2-/9))
while the surface elevation is as: usinc: .,t =
H
D
t)
=,-
where
H)
.2
3
expressed by
(0.=,0,2
=
n
Ho
Ci)
(A.15) (A.16)
Furthermore, the Kochin function corresponding to the diffraction wave is described by that of radiation wave as:
3)
C0A.
-
C)
-
H(c)
(A.17)
where l1i+(K) .eans the conjugate of H*(K) The incident wave and the diffraction wave velocity potentials are written as: I €ssion (.5) ZC ) - -
Substituting (A.10), (A.8) into expre(A.4), the progressing waves propaas: gate to outward are described at ixi
(e-&
i = 4 : diffraction wave
From.expression (A.18), the arplitude of the wave propagates in the +X direction (reflected wave) is
(A.6)
=
On the other hand, the radiation wave velocity potential is written as: (A.7) )
A
= &K
where i =1I: swaying
i KL
Ar
=2 : heaving = 3 : rolling
(A.8)
= can be Here the Kochin function II(K) introduced as:
Hence, the proS, essing wave at x expressed asymptotically as: -4,C
(.20)
This appendix describes the calculation metho'- of reflected wave heicht[161. Generally, using the Fourier Analysis, tire dorain data 3 (t) expands in a series of wave components with circular frequency WJarad/sec). The wave amplitude Ja, of which wave frequency is u-(rad/sec), can be described by the following, equations: ea by the f=
Assuming that the rotions are linear and harmonic, the total velocity potential 4?(x, y) can be written as:
N~cK~jCU
XjH-(K)
APPENDIX II
: amplitude of i-th node Xi and oscillation
Xik ,)
X
(KA. 19) HCK) transmitted the of Similarly that wave, including the incident wave, is
e(X,1)= lwXi(Xi
WXX. ) = ;4) ice where X. = X, = 1/K
e;K(.18).
(,
and (
Xi H(
=-() i K
where i = 0 : incident wave
3,=
_VA _B_(A.21)
StcS~d
(A.9)A l is
.
,.where
-852-
(A.23)
to is the time duration of S(t). the Fourier coefficients A and B are known, J, 4(t), wave component having circular frequency tJ,, can be expressed as the following equation:
ten
Let us devide the velocity potential into real part with subscript c, and imaginary part with subscript s, and is
f.Jf)SIF&..tdt
(A.22)
It should be noted that if the Fourier coefficients of the wave data measured by means of one set of height meters located in front of the floating body (see Fig. 4), incident wave amplitude a and reflected wave amplitude j. are calculated by using the following equations:
Jax-- t
rA [CA"ACOSk*4-S1MkINfj _AI
+
(Bl+MA,,*e--BcoskA! )Y "a (A.2s) J~xZISkTsi. CAa-rCjt"MSPka" (B.-,Aid
- B, COS kAf)'
(A.26)
where A, and B, are the Fourier coefficients derived fror the reasured value of wave height meter WI, and A2 and B are the
Fourier coefficients of W2, respectively. 1-ave reflection coefficient CV can be described as follow:
T&~ iJ'1
CR
I
_-
(A.27)
853 -
Discussion
K.Ikegami
-IHI)
At first, I would like to show my respect to the first study on practical application of Prof. Bessho's theory on com-lete wave energy absorption. I would like to ask some questions about the model test of "Chain type" device. (1) The characteristics of model used in the 'Chain type" device test is different from the ones in "Dolphin type" device test. it seems to me that it is more advantageous to use the same model in both tests in order to compare the characteristics of two types of devices. How did the authors decide the characteristics of model ? (2) Natural periods of heaving motion and rolling motion are not equal in "Chain type" device test. In order to attain a complete wave energy absorption, however, it is necessary to equalize both the natural periods of heaving and rolling motions as in "Dolphin type" device test. And it seems to me that the natural period of rollina motion can be easily attuned to a given period. Why did not the authors equalize both the natural periods of heaving and rolling motions ? (3) The rolling amplitude of fl. body is shown in Fig.16. i can underst.-d that the tendency of rolling amplitude to wave length shows the characteristics of motion influenced by larger damping. However, it seems to me that the values are very large in longer wave length range. May I understand that this is caused by coupling effect of rolling and swaying motions ? If we could ask the authors for further explanation of the characteristics of the motion of the floating body " rather detail, especially the rolling motion, it would be highly informative, and appreciated very mach. (4) In the case of "Chain type" device test, swaying motion of floating body was not controlled. To the extent of my understanding, the condition for a complete wave energy absorption is to make both the symmetrical motion and either one of the asym-
generated in far field by the transverse motion of the point which lies at the depth iw below the water plane. It should be noted that rotational motion around the point is waveless motion at the specified frequency. Therefore, we can pay attention only to the transverse motion of that point. I think it is important to design a control system for the purpose to control the transverse motion of that point composed of swaying and rolling motions. From this viewpoint, I think, changing the direction of tension dire might result in heigher efficiency of wave energy absorption. Have the authors made such experiments or calculations ? If I could hear the authors' opinion on these points, I would appreciate it very much.
R M.BeSShO dawmlDeeACad
We thank Mr. Ikegami for his remarks on the model test of "the Chain type". At first, we decided the configuration as shown in Fig.14 on the standpoint of practical use and carried out parametric study with varying Ho and a of Lewis Form. Then we chose the sectional form of Fig.13 because it showed better efficiency than othersIn the case of Fig.14, as the tension wires are installed vertically, some coefficients of (13) become U1
(R.1) C
where : Sway-roll coupling lever for damping This formula shows that asymmetric wave is
0
(H1 t I') CI.R +
C21 R
+_+.
quency, and to give sufficient damping to each motions. Let me call your attention to a well-known formula, for further explanation, (X + lX ) H+(k)
= C31
Substituting eq. (R.1) into eq. (19), we obtain following equation.
metrical motions resonant at the same fre-
rad,ASy
-i0
+
2
2C13RHHlw t-
(R.2) R
where CijR = Re tCij}M The right hand of eq.(R.2) is not equal to zero in general. Therefore the natural period of rolling motion should differ from that of heaving motion for complete absorption of the asymmetrical scattered wave. But in our model test, eq.(R-2) could not be satisfied completely.
.854T
On the thi-d question, it was show.
eq.(l9) or rewritten as
by Pxof. Tasai that nondimensional rolling response becomes to slightly less than
unity ii.long wave range if
(C-M,+ i2Hs.H '(C33 + + ~h( 2 3+C-+3
additional
restoring forces are attached. This fact must be also true in the case of "the Chain type". Unfortunately we did not calculate the response function in the long wave range. Thus we can not say how high the
-(:+ 3 +
+ H3 i2HH
-
) = 0
i2H+il) (C, + i2H+l
(R3)
If we adopt some restrictions on the ccefficients of (13), that is
respo.tse function growth or where it begins
u:: =
to reduce. On the last question, we suppose that there may be such a case as Mr. Ikegami pointed out. In the case where bol-h swaying and rolling motions exist, the condition for complete absorption of the asyMretrical scattered wave is expressed by
-,
C 3 R= CI
=
3Z
1.I,
=
(R
e can find moe have retates th eq.(R.3) but we have not yet tested such a case.
-855-
Theoretical and Experimental Study on Wave Power Absorption Hisaaki Maeda
Hwro4~sa Tanaka Takteshi Kims.hita Tne wifinets-4v of Tokro
'yok
jamd-
ABSTRACT In this paper, we show theoretical analyses and syntheses for floating-type wave power absorbers by comparing theoretical characteristics wsith experimental ones of the platotype whicrdhtinfoc conists o ahsDcnd hydraulicpoe ovsinmcagndrzdrdatnfre degrees of freedom in finite water depth are discussed. F1, We iniestipte theories and experiments with regard to hydrodyramic iorces. absorb~ed wave po*-tr. lc'-d reaction forcces FLi which act on the two-dinientional or three,4irnentional slender ggaiyrelrto bodies with asynmnetric sections oscillating in regular or irregulu &j waves, and discuss about the effects of oblique waves. coupled motions and hydraulic load characteristics on the wave power absorption efficiencies. Fron, the test results, we c.)nfirm the following conV 61 clusions: the linear theory is applic3ble to the prediction of the h performances of the absorber. the Sahez3 Duck absorbs efficiently H,(1 both regualr and irregular wave power. and the hydraulic power co.nversion mechanism is successful in convring wav energy to H, (P.0) mechanical energy due to its easy control and storage of energy.HWaehit
F
=
liw
I (c,,) Hl,! (CO) Hpw (w) K k, Ic
Nomenclture incident w*av amplitude a body width 8,floating C.hydrostatic restotinc force coefficient force coefficient mat rix crestoring draft of floating body D0 sliallow water parameter D) coefficient of Lnad of the hydraulic ddamping system damping coefficient matrix of load d Kroneckces delta excitation Lwav column %vctorof wave excitation dF~dt time averaged energy per unit time
drift force normalized load force (torque) linear restoring force coefficient of moorin system !inear restoring force coefficient matrixc of mnooring system wetted hull surface water depth radiation Kochin function diffraction IKocltin function vsual wamv height or significant wave height poris function of absorbed wave power response funciton of i-tb mode motion response function of drift fore wave number restoring force cofficientt of hydraulic system restoring force coefficient matrx of hydraulic
mn, 14.
amplitude of i-th mode motion column v-ctior of amplitude of motion !aminar tube length distance between the origin and rolling axis generalized added mass generalized mnass
in
geerlie masmti generalized -Aded mass Tnatrix
I.
-857-
wave power absorption efficiency (2-I). wave power absorptioni coefficient or wn-drnensiorialA absorption width ratio (34)
wa- damping coeflicient
prcdi-tion methods of p-ormrics of absorbers.
N
wave damping coefficient matrix
v
dispk-cd s.--tje circular frequency fnid domain _ dobtained
Thc theory of responscs of floaitin-tyte wave power absorbers can be interpreted by the appli-ntion af -atcr w3r theories in 2-D or J-D without forward spec-i. Thcor-etical an-ah of wave power aborbers was investigated f' ir by Bessho I I who % showed that the optimum efficiency of absorbcd wav power -was wle it jseili.atcd at resonant ard damring of load was equal to radiation wave dampingr The ticn of a_wa1rower absorber was completed by the works of Besho [21. Mei .IIL. Newmans 1141 and Evans 141. Hlower, t..here hav bCn few reports which comparxd thzse linear theories with expev.ents and invcstigated the applicability of the theory in order to -ake the theory p-acticl. n T here are three_ tfl'pcs of leading wave poser zbsurber a among the tloatlk--y absorbers 151. Thos zre ai-turbine r pe. Sallers duck type and raft t!pe absorbers, lydrod namic cacteristic of these floating bodies ae not aways smenc. as those of" ships or floating offshore structures beccuis of their special shapes however. th ir chartacristics can h. studed by t-- vt'ilar cxperimenta r.thods to those for shipa Dynamics of wae powvr
potential with unflit stonty amplitude cident wave potental with unit amplitude diffraction potential corresponding to . oo + ;: stteluring potential ° time-rever-ed velocity potewial .0,* potential Svelocity 4ty tl P pressure of fluid pill,, Tvi joint probability density function 01radiation
0
p
=
:
variance of spectrum of time averaged absorbed wave power
RtW7i1
< Ettti it-r)> correlation function
absorver depend not only on the charactcristics of floatin bodies but also on the charactcristics of power takt tnaff n..-s In this paper. we show the theory of a wave ;o-... absobcr by enmining the dynamic characteristics of floating shdieS and of power taking off mchanisms and can out some spmial experients for confirming these theoretical results- We investgated the Wave power absorption characristics by makirmg use of Salters Duck of 32 an in diameter of submerged circular cylinder and 167 cm in width, and oi-hydrauNc power -orion m-,cha nis m.. because the Salters Duck has the2adntage of simple shape forces on the body. concerning the calculation of hydro and because the hydraulic power conversion mechanism has fratures of cas cont.-ol and storage of absored enegmy. This si t-m is or example of Arlc power absotbers and we do not insist
density nf water variance of spectrum of roll motion Pi-tson-Moskowitz type wave spectrum spectrum of absorbed wave power opode motion fa drift fore- spectrum
=o.14i
5' 1w)
51w) (4a R
Srjiitl(w)
cross spectrum of fore (torque) arnd vei%-city (angular velocity) ofj-th mode motion
"atoju ( as spectrum of force (torqte) an disphacement (roling angle) of ,hmode motion
S S 11i()
spectrum of velocity tangular velocity) of jih mode motion
T
period natural period of floating lodY
the superiority of this system. In orlder to di the applicability of linear th.ony to the desip of wave power absorbrs we invsigatc the characteristics of hydraulic power conversion me-chaniuns. of wave excitation. of body motions in regular and irregular wares. and the forced
Tw
visual wave period or mean wave period
T, T
peak period of wave spectrum
osillat o,t tests in calr water are carried out. Tbese testh been done in the two-dirnetional model basin or three-dtmentionsl seaAkeping basin. " efiects of obliquwa and -I) effects are also examined. The effe t of moring sterr.is not discussed here because that effect ran be substituted for that W-
column vector of torque
coupied motions.
V,
group velocity normalized velocity (angular veocity) displa---ment of floating body in i-th mode motion column vector of disphizetnt of motion amplitude of reflected waves oy floating-body amplitude of transmitted waves through floating
Take a rectangular coordinate system with z=O P Of the undisturbed free surface and z positive upuod-.An ale O adenoted direcion of progressmg incident pla e wa- from x-axis bvilasshowninF-t l.!tisasu-ndthatabodrisfloa
amplitude of reflected waves by fixed body amplitude of transmitted wares through fixed body
ing on the free surface of an in-isci. incompr-ssOe fluid wit con4ant depth h an all ow-ilator) motiom of the body and thse flu are so small that the .wobk-m can be s-ohd within t& frnwork of lineri r el waterway theory. To shape of the body -s
amplitude of radiation wavm of-h t -ode motion
rot n- rily
Tq:= 2W
C
i
t
r
02
Zero cross mean period and its correspondir circular frequency
_TIEORY
sm t.t,.
1. INTRODUCTION
r
4
I, .
Nlasuda 1101 succeocsd to generate electlic power from ocan wa e for the first tirw by making use of air tubines rotated by compressed air in resonant water tubs. His Frit proW t-pe was applied to 1*1ht-bouys, and rccently, wave absorbing ship Kaimnie succeeded to sapply electric pcower to the power station at shor. At this -tage in order to design mcrc economical or pactical wave power absorbem it is n "ecerary to ake clear the wave power absorption mechanism and also to estsalish theoretical -858-
n
"
N
Fig. I Coordinate stem
-
~2
~
~
qo
tintas show 'irslybkw essho IIA ful to desire the ritior revt-.ned rno',t _r.~rspow&ic to the tinie-rerersed relOcit-iU. f=e :xarnpk. a mion of rn-erid wtatinl movies
tIywvnn orces
Thaiao
~
ore
F,
ite
spotenu2z
.3 oci w -y ial otcilal.Anditis also expresse-d in another fv m. by tatkini account of a- uniq-ueness of a soution o. the boundary-ratue problem Namely the- tine-ererd vcebel pot-cmal ofa rdiatiritnfor a cattert ctnaiksu of otgil radiation potential 1or at rstmerng po'nh cm t revrs incidenit watre and a 5cate.In Potential dme :0 the KinKreversed welociry motenti. L4t us derive this -elationa in cai of D r-dntion pro'-Leim asan in pkn. Anatymptouic bhaviour of th ra~ ien coetilz
'here f,, is a force co-nponenz ut direction i due to a unit vscocityv of the body in the direction i and'1 is a complex zenwlt*ade of i-tb god mtion. Hlere p is dentt, offlMid and wisae ircula frequeny. f,,sgireny 11
3OL dsf
S'r
1
In'.-
'5
wL.,. C)0. is a radiation rinentiai due to a unflt eioc.:y of the-body in the dire-ction j. The suffim deno!tes the dxiretion of six riidhods. deuces ot fredomn. =nzily. suw. swat-. *.--c.roll, pitch am.' yaw for j1 6 respc-ir. Toec tem lorecis usce here in a cencralIized case. to include the mocm Thwe added mss n%, and thec wave daming eoettsernt Nare g-iven by it
fq
f,I2
.
where arpurnents =172 correspo-d to y coonzteois
Hrg
An Ow. cthe
--
(41-
resp cd. A rnpecript* indimcats a eomrPies conjugate. From the- flas.r ttiom.. the ware excitation is T+)
z,: ppl,
I
.
wnhd~e noin ana niiy osdun h he drecn&-t fator cxp fiattu Now introduc a wera -otenti" 0' which has out-goin;: pwt-jcfe navrs atv fiiy
0,:
3grvrleraion.a is an anpbtakofanimci
0f
v,.o~t
ware. D is a shallow water parmete defined_ br D and Ii,
infl
6
tath Kht+K1.1cW Kh. lis 1 a n~diaen Koduin tuncto derined by H,-~3~ (4:
,, u
is a scaiug potential &-finted bqy
-a
v-
whereC r. is the wetted hut surfa7Ce. From a zniquecens of the %cuvr-nbe2 the radiation cmat kndefined by gij n4 :-moa'to
Here 9-.($t is an inidert ware potentil defined hr
oj ($! is a diffraction pot.-ntial duc !o the body. wre Ks MA is a tare numer N i a waw. length and i N~zm Tar dis7 r"eltion is
=lg jj
in ther ful Rluid dormain 1Mnot onl, or.the bT-- rerfac r .Sothat iohr om -- tie rrrrtto otJr.e.-ve atral C) s nrra toe aMS ea-e 011 da...i bxms In the se manner, the- following resahsN ar ic er.s of -Dicstr Foir ing jwnbicms- and 4)pro1lms as well as :-n raiation ,.voblr=L
and
K: tanhit K9) snd acrurbct
lsut,.hoarcrd-
iath!dminUsgte
tion cf a scaterancprob_-_ ar-d eorzsder- t-' Z2:-Fan isa2 real functionanthe full,.the boundary condition oft-be u-wtmroary potential 0. is Writteg as
..
rH
1Ohere
0
wc
i14-
is
.
.10
.)~
A diffiraction Kochsn hinaction is defind by 11(59M.
iiJ~
®
an
-
(11)-sa
0
f
. M1 -
lb
Ztw,~nl
-o:n~w
-
2 'LVo
*iC'5
where, the argumewnts 5anO 04 de-a IKochin functizn of diretion 9 diur to an. i ckIdent ware of direct in AThcdo W.hneintqaal shouhi he recplaced by a rJe integral in 2-D) problemx i & ('1'
O .1
It is knoln that some usefull rrtatiom c-ta beitrn Ng ill 41 and W '5.f 0). A tirne--rrnerd revU; pmtential is hep-u
litr 2nd irnwrtera andI 549.
~i4tt~tntz.'fla
49t1(544m
Samrseo
.
the ectms
cte
r
2-D case and 3-D case, respectively. The equations (16) and (17) were derived firstly by llessho 121 for a case of' an infinite depth of water. Thiese ielations are albo field in case of a finite depth, if an incident wale and K-iin functions are defined by (8), and--(7) and ( 11). respectively. Substitution of (2) and (16) into (4) yields I)
2
~IIt~ I ,(~+Il
N.,
~The
(-111,(
2
2
~)
.
2
Ia)
(0 11,D (018
.f" ,*(18b)
The right-hand side of (18) denotes the energy dissipation 121 I
Hj ()
- i, 4
1 ,WrH
(+70a 4
+11,*(0
001
(19b)
27 Hd (0+27 0+7r) H,- (0) dO ,
i
1)
R
~(; j2, 3, 4.
(26)
linear equation (26) can be solved for ~,~and T rT the uniqueness of the solutions, the modes of motion j are got indep'endent each other. There are two linear independcuet modes. We can arbitrarily choose any two mocdes out of' three modes of sway, heave and roll in case of an asymmetric shape of body about y = 0. But .iote that roll and sway are linear dependent modes each, other in case of a symmetric shape of body.
~rFrom
Equations of motion Alna hoyo h rdcino h oino h floating body of wave power absorption is following. Without we can define an incident wave 00 (ff2) which progresses to positive y-axis. Let M,. di. ki, gij, c,,, in,,. N,, and E, be mass of a floating body, a damping and a restoring force coefficient of an energy absorbing device wvhich is a hyiraulic system
in this case, a linear restoring force coefficient of a mooring
71-+rP7)
1I9a)
(0,0)
Hf,(
141.2-2
Relations between Kochin functions are also (derived t'rcmn (16) and (I 17).Considering the conditions of original potentials aiid their time-reversed potentials at in finity like as (12) and (13;. (16 an (1) yeldgenerality
Hd
and (I 9a) yields
system, a hydrostatic restoring force coefficient, an added mass au-i a wave damping coefficient due to radiation and awave excitation. respectively The term "mass" in a generahzed sense includes the moment of inertia, and similarly the term "force" includes the moment.
rhe equations of motion are
- Hd kP+ir.0+7f)
ad (0-0+27) Ild (0-0+7r) I +H;(0.0) H * (0+i27.04T I7
(20am
whlere I eead0aeete of
AdFromGr
e
(13. ca
in
11 s1.a~
:
(27)
eaiadGren's wem theorem, haveation.
6,
Fromee(21)ecand 22). (ad yieldsmitdwvsa ela eain whichare ar deivd checki of +e and
be a aves transmitte o a ixed
(23 corsodn respectively.
i
2
2
andton
iJ arefu sqar
sewhicrac epn t
(31)
of
eomple ofliud (27 ofiete anaclr ,avlciyadadipaeet 7 r erthey aredectromsaolos
BN~d
33
device explicitly. Both (42) and (43) are available in 3-D cases as well as in 2-D cases. The optimal characteristics of the load, which are the daiping coefficient d, and the restoring force coefficient k,. can be determined by the following condit.ns, (44) O a d (44) = 1= 0 . 9 -" Dk- I 7 1
ti -Aluare matrices o. mass M. damping coefficient of a w1;r thc load d_ acid its restoring force coef icient t are all diagonal, and those of added mass.0, wave damping coefficient N, hydroqtatic restoring force coefficient c and linear icstoring foi,:e coefficient of a mooring system$ are symmetric, From (30) and (29). the response in regular waves is (34) oA + i B ]-'E , .. 1- 1 denotes an inverse matrix. From (30) and (33), where I the hydrodynamic force defined by a sum of radiation forces and a wave excitation is represented in two forms, as = ([C-
when a geometry of a floating body, a mooring system and a wave frequency o are given. If there are no coupling terms among each mode in the equations of motion, dr7d-t is maximized when
.
k, = W, (NI,
)(35)
+P+(c +
NZ
An absorbed wave energy is the work done by the hydrodynamic forces acting on the floatinlbody. The time-averaged absorbed wave er.ergy per unit time dE/dt is:
l(-2) f
(36 PI- d
.P~dsldt,(6
'
(.ij
i
where the fluid pressure P and the velocity potential '4 include
the time-dependent factor, as follows P = ReI-iow(Z4, + 4s)e 6 q, = Re 1(ylq,)e
r
P a 2l 21d
(39)
i,
a'I'2
L- [ E ..
- E,l I ..
=-'!- Wn (Im61 ,1, * (- T ji
2
dt
-5 IN I 2 .. W
Z I,lIjN
,
(42)
(48b)
dT/di
,fti
1 oga2 V, 2 .
I
dt
(49a)
(49b)
;i pga' ,B,,(4b
2 24
2v2r
/
of the indemodes. where most.A number 2-Dindependent cases is 2 at the pendentj denotes modes inthe For general coupling motions, d, and k, which are given by the conditions (44) are different from (45) and (46). respectively. The conditio.s (44) give simultaneous higher order algebraic equations. The time-averaged energy per unit ine of an incident wave per unit length of the wave crest is pga-V, /2. Now the wave energy absorptio efficiency / coefficient (in 3-D cases) 17is defined by
2
_1
l()I
(41)
,1
whicn was derived by Newman 1141 and Mei & Newman 1121. From the lower form of (35), which is a representation of the force from apoint of vi"w of a body, (36) yields dtE
8
)(
I4,H)
k~
2
where the superscript T denotes a transposed matrix. Substituting (41) and (35) to (36), we have representations of JE i in two different forms, which are corresponding to the two forms of the hydrodynamic force column vector F of (35). From the upper form of (35), whicl" a representation of the foTce' from a point of view of a fluid. (3t), yidds dt
(47b,
( -
max- ---
And we can derive the following relation,
t f , P- nds=RetF I ReIf
__(-_)
And the maximized absorbed wave power is
(38)
1,
ice te fo n
(47a)
f
j "(37)
He r e
Ad
+ Il
w 2 k fHii)
t t t
(46)
min)- (c) I, gj)) ,
namely the damping coefficient is equal to th wave damping coefficient and the restoring force coefficient ;s chosen for its motion to be resonant 111. 141. At this time. the amplitude of the motion is
2-3 Absorbed wave power and power characteristics
Ij=rf TS/T,.
(45)
d =Nil
F- Eel"' -mE -Ni
dZl'l,
,
(43) wher.e P. is the width of the floating body. The wave energy dbForitionl coefficient is "the absorption width" non-dimensiiiualized by Bo, which is able to be called "absorption width .atio". It is convenient to express the dynamic characteristics of the hydraulic system as an energy absorbing device in terms of
which is an extention of the representation derived by Evans 141 to a case of 6 modes of motion. (43) is more convenient than ,42) from a point of view of optimization of a wave energy absorption system, since (43) includes the characteristics of the load of an energy absorbing -861
-
Sinc-
tile power characteristics,' namely the relation between the load force (or torque), which corresponds to the pressure in the hydraulic cylinder, and the velocity (or angular velocity), which curresponds to the flow flux. -et FL, and VLI be the component of load force (or torque) with the same phase as the velocity (or angular velocity) noimalized by a wave excitation and the velocity (or angular velocity) no.ralized by the velocity at the so-called natural frequency wo of a j-mode motion, respectively. They are
d I? -/
[IH (#)+l
.
1
H,(l) ] B
j
g
2
KB0 2g
given by
-
I 2 [ -pga VBo]
Bo)
")12
0
d/B4
(58)
(
0
-f 27-,I
dolljl/IEl
FL,
then
(50)
.
W
!2
(SI)
I-
.
g'
W By
*=KBO
l 21,H,(O)12 (sin3P
f
sin 0) dO /B
where o= '(k,+ c,,+ g,,,)/(M,
+
)
2-5
2-4 Characteristics of a floating breakwater and its drift force
(59)
sin
7
3-D effect
The high efficiency of energy extraction mleans good
For calculations of 3-D hydrodynamic forces, the inter-
characteristics as a floating breakwater. Not only 2-D devices but also 3-D devices with proper arrays canl be used as a floating breakwater. An infinite array at regular intervals is one of the examples, since both its reflected and transmitted waves are sums of finite number of 2-D plane waves. Firstly consider 2-D problem. Let the transmitted and reflected waves be tT and trespectively, corresponsponding of a unit amplitude. t - (or G -) to the incident wave 00 (7r/2) is a sum of the transmitted (or reflected) waves 'T- (or r - ) of a fixed body and the radiation waves, those are.
polation solution of Maruo et al 191 is expanded to an asymmetric shape of body about y=O (71. According to the solution, hydrodynamic force f,defined by (2) is
'
= -
i,li ) +-
.
f,
f U, (x) h,,'(x) dx
(60)
.
in a deep sea. Here j =j
U,1x)=1
for j 2-4
(52)
U,(x) =(-IPx. j'=8-i forj=5,6,
(53)
h,,'(x)
g 'and
g- .Ot 7 1-,H,(--
Ra
)+'
hij'
(2D) (2)
body due to where 1,is an amplitude of tie motion of tile ¢ Or!2). which is given by (34) The drift force is I 1 )/l "-"121 FI) /(/p.a V; FkaV' 5-a ( l 11i2 -I R.2 )l
(2pga 2 V)
=1
(54)
(I~
+ltH 12) ,
~ + hii
(3D) ,
,(2
(63)
for i=2-4, i'i i'=8-i for i=5,6.
(64 (64)
h,. tn)=W,(x)I-S. (x)+B. (x)/K 1,
(65)
and
S 2 (x)=0,S 3 (x)=Swhichisa sectionalarea, S 4 (x) = ffrt ydydz. which is a moment of
(55)
(54) yields
sectional area, B2 (x) = 0, B3 (x)
=y
which is width of water line, B4 (x)
F.)/(/Pga 2
K
W,(x)
The drift force in 3-D cases is 2 Fni/(;pga Vg ~Bo)
+---
)
2
-
--Y', -
(y2
({66)
y])12.i
Here W, (x)is defined by
(56)
It 12 + "
=
(62)
f',.U, (x) - f3j' W(x)
where f',.,. is a sectional hydrodynamic force and
Since =t
=
0)
2 -ao (x)IN(KIBo/2+xl) +N(KIB 0 /2-xl)] fao(x') -ao(x)] N'(Klx-x'l)dx'. (67)
-
fB-
; -2 w lB0 sinP.~^r, I IH.(O) +IH Ol /B 2
2 Bwhere U
KB0 where 17-
B
.o
__(2
and H7 (0)
N(u) =- i -i-ln 2u+ - fIHo(u')+Yo(u') -0 +2iJf (t0')s du',
=
H
0,
N"u)=
0).
+ "
-
Ho (u) + Yo (u) + 2iJo(u)
u -862-
-___
__
--
_____m
~==-'~-=-~~:-
(68)
F1
The d,, in the following andis aEuk1r's HO is a ccnstant. Struve function. YO(67) andisJOa solution are Besselof functions andU,0
27a 0
r
=fV7fr
integral equation.
10.09 + HI,.
=III.(
l13 (4)+
-
113(
)
(
~)
(69)
.
dt
-f
(,i + 'y2 sinO)elkt iiB./2
(70)
o1d~
It,1( a
-
+11
l-,. F )if,.( fit
a 2 22
,,X
1)
WX
energy per unit time in irregular waves is
- ) I L',
where If.
is a response lunction of the tie veraged absorbed
wase energy per unit time. which is ex.pressed as
of 1I, (0!. we have a wave damping coefficienlt N,, as
n
I:
Note that the approximate hydrodynamie force f,, gisen
dt 2
-
(82) ,.oefficient (it) 3-1) cases)
.I
183)
... ... = f'
where B0 is a unit in 2-1) cases. The ex~pected time-as eraged wase energy i)er unit time at in area %,,iereabsorbers are set is EiIPI = ffr'- pffl )(ill,, JT , (84) 1,.T
-a
afYjiii of
which is comparable with (79). ie use of a response function ltiwto of an amplitude of motions v.liich is given by (34) corresponding to an incident wave of a unit amplitude. wvehave a response spectrum Stitw) of the mlotion inirregular waxes. as follows
I 2By
11 HT(73)
Ji
-L
dt
Is W) wlw = Ill (0.(w) 12 2S'(., dw
Sr 87
2S'(o)dw
I -0.44(Tr
w/27r
J.
3
-41 CXP k,07 HWTWT.. /ir~x -0.44(T w/2irfI a(
2w T
27Zr
26%~ -12i - WIby
1t
T.
2
XEIET
In order to clarify the applicability of the linear theories to the prediction of the performance of the wave po%%er absorber. we investigated experimentally tl- characteristics of absorbed wave power, wave excitation, and iotion of tile body in regular r irregular waves, and characteristi s of radiation waves generated teforced oscillation test in calm water. Their characteristics are tested in the two-dimentional model basin with 20 x 1.8 x I m (length x widthl x depth) or in the sea-keeping basin with x 30 x 2.5 im. both of whichl belong to University of Tokyo.
(75)
and .JONSWAP spectrunm, as
(85)
we have a respcnse spectrum Si) (w) of a drift force in irregular waves, as follows is 1 ) S (~w lu w ''wdo(6
S' (w) is a sea spectrum. ISSC Spectrum S' (w) based on the Pierson-Moskowitz sea spectrum with visual wave height H= and visual wave period T.~ is
-tHw Tw (Tw w/2irY'ecxp
.
And using a response function IIiw ( w) of a drift force Fn, which is given by 056) or (S9) devidcd by a-' in the sanle way as in (81).
(74)
0
IE
3.3e'
to an incident wave of
isdfndb
T27r~~
S'i
orsodn
The wave cnergs absorption cffiktinc sdfndb
The time-averaged wave cnergyyper unit time per unit lengthl of long crested irregular waves dli/dt is
S'(w)
41
f
P=
regular waves with wave height l12a and period T in a deep sea is written as
Vwhere
w 4)o
I l
wave_____ powe in ireua
rhe time-averaged vase energy per unit time
. pL V
ie
l
unit time in irreitular waves is
b).per
(I
by (60) is not necessarily symmetric andi that a wave damping coeffieieni N,, given by t60) and (4) is not necessarily equal to N,, given by (72). because this approximation is inconsistent. But the discrepancy amiong reciprocal relations is proted to be negligible numerically. 2-6____ Abore
dtit)
=
fron
(80)
2S,,(WdW=H,, (w,) 2S'(w)d(4
-
~~~By the use
kwv/m (JONSWAP)
00 dt dt A responsc spectrum of the time-averaged absorbed wave
--
=
II,'T.
(78)
wave energy per unit time per unit length is derived as lI-f(L( T.)~T..(9
(71) 72
wmdS)
I~
=05
respectively. From the joint probability density function p~lI.. T ) at an area where wave absorbers are set, the expected time-averaged
where 71
w> 4.85TL
10.57
By making use of W, Wx.we have a Kochin function I-, (0) in 3-D cases a-, 1,(0)
for
Substitution of (75) or (76) into (74) yields
;Lt (X) W,(X)
T-
(76)
where where50 -863-
-
-
_I
--
A
--
--
~
Water depth effect on Salters tDuck is so small that tc experiments are carried out only in deep water 1131. 3-1
We test first the reaction-torque characteristics of this mechanism, because it has effects on both the motion and the absorbed wave power efficiency of the floating body. Especially it is important to clarify the effect of the suction capability of cylinder and of the volumetric efficiency of the rectifying valve on the reaction torque characteristics. The reaction torque is measured by the strain-gauge with capacity of 6.0 kg-m fixed on the Lantilever which supports the cylinder and the body.
Shape of the model (Salters Duck)
Principal particulars of the model ire shown in Fig. 2. Its radius of the submerged Lircular .lindcr is Ro= 15.9 ,m. honrizontal distance between rolling axis and the edge of the water plane is 30 cm, and width of the body is 163 cm. The slenderness ratio is I 3.55. The sectional shape and the draft of the model were decided by the results of theoretical calculations in order to absorb wave power efficiently in our model basin (See Appendix A). The moment of inertia of the model about rolling axis with hydraulic power conversion mechanism is 0.494 kg-in-sec 2 , and the one without the hydraulic power conersion ineclanis i is 0 346 kg-in-sec. Restoring force coefficients in roll mode and heave mode are 13.9 kg-in/rad and 489 kg/m. respectively. Tie vertical distance between the rolling axis and the origin is I = 15.9 cm.
4_
_
-
The body is supported at both ends of the body through rotation.l bearings by the frame which permits the heave and suay motions (Fig. 4!. The weights of the model in heave and sway direction are 176 kg and 199 kg. respectiveh.
,9.2
--
l
-
6..-
3-2 Forced oscillation test
roll.omen is easredbytheblok aug wih
The model oscillates sinusoidaly in roll mode by an oilservomechamsm waterb iny the the block 2-I) model Thyuraulic he reaction roll m om ent in is caln m easured ca gebasin. w ith
:L
... .- -....
-..
-'.. ..
.
capaciy of 100 kg. which is used for the measurement of .ae excitation force, too. Radiation hydrodynamie force is decompo-: opredicted added moment of inertia and wave damping by analyzing the relation between nmagn;tude and phase of this force. 3-3
Incident wave
Wave powcr atbsorptioi system
Wave power is coiiverted to hydraulic power by hydrostatic power conversion mechanism which consists of a hydraulic cylinder, a flow rectifying vahe and laninar tubes for loads. This mechanism is compact and has the advantage of easy control and storage of energy (Fig. 4).
23.8"
'
V__
[--5.9 ji ,:5.9
"-
-,!\
Fig. 4 Schematic drawing of wave power absorber. I. Floating body 2. Torque measuring qauge 3. Hydraulic cylinder
4. Hydraulic rectifying valve
5. Laminar tube
6. Spring for sway
\ "
3
d318 -a.1
-.Y00_ /
k
'Kgm/racd)lm IPo (0)
30.
L
4_ .0-
L a, t.to 0
Fig. 2 Section of Salter Duck
t*40
.G l .6
t.0oj-
1.0-I
-25
o*
302"
I.
B
I
0.6 a -
V
2A
8
tO
0 I
Fig. 3 htydrostatic restoring force coefficient
2 1
6
4
8
1
4
Fig. 5 Load damping coefficient and added spring constant .864 -
a
1.__0
Reaction torque characteristics are examined by forced oscillation test of the hydraulic cylinder, and analyzed by Fourier analysis. We get the test results that the mechanism has not only damping characteristics but also spring one due to cavitation or aeration of fluid in tile cylinder chanber. The both characteristics are shown in Fig. 5. In these figures. d and k represent the load damping coefficient and restoring force coefficient of the hydraulie power mechanism. power i conversion e conversion smeais onitsi
f34G =
where. I is the vertical distance between the origin and the rolling axis G. The Kochin function about G is as follows,
approximate solution in Lasw of K 0 and K -. oThe remainder is substituted into (67. then , -- I,, - \,, = if x I N(K l(2--xi + N(K.1I(, 2
2.3 i-t-
,
-
2 1911
= 4. 4(;
Fourier anal sis is used fIor the analsm' ot the experIniental data in regular wasc The in-phase and out-ol-ph..se coilponents of first to fourth order are anal sed from data of fise to ten periods. FFT (Fast Fourier Transform I is ued to anai /e the data in irregular wasc. The number of samples is 2048 and the sampling time is 0.07 sec. Q %,indow, is applied. The results of the experiment and nuinerical calculation are classified and listed under the following items. hy drod nialmic force, motion, absorbed energy and power characteristics. c'larac teristics of a floating breakwater and drift force, results in oblique waves, and results in irregular waves. 4-1
lldrodynamic forces
Fig. 6 shows tile added moment of inertia about the rolling axis G tested by forced oscillation of the body in 2-1) model basin. Experiments of both wall-sided and inclined bull surface are carried out. In tile figure. results of numerical calculation are shown in case of both 2-1) and 3-I) Fig. 7 sll\wS tile tests results and calculation results of damping coefficient. In the figure. not only dalping measured by torque gauge but also damping calculated by measured radiation wave3 are shown. Fig 8 shows the test results and calculated results of wave excitation. In the figure. experiments of exciting moment about a rolling axis G and of amplitude ratios of radiation wa,cs are shown with tle expression of Kochmn function. 4-2 \Motions
2 ff 4 ,
= f24 - Tf722
(o0t
-
=
,
__-hydrodynamic
22
9)
.
.,-=
1
Tile theoretical calculations of hydrodynamic forces are determined by the source distribution method making use of stream functions The evaluation of tile accuracy of the calculation depends on three different methods which use the ltaskind relation of (18a). the reciprocal theorem of hydrodynamic forces of (2) and the relations between reflected and tranmmitted waves of (22. 24. 25). The number of segments on the wetted hull surface is 90 - 100. The hydrodynamic force f 44 6 etc. of rolling mode about die rolling axis G is determined by tle following equations. if
i9
The relation between Kochin lunction- and radiation wase amplitude ratio A,' in 2-1) problem is as follo%%.
THE RESULTS OF CALCULATION AND EXPERIMENT
+
t -
Because the section of the floating bod, is uniform and its slendcrness ratio is I : 3.55. there are two kinds of problem left with regard to the application of the slender ship theory. One is that the assumption of the end of the slender body is no* satisfied and the other is the slenderness ratio The precise discuss'on should be necessary. but it is left in future. -T'he end effect is treat-J in the manner The second term of the (69) is neglected as the
Test results of 2-D model experimented in the model basin is called "2-1) Experiment' in this paper, and those of 3-1) model in tle sea-keeping basin is called '3-) Experient"_ The motion of the model is tested first %ithout hydraulic power conversion mechanism with the incident wave of height of If,, =2. 4 and 6 cm and period of I to 2 seconds, and secondl tested with hydraulic system with the wave of height of 4 and 6cm. 'I lie length of laminar tubes for hvdraulic load is selected t) I.=O 65. 0.7. 1.1, 1.4 and 1.6 in long. lffect of oblique wase is tested at the encounter angles of incident wave 0=650 and 90' . The motions of the body are classified into four categories. One is I mode (roll) which denotes only roll motion, the second is 2 mode (roll-sway) which denotes the roll-sway coupled motion. similarly tile third is 2 mode (roll-heave) and the fourth is 3 mode (roll-sway-heave) denotes the roll-sway-heave coupled motion. For tile sway-mode motion, the body is supported by additional rings. The sluing constant is g2 =41.7 kg/iii which makes tile drift di-placernent be one tenth of the supposed water depth t200 m) in regular waves of seven times of significant wa%e height of operating condition when the drift force coefficient is assumed to be I. In case of 2 mode (roll-heaset. mechanical friction of heave mode which is estimated to be 1.3 kg is added, in the forced oscillation test. the model oscillates in I mode (roll) of amplitude 114(; I = 3' in tile 2-D model basin, In this experiments, when the model rolls, water runs tip the slope from the lee wave side. Tile effect of water running up the slope is examined by using two types of bodies, one %%ith %all-sided hull surface and the other with inclined surface. Wave excitations of 2-D an~d 3-D experiments are measured in regular waves with height H% = 2. 4 and 6 cm and period Tw 1 to 2 sec. Length of laminar tube is 1,=0.7. 1.1 and 1.4 i. 3-1) experiments in irregular waves of JONSWAP with significant wave height Hw=5.16 cm and peak pe~iods of Tr=l.2. 1.4. 1.6 and 1.8 seconds are carried out. Duration time of irregular waxes are about three minutes. The scale ratio of JONSWAP is settled to be 1/30.98. Wave power is absorbed only in roll mode-
= ff 4,
+
_ 1r _ , 2 = f (88 Fquations (87) and (88) are available to 3-) case. But In tis case 3-1) modification should be applied to hidrod~iaic forces.
114
The owe co~erionmecanis cosiss o a ydruh~ cylinder of 20 mm in diameter and 150 mm i stroke, a flowinch ports. and lamnar tubes of 0.65 to recuifying salve w~ithi 3 ~8 1 6 m long and 1.55 m in diameter of which pressure and flow characteristics are in linear relation. Working fluid is turbine oil with kinematic viscosity of 0.2 cn :sec at 30TC. 3.4. Kperineintal condition .. cfollowing ..
4.
fi 3
(87)
Fig. 9 shows the roll amplitudes of I mode (roll) in case of wave power absorbed and not absorbed. Magnitude of hydraulic load is adjusted by the length of laminar tubes Fig. 51. In this experiments, the natural period of the body is adjusted to be 1 about To= .3sec. The length of laminar tube 1, isl = 1.4 m.
-865.
.
+
-
-
_+
.=
.+~=-~=
-~
~
]2.0
'Th M443
2
D1f Slope
K'
7IH~- -1.
'NatI a
ae
2-D016.
053-0
I
0
_
_
_
o ieri __mn
0.5
K.
Fig. 8 Wave exciting moment
________
0
-
.
100
~
0
1A
0
6_
S-e
Wria~
0.5
ig.
15ope 0
2-D{-
A
0-
_
R
4
1.0
cc"
Fig. 7 Hydrodynamic damping coefficient
I
Ka
The coupled motions of roll-heave and rolk-way are shown in Fig. 10 and 11, respectively. Each of them shows the rocking amplitude characteristics in which heave or sway motion of the body is allowed. In the calculation of Fig. 9, 10 and 11, the added moment of inertia is estimated by the experimnrt.
2.Dl' -ae (Il
EXPCAI ELP CAL. 'd---j
-lvdes(R-m) A
1lo4
1
4-3 Absorbed wave power Fig. 12 shows the wavc power absorption efficiency or ratio of 2-0 or 3-1) experiments of I mode (roll). Nz.ural period is tuned to be about To= 1.3 see and length of lamina-r tubes for hydraulic load is adjusted to lp=0.7, 1.1 and ..4 in. The added moment of inertia is estimated by the theory with regzard to the results of the calculation of CAL". and by the experiment M44GP
I~
texp.i with regard to CAL.
_________________
0cs10
FiR. 13 shows thle comparison of efficiencies affected by allowance of coupled motions. Wave power is ab~sorbed in roll mode of 2-D experiment with natural period about To= 1.1 sec.. of 3-0 experiment with about Tp = 1.3 sec and hydraulic load of V 1.4 m tube length. In the calculation, the added moment of inertia is estimated by tile experiment In44G (eXpj.)
Fig. 9 Roll amplitude
-866-
1.0
lp D.01ALCA-
EXP CAL
?1
9o
23Droll heave
I1.
A
Un A
101 ;0 L
to-
0
0.5
1.0
0
Fig.10 o copledmoton mpltuds rol-heve)Fig.
0.5
hn~de
'V~~~~
Ka
1.0
12 Absorption width ratio (pure roll)
[2-1,
It
-' KR
3-
E
EXPAI
2modes(R-H)--------A
3-D roll o
050
a
-u
..
~
05~
-
US;10
0.
0.5
0
0
0
10
Fig. 13 Absorption width ratio (coupled motion)
LID 0.5
06R.0
10 S 2-D)
0
r
-body
Fig. 14 shows the power characteristics of relation bhetween torque and angular velocity of 2- experiment, moment of inertia of which is adjusted to be M4G =0.494 kg-in-sec'. The oscillates in one roll mode in sinusoidal wave trains, and the magnitude of hydraulic load is adjusted by the length of laminar tubes. In the figure. the soaid linei anti broken lines show the. calculated power characteristics and efficiencies. respectively. Fig. 15 shows the characteristics of wave power absorbing torque and velocity of 3-D experiment with moment of inertia of M r, =0.494 kg-m.SeC2 .
1
FL4G"
KR(Tsecl Kit.s
Fig. II Amplitudes of cotipled motion (roll-sway)
o'
0 4. (1.10*
-AA-
(1
/ -
CAL
0I -
C
CAL .
0
as
1
Is0
z
.
Fig. 14 Power characterstics (2-Dl)
I
Itzlim 11.=0159m
3-0
M~~O4~e(msc'2-D j,
CAPLA
'V 2jI ~~ 2p
7, 1.0
A
X
osobed (R) notabsorWedR)
(R-5 absorbed (R-S) 101
0
02
H
04
02
10
Fli It 15 Ponwer ciracteristics (3-D)
i1-
F-ig. IS D~rift force Q5;.
nR
1.0
10'-
Fig. 16 Transmission coefficient (load damping effect)
2-D
absrbd
0.5-
L
A
\.-.
A
t
*
0.5
KR.
k-
05
*-
LFig. 0
5
0'"f
LIP~AL05'-
2 i~dsR~
KR.
1.0
19 Absorption width ratio (oblique waves)
1.0
The experimental results of drift force of 2 mode (rollsway) ca.--s are plotted in Fig. 18. The conditions are as follows: wit~i absorning energy in 3-D cases. the length of laminar tube is lp= 1.4 m and designed natural period is T, 1.3 sec. In- the calculation of Fig. 16. 17 and IS. the added moment of inertia is estimated by the experiment M44G xp)
Fig. 17 Transmission coefficient (coupled motien effect)
4-4 Cliaracteris' ics of a loating breakwater.-nd drift force e
n byu theuse ftemdlwt hlnes in Obliquete waesiv2Dese 3-D experiments are carsied out in the sea-keeping basin
absobin aborbig enrgy.wit eerg andresraied -ondtio in waves- The corresponding numerical results are shown in the samze figure. The resut of (lie calculation 01 not-absorbed case in Iei. !6 coincides with that of absorbed I mode troll) case. The couplet! inotior effect on transmission coefficient is shown in
rato o 1.3 mx 0.46 mn.The encounter angles of incident waves are =90* (beanm sea) and 65'. Th~e length of lamina, tub is I~ 1.4 mnand the designed roll naturat period is To= 1.3 sec. The experimental and numerical results of absorbed energy and motion are shown in Fig. 19. The 3dded moment of inertia in the calcula-
Fig. 17.
tion is esrimm~ed by (he experiment M44G (exp.).
4-5 Reslt Lxperimental~~4results
Fxeietlcniin
arete thre
r
-868-j
JNSWA 0
x 0215
o07
IMNaSWAP
14
ia~~
0,-
0 21- Nkgmr)
_6
40
10
70
60
so
4C
EXP-AI. 6 a(roe? Sec)
T-ry Ttr
so
60
70
(J0rdise4J
Ut~d~)-
Fig. 23 Absorbt~d wave power (load damping effect)
Pig. 20 Wave spectra (JONSWAP & ISSC) to.
0233
0
0
30
0
0
'd Ia66n
V
Q663-1
-6113
OtSWAP WOZ EXP
--- 0 188 S757--0 233 282 -0
ri.-.
L
1
208
M,
Fig
30 Tole spectramofh
oi-
!.II
\~.xThewe lztz \111which 80of
rrgua wve
f OWWP nd
wae poer oIsctra~ 2 0. Tbore arhw nFig. those of ONSW(efae of=mea 1.4. 1.6raod)18sc
.4s h e
waerimnthe correspondin maeasredcres' ult r
h incaeo
tefigure. The results of experiments and two kinds of calcula21. tion ofxthe roll spectrum of I mode (roll) are shown in Fig.wave one in which design are theoretical of calculatrions theoreload and hydraulic damping coefficient design, specturm. and the one is are tised. added M4-(; (exp.)spectrum, function with tical response the experimental results of wave moment inertia and damping coefficient of hydraulic load are used. The spectra of abborbed wave power are shown in Fig.22
I
Fir PThe
analytical methods of absorbed wave power from :he
EXP C.L d.
time series of the experimental data in irregular waves are described as follows. We assume that the column vector of load forces T can be expressed linearly like
T = di + kt
m) tS,
I
[
02,.99
-
-M
dd,j
.
02867
--
Theory
l
as compared with (35). The virtual absorbed wave power is written d-F: = Rie IT ] Re It~I
*
(t)
;,, -fCJ)S(Lj)(EXP)
',)
(92) (92)*,
T~d~ +~
I
02,5
I
where
+i
,,
.(94)
&
Then.
A
T
I'
.in
"dt
dZNt,
0, 0.32 0.36 0.32 0.37 0.26 0.37
,
C. > 0 i
C.32 0.36 0.32 0.37 0.26 0.37
.170 .386 .586 .180 .267 .347
0.00 0.00
Four-Raft Train
Table IV gives the results for seven trains with the same total length and width. The optimum efficiency is somewhat improved over the three-raft trains, but the gain is probably not large enough to justify the added cost of the third hinge. Note again that all Ci's are negative and the downwave
.337 .089 11.49 -0.30 -0.21 .346 .069 12.06 -0.25 -0.21
Restrained Optimization
is predominated by the
R radiated wave A (E), giving further evidence have ample motion to must that the train radiate sufficiently large waves toward 6 = 0.
C3
Unrestrained Optimization 0.32 0.36 0.32 0.37 0.26 0.32
,(O)
0.00 0.00
hinge must hrve negative damping a4 4 EFFZCT OF RAFT-WIDTH
It is .:Leresting to note that in the last case of non-negative Ci, he optimal a. pSo and the cptival C. is zero. Comparing the three alternatives above, it seems necessary to design for negative C. for good efficiency. For better physical understanding of these optimum impedances we turn to a general Let the farformula derived by Newman 1i]. field of the total radiated o
< 0.
-881.
far all results presented are for a 0.1 slender rafts T1 and T2 with B/L Now a slender raft is expected to have a large displacement at optimum which may cause difficulties in structural design, not to mention the loss of relevance of the linearized theory- Newman's choice is to impose a limit of the vertical displacement, possibly by changing the hinge impedance away from optimum. An alternative is to use
MBLE IV Optium ixtpedances for four-raft trains L=A and B/=O.]IL 1 +L2 +L3+L4
wider rafts. Table V shows the optimum results of trains with the same length distributions as Tl and T2, to be called classes I and II respectively, but with different widths. The dimensionless displacement of the unrestrained bow* 1alI, i.e.
h
the ratio amplifying the incident amplitde A, is also listed along with the octim,un impedances. It may be seen that for B/I. = 0.1. 'a,' = 4.75 which is probably too large f&r structural safety and is expected to be still larger for longer waves. Increasing the width not only reduces the displazement, but also increases the efficiency. Figure 2plots La/ vs B/L for class I trains.
.21 .26 .32 .37 .21 .22 .21
0
2
3
4
5
6
149)1'
.21 .21 .21 .21 .26 .26 .37
.21 .21 .21 .21 .21 .?6 .21
.37 .32 .26 .21 .32 .26 .21
.742 .752 .755 .741 .758 .760 .766
C3
.100 .110 .115 .116 .072 .063 .048
.131 .114 .101 .125 .097 .065 .076
-.084 -.075 -. 059 -. 033 -. 069 -. 075 -. 054
-.073 -.053 -. 043 -. 067 -. 058 -. 019 -. 052
-.105 -.139 -. 154 -. 176 -. 118 -. 154 -. 149
B/L
1all
La/k
02
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
4.75 3.23 2.65 2.40 2.24 2.'.2 2.35 2.01 1.97 2.04
.73; .750 .790 .825 .883 .952 1.03 1.12 1.20 1.30
.078 .137 .190 .241 .298 .358 .423 .491 .559 .628
LElCGTH CLASS B/L La/A a 0.1 .728 0.2 .739 0.3 .768 0.4 .813 0.5 .071 0.6 .941 0.7 1.02 0.8 1.11 0.9 1.19 1.0 1.29
A
I
C2
C4 -.100 -.092 -. 082 -. 066 -. 081 -. 076 -. 049
h
D
1
a4
Optimum impedances and efficiencies for Thre--raft LENGTH CLASS I: = 0.37 L, L2=0.26 L, L=0.37L
little for increasing but small B/L. Clearly economical consideration2 accounting for structural strength and converter design must enter for choosing the proper width.
a
03
TABLE V
Note that the curve has a horizontal tangent at B/L = 0 showing that La/A increases very
=
02
L2 L3 L4 La/A F F F
F
7
e3
C2
-.063 -.104 -.132 -.160 -.191 -.224 -.261 -.302 -.343 -.384
C3
-.065 -.101 -.120 -.138 -.153 -.164 -.173 -.183 -.194 -. 203
-.101 -.172 -.243 -.314 -.389 -.470 -.555 -.642 -.729 -.819
IIL=O.32 L, L2 =0.36 L, L3 =0-32L a2 2 .074 .140 .199 .261 .327 .395 .468 .543 .621 .697
a C3 -.118 -.209 -.290 -.376 -.467 -.562 -.659 -.760 -.865 -.969
C2 -.095 -.137 -.174 -.201 -.225 -.245 -.269 -.292 -.514 -.236
3 -.058 -.091 -.118 -.141 -.166 -.192 -.224 -.259 -.295 -.331
Fig. 1: Angu'ar variation of total radiation W p -, -*2 intensity 1$(0)2 IP Is
79'
A: Scattering only 1
0
B: Optimu, under Ci C: Optimum under a,
0
[.x
D: Unconstrained optimum C
0!
This
end has the largest displa~ement
Fig. 2:
among all joints.
02
6.
03
04
05
06
'
08
09
'0
Optimal efficiency for three-raft
Trains of Clpss I with different widths.
-882-
.
++
++
-
_ -_-- +
_
-
--
+
+
_
_
_
.
J..
5.
EFFECTS OF CHANGING WAVELENGTH ON La/A
in this section two widths are considered B/L D/L = 0.1 and B/L The = 0.3; draft are is of, lengths raft the = 0.016. still The optimum data for B/L--0.3 are class I. given in Table VI. The optimum LaA for
.61
B/L = 0.1 and 0.3 are also plotted as curves C and A in Figure 3. TABLE VI Optimum results for variable L/A. 0.3 D/L = 0.016, for two trains B/L of different length ratios. TRAIN T3:
5LA
La a %L' La
0.3 0.414
= 0.37L, L 2
L
=
0.26L, L3
Lal1
C2
2
1.38 54.54 2.15 10
0.4 0.415 1.04 19.47 2.86 10 0.5 0.487 0.97 10.39 .039 .052 0.6 0.730 1.22 7.97 .076 0.7 0.75 1.12 5.57 .064 0.8 C.819 1.02 4.89 .115 0.9 0.803 0.89 3.23 .190 1.0 0.780 0.78 2.65 .281 1.1 0.752 0.68 2.32 .400 1.2 0.718 0.60 2.11 1.3 0.675 0.52 1.94 0.560 1.5 0.565 0.38 1.61 1.080 1.7 0.438 0.26 1.32 2.160 2.0 0.292 0.15 1.19 6.680 2.3 0.238 0.10 0.68 16.51 L1 r.32,, L TRAIN T4L TRa
=
C
F
0-37L Fig. 3: 4
-.089 -.013 -.200 -. 088 -.040 -.085 -.070 -. 051 -. 049 -. 070 -. 070 -. 062 -. 096 -. 094 -. 111 -. 133 -. 120 -. 164 -. 10 -. 193 -. 007 -. 220 0.000 -0.31 0.380 -0.82 1.000 -5.33 3.070 -21.34 12.21 0.16L. L
-.170 -.067 -. 050 -. 080 -. 094 -. 160 -. 243 -. 345 -. 469 -0.61 -0.92 -1.07 -1.14 -13.85 = 0. 32L
Efficiency curves.
A: B/L = 0.3
Optimum at all frequencies
B: B/L = 0.3
Optimum for L =
C: B/L = 0.1
Optinum at all frequencies
optimum for B/L = 0.1 agrces well with his and is somewhat higher for B/L = u.3 due to However, for longer the greater raft width. waves convergence towards Newman's optimum was too slow; the values of La/A renorted here are only half of Newman's. Upon closer examination of our calculations it is found that the discrepancy is associated with the fact that only the even mode is fully excited but not the odd mode; the modes being those defined in Ref. [1I Since Newman already showed that both even and odd modes give comparable efficiency, the nearly 50% reduction of La/A, therefore, corroborates with the weak excitation of the odd mode.
La a
LaC a
0.3 0.414
1.38
1.03 10- 4
-. 089
-. 010
-. 200
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.5 1.7 2.0 2.3
1.04 0.98 1.28 1.12 1.02 0.88 0.77 0.67 0.58 0.51 0.37 0.26 0.16 0.13
1.33 0 .020 .035 .051 .066 .120 .200 .306 .452 0.651 1.280 2.530 7.070 15.47
-. 088 -.075 -. 042 -. 064 -.056 -. 087 -. 118 -.141 -.154 -. 160 -0.24 -0.75 -5.75 -17.2
1.030 -.063 -. 061 -. 101 -.112 -. 147 -. 174 -.181 -.155 -0.09 0.190 0.730 2.950 11.22
-.170 -.069 -. 052 -. 078 -.129 -.)99 -. 290 -.404 -.537 -0.68 -0.92 -0.94 -1.19 -17.3
L/A
0.415 0.488 0.768 0.787 0.817 0.794 0.768 0.737 0.701 0.658 0.552 0.438 0.323 0.294
C4
"3
First we note that for very long waves the displacement is very large, in qualitaFor comparison Li:, meffcieny we clcuatedtheoptm, with Newman. tive agreement we calculated the optimum efficiency L. a.P, 0 according to Newman's theory for B/L 1 our as shown by dashed lines. When tA
Haren [2) has
L =A/2 that Newman's theory calculated forrotimal amplitudes of the even predicts the and odd modes co be 8 and 30 respectively, while our computation only gives 8.45 and 3.33 respectively. For small EL it can also be inferred from [I] that the optimum displacements of the two modes are in the ratio of A e Ao
5KL 4
a (2 +
(1+
a
)
(5.1)
aL is the len-th which is very small where = The 0.26 for TI). of the middle raft !a vastly larger " implies that the odd mode is much more sharpi. tuned than the even mode, and demands much higher numerical accuracy in the present optimization program. In another example for a train in a channel. the two modes have comparable and moderate atioderapre e present desanda the t prooptimizatin a-.plituds gram gives the same result as Newman [Ii. While a more efficient optimization technique can in principle oring further aareement with Newman, the present experience suggests that
-883M-
the same difficulty must also exist in designing a control system which would optimize both even and odd modes. In any case, the extraordinarily large amplitudes for long waves should be avoided, Finally we show in Curve B the efficiency when the dimensional i and C take the The sharpness optimal values only at L = A. of the peak means that the system is rather highly tuned, and an inertial converter with electronic control to optimize according to incident seas is highly desirable. From preceeding discussions it is clear that proper design of power converters whose impedances are automatically adjustable with respect to hinge location and incident wave characteristics is essential to good efficiency. Using the same converter of fixed impedance at all hinges is unwise. This point appears not to have been appreciated by many inventors with the notable exceptions of Salter for beam-sea absorbers and Buda! and Falnes for point absorbers.
for L =A is shown in Figure 5 optimal L aA as a function of a. The curve is reasonably flat for small B. For example for 8 = 150 the reduction from head sea incidence is 10% while for B = 250 the reduction is 20%. Since it is typical that 95% of the energy of efficien250 sea the is reduction is within B rot serious. to oblique cy due
9
070-
L4b
N 03 02
6. WAVE FORCES
r
in Figure 4 we plot the vertical forces and the moment on the up-wave hinge (m = 2) corresponding to Table VI (hence the values for L/A < 1 are imperfectly optimal). The force and moment on the down-wave hinge are similar and are not presented here. In addition the second order drift force can be calculated from the far field amplitude according tL the theory of Maruo !8] 2 = . i 2 V A . 2Kq
Fig. 4: A: B:
and is shown as curve C. Note that all forces become lar e for lona waves implying in practice that ?he system'should be detuned to sacrifice the theoretical optimum. 7.
e
,5
21
Wave forces on three-raft trazn T3 force at hing Normalized vertical 2 2: 10"R 2 /!ogAL at hince 2: Normalized moment 3 101M2/A
C: Normalized drift force: /A2L_ Results for L/. 0. The variations of iCn - in t;) with w are very different if
condition. optimal be better to define the hinge (3) It will but byyshift impedance not by f i-iW -Cn/i w) because ofn the correspondence
0 or if Cn = 0 and in > 0. Cn < 0 and in a fixed Cn leads to a that it appears bandwidth than a fixed in broader All this only corresponds to a phase betweenwhich the resisting moment and displacement could eventually be the
to mechanical impedance. (4) The problem of negative spring
dealt with by a "smart" electronically controlled hinge.
at L
=/L0.21
and 0.47, where(*) means
cn will
NE
tam
I
The Sea Trials and Discussions on the Wave Power Generator Ship "KAIMEI" Yoshio Masuda. Gent',ro Kai Takeaki Mtyazaki JamanMa;°,'e Sc en-ce a-v Tcric-gtCer. ,J-"S ECj YO---t,.d
Yoshiyuki Inoue
Japan
1. INTRODUCTION Very large amounts of energy are stored in the Oceans of the world in the foram of waves. In principle, this energy could replace a significant part of that currently being provided through exhaustible fuels. however, the potential of wave Dower generation's only practical anp)i--ation in the world today is in a navigauion light buoy. Thus, the utilization of wa%3 energy is still limited to a small sca .The Japan Marine Science and r.chno-
exiting flow of air is actuated by the unward and downward movement of the water in the box. An air turbne and generator unit are installed on the box and utilize these air currents to generate electricity. After various tyPes of model tests, the steelnade 80-r lnng, 12-r wide 'Kainei was constructed as shown 4i Fioure I. The hull of the Kaimei consits of four buoyancv rooms and a number of air pump rooms where wave enercv is converted into air flow.
logy Center (JANSTEC) has been carrying
out a five-year development research project on a large output wave powe-- cenerator since 1976. Since the objective of the researcn is the conversion of wave energy into electricity, a wave power generator, "Kaimei," was constr-tcc n ce r t-n perrform two series of cpen-sea trials during 1978 - 1980. From these tests, much of the results obtained through experiments us in m-del vessels have been Proven cn the actual-. -. the. seas. A--ng 'he various finings, generator No.9 has recorded a maximum output of about 291KW, exceeding all expectation.
,
.
--
Z '
-
,.. --
il
-.-. I=
j
-)° .....
-
2. PRINCIPLE OF T1HE SYSTEM Kaiii 1979 Yhe Second Trials F;g. Various anolicable ,ethds for the con-ersion of wave power into electrical3. O-E-S h TIA power have been proposed by experts throuchout the wor=d. However, JAXSTEC The omen-sea trials of the ,aimei were has been concentratino its efforts on the P~rformed twice. T1-he first series were 0 - s--ar~ of a wave power generator eroyc..rried out during the winter of 1972°-79, ig air- turbines. in this type of cenerasince wave activit- at the test site in thc ,bottomless box is ilo~c on the Janan Sea is at its peak during this sur of the sea, an.d an enter-. and period- During the trials, the Kaimei
length of the Kaimei was designed to be 80 m. Estimating maximum wave height at 12m, the strength of each section of the hull was calculated based on this assumption. For the first open-sea trials, the displacement of the Kaimei including the three generator units was found to be 599 tons, and GM was 7.25M. Through forced pitch and roll tests mada under calm water conditions, the natural period of rolling was found to be T = 5.2sec. For the second open-sea trials, displacement including the eight generator units was 723.6 tons, GM was 4.49M, and the natural period of rolling was T = 6.3 sec. Since the natural period of rolling was so close to the wave period, there were apprehensions of resonance. However, after confirming safety through studies, 52.4 tons of concrete ballast was placed at the bottom of the ship for trim adjustment. This increased the displacement to 776 tons under normal conditions, and 821.2 tons during the open-sea experiments with the weight of the mooring chains -- 45.2 tons added (the weight of generator unit -No.1 was not included).
carried three Japanese made generators. Following the trials, the Kaimei was returned to harbour to allow refitting with a new group of generators, seven of Japanese design and manufacture and one from the U.K. With the exception of the nonarrival of the generator from the USA, the work was completed on schedule and in time to allow the Kaimei to be returned to the test site for the second open-sea trials during the winter of 1979-80. The second sea trial was a joint research project involving five countries as an International Energy Agency undertaking. The participants are as follows. Canada -The National Research Council of Canada Ireland-The Government of Ireland Japan -The Japan Marine Science and Technology Center U.K. -The Secretary of State of Energy -The Department of Energy USA 3.1 The Test Site
3.3 Design of Mooring
The test site was situated 3km off
' Yura (N: 38044 , E: 1390391) on the West
The primary concern in conducting the open-sea trials was related to the mooring. The Kaimei was moored at four points at the bow and one point at the stern with a buoy in the middle of the chain. As a result, a slack mooring method using heavy chains was confirmed to be the safest. The two main lines are 95mm super grade 4 chain chains and the other three lines are 76mm grade 3 chains.
coast of Japan as shown in Figure 2. Wave activity at the test site in the Japan Sea is at its peak during the winter. The Kaimei was moored heading to WNW since the maximum winds are predominantly out of between W and NW. The fetch is open to the W for about 1000km and to the NW for about 700km.
45'
Sea
TJApan
I
II I
40'
Ei439*39*48 The
I1
1-.;..n
e.,
VURA
e
2
TOKYO
t
//
The Pacific OceanI
e.
.......
All
I
VT TestnfoSit of wihte the Kaimeivrgewv 3.2 Construction average wavethe with the Japan Sea, in the periodIn ofconformity 6 - 7 sec.
Fig. 3 Outline of Mooring System
-888-
T-
Fig.3 illustrates the mooring arrangement. This system was applied to restrict the range of movement due to the use of transmission cables.
-.
DUr..k
.,
Although the direc-
tion of the bow changed in accordance with the direction of the wind and waves, actual observations showed that its movement was within a 120' range, almost the same figure as had been projected. The method of mooring calculations was the same as in reference [1]. Design considerations included waves with 1/3 significant wave height of 8m and period of 10 seconds, wind velocity of 100kt, and currents of 3kt. The wind and waves are out of the head of the Kaimei, and a maximum tension of 87 tons wa expected on the main mooring lines.
I L
_f_
vp .. ,) diftWt of the
o
'-.-, 1
T...-.
Vvo
I Ale
~~ ~
i
Jinstalled
b
j \Figure
-turbines
-\I" 7-for V.I-
N
,A t C
-t
AaoPC
3 ..... .-
" -A
th
e
.
was applied to unit No.2. While there is an advantage in that tie number of turbines and generators are reduced in accordance with the number of air pump rooms, there is also a disadvantage in that the number of valves must be increased to 120 thus requiring the use of a large valve box. For damper tanks, the buoyancy rooms on the left and right sides of the Kaimei arE utilized. Although the tanks in both the plus and minus side have a capacity of approximately 400m 3 each, as a result of computer simulation , it was desirable to double the capacity. aboard the Kaimei is shown in 7. All seven of the Japanese-made units employ the same type of air turbine. As a result of the model tests, impulse with round frontal blades were developed taking int. consideration the fluctuations of air flow. Anti-corrosion aluminum alloy (hydronalium) was used the turbines in order to prevent corrosion by sea water. The turbines were inspected after the first open-sea trials, generator is directly connected to The six Japanese-made generators, the one employing the amper tank system, are synchronous types and produce 125KI of electrical power at 840rpm. Since +--e rotation speed of the turbines changes accordance with the fluctuations of the air flow, generated output also changes. The generated electrical power is dissipated in a resistor aboard the Kaimei, and measurements of the electrical current voltage are observed and recorded. tank admer The genare The generator employing a damper tank system differs in that it is designed to transmit electricity to electrical power on land. As a result, it requires use of an induction generator which the turbine and made to rotate at the same
speed.
S.......excluding of
_ M.t of th.
but no abnormality was observed. /
M--,nt
Oo Dcu-) rw.j
An air turbine with a diameter of 1.4m
ent
Fig. 4 2-valve System
1) up-,d
W..S
pump rooms are collected to turn one turbine. However, in order to reduce the flucmuch as possible, of damper air pressure tuation one large tank is asemployed for each of the plus and minus sides. This system
valve box causes the flap valves to open and to close. FRP material which is light and strong is generally used for the valve board. Figure 5 illustrates a 2-valve system in which two air pump rooms are used in pair: one reacting to the upward movement of the waves, and the other, to the downward. Figure 6 shows a 4-valve system whose air pump room reacts to both periods of the waves,
1I
_ of U..
Fig. 6 Damper Tank System
The alternating air flow generated in the air pump rooms is made to flow in the same direction through the air turbine by box difference as illusthe flapinvalves in 4the trated Figures and valve 5. The in air pressure inside and outside the
V VI v.1v:j
I
.
Ai...
3.4 Generator Units
.
h
/
1) Upiard X-o...ent
....
. -
A__
b
i)
booo..8
now"A,
of
theto.s
St According to comparative studies made in the water tank tests, it was found that with long-period waves a 4-valve system wihln-pro-aesa4vlv ytmand obtained an output almost twice that of a 2-valve system. Tests are being carried out with a 4-valve system plus a damper tank system as shown in Figure 6. In this cotsystems ce tthe -889-
6
facilitates hook-up to the land power system.
L
ence of the air pump room, decrement of wave height, motion of the model, and distribution of air pressure and air outpu t. The method of theoretical analysis was the same as in reference (2] and was compared with the tank test results. It clear that air output changed depending on the position of the air pump room. Figure 8 shows that the results of theoretical analysis agree well to the experimental results. By using this theory, the total air output power of the 80 meter Kaimei was found to be 4,000 KW. If efficiency of air output to electrical output was 50%, 2,000 KW of total electrical output would be expected. This value represents the first stage estimate at the beginning of the Kaimei study.
-=was
-u-..calculated
I
/
ISO-
Fig. 7 Turbine and Generator Japan employed 2 and 4-valve systems, and the U.K., a 4-valve system. Table 1 illustrates the specifications of the generating units of each country participating in the test on the Kaimei.
E"100
4 50-
Table 1 Turbines and Generators on the Kaimei LSTERN
Country
Itm
U.K
Japn
Fig. 8 Variation of Output
3.6 Measurement Sstem
te Vah'esystem
BOW
2- ahe
e
.
4-valv
4-vah.-
!4-valvc
(air-dumper
-
re e
s
t
S
s
e
Since the Kaimei was designed for un-
method)
manned power generation tests in rough Turbinnipp
I
Turbine outer
diameter
Impls 400
1
mim
-Materialof tubine l.dronalium
Impule
Impulse 4
4
reliable apparatus was adopted. The measurement items can generally be classified as: weather, waves, hull movements, mooring forces, relative wave heights in aid out of the air pump room,
1 00 mm
1 00mm
990mm
Ilydtronalhum
Ilydronatium
Atcoralloum mun-
Alternating-
Squirrel-cage
Altcrnatng-
air pressure in the air pump room, differ-
induction generator
current synchronous generator
ential pressure of the turbines, revolu-
Generator
Altetnating-
type
current curren synchronous s.ncthronous generator generator
(vertical)
seas, an automated measurement system using
0% reaction
|hoIrontal
tions of the turbines, electrical current,
('orizontal)
voltage, and electrical power.
Ratedoutput voltae
I
Unit nunber
Numberof generators
data items were
a wireless telemeter to the shore
2.through Nos3.Sand 6 Nos 4,7 and9
station where they were monitored and
No. 8
o 2
recorded. Although the major measurement items were under continuous observation, other items, as a rule, were monitored
3
[Ithe
There are also turbines which utilize alternating air flow to rotate in one direction without the use of valve mechanisp:s. A non-valve special turbine, called the Wells turbine, was tested on the Kaimei from April 1980.
for 20 minutes (actual time 19 minutes) for every three hours. The monitored data were input into computers at a sampling interval of 0.5 seconds, recorded in magnetic tape, and preserved. In addition, a control function for transferring the generated electricity to the main land network was also incorporated into the telemeter system.
3.5 Theoretical Analysis
4. TEST RESULTS
Theoretical analysis was made tc clarify essenLial phenomena of the shiptype model Raimea, especially the influ-
4.1 Waves Four types of wave height meters
-890-
;
The 82
simultaneously transmitted
-
-
-
~
.
-
--
-
--
_-
that in summer. Moreover, wave power recorded during the winter of the second opensea trial was considerably higher than that in the first. Figure 11 shows the scatter diagram of the second trial. Every point was taken from an analysis of 19 minutes every 3 hours. Wave distribution was mainly 4 - 10 seconds in period and 1 - 4 in wave height.
(wave rider, ship-borne wave height meter, ultrasonic wave neight meter, and buoytype wave height meter) were employed in the open-sea tests. The measured data obtained by the wave rider and the ultrasonic wave height meter were very close, Satisfactory results were also obtained from the other wave height meters. All wave data were analyzed as follows: max, mean and significant wave heights and wave periods were lead by the zero crossing method, spectral density, wave periods defined by wave spectrum and others.
H.o
I
.
0
Figure 9 shows wave spectrum in storm sea.
Figure 10 shows wave power recorded during the first and second open-sea trials.
2
P.F.
0.06 HZ
Ku.
6.1m:
;
20 10
0.3(Hz)
0.2
0.1
0
Frequency
Fig. 9 Wave Spectrum [t,/a)
S(Sep
SB
'S|p-i'O(
I
___
'"'
_
-
_ IHI
Fig._
Fig. 1
S11
s,,. Oct.
Nov. 1ec. Ill. F.
m
as
1/3 significant wave height computed by the zero-up crossing method with H:,1 as computed from the variance spectra. density. Hm0 can be obtained by the equatior (2):
_the
All.
Scatter Diagram of the Second Trial 12 shows comparison of H
iI 'Figure
"i
-
.Apt.
Fig. 10 Wave Power Record
4.004fi .................... (2) HmO mn = JfnS(f)df .................. (3)
Wave power, P, can be obtained by the following equation: .................... (1) .T P = Pgf-B-R p 64r where fi/ is the significant wave height
U
where m n is the n-th moment of frequency spectrum defined by Eq. (3) From Fiq. 12, the relation of Hm0 = 1.05 Hi/3 can be obtained. Then, it considered that the constant number, be uest t 40 o o that r requires to be 4.004 of Eq.(2) for Hm corrected to 3.81. It has been reported 3.81 is applicable to the the large number that of area waves. a
and T is peak period of wave spectral density. In Figure 10, the waves recorded he avesrecodedis In Fgure10, during the first open-sea trial are indicated by the dotted line; that of the second open-sea trial, up until July, is indicated by the solid line. In general, wave power in the winter is higher than
I i
-
-891-
-
0
-~--~
_A
=~-- -
a=
3
too 6 8.0
4
-----
-
'20
40
____
Fig.
~~seas.
1
3 4 5 Hz35 (Z,,,-u, c,,,) zs
80
~~To
100
120
.Im
_____
12 Comparison of Hm, and
6
(
Cr s'
lur-l7|
/2
In the sea of Japan, this relationship was established from calm seas to storm
so
According to the statistical theory, the mean zero crossing wave period can be
40
T;m2 m / T 2 ............ (4)
1.Ime 1110 it "
- $2 To
20. 0
a
in the wave record usually shows some
deviation from the value of Eq.(4) due to wave and othertoreasons, symbolnonlinearity of Tm, is employed denote the value calculated by Eq.(4). waepro sdfndb:4.2
.
-
estimated from the frequency spectrum as: As the mean wave period actually observed q
00
Fig. 13 Comparison ,f Spectral Moment T m to TID
0 . 0
~
20
40
60
so
100
12.0
(StO
oprio offT 02.1t Fig. 14 Comparison t Tp to Tension of Mooring Line
Another mean
. .Tm. = m./m. .................... (5) and m TStrain are calculated using the formulas by Mitsuyasu(1970) and Ferdinande ez 1.(1975) as in the following: =0.7718 Tp Tm = 0.7104 Tp Actual wave measurements and spectral analyses at the test site showed the difference of the theoretical values, Fig. 13 and Fig. 14 show comparison of spectral moments with spectral peak. From these figures, Tmo , Tmm in terms of T, will be Tm, = 0.69Tp, Tm = 0.62Tp. These values will show little differences against the theoretical values,
-892.-
Therporig
ensonwas mesrdby
aA
stall tension meter in the first trials of 1978, and maximum tension was found to be only 50 tons. Since the Kaimei in the second trials was approximately 200 tons heavier than in the first, the surface area subjected to wind pressure was larger. Consequently, an increase in the mooring force was expected. The maximxum mooring tension recorded during the second trial was 84.4 tons on the NW line, recorded on October 19. Figure 15 shows the data of mooring tension. On that day, wind direction was 2830, average velocity was 32.4kt, H s was 5.9m, and Hmax was 9.1m. Fortunately, this recorded tension value was very close to the estimated 87 ton value. The fluctuations of tension include long period fluctuations of 104 seconds and short period fluctuations of 7.7 seconds. The 7.7 second short period coincides with the pitching period of the Kaimei, while the motion of the 1'4 second long period is considered as the low frequency drift of the Kaimei.
)
ture, it makes pitching, rolling, and heaving movements in conjunction with the elevation of the waves. These movements influence relative wave height in the air pump rooms. During the second open-sea trials, heightwhich gaugesexert measured relativeofwave the waves actualtheinfluence on the air pump rooms. Two meters each was installed on both sides of the bow, the midship, and the stern.
Reords of Tension
oct.i9
6.0 4.0
:S2.0
!height
(I
j
C
."
21:40
Shipbo,,
in Figure are ratios shown are measured 17.The Relative wave data height derived by dividing relative wave heights by significant wave heights at the location of a wave rider.
-2.0 4.0 -6.0-
I
Tension-
NW
Relative wave Height Ratios
0.9
60 -7,
Be. Midship
20_
0.7
0
0
0
100
150
200
ii
250
Time in sec
0.6
o__
Fig. 15 Record of Wave and Tension(N)
zL
-
4.3 Transmission Cable
_
The electrical power generated by the No.2 induction generator is transmitted to land through a transmission cable laid on the sea bottomn. The installation of such a cable to a floating structure is difficult, especially to a floating vessel as the Kaimei which can move freely within a limited range. With the cooperation of an electrical manufacturing company, we have produced a special FRP armored cable which is effective against kinking. The cable between the sea bottom and Kaimei, installed with the combination of weights and buoys shown in Figure 16, gave good results during the first open-sea trials. Consequently, the same method was applied for the second trials. A widely used iron armired cable was employed as the undersea cab-re, ____---_---_-__--_
Kaimei
S•
.Ch ainZ
#'F~RP-Cae
-1 o,
I-
Stein
-.
-H o
04
im : 0.3 -
,,Q,
.0
I
0.2
06 1
2
3
4
S.gnificmnt WVav Height
5
6
H (m)
Fig. 17 Relative Wave Height by Hs
When wave height, Rii3 increases, wave period and length also increase. As a consequence, the Kaimei begins to pitch, and the relative wave heights at the bow and stern become as low as 50% - 40% below that of the wave height outside. Particubuoy larly at midship, wave height is about 3/10 of the wave height outside. Such .,unexpectedly low relative wave height is a negative factor for increased generated output of the Kaimei. Thus, some improvement must be done to raise relative wave height. 4.5 Generated Output
Fig. 16 Installation of Cable
The major objectives of the experiment were to learn the Kaimei's generated output characteristics and areas for improvement. In this section, the following subjects will be discussed: improvement of
4.4 Relative Wave Height Since the Kaimei is a floating struc*893-
-
--
T
-
generating output through use of a 4-valve system; output and efficiency of the No.9
and Tmean = 6.3 secs, and maximum, and average output were 290 KW and 50 KW respec-
linkage using the No.2 generator; and comparisons between actual output of the generators and results of the water-tank tests.
The generating efficiency of the No.9 unit can be calculated by obtaining the air output from air pressure, P, the inside wave height, Hin, of the air chamber, and the generated electricity from electric current and voltage. The resulting efficiency of impulse turbine (No.9), as shown in Figure 20, was 45% - 75%. Efficiency, of course, increased along with air pressure.
4.5.1 Improvement of generated output employing the 4-valve s.stem The 2-valve system operates with a single cycle of the air flow, whereas the 4-valve system operates with alternating air flows. Because it was desirable to reduce the number of valves in designing a valve box for the Kaimei, three 2-valve systems, with only half the number of valves as the 4-valve system, were tested in the first trials. The results of the water-tank test indicated that the generated output of a 4-valve system, as compared with the 2valve system, could be increased two-fold with long wave lengths. Based on the above results, in the second trials three 4-valve systems were tested for comparison. Comparison between the generating outputs of the 2-valve and 4-valve systems is shown in Figure 18. At the start, there was almost no difference between the systems. Later, with an increase in the period, the output of the 4-valve system becdme twice that of the 2-valve system.
A
300
No.9 Geneator Output -
250 200 -
0 0
151
100
0 so
too Time Ia
Iso sec
200
250
Fig. 19 Generating Conditions of the No.9, 4-Valve Synchronous Generator
04 80
a
No.4 (.)
0-0-
2, 0
Wels Turline
,
.06
N.3 12,ok59) X0'
is
(1500
%
0a
Fitore N. Wills Turbiet
Ii-40 " NZ
S v, N,.2
Impulse Totbi N0.9
"Jk4
13
IIis
N8.16
Date A
Fig.
0 C.*. bf,
2
0.
18 A Comparison of Output between 2 Valves and 4 Valves System
4.5.2 Output and efficiency of No.9 generator The generated outputs were highest at the bow and stern of the Kaimei. Since the U.S. unit was not installed at the bow, the No.9 unit installed at the stern generated maximum output. Figure 19 shows the output of the No.9 generator recorded under th. large waves of October 19. The waves were recorded at HI, = 5.9m, Hmax = 9.8m,
-
_
_
_-894-
0~0 0
200
400
IA" 600
800 (umAq)
Air Pressure Fig. 20 Turnine Efficiency of S Iii4pulse and Wells 4.5.3 Power transmission test (No.2 generator) The highlight of the second trials was the power transmission test. This was an epoch-making event, unpreceded in the world. Electricity generated from the No.2 induction generator was transmitted the electricity system on land through tocables.
The experiment on the No.2 generator also included a test on the stabilization of air output employing an air damper tank. The test of the induction generator linkage to the land system was conducted in December, January, and the middle of April. At first, there were technical problems which included rush currency upon the input of the induction generator, voltage fluctuation of the main power system due to the nature of waves, and selfexcitation in case of disconnection. Consequently, the linkage tests were conducted very carefully under joint research with the Tohoku Electric Power Co., Inc. Every test results were below the permissible limit, and thus the tests were completed
same generated output for the units installed at the bow and stern of the Kaimei, and one-half of the output value for the one installed at midship. Comparison of the output of the generators during the second trials were made in terms of average output. Figure 21 shows the changes in output of each generator by significat.t wave heights. kw)
I
i
N3.9G
i
30
__/
j No.2/
successfully wihtout any problems.
Results of the long-term continuous transmission test conducted from January 14 to 31 are shown in Table 2. Comparisons are made with results taken from the No.9 unit (synchronous generator) at different significant wave height.
2 G . -
Table 2 Generated Outputs of the No.2 and No.9 Units in the Latter Half of January twalcnt) (m)
(113 ,o.2
Induction
0.3
717 1
3
Nfaxum 02 Output tKW 20 35 66 9 IGenerate, Arise ut', 3 SA4 104 14,7 34put 1K.)
Ii
4
5
125
ISO
20
o
0
output. .$27$7
-3 H
4
I 0 GNOG 5(m)
4.7 Mean-maximum Generating Output Rates The biggest problem in improvement of wave power generation is fluctuation of
No at.
due to I-"
-actionof 25 the brca~kct
The cycle of fluctua-
over, ratio isof about average maximumthe output 1/6 output - 1/7. over
235so
4-valve system observed a smaller
fluctuation margin than the 2-valve system. The mean-maximum output ratio of 1/6 - 1/? can be understood as follows± 1/6, mean/maximum = (0.72)/(1.8)2 where 0.72 i is average energy
In order to avoid any influence of voltage fluctuation on the general power system caused by excessively large waves, maximum voltage for the test was limited to 178KW. In fact, when output exceeded 150KW, the breaker would react, and the safety valves in the valve box would open and halt generation. This may have been attributable to a discrepancy in data sampling. During the period of there experiment which lasted for half a month, were
height, and 1.8H/v is maximum wave height. During the second trials, the only experiment conducted on normalization of generated output was on the air damper tank method. However the normalization problmisoa v fie o study uaito problem is a vital field of as it
seven occasions when generation was stopped automatically because generating output exceeded the limit. The rated output of the induction generator is 125KV.5 Its air turbine is the same as other Japanese-made synchronous generators, with a diameter of 1.4m. Voltage of the generator was 200V, and rotation speed was set at 600RPM in accordance with the 50-cycle main power increased to 6000V, was i isystem. trnsmtteOutput, tothelan poer ystmfound i . i ..4.6 .. , .. Differences the Generated at DifferentinLocations of theOutput Kaimei ; _ The first open-sea trials recorded the
_U
2
by Wave Height
8The
I
1
3G .No: G
Fig. 21 Average Output for each Generator 6
.generated
S'nchtenous Output(K%')
-.
18
-895-
may determine whether wave power generation can be utilized commercially.
The two series of open-sea trials on the wave power generator 1,aimei were performed in safety. a) From actual wave measurements and spectral analyses at the test sites, mean wave periods of Tm. , Tm. were to be 0.69Tp and 0.62Tp respectively. b) The maximum mooring tension ed during the second trials was recordddrn h eodtil a 84.4 tons agreeing to t ne tneoretical results.
a) Relative wave height must be increased in order to increase air output, NEL's idea of a side opening with a bottom plate will be one of the promising ideas. The lencth and depth of the Kaimei, etc., must be reconsidered from this point of view. b) Side opening with bottom plate (Figure 22) This idea was proposed by NEL, U.K. A model Kaimei with a botto.n plate showed an increased air pressure ratio of about 1.6 - 2 times of that without a bottom plate. Therefore, a 2.5 - 4 fold increase in air output can be expected through this method. 4-''": i Ii
c) The occurence of power cable kinking between the floating vessel and the sea bottom could be avoided by using an FRP armored cable. d) The output of the 4-valve system was twice that of the 2-valve system with lop,, wave periods. e) Li- age tests on the No.2 induction generator were completed successfully ana without any problem. ACKNOWLEDGE-ENT The second open sea trial was an International Energy Agency project supported by Japan, UK, USA, Canada and Ireland. The authors are grateful for the permisto of Committee Executive of the sion solely which consists paper its IEA publish their own interpretations of the data and
it
views which are not necessarily shared by the other participants.
'litL,-
We would like to extend our greatest
appreciation to the members of JA4STEC, and to the researchers both in Japan and
______
abroad for the cooperation and valuable advice which have made the experiments and studies with the Kaimei possible.
.1.1
-\
62
-
l
1. Inoue, Y. "The Design Criteria for the Mooring Systems of Offshore Structures", Journal of the Society of Naval Architects of Japan, Vol. 145, June 1979.
,
I
-
REFERENCES
i !
U
2. Miyazaki, T. and Masuda, Y. "Research and Development of Wave Power Generation System", Oceanology International '2, Brighton 1978, pp.45 - 49.
I
-,
i
I $
Fig. 22 Comparison of Air Pressure Ratio of a Kaimei Model with and without bottom plate
3. Hiramoto, A. "The Theoretical Analysis of an Air Turbine Generation System", Symposium on Wave and Tidal Energy, BHRA Fluid Engineerinq, Vol. 1. IT78, pp. 7 3 - 84. 4. Masuda, Y. and Miyazaki, T. "Wave Power Electric Generation Study in Japan", Symposium on Wave and Tidal Energy, BHRA 85 - 92. Fluid Engineering, Vol. 1, 1978, pp. 5. Masuda, Y. "Experimental Full Scale Result of Wave Power Machine Kaimei in 1978", Symposium on Wave Enerov Utilization, Chalmers University of Technology, Sweden, November, 1979, pp.349 - 363. 6. Miyazaki, T. and Masuda, Y. "Tests on the Wave Power Generator Kaimei", Offshore Technoloqv Conference 1980 OTC 3689, pp.101 - 111. TECHNICAL REFERENCE In order to evaluate characteristics related to the Kaimei, much of the data were analyzed; and in order to make as c z . r-q. Qel,-r. turbine tests were carried out. The main results are as follows,
-896-
c) Turbine nozzle ratio (turbine nozzle area/air pump room area) A 1/230 nozzle throttle ratio was adopted for Kaimei's turbine, but the tank test with the 8m model indicated that a 1/75 nozzle throttle ratio was best and would increase power 1.8 times greater than that with a 1/230 nozzle throttle ratio. d) Wells turbine test result A non-valve wells turbine of 0.6m diameter with 4 wings operated on the Kaimei. its energy conversion efficiency was found to be relatively high (more than 60%) with very :small waves, and its features of high speed rotation and simpile construction may greatly decrease the cost of turbine and generator units. Figure 20 shows a comparison of turbine efficiency between the wells turbine and the impulse turbine. The impulse turbine shows low efficiency with low air pressure and high efficiency,with high air pressure. The wells turbine, however, showed high efficiency with low air pressure and low efficiency with high air pressure. This different characteristic may give a safety factor to the wells turbine in very high seas.
e) Damper tank model test The damper tank method represents one way of smoothing wave power. It was tested using the 8 m model and the entire buouancy room as the damper tank. Smoothing effect was observed. A main feature of this method entailed decreasing the nozzle throttle ratio. A ratio of 1/400 gave maximum output, therefore turbine dia.eter will be relatively small with Damper tank method. f) Output normalization methods Output normalization methods, and its water tank testing must be researched. Since the damper tank is not sufficient for output normalization of 1 - 2 minute variations, use of a fly wheel, etc. must be researched.
F--
-897-
Discussion OM Count (uK
,ii
stuady air flow and wolcomoo tho actiVin itios of JAMSTEC in this aran.
(,uciciooy,,,
utoai,,joa,)
I would like to congratulate Mr.Masuda
on ostablishing the feasibility of wave power with the construction of KAIMEI. I would just like to add that in the UK we are very interested in using nir turbines
for wave energy absorption. We have analyzod and tested the Well's Turbine which requires no valves to operate in a cyclic flow. Our toots confirm our theory that efficiencies of about 70% can be obtained
Author's
Reply
Y, Masuda(JAMSTEC) Our test result of the Wells turbine is shown in Fiq.20.
Efficiency was more than 60%, it agreos to UK's theoretical analysis, Non valve turbine is very useful for improvement of KAIMEI, but further study incloiding hydrodynamic is necessary to develop this turbine.
LIST OF PARTICIPANTS
I
BRASIL
FRANCF
Nishimoto. Kazuo. Univ of Sio Pauzlo
Aucher, Max. Ingr. Bassin d'Essis des Carenes
Baiquet. Bassin dEssais des Carenes
IIULGAR 14
Cordonnici,
.LP. .. Ecolc
National Supiricure de Micaniqu
Dern. Jean-Claude. Bassin d'Essais des Carenes Khadjimihalev. P.. Bulgarian Ship Hydrodynamics
I I
Lecoffre. Yves. NEYRTEC Le Goff, Jean Pierre. O.-re.tiozi des Recherches, Eludes et Tech-
~
~CAN-ADA Miles, M.D., National Res Council
fqe Rowe. Alain Robert. Inst. de Mcinque de Grenoble2
Murdey. David C.. National Res, Council
Schmitke. R-T.. Defence Res. Est. Attantic
FEDERAL REPUBLIC UF GER.MANY
CHINA
Albrecht. Klaus. Fraunof~r-lnstwut tiur Ifydroakustik Blumec. Peter, Ifamburgische Schifibau-Versuchsanstal:
Chen. Chiu- Si China Ship SeE. Res. Ctt.
I
Far~g
~-Chun. China Ship Set. Res. Ctr.
u. 'tao-Xianvg. China Shit, Sci Re-- Ctr.
Fleischer. Klaus Peter. XVersuchsanstalt fur Wasserbou sand Schiffbau Krappinger. Ode. llambur_4sche Schjlfhaa-Versuchmnstalt
k'ux. Jjreecn H., Institut fur Schiflban
Hsueh, Chun#-Ciuan, Huazhong Inst. of Tech.
Laudan. Jochen. llarnburgiscbe Scbiffau-Vetsuchsanstalt
Jiang. Ci-Ping. Shanghai Jiao Tong Univ.
Merbt. Hiorst. Ftaunhofer-Institut fu-r Hydroakustik
Liua. Da-KaL. East China College of Hydraulic Eng.
Miler-Graf. Bsarkhard. Versuebsanstalt fu-r Wasserbaa und Schiff-
Ma. Tao. Shanghai Ship Design and Re%. !nst. Xsa, Wei-De. Hlarbin Shipbuilding Eng. Inst.
i
basa Nowacki. Horst. Teci;. Umnv. Berlin Oltmann. Peter. Institut fEE:SchiPfa
DENMARK
Papanikolsoa. Apostolos. Tech, Vmrh. Berli Remnmers. Betnd. Kemnpf& Remnwes
Munk. Torben. Danish Ship Res. Lab. Schmicduen. MichaeL Verstuchsanstalt f-ur Wissetbws sand Schifli'mu Weitendorf. Ernst-August. Harnburtische Schilfha-Vecrsua~haanstaft
EGYPTWolff, Edvis. Sakti. Egyptian Shipbuilding & Repair Co.. Ltd.
Karsten. Institut fr Schiffbau
IRE 1.5,%4
FINLAND Dagan. Gedeon. Tel-Aviv Un-VV. Kossilainen. Valter. Helsinki Univ. of Tech.
Mil-A Touvi. Tel-Aviv Univ.
E
U
Ikebuchi. T. Kawasaki Heavy Ind.
ITALI Y
Ikeda, Yoshiho, Univ. of Osaka Prefecture
lh'llone Guglielmo. Cantier! Navali Riuniti
Ikegami, Kunihiro, Mitsubishi Heavy Ind. Ikehata, Mitsuhisa, Yokohama National Univ.
Pdenzana-lonvino. Carlo CE TE. NA. S.p.A. Genova
Inizu, Hayama, Tokyo Univ. of Mercantile Marine Inoue, Shosuke, Kyushu Univ.
XIF
Inoue, Yoshiyuki, Yokohama National Univ.
JA PAN
lnui, Takao, Univ. of Tokyo lsshiki, Hiroshi, Hitachi S & E Co.
Abe, Mitsuhiro, Mitsui E &S Co. Ito, M., Kobe Steel Co. Adachi, Hiroyuki Ship Research Inst. Iwata, Tatsuzo, Mitsui E& S Co. Aizawa, Koji, Niigata Eng. Co.
Jinnaka, Tatsuo, Nagasaki Inst. of Applied Science
Ando, Noritaka, Shipbuildizg Reb. Ass. of Japan iingu. Norio, Nippon Kokan K.K. Arai, Makoto, Ishikawajima-Harima Heavy Ind. Jingu, Norio, Nippon Koban K.K. Arakawa, Chuichi, Univ. of Tokyo Ashidate, Isao, Nippon Kokan K.K. 4
Azuma, Tetsuro, Osaka Shipbuilding Co.
kadoi, Hiroyuki, Ship Research Inst.A Kagemoto, Hiroshi, Ship Research Inst. Kai. G,, Japan Marine Sdi. &Tech. Ctr,
Bessho, Masatoshi, National Defense Academy
Kaiho, Toshimitsu, Japan Defense Agency
Chiba, Noritane, Mitsubishi Heavy Ind. Kaizu, Genji, Nakashima Propeller Co. Fujii, Hitoshi, Mitsubishi Heavy Ind. Kajitani, Hisashi, Univ. of Tokyo Fujimoto, Ryosuke, Nippon Kokan K.K. Kan, Makoto, Ship Research Inst. Fujimura. Hiroshi, Mitsubishi Heavy Ind. Fujino, Masataka. Univ. of Tokyo Fujino, Ryosuke, Ishikawajima-Harima Heavy Ind. Fukuda, Jun-ichi, Kyushu Univ.
Kano, Masayoshi, Hitachi S & E Co. Karasuno, Keiiehi, Kobe Univ. of Mercantile Marine Kasaliora, Yoshikazu. Nippon Kokan K.K. Kashiwadani, Tatsuo, Japan Defense Agency
Fu~xa, Takeshi, Ship Research Inst. Kato, Hiroharu, Univ. of Tokyo Hamano. Kazuo, Mitsui E&S Co. Hasegawa, Kazuhiko, Hiroahima Univ.
Kato, Naomi, Tokai Univ. Kawaguchi, Noboru, Mitsubishi Heavy Ind.
Hatano, Shuji, Hiroshima Univ. HayahiNobuuki NiiataEng.Co.Kaxvasliima, Himeno, Yoji, Univ. of Osaka Prefecture
Toshihiko, National Research Inst. of Fishery's Eng. KyYsiMtuih ev Kijima, Katsuro, Kyushu Univ.
d
Hirano, Masayoshi, Mitsui E& S Co. Kinoshita, Masao, Hitachi S & E Co. H-irano, Susumu, U.m.. of Osaka Prefecture Kinoshita, Takeshi, Univ. of Tokyo Hirayama, Tsugukiyo, Yokohama National Univ. Kishi, Yasutaro, Mtsui E & S Co. Hoshino. Tetsuji, Mitsubishi Heavy Ind. Kishimoto, Masahiro, Nippon Kokan K.K. Kishimoto. Osanmu, Mitsui E & S Co. Hotta, Takio, Hiroshima Univ
Kitazawa, Takamune, Hitachi S & ECo.
-900-f
~-
---
N
Kobayamhi. Masanori, Mitsui E & S Co.
Murakami, Mitsunori, Hitachi S & I. Co.
Kodarna. Yoshiaki. Ship Research Inst.
Murakami, T., Japan Defense Agency
Kodan. Norihisa, Nippon Kokan K.K.
Murao Rin- ichi, Aoyama Gakuin Uiniv.
Koga. Shigeichi, Shipbuilding Res. Ass. of Japan
Muraoka, Kenji, Ishikawajima-Harima Heavy Ind.
Kojima, Ryoichi, Hitachi S & EiCo.
Nagainatsu, Nobuo, Kawasaki Heavy Ind.
Komatsu, Masahiko, Japan Defense Agency
Nagainatsu, Shuichi, Sumitomo Heavy Ind.
Koyama, Koichi, Ship Research Inst.
Nagainatsu, Tetsuo, Mitsubishi Heavy Ind.
Koyama, Takeo, Univ. of Tokyo
Naito, Shigeru, Osaka Unvi.
Kudo, Kimaki. Hiroshima Univ.
V
Nakamura, Shoichi, Osaka Univ.
Kunitake, Yoshikuni, Mitsui E & S Co.
Nakaslima, Minoru, Nakasisima Propeller Co.
Kuroi, Masaaki, Hitachi S & E Co.
Nakatake, Kunilsaru, Kyushu Univ.
Kusaka, Mikio, Osaka Shipbuilding Co.
Nakato, Michio. Hiroshima Univ.
Ku~ak, & Co.N'kazaki. Yuo, MtruiE Kyozuka Yusaku, National Defense Academy
Narita, Hitoshii, Mitsui Eia S Co.
Maeda, Hisaaki, Univ. of Tokyo
Narita, Shuumei. Nippon Kokan K.K.
Masatosui, Osaka Shipbuilding Co.
Makizonti, Masataka. Yamashita Shin Nihon Steamship Co.
1